# st: wald test

 From Kit Baum <[email protected]> To [email protected] Subject st: wald test Date Mon, 22 Jan 2007 06:25:34 -0500

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
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
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,
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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