# How do I decide whether to use the Wald or F-test in testing the significance of slope coefficient?

I have a multiple linear regression model, and need to test the significance of the slopes. My professor says that there are two ways to go about it - so which should I use? The questions in the exam would have the standard error of the regressors along with the estimated value of the regressors as well.

• use whichever is easiest to implement! They are asymptotically equivalent, so no need to get fancy. For a single coefficient, you can also use the t-test, which allows you to make 1-sided hypothesis tests. Dec 14, 2011 at 0:54

I am certainly not the greatest person to answer this, but here it goes!

Wald test has only an asymptotic distribution that is known in general. In some specific cases, one can do much better than that. Let me give you a simple example.

For instance, suppose that you are interested in estimating the model $y_i = \beta_0 + \beta_1 x_i + u_i, \; \; i = 1, \dots, N$, where $u_i \sim \mbox{N}(0, \sigma^2)$, so that the error terms are normally distributed. Using ordinary least squares (OLS), one can obtain estimates $\hat{\beta_0}$ and $\hat{\beta_1}$. If you want to test whether the estimate $\hat \beta_1$ is significantly different from some value, say, $\beta_{10}$, you can use a simple $t$-test, square of which is equal to the $F$ test on $\hat \beta_1$.

In this perhaps slightly unrealistic setting, distribution of both of these tests is known exactly. That is to say that one knows the true distribution of the $t$ and $F$ statistics under the null hypothesis (incidentally, one can also find the distributions under the alternative). Hence, one does not need to resort to asymptotic distributions, which, in general, are only approximately true. As a corollary, for instance, confidence intervals constructed using the $F$ test will have better coverage. Hence, in this particular setting you would be better advised to use the $F$ test.

If the error terms are not normally distributed, $t$ and $F$ statistics no longer have exact finite sample distributions. So the choice no longer seems clear-cut. However, it can be shown (e.g., Davidson & MacKinnon, "Econometric Theory and Methods", p. 244) that $F$ and Wald tests are asymptotically equivalent, so that the choice is not really that important.

You may also be interested in taking a look at this reference.

Best answer I have found is at Stata FAQ: Chi square vs F Essentially, as denominator degrees of freedom increase, the F distribution approaches a chi square distribution.

• Perhaps not surprising, because the F distribution arises as a ratio of $\chi^2$ distributions.
– Sycorax
Apr 7, 2016 at 17:53
• Is there any chance you could fill in some more details? We prefer answers that are largely self-contained, in case external links go dead. Apr 7, 2016 at 18:34