F Test Stata Handout

. /* This is an example of hypothesis testing. > See chapter 11 in GHJ (1993) for more discussions on this example. > The data file can be downloaded from our class web page. > You are encouraged to replicate this handout with the data. > */ . . infile q pb pl pr m using table11.1, clear (30 observations read) . /* read the ascii file. Because the > ascii file doesn't have variable > names in it, we add the names in > the command. > */ . . /* q: quantity demanded for beer > pb: price of beer > pl: price of liquor > pr: price of remaining goods > m: income > */ . . gen lnq = ln(q) . gen lnpb = ln(pb) . gen lnpl = ln(pl) . gen lnpr = ln(pr) . gen lnm = ln(m) . . sum /* find summary statistics */ Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------q | 30 56.11333 7.857381 44.3 81.7 pb | 30 3.08 .6421945 1.78 4.07 pl | 30 8.367333 .7696347 6.95 9.52 pr | 30 1.251333 .298314 .67 1.73 m | 30 32601.8 4541.966 25088 41593 lnq | 30 4.018531 .1332578 3.790985 4.403054 lnpb | 30 1.102373 .220314 .5766134 1.403643 lnpl | 30 2.120151 .0936158 1.938742 2.253395 lnpr | 30 .1947126 .2525011 -.4004775 .5481214 lnm | 30 10.38263 .1406034 10.13015 10.63569 . . . . . * Remember: you have to run the regression before * you run the tests in Stata * regress lnq lnpb lnpl lnpr lnm Source | SS df MS ---------+-----------------------------Model | .425051343 4 .106262836 Residual | .08992006 25 .003596802 ---------+-----------------------------Total | .514971403 29 .017757635 Number of obs F( 4, 25) Prob > F R-squared Adj R-squared Root MSE = = = = = = 30 29.54 0.0000 0.8254 0.7975 .05997 -----------------------------------------------------------------------------lnq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------lnpb | -1.020419 .239042 -4.269 0.000 -1.512735 -.5281025 1 lnpl | -.58293 .5601505 -1.041 0.308 -1.736582 .5707216 lnpr | .2095452 .0796927 2.629 0.014 .0454151 .3736753 lnm | .9228608 .4155145 2.221 0.036 .0670928 1.778629 _cons | -3.243215 3.743001 -0.866 0.394 -10.95207 4.46564 -----------------------------------------------------------------------------. . * (a) test several zero coefficients h0: b3=0 & b4=0 . test lnpl lnpr ( 1) ( 2) lnpl = 0.0 lnpr = 0.0 F( . . . . . 3.63 0.0413 * (b) test all the slope coefficients * The F statistic should be the same as that in * the regression output. test lnpb lnpl lnpr lnm ( ( ( ( 1) 2) 3) 4) lnpb = 0.0 lnpl = 0.0 lnpr = 0.0 lnm = 0.0 F( . . . . . . 2, 25) = Prob > F = 4, 25) = Prob > F = 29.54 0.0000 * (c) redo (a) with restricted regression and * unrestricted regressions * Unrestricted model regress lnq lnpb lnpl lnpr lnm Source | SS df MS ---------+-----------------------------Model | .425051343 4 .106262836 Residual | .08992006 25 .003596802 ---------+-----------------------------Total | .514971403 29 .017757635 Number of obs F( 4, 25) Prob > F R-squared Adj R-squared Root MSE = = = = = = 30 29.54 0.0000 0.8254 0.7975 .05997 -----------------------------------------------------------------------------lnq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------lnpb | -1.020419 .239042 -4.269 0.000 -1.512735 -.5281025 lnpl | -.58293 .5601505 -1.041 0.308 -1.736582 .5707216 lnpr | .2095452 .0796927 2.629 0.014 .0454151 .3736753 lnm | .9228608 .4155145 2.221 0.036 .0670928 1.778629 _cons | -3.243215 3.743001 -0.866 0.394 -10.95207 4.46564 -----------------------------------------------------------------------------. . scalar r2_ur = e(r2) . scalar rss_ur = e(rss) . . * Restricted model . regress lnq lnpb lnm 2 Source | SS df MS ---------+-----------------------------Model | .398947679 2 .19947384 Residual | .116023724 27 .004297175 ---------+-----------------------------Total | .514971403 29 .017757635 Number of obs F( 2, 27) Prob > F R-squared Adj R-squared Root MSE = = = = = = 30 46.42 0.0000 0.7747 0.7580 .06555 -----------------------------------------------------------------------------lnq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------lnpb | -1.221246 .2313224 -5.279 0.000 -1.69588 -.7466114 lnm | 1.160383 .3624633 3.201 0.003 .4166696 1.904096 _cons | -6.683032 3.516249 -1.901 0.068 -13.89778 .5317156 -----------------------------------------------------------------------------. . scalar r2_r = e(r2) . scalar rss_r = e(rss) . . * Now calculate the F statistic . * (1) the RSS form . scalar f_rss = ((rss_r-rss_ur)/2)/(rss_ur/(30-5)) . * (2) the R^2 form . scalar f_r2 = ((r2_ur - r2_r)/2)/((1-r2_ur)/(30-5)) . * Compare the two results with (a) . display f_rss 3.628732 . display f_r2 3.628732 . . . . . . . . . * * * * Question: Use the above restricted model and unrestricted model method to test (b) and compare the results. The restrcited model is y=b1. In stata "regress lnq" Notice that in the restricted model R2=0. * (d) test linear combinations of coefficients * b2=b3, b3=b4 regress lnq lnpb lnpl lnpr lnm Source | SS df MS ---------+-----------------------------Model | .425051343 4 .106262836 Residual | .08992006 25 .003596802 ---------+-----------------------------Total | .514971403 29 .017757635 Number of obs F( 4, 25) Prob > F R-squared Adj R-squared Root MSE = = = = = = 30 29.54 0.0000 0.8254 0.7975 .05997 -----------------------------------------------------------------------------lnq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------lnpb | -1.020419 .239042 -4.269 0.000 -1.512735 -.5281025 lnpl | -.58293 .5601505 -1.041 0.308 -1.736582 .5707216 lnpr | .2095452 .0796927 2.629 0.014 .0454151 .3736753 lnm | .9228608 .4155145 2.221 0.036 .0670928 1.778629 _cons | -3.243215 3.743001 -0.866 0.394 -10.95207 4.46564 -----------------------------------------------------------------------------. * test command in stata always bases the test on the . * most recent regression (which is regress lnq lnpb lnm) . * so rerun the full regression for the test. 3 . test lnpb=lnpl ( 1) lnpb - lnpl = 0.0 F( 1, 25) = Prob > F = 0.39 0.5370 . test lnpl=lnpr, accumulate ( 1) ( 2) lnpb - lnpl = 0.0 lnpl - lnpr = 0.0 F( 2, 25) = Prob > F = 19.93 0.0000 . . . . . * (e) use the un/restricted model method to repeat (d) * Under the restrictions the model becomes * lnq = b1+b2*(lnqb+lnql+lnpr)+b5*lnm gen newvar = lnpb+lnpl+lnpr . . . . . * You should also run the unrestricted model. * Since the unrestricted model is the same as that in (c), * we will just re-use the results for unrestricted model * and only run the restricted model. regress lnq newvar lnm Source | SS df MS ---------+-----------------------------Model | .281701355 2 .140850677 Residual | .233270049 27 .008639631 ---------+-----------------------------Total | .514971403 29 .017757635 Number of obs F( 2, 27) Prob > F R-squared Adj R-squared Root MSE = = = = = = 30 16.30 0.0000 0.5470 0.5135 .09295 -----------------------------------------------------------------------------lnq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------newvar | .0615971 .1139342 0.541 0.593 -.1721766 .2953709 lnm | -.9222188 .4328337 -2.131 0.042 -1.81032 -.0341174 _cons | 13.3831 4.122115 3.247 0.003 4.925218 21.84098 -----------------------------------------------------------------------------. scalar r2_r = e(r2) . scalar rss_r = e(rss) . * Now calculate the F statistic . * (1) the RSS form . scalar f_rss = ((rss_r-rss_ur)/2)/(rss_ur/(30-5)) . * (2) the R^2 form . scalar f_r2 = ((r2_ur - r2_r)/2)/((1-r2_ur)/(30-5)) . * Compare the two results with (d) . display f_rss 19.927421 . display f_r2 19.927421 4