ders 7.1 VAR.pptx

Assoc Prof Dr Ergin Akalpler
VAR Model

VAR Model
 VECTOR auto-regressive (VAR) integrated model
comprises multiple time series and is quite a useful
tool for forecasting. It can be considered an
extension of the auto-regressive (AR part of
ARIMA) model.

VAR Model
 VAR model involves multiple independent variables and
therefore has more than one equations.
 Each equation uses as its explanatory variables lags of all
the variables and likely a deterministic trend.
 Time series models for VAR are usually based on applying
VAR to stationary series with first differences to original
series and because of that, there is always a possibility of
loss of information about the relationship among integrated
series.

VAR model
 Differencing the series to make them stationary is
one solution, but at the cost of ignoring possibly
important (“long run”) relationships between the
levels. A better solution is to test whether the levels
regressions are trustworthy (“cointegration”.)

VAR Model
 The usual approach is to use Johansen’s method for
testing whether or not cointegration exists. If the answer is
“yes” then a vector error correction model (VECM),
which combines levels and differences, can be estimated
instead of a VAR in levels. So, we shall check if VECM is
been able to outperform VAR for the series we have.

What is the difference between VECM and
VAR?
 Through VECM we can interpret long term and short
term equations.
 We need to determine the number of co-integrating
relationships.
 The advantage of VECM over VAR is that the resulting
VAR from VECM representation has more efficient
coefficient estimates.

Introduction
The basics of the vector autoregressive model.
We lay the foundation for getting started with this crucial multivariate time
series model and cover the important details including:
•What a VAR model is.
•Who uses VAR models.
•Basic types of VAR models.
•How to specify a VAR model.
•Estimation and forecasting with VAR models.

To determine whether VAR model in levels is possible or not, we need to transform
VAR model in levels to a VECM model in differences (with error correction terms),
to which the Johansen test for cointegration is applied.
In other words, we take the following 4 steps
1. construct a VECM model in differences (with error correction terms)
2. apply the Johansen test to the VECM model in differences to find out the
number of cointegration (r) (none or Atmost)
3. if r = 0, estimate VAR in differences
4. if r > 0, estimate VECM model in differences or VAR in levels (at least one
cointegration equation exist)

Its identification depends on the number of cointegration in the following
way.
(none) or 0, r = 0 (no cointegration)
In the case of no cointegration, since all variables are non-stationary in level,
the above VECM model reduces to a VAR model with growth variables.
At most 1, r = 1 (one cointegrating vector)
At most 2, r = 2 (two cointegrating vectors)
At most 3) r = 3 (full cointegration)
In the case of full cointegration, since all variables are stationary, the above
VECM model reduces to a VAR model with level variables.

How to determine Restricted VAR –VECM- or
Unrestricted VAR
 If all variables converted to first difference then they become
stationary (integrated in same order)
 Null hypo: variables are stationary
 Alt Hypo: Variables are not stationary
 If the variables are cointegrated and have long run association
then we run restricted VAR (that is VECM),
 But if the variables are not cointegrated we cannot run VECM
rather we run unrestricted VAR.

RESTRICTED VAR
 After performing cointegration test results will
shows following estimations:
 Trace STATS > TCV
 Null: there is no cointegration
 Alt: There is cointegration
 When the Trace stats is more than TCV we can
reject null hypo there is cointegration
 Probability values are less than 0.05

Lets go UNRESTRICTED VAR
 After performing cointegration test results will
shows following estimations:
 Trace STATS < TCV
 Null: there is no cointegration
 Alt: There is cointegration
 When the Trace stats is less than TCV we cannot
reject null hypo there is no cointegration
 Probability values are more than 0.05
 >

Guideline to VAR
1.Variable selection & purpose
The initial step is to understand the task at hand –
2.what kind of variables should be included,
3.what maximum lag order would make sense based on the frequency and
description of the data.
4.What is the underlying economic or financial theory that is assumed
beforehand?

Var guideline
1.Data analysis and transformations
After selecting the data, the second step would be examine the data plots and summary
statistics and answer these questions:
•Are there any outliers in the data?
•Is there any missing data?
•Do you need to transform the data, e.g. take logarithms?
•Do you need to create any new variables - e.g. GDP per capita, number of children per
household etc.?

Var guideline
Unit root tests
After determining the final set of variables, Yt, we need to
test, whether they have a unit root (I(1), I(2), …) or are
stationary (I(0)).
To do this, use the ADF test and others
Probability values are more than 0.05

Var guideline
 VAR(p) model order selection
 Usually model selection is done based on some kind of
information criteria - usually AIC or BIC sometimes other
information criteria. This can be done using VARselect()
from the vars package. BIC penalizes model complexity
more heavily.

Var guideline
 Cointegrating relationship test for unit root series
If the time series has a unit root, we should check, whether there are any
cointegrating relationships between the series. There are three types of
tests, which can be performed:

Maximum Eigenvalue Test
The maximum eigenvalue test examines whether the largest eigenvalue is
zero relative to the alternative that the next largest eigenvalue is zero.
Maxeigen stats are smaller than TCV for VAR
 Trace Test
 The trace test is a test whether the trace stats are smaller than TCV for
VAR

Var guideline :Estimating the model
 If the series is not cointegrated, we can estimate the model via
VAR() function from package vars for the differences of the series,
ΔYt (if a unit root is present).
 If the series are cointegrated, we need to consider the long-run
relationship by estimating a VECM using either VECM() from
specifying the number of cointegrating relations, which we found
from the previous step.
 Depending on the function, we may also need to specify the lag
order of the VECM representation.

Var guideline : Model diagnostics tests
 Now that we have the model estimated, we need to verify if it is well
specified. This is usually done by examining the residuals from the model.
 The most important test for time series data are tests for autocorrelations (or
serial correlations) of the residuals, also known as Portmanteu test.
 Two most well-known versions of this test are the Ljung-Box and the Box-
Pierce tests, which are implemented in the Box.test() function from the stats
package.
 For multivariate time series alongside autocorrelation, another problem is the
cross-correlation of the residuals,
 i.e., when cor(ϵ1,t,ϵ2,t+s)≠0,s>0.
 For this reason, we may use the serial.test() function from the vars package,
which computes the multivariate test for serially correlated errors.
 A multivariate Ljung-Box test is implemented in the mq() function from the
MTS package.

VAR guideline : Results & conclusions
 After we verify that the model is adequate, we can either
predict() future values, or examine the impulse-response
functions via if r() from the vars package in order to check
how the variables respond to a particular shock.

VAR modeling is a multi-step process and a complete VAR
analysis involves:
1.Specifying and estimating a VAR model.
2.Using inferences to check and revise the model (as needed).
3.Forecasting.
4.Structural analysis.

What are VAR models used for?
 VAR models (vector autoregressive models) are used
for multivariate time series. The structure is that each
variable is a linear function of past lags of itself and past
lags of the other variables.

Who uses VAR models?
VAR models are traditionally widely used in finance and
econometrics because they offer a framework for accomplishing
important modeling goals, including (Stock and Watson 2001):
•Data description.
•Forecasting.
•Structural inference.
•Policy analysis.

The reduced form, recursive, and structural VAR
There are three broad types of VAR models, the reduced form, the
recursive form, and the structural VAR model.
Reduced form VAR models consider each variable to be a function of:
•Its own past values.
•The past values of other variables in the model.

 Recursive VAR models contain all the components of the
reduced form model, but also allow some variables to be
functions of other concurrent variables. By imposing these
short-run relationships, the recursive model allows us to model
model structural shocks.
 Structural VAR models include restrictions that allow us to
identify causal relationships beyond those that can be identified
identified with reduced form or recursive models. These causal
relationships can be used to model and forecast impacts of
individual shocks, such as policy decisions

While reduced form models are the simplest of the VAR models, they do
come with disadvantages:
• variables are not related to one another.
•The error terms will be correlated across equations. This means we cannot
consider what impacts individual shocks will have on the system.

 What makes up a VAR model?
 A VAR model is made up of a system of equations that represents the
relationships between multiple variables. When referring to VAR models,
we often use special language to specify:
• How many endogenous variables there are included.
• How many autoregressive terms are included.
 For example, if we have two endogenous variables and autoregressive
terms, we say the model is a Bivariate VAR(2) model. If we have three
endogenous variables and four autoregressive terms, we say the model is
a Trivariate VAR(4) model.
 In general, a VAR model is composed of n-equations
(representing n endogenous variables) and includes p-lags of the variables.

Specification
 What is the appropriate lag length in the VAR?
 Three criterions:
i. Akaike information criterion (AIC)
ii. Schwarz criterion (SIC)
iii. Hannan-Quinn criterion (HQC)
( all functions of m, T, and variance-covariance matrix)
 In practice: Fix an upper bound of lag length q (12), choose the q which
minimizes one of the information criterion
 AIC is inconsistent
 For T>20, SIC and HQC will always choose smaller models than AIC

Estimation
 Multivariate Generalized Least Squares (GLS) estimates are the
same as equation by equation OLS estimates.
 For unrestricted VAR models: Maximum likelihood (ML)
estimates and equation by equation OLS estimates coincide.
 When a VAR is estimated under some restrictions, ML estimates
are different from OLS estimates;
ML estimates are consistent and efficient if the restrictions are
true.

Presentation of Results
 It is rare to report estimated VAR coefficients.
Instead:
 Impulse responses
 Forecast error variance decomposition: assess the relative
contribution of different shocks to fluctuations in
variables
 Historical Decomposition: given the path of one specific
shock, how will the variables evolve?

How do we decide what endogenous variables to include in our VAR
model?
 From an estimation standpoint, it is important to be deliberate about how
many variables we include in our VAR model. Adding additional variables:
• Increases the number of coefficients to be estimated for each equation
and each number of lags.
• Introduce additional estimation error.
 Deciding what variables to include in a VAR model should be founded in
theory, as much as possible.
 We can use additional tools, like Granger causality or Sims causality, to
test the forecasting relevance of variables.
o

UNRESTRICTED VAR
 Assess the selection of the optimal lag length in a VAR
 Evaluate the use of impulse response functions with a
VAR
 Assess the importance of variations on the standard VAR
 Critically appraise the use of VAR s with financial
models.
 Assess the uses of VECMs

What is a vector autoregressive model?
The vector autoregressive (VAR) model is a workhouse multivariate time
series model that relates current observations of a variable with past
observations of itself and past observations of other variables in the
system.
VAR models differ from univariate autoregressive models because they
allow feedback to occur between the variables in the model. For example,
we could use a VAR model to show how real GDP is a function of policy rate
and how policy rate is, in turn, a function of real GDP.

• A systematic but flexible approach for capturing complex real-world
behavior.
• Better forecasting performance.
• Ability to capture the intertwined dynamics of time series data.
Advantages of VAR models

VAR modeling is a multi-step process and a complete VAR analysis
involves:
1.Specifying and estimating a VAR model.
2.Using inferences to check and revise the model (as needed).
3.Forecasting.
4.Structural analysis.

 How do we choose the number of lags in a VAR model?
 Lag selection is one of the important aspects of VAR model specification. In
practical applications, we generally choose a maximum number of lags, p
max, and evaluate the performance of the model including p=0,1,…,p max.
 The optimal model is then the model VAR(p) which minimizes some lag
selection criteria.
 These methods are usually built into software and lag selection is almost
completely automated now.

Estimating and inference in VAR models
 Despite their seeming complexities, VAR models are quite easy to
estimate. The equation can be estimated using ordinary least
squares given a few assumptions:
• The error term has a conditional mean of zero.
• The variables in the model are stationary.
• Large outliers are unlikely.
• No perfect multicollinearity.

 Under these assumptions, the ordinary least squares
estimates:
• Will be consistent.
• Can be evaluated using traditional t-statistics and p-
values.
• Can be used to jointly test restrictions across multiple
equations.

Forecasting
 One of the most important functions of VAR models is to generate
forecasts. Forecasts are generated for VAR models using an
iterative forecasting algorithm:
1. Estimate the VAR model using OLS for each equation.
2. Compute the one-period-ahead forecast for all variables.
3. Compute the two-period-ahead forecasts, using the one-period-
ahead forecast.
4. Iterate until the h-step ahead forecasts are computed.

Reporting and evaluating VAR models
 Often we are more interested in the dynamics that are predicted by
our VAR models than the actual coefficients that are estimated. For
this reason, it is most common that VAR studies report:
• Granger-causality statistics.
• Impulse response functions.
• Forecast error decompositions

Lag Length in VAR
 When estimating VARs or conducting ‘Granger causality’ tests, the
test can be sensitive to the lag length of the VAR
 Sometimes the lag length corresponds to the data, such that
quarterly data has 4 lags, monthly data has 12 lags etc.
 A more rigorous way to determine the optimal lag length is to use the
Akaike or Schwarz-Bayesian information criteria.
 However the estimations tend to be sensitive to the presence of
autocorrelation, in this case following the use of information
criteria, if there is any evidence of autocorrelation, further lags are
added, above the number indicated by the information criteria, until
the autocorrelation is removed.

Information Criteria
 The main information criteria are the Schwarz-Bayesian criteria
and the Akaike criteria.
 They operate on the basis that there are two competing factors
from adding more lags to a model. More lags will reduce the RSS,
but also means a loss of degrees of freedom (penalty from adding
more lags).
 The aim is the minimise the information criteria, by adding an
extra lag, it will only benefit the model if the reduction in the RSS
outweighs the loss of degrees of freedom.
 In general the Schwarz-Bayesian (SBIC) has a harsher penalty term
than the Akaike (AIC), which leads it to indicate a parsimonious
model is best.

The AIC and SIC
 The two can be expressed as:
parameters
of
No.
-
k
size,
sample
-
variance
residual
ˆ
:
ln
)
ˆ
ln(
2
)
ˆ
ln(
2
2
2
T
Where
T
T
k
SBIC
T
k
AIC









Multivariate Information Criteria
 The multivariate version of the Akaike information criteria
is similar to the univariate:
equations
all
in
regressors
of
number
total
ns
observatio
of
number
matrix)
the
of
diagonal
main
the
off
residuals
the
between
s
covariance
and
diagonal
main
on the
variances
the
gives
(This
residuals.
the
of
matrix
var
ˆ
)
(
/
2
ˆ
log










k
T
iance
Co
Variance
Akaike
T
k
MAIC

Multivariate SBIC
 The multivariate version of the SBIC is:
equations
all
in
regressors
of
number
total
ns
observatio
of
number
residuals
the
of
matrix
var
ˆ
)
log(
ˆ
log










k
T
iance
Co
Variance
T
T
k
MSBIC

The best criterion
 In general there is no agreement on which criteria is best
(some recommends the SBIC).
 The Schwarz-Bayesian is strongly consistent but not
efficient.
 The Akaike is not consistent, generally producing too large
a model, but is more efficient than the Schwarz-Bayesian
criteria.

Criticisms of Causality Tests
Granger causality test, much used in VAR modelling, however do not
explain some aspects of the VAR:
 It does not give the sign of the effect, we do not know if it is positive or
negative
 It does not show how long the effect lasts for.
 It does not provide evidence of whether this effect is direct or indirect.

VARs and seemingly unrelated regression SUR
 In general the VAR has all the lag lengths of the individual
equations the same size.
 It is possible however to have different lag lengths for different
equations, however this involves another estimation method.
 When lag lengths differ, the) seemingly unrelated regression (SUR
approach can be used to estimate the equations, this is often
termed a ‘near-VAR’.

Alternative VARs
 It is possible to include contemporaneous terms in a VAR, however in
this case the VAR is not identified.
 It is also possible to include exogenous variables in the VAR, although
they do not have separate equations where they act as a dependent
variable.
 (They simply act as extra explanatory variables for all the equations in
the VAR.)
 It is worth noting that the impulse response functions can also produce
confidence intervals to determine whether they are significant, this is
routinely done by most computer programmes.

VECMs
 Vector Error Correction Models (VECM) are the basic VAR,
with an error correction term incorporated into the model.
 The reason for the error correction term is the same as
with the standard error correction model, it measures any
movement away from the long-run equilibrium.
 These are often used as part of a multivariate test for
cointegration, such as the Johansen Maximum likelihood
(ML) test.

VECMs
 However there are a number of differing approaches to modelling VECMs,
for instance how many lags should there be on the error correction term,
usually just one regardless of the order of the VAR
 The error correction term becomes more difficult to interpret, as it is not
obvious which variable it affects following a shock

What is Wald test
 The Wald statistic explains the short run causality
between variables whiles the statistics provided by the
lagged error correction terms explain the intensity of the
long run causality effect.
 Short run Granger causalities are determined by Wald
statistic for the significance of the coefficients of the
series.

Criticisms of the VAR
 Many argue that the VAR approach is lacking in theory.
 There is much debate on how the lag lengths should be determined
 It is possible to end up with a model including numerous explanatory
variables, with different signs, which has implications for degrees of
freedom.
 Many of the parameters will be insignificant, this affects the efficiency
of a regression.
 There is always a potential for multicollinearity with many lags of the
same variable

Stationarity and VARs
 Should a VAR include only stationary variables, to be valid?
 Sims argues that even if the variables are not stationary, they should
not be first-differenced.
 However others argue that a better approach is a multivariate test for
cointegration and then use first-differenced variables and the error
correction term

Sample VAR Result
 OLS estimation of a single equation in the Unrestricted VAR
 ******************************************************************************
 Dependent variable is TBILL
 127 observations used for estimation from 1960Q2 to 1991Q4
Regressor Coefficient Standard Error T-Ratio [Prob]
 TBILL(-1) .96200 .067845 14.1795 [.000]
 R10(-1) -.015333 .068439 -.22404 [.823]
 K .36563 .23386 1.5635 [.120]
 R-Squared .90159 R-Bar-Squared .90000
 Akaike Info. Criterion -165.9593 Schwarz Bayesian Criterion -170.22
 Serial Correlation*CHSQ( 4)= 22.3179[.000]
 Dependent variable is R10
 ******************************************************************************
 Regressor Coefficient Standard Error T-Ratio[Prob]
 TBILL(-1) .11106 .039920 2.7821[.006]
 R10(-1) .87432 .040269 21.7117[.000]
 K .26981 .13760 1.9608[.052]
 R-Squared .96507 R-Bar-Squared .96451
 Akaike Info. Criterion -98.6049 Schwarz Bayesian Criterion -102.8712
 Serial Correlation*CHSQ( 4)= 8.6481[.071]

Granger-causality statistics
As we previously discussed, Granger-causality statistics test whether
one variable is statistically significant when predicting another variable.
The Granger-causality statistics are F-statistics that test if the
coefficients of all lags of a variable are jointly equal to zero in the
equation for another variable. As the p-value of the F-statistic
decreases, evidence that a variable is relevant for predict another
variable increases.

Granger causality
 Granger causality tests whether a variable is “helpful” for
forecasting the behavior of another variable.
 It’s important to note that Granger causality only allows us to make
inferences about forecasting capabilities -- not about true causality.

Granger Causality Test
 ******************************************************************************
 Dependent variable is R10
 List of the variables deleted from the regression: TBILL (-1)
 127 observations used for estimation from 1960Q2 to 1991Q4
 ******************************************************************************
 Regressor Coefficient Standard Error T-Ratio [Prob]
 R10(-1) .97627 .017142 56.9508 [.000]
 K .20365 .13914 1.4637 [.146]
 ******************************************************************************
 Joint test of zero restrictions on the coefficients of deleted variables:
 F Statistic F( 1, 124)= 7.7400[.006]

 Dependent variable is TBILL
 List of the variables deleted from the regression: R10(-1)
 Regressor Coefficient Standard Error T-Ratio [Prob]
 TBILL(-1) .94817 .028025 33.8328 [.000]
 K .33727 .19589 1.7217 [.088]
 ******************************************************************************
 Joint test of zero restrictions on the coefficients of deleted variables:
 F Statistic F( 1, 124)= .050192[.823]
 *****************************************************************************

 For example, in the Granger-causality test of X on Y, if the p-
value is 0.02 we would say that X does help predict Y at the
5% level. However, if the p-value is 0.3 we would say that
there is no evidence that X helps predict Y.

Granger Causality Tests Continued
 According to Granger, causality can be further sub-divided
into long-run and short-run causality.
 This requires the use of error correction models or VECMs,
depending on the approach for determining causality.
 Long-run causality is determined by the error correction
term, whereby if it is significant, then it indicates
evidence of long run causality from the explanatory
variable to the dependent variable.
 Short-run causality is determined as before, with a test on
the joint significance of the lagged explanatory variables,
using an F-test or Wald test.

Impulse Response and Variance
decomposition
 the impulse responses are the relevant tools for
interpreting the relationships between the variables
 Variance decompositions examine how important each of
the shocks is as a component of the overall
(unpredictable) variance of each of the variables over
time.

 The impulse response function traces the dynamic path of variables in the
system to shocks to other variables in the system. This is done by:
• Estimating the VAR model.
• Implementing a one-unit increase in the error of one of the variables in
the model, while holding the other errors equal to zero.
• Predicting the impacts h-period ahead of the error shock.
• Plotting the forecasted impacts, along with the one-standard-deviation
confidence intervals.

Impulse Response Functions
 Given:























 
0
1
0
at time
shock to
unit
a
Given
:
20
10
0
1
2
1
2
1
1
1
1
u
u
y
t
y
A
Where
u
y
A
y
t
t
t
t





Impulse Response Functions
 These trace out the effect on the dependent variables in the VAR to
shocks to all the variables in the VAR
 Therefore in a system of 2 variables, there are 4 impulse response
functions and with 3 there are 9.
 The shock occurs through the error term and affects the dependent
variable over time.
 In effect the VAR is expressed as a vector moving average model
(VMA), as in the univariate case previously, the shocks to the error
terms can then be traced with regard to their impact on the
dependent variable.
 If the time path of the impulse response function becomes 0 over
time, the system of equations is stable, however they can explode if
unstable.

 The impulse response function traces the dynamic path of variables in the system
to shocks to other variables in the system. This is done by:
• Estimating the VAR model.
• Implementing a one-unit increase in the error of one of the variables in the model,
while holding the other errors equal to zero.
• Predicting the impacts h-period ahead of the error shock.
• Plotting the forecasted impacts, along with the one-standard-deviation confidence
intervals.

 The results show IR (Impulse
response) to dependent variables. Only
for NIR IR function is illustrated on the
table and
 as on the table seen only NIR has
positive response to CPI. But against to
this all other variables have negative
response to NIR
 Impulse Response positive values
have positive negative values have
negative effects on dependent (here
CPI)
R. of
DCPI:
Period RGDP DCPI DNIR DREER
1 -3.870022 10.52160 0.000000 0.000000
2 4.350339 0.388418 0.635650 -3.964539
3 2.581088 -0.057747 1.343376 -0.210536
4 -1.406336 0.760648 0.709599 -0.485223
5 -1.189040 0.131412 0.477037 -0.098667
6 0.043845 -0.346002 0.243500 0.050212
7 0.401353 -0.000346 0.078936 0.053059
8 -0.003204 0.089603 0.006877 -0.037810
9 -0.052022 0.019648 -0.044851 -0.027014
10 -0.032278 -0.017211 0.007166 0.004444
Impulse response sample estimation and interpretation

Variance decomposition estimation and interpretation
 On the table, the variance
decomposition results for CPI
illustrated.
 RGDP and REER affects CPI
more than NIR.
 Higher values have more
effects than smaller values
VD of
DCPI:
Period S.E. RGDP DCPI DNIR DREER
1 11.21076 11.91672 88.08328 0.000000 0.000000
2 12.68381 21.07330 68.90575 0.251152 9.769804
3 13.01512 23.94694 65.44426 1.303893 9.304905
4 13.14111 24.63526 64.53047 1.570594 9.263682
5 13.20444 25.21040 63.92289 1.686082 9.180623
6 13.21138 25.18501 63.92429 1.718280 9.172418
7 13.21782 25.25268 63.86205 1.720173 9.165098
8 13.21818 25.25131 63.86316 1.720107 9.165417
9 13.21840 25.25202 63.86125 1.721200 9.165528
10 13.21845 25.25241 63.86091 1.721216 9.165466

Long-run Causality
 Before the ECM can be formed, there first has to be evidence of
cointegration, given that cointegration implies a significant error
correction term, cointegration can be viewed as an indirect test of long-
run causality.
 It is possible to have evidence of long-run causality, but not short-run
causality and vice versa.
 In multivariate causality tests, the testing of long-run causality between
two variables is more problematic, as it is impossible to tell which
explanatory variable is causing the causality through the error correction
term.

A simple example
As an example, let's consider a VAR with three endogenous variables,
the unemployment rate, the inflation rate, and interest rates.
To estimate the structural VAR model of the system, we have to put
restrictions on our model. For example, we may assume that the
Fed follows the inflation targeting rule for setting interest rates. This
assumption would be built into our system as the equation for
interest rates.

Specification
 What is the appropriate lag length in the VAR?
 Three criterions:
i. Akaike information criterion (AIC)
ii. Schwarz criterion (SIC)
iii. Hannan-Quinn criterion (HQC)
( all functions of m, T, and variance-covariance matrix)
 In practice: Fix an upper bound of lag length q (12), choose the q
which minimizes one of the information criterion
 AIC is inconsistent
 For T>20, SIC and HQC will always choose smaller models than AIC

Estimation
 Multivariate generalized least squares (GLS) estimates are the same as
equation by equation OLS estimates.
 For unrestricted VAR models: Multıvariate linera model – (ML)
estimates and equation by equation OLS estimates coincide.
 When a VAR is estimated under some restrictions, ML estimates are
different from OLS estimates;
ML estimates are consistent and efficient if the restrictions are true.

Presentation of Results
 It is rare to report estimated VAR coefficients.
 Impulse responses
 Forecast error variance decomposition: assess the relative contribution
of different shocks to fluctuations in variables
 Historical Decomposition: given the path of one specific shock, how will
the variables evolve?

How do we decide what endogenous variables to include in our VAR
model?
 From an estimation standpoint, it is important to be deliberate about how
many variables we include in our VAR model. Adding additional variables:
• Increases the number of coefficients to be estimated for each equation
and each number of lags.
• Introduce additional estimation error.
 Deciding what variables to include in a VAR model should be founded in
theory, as much as possible.
 We can use additional tools, like Granger causality or Sims causality, to test
the forecasting relevance of variables.

Estimating and inference in VAR models
 Despite their seeming complexities, VAR models are quite easy to
estimate. The equation can be estimated using ordinary least squares
given a few assumptions:
• The error term has a conditional mean of zero.
• The variables in the model are stationary.
• Large outliers are unlikely.
• An outlier is an unusually large or small observation. Outliers can
have a disproportionate effect on statistical results, such as the
mean, which can result in misleading interpretations.
• No perfect multicollinearity.

 Under these assumptions, the ordinary least squares estimates:
• Will be consistent.
• Can be evaluated using traditional t-statistics and p-values.
• Can be used to jointly test restrictions across multiple equations.

Forecasting
 One of the most important functions of VAR models is to generate
forecasts. Forecasts are generated for VAR models using an iterative
forecasting algorithm:
1. Estimate the VAR model using OLS for each equation.
2. Compute the one-period-ahead forecast for all variables.
3. Compute the two-period-ahead forecasts- like times series for one
period of time- , using the one-period-ahead forecast.
4. Iterate until the n period or h-step ahead forecasts are computed.

 Forecast error decomposition separates the forecast error variance into
proportions attributed to each variable in the model.
 Intuitively, this measure helps us judge how much of an impact one
variable has on another variable in the VAR model and how intertwined
our variables' dynamics are.
 For example, if X is responsible for 85% of the forecast error variance of Y,
it is explaining a large amount of the forecast variation in X. However,
if X is only responsible for 20% of the forecast error variance of Y, much of
the forecast error variance of Y is left unexplained by X.

What is the difference between VAR and VEC
model?
 Through VECM we can interpret long term and short term
equations. We need to determine the number of co-integrating
relationships. The advantage of VECM over VAR is that the
resulting VAR from VECM representation has more efficient
coefficient estimates.

When to use VAR/VECM?
You should use VECM if 1) your variables are nonstationary
and 2) you find a common trend between the variables
(cointegration).

Why VAR is better than AR?
 VAR (vector autoregression) is a generalization of
AR (autoregressive model) for multiple time series,
identifying the linear relationship between them. The
AR can be seen as a particular case of VAR for only one
series

Conclusion
 VAR models are an essential component of multivariate time series
modeling. You should have a better understanding of the
fundamentals of the VAR model including:
• What a VAR model is.
• Who uses VAR models.
• Basic types of VAR models.
• How to specify a VAR model.
• Estimation and forecasting with VAR models.

Conclusion
 VARs have a number of important uses, particularly
causality tests and forecasting
 To assess the affects of any shock to the system, we
need to use impulse response functions and variance
decomposition
 VECMs are an alternative, as they allow first-
differenced variables and an error correction term.
 The VAR has a number of weaknesses, most
importantly its lack of theoretical foundations

Thank you
erginakalpler@csu.edu.tr

ders 7.1 VAR.pptx

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ders 7.1 VAR.pptx