7.5 Cointegrating VAR options
7.5.4 Impulse Response Analysis and Forecasting Menu
This menu appears on the screen after a successful implementation of options 1 to 6 in the Long-Run Structural Modelling Menu (see Section 7.5.3). It contains the following option:
0. Return to identify/test cointegrating vectors
1. Impulse Response of variables to shocks in equations 2. Forecast Error Variance Decomposition analysis 3. Impulse Response of CVs to shocks in equations 4. Persistence Pro…le of CVs to system-wide shocks 5. Trend/Cycle Decomposition
6. Compute multivariate dynamic forecasts 7. Display restricted/…xed CVs again 8. Display error correction equations 9. Save error correction terms
10. Display system covariance matrix of errors 11. Save the cointegrating V AR model in a CSV …le
Option 0 enables you to estimate/test (further) over-identify restrictions on the coin- tegrating or long-run coe¢ cients. The restrictions could involve parameters from di¤erent long-run relations (see Section 22.7 and option 4 in the Long-Run Structural Modelling Menu described in Section 7.5.3). When you choose this option you will be asked to con- …rm whether you wish to test over-identifying restrictions on the long-run relations. If you say ‘No’, you will be returned to the Long-Run Structural Modelling Menu (see Sections
22.9.1-22.9.3). If your answer is in the a¢ rmative you will be presented with a box-editor to specify your over-identifying restrictions. Our recommendation is to introduce these restric- tions gradually (ideally one by one), starting from those that are less likely to be rejected. The asymptotic standard errors reported below the just-identi…ed estimates could provide a good guide as to which over-identifying restrictions to impose …rst, second and so on. Once your over-identifying restrictions are added successfully to the existing set of restric- tions (including the just-identifying ones), you will be presented with a screen containing initial values for all the long-run coe¢ cients. These are the estimates obtained under the previous set of restrictions. We recommend that you accept these initial estimates.6 If you
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You can, of course, edit these initial estimates if you experience di¢ culties with the convergence of the iterative algorithm.
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now click to accept the initial values you will be presented with a small menu giving you a choice of the ‘Back substitution algorithm (A) as in Micro…t 4’, the ‘Back substitu- tion algorithm (B) new to Micro…t 5’, and the ‘Modi…ed Newton-Raphson algorithm’. The highlighting is always on the ‘Back substitution algorithm (B) new to Micro…t 5’, which is the one that we recommend. If you choose the modi…ed Newton-Raphson algorithm7 you will also be given a choice of a damping factor in the range [0.01 to 2.0]. We recommend starting with the value of 0.01, unless you experience di¢ culties with getting the algorithm to converge. Once you have chosen the algorithm, the program starts the computations and, if the iterative process converges successfully, presents you with the M L estimates of the long-run relations subject to the over-identifying restriction, together with their asymptotic standard errors in round brackets. Micro…t also generates log-likelihood ratio statistics for testing the over-identifying restrictions, which are asymptotically distributed as 2 variates with degrees of freedom given by k r2, where k is the total number of restrictions and r2
is the number of just-identifying restrictions (see Section22.9.3). If you click , you will be presented with a window asking whether you want bootstrapped critical values of overi- dentifying restrictions on long-run relationships. You can choose the number of replications and two di¤erent signi…cance levels. If you click Micro…t starts the computation and presents you with an output window which reports the bootstrapped critical values of the log-likelihood ratio statistics. If you click you return to the Impulse Response Analysis and Forecasting Menu.
Option 1 computes and displays orthogonalized and generalized impulse responses of variable-speci…c shocks on the di¤erent variables in the cointegrating V AR model, (possibly) subject to over-identifying restrictions on the long-run coe¢ cients. Once the results are ob- tained, it is also possible to compute bootstrapped con…dence intervals of impulse responses, for any desired con…dence level.
Option 2 computes and displays orthogonalized and generalized forecast error variance decompositions for the cointegrating V AR model, (possibly) subject to restrictions on the long-run relationships. You can then obtain bootstrapped con…dence intervals for the error variance decomposition at a given con…dence level.
Option 3 computes and displays orthogonalized and generalized impulse responses of the e¤ect of variable-speci…c shocks on the r cointegrating relations.
Option 4 computes and displays the time pro…le of the e¤ect of system-wide shocks on the cointegrating relations, referred to as ‘persistence pro…les’. Selecting options 3 or 4 allows you to obtain bootstrapped con…dence intervals for persistence pro…les, for a given con…dence level. The algorithms used to carry out the computations for options 3 and 4 are set out in Section22.9.5and 22.9.6, where references to the literature can also be found.
Option 5 allows you to perform the multivariate Beveridge Nelson trend/cycle decom- position (see Section22.11).
Option 6 enables you to compute multivariate, multi-step ahead forecasts (of levels and of …rst-di¤erences) of yt conditional on values of xt and wt. The forecasts obtained using
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For a detailed account of the Back substitution algorithm (A) and of the Modi…ed Newton-Raphson algorithm see Section22.9.2.
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this option and those obtained using option 7 in the Cointegrating V AR Post Estimation Menu will be identical under just-identifying restrictions on the cointegrating relations, and di¤er only when there are over-identifying restrictions on (see Section7.4.4, and option 7 in Section 7.5.2). In the case of cointegrating V ARX (option 3 from the System Estimation Menu), you can choose between conditional and unconditional or ex ante forecasts, depending on whether you wish to use the realized values of the exogenous variables or their forecast values. For the conditional forecasts the values of the exogenous variables during the forecast period are treated as known. The unconditional forecasts use forecasts of the exogenous variables, obtained using the marginal model.
Option 7 displays the M L (or …xed) estimates of the cointegrating vectors again. This option also allows you to obtain bootstrapped con…dence intervals for M L estimates, for two di¤erent signi…cance levels.
Option 8 displays error correction equations for each of the jointly determined I(1) variables in the model. These estimates are followed by diagnostic statistics and the other options available after the OLS option. See section 6.6.
Option 9 saves error correction terms in the workspace.
Option 10 displays the degrees-of-freedom adjusted system covariance matrix of the errors in the underlying V AR model, (7.11). The adjustments are made by dividing the cross-product of residuals from di¤erent equations by n s, where s is the total number of coe¢ cients estimated for each equation in the V AR. This adjustment does not take account of the cross-equation restrictions on the long-run coe¢ cients, y, implicit in the
cointegrating restrictions. These estimates will be identical to those obtained using option 4 in the Cointegrating V AR Post Estimation Menu, if the cointegrating vectors, , are not subject to over-identifying restrictions. See Section 7.5.2.
Option 11 allows you to save the estimated cointegrating V AR model as a CSV …le.