Introduction to VARs

and Structural VARs:

Estimation & Tests Using Stata

Bar-Ilan University 26/5/2009

Avichai Snir

Background: VAR

• Background:

• Structural simultaneous equations

– Lack of Fit with the data

– Lucas Critique (1976)

• VAR: Vector Auto Regressions

– Simple

– Non Structural

– All Variables are treated identically

– Better Fit with the Data

Simple VAR: Sims (1980)

• Symmetric

– Lags of the dependent variables

– Same Number of Lags

tttttttt

tttttttt

tttttttt

yyyyyyy

yyyyyyy

yyyyyyy

,32,332,222,141,331,221,110,31

,22,332,222,141,331,221,110,2

,12,332,222,141,331,221,110,1

...

...

...

εγγγγγγγεβββββββεααααααα

++++++++=

++++++++=

++++++++=

−−−−−−

−−−−−−

−−−−−−

( )3,2,1,

0),(

,,,

∈

≠

==

ji

tif

tifE

jijti

τ

τσεε τ

Simple VAR: Matrix Form• In Matrix Form:

• is a vector of the Dependent Variables

• is a Matrix of Coefficients

• is a Matrix in Lagged Variables

• is a Vector of White Noise Errors

• is a Matrix of exogenous variables (constant,…)

( )[ ] tt

tttt

yI

yyy

ε

εα

+Α=Γ−

++Γ+Γ+= −−−−

L

Simplyor

tt

:

...2211

ty

i−Γt( )LΓ

tε

Α

( )

=×

=×

:

00...0...00

...0...0...00

::::

00...00...0

...0......00

...00

::::

...00...00...00

...0...0...00

...0...0...00

.........

:

:

:

3,32,31,3

3,32,31,3

3,21,2

3,22,21,2

2,21,2

1,1

3,12,11,1

3,12,11,1

3,32,31,33,22,21,23,12,11,1

1,1

2,3

1,3

1,2

2,2

1,2

3,1

2,1

1,1

σσσσσσ

σσσσσ

σσ

σσσσ

σσσ

εεεεεεεεε

ε

εε

εεε

εεε

'εε tt

Covariance Matrix

Contemporary Variance Matrix

=Ω

3,32,31,3

3,22,21,2

3,12,11,1

σσσσσσσσσ

Issues Before Estimation

• Stationarity:

–Constant expected value

–Constant Variance

–Constant Covariances

• Granger Exogeneity:

–Order of variables

• Lag Length

–Optimal lag length

Testing Stationarity

• We have data on Canada 1966Q1-2002Q1

– GDP

– Consumer Price Index (CPI)

– Household Consumption (consumption)

1966Q365.4718.4138.46

1966Q264.5818.2437.47

1966Q162.5318.0436.91

DescriptorGDPCPIConsumption

Declare: Time Series• Define and format: time variable

– date(var_name,”dmy”) or

– Quarterly(var_name, “yq”)

– format: format var_name %d

• Declare database as time series

– Menu: statistics time series setup & utilities

declare dataset to be time series data

Declaring Time series

Declare Time Series

Transformations

• Convenience

– taking logs: gen var_name=log(var_name)

• Differences are in percentage

Plots• Menu: Graphics easy graphs line graph

• Follow the wizard…

GDP4

56

7log of gdp

01jan1960 09sep1973 19may1987 25jan2001date

log of gdp

34

56

7log of household consumption

01jan1960 09sep1973 19may1987 25jan2001date

Log of household consumption

33.5

44.5

5log of cpi

01jan1960 09sep1973 19may1987 25jan2001date

log of CPI

Stationarity• Data don’t look stationary

• Formal test required

• Common tests (Greene, 636-646):

– Dickey Fuller:

• H0: Variable has a unit root

– Philips Peron

• H0: Variable has a unit root

– Dickey Fuller – GLS

• H0: Variable has a unit root

Testing:

• Menu: StatisticsTime SeriesTests

• Choose a test and follow the menu

– Augmented Dickey Fuller

– DF-GLS for a Unit root

– Phillips-Peron unit root

Choosing a test

Running a Test• Augmented Dickey-Fuller Test

– 6 lags

– Including Trend

Result

• Cannot reject the null at 5%

Result Critical Values

Create First Differences

• Cannot Reject Unit root: Data is I(1)

• Create First Differences of the data:

Check the new graphs-.02

0.02

.04

.06

log of differenced GDP

01jan196 0 09sep1973 19may1987 25jan2001date

Log o f d if fe renced GDP

-.01

0.01

.02

.03

inflation

01jan1960 09sep1973 19may1987 25jan2001date

Inflation

0.01

.02

.03

.04

.05

difference household consumption

01jan1960 09sep1973 19may1987 25jan2001date

differenced household consumptionLog

Is it stationary now? (PP test)

The differenced data seems to be stationary

Granger Causality

• X Does not Granger Cause Y if:

• Test (Greene, p.592):

– Regress Y on lags of X and Y

– Regress Y on lags of Y

– Test if the restricted model is significantly outperformed by the non restricted model

– Either χ2 or F test

...),|(...),,...,,|( 212121 −−−−−− = tttttttt yyyExxyyyE

Granger Test• Run simple VAR between the variables of

interest

• Menu: Statisticsmultivariate time

seriesBasic Vector Autoregression Model

• Choose

– Variables

– Lag Length

Granger Test: Running VAR

Testing in Stata

• Statisticsmultivariate time series

var diagnostics and tests

Granger causality test

Granger Test

• Choose variables

Granger Test: Results

• We can reject that Inflation Granger Cause

Household Consumption

• We cannot reject that Household Consumption

Granger Cause Inflation

Optimal Lag Length• Sometimes, we have theory to guide us

• Often, we do not

• Three common tests (Greene, 589):

– Likelihood Ratio Test (LR)

– Akaike Information Criterion

– Bayesian (Schwartz) Information Criterion

•

Likelihood Ratio (LR) test

General to simple approach: Run VAR

with p lags. Use the LR test. If the test

rejects the null, then stop. Otherwise

run p-1 lags and compare with p-2…

( )

equationsofNumberM

matrixarianceedunrestrictW

matrixariancerestrictedW

MWWT

unres

res

unresres

−

−

−

→−=

cov

cov

lnln22

χλ

Information Criteria

• Two information Criteria: Akaike (AIC) and

Bayesian (BIC). Find the information criteria

for lag length 1 to p. Choose the lag length

that minimizes the information criteria that

you chose.

,,2)(

,

,,cov

)()(|)ln(|

2

BICforTAICforTIC

equationsofnumberMnsobservatioofnumberT

lagsofnumberpMatrixarianceTheW

T

TICMpMW

−

−−

−−

++=λ

Tests in Stata• Menu: Statisticsmultivariate time series

var diagnostics and tests

Lag-Order Selection statistics

Running test

• Choose Variables

• Choose maximum lags

Lag Length: Results

LR AIC BIC

We go with the LR and AIC and say 6

(why not?)

Run Simple VAR

• We run a simple VAR (not structural, no

assumptions on order of variables)

between Household Consumption,

Inflation and GDP

• To do so:

• Menu: Statisticsmultivariate time

seriesBasic Vector

Autoregression Model

Simple VAR• Choose

–Variables

–Lag Length

• Choose how to plot the response functions:

– Irf (simply uses the covariance matrix, minimum

order)

– Orf (orthogonalized the Covariance matrix to set

order)

– FEVD: Variance Decomposition Tables (In a

graph form)

Simple VAR

Results: Table of Coefficients

Impulse Response Function

- .005

0

.005

.01

-.005

0

.005

.01

-.005

0

.005

.01

0 5 10 15 0 5 10 15 0 5 10 15

v arbas ic , dcons , dcons v arbas ic , dcons , inf lat ion v arbas ic , dcons , y

v arbas ic , inf lat ion, dcons v arbas ic , inf lat ion, inf lat ion v arbas ic , inf lat ion, y

v arbas ic , y , dcons v arbas ic , y , inf lat ion v arbas ic , y , y

95% CI orthogonali zed irf

s tep

Graphs by irf name, impulse variable, and response variable

Simple VAR: Variance

Decomposition Table

Generating After Estimation

• generate after estimation:

– Choose:

• Menu: StatisticsMultivariate time series IRF &

Variance Decomposition Analysis

• Choose the table or impulse response function that

you need

To get the results

• If you want to use some of the results:• Coefficients

• Number of observations

• Etc…

– Stata keeps them under the ereturn command

– To get them type e(variable_name)

– To see all the variables that you can choose

from:

• ereturn list

Examples

More than simple VAR• More than a simple VAR:

– Adding Exogenous Variables

– Constraining blocks of variables to equal zero

• Use Menu: Statisticsmultivariate

time series Vector Autoregression

Model

• Generating Impulse Responses:

• Menu: StatisticsMultivariate time series

IRF & Variance Decomposition Analysis

More than simple VAR• Adding constraints on the A or B matrix

– A: y Matrix, B: errors matrix

– Short and long run constraints

• skip lags

• Menu: Statisticsmultivariate time seriesStructural Autoregression Model

• Stata runs the VAR with the restrictions

• Caveat 1: Too many constraints can lead to failures in the convergence process

• Caveat 2: You need enough constraints to allow identification.

Structural VAR

Defining a matrix

of constraints:

the ‘.’ imply a free

parameter

Using the

constraints:

Forcing the

values in the

constrained

matrix

Structural VAR: Results

Structural VAR: Results

- .005

0

.005

.01

-.005

0

.005

.01

-.005

0

.005

.01

0 5 10 15 0 5 10 15 0 5 10 15

v arbas ic , dcons , dcons v arbas ic , dcons , inf lat ion v arbas ic , dcons , y

v arbas ic , inf lat ion, dcons v arbas ic , inf lat ion, inf lat ion v arbas ic , inf lat ion, y

v arbas ic , y , dcons v arbas ic , y , inf lat ion v arbas ic , y , y

95% CI orthogonali zed irf

s tep

Graphs by irf name, impulse variable, and response variable

Structural VARs

• Structural VAR: VAR that is the result

of a structural model

• Goal: Obtaining the Structural

parameters out of the Estimated

Reduced Form

• Required: Number of Constraints

Model: Inflation and GDP

• Assume we have a simple model of the

form:

iablesrandomtindependennoiseWhite

lation

GDPy

yy

yy

tt

t

t

ttttt

tttt

var,,

inf

13210

12110

−

−

−

++++=

+++=

−

−−

ευπ

υπββββπεπααα

We can write it:

ttttt

tttt

yy

yy

υπββββπεπααα+++=−

+++=

−

−−

13201

12110

In Matrix Form:

+

+

=

− −

−

t

t

t

tt

t

t yy

υ

ε

πβ

α

πβ 1

1

0

0

1 1

01

OR:

−+

−+

−=

−

−

−−−

t

t

t

tt

t

t yy

υ

ε

βπββ

α

βπ

1

11

11

10

01

1 1

01

1

01

1

01

Inverting the Matrix gives

=

−

−

1

01

1

01

1

1

1 ββ

So we can substitute this in the equations:

We find:

( ) ( ) ( ) ( )ttttt

tttt

y

yy

υεβπβαββαββαβπεπααα

+++++++=

+++=

−−

−−

113211211001

12110

So we can write in VAR form:

tttt

tttt

y

yy

ηπθθθπεπααα

+++=

+++=

−−

−−

12110

12110

Almost there

• After estimating the VAR we can find:

( )( )( ) 2321

1211

0001

θβαβ

θβαβ

θβαβ

=+

=+

=+

So we have three equations and four

unknowns…

Hakuna Matata

• We also have the covariance matrix:

• So we have a fourth equation:

( )

+=

2

,1,1

,1,

,,

,,

tυσβσβ

σβσ

σσσσ

εεεε

εεεε

ηηηε

ηεεε

ηεεε σσβ ,,1 =

Run the VAR• Note that because we assume that the “real”

covariance matrix has the triangular form:

• We can use the OIRF that Stata gives us (Cholesky

factorization) to watch the Structural impulse

functions.

εεεε

εεσσβ

σ

,,1

, 0

Run the VAR (1 lag)

Study the Impulse Responses

0

.00 5

.01

0

.00 5

.01

0 2 4 6 8 0 2 4 6 8

v arbas ic , inf lat ion, inf lat ion v arbas ic , inf lat ion, y

v arbas ic , y , inf lat ion v arbas ic , y , y

95% CI orthogonali zed irf

s tep

Graphs by ir f name, impulse var iable, and response variable

Get the coefficients

1α

2θ1θ

0α2α

0θ

Get the Errors matrix

εεσ ,

εεσβ ,1

We find:

614.0142.0086.0625.0

142.0622.0086.0195.0

0043.00574.0086.00005972.0

086.000008145.0

000007018.0),cov(

2123

1112

0100

,1

=×−=−=

=×−=−=

−=×−=−=

===

αβθβ

αβθβ

αβθβ

σηε

βεε

Conclusion

• Enough restrictions

• Exact Identification

• Possible to deduce the Structural

Parameters

To test a restricted Model

• Run a non restricted model

• Test the null by using the LR test on the

difference between the restricted and

unrestricted model

( ) ( )

nsrestrictioofNumberM

matrixarianceedunrestrictW

matrixariancerestrictedW

MWWT

unres

res

unresres

−

−

−

→−=

cov

cov

lnln2

χλ

Caveat

• With the data we used, it is likely that

the variables are cointegrated

(consumption and GDP)

• One should (theoretically) check for

that option