[Gretl-users] Bootstrap news

[Allin Cottrell]

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.