[Gretl-users] Bootstrap news

Andreas K suggested a short while ago that gretl should offer various bootstrap options via the graphical interface (GUI). I agreed at the time that this was a good idea. Here's a note on progress so far (in CVS, and the current gretl snapshot for Windows). First, the limitation: all of the following new stuff applies only to single-equation models estimated via OLS. (Though please note that we already offer bootstrapped confidence intervals for impulse response functions in relation to VARs.) When you estimate a model via OLS in the GUI, the model viewer window has a menu bar, including items labeled "Analysis" and "Tests". New under "Analysis": there's a "Bootstrap..." item. This opens a dialog box where you get to make five choices: (1) The variable/coefficient to examine. (2) "Confidence interval" vs "Studentized confidence interval" vs "P-value". (3) "Resampled residuals" vs "Simulate normal errors". (4) Number of replications (default 1000). (5) Show graph of bootstrap sampling distribution (no/yes). Notes: In relation to (1): you only get to examine one coefficient at a time by this particular means. On (2): the default (95%) confidence interval is based directly on the quantiles of the bootstrap coefficient estimates. The "studentized" version is as per Davidson and MacKinnon's "Econometric Theory and Methods" (ETM), chapter 5: at each bootstrap replication a t-ratio is formed as (a) the difference between the current and the baseline coefficient estimate, divided by (b) the baseline estimated standard error. Then the confidence interval is formed based on the quantiles of this t-ratio, as explained in ETM. The "P-value" variant is, again, as explained in ETM. On (3): you get to choose between resampling with replacement of the original residuals (rescaled as suggested in ETM), and simulated normal errors with the empirically given variance. Andreas suggested including the option of "case resampling" (that is, resampling the (y, X) pairs rather than the residuals. I have not implemented this to date for two reasons: first, it seems statistically dodgy, and second it is considerably more burdensome from the computational viewpoint. (You can economize substantially if the X matrix is treated as constant across the bootstrap replications.) Point (4) should be mostly self-explanatory. However, when you're doing a (1 - alpha) confidence interval, then, as explained in ETM, it is desirable that alpha(B + 1)/2 is an integer (where B is the number of replications). So gretl adjusts the user-chosen B value to ensure this is the case. Point (5) again should be self-explanatory: you can get gretl to make a graph of the density of the bootstrapped coefficient or t-ratio. This option employs gretl's kernel density estimation facility. The above all pertains to the "Analysis/Bootstrap" menu item. In addition you have options under "Tests/Linear restrictions". The restrictions dialog now has a "Use bootstrap" check box. If you check this, you get a bootstrapped F-test for whatever set of linear restictions you have entered. The methodology is as described in ETM for bootstrapped P-values. Autoregressive models: If the set of regressors includes the first lag of the dependent variable this should be handled correctly: the bootstrap data sets are calculated recursively, taking into account the autoregression. Please note that higher-order autoregressions are _not_ currently recognized and handled appropriately. In script mode: For single-equation models estimated via OLS, you can append the --boot flag to the "restrict" command to get bootstrapped tests. You can also set the default number of bootstrap replications using the "set" command with "bootrep" parameter. For example: set bootrep 10000 Testing and comments welcome! Allin.