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Time Series Forecasting

         Siem Jan Koopman
http://personal.vu.nl/s.j.koopman

       Department of Econometrics
        VU University Amsterdam
           Tinbergen Institute
                  2012
Unobserved components: decomposing time series
A basic model for representing a time series is the additive model

               yt = µt + γt + εt ,        t = 1, . . . , n,

also known as the classical decomposition.

             yt = observation,
            µt = slowly changing component (trend),
             γt = periodic component (seasonal),
             εt = irregular component (disturbance).

In a Structural Time Series Model (STSM)
or a Unobserved Components Model (UCM),
the components are modelled explicitly as stochastic processes.

Basic example is the local level model.
                                                                     2 / 91
Illustrations


We present various illustrations of time series analysis and
forecasting:
 1. European business cycle
 2. Bivariate analysis: decomposing and forecasting of Nordpool
    daily (average) of spot prices and consumption.
 3. Periodic dynamic factor analysis: joint modeling of 24 hours
    in a daily panel of electricity loads.
 4. Modelling house prices in Europe.
 5. Modelling the U.S. Yield Curve.




                                                                   3 / 91
Illustration 1: European business cycle


Azevedo, Koopman and Rua (JBES, 2006) consider European
business cycle based on
  • a multivariate model consisting of generalised components for
    trend and cycle with band-pass filter properties;
  • data-set includes nine time series (quarterly, monthly) where
    individual series that may be leading/lagging GDP;
  • a model where all equations have individual trends but share
    one common “business cycle” component.
  • a common cycle that is allowed to shift for individual time
    series using techniques developed by R¨nstler (2002).
                                          u




                                                                    4 / 91
Shifted cycles

 0.2


 0.0


−0.2
       estimated cycles
       gdp (red) versus
       cons confidence (blue)
−0.4
                1980                    1985      1990   1995


 0.2


 0.0


−0.2
       estimated cycles
       gdp (red) versus
       shifted cons confidence (blue)
−0.4
                1980                    1985      1990   1995



                                                                5 / 91
Shifted cycles

In standard case, cycle ψt is generated by

         ψt+1            cos λ sin λ           ψt        κt
          +       =φ                            +   +
         ψt+1            − sin λ cos λ         ψt        κ+
                                                          t

The cycle
                                           +
                       cos(ξλ)ψt + sin(ξλ)ψt ,
is shifted ξ time periods to the right (when ξ > 0) or to the left
(when ξ < 0).
Here, − 1 π < ξ0 λ < 2 π (shift is wrt ψt ).
        2
                     1

More details in R¨nstler (2002) for idea of shifting cycles in
                 u
multivariate unobserved components time series model of
Harvey and Koopman (1997).


                                                                     6 / 91
The basic multivariate model


Panel of N economic time series, yit ,
               (k)                (m)                   +(m)
        yit = µit + λi cos(ξi λ)ψt         + sin(ξi λ)ψt        + εit ,

where
  • time series have mixed frequencies: quarterly and monthly;
                                     (k)
  • generalised individual trend µit       for each equation;
                                                 (m)        +(m)
  • generalised common cycle based on ψt               and ψt      ;
  • irregular εit .




                                                                          7 / 91
Business cycle
Stock and Watson (1999) states that fluctuations in aggregate
output are at the core of the business cycle so the cyclical
component of real GDP is a useful proxy for the overall business
cycle and therefore we impose a unit common cycle loading and
zero phase shift for Euro area real GDP.

Time series 1986 – 2002:
quarterly GDP
industrial production
unemployment (countercyclical, lagging)
industrial confidence
construction confidence
retail trade confidence
consumer confidence
retail sales
interest rate spread (leading)
                                                                   8 / 91
Eurozone Economic Indicators

14.30   GDP                                 Retail sales
        IPI                                 unemployment
        Interest rate spread                Industrial confidence indicator
14.25   Construction confidence indicator   Retail trade confidence indicator
        Consumer confidence indicator

14.20


14.15


14.10


14.05


14.00


13.95


13.90


                         1990                      1995                         2000



                                                                                       9 / 91
Details of model, estimation


• we have set m = 2 and k = 6 for generalised components
• leads to estimated trend/cycle estimates with band-pass
  properties, Baxter and King (1999).
• frequency cycle is fixed at λ = 0.06545 (96 months, 8 years),
  see Stock and Watson (1999) for the U.S. and ECB (2001)
  for the Euro area
• shifts ξi are estimated
• number of parameters for each equation is four (σi2,ζ , λi , ξi ,
  σi2,ε ) and for the common cycle is two (φ and σκ )
                                                  2

• total number is 4N = 4 × 9 = 36




                                                                      10 / 91
Decomposition of real GDP

 14.2                                        0.003


 14.1
                                             0.002

 14.0
                                             0.001
               GDP Euro Area       Trend                           slope
 13.9

        1990      1995         2000                  1990   1995     2000

 0.01                                       0.0050

                                            0.0025
 0.00
                                            0.0000

                                           −0.0025
−0.01
                                           −0.0050
                                Cycle                                irregular


        1990      1995         2000                  1990   1995     2000



                                                                                 11 / 91
The business cycle coincident indicator

Selected estimation results
              series         load  shift R2d
              gdp             −−    −− 0.31
              indutrial prod 1.18  6.85 0.67
              Unemployment −0.42 −15.9 0.78
              industriual c  2.46  7.84 0.47
              construction c 0.77  1.86 0.51
              retail sales c 0.26 −0.22 0.67
              consumer c     1.12  3.76 0.33
              retail sales   0.11 −4.70 0.86
              int rate spr   0.57  16.8 0.22




                                                  12 / 91
Coincident indicator for Euro area business cycle

 0.010




 0.005




 0.000




−0.005




−0.010




−0.015


                1990         1995         2000



                                                         13 / 91
Coincident indicator for growth



• tracking economic activity growth is done by growth indicator
• we compare it with EuroCOIN indicator
• EuroCOIN is based on generalised dynamic factor model of
  Forni, Hallin, Lippi and Reichlin (2000, 2004)
• it resorts to a dataset of almost thousand series referring to
  six major Euro area countries
• we were able to get a quite similar outcome with a less
  involved approach by any standard




                                                                   14 / 91
EuroCOIN and our growth indicator

0.0150

0.0125

0.0100

0.0075

0.0050

0.0025

0.0000

−0.0025

−0.0050

−0.0075                                    Coincident   Eurocoin

          1990               1995           2000



                                                                   15 / 91
Illustration 2: Nord Pool data
• we consider Norwegian electricity prices and consumption
  from Nord Pool.
• mostly hydroelectric power stations; supply depends on
  weather.
• Norway’s yearly hydro power plant capacity is 115 Tw hours.
• Nord Pool is day ahead market: daily trades for next day
  delivery.
• daily series of average of 24 hourly price and consumption.
• spot prices measured in Norwegian Kroner (8 NOK ≈ 1 Euro).
• sample: Jan 4, 1993 to April 10, 2005; 640 weeks or 4480
  days.
• data are subject to yearly cycles, weekly patterns, level
  changes, and jumps.

                                                                16 / 91
Bivariate analysis: daily spot prices and consumption
Our unobserved components model is given by
                                                        2
        yt = µt + γt + ψt + xt′ λ + εt ,   εt ∼ NID(0, σε ),

where
  • yt is bivariate: electricity spot price and load consumption;
  • µt is long term level;
  • γt is seasonal effect with S = 7 (day of week effect);
  • ψt is yearly cycle changes (summer/winter effects);
     ′
  • xt λ has regression effects, mainly dummies for special days;
  • εt is the irregular noise.
Parameter estimation and forecasting of observations have been
carried out by the STAMP 8 program of Koopman, Harvey,
Doornik and Shephard (2008, stamp-software.com):
user-friendly but still flexible, also for multivariate models.
                                                                    17 / 91
Daily spot electricity prices from the Nord Pool
                     (i)                                               (ii)
                                                                 0.1
               6

                                                                 0.0
               5


               4                                                −0.1


               3
                                                                −0.2
                           100   200   300   400   500   600              100   200   300   400   500   600
                     (iii)                                             (iv)
                                                                0.50
               1


                                                                0.25
               0

                                                                0.00
             −1

                                                               −0.25
                           100   200   300   400   500   600              100   200   300   400   500   600

                   Univariate decomposition of Nord Pool daily prices January 4, 1993 to April 10, 2005:

(i) data and estimated trend plus regression; (ii) seasonal component (S = 7, the day-of-week effect); (iii) yearly

cycle; (iv) irregular.


                                                                                                                     18 / 91
Joint decomposition of electricity prices & consumption
               7
                     (i−a)                                  0.50
                                                                   (i−b)

               6                                            0.25
                                                            0.00
               5                                           −0.25

                      450      500      550       600               450      500     550      600
                     (ii−a)                                0.010
                                                                   (ii−b)
             0.05                                          0.005
                                                           0.000
           −0.05
                                                          −0.005

                      450      500      550       600               450      500     550      600
                     (iii−a)                                0.50
                                                                   (iii−b)
               1                                            0.25
                                                            0.00
               0
                                                           −0.25

                      450      500      550       600               450      500     550      600
                     (iv−a)                                        (iv−b)
              0.2                                          0.025
              0.0
                                                          −0.025
             −0.2
                      450      500      550       600               450      500     550      600

                    Bivariate decomposition of prices and consumption: Feb 19, 2001 to April 10, 2005:

 (ia,b) data and estimated trend plus regression; (iia,b) seasonal component (S = 7, the day-of-week effect); (iiia,b)

 yearly cycle; (iva,b) irregular.


                                                                                                                        19 / 91
Forecasting results
We present MAPE for forecasting of one- to seven-days ahead prices for both uni- and bivariate models. The one-

to seven-days ahead forecasts are for the next seven days. The first forecast is for Monday, March 14, 2005 in

Week 637. The last forecast is for Sunday, April 10, 2005 in Week 640. The weeks 638 and 639 contain calendar

effects for Maundy Thursday (March 24, 2005) and the days until Easter Monday (March 28, 2005).

                   week 637                week 638               week 639                week 640
                   uni   biv               uni   biv              uni   biv               uni   biv
  horizon
  1M              0.83        1.11        0.15        0.07       0.83        1.01        0.92       0.27
  2T              0.86        0.94        0.51        0.53       1.20        1.36        0.74       0.20
  3W              1.43        1.55        0.67        0.79       1.40        1.52        0.62       0.16
  4T              1.94        2.09        0.64        0.88       1.71        1.75        0.60       0.14
  5F              1.69        1.93        0.65        0.72       2.01        2.00        0.60       0.30
  6S              1.62        1.95        0.58        0.69       2.26        2.17        0.67       0.43
  7S              1.61        2.05        0.68        0.90       2.44        2.27        0.79       0.56

                                                                                                                  20 / 91
Illustration 3: periodic dynamic factor analysis

Aim: the joint modeling of 24 hours in a daily panel of electricity
loads for EDF.
Focus: modelling and short-term forecasting of hourly electricity
loads, from one day ahead to one week ahead.
  • EDF provides a long time series: 9 years of hourly loads
  • We can establish a long-term trend component but also
  • different levels of seasonality (yearly, weekly, daily)
  • special day effects (EJP)
  • weather dependence (temperature, cloud cover)
  • We look at the intra-year as well as the long-run dynamics by
    using these different components.



                                                                      21 / 91
Periodic dynamic factor model specification
The adopted methodology builds on Dordonnat, et al (2008, IJF):
  • Model is for high-frequency data, for hourly data);
  • Hours are in the cross-section (yt is 24 × 1 vector);
  • The model dynamics are formulated for days: a multivariate
    daily time series model;
  • In effect, we adopt a periodic approach to time series
    modelling;
  • The right-hand side of the model is set-up as a multiple
    regression model;
  • We let the regression parameters evolve over time (days);
  • We have a time-varying regression model, written in
    state-space form;
  • The 24-dimensional time-varying parameters are subject to
    common dynamics (random walks);
  • Novelty: dynamic factors in the time-varying parameters.
                                                                  22 / 91
Daily National Electricity Load, 1995-2004

                                     80000




                                                                                                                   80000
         National Load (MegaWatts)




                                                                                       National Load (MegaWatts)
                                     60000




                                                                                                                   60000
                                     40000




                                                                                                                   40000
                                                         2000                      2005                              2002                                              2003
                                                          Year     (a)                                                                      Date    (b)




                                                                                                                   80000
                                     50000
         National Load (MegaWatts)




                                                                                       National Load (MegaWatts)
                                                                                                                   60000
                                     40000




                                                                                                                   40000
                                     30000




                                         0      5           10         15         20                                       −5   0    5      10   15      20     25      30
                                             Days elapsed since August 8th,2004 (c)                                                 National Temperature (°C)    (d)

Time series and temperature effects at 9 AM




                                                                                                                                                                              23 / 91
Daily National Electricity Load, 1995-2004


                                      60000




                                                                                                                  60000
              Mean Load (MegaWatts)




                                                                                          Mean Load (MegaWatts)
                                      50000




                                                                                                                  50000
                                      40000




                                                                                                                  40000
                                              1   2   3   4   5   6    7   8     9 10 11 12                               4   8   12     16    20   24
                                      60000




                                                                                                                  60000
                                                              Month        (a)                                                    Hour (b)
              Mean Load (MegaWatts)




                                                                                          Mean Load (MegaWatts)
                                      50000




                                                                                                                  50000
                                      40000




                                                                                                                  40000




                                                  4       8       12       16      20    24                               4   8   12     16    20   24
                                                                  Hour     (c)                                                    Hour   (d)
                                                                                   Average patterns

(c) Oct-Mar, (d) Apr-Sep




                                                                                                                                                         24 / 91
Multivariate Time Series Model
A periodic approach: from a univariate hourly to a daily 24 × 1
vector:
yt = (y1,t . . . yS,t )′ ,       S = 24 hours per day,         t = 1, . . . , T days.
Our multivariate time-varying parameter regression model is given
by:
                   K
    yt = µ t +          Btk xtk + εt ,   εt ∼ IIN (0, Σε ) ,    t = 1, . . . , T ,
                  k=1

   • Trends: µt = (µ1,t . . . µS,t )′
                                             k     k          k
   • Daily vectors of explanatory variables xt = (x1,t . . . xS,t )′ ,
     k = 1, . . . , K , depending only on the day or on the hour of the
     day.
                                   k       k        k
   • Regression coefficients βt = (β1,t . . . βS,t )′ , k = 1, . . . , K . In
     matrix form: Bt     k = diag (β k ), k = 1, . . . , K
                                     t
   • Irregular Gaussian white noise εt = (ε1,t . . . εS,t )′ .
                                                                                     25 / 91
Time-varying regressions and dynamic factors

                 K
   yt = µ t +         Btk xtk + εt ,   εt ∼ IIN (0, Σε ) ,      t = 1, . . . , T ,
                k=1

where the time-varying regression parameters are given by
        µt    = c 0 + Λ0 ft0 ,
         k
        βt    = c k + Λk ftk ,         k = 1, . . . , K ,    t = 1, . . . , T ,
                      j        j
with constant c j = (c1 . . . cS )′ , S × R j factor loading matrices Λj
and R j dynamic factors ftj = (f1,t . . . fR j ,t )′ , for j = 0, . . . , K and
                                      j      j

0 ≤ R j ≤ S.
  • Factor structure requires 0 < R j < S;
  • Constant parameter component for R j = 0;
  • Model has unrestricted component when R j = S;
  • Identification restrictions apply.
                                                                                     26 / 91
Dynamic factor specfications

Local linear trend model for factors in trends µt :

          ft+1 = ftj
            j
                           + gtj + v jt ,       v jt ∼ IIN(0, Σj )
                                                               v
            j
          gt+1 =             gtj + w jt ,       w jt ∼ IIN(0, Σj )
                                                                w



  • vector of dynamic factors ftj ,
  • slope or gradient vector gtj ,
  • level disturbance v jt and slope disturbance w jt .

Random walk model for factors in the regression coefficients:

ft+1 = ftj + e jt ,
  j
                      e jt ∼ IIN(0, Σe ),
                                     j
                                            j = 1, . . . , K ,   t = 1, . . . , T ,

with regression coefficient disturbance e jt .

                                                                                      27 / 91
Empirical application to French national hourly Loads
  • French national hourly electricity loads from Sept-95 until
      Aug-04
  •   Estimation of trivariate models for neighbouring hours
  •   Smooth trends
  •   Intentional missing values for special days (EJP) and turn of
      the year. No problem for state space models.
  •   Yearly pattern regressors: sine/cosine functions of time are
      used (2 frequencies)
  •   Day-of-the-week effects: day-type dummy regressors
  •   Weather dependence: heating degrees, smoothed-heating
      degrees and cloud cover
  •   Heating degrees beneath treshold temperature of 15 C
  •   Exponentially smoothed temperature
  •   Cooling degrees above treshold temperature of 18 C
  •   Implemention: SsfPack 3 of Koopman, Shephard and
      Doornik (2008, ssfpack.com) for Ox 6 (2008, doornik.com)
                                                                      28 / 91
Yearly pattern estimates per hour




         4     ˆk k
µs,t +
ˆ        k=1   βs,t xs,t   for hours (a) s = 0, 1, 2 ; (b) s = 3, 4, 5 ; (c) s = 6, 7, 8 (with extra component), etc.

Estimation: Jan 1997 - Aug 2003, Graph: Jan 1998 - Aug 2003.

                                                                                                                        29 / 91
Components 9 AM, factor model (blue) and univariate
                  Factor   Univariate
                                                         1500     Factor   Univariate
       500                                               1000

         0                                                500

                            2000        (a)   2002                          2000        (b)   2002
       300        Factor   Univariate
                                                         −500     Factor   Univariate

       200                                              −1000
       100                                              −1500

                            2000        (c)   2002                          2000        (d)   2002
         0        Factor   Univariate
                                                        −6000     Factor   Univariate

                                                        −8000
      −200
                                                       −10000
                            2000        (e)   2002                          2000        (f)   2002
    −10000        Factor   Univariate
                                                        60000     Factor   Univariate

    −12500
                                                        50000
    −15000

                            2000        (g)   2002                          2000          (h) 2002

             ˆ9
 (a) heating β9,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h) trend +

 yearly pattern

                                                                                                                       30 / 91
9 AM st. errors, factor model (blue) and univariate
            Factor    Univariate                                          Factor    Univariate
  300                                                           500
  200
                                                                250
  100

                         2000      (a)    2002                                       2000              (b)   2002
   75       Factor    Univariate
                                                                300       Factor    Univariate

   50                                                           200
   25                                                           100

                         2000      (c)    2002                                       2000              (d)   2002
   80       Factor    Univariate                                500       Factor    Univariate
   70
   60                                                           300
   50
                         2000      (e)    2002                                       2000              (f)   2002
  750       Factor    Univariate
                                                               2000       Factor    Univariate

  500
                                                               1000
  250
                         2000      (g)    2002                                     2000          (h)                   2005


                 ˆ9
(a) heating s.e. β9,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h)

trend + yearly pattern

                                                                                                                    31 / 91
Sample ACFs of residuals (daily lags)
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300
    0.2                     0.2                     0.2                     0.2
    0.0                     0.0                     0.0                     0.0

          0   100 200 300         0   100 200 300         0   100 200 300         0   100 200 300


Selecting the right model requires experience and stamina!

                                                                                                    32 / 91
Conclusions

• A general, flexible and insightful methodology is developed.
• Many dynamic features of load and price data can be
  captured.
• We can detect many interesting signals which are not
  discovered before.
• Decent forecasts.
• Decent diagnostics.
• Many possible extensions.
• Remaining challenge: a full multivariate unobserved
  components model for all 24 hours to capture evolutions of
  complete intradaily load pattern.
• More work is required !


                                                                33 / 91
Short Bibliography
• “Multivariate structural time series models” by Harvey and
  Koopman (1997), Chapter in Heij et al. (1997) Wiley.
• “Time-series analysis by state-space methods” by Durbin and
  Koopman (Oxford, 2001)
• “Periodic Seasonal Reg-ARFIMA-GARCH Models for Daily
  Electricity Spot Prices” by Koopman, Ooms and Carnero
  (JASA, 2007).
• “An hourly periodic state-space for modelling French national
  electricity load” by Dordonnat, et.al. (International Journal of
  Forecasting, 2008)
• “Forecasting economic time series using unobserved
  components time series models” by Koopman and Ooms
  (2011), Chapter in Clements and Hendry, OUP Handbook of
  Forecasting.

                                                                     34 / 91
Illustration 4: The macroeconomy in the euro area
Quarterly time series, 1981 – 2008, GDP in volumes,
for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.
         (i)                                              (ii)
                                                 13.2
12.8

                                                 13.0
12.6

                                                 12.8
12.4
  1980  1985      1990   1995   2000   2005   2010 1980  1985       1990   1995   2000   2005   2010
12.7 (iii)                                      12.25 (iv)

12.6
                                                12.00
12.5

                                                11.75
12.4

12.3                                            11.50

  1980     1985   1990   1995   2000   2005   2010 1980      1985   1990   1995   2000   2005   2010




                                                                                                       35 / 91
Illustration 4: The housing market in the euro area
Quarterly time series, 1981 – 2008, real house prices (HP),
for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.
          (i)                                          0.3     (ii)

  5.0
                                                       0.2


  4.5                                                  0.1


  4.0                                                  0.0

   1980         1985   1990   1995   2000   2005   2010 1980      1985   1990   1995   2000   2005   2010
          (iii)                                                (iv)
                                                       3.0
 0.25


 0.00                                                  2.5



−0.25                                                  2.0



   1980         1985   1990   1995   2000   2005   2010 1980      1985   1990   1995   2000   2005   2010




                                                                                                            36 / 91
Any (common) cyclical dynamics in the data ?
Autocorrelograms and sample spectra, based on first differences...
           GDP−Correlogram           GDP−Spectrum              HP−Correlogram            HP−Spectrum
   1                                                      1
 (i)                         0.4                                                 0.4
   0                                                      0
                             0.2                                                 0.2

    0          10      20      0.0        0.5       1.0    0       10      20      0.0        0.5      1.0
   1                                                      1                     0.75
 (ii)                        0.2
                                                                                0.50
   0                                                      0
                             0.1                                                0.25

    0          10      20      0.0        0.5       1.0    0       10      20      0.0        0.5      1.0
   1                                                      1
 (iii)                       0.4                                                0.50
   0                                                      0
                             0.2                                                0.25

    0          10      20      0.0        0.5       1.0    0       10      20      0.0        0.5      1.0
   1                                                      1
 (iv)                        0.4                                                 1.0
   0                                                      0
                             0.2                                                 0.5

       0       10      20      0.0        0.5       1.0   0        10      20      0.0        0.5      1.0


                                                                                                             37 / 91
The basic multivariate model
Multiple set of M economic time series, yit , is collected in
yt = (y1t , . . . , yMt )′ and model is given by
                                 (1)      (2)
                    yt = µt + ψt       + ψt     + εt ,
where the disturbance driving each vector component is a vector
too, with a variance matrix. The structure of the variance matrix
determines the dynamic interrelationships between the M time
series.
For example, if trend component µt follows the random walk,
µt+1 = µt + ηt with disturbance vector ηt , with variance matrix
Ση :
  • diagonal Ση , independent trends;
  • rank(Ση ) = p < M, common trends (cointegration);
  • rank(Ση ) = 1, single underlying trend;
  • Ση is zero matrix, constant.
Similar considerations apply to other components.
                                                                    38 / 91
Dynamic factor representations

We can formulate the multivariate unobserved components model
also by
                                   (1)           (2)
          yt = µ∗ + Aη µt + A(1) ψt
                             κ           + A(2) ψt
                                            κ          + Aε εt ,

where, for the trend component, for example, the loading matrix
Aη is such that
                                     ′
                          Ση = Aη Aη ,
and, similarly, loading matrices are defined for the other variance
matrices of disturbances that drive the components.
                                                                   (1)   (2)
Here the dynamic factors or unobserved components µt , ψt , ψt
and εt are ”normalised”.


                                                                               39 / 91
STAMP
Model is effectively a state space model: Kalman filter methods
can be applied for maximum likelihood estimation of parameters
(such as the loading matrices).
Kalman filter methods are employed for the evaluation of the
likelihood function and score vector.
Kalman filter and smoothing methods are employed for signal
extraction or the estimation of the unobserved components.
User-friendly software is available for state space analysis.
We have used S T A M P for this research project: a
multi-platform, user-friendly software: econometrics, time series
and forecasting by clicking.
It can treat multivariate unobserved components time series
models...
                                                                    40 / 91
Motivation of our study
  • Evidence of any relationship between housing prices and GDP
    in the euro area.
  • Focus on more recent developments...
  • We prefer to model the time series jointly and establish
    interrelationships between the time series
  • Focus on cyclical dynamics, long-term and short-term
  • Emphasis on real housing prices: relevant for the monetary
    policy
  • We also like to discuss synchronisation of housing markets in
    euro area

Empirical results are based on our data-set with two variables:
GDP and real house prices (HP); and for four euro area countries:
France, Germany, Italy and Spain.
                                                                    41 / 91
Relevant literature


• Unobserved components model: Harvey (1989)
• State space methods: Durbin and Koopman (2001)
• Multivariate unobserved components: Harvey and Koopman
  (1997), Azevedo, Koopman and Rua (2006);
• Parametric approaches for house prices:
    • Probit regressions: Borio and McGuire, 2004, van den Noord,
      2006;
    • Dynamic Factor models: Terrones, 2004, DelNegro and Otrok,
      2007, Stock and Watson, 2008;
    • VAR: Vargas-Silva, 2008, Goodheart and Hofmann, 2008.




                                                                    42 / 91
Univariate analysis
Objectives:
  • Verify the trend-cycle decomposition for each series
  • Verify whether possible restrictions are realistic

Results for GDP:
  • two short cycles in France and Italy are detected (¡6 years);
  • Germany and Spain contain both a short cycle (5.42 and 3.62
    years, resp.) and a long cycle (13.5 and 9.11 years)
  • Various cycles are deterministic (fixed sine-cosine wave)
Results for HP:
  • Results are quite different for each series
  • Two cycles for Germany (5.4 and 13.5 years)
  • Two short cycles for Italy (3.0 and 5.5 years) and France (3.1
    and 5.8 years)
  • For Spain a cycle reduces to an AR(1) process
                                                                     43 / 91
Univariate results for GDP

               France        Germany      Italy         Spain
GDP                     R           R             R             R
Trend var     0.65   0.03   0.01 0.03   0.48   0.03   0.10   0.03
Cycle 1 var   0.81   0.17   0.00 0.15   3.85   5.75   0.07   0.00
Cycle 1 ρ     0.94   0.90    1.0 0.90   0.87   0.90   0.95   0.90
Cycle 1 p     3.04      5   5.42    5   2.97      5   3.62      5
Cycle 2 var   0.00      1   1.81 2.86   0.00   7.79   0.00   2.31
Cycle 2 ρ      1.0   0.95   0.95 0.95   1.00   0.95   1.00   0.95
Cycle 2 p      5.8     12   13.5   12   5.50     12   9.11     12
Irreg var        1    0.0      1    1      1      1      1      1
N              7.2   11.4   3.23 5.23   6.58   11.1   27.1   34.9
Q             14.5   24.9   15.1 14.6   9.26   13.3   22.1   24.8
R2            0.31   0.24   0.11 0.02   0.23   0.12   0.22   0.12

                                                                    44 / 91
Univariate results for HP

               France        Germany      Italy         Spain
RHP                     R           R             R             R
Trend var     0.59   0.03   0.34 0.03   0.00   0.03   0.39   0.03
Cycle 1 var   0.00   0.01   0.31 1.51   0.04   0.02      1   0.01
Cycle 1 ρ      1.0   0.90   0.97 0.90   0.96   0.90   0.34   0.90
Cycle 1 p     6.34      5   4.48    5   1.11      5      –      5
Cycle 2 var   0.00   2.19      1 19.9      1   49.4   0.00   39.5
Cycle 2 ρ      1.0   0.95   0.61 0.95   0.99   0.95   0.99   0.95
Cycle 2 p     8.37     12   2.82   12   13.3     12      –     12
Irreg var        1      1      0    1      0      1      0      1
N             23.8   0.59   5.89 9.95   7.03   8.32   36.1   11.9
Q             10.6    187   55.5 111    13.7   68.4   29.3    127
R2            0.61   0.25   0.35 0.15   0.56   0.22   0.47   0.28

                                                                    45 / 91
Cycle correlations from univariate analysis
                                   (1)      (2)
Correlations for combined cycles (ψt     + ψt ):
  • Strong correlations between GDP of four countries
    (correlations range from 0.52 to 0.94)
  • The correlations with German GDP are the lowest
  • Correlations between HP of four countries range from 0.42 to
    0.94
  • The highest correlation is between Spain and France HP’s
  • Correlation on combined cycle are mostly due to long-term
    cycle, not to the short-term cycle
  • Correlations between GDP and HP for each country range
    from 0.06 for Germany to 0.76 for Spain
  • Overall low cross-correlations between GDP of one country
    and HP of another country

                                                                   46 / 91
Correlations between combined cycles for eight series



                                              (1)   (2)
                              Combined cycle (ψt + ψt )
         F GDP     F HP    G GDP G HP I GDP I HP            S GDP     S HP
 F GDP      1.00    0.51      0.52   0.23      0.83  0.15      0.89    0.61
 F HP       0.51    1.00      0.44   0.44      0.52  0.68      0.68    0.94
 G GDP      0.52    0.44      1.00   0.50      0.54  0.47      0.61    0.44
 G HP       0.23    0.44      0.50   1.00      0.08  0.80      0.22    0.42
 I GDP      0.83    0.52      0.54   0.08      1.00  0.06      0.84    0.64
 I HP       0.15    0.68      0.47   0.80      0.06  1.00      0.29    0.64
 S GDP      0.89    0.68      0.61   0.22      0.84  0.29      1.00    0.76
 S HP       0.61    0.94      0.44   0.42      0.64  0.64      0.76    1.00




                                                                      47 / 91
Correlations between short cycle for eight series



                                                 (1)
                                    Short cycle ψt
        F GDP     F HP    G GDP      G HP I GDP        I HP    S GDP     S HP
F GDP      1.00    0.46      0.40      0.24     0.64   -0.46      0.57    0.42
F HP       0.46    1.00      0.29      0.62     0.33   -0.51      0.35    0.39
G GDP      0.40    0.29      1.00      0.32     0.75   -0.16      0.67    0.58
G HP       0.24    0.62      0.32      1.00     0.18   -0.52      0.06    0.13
I GDP      0.64    0.33      0.75      0.18     1.00   -0.13      0.61    0.65
I HP      -0.46   -0.51     -0.16     -0.52    -0.13    1.00     -0.25   -0.19
S GDP      0.57    0.35      0.67      0.06     0.61   -0.25      1.00    0.75
S HP       0.42    0.39      0.58      0.13     0.65   -0.19      0.75    1.00




                                                                         48 / 91
Correlations between long cycle for eight series



                                                (2)
                                    Long cycle ψt
        F GDP     F HP    G GDP     G HP I GDP        I HP    S GDP     S HP
F GDP      1.00    0.51      0.53     0.23     0.89    0.16      0.90    0.63
F HP       0.51    1.00      0.46     0.44     0.58    0.68      0.68    0.94
G GDP      0.53    0.46      1.00     0.52     0.44    0.49      0.62    0.46
G HP       0.23    0.44      0.52     1.00     0.07    0.82      0.22    0.43
I GDP      0.89    0.58      0.44     0.07     1.00    0.08      0.90    0.72
I HP       0.16    0.68      0.49     0.82     0.08    1.00      0.29    0.64
S GDP      0.90    0.68      0.62     0.22     0.90    0.29      1.00    0.76
S HP       0.63    0.94      0.46     0.43     0.72    0.64      0.76    1.00




                                                                        49 / 91
Bivariate analysis
For each country, we carry out a bivariate analysis between GDP
and RHP:
                                (1)    (2)
                   yt = µt + ψt + ψt + εt ,
where yt is a 2 × 1 vector for two series: GDP and HP.
We can conclude that
  • highest correlation is found for cycle components (except
    Italy)
  • for France, high correlation for medium-term cycle (8 years)
    but no dependence for long-term cycle (15.6 years)
  • for Spain, strong correlations for both medium-term (8.2
    years) and long-term (14.4 years)
  • for Germany, correlations for both cycles, but with low periods
    (4.3 and 7 years)

                                                                      50 / 91
Bivariate results for GDP and HP


        GDP    RHP      corr    per     ρ    diag   GDP    RHP
FRA
trend    0.0    0.0      0.0      –      –   N      3.25   13.4
cyc 1    3.0    3.3     0.88    8.0   0.98   Q      17.0   17.4
cyc 2    1.0    126     0.07   15.6   0.99   R2     0.38   0.63
irreg    0.6    1.6    -0.19      –      –
GER
trend    0.0   0.003    0.0       –      –   N      8.52   1.08
cyc 1    2.5     5.4   -0.6     4.3   0.90   Q      6.86   42.1
cyc 2    3.1     0.5    1.0     7.0   0.98   R2     0.39   0.29
irreg    4.3     1.1   0.58       –      –




                                                                  51 / 91
Bivariate results for GDP and HP


        GDP    RHP     corr   per      ρ    diag   GDP    RHP
ITA
trend    0.1    0.9   -0.15      –      –   N      4.19   4.57
cyc 1    4.3   16.2   -0.08    6.0   0.92   Q      10.1   8.60
cyc 2    0.0    8.4     0.0    1.1   0.94   R2     0.14   0.47
irreg    0.8    1.2    0.96      –      –
SPN
trend    0.0    0.0     0.0      –      –   N      9.05   21.7
cyc 1    3.3   11.9    0.95    8.2   0.98   Q      17.5   43.0
cyc 2    0.0   83.3    0.82   14.4   0.99   R2     0.45   0.73
irreg    3.9    7.7   -0.35      –      –




                                                                 52 / 91
Four-variate cross-country analysis of GDP and RHP
Now we incorporate earlier findings and impose a strict short- and
long-term cycle decomposition for our analysis.
In particular, we have
  • an independent trend µt (i.e. diagonal variance matrix Ση for
    disturbance vectors of µt+1 = µy + ηt )
  • similarly, an independent irregular component εt (i.e. diagonal
    variance matrix Σε )
  • a two-cycle parametrization with restricted periods of 5 and
    12 years
  • the rank of the 4 × 4 cycle variance matrices Σκ is 2:
    common cyles ...
  • we load the two ”independent” cycles on France and Germany,
    i.e. cyclical dynamics of Spain and Italy are obtained as linear
    functions of the two times two (short and long) cyclical factors
                                                                       53 / 91
Four-variate decomposition for GDP, cross-country
 13.00     LFRA_GDP       Level               LGER_GDP       Level               LITA_GDP       Level
                                                                                                          12.25     LSPA_GDP       Level
                                    13.25                               12.6
 12.75                                                                                                    12.00
                                    13.00                                                                 11.75
 12.50                                                                  12.4
                                    12.75                                                                 11.50
    1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010
           LFRA_GDP−Cycle 1                   LGER_GDP−Cycle 1
                                                                        0.02     LITA_GDP−Cycle 1
                                                                                                          0.010     LSPA_GDP−Cycle 1
  0.01                               0.02                               0.01                              0.005
  0.00                               0.00                               0.00                              0.000
 −0.01                              −0.02                              −0.01                             −0.005

    1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010
           LFRA_GDP−Cycle 2                   LGER_GDP−Cycle 2                   LITA_GDP−Cycle 2
                                                                                                          0.050     LSPA_GDP−Cycle 2
 0.025                              0.025                               0.02                              0.025
 0.000                              0.000                                                                 0.000
                                                                        0.00
−0.025                             −0.025                                                                −0.025
    1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010
           LFRA_GDP−Irregular                 LGER_GDP−Irregular
                                                                      0.0050     LITA_GDP−Irregular
                                                                                                           0.02     LSPA_GDP−Irregular
 0.001                               0.01
                                                                      0.0025                               0.01
 0.000                               0.00                             0.0000                               0.00
−0.001                              −0.01                            −0.0025                              −0.01
    1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010 1980    1990       2000       2010



                                                                                                                                                  54 / 91
Four-variate results for cross-country: GDP


            Fra     Ger       Ita     Spn        Fra       Ger
  Cycle   short (cov ×10−6 )                   factor loadings
  Fra       4.11   0.25 ∗   0.77 ∗   -0.40 ∗       1         0
  Ger       1.77    11.8    0.81 ∗    0.78 ∗       0         1
  Ita       5.65    10.1     13.1    0.27 ∗     1.08      0.69
  Spn      -1.04    3.50     1.27      1.65    -0.41      0.35
  Cycle   long (cov ×10−6 )
  Fra       8.08   0.79 ∗   0.48 ∗   0.98 ∗       1         0
  Ger       7.94    12.5   -0.16 ∗   0.64 ∗       0         1
  Ita       3.43   -1.39     6.28    0.66 ∗    1.42     -1.02
  Spn       11.2    9.11     6.73     16.4     1.79     -0.41




                                                                 55 / 91
Four-variate results for cross-country: GDP



• Diagnostic statistics are satisfactory
• Strong correlation France-Germany for long-term cycle
• Business cycles for Italy and Spain are closely connected with
  the one for France (however, negative ??? marginal
  correlation Fra-Spa for short-term cycle)
• German cycles strongly affect business cycles in Italy and
  Spain (however, their marginal correlations for longer cycle are
  negative)




                                                                     56 / 91
Four-variate decomposition for HP, cross-country
  5.5     LFRA_RHprice       Level
                                       0.3     LGER_RHprice       Level             LITA_RHprice       Level               LSPA_RHprice       Level
                                                                           0.25                                    3.0
  5.0                                  0.2
  4.5                                                                      0.00                                    2.5
                                       0.1
  4.0                                                                     −0.25                                    2.0
                                       0.0
   1980    1990        2000        2010 1980    1990       2000         2010 1980    1990          2000        2010 1980    1990        2000          2010
 0.02     LFRA_RHprice−Cycle 1
                                      0.02     LGER_RHprice−Cycle 1                 LITA_RHprice−Cycle 1
                                                                                                                  0.04     LSPA_RHprice−Cycle 1
                                      0.01                                 0.05                                   0.02
 0.00
                                      0.00                                                                        0.00
                                                                           0.00
−0.02                                −0.01
                                                                                                                 −0.02
   1980    1990        2000        2010 1980    1990       2000         2010 1980    1990          2000        2010 1980    1990        2000          2010
          LFRA_RHprice−Cycle 2
                                     0.050     LGER_RHprice−Cycle 2
                                                                            0.2     LITA_RHprice−Cycle 2                   LSPA_RHprice−Cycle 2
  0.1                                                                       0.1                                    0.2
                                     0.025
  0.0                                0.000                                  0.0                                    0.0
 −0.1                                                                      −0.1                                   −0.2
                                    −0.025
   1980    1990        2000        2010 1980    1990       2000         2010 1980    1990          2000        2010 1980    1990        2000          2010
          LFRA_RHprice−Irregular               LGER_RHprice−Irregular
                                                                          0.002     LITA_RHprice−Irregular                 LSPA_RHprice−Irregular
 0.01                                 5e−5                                                                        0.01
                                                                          0.001
 0.00                                    0                                0.000                                   0.00
−0.01                                −5e−5                               −0.001
   1980    1990        2000        2010 1980    1990       2000         2010 1980    1990          2000        2010 1980    1990        2000          2010



                                                                                                                                                             57 / 91
Four-variate results for cross-country: HP


           Fra      Ger       Ita     Spn        Fra       Ger
 Cycle   short (cov ×10−6 )                    factor loadings
 Fra       15.5   0.37 ∗ -0.89 ∗      0.05 ∗       1         0
 Ger       4.73    10.8    0.10 ∗    -0.91 ∗       0         1
 Ita      -21.0     1.97     36.2    -0.50 ∗   -1.64      0.90
 Spn       0.89    -14.6    -14.6      23.8     0.55     -1.60
 Cycle   long (cov ×10−6 )
 Fra       44.5   0.38 ∗    0.70 ∗   0.93 ∗       1         0
 Ger       4.43    3.13    -0.40 ∗   0.69 ∗       0         1
 Ita       66.9    -10.3    207.1    0.38 ∗    2.13     -6.30
 Spn      100.4    19.9      88.3    262.8     1.89      3.69




                                                                 58 / 91
Four-variate results for cross-country: HP


• Overall, these results seem to indicate that there is less
  evidence of common (cyclical) dynamics in HP series
• Low correlations between France and Germany
• Strong negative correlations for the 5-year cycle between
  Fra-Ita and Ger-Spa
• However, more commonalities for the 12-year cycle (Fra-Spa,
  Fra-Ita, Ger-Spa)
• Similarities between correlation matrices for the 12-year HP
  and GDP cycles, except that relationship Fra-Ger is stronger
  for GDP (0.79 against 0.38 for HP)




                                                                 59 / 91
Eight-variate results: HP and GDP for four countries

Similar restrictions apply as in four-variate analyses.

We conclude that
  • strong correlations among GDPs for short-term cycles but less
     evidence for long-term cycles, especially for Germany
  • low correlations among HP series.
  • for short-term cycle, these correlations for HP Fra-Ger is 0.65
     and for HP Spa-Ger is -0.95.
  • only a few positive correlations for the long-term cycle in HP
     have been found: Fra-Spa (0.58) and Ger-Ita (0.57)
  • correlations HP-GDP are only found for long-term cycle,
     especially for France and Spain.


                                                                      60 / 91
Eight-variate results: short cycle correlations


          France       Germany           Italy            Spain

       GDP     HP     GDP     HP     GDP       HP     GDP       HP
F-G     1     -0.33    0.67   0.10    0.81    -0.59    0.77     0.13
F-H             1     0.075   0.65   -0.35    -0.13   -0.12    -0.64
G-G                      1    0.17    0.80    -0.27    0.88   -0.011
G-H                             1    0.055    -0.26   -0.10    -0.95
I-G                                     1    -0.037    0.66    0.034
I-H                                             1     -0.55   -0.040
S-G                                                      1      0.34
S-H                                                               1




                                                                       61 / 91
Eight-variate results: long cycle correlations


         France      Germany            Italy          Spain

       GDP   HP     GDP     HP      GDP      HP     GDP      HP
F-G     1    0.95   0.19   0.043    0.72     0.41   0.54     0.50
F-H            1    0.44    0.24    0.63     0.43   0.57     0.58
G-G                   1    0.41     -0.31    0.26   0.44     0.21
G-H                           1    -0.005    0.57   0.036    0.29
I-G                                   1     0.045   0.12     0.37
I-H                                            1    0.13    0.099
S-G                                                   1     0.61
S-H                                                            1




                                                                    62 / 91
Illustration 5 : Modelling U.S. Yield Curve

       Yield (in %)
6.50



6.25



6.00



5.75



5.50



5.25
                                                              Maturity (in months)
   0       10         20   30   40   50   60   70   80   90     100      110     120



                                                                                       63 / 91
Time Series of Four Maturities
     Yield (in %)            Time to maturity: 3 month
                                                          10 Yield (in %)                Time to maturity: 1 year

8
                                                           8

6
                                                           6

4
                                                           4
                                              Date                                                      Date
 1985           1990       1995              2000           1985               1990   1995             2000
   Yield (in %)                                                 Yield (in %)
                               Time to maturity: 3 year                                  Time to maturity: 10 year
10
                                                         10.0

8

                                                          7.5
6

                                                          5.0
                                               Date                                                     Date
 1985               1990   1995              2000           1985               1990   1995             2000



                                                                                                                     64 / 91
Term Structure of Interest Rates over Time
        10.0
Yield (Percent)
       7.5
        5.0




          125
                   100
                             75                                                                     2000.0
                         Mat                                                               1997.5
                            urity        50                                       1995.0
                                    (Mo
                                       nths                            1992.5
                                            )   25            1990.0       Time
                                                     1987.5




                                                                                                             65 / 91
Literature Review
Earlier analyses of this data:
  • Affine Term Structure Models (ATSM):
     Vasicek (1977), Cox, Ingersoll, and Ross (1985), Duffie and
     Kan (1996), Dai and Singleton (2000), and De Jong (2000)
  • Nelson-Siegel Model (NS):
     Nelson and Siegel (1987), Diebold and Li (2006), Diebold,
     Rudebusch and Aruoba (2006), De Pooter (2007), and
     Koopman, Mallee, and Van der Wel (2009)
  • Arbitrage-Free Nelson-Siegel Model (AFNS):
     Christensen, Diebold, and Rudebusch (2007)
  • Functional Signal plus Noise (FSN):
     Bowsher and Meeks (2008)
In all cases: a dynamic factor model set-up !

                                                                 66 / 91
Still Some Outstanding Issues...

• Which of these models provides an accurate description of the
  data?
    • Duffee (2002) and Bams and Schotman (2003) provide
       evidence against affine term structure models
• What are the dynamics of the underlying factors:
  Stationary or Nonstationary?
    • Stationary: Affine Term Structure Models, Nelson-Siegel,
       Arbitrage-Free Nelson-Siegel
    • Nonstationary: Campbell and Shiller (1987), Hall, Anderson
       and Granger (1992), Engsted and Tanggaard (1994) and
       Bowsher and Meeks (2008)
• What are the dynamics of the underlying factors: #lags?
   • Almost all models take a VAR(1) for the factor dynamics
   • Exception: Bowsher and Meeks (2008)



                                                                   67 / 91
Further Outline


• (General) Dynamic Factor Model (DFM)
    • General Set-Up
    • Stationary and Nonstationary
• Smooth Dynamic Factor Model (SDFM)
    • Specification
    • Knot Selection
• Other Restrictions of the DFM
• Results
• Conclusion




                                         68 / 91
The Dynamic Factor Model (DFM)
• Time series panel of N monthly yield observations
  yt = (yt (τ1 ), . . . , yt (τN ))′ with yt (τi ) the yield at time t with
  maturity τi
• The general dynamic factor model is given by:

             yt    = µy + Λft + εt ,              εt ∼ N(0, H),
              ft   = Uαt
          αt+1 = µα + T αt + R ηt ,                ηt ∼ N(0, Q),

  for t = 1, . . . , n
• In here ft is an r -dimensional stochastic process that is
  generated by the p-dimensional state vector αt and ηt is a
  q × 1 vector
• We take H diagonal and have for the p × 1 initial state
  α1 ∼ N(a1 , P1 ), and assume N >> r , r ≤ p and p ≥ q
                                                                              69 / 91
The Dynamic Factor Model (DFM) – Cont’d


• Vectors µy and µα , and matrices Λ, H, U, T and Q are
  system matrices, R is selection matrix of ones and zeros
• Special case of state space model.
• All vector autoregressive moving average processes can be
  formulated in this framework (see, e.g., Box, Jenkins and
  Reinsel (1994))
• In this paper: VAR and Cointegrated VAR (CVAR) for ft .
  Obtained by suitable specification of U, T and R
• Elements of system matrices µy , Λ, H, µα , T and Q generally
  contain parameters that need to be estimated




                                                                  70 / 91
The Dynamic Factor Model (DFM) – Cont’d

• Need to impose restrictions on loading and variance matrices
  to ensure identification, see Jungbacker and Koopman (2008):

    • Vectors µy and µα not both estimated without restrictions
      =¿ Restrict µα = 0 to focus on loading matrix Λ
    • Impose restrictions on Λ, T & Q that govern covariances
      =¿ Restrict r rows in Λ to be r × r identity matrix:
                                                   
                              1        0         0
                          0           1         0 
                                                   
                          0           0         1 
                                                   
                     Λ =  λ4,1       λ4,2     λ4,3 
                                                   
                          .  .        .
                                       .         . 
                                                 . 
                          .           .         .
                            λN,1      λN,2     λN,3



                                                                  71 / 91
Dynamic Factor Model (DFM) – Stationary Case
• Take a VAR(k) model for the r × 1 vector ft :

                   k−1
          ft+1 =         Γt−j ft−j + ζt ,     ζt ∼ NID(0, Qζ )
                   j=0

• Stationarity imposed by restriction that |Γ(z)| = 0 has all
  roots outside the unit circle
• Can write this ft in DFM. For example, for k = 1 have

                                αt   = ft ,
                                U = Ir ,
                                R = Ir ,
                                T    = Γ0 ,
                                Q = Qζ

                                                                 72 / 91
Dynamic Factor Model (DFM) – Nonstationary Case
 • Now the ft are generated by a cointegrated vector
   autoregressive process:
                        k−1
    ∆ft+1 = βγ ′ ft +          Γj ∆ft−j + ζt ,            ζt ∼ N(0, Qζ ),
                         j=0

 • The r × s matrices β and γ have full column rank; in the
   nonstationary case s < r and all ft nonstationary
 • We propose an alternative but observationally equivalent
   specification for ft via factor rotation:

                               f¯N                   ′
                  f¯ =          t
                                     =     β⊥    γ       ft ,
                   t
                               f¯S
                                 t
                                             ¯
   also need to construct new loading matrix Λ =                ¯
                                                                ΛN   ¯
                                                                     ΛS

                                                                            73 / 91
Dynamic Factor Model (DFM) – Estimation

• As noted earlier, the Dynamic Factor Model (DFM) can be
  seen as a special case of state space model
• Generally we can use likelihood-based methods: direct ML
  and/or EM methods
• However. . .
    • . . . the dimension of the observations vector is much larger
      than the state vector
    • . . . there is a large number of parameters (DFM-VAR(1),
      N = 17, r = 3: 91 parameters)
• We therefore estimate the models using the methodology of
  Jungbacker and Koopman (2008) and estimate parameters by
  direct ML using analytical score expressions



                                                                      74 / 91
Smooth Dynamic Factor Model (SDFM)
• For cross-sectional observation i we can write the DFM as:
                        r
  yt (τi ) = µy ,i +         λij fjt +εit ,     t = 1, . . . T ,   i = 1, . . . , N,
                       j=1

  where λij is the loading of factor j on maturity i
• We propose to let the loading parameter be an unknown
  function gj (·) for each factor j, where the argument of the
  function relates to i
• Assume these functions g1 (·), . . . , gr (·) smooth functions of
  time to maturity:
                                       λij = gj (τi )
• In practice: cubic spline for each gj (·)
• Call this Smooth Dynamic Factor Model (SDFM)

                                                                                   75 / 91
Smooth Dynamic Factor Model (SDFM) – Spline
• In a spline the location of the knots determines how the factor
  loadings behave for varying maturities
                        j
• Knot k for column j: sk
• Order the knots by time to maturity:
                   j             j
           τ 1 = s 1 < · · · < s Kj = τ N ,        j = 1, . . . , r

• Get following loading function:
                                           j 
                                    gj (s1 )
                                       .
                                        .     
     gj (τi ) = wij λj ,     λj =      .     ,        j = 1, . . . , r ,
                                          j
                                    gj (sKj )

  with wij a 1 × Kj vector (only depends on the knot locations)
  and λj a Kj × 1 parameter vector

                                                                             76 / 91
Smooth Dynamic Factor Model (SDFM) – Select Knots


   • But how many knots Kj to select in the spline W λ?
       • Small number of knots: Loadings lie on same polynomial for
         considerable number of maturities
       • Large number of knots: Get closer to unrestricted DFM
   • Propose using a Wald test procedure to determine the knots
   • This is standard as we are testing linear restrictions
   • Amounts to an iterative general-to-specific approach:
      1. Start with all knots
      2. Calculate test statistic for all knots
      3. Remove knot with smallest non-significant statistic
      4. Continue with 2 and 3 until all knots are significant




                                                                      77 / 91
Dynamic Factor Models for the Term Structure

• The general dynamic factor model is given by:

            yt    = µy + Λft + εt ,          εt ∼ N(0, H),
             ft   = Uαt
         αt+1 = µα + T αt + R ηt ,           ηt ∼ N(0, Q),

• It nests the term structure models mentioned earlier
• Functional Signal plus Noise – Bowsher and Meeks (2008)
     • Rather than a spline for the factor loadings they adopt the
       Harvey and Koopman (1993) time-varying spline for the yield
       curve:
                  yt = µy + Wft + εt ,      εt ∼ NID(0, H),
       with W as before and ft time-varying knot values
     • Take a CVAR(k) for ft and have restrictions on Λ


                                                                     78 / 91
Dynamic Factor Models for the Term Structure – Cont’d

   • Nelson-Siegel – Nelson and Siegel (1987), Diebold and Li
     (2006), Diebold, Rudebusch and Aruoba (2006)
       • The yield curve is expressed as a linear combination of smooth
         factors
                             1 − e −λτ             1 − e −λτ
           gns (τ ) = ξ1 +               · ξ2 +              − e −λτ    · ξ3
                                λτ                    λτ

         which gives

                    yt = µy + Λns ft + εt ,       εt ∼ NID(0, H)

       • Interpretation as level, slope and curvature for the factors
       • Typically: (restricted) VAR(1) for the state, µy = 0
       • Restrictions on Λ



                                                                               79 / 91
Dynamic Factor Models for the Term Structure – Cont’d

   • Arbitrage-Free NS – Christensen, Diebold and Rudebusch
     (2007)
       • The NS model is not arbitrage free
       • CDR employ “reverse engineering” and obtain an
          Arbitrage-Free NS model
       • Dynamics of latent factors now coming from solution of SDE,
          plus ‘correction’ term for µy
       • Restrictions on Λ, T and µy
   • Affine Term Structure Models – Duffie and Kan (1996)
      • Term structure can be explained by dynamics of unobserved
        short rate
      • Short rate depends on unobserved factors
      • Focus on Gaussian case
      • Restrictions on Λ, T and µy



                                                                       80 / 91
Results
Strategy:
  • Following, e.g., Litterman and Scheinkman (1991) we only
    look at 3-factor models
  • Restrict ourselves to Gaussian models
  • Use an existing dataset: unsmoothed Fama-Bliss
  • For DFM, SDFM, FSN and NS: VAR and CVAR
  • For CVAR case focus on 1 random walk
We show the following results:
  • VAR and CVAR for DFM
  • Results for SDFM
  • Estimation results NS, FSN, AFNS and ATSM
  • In-sample fit of all models
  • Validity of restrictions

                                                               81 / 91
DFM Likelihoods and AIC


Below we show the value of the loglikelihood at the ML value
(ℓ(ψ)) and AIC (AIC) for the Dynamic Factor Model (DFM):


  Model         ℓ(ψ)     AIC      Model           ℓ(ψ)     AIC
  VAR(1)      3894.5   -7595      CVAR(1)       3899.0   -7606
  VAR(2)      3918.5   -7625      CVAR(2)       3923.7   -7637
  VAR(3)      3922.6   -7615      CVAR(3)       3927.7   -7627
  VAR(4)      3932.2   -7616      CVAR(4)       3937.3   -7628


Note: Similar results hold for the NS and FSN model



                                                                 82 / 91
DFM-CVAR Influence of Factor Dynamics on Loadings

                             CVAR(1)   CVAR(2)
         Panel A             CVAR(3)   CVAR(4)             Panel B
 1.0

                                                 1.50
 0.5
                                                 1.25

 0.0
    0         25   50   75      100      125           0        25   50   75   100   125
                                                 1.0
 1.0

                                                 0.5
 0.5

                                                 0.0
 0.0
     0        25   50   75       100     125           0        25   50   75   100   125
 1.0                                               1

                                                   0
 0.5
                                                 −1

 0.0                                             −2

    0         25   50   75      100      125        0          25    50   75   100   12583 / 91
Smooth Dynamic Factor Model – Knot Selection


            Maturity               Unrestricted model                    Final result
            6               2.65       4.22∗     6.08∗        59.08∗∗       6.50∗         5.24∗
            9               0.79       2.40      5.59∗            -         6.58∗        8.92∗∗
            12              0.23       1.35      4.29∗            -       16.25∗∗       19.62∗∗
            15              0.04       0.33      1.51             -       24.17∗∗       26.83∗∗
            18              0.00       0.02      0.28             -           -             -
            21              0.95       0.74      1.52         18.55∗∗         -             -
            24              3.51       2.37      3.99∗        23.13∗∗         -          7.35∗∗
            36              1.14       1.50      6.68∗∗           -           -         26.88∗∗
            48              0.44       2.88      13.47∗∗          -       30.07∗∗       52.87∗∗
            60              1.19       4.99∗     18.04∗∗          -       26.79∗∗       54.39∗∗
            72              2.59       5.74∗     15.67∗∗          -       22.80∗∗       43.00∗∗
            84              2.59       4.57∗     8.81∗∗           -           -         17.85∗∗
            96              0.76       1.68      1.79         7.68∗∗          -           5.10∗
            108             0.01       0.05      0.00             -           -             -


Note: 3, 30 and 120 months not shown as these knots can not be removed




                                                                                                  84 / 91
SDFM Factor Loadings – CVAR

      Loading 1                                                 SDFM   DFM
1.0


0.5


0.0

          10      20   30   40   50   60   70   80   90   100    110   120
      Loading 2
1.0


0.5



        10        20   30   40   50   60   70   80   90   100    110   120
1.0 Loading 3


0.5


0.0

          10      20   30   40   50   60   70   80   90   100    110   120
                                                                        85 / 91
VAR coefficient matrix estimates


              Panel A: Stationary models
    SDFM                NS                 FSN
      real    img.       real     img.       real    img.
1    0.164    0.159     0.156 0.166         0.216    0.143
2    0.164   -0.159     0.156 -0.166        0.216   -0.143
3    0.607    0.134     0.593 0.056         0.642    0.259
4    0.607   -0.134     0.593 -0.056        0.642   -0.259
5    0.965      -       0.964       -       0.969      -
6    0.992      -       0.992       -       0.993      -




                                                             86 / 91
VAR coefficient matrix estimates (cont’)


              Panel B: Nonstationary models
     SDFM                NS                 FSN
       real     img.       real    img.       real    img.
1     0.155     0.162     0.151 0.165        0.206    0.143
2     0.155    -0.162     0.151 -0.165       0.206   -0.143
3     0.601     0.123     0.594 0.099        0.649    0.258
4     0.601    -0.123     0.594 -0.099       0.649   -0.258
5     0.973       -       0.972      -       0.970      -
6       1         -         1        -         1        -




                                                              87 / 91
NS Influence of Factor Dynamics on Loadings

To get a feeling how the choice of factor dynamics affects the
factor loadings we estimate the factor loadings parameter λ in the
Nelson-Siegel model for different choices of factor dynamics:


      Model            p=1         p=2        p=3        p=4
      VAR(p)          0.07303     0.07211    0.07216    0.07193
      CVAR(p)         0.07302     0.07210    0.07213    0.07191


Recall that for the Nelson-Siegel model we have

                      1 − e −λτ             1 − e −λτ
    gns (τ ) = ξ1 +               · ξ2 +              − e −λτ   · ξ3
                         λτ                    λτ


                                                                       88 / 91
All Models - Overview
Finally, we provide an overview of all models and test the
restrictions imposed on the DFM by the various models

            Model                  ℓ(ψ)    nψ     AIC
            DFM-VAR(2)            3918.5   106   -7625
            DFM-CVAR(2)           3923.7   105   -7637
            SDFM-VAR(2)           3906.8   85    -7644
            SDFM-CVAR(2)          3913.6   85    -7657
            FSN-VAR(2)            3479.0   64    -6830
            FSN-CVAR(2)           3483.7   63    -6841
            NS-VAR(2)             3808.4   65    -7487
            NS-CVAR(2)            3813.5   64    -7499
            AFNS                  3253.3   42    -6423
            ATSM                  3393.0   30    -6726


                                                             89 / 91
All Models - Ljung-Box Statistics



                                    CVAR(2)   factors                             VAR(1) factors
      Maturity         DFM         NS          FSN       SDFM            NS          AFNS        ATSM
      3                5.8         6.2         84.3∗∗    6.0             11.6        10.5        17.9
      6                7.1         7.4         11.2      7.4             12.6        11.8        33.1∗∗
      9                19.2∗       19.3∗       11.6      18.7∗           31.8∗∗      39.9∗∗      55.1∗∗
      12               22.5∗∗      29.2∗∗      16.7      23.1∗∗          36.2∗∗      53.1 ∗∗
                                                                                                 52.6∗∗
      18               12.8        13.0        13.2      12.7            22.2∗∗      28.1∗∗      22.2∗∗
      21               12.2        12.0        13.8      12.2            18.8∗       22.4∗∗      19.1∗
      24               10.2        11.2        15.3      10.6            18.9∗       21.6∗∗      22.0∗∗
      30               9.3         9.4         10.5      9.1             17.2        15.8        16.0
      60               5.9         5.7         9.2       5.6             11.5        10.3        11.2
      84               8.4         9.4         10.4      9.1             15.3        12.9        17.4
      120              10.1        9.7         6.9       11.0            9.7         9.3         12.2


Note: To preserve space 15, 36, 48, 72, 96 and 108 months omitted here




                                                                                                          90 / 91
All Models - Tests of Restrictions


        Stationary Models      Nonstationary Models
Model     LR    k p-value       LR     k    p-value
NS      220.2 41 0.000         220.4 41      0.000
FSN     879.0 42 0.000         879.8 42      0.000
SDFM     23.4 21 0.320         20.2 20       0.450


         Panel C: Arbitrage-Free Models
         Model        LR      k p-value
         AFNS       1330.4 64 0.000
         ATSM       1051.0 76 0.000




                                                      91 / 91

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Paris2012 session3b

  • 1. Time Series Forecasting Siem Jan Koopman http://personal.vu.nl/s.j.koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2012
  • 2. Unobserved components: decomposing time series A basic model for representing a time series is the additive model yt = µt + γt + εt , t = 1, . . . , n, also known as the classical decomposition. yt = observation, µt = slowly changing component (trend), γt = periodic component (seasonal), εt = irregular component (disturbance). In a Structural Time Series Model (STSM) or a Unobserved Components Model (UCM), the components are modelled explicitly as stochastic processes. Basic example is the local level model. 2 / 91
  • 3. Illustrations We present various illustrations of time series analysis and forecasting: 1. European business cycle 2. Bivariate analysis: decomposing and forecasting of Nordpool daily (average) of spot prices and consumption. 3. Periodic dynamic factor analysis: joint modeling of 24 hours in a daily panel of electricity loads. 4. Modelling house prices in Europe. 5. Modelling the U.S. Yield Curve. 3 / 91
  • 4. Illustration 1: European business cycle Azevedo, Koopman and Rua (JBES, 2006) consider European business cycle based on • a multivariate model consisting of generalised components for trend and cycle with band-pass filter properties; • data-set includes nine time series (quarterly, monthly) where individual series that may be leading/lagging GDP; • a model where all equations have individual trends but share one common “business cycle” component. • a common cycle that is allowed to shift for individual time series using techniques developed by R¨nstler (2002). u 4 / 91
  • 5. Shifted cycles 0.2 0.0 −0.2 estimated cycles gdp (red) versus cons confidence (blue) −0.4 1980 1985 1990 1995 0.2 0.0 −0.2 estimated cycles gdp (red) versus shifted cons confidence (blue) −0.4 1980 1985 1990 1995 5 / 91
  • 6. Shifted cycles In standard case, cycle ψt is generated by ψt+1 cos λ sin λ ψt κt + =φ + + ψt+1 − sin λ cos λ ψt κ+ t The cycle + cos(ξλ)ψt + sin(ξλ)ψt , is shifted ξ time periods to the right (when ξ > 0) or to the left (when ξ < 0). Here, − 1 π < ξ0 λ < 2 π (shift is wrt ψt ). 2 1 More details in R¨nstler (2002) for idea of shifting cycles in u multivariate unobserved components time series model of Harvey and Koopman (1997). 6 / 91
  • 7. The basic multivariate model Panel of N economic time series, yit , (k) (m) +(m) yit = µit + λi cos(ξi λ)ψt + sin(ξi λ)ψt + εit , where • time series have mixed frequencies: quarterly and monthly; (k) • generalised individual trend µit for each equation; (m) +(m) • generalised common cycle based on ψt and ψt ; • irregular εit . 7 / 91
  • 8. Business cycle Stock and Watson (1999) states that fluctuations in aggregate output are at the core of the business cycle so the cyclical component of real GDP is a useful proxy for the overall business cycle and therefore we impose a unit common cycle loading and zero phase shift for Euro area real GDP. Time series 1986 – 2002: quarterly GDP industrial production unemployment (countercyclical, lagging) industrial confidence construction confidence retail trade confidence consumer confidence retail sales interest rate spread (leading) 8 / 91
  • 9. Eurozone Economic Indicators 14.30 GDP Retail sales IPI unemployment Interest rate spread Industrial confidence indicator 14.25 Construction confidence indicator Retail trade confidence indicator Consumer confidence indicator 14.20 14.15 14.10 14.05 14.00 13.95 13.90 1990 1995 2000 9 / 91
  • 10. Details of model, estimation • we have set m = 2 and k = 6 for generalised components • leads to estimated trend/cycle estimates with band-pass properties, Baxter and King (1999). • frequency cycle is fixed at λ = 0.06545 (96 months, 8 years), see Stock and Watson (1999) for the U.S. and ECB (2001) for the Euro area • shifts ξi are estimated • number of parameters for each equation is four (σi2,ζ , λi , ξi , σi2,ε ) and for the common cycle is two (φ and σκ ) 2 • total number is 4N = 4 × 9 = 36 10 / 91
  • 11. Decomposition of real GDP 14.2 0.003 14.1 0.002 14.0 0.001 GDP Euro Area Trend slope 13.9 1990 1995 2000 1990 1995 2000 0.01 0.0050 0.0025 0.00 0.0000 −0.0025 −0.01 −0.0050 Cycle irregular 1990 1995 2000 1990 1995 2000 11 / 91
  • 12. The business cycle coincident indicator Selected estimation results series load shift R2d gdp −− −− 0.31 indutrial prod 1.18 6.85 0.67 Unemployment −0.42 −15.9 0.78 industriual c 2.46 7.84 0.47 construction c 0.77 1.86 0.51 retail sales c 0.26 −0.22 0.67 consumer c 1.12 3.76 0.33 retail sales 0.11 −4.70 0.86 int rate spr 0.57 16.8 0.22 12 / 91
  • 13. Coincident indicator for Euro area business cycle 0.010 0.005 0.000 −0.005 −0.010 −0.015 1990 1995 2000 13 / 91
  • 14. Coincident indicator for growth • tracking economic activity growth is done by growth indicator • we compare it with EuroCOIN indicator • EuroCOIN is based on generalised dynamic factor model of Forni, Hallin, Lippi and Reichlin (2000, 2004) • it resorts to a dataset of almost thousand series referring to six major Euro area countries • we were able to get a quite similar outcome with a less involved approach by any standard 14 / 91
  • 15. EuroCOIN and our growth indicator 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 −0.0025 −0.0050 −0.0075 Coincident Eurocoin 1990 1995 2000 15 / 91
  • 16. Illustration 2: Nord Pool data • we consider Norwegian electricity prices and consumption from Nord Pool. • mostly hydroelectric power stations; supply depends on weather. • Norway’s yearly hydro power plant capacity is 115 Tw hours. • Nord Pool is day ahead market: daily trades for next day delivery. • daily series of average of 24 hourly price and consumption. • spot prices measured in Norwegian Kroner (8 NOK ≈ 1 Euro). • sample: Jan 4, 1993 to April 10, 2005; 640 weeks or 4480 days. • data are subject to yearly cycles, weekly patterns, level changes, and jumps. 16 / 91
  • 17. Bivariate analysis: daily spot prices and consumption Our unobserved components model is given by 2 yt = µt + γt + ψt + xt′ λ + εt , εt ∼ NID(0, σε ), where • yt is bivariate: electricity spot price and load consumption; • µt is long term level; • γt is seasonal effect with S = 7 (day of week effect); • ψt is yearly cycle changes (summer/winter effects); ′ • xt λ has regression effects, mainly dummies for special days; • εt is the irregular noise. Parameter estimation and forecasting of observations have been carried out by the STAMP 8 program of Koopman, Harvey, Doornik and Shephard (2008, stamp-software.com): user-friendly but still flexible, also for multivariate models. 17 / 91
  • 18. Daily spot electricity prices from the Nord Pool (i) (ii) 0.1 6 0.0 5 4 −0.1 3 −0.2 100 200 300 400 500 600 100 200 300 400 500 600 (iii) (iv) 0.50 1 0.25 0 0.00 −1 −0.25 100 200 300 400 500 600 100 200 300 400 500 600 Univariate decomposition of Nord Pool daily prices January 4, 1993 to April 10, 2005: (i) data and estimated trend plus regression; (ii) seasonal component (S = 7, the day-of-week effect); (iii) yearly cycle; (iv) irregular. 18 / 91
  • 19. Joint decomposition of electricity prices & consumption 7 (i−a) 0.50 (i−b) 6 0.25 0.00 5 −0.25 450 500 550 600 450 500 550 600 (ii−a) 0.010 (ii−b) 0.05 0.005 0.000 −0.05 −0.005 450 500 550 600 450 500 550 600 (iii−a) 0.50 (iii−b) 1 0.25 0.00 0 −0.25 450 500 550 600 450 500 550 600 (iv−a) (iv−b) 0.2 0.025 0.0 −0.025 −0.2 450 500 550 600 450 500 550 600 Bivariate decomposition of prices and consumption: Feb 19, 2001 to April 10, 2005: (ia,b) data and estimated trend plus regression; (iia,b) seasonal component (S = 7, the day-of-week effect); (iiia,b) yearly cycle; (iva,b) irregular. 19 / 91
  • 20. Forecasting results We present MAPE for forecasting of one- to seven-days ahead prices for both uni- and bivariate models. The one- to seven-days ahead forecasts are for the next seven days. The first forecast is for Monday, March 14, 2005 in Week 637. The last forecast is for Sunday, April 10, 2005 in Week 640. The weeks 638 and 639 contain calendar effects for Maundy Thursday (March 24, 2005) and the days until Easter Monday (March 28, 2005). week 637 week 638 week 639 week 640 uni biv uni biv uni biv uni biv horizon 1M 0.83 1.11 0.15 0.07 0.83 1.01 0.92 0.27 2T 0.86 0.94 0.51 0.53 1.20 1.36 0.74 0.20 3W 1.43 1.55 0.67 0.79 1.40 1.52 0.62 0.16 4T 1.94 2.09 0.64 0.88 1.71 1.75 0.60 0.14 5F 1.69 1.93 0.65 0.72 2.01 2.00 0.60 0.30 6S 1.62 1.95 0.58 0.69 2.26 2.17 0.67 0.43 7S 1.61 2.05 0.68 0.90 2.44 2.27 0.79 0.56 20 / 91
  • 21. Illustration 3: periodic dynamic factor analysis Aim: the joint modeling of 24 hours in a daily panel of electricity loads for EDF. Focus: modelling and short-term forecasting of hourly electricity loads, from one day ahead to one week ahead. • EDF provides a long time series: 9 years of hourly loads • We can establish a long-term trend component but also • different levels of seasonality (yearly, weekly, daily) • special day effects (EJP) • weather dependence (temperature, cloud cover) • We look at the intra-year as well as the long-run dynamics by using these different components. 21 / 91
  • 22. Periodic dynamic factor model specification The adopted methodology builds on Dordonnat, et al (2008, IJF): • Model is for high-frequency data, for hourly data); • Hours are in the cross-section (yt is 24 × 1 vector); • The model dynamics are formulated for days: a multivariate daily time series model; • In effect, we adopt a periodic approach to time series modelling; • The right-hand side of the model is set-up as a multiple regression model; • We let the regression parameters evolve over time (days); • We have a time-varying regression model, written in state-space form; • The 24-dimensional time-varying parameters are subject to common dynamics (random walks); • Novelty: dynamic factors in the time-varying parameters. 22 / 91
  • 23. Daily National Electricity Load, 1995-2004 80000 80000 National Load (MegaWatts) National Load (MegaWatts) 60000 60000 40000 40000 2000 2005 2002 2003 Year (a) Date (b) 80000 50000 National Load (MegaWatts) National Load (MegaWatts) 60000 40000 40000 30000 0 5 10 15 20 −5 0 5 10 15 20 25 30 Days elapsed since August 8th,2004 (c) National Temperature (°C) (d) Time series and temperature effects at 9 AM 23 / 91
  • 24. Daily National Electricity Load, 1995-2004 60000 60000 Mean Load (MegaWatts) Mean Load (MegaWatts) 50000 50000 40000 40000 1 2 3 4 5 6 7 8 9 10 11 12 4 8 12 16 20 24 60000 60000 Month (a) Hour (b) Mean Load (MegaWatts) Mean Load (MegaWatts) 50000 50000 40000 40000 4 8 12 16 20 24 4 8 12 16 20 24 Hour (c) Hour (d) Average patterns (c) Oct-Mar, (d) Apr-Sep 24 / 91
  • 25. Multivariate Time Series Model A periodic approach: from a univariate hourly to a daily 24 × 1 vector: yt = (y1,t . . . yS,t )′ , S = 24 hours per day, t = 1, . . . , T days. Our multivariate time-varying parameter regression model is given by: K yt = µ t + Btk xtk + εt , εt ∼ IIN (0, Σε ) , t = 1, . . . , T , k=1 • Trends: µt = (µ1,t . . . µS,t )′ k k k • Daily vectors of explanatory variables xt = (x1,t . . . xS,t )′ , k = 1, . . . , K , depending only on the day or on the hour of the day. k k k • Regression coefficients βt = (β1,t . . . βS,t )′ , k = 1, . . . , K . In matrix form: Bt k = diag (β k ), k = 1, . . . , K t • Irregular Gaussian white noise εt = (ε1,t . . . εS,t )′ . 25 / 91
  • 26. Time-varying regressions and dynamic factors K yt = µ t + Btk xtk + εt , εt ∼ IIN (0, Σε ) , t = 1, . . . , T , k=1 where the time-varying regression parameters are given by µt = c 0 + Λ0 ft0 , k βt = c k + Λk ftk , k = 1, . . . , K , t = 1, . . . , T , j j with constant c j = (c1 . . . cS )′ , S × R j factor loading matrices Λj and R j dynamic factors ftj = (f1,t . . . fR j ,t )′ , for j = 0, . . . , K and j j 0 ≤ R j ≤ S. • Factor structure requires 0 < R j < S; • Constant parameter component for R j = 0; • Model has unrestricted component when R j = S; • Identification restrictions apply. 26 / 91
  • 27. Dynamic factor specfications Local linear trend model for factors in trends µt : ft+1 = ftj j + gtj + v jt , v jt ∼ IIN(0, Σj ) v j gt+1 = gtj + w jt , w jt ∼ IIN(0, Σj ) w • vector of dynamic factors ftj , • slope or gradient vector gtj , • level disturbance v jt and slope disturbance w jt . Random walk model for factors in the regression coefficients: ft+1 = ftj + e jt , j e jt ∼ IIN(0, Σe ), j j = 1, . . . , K , t = 1, . . . , T , with regression coefficient disturbance e jt . 27 / 91
  • 28. Empirical application to French national hourly Loads • French national hourly electricity loads from Sept-95 until Aug-04 • Estimation of trivariate models for neighbouring hours • Smooth trends • Intentional missing values for special days (EJP) and turn of the year. No problem for state space models. • Yearly pattern regressors: sine/cosine functions of time are used (2 frequencies) • Day-of-the-week effects: day-type dummy regressors • Weather dependence: heating degrees, smoothed-heating degrees and cloud cover • Heating degrees beneath treshold temperature of 15 C • Exponentially smoothed temperature • Cooling degrees above treshold temperature of 18 C • Implemention: SsfPack 3 of Koopman, Shephard and Doornik (2008, ssfpack.com) for Ox 6 (2008, doornik.com) 28 / 91
  • 29. Yearly pattern estimates per hour 4 ˆk k µs,t + ˆ k=1 βs,t xs,t for hours (a) s = 0, 1, 2 ; (b) s = 3, 4, 5 ; (c) s = 6, 7, 8 (with extra component), etc. Estimation: Jan 1997 - Aug 2003, Graph: Jan 1998 - Aug 2003. 29 / 91
  • 30. Components 9 AM, factor model (blue) and univariate Factor Univariate 1500 Factor Univariate 500 1000 0 500 2000 (a) 2002 2000 (b) 2002 300 Factor Univariate −500 Factor Univariate 200 −1000 100 −1500 2000 (c) 2002 2000 (d) 2002 0 Factor Univariate −6000 Factor Univariate −8000 −200 −10000 2000 (e) 2002 2000 (f) 2002 −10000 Factor Univariate 60000 Factor Univariate −12500 50000 −15000 2000 (g) 2002 2000 (h) 2002 ˆ9 (a) heating β9,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h) trend + yearly pattern 30 / 91
  • 31. 9 AM st. errors, factor model (blue) and univariate Factor Univariate Factor Univariate 300 500 200 250 100 2000 (a) 2002 2000 (b) 2002 75 Factor Univariate 300 Factor Univariate 50 200 25 100 2000 (c) 2002 2000 (d) 2002 80 Factor Univariate 500 Factor Univariate 70 60 300 50 2000 (e) 2002 2000 (f) 2002 750 Factor Univariate 2000 Factor Univariate 500 1000 250 2000 (g) 2002 2000 (h) 2005 ˆ9 (a) heating s.e. β9,t , (b) smoothed-heating, (c) cooling, (d) Monday, (e) Friday, (f) Saturday, (g) Sunday, (h) trend + yearly pattern 31 / 91
  • 32. Sample ACFs of residuals (daily lags) 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 Selecting the right model requires experience and stamina! 32 / 91
  • 33. Conclusions • A general, flexible and insightful methodology is developed. • Many dynamic features of load and price data can be captured. • We can detect many interesting signals which are not discovered before. • Decent forecasts. • Decent diagnostics. • Many possible extensions. • Remaining challenge: a full multivariate unobserved components model for all 24 hours to capture evolutions of complete intradaily load pattern. • More work is required ! 33 / 91
  • 34. Short Bibliography • “Multivariate structural time series models” by Harvey and Koopman (1997), Chapter in Heij et al. (1997) Wiley. • “Time-series analysis by state-space methods” by Durbin and Koopman (Oxford, 2001) • “Periodic Seasonal Reg-ARFIMA-GARCH Models for Daily Electricity Spot Prices” by Koopman, Ooms and Carnero (JASA, 2007). • “An hourly periodic state-space for modelling French national electricity load” by Dordonnat, et.al. (International Journal of Forecasting, 2008) • “Forecasting economic time series using unobserved components time series models” by Koopman and Ooms (2011), Chapter in Clements and Hendry, OUP Handbook of Forecasting. 34 / 91
  • 35. Illustration 4: The macroeconomy in the euro area Quarterly time series, 1981 – 2008, GDP in volumes, for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain. (i) (ii) 13.2 12.8 13.0 12.6 12.8 12.4 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010 12.7 (iii) 12.25 (iv) 12.6 12.00 12.5 11.75 12.4 12.3 11.50 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010 35 / 91
  • 36. Illustration 4: The housing market in the euro area Quarterly time series, 1981 – 2008, real house prices (HP), for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain. (i) 0.3 (ii) 5.0 0.2 4.5 0.1 4.0 0.0 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010 (iii) (iv) 3.0 0.25 0.00 2.5 −0.25 2.0 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010 36 / 91
  • 37. Any (common) cyclical dynamics in the data ? Autocorrelograms and sample spectra, based on first differences... GDP−Correlogram GDP−Spectrum HP−Correlogram HP−Spectrum 1 1 (i) 0.4 0.4 0 0 0.2 0.2 0 10 20 0.0 0.5 1.0 0 10 20 0.0 0.5 1.0 1 1 0.75 (ii) 0.2 0.50 0 0 0.1 0.25 0 10 20 0.0 0.5 1.0 0 10 20 0.0 0.5 1.0 1 1 (iii) 0.4 0.50 0 0 0.2 0.25 0 10 20 0.0 0.5 1.0 0 10 20 0.0 0.5 1.0 1 1 (iv) 0.4 1.0 0 0 0.2 0.5 0 10 20 0.0 0.5 1.0 0 10 20 0.0 0.5 1.0 37 / 91
  • 38. The basic multivariate model Multiple set of M economic time series, yit , is collected in yt = (y1t , . . . , yMt )′ and model is given by (1) (2) yt = µt + ψt + ψt + εt , where the disturbance driving each vector component is a vector too, with a variance matrix. The structure of the variance matrix determines the dynamic interrelationships between the M time series. For example, if trend component µt follows the random walk, µt+1 = µt + ηt with disturbance vector ηt , with variance matrix Ση : • diagonal Ση , independent trends; • rank(Ση ) = p < M, common trends (cointegration); • rank(Ση ) = 1, single underlying trend; • Ση is zero matrix, constant. Similar considerations apply to other components. 38 / 91
  • 39. Dynamic factor representations We can formulate the multivariate unobserved components model also by (1) (2) yt = µ∗ + Aη µt + A(1) ψt κ + A(2) ψt κ + Aε εt , where, for the trend component, for example, the loading matrix Aη is such that ′ Ση = Aη Aη , and, similarly, loading matrices are defined for the other variance matrices of disturbances that drive the components. (1) (2) Here the dynamic factors or unobserved components µt , ψt , ψt and εt are ”normalised”. 39 / 91
  • 40. STAMP Model is effectively a state space model: Kalman filter methods can be applied for maximum likelihood estimation of parameters (such as the loading matrices). Kalman filter methods are employed for the evaluation of the likelihood function and score vector. Kalman filter and smoothing methods are employed for signal extraction or the estimation of the unobserved components. User-friendly software is available for state space analysis. We have used S T A M P for this research project: a multi-platform, user-friendly software: econometrics, time series and forecasting by clicking. It can treat multivariate unobserved components time series models... 40 / 91
  • 41. Motivation of our study • Evidence of any relationship between housing prices and GDP in the euro area. • Focus on more recent developments... • We prefer to model the time series jointly and establish interrelationships between the time series • Focus on cyclical dynamics, long-term and short-term • Emphasis on real housing prices: relevant for the monetary policy • We also like to discuss synchronisation of housing markets in euro area Empirical results are based on our data-set with two variables: GDP and real house prices (HP); and for four euro area countries: France, Germany, Italy and Spain. 41 / 91
  • 42. Relevant literature • Unobserved components model: Harvey (1989) • State space methods: Durbin and Koopman (2001) • Multivariate unobserved components: Harvey and Koopman (1997), Azevedo, Koopman and Rua (2006); • Parametric approaches for house prices: • Probit regressions: Borio and McGuire, 2004, van den Noord, 2006; • Dynamic Factor models: Terrones, 2004, DelNegro and Otrok, 2007, Stock and Watson, 2008; • VAR: Vargas-Silva, 2008, Goodheart and Hofmann, 2008. 42 / 91
  • 43. Univariate analysis Objectives: • Verify the trend-cycle decomposition for each series • Verify whether possible restrictions are realistic Results for GDP: • two short cycles in France and Italy are detected (¡6 years); • Germany and Spain contain both a short cycle (5.42 and 3.62 years, resp.) and a long cycle (13.5 and 9.11 years) • Various cycles are deterministic (fixed sine-cosine wave) Results for HP: • Results are quite different for each series • Two cycles for Germany (5.4 and 13.5 years) • Two short cycles for Italy (3.0 and 5.5 years) and France (3.1 and 5.8 years) • For Spain a cycle reduces to an AR(1) process 43 / 91
  • 44. Univariate results for GDP France Germany Italy Spain GDP R R R R Trend var 0.65 0.03 0.01 0.03 0.48 0.03 0.10 0.03 Cycle 1 var 0.81 0.17 0.00 0.15 3.85 5.75 0.07 0.00 Cycle 1 ρ 0.94 0.90 1.0 0.90 0.87 0.90 0.95 0.90 Cycle 1 p 3.04 5 5.42 5 2.97 5 3.62 5 Cycle 2 var 0.00 1 1.81 2.86 0.00 7.79 0.00 2.31 Cycle 2 ρ 1.0 0.95 0.95 0.95 1.00 0.95 1.00 0.95 Cycle 2 p 5.8 12 13.5 12 5.50 12 9.11 12 Irreg var 1 0.0 1 1 1 1 1 1 N 7.2 11.4 3.23 5.23 6.58 11.1 27.1 34.9 Q 14.5 24.9 15.1 14.6 9.26 13.3 22.1 24.8 R2 0.31 0.24 0.11 0.02 0.23 0.12 0.22 0.12 44 / 91
  • 45. Univariate results for HP France Germany Italy Spain RHP R R R R Trend var 0.59 0.03 0.34 0.03 0.00 0.03 0.39 0.03 Cycle 1 var 0.00 0.01 0.31 1.51 0.04 0.02 1 0.01 Cycle 1 ρ 1.0 0.90 0.97 0.90 0.96 0.90 0.34 0.90 Cycle 1 p 6.34 5 4.48 5 1.11 5 – 5 Cycle 2 var 0.00 2.19 1 19.9 1 49.4 0.00 39.5 Cycle 2 ρ 1.0 0.95 0.61 0.95 0.99 0.95 0.99 0.95 Cycle 2 p 8.37 12 2.82 12 13.3 12 – 12 Irreg var 1 1 0 1 0 1 0 1 N 23.8 0.59 5.89 9.95 7.03 8.32 36.1 11.9 Q 10.6 187 55.5 111 13.7 68.4 29.3 127 R2 0.61 0.25 0.35 0.15 0.56 0.22 0.47 0.28 45 / 91
  • 46. Cycle correlations from univariate analysis (1) (2) Correlations for combined cycles (ψt + ψt ): • Strong correlations between GDP of four countries (correlations range from 0.52 to 0.94) • The correlations with German GDP are the lowest • Correlations between HP of four countries range from 0.42 to 0.94 • The highest correlation is between Spain and France HP’s • Correlation on combined cycle are mostly due to long-term cycle, not to the short-term cycle • Correlations between GDP and HP for each country range from 0.06 for Germany to 0.76 for Spain • Overall low cross-correlations between GDP of one country and HP of another country 46 / 91
  • 47. Correlations between combined cycles for eight series (1) (2) Combined cycle (ψt + ψt ) F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP 1.00 0.51 0.52 0.23 0.83 0.15 0.89 0.61 F HP 0.51 1.00 0.44 0.44 0.52 0.68 0.68 0.94 G GDP 0.52 0.44 1.00 0.50 0.54 0.47 0.61 0.44 G HP 0.23 0.44 0.50 1.00 0.08 0.80 0.22 0.42 I GDP 0.83 0.52 0.54 0.08 1.00 0.06 0.84 0.64 I HP 0.15 0.68 0.47 0.80 0.06 1.00 0.29 0.64 S GDP 0.89 0.68 0.61 0.22 0.84 0.29 1.00 0.76 S HP 0.61 0.94 0.44 0.42 0.64 0.64 0.76 1.00 47 / 91
  • 48. Correlations between short cycle for eight series (1) Short cycle ψt F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP 1.00 0.46 0.40 0.24 0.64 -0.46 0.57 0.42 F HP 0.46 1.00 0.29 0.62 0.33 -0.51 0.35 0.39 G GDP 0.40 0.29 1.00 0.32 0.75 -0.16 0.67 0.58 G HP 0.24 0.62 0.32 1.00 0.18 -0.52 0.06 0.13 I GDP 0.64 0.33 0.75 0.18 1.00 -0.13 0.61 0.65 I HP -0.46 -0.51 -0.16 -0.52 -0.13 1.00 -0.25 -0.19 S GDP 0.57 0.35 0.67 0.06 0.61 -0.25 1.00 0.75 S HP 0.42 0.39 0.58 0.13 0.65 -0.19 0.75 1.00 48 / 91
  • 49. Correlations between long cycle for eight series (2) Long cycle ψt F GDP F HP G GDP G HP I GDP I HP S GDP S HP F GDP 1.00 0.51 0.53 0.23 0.89 0.16 0.90 0.63 F HP 0.51 1.00 0.46 0.44 0.58 0.68 0.68 0.94 G GDP 0.53 0.46 1.00 0.52 0.44 0.49 0.62 0.46 G HP 0.23 0.44 0.52 1.00 0.07 0.82 0.22 0.43 I GDP 0.89 0.58 0.44 0.07 1.00 0.08 0.90 0.72 I HP 0.16 0.68 0.49 0.82 0.08 1.00 0.29 0.64 S GDP 0.90 0.68 0.62 0.22 0.90 0.29 1.00 0.76 S HP 0.63 0.94 0.46 0.43 0.72 0.64 0.76 1.00 49 / 91
  • 50. Bivariate analysis For each country, we carry out a bivariate analysis between GDP and RHP: (1) (2) yt = µt + ψt + ψt + εt , where yt is a 2 × 1 vector for two series: GDP and HP. We can conclude that • highest correlation is found for cycle components (except Italy) • for France, high correlation for medium-term cycle (8 years) but no dependence for long-term cycle (15.6 years) • for Spain, strong correlations for both medium-term (8.2 years) and long-term (14.4 years) • for Germany, correlations for both cycles, but with low periods (4.3 and 7 years) 50 / 91
  • 51. Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHP FRA trend 0.0 0.0 0.0 – – N 3.25 13.4 cyc 1 3.0 3.3 0.88 8.0 0.98 Q 17.0 17.4 cyc 2 1.0 126 0.07 15.6 0.99 R2 0.38 0.63 irreg 0.6 1.6 -0.19 – – GER trend 0.0 0.003 0.0 – – N 8.52 1.08 cyc 1 2.5 5.4 -0.6 4.3 0.90 Q 6.86 42.1 cyc 2 3.1 0.5 1.0 7.0 0.98 R2 0.39 0.29 irreg 4.3 1.1 0.58 – – 51 / 91
  • 52. Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHP ITA trend 0.1 0.9 -0.15 – – N 4.19 4.57 cyc 1 4.3 16.2 -0.08 6.0 0.92 Q 10.1 8.60 cyc 2 0.0 8.4 0.0 1.1 0.94 R2 0.14 0.47 irreg 0.8 1.2 0.96 – – SPN trend 0.0 0.0 0.0 – – N 9.05 21.7 cyc 1 3.3 11.9 0.95 8.2 0.98 Q 17.5 43.0 cyc 2 0.0 83.3 0.82 14.4 0.99 R2 0.45 0.73 irreg 3.9 7.7 -0.35 – – 52 / 91
  • 53. Four-variate cross-country analysis of GDP and RHP Now we incorporate earlier findings and impose a strict short- and long-term cycle decomposition for our analysis. In particular, we have • an independent trend µt (i.e. diagonal variance matrix Ση for disturbance vectors of µt+1 = µy + ηt ) • similarly, an independent irregular component εt (i.e. diagonal variance matrix Σε ) • a two-cycle parametrization with restricted periods of 5 and 12 years • the rank of the 4 × 4 cycle variance matrices Σκ is 2: common cyles ... • we load the two ”independent” cycles on France and Germany, i.e. cyclical dynamics of Spain and Italy are obtained as linear functions of the two times two (short and long) cyclical factors 53 / 91
  • 54. Four-variate decomposition for GDP, cross-country 13.00 LFRA_GDP Level LGER_GDP Level LITA_GDP Level 12.25 LSPA_GDP Level 13.25 12.6 12.75 12.00 13.00 11.75 12.50 12.4 12.75 11.50 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 LFRA_GDP−Cycle 1 LGER_GDP−Cycle 1 0.02 LITA_GDP−Cycle 1 0.010 LSPA_GDP−Cycle 1 0.01 0.02 0.01 0.005 0.00 0.00 0.00 0.000 −0.01 −0.02 −0.01 −0.005 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 LFRA_GDP−Cycle 2 LGER_GDP−Cycle 2 LITA_GDP−Cycle 2 0.050 LSPA_GDP−Cycle 2 0.025 0.025 0.02 0.025 0.000 0.000 0.000 0.00 −0.025 −0.025 −0.025 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 LFRA_GDP−Irregular LGER_GDP−Irregular 0.0050 LITA_GDP−Irregular 0.02 LSPA_GDP−Irregular 0.001 0.01 0.0025 0.01 0.000 0.00 0.0000 0.00 −0.001 −0.01 −0.0025 −0.01 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 54 / 91
  • 55. Four-variate results for cross-country: GDP Fra Ger Ita Spn Fra Ger Cycle short (cov ×10−6 ) factor loadings Fra 4.11 0.25 ∗ 0.77 ∗ -0.40 ∗ 1 0 Ger 1.77 11.8 0.81 ∗ 0.78 ∗ 0 1 Ita 5.65 10.1 13.1 0.27 ∗ 1.08 0.69 Spn -1.04 3.50 1.27 1.65 -0.41 0.35 Cycle long (cov ×10−6 ) Fra 8.08 0.79 ∗ 0.48 ∗ 0.98 ∗ 1 0 Ger 7.94 12.5 -0.16 ∗ 0.64 ∗ 0 1 Ita 3.43 -1.39 6.28 0.66 ∗ 1.42 -1.02 Spn 11.2 9.11 6.73 16.4 1.79 -0.41 55 / 91
  • 56. Four-variate results for cross-country: GDP • Diagnostic statistics are satisfactory • Strong correlation France-Germany for long-term cycle • Business cycles for Italy and Spain are closely connected with the one for France (however, negative ??? marginal correlation Fra-Spa for short-term cycle) • German cycles strongly affect business cycles in Italy and Spain (however, their marginal correlations for longer cycle are negative) 56 / 91
  • 57. Four-variate decomposition for HP, cross-country 5.5 LFRA_RHprice Level 0.3 LGER_RHprice Level LITA_RHprice Level LSPA_RHprice Level 0.25 3.0 5.0 0.2 4.5 0.00 2.5 0.1 4.0 −0.25 2.0 0.0 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 0.02 LFRA_RHprice−Cycle 1 0.02 LGER_RHprice−Cycle 1 LITA_RHprice−Cycle 1 0.04 LSPA_RHprice−Cycle 1 0.01 0.05 0.02 0.00 0.00 0.00 0.00 −0.02 −0.01 −0.02 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 LFRA_RHprice−Cycle 2 0.050 LGER_RHprice−Cycle 2 0.2 LITA_RHprice−Cycle 2 LSPA_RHprice−Cycle 2 0.1 0.1 0.2 0.025 0.0 0.000 0.0 0.0 −0.1 −0.1 −0.2 −0.025 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 LFRA_RHprice−Irregular LGER_RHprice−Irregular 0.002 LITA_RHprice−Irregular LSPA_RHprice−Irregular 0.01 5e−5 0.01 0.001 0.00 0 0.000 0.00 −0.01 −5e−5 −0.001 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 57 / 91
  • 58. Four-variate results for cross-country: HP Fra Ger Ita Spn Fra Ger Cycle short (cov ×10−6 ) factor loadings Fra 15.5 0.37 ∗ -0.89 ∗ 0.05 ∗ 1 0 Ger 4.73 10.8 0.10 ∗ -0.91 ∗ 0 1 Ita -21.0 1.97 36.2 -0.50 ∗ -1.64 0.90 Spn 0.89 -14.6 -14.6 23.8 0.55 -1.60 Cycle long (cov ×10−6 ) Fra 44.5 0.38 ∗ 0.70 ∗ 0.93 ∗ 1 0 Ger 4.43 3.13 -0.40 ∗ 0.69 ∗ 0 1 Ita 66.9 -10.3 207.1 0.38 ∗ 2.13 -6.30 Spn 100.4 19.9 88.3 262.8 1.89 3.69 58 / 91
  • 59. Four-variate results for cross-country: HP • Overall, these results seem to indicate that there is less evidence of common (cyclical) dynamics in HP series • Low correlations between France and Germany • Strong negative correlations for the 5-year cycle between Fra-Ita and Ger-Spa • However, more commonalities for the 12-year cycle (Fra-Spa, Fra-Ita, Ger-Spa) • Similarities between correlation matrices for the 12-year HP and GDP cycles, except that relationship Fra-Ger is stronger for GDP (0.79 against 0.38 for HP) 59 / 91
  • 60. Eight-variate results: HP and GDP for four countries Similar restrictions apply as in four-variate analyses. We conclude that • strong correlations among GDPs for short-term cycles but less evidence for long-term cycles, especially for Germany • low correlations among HP series. • for short-term cycle, these correlations for HP Fra-Ger is 0.65 and for HP Spa-Ger is -0.95. • only a few positive correlations for the long-term cycle in HP have been found: Fra-Spa (0.58) and Ger-Ita (0.57) • correlations HP-GDP are only found for long-term cycle, especially for France and Spain. 60 / 91
  • 61. Eight-variate results: short cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HP F-G 1 -0.33 0.67 0.10 0.81 -0.59 0.77 0.13 F-H 1 0.075 0.65 -0.35 -0.13 -0.12 -0.64 G-G 1 0.17 0.80 -0.27 0.88 -0.011 G-H 1 0.055 -0.26 -0.10 -0.95 I-G 1 -0.037 0.66 0.034 I-H 1 -0.55 -0.040 S-G 1 0.34 S-H 1 61 / 91
  • 62. Eight-variate results: long cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HP F-G 1 0.95 0.19 0.043 0.72 0.41 0.54 0.50 F-H 1 0.44 0.24 0.63 0.43 0.57 0.58 G-G 1 0.41 -0.31 0.26 0.44 0.21 G-H 1 -0.005 0.57 0.036 0.29 I-G 1 0.045 0.12 0.37 I-H 1 0.13 0.099 S-G 1 0.61 S-H 1 62 / 91
  • 63. Illustration 5 : Modelling U.S. Yield Curve Yield (in %) 6.50 6.25 6.00 5.75 5.50 5.25 Maturity (in months) 0 10 20 30 40 50 60 70 80 90 100 110 120 63 / 91
  • 64. Time Series of Four Maturities Yield (in %) Time to maturity: 3 month 10 Yield (in %) Time to maturity: 1 year 8 8 6 6 4 4 Date Date 1985 1990 1995 2000 1985 1990 1995 2000 Yield (in %) Yield (in %) Time to maturity: 3 year Time to maturity: 10 year 10 10.0 8 7.5 6 5.0 Date Date 1985 1990 1995 2000 1985 1990 1995 2000 64 / 91
  • 65. Term Structure of Interest Rates over Time 10.0 Yield (Percent) 7.5 5.0 125 100 75 2000.0 Mat 1997.5 urity 50 1995.0 (Mo nths 1992.5 ) 25 1990.0 Time 1987.5 65 / 91
  • 66. Literature Review Earlier analyses of this data: • Affine Term Structure Models (ATSM): Vasicek (1977), Cox, Ingersoll, and Ross (1985), Duffie and Kan (1996), Dai and Singleton (2000), and De Jong (2000) • Nelson-Siegel Model (NS): Nelson and Siegel (1987), Diebold and Li (2006), Diebold, Rudebusch and Aruoba (2006), De Pooter (2007), and Koopman, Mallee, and Van der Wel (2009) • Arbitrage-Free Nelson-Siegel Model (AFNS): Christensen, Diebold, and Rudebusch (2007) • Functional Signal plus Noise (FSN): Bowsher and Meeks (2008) In all cases: a dynamic factor model set-up ! 66 / 91
  • 67. Still Some Outstanding Issues... • Which of these models provides an accurate description of the data? • Duffee (2002) and Bams and Schotman (2003) provide evidence against affine term structure models • What are the dynamics of the underlying factors: Stationary or Nonstationary? • Stationary: Affine Term Structure Models, Nelson-Siegel, Arbitrage-Free Nelson-Siegel • Nonstationary: Campbell and Shiller (1987), Hall, Anderson and Granger (1992), Engsted and Tanggaard (1994) and Bowsher and Meeks (2008) • What are the dynamics of the underlying factors: #lags? • Almost all models take a VAR(1) for the factor dynamics • Exception: Bowsher and Meeks (2008) 67 / 91
  • 68. Further Outline • (General) Dynamic Factor Model (DFM) • General Set-Up • Stationary and Nonstationary • Smooth Dynamic Factor Model (SDFM) • Specification • Knot Selection • Other Restrictions of the DFM • Results • Conclusion 68 / 91
  • 69. The Dynamic Factor Model (DFM) • Time series panel of N monthly yield observations yt = (yt (τ1 ), . . . , yt (τN ))′ with yt (τi ) the yield at time t with maturity τi • The general dynamic factor model is given by: yt = µy + Λft + εt , εt ∼ N(0, H), ft = Uαt αt+1 = µα + T αt + R ηt , ηt ∼ N(0, Q), for t = 1, . . . , n • In here ft is an r -dimensional stochastic process that is generated by the p-dimensional state vector αt and ηt is a q × 1 vector • We take H diagonal and have for the p × 1 initial state α1 ∼ N(a1 , P1 ), and assume N >> r , r ≤ p and p ≥ q 69 / 91
  • 70. The Dynamic Factor Model (DFM) – Cont’d • Vectors µy and µα , and matrices Λ, H, U, T and Q are system matrices, R is selection matrix of ones and zeros • Special case of state space model. • All vector autoregressive moving average processes can be formulated in this framework (see, e.g., Box, Jenkins and Reinsel (1994)) • In this paper: VAR and Cointegrated VAR (CVAR) for ft . Obtained by suitable specification of U, T and R • Elements of system matrices µy , Λ, H, µα , T and Q generally contain parameters that need to be estimated 70 / 91
  • 71. The Dynamic Factor Model (DFM) – Cont’d • Need to impose restrictions on loading and variance matrices to ensure identification, see Jungbacker and Koopman (2008): • Vectors µy and µα not both estimated without restrictions =¿ Restrict µα = 0 to focus on loading matrix Λ • Impose restrictions on Λ, T & Q that govern covariances =¿ Restrict r rows in Λ to be r × r identity matrix:   1 0 0  0 1 0     0 0 1    Λ =  λ4,1 λ4,2 λ4,3     . . . . .  .   . . . λN,1 λN,2 λN,3 71 / 91
  • 72. Dynamic Factor Model (DFM) – Stationary Case • Take a VAR(k) model for the r × 1 vector ft : k−1 ft+1 = Γt−j ft−j + ζt , ζt ∼ NID(0, Qζ ) j=0 • Stationarity imposed by restriction that |Γ(z)| = 0 has all roots outside the unit circle • Can write this ft in DFM. For example, for k = 1 have αt = ft , U = Ir , R = Ir , T = Γ0 , Q = Qζ 72 / 91
  • 73. Dynamic Factor Model (DFM) – Nonstationary Case • Now the ft are generated by a cointegrated vector autoregressive process: k−1 ∆ft+1 = βγ ′ ft + Γj ∆ft−j + ζt , ζt ∼ N(0, Qζ ), j=0 • The r × s matrices β and γ have full column rank; in the nonstationary case s < r and all ft nonstationary • We propose an alternative but observationally equivalent specification for ft via factor rotation: f¯N ′ f¯ = t = β⊥ γ ft , t f¯S t ¯ also need to construct new loading matrix Λ = ¯ ΛN ¯ ΛS 73 / 91
  • 74. Dynamic Factor Model (DFM) – Estimation • As noted earlier, the Dynamic Factor Model (DFM) can be seen as a special case of state space model • Generally we can use likelihood-based methods: direct ML and/or EM methods • However. . . • . . . the dimension of the observations vector is much larger than the state vector • . . . there is a large number of parameters (DFM-VAR(1), N = 17, r = 3: 91 parameters) • We therefore estimate the models using the methodology of Jungbacker and Koopman (2008) and estimate parameters by direct ML using analytical score expressions 74 / 91
  • 75. Smooth Dynamic Factor Model (SDFM) • For cross-sectional observation i we can write the DFM as: r yt (τi ) = µy ,i + λij fjt +εit , t = 1, . . . T , i = 1, . . . , N, j=1 where λij is the loading of factor j on maturity i • We propose to let the loading parameter be an unknown function gj (·) for each factor j, where the argument of the function relates to i • Assume these functions g1 (·), . . . , gr (·) smooth functions of time to maturity: λij = gj (τi ) • In practice: cubic spline for each gj (·) • Call this Smooth Dynamic Factor Model (SDFM) 75 / 91
  • 76. Smooth Dynamic Factor Model (SDFM) – Spline • In a spline the location of the knots determines how the factor loadings behave for varying maturities j • Knot k for column j: sk • Order the knots by time to maturity: j j τ 1 = s 1 < · · · < s Kj = τ N , j = 1, . . . , r • Get following loading function:  j  gj (s1 )  . .  gj (τi ) = wij λj , λj =  . , j = 1, . . . , r , j gj (sKj ) with wij a 1 × Kj vector (only depends on the knot locations) and λj a Kj × 1 parameter vector 76 / 91
  • 77. Smooth Dynamic Factor Model (SDFM) – Select Knots • But how many knots Kj to select in the spline W λ? • Small number of knots: Loadings lie on same polynomial for considerable number of maturities • Large number of knots: Get closer to unrestricted DFM • Propose using a Wald test procedure to determine the knots • This is standard as we are testing linear restrictions • Amounts to an iterative general-to-specific approach: 1. Start with all knots 2. Calculate test statistic for all knots 3. Remove knot with smallest non-significant statistic 4. Continue with 2 and 3 until all knots are significant 77 / 91
  • 78. Dynamic Factor Models for the Term Structure • The general dynamic factor model is given by: yt = µy + Λft + εt , εt ∼ N(0, H), ft = Uαt αt+1 = µα + T αt + R ηt , ηt ∼ N(0, Q), • It nests the term structure models mentioned earlier • Functional Signal plus Noise – Bowsher and Meeks (2008) • Rather than a spline for the factor loadings they adopt the Harvey and Koopman (1993) time-varying spline for the yield curve: yt = µy + Wft + εt , εt ∼ NID(0, H), with W as before and ft time-varying knot values • Take a CVAR(k) for ft and have restrictions on Λ 78 / 91
  • 79. Dynamic Factor Models for the Term Structure – Cont’d • Nelson-Siegel – Nelson and Siegel (1987), Diebold and Li (2006), Diebold, Rudebusch and Aruoba (2006) • The yield curve is expressed as a linear combination of smooth factors 1 − e −λτ 1 − e −λτ gns (τ ) = ξ1 + · ξ2 + − e −λτ · ξ3 λτ λτ which gives yt = µy + Λns ft + εt , εt ∼ NID(0, H) • Interpretation as level, slope and curvature for the factors • Typically: (restricted) VAR(1) for the state, µy = 0 • Restrictions on Λ 79 / 91
  • 80. Dynamic Factor Models for the Term Structure – Cont’d • Arbitrage-Free NS – Christensen, Diebold and Rudebusch (2007) • The NS model is not arbitrage free • CDR employ “reverse engineering” and obtain an Arbitrage-Free NS model • Dynamics of latent factors now coming from solution of SDE, plus ‘correction’ term for µy • Restrictions on Λ, T and µy • Affine Term Structure Models – Duffie and Kan (1996) • Term structure can be explained by dynamics of unobserved short rate • Short rate depends on unobserved factors • Focus on Gaussian case • Restrictions on Λ, T and µy 80 / 91
  • 81. Results Strategy: • Following, e.g., Litterman and Scheinkman (1991) we only look at 3-factor models • Restrict ourselves to Gaussian models • Use an existing dataset: unsmoothed Fama-Bliss • For DFM, SDFM, FSN and NS: VAR and CVAR • For CVAR case focus on 1 random walk We show the following results: • VAR and CVAR for DFM • Results for SDFM • Estimation results NS, FSN, AFNS and ATSM • In-sample fit of all models • Validity of restrictions 81 / 91
  • 82. DFM Likelihoods and AIC Below we show the value of the loglikelihood at the ML value (ℓ(ψ)) and AIC (AIC) for the Dynamic Factor Model (DFM): Model ℓ(ψ) AIC Model ℓ(ψ) AIC VAR(1) 3894.5 -7595 CVAR(1) 3899.0 -7606 VAR(2) 3918.5 -7625 CVAR(2) 3923.7 -7637 VAR(3) 3922.6 -7615 CVAR(3) 3927.7 -7627 VAR(4) 3932.2 -7616 CVAR(4) 3937.3 -7628 Note: Similar results hold for the NS and FSN model 82 / 91
  • 83. DFM-CVAR Influence of Factor Dynamics on Loadings CVAR(1) CVAR(2) Panel A CVAR(3) CVAR(4) Panel B 1.0 1.50 0.5 1.25 0.0 0 25 50 75 100 125 0 25 50 75 100 125 1.0 1.0 0.5 0.5 0.0 0.0 0 25 50 75 100 125 0 25 50 75 100 125 1.0 1 0 0.5 −1 0.0 −2 0 25 50 75 100 125 0 25 50 75 100 12583 / 91
  • 84. Smooth Dynamic Factor Model – Knot Selection Maturity Unrestricted model Final result 6 2.65 4.22∗ 6.08∗ 59.08∗∗ 6.50∗ 5.24∗ 9 0.79 2.40 5.59∗ - 6.58∗ 8.92∗∗ 12 0.23 1.35 4.29∗ - 16.25∗∗ 19.62∗∗ 15 0.04 0.33 1.51 - 24.17∗∗ 26.83∗∗ 18 0.00 0.02 0.28 - - - 21 0.95 0.74 1.52 18.55∗∗ - - 24 3.51 2.37 3.99∗ 23.13∗∗ - 7.35∗∗ 36 1.14 1.50 6.68∗∗ - - 26.88∗∗ 48 0.44 2.88 13.47∗∗ - 30.07∗∗ 52.87∗∗ 60 1.19 4.99∗ 18.04∗∗ - 26.79∗∗ 54.39∗∗ 72 2.59 5.74∗ 15.67∗∗ - 22.80∗∗ 43.00∗∗ 84 2.59 4.57∗ 8.81∗∗ - - 17.85∗∗ 96 0.76 1.68 1.79 7.68∗∗ - 5.10∗ 108 0.01 0.05 0.00 - - - Note: 3, 30 and 120 months not shown as these knots can not be removed 84 / 91
  • 85. SDFM Factor Loadings – CVAR Loading 1 SDFM DFM 1.0 0.5 0.0 10 20 30 40 50 60 70 80 90 100 110 120 Loading 2 1.0 0.5 10 20 30 40 50 60 70 80 90 100 110 120 1.0 Loading 3 0.5 0.0 10 20 30 40 50 60 70 80 90 100 110 120 85 / 91
  • 86. VAR coefficient matrix estimates Panel A: Stationary models SDFM NS FSN real img. real img. real img. 1 0.164 0.159 0.156 0.166 0.216 0.143 2 0.164 -0.159 0.156 -0.166 0.216 -0.143 3 0.607 0.134 0.593 0.056 0.642 0.259 4 0.607 -0.134 0.593 -0.056 0.642 -0.259 5 0.965 - 0.964 - 0.969 - 6 0.992 - 0.992 - 0.993 - 86 / 91
  • 87. VAR coefficient matrix estimates (cont’) Panel B: Nonstationary models SDFM NS FSN real img. real img. real img. 1 0.155 0.162 0.151 0.165 0.206 0.143 2 0.155 -0.162 0.151 -0.165 0.206 -0.143 3 0.601 0.123 0.594 0.099 0.649 0.258 4 0.601 -0.123 0.594 -0.099 0.649 -0.258 5 0.973 - 0.972 - 0.970 - 6 1 - 1 - 1 - 87 / 91
  • 88. NS Influence of Factor Dynamics on Loadings To get a feeling how the choice of factor dynamics affects the factor loadings we estimate the factor loadings parameter λ in the Nelson-Siegel model for different choices of factor dynamics: Model p=1 p=2 p=3 p=4 VAR(p) 0.07303 0.07211 0.07216 0.07193 CVAR(p) 0.07302 0.07210 0.07213 0.07191 Recall that for the Nelson-Siegel model we have 1 − e −λτ 1 − e −λτ gns (τ ) = ξ1 + · ξ2 + − e −λτ · ξ3 λτ λτ 88 / 91
  • 89. All Models - Overview Finally, we provide an overview of all models and test the restrictions imposed on the DFM by the various models Model ℓ(ψ) nψ AIC DFM-VAR(2) 3918.5 106 -7625 DFM-CVAR(2) 3923.7 105 -7637 SDFM-VAR(2) 3906.8 85 -7644 SDFM-CVAR(2) 3913.6 85 -7657 FSN-VAR(2) 3479.0 64 -6830 FSN-CVAR(2) 3483.7 63 -6841 NS-VAR(2) 3808.4 65 -7487 NS-CVAR(2) 3813.5 64 -7499 AFNS 3253.3 42 -6423 ATSM 3393.0 30 -6726 89 / 91
  • 90. All Models - Ljung-Box Statistics CVAR(2) factors VAR(1) factors Maturity DFM NS FSN SDFM NS AFNS ATSM 3 5.8 6.2 84.3∗∗ 6.0 11.6 10.5 17.9 6 7.1 7.4 11.2 7.4 12.6 11.8 33.1∗∗ 9 19.2∗ 19.3∗ 11.6 18.7∗ 31.8∗∗ 39.9∗∗ 55.1∗∗ 12 22.5∗∗ 29.2∗∗ 16.7 23.1∗∗ 36.2∗∗ 53.1 ∗∗ 52.6∗∗ 18 12.8 13.0 13.2 12.7 22.2∗∗ 28.1∗∗ 22.2∗∗ 21 12.2 12.0 13.8 12.2 18.8∗ 22.4∗∗ 19.1∗ 24 10.2 11.2 15.3 10.6 18.9∗ 21.6∗∗ 22.0∗∗ 30 9.3 9.4 10.5 9.1 17.2 15.8 16.0 60 5.9 5.7 9.2 5.6 11.5 10.3 11.2 84 8.4 9.4 10.4 9.1 15.3 12.9 17.4 120 10.1 9.7 6.9 11.0 9.7 9.3 12.2 Note: To preserve space 15, 36, 48, 72, 96 and 108 months omitted here 90 / 91
  • 91. All Models - Tests of Restrictions Stationary Models Nonstationary Models Model LR k p-value LR k p-value NS 220.2 41 0.000 220.4 41 0.000 FSN 879.0 42 0.000 879.8 42 0.000 SDFM 23.4 21 0.320 20.2 20 0.450 Panel C: Arbitrage-Free Models Model LR k p-value AFNS 1330.4 64 0.000 ATSM 1051.0 76 0.000 91 / 91