Análisis económico de ciclo de vida (AECV)

7.4.2 Unrestricted VAR Dynamic Response Analysis Menu

This menu has the following options

0. Return to V AR Post Estimation Menu

1. Orthogonalized IR of variables to shocks in equations 2. Generalized IR of variables to shocks in equations 3. Orthogonalized forecast error variance decomposition 4. Generalized forecast error variance decomposition

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When you choose any one of the above options 1 to 4, you will be asked to choose the equation to be shocked. Each equation is designated by its left-hand-side variable. Move the cursor to the desired variable name (equation) and click . You will now be asked to specify the horizon (denoted in Chapter 22 by N ) for the impulse responses (or forecast error variance decomposition). The default value is set to 50, otherwise you need to type your desired value of N and then click (with N 150). Once you have speci…ed the horizon the program carries out the computations and presents you with a list of impulse responses (or forecast error variance decompositions) at di¤erent horizons. To plot or save the results click to move to the Impulse Response Results Menu. This menu has the following options

0. Move back/bootstrap con…dence intervals 1. Display results again

2. Graph

3. Save in a CSV …le

Option 0allows you to compute the empirical distribution of impulse responses for one or more variables by applying the bootstrap method. Choose the desired number of replications and the con…dence level (1 ) and click , then select the variable you want to inspect. You will be presented with the list of impulse responses at di¤erent horizons for the selected variable, together with their bootstrapped 1 ₂ and ₂ percentiles, their median and mean. Click to return to the Impulse Response Results Menu.

Option 1 enables you to see the results of the impulse response analysis and forecast error variance decompositions again.

Option 2 enables you to plot the impulse responses (or the forecast error variance decompositions) for one or more of the variables in the V AR at di¤erent horizons. If you have previously used option 0 to compute the bootstrapped con…dence intervals at a con…dence level (1 ), you can also plot the mean, median, 1 ₂ and ₂ percentiles of the impulse responses bootstrapped empirical distributions.

Option 3 allows you to save the impulse responses (or the forecast error variance de- compositions) for all the variables in a CSV …le for subsequent analysis.

It is worth noting that the orthogonalized impulse responses and the orthogonalized forecast error variance decompositions usually depend on the ordering of the variables in the V AR, but their generalized counterparts do not. The orthogonalized and the generalized impulse responses exactly coincide either for the …rst variable in the V AR or if is diagonal. An account of these concepts and the details of their computation are set out in Sections

22.5 and22.6.

7.4.3 VAR Hypothesis Testing Menu

This menu appears on the screen when option 4 in the Unrestricted V AR Post Estimation Menu is chosen. (see Section 7.4.1), and has the following options

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0. Return to V AR Post Estimation Menu

1. Testing and selection criteria for order (lag length) of the V AR 2. Testing for deletion of deterministic/exogenous variables in the V AR 3. Testing for block non-causality of a subset of variables in the V AR

Option 0 returns you to the Unrestricted V AR Post Estimation Menu (see Section

7.4.1).

Option 1 computes Akaike information and Schwarz Bayesian model selection criteria for selecting the order of the V AR(p), for p = 0; 1; 2; :::; P , where P represents the maximum order selected by the user (see Section 7.4). The selection procedure involves choosing the V AR(p) model with the highest value of the AIC or the SBC. In practice, the use of SBC is likely to result in selecting a lower order V AR model, as compared to the AIC. But in using both criteria it is important that the maximum order chosen for the V AR is high enough, so that high-order V AR speci…cations are given a reasonable chance of being selected, if they happen to be appropriate. This option also computes log-likelihood ratio statistics and their small sample adjusted values which can be used in the order-selection process. The log-likelihood ratio statistics are computed for testing the hypothesis that the order of the V AR is p against the alternative that it is P , for p = 0; 1; 2; ::; P 1. Users interested in testing the hypothesis that the order of the V AR model is p against the alternative that it is p + 1, for p = 0; 1; 2; :::; P 1, can construct the relevant log-likelihood statistics for these tests by using the maximized values of the log-likelihood function given in the …rst column of the result table corresponding to this option. For example, to test the hypothesis that the order of the V AR model is 2 against the alternative that it is 3, the relevant log-likelihood ratio statistic is given by

LR(2 : 3) = 2 (LL3 LL2) (7.9)

where LLp, p = 0; 1; 2; :::; p refers to the maximized value of the log-likelihood function for

the V AR(p) model. Under the null hypothesis, LR(2 : 3) is distributed asymptotically as a chi-squared variate with m2(3 2) = m2 degrees of freedom, where m is the dimension of zt

in equation (7.1). For further details and the relevant formulae see Section 22.4.1.

Option 2 computes the log-likelihood ratio statistic for testing zero restrictions on the coe¢ cients of a sub-set of deterministic/exogenous variables in the V AR. For example, to test the hypothesis that the V AR speci…cation in (7.1) does not contain a deterministic trend the relevant hypothesis will be a1= 0: In general, this option can be used to test the validity

of deleting one or more of the exogenous/deterministic variables from the V AR. When you choose this option you will be asked to list the deterministic/exogenous variable(s) to be dropped from the V AR model. Type in the variable name(s) in the box editor and click to process. The test results should now appear on the screen; they give the maximized values of the log-likelihood function for the unrestricted and the restricted model, and the log-likelihood ratio statistic for testing the restrictions. The degrees of freedom and the rejection probability of the test are given in round ( ) and square [ ] brackets, respectively. For further details and the relevant formulae see Section22.4.2.

Option 3 computes the log-likelihood ratio statistic for testing the null hypothesis that the coe¢ cients of a sub-set of jointly determined variables in the V AR are equal to zero.

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This is known as Block Granger Non-Causality test and provides a statistical measure of the extent to which lagged values of a set of variables (say z2t) are important in predicting

another set of variables, (say z1t) once lagged values of the latter set are included in the

model.

More formally, in (7.1), let zt= (z01t; z2t0 ) where z1tand z2tare m1 1 and m2 1 sub-sets

of zt, and m = m1+ m2. Consider now the following block decomposition of (7.1)

z1t= a10+ a11t + p X i=1 i;11z1;t i+ p X i=1 i;12z2;t i+ 1wt+ u1t; (7.10) z2t= a20+ a21t + p X i=1 i;21z1;t i+ p X i=1 i;22z2;t i+ 2wt+ u2t:

The hypothesis that the subset z2t does not ‘Granger-cause’z1t is de…ned by

HG : 12= 0;

where 12 = ( 1;12; 2;12:::; p;12) : When you choose this option you will be asked to list

the subset of variable(s) on which you wish to carry out the block non-causality test, namely z2t, in the above formulation. The program then computes the relevant log-likelihood ratio

statistic and presents you with the test results, also giving the maximized log-likelihood values under the unrestricted ( 12 6= 0) and the restricted model ( 12 = 0). For further

details and the relevant formulae see Section 22.4.3. Note that the Granger non-causality tests may give misleading results if the variables in the V AR contain unit roots (namely when one or more roots of (22.34) lie on the unit circle). In such a case one must ideally either use V AR models in …rst di¤erences, or cointegrating V AR models if the underlying variables are cointegrated. See the discussion in Canova (1995) p. 104, and the references cited therein.

7.4.4 Multivariate Forecast Menu

This menu appears on the screen when option 5 in the Unrestricted V AR Post Estimation Menu is selected. (See Section 7.4.1). It contains the following options

0. Choose another variable

1. Display forecast and forecast errors

2. Plot of in-sample …tted values and out of sample forecasts 3. Save in-sample …tted values and out of sample forecasts

Option 0 enables you to inspect forecasts of the level or …rst-di¤erences of another variable in the V AR.

Option 1 lists the actual values, multivariate forecasts and the forecast errors. In cases where actual values for the jointly determined variables over the forecast period are not

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available, it is still possible to generate multi-step ahead forecasts so long as observations on the exogenous/deterministic variables in the V AR (namely wt, intercepts and trends)

are available over the forecast period. In addition to listing the forecasts, this option also computes a number of standard summary statistics for checking the adequacy of the forecasts over the estimation and the forecast periods.

Option 2 enables you to see plots of the actual and forecast values for the selected variable. In the graph window you can specify a di¤erent period over which you wish to see the plots. Click the Start and Finish …elds and scroll through the drop-down lists to select the desired sample period, and then press the button ‘Refresh graph over the above sample period’.

Option 3 allows you to save the …tted and forecast values of the selected variable in the workspace in a new variable to be used in subsequent analysis.