st: bootstrap test, combined bootstrap datasets, statistical properties of the bootstrap

[Gianluca Cafiso <gcafiso@unict.it>]

Dear Statalisters,

I have a question about a bootstrap test I am developing. My doubt  
concerns the statistical properties of the bootstrap test as I have  
envisaged it, and not how to technically implement it into Stata.

____________________________________

My statistic of interest is the product of two sub-statistics; it is:
dif_L= dif_TF * dif_GCI

where the ?dif? suffix denotes a time difference, TF is a mean value,  
and GCI an Entropy Index.

I am interested in testing ?Ho: dif_L>0?, against ?H1: dif_L<=0?.

The series of data used to compute ?dif_TF? has a different size (N2  
observations) than the series used for ?dif_GCI?(N1 observations).

If the two sub-statistics TF and GCI were generated from two series of  
the same size,
this would bring me to the general case and I would simply write a  
programme to
generate the ?dif_L? overall statistic and bootstrap it as usual.  But  
since this is not the case, I have thought to do the following:

1-	First Bootstrap for ?dif_TF?: I generate the ?dif_TF? statistic,  
bootstrap it (R repetitions) and store the dataset generated by the  
bootstrap (generates R samples of size N2, but one dataset with R  
observations for the estimated dif_TF, 1st dataset).

2-	Second Bootstrap for ?dif_GCI?:  I generate the ?dif_GCI?  
statistic, bootstrap it (R repetitions) and store the dataset  
generated by the bootstrap (generates R samples of size N1, but one  
dataset with R observations for the estimated dif_GCI, 2nd  dataset).

3-	Multiply the two dataset:  Since each of the R observations in each  
dataset (1 and 2) is an estimate of the statistic dif_TF and dif_GCI,  
by multiplying the two dataset I generate a unique dataset with R  
observations for the combined statistics:

dif_Lj= dif_TFj * dif_GCIj  where j=1,?,R.

4-	Use the Percentile Bootstrap of the dataset generated at point 3  
for the combined statistic ?dif_Lj? to the test the null hypothesis.

My doubt is the following:
Is the test based on the unique dataset (as generated at point 3)  
still valid? Or, for the so-generated dataset, do the usual  
distributional properties -on which Bootstrap-based tests are based-  
not hold?


Any help, suggestion, reference on this is really welcome.

Many thanks. Gianluca Cafiso

___________________________________________________________
Dr. Gianluca Cafiso
Research Fellow, Economics Department-University of Catania.
Corso Italia 55, Catania, Italy.

e-mail: gcafiso@unict.it
tel.:  +39 0957537745


----------------------------------------------------------------
Universita' di Catania - A.P.Se.Ma.
Servizio di Posta Elettronica


*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/