Unformatted text preview: ncepts
• One important feature of VARs is the compactness with which we can
write the notation. For example, consider the case from above where k=1. • We can write this as or or even more compactly as
yt
g×1 = β0
g×1 + β1 yt1
g×g g×1 + ut
g×1
40 Vector Autoregressive Models:
Notation and Concepts (cont’d) • This model can be extended to the case where there are k lags of each
variable in each equation:
yt = β0 + β1 yt1
g×1 g×1 g×g g×1 • + β2 yt2
g×g g×1 +...+ βk ytk + ut g×g g×1 g×1 We can also extend this to the case where the model includes first
difference terms and cointegrating relationships (a VECM). 41 Vector Autoregressive Models Compared with Structural
Equations Models
• • Advantages of VAR Modelling
 Do not need to specify which variables are endogenous or exogenous  all are
endogenous
 Allows the value of a variable to depend on more than just its own lags or
combinations of white noise terms, so more general than ARMA modelling
 Provided that there are no contemporaneous terms on the right hand side of the
equations, can simply use OLS separately on each equation
 Forecasts are often better than “traditional structural” models.
Problems with VAR’s
 VAR’s are atheoretical (as are ARMA models)
 How do you decide the appropriate lag length?
 So many parameters! If we have g equations for g variables and we have k lags of each
of the variables in each equation, we have to estimate (g+kg2) parameters. e.g. g=3, k=3,
parameters = 30
 Do we need to ensure all components of the VAR are stationary?
 How do we interpret the coefficients?
42 Choosing the Optimal Lag Length for a VAR
2 possible approaches: crossequation restrictions and information criteria
CrossEquation Restrictions
In the spirit of (unrestricted) VAR modelling, each equation should have
the same lag length
Suppose that a VAR(2) estimated using quarterly data has 8 lags of the two
variables in each equation, and we want to examine a restriction that the
coefficients on lags 5 through 8 are jointly zero. This can be done using a
likelihood ratio test
Denote the variancecovariance matrix of residuals (given by ), as . The
likelihood ratio test for this joint hypothesis is given by 43 Choosing the Optimal Lag Length for a VAR
(cont’d)
where
is the variancecovariance matrix of the residuals for the restricted
model (with 4 lags),
is the variancecovariance matrix of residuals for the
unrestricted VAR (with 8 lags), and T is the sample size.
• The test statistic is asymptotically distributed as a χ2 with degrees of freedom
equal to the total number of restrictions. In the VAR(2) case above, we are
restricting 4 lags of two variables in each of the two equations = a total of 4 *
2 * 2 = 16 restrictions.
• In the general case where we have a VAR with p equations, and we want to
impose the restriction that the last q lags have zero coefficients, there would
be p2q restrictions altogether
• Disadvantages: Conducting the LR test...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
 Summer '13
 JaneBargers
 Financial Markets

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