VAR and SVAR WITH STATA

Declare: Time Series• Define and format: time variable – datevar_name,”dmy” or – Quarterlyvar_name, “yq” – format: format var_name %d • Declare database as time series – Menu: statistics

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

Simple VAR: Sims (1980)

• Symmetric

– Lags of the dependent variables

– Same Number of Lags

t t

t t

t t

t t

t t

t t

t t

t t

t t

t t

t t

t t

y y

y y

y y

y

y y

y y

y y

y

y y

y y

y y

y

, 3 2

, 3 3 2

, 2 2 2

, 1 4 1

, 3 3 1

, 2 2 1

, 1 1 0

,

31

, 2 2

, 3 3 2

, 2 2 2

, 1 4 1

, 3 3 1

, 2 2 1

, 1 1 0

,

2

, 1 2

, 3 3 2

, 2 2 2

, 1 4 1

, 3 3 1

, 2 2 1

, 1 1 0

γ γ

γ γ

γ γ

ε β

β β

β β

β β

ε α

α α

α α

α α

+ + +

+ +

+ +

+

=

+ + +

+ +

+ +

+

=

+ + +

+ +

+ +

t if

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,…)

( )

t t

t t

y I

y y

y

ε

ε α

+ Α

= Γ

−

+ +

Γ + Γ

+

L

Simply or

1 1

0

0 0

0

0

0 0

: :

: :

0 0

0 0

0

0

0 0

0 0

: :

: :

0 0

0 0

0 0

0

0

0 0

0

0

0 0

: : :

3 , 3 2

, 3 1

, 3

3 , 3 2

, 3 1

,

3

3 , 2 1

, 2

3 , 2 2

, 2 1

, 2

2 , 2 1

,

2

1 , 1

3 , 1 2

, 1 1

, 1

3 , 1 2

, 1 1

,

1

3 , 3 2 , 3 1 , 3 3

, 2 2

, 2 1 , 2 3

, 1 2 , 1 1 , 1

1 , 1

2 , 3

1 , 3

1 , 2

2 , 2

1 , 2

3 , 1

2 , 1

1 , 1

σ σ

σ

σ σ

σ

σ σ

σ σ

σ

σ σ

σ

σ σ

σ

σ σ

σ

ε ε

ε ε

ε ε

ε ε

ε

ε ε ε

ε ε ε

ε ε ε

'

ε

ε t t

Covariance Matrix

Contemporary Variance Matrix

3 , 3 2

, 3 1

, 3

3 , 2 2

, 2 1

, 2

3 , 1 2

, 1 1

, 1

σ σ

σ

σ σ

σ

σ σ

σ

Issues Before Estimation

Testing Stationarity

• We have data on Canada 1966Q1-2002Q1

– GDP

– Consumer Price Index (CPI)

– Household Consumption (consumption)

1966Q3

65.47 18.41

38.46

1966Q2

64.58 18.24

37.47

1966Q1

62.53 18.04

36.91

Descriptor GDP

CPI Consumption

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

• Convenience

• Differences are in percentage

• Menu: Graphics
easy graphs
line graph

• Follow the wizard…

• Data don’t look stationary

• Formal test required

• Common tests (Greene, 636-646):

• 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

• Cannot reject the null at 5%

Create First Differences

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

Check the new graphs

Is it stationary now? (PP test)

The differenced data seems to be stationary

– Test if the restricted model is significantly

outperformed by the non restricted model

),

|(

),

, ,,

|( yt yt−1 yt−2 xt −1 xt−2 = E yt yt−1 yt−2

E

Granger Test

• Run simple VAR between the variables of interest

• Choose

– Variables

– Lag Length

Granger Test: Running VAR

Testing in Stata

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

– 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…

equations of

Number M

matrix ariance

ed unrestrict

W

matrix ariance

restricted W

M W

W T

unres

res

unresres

ln

λ

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

BICfor

TAICfor

T

IC

equationsof

numberM

nsobservatioof

number

T

lagsof

numberp

Matrixariance

The

W

T

TICM

pMW

=

λ

Tests in Stata

Lag-Order Selection statistics

Running test

• Choose Variables

• Choose maximum lags

Lag Length: Results

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:

series


Basic 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

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

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

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

95% CI orthogonali zed irf

s tep

Graphs by irf name, impuls e v ariable, and res pons e v ariable

Simple VAR: Variance Decomposition Table

Generating After Estimation

• generate after estimation:

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: Statistics


multivariate

time series


Vector Autoregression Model

• Generating Impulse Responses:

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

Structural 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

Structural VAR: Results

Structural VAR: Results

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

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

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

95% CI orthogonali zed irf

s tep

Graphs by irf name, impuls e v ariable, and res pons e v ariable

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:

iables random

t independen noise

White

lation

GDP y

y y

y y

t t

t t

t t

t t

t

t t

t t

var ,

,

inf

1 3

2 1

0

1 2

1 1

+ +

=

+ +

β β

β β

π

ε π

α α

α

We can write it:

t t

t t

t

t t

t

t

y y

y

y

υ π

β β

β β

π

ε π

α α

α

+ +

+

=

−

+ +

2 0

1

1 2

1 1

0

t t

β

α π

1 0

0

1 1

0 1

t t

π β

β

α β

π

1

1 1

1 1

1 0

0 1

0 1

1

0 1

1 0 1

Inverting the Matrix gives

0 1

We find:

( ) ( ) t ( ) t ( t t )

t

t t

t t

y

y

y

υ ε

β π

β α

β β

α β β

α β

π

ε π

α α

α

+ +

+ +

+ +

+

=

+ +

3 2

1 1

2 1

1 0

0 1

1 2

1 1

0

So we can write in VAR form:

t t

t t

t t

θ θ

θ π

ε π

α α

α

+ +

+

=

+ +

1 1

0

1 2

1 1

0

0 1

θ β

α β

θ β

α β

θ β

α β

= +

= +

= +

So we have three equations and four unknowns…

Hakuna Matata

• We also have the covariance matrix:

• So we have a fourth equation:

1 ,

1

, 1

, ,

,

, ,

t

υ σ

β σ

β

σ β

σ σ

σ

σ

σ

ε ε ε

ε

ε ε ε

ε η

η η

ε

η ε ε

ε

η ε

ε

σ

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

ε

ε

ε

σ σ

β

σ

, ,

1

Run the VAR (1 lag)

Study the Impulse Responses

Get the coefficients

0

α2α

0

θ

Get the Errors matrix

ε

ε

σ ,

ε ε

σ

β1 ,

We find:

614

0 142

0 086

0 625

0

142

0 622

0 086

0 195

0

0043

0 0574

0 086

0 0005972

0

086

0 00008145

0

000007018

0 )

, cov(

2 1 2

3

1 1 1

2

0 1 0

0

, 1

β

α β θ

β

α β θ

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

ns restrictio of

Number M

matrix ariance

ed unrestrict

W

matrix ariance

restricted W

M W

W T

unres

res

unres res

ln

λ