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
g×1 = β0
g×1 + β1 yt-1
g×g g×1 + ut
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 yt-1
g×1 g×1 g×g g×1 • + β2 yt-2
g×g g×1 +...+ βk yt-k + 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
• • Advantages of VAR Modelling
- Do not need to specify which variables are endogenous or exogenous - all are
- 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 a-theoretical (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: cross-equation restrictions and information criteria
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 variance-covariance 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
is the variance-covariance matrix of the residuals for the restricted
model (with 4 lags),
is the variance-covariance 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
- Financial Markets