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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 yt-1 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 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 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 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 Cross-Equation 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 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 (cont’d) where 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.

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