st: wald test

[Kit Baum <baum@bc.edu>]

Joe suggested

Stata reports F statistics after -regress- (linear models) when testing
multiple predictors jointly.  I'm not sure why you'd want a chi- 
square test
statistic if you can obtain an F statistic, but you can use -glm- in  
order
to obtain the chi-square statistic and associated p-value if you're  
working
with linear models.  It's illustrated below.

sysuse auto
regress price mpg headroom trunk
test mpg headroom trunk // F statistic
glm price mpg headroom trunk, nolog
test mpg headroom trunk // chi-square statistic
exit


Although it is certainly true that the Wald F is the ratio of two chi- 
squares, if you want a "large-sample" test of the joint hypothesis, I  
would recommend using large-sample statistics throughout. In this  
example, glm will produce "OIM" standard errors identical to those of  
OLS (regress), but it calls the ratio of coefficient to s.e.  "z" and  
evaluates it as "z", that is, N(0,1).

If you do the same regression using
ivreg2 price mpg headroom trunk
you get, by default, z-statistics on the output (unless you use  
'small', which reproduces -regress-). The standard errors and z- 
statistics are different (slightly smaller) because of the division  
by N rather than (N-k). Likewise,
test mpg headroom trunk
after ivreg2 will yield a chi-square which is somewhat larger than  
that of glm.

I am not sure how the ratio of coefficients to glm standard errors  
should be considered 'z'. These are maximum likelihood estimates,  
which customarily use a divisor of N, and the estimate corresponding  
to sigma^2 from glm, the 'scale parameter', is  6874339. You will see  
that this is exactly the square of 2622, ivreg2's Root MSE, and NOT  
the square of 2550, regress's Root MSE. So ivreg2 and glm use the  
same estimate of sigma^2 in calculating the vce, but they produce  
different vce estimates.

In this regard I think Joanne's request for a 'large-sample joint  
test' is better served by ivreg2 followed by test. I don't know much  
about glm, but I find it somewhat odd that (a) its sigma^2 is the MLE  
estimate (divisor of N) but its vce is the small-sample version  
corresponding to (N-k), and (b) what appear to be statistics that are  
t_N-k under their null are being quoted as z statistics. Perhaps  
someone familiar with glm's methodology could comment on these two  
points.

Kit Baum, Boston College Economics
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html


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