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# Paris2012 session3b

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### Paris2012 session3b

1. 1. Time Series Forecasting Siem Jan Koopmanhttp://personal.vu.nl/s.j.koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2012
2. 2. Unobserved components: decomposing time seriesA 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. 3. IllustrationsWe present various illustrations of time series analysis andforecasting: 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. 4. Illustration 1: European business cycleAzevedo, Koopman and Rua (JBES, 2006) consider Europeanbusiness 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. 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. 6. Shifted cyclesIn standard case, cycle ψt is generated by ψt+1 cos λ sin λ ψt κt + =φ + + ψt+1 − sin λ cos λ ψt κ+ tThe 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 1More details in R¨nstler (2002) for idea of shifting cycles in umultivariate unobserved components time series model ofHarvey and Koopman (1997). 6 / 91
7. 7. The basic multivariate modelPanel 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. 8. Business cycleStock and Watson (1999) states that ﬂuctuations in aggregateoutput are at the core of the business cycle so the cyclicalcomponent of real GDP is a useful proxy for the overall businesscycle and therefore we impose a unit common cycle loading andzero phase shift for Euro area real GDP.Time series 1986 – 2002:quarterly GDPindustrial productionunemployment (countercyclical, lagging)industrial conﬁdenceconstruction conﬁdenceretail trade conﬁdenceconsumer conﬁdenceretail salesinterest rate spread (leading) 8 / 91
9. 9. Eurozone Economic Indicators14.30 GDP Retail sales IPI unemployment Interest rate spread Industrial confidence indicator14.25 Construction confidence indicator Retail trade confidence indicator Consumer confidence indicator14.2014.1514.1014.0514.0013.9513.90 1990 1995 2000 9 / 91
10. 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. 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. 12. The business cycle coincident indicatorSelected 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. 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. 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. 15. EuroCOIN and our growth indicator0.01500.01250.01000.00750.00500.00250.0000−0.0025−0.0050−0.0075 Coincident Eurocoin 1990 1995 2000 15 / 91
16. 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. 17. Bivariate analysis: daily spot prices and consumptionOur 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 beencarried 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. 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) yearlycycle; (iv) irregular. 18 / 91
19. 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. 20. Forecasting resultsWe 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 inWeek 637. The last forecast is for Sunday, April 10, 2005 in Week 640. The weeks 638 and 639 contain calendareﬀects 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. 21. Illustration 3: periodic dynamic factor analysisAim: the joint modeling of 24 hours in a daily panel of electricityloads for EDF.Focus: modelling and short-term forecasting of hourly electricityloads, 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. 22. Periodic dynamic factor model speciﬁcationThe 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. 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. 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. 25. Multivariate Time Series ModelA periodic approach: from a univariate hourly to a daily 24 × 1vector:yt = (y1,t . . . yS,t )′ , S = 24 hours per day, t = 1, . . . , T days.Our multivariate time-varying parameter regression model is givenby: 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. 26. Time-varying regressions and dynamic factors K yt = µ t + Btk xtk + εt , εt ∼ IIN (0, Σε ) , t = 1, . . . , T , k=1where 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 jwith constant c j = (c1 . . . cS )′ , S × R j factor loading matrices Λjand R j dynamic factors ftj = (f1,t . . . fR j ,t )′ , for j = 0, . . . , K and j j0 ≤ 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. 27. Dynamic factor specﬁcationsLocal 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. 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. 29. Yearly pattern estimates per hour 4 ˆk kµs,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. 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. 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. 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 300Selecting the right model requires experience and stamina! 32 / 91
33. 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. 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. 35. Illustration 4: The macroeconomy in the euro areaQuarterly time series, 1981 – 2008, GDP in volumes,for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain. (i) (ii) 13.212.8 13.012.6 12.812.4 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 201012.7 (iii) 12.25 (iv)12.6 12.0012.5 11.7512.412.3 11.50 1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010 35 / 91
36. 36. Illustration 4: The housing market in the euro areaQuarterly 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. 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. 38. The basic multivariate modelMultiple set of M economic time series, yit , is collected inyt = (y1t , . . . , yMt )′ and model is given by (1) (2) yt = µt + ψt + ψt + εt ,where the disturbance driving each vector component is a vectortoo, with a variance matrix. The structure of the variance matrixdetermines the dynamic interrelationships between the M timeseries.For example, if trend component µt follows the random walk,µt+1 = µt + ηt with disturbance vector ηt , with variance matrixΣη : • 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. 39. Dynamic factor representationsWe can formulate the multivariate unobserved components modelalso 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 matrixAη is such that ′ Ση = Aη Aη ,and, similarly, loading matrices are deﬁned for the other variancematrices of disturbances that drive the components. (1) (2)Here the dynamic factors or unobserved components µt , ψt , ψtand εt are ”normalised”. 39 / 91
40. 40. STAMPModel is eﬀectively a state space model: Kalman ﬁlter methodscan be applied for maximum likelihood estimation of parameters(such as the loading matrices).Kalman ﬁlter methods are employed for the evaluation of thelikelihood function and score vector.Kalman ﬁlter and smoothing methods are employed for signalextraction 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: amulti-platform, user-friendly software: econometrics, time seriesand forecasting by clicking.It can treat multivariate unobserved components time seriesmodels... 40 / 91
41. 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 areaEmpirical 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. 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. 43. Univariate analysisObjectives: • Verify the trend-cycle decomposition for each series • Verify whether possible restrictions are realisticResults 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. 44. Univariate results for GDP France Germany Italy SpainGDP R R R RTrend var 0.65 0.03 0.01 0.03 0.48 0.03 0.10 0.03Cycle 1 var 0.81 0.17 0.00 0.15 3.85 5.75 0.07 0.00Cycle 1 ρ 0.94 0.90 1.0 0.90 0.87 0.90 0.95 0.90Cycle 1 p 3.04 5 5.42 5 2.97 5 3.62 5Cycle 2 var 0.00 1 1.81 2.86 0.00 7.79 0.00 2.31Cycle 2 ρ 1.0 0.95 0.95 0.95 1.00 0.95 1.00 0.95Cycle 2 p 5.8 12 13.5 12 5.50 12 9.11 12Irreg var 1 0.0 1 1 1 1 1 1N 7.2 11.4 3.23 5.23 6.58 11.1 27.1 34.9Q 14.5 24.9 15.1 14.6 9.26 13.3 22.1 24.8R2 0.31 0.24 0.11 0.02 0.23 0.12 0.22 0.12 44 / 91
45. 45. Univariate results for HP France Germany Italy SpainRHP R R R RTrend var 0.59 0.03 0.34 0.03 0.00 0.03 0.39 0.03Cycle 1 var 0.00 0.01 0.31 1.51 0.04 0.02 1 0.01Cycle 1 ρ 1.0 0.90 0.97 0.90 0.96 0.90 0.34 0.90Cycle 1 p 6.34 5 4.48 5 1.11 5 – 5Cycle 2 var 0.00 2.19 1 19.9 1 49.4 0.00 39.5Cycle 2 ρ 1.0 0.95 0.61 0.95 0.99 0.95 0.99 0.95Cycle 2 p 8.37 12 2.82 12 13.3 12 – 12Irreg var 1 1 0 1 0 1 0 1N 23.8 0.59 5.89 9.95 7.03 8.32 36.1 11.9Q 10.6 187 55.5 111 13.7 68.4 29.3 127R2 0.61 0.25 0.35 0.15 0.56 0.22 0.47 0.28 45 / 91
46. 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. 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. 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 HPF GDP 1.00 0.46 0.40 0.24 0.64 -0.46 0.57 0.42F HP 0.46 1.00 0.29 0.62 0.33 -0.51 0.35 0.39G GDP 0.40 0.29 1.00 0.32 0.75 -0.16 0.67 0.58G HP 0.24 0.62 0.32 1.00 0.18 -0.52 0.06 0.13I GDP 0.64 0.33 0.75 0.18 1.00 -0.13 0.61 0.65I HP -0.46 -0.51 -0.16 -0.52 -0.13 1.00 -0.25 -0.19S GDP 0.57 0.35 0.67 0.06 0.61 -0.25 1.00 0.75S HP 0.42 0.39 0.58 0.13 0.65 -0.19 0.75 1.00 48 / 91
49. 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 HPF GDP 1.00 0.51 0.53 0.23 0.89 0.16 0.90 0.63F HP 0.51 1.00 0.46 0.44 0.58 0.68 0.68 0.94G GDP 0.53 0.46 1.00 0.52 0.44 0.49 0.62 0.46G HP 0.23 0.44 0.52 1.00 0.07 0.82 0.22 0.43I GDP 0.89 0.58 0.44 0.07 1.00 0.08 0.90 0.72I HP 0.16 0.68 0.49 0.82 0.08 1.00 0.29 0.64S GDP 0.90 0.68 0.62 0.22 0.90 0.29 1.00 0.76S HP 0.63 0.94 0.46 0.43 0.72 0.64 0.76 1.00 49 / 91
50. 50. Bivariate analysisFor each country, we carry out a bivariate analysis between GDPand 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. 51. Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHPFRAtrend 0.0 0.0 0.0 – – N 3.25 13.4cyc 1 3.0 3.3 0.88 8.0 0.98 Q 17.0 17.4cyc 2 1.0 126 0.07 15.6 0.99 R2 0.38 0.63irreg 0.6 1.6 -0.19 – –GERtrend 0.0 0.003 0.0 – – N 8.52 1.08cyc 1 2.5 5.4 -0.6 4.3 0.90 Q 6.86 42.1cyc 2 3.1 0.5 1.0 7.0 0.98 R2 0.39 0.29irreg 4.3 1.1 0.58 – – 51 / 91
52. 52. Bivariate results for GDP and HP GDP RHP corr per ρ diag GDP RHPITAtrend 0.1 0.9 -0.15 – – N 4.19 4.57cyc 1 4.3 16.2 -0.08 6.0 0.92 Q 10.1 8.60cyc 2 0.0 8.4 0.0 1.1 0.94 R2 0.14 0.47irreg 0.8 1.2 0.96 – –SPNtrend 0.0 0.0 0.0 – – N 9.05 21.7cyc 1 3.3 11.9 0.95 8.2 0.98 Q 17.5 43.0cyc 2 0.0 83.3 0.82 14.4 0.99 R2 0.45 0.73irreg 3.9 7.7 -0.35 – – 52 / 91
53. 53. Four-variate cross-country analysis of GDP and RHPNow we incorporate earlier ﬁndings and impose a strict short- andlong-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. 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. 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. 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. 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. 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. 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. 60. Eight-variate results: HP and GDP for four countriesSimilar 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. 61. Eight-variate results: short cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HPF-G 1 -0.33 0.67 0.10 0.81 -0.59 0.77 0.13F-H 1 0.075 0.65 -0.35 -0.13 -0.12 -0.64G-G 1 0.17 0.80 -0.27 0.88 -0.011G-H 1 0.055 -0.26 -0.10 -0.95I-G 1 -0.037 0.66 0.034I-H 1 -0.55 -0.040S-G 1 0.34S-H 1 61 / 91
62. 62. Eight-variate results: long cycle correlations France Germany Italy Spain GDP HP GDP HP GDP HP GDP HPF-G 1 0.95 0.19 0.043 0.72 0.41 0.54 0.50F-H 1 0.44 0.24 0.63 0.43 0.57 0.58G-G 1 0.41 -0.31 0.26 0.44 0.21G-H 1 -0.005 0.57 0.036 0.29I-G 1 0.045 0.12 0.37I-H 1 0.13 0.099S-G 1 0.61S-H 1 62 / 91
63. 63. Illustration 5 : Modelling U.S. Yield Curve Yield (in %)6.506.256.005.755.505.25 Maturity (in months) 0 10 20 30 40 50 60 70 80 90 100 110 120 63 / 91
64. 64. Time Series of Four Maturities Yield (in %) Time to maturity: 3 month 10 Yield (in %) Time to maturity: 1 year8 86 64 4 Date Date 1985 1990 1995 2000 1985 1990 1995 2000 Yield (in %) Yield (in %) Time to maturity: 3 year Time to maturity: 10 year10 10.08 7.56 5.0 Date Date 1985 1990 1995 2000 1985 1990 1995 2000 64 / 91
65. 65. Term Structure of Interest Rates over Time 10.0Yield (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. 66. Literature ReviewEarlier 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 81. ResultsStrategy: • 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 walkWe 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. 82. DFM Likelihoods and AICBelow 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 -7628Note: Similar results hold for the NS and FSN model 82 / 91
83. 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. 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. 85. SDFM Factor Loadings – CVAR Loading 1 SDFM DFM1.00.50.0 10 20 30 40 50 60 70 80 90 100 110 120 Loading 21.00.5 10 20 30 40 50 60 70 80 90 100 110 1201.0 Loading 30.50.0 10 20 30 40 50 60 70 80 90 100 110 120 85 / 91
86. 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 0.1432 0.164 -0.159 0.156 -0.166 0.216 -0.1433 0.607 0.134 0.593 0.056 0.642 0.2594 0.607 -0.134 0.593 -0.056 0.642 -0.2595 0.965 - 0.964 - 0.969 -6 0.992 - 0.992 - 0.993 - 86 / 91
87. 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 0.1432 0.155 -0.162 0.151 -0.165 0.206 -0.1433 0.601 0.123 0.594 0.099 0.649 0.2584 0.601 -0.123 0.594 -0.099 0.649 -0.2585 0.973 - 0.972 - 0.970 -6 1 - 1 - 1 - 87 / 91
88. 88. NS Inﬂuence of Factor Dynamics on LoadingsTo get a feeling how the choice of factor dynamics aﬀects thefactor loadings we estimate the factor loadings parameter λ in theNelson-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 0.07191Recall 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. 89. All Models - OverviewFinally, we provide an overview of all models and test therestrictions 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. 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 12.2Note: To preserve space 15, 36, 48, 72, 96 and 108 months omitted here 90 / 91
91. 91. All Models - Tests of Restrictions Stationary Models Nonstationary ModelsModel LR k p-value LR k p-valueNS 220.2 41 0.000 220.4 41 0.000FSN 879.0 42 0.000 879.8 42 0.000SDFM 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