Going from derived estimators to their implementation in software

Estimation and Inference in Econometrics by Davidson and MacKinnon (1993 edition, the older one) on page 552, ch 16.3 'Covariance Matrix Estimation' states:

"Consequently, the matrix $$\begin{eqnarray} (n^{-1} \hat{X}'\hat{X})^{-1}(n^{-1}\hat{X}' \hat{\Omega} \hat{X})(n^{-1}\hat{X}'\hat{X})^{-1} \end{eqnarray}$$ consistently estimates the probability limit of the OLS heteroskedastic covariance matrix. Of course, in practice one ignores the factors of $$n^{-1}$$ and uses the matrix $$\begin{eqnarray} ( \hat{X}'\hat{X})^{-1}(\hat{X}' \hat{\Omega} \hat{X})(\hat{X}'\hat{X})^{-1} \end{eqnarray}$$ to estimate the covariance of $$\hat{\beta}$$.

Now, why can we ignore the factors of $$n^{-1}$$? (They don't even cancel out.)

As a second point, what do the authors mean by $$\hat{X} \equiv X(\hat{\beta})$$?

Can someone help me understand this?

• It would help if you mentioned what acclaimed text you had in hand, preferably with a page number. – Dimitriy V. Masterov Mar 3 '15 at 19:53
• You might also want to give it a more informative title. Ed Leamer had something completely different in mind when he wrote Let's take the con out of econometrics :-) – Andy Mar 3 '15 at 20:01
• Oh ok what would be a better title? Well, I remember reading about the "whimsical" by which Leamer I think referred to something including this variable and sometimes not. I find the dropping of the n just as random and frustrating as all the handwaviness that abounds. But am happy for a better title! ;-) – Hirek Mar 3 '15 at 20:04
• It will be necessary to have a clear account of what "$\hat X$" and "$\hat \Omega$" are in order to answer this question. (My suspicion is that the "factors of $n^{-1}$" to be ignored are involved in asymptotic expressions for these estimates, and not the ones in the matrix given in the first formula.) – whuber Mar 3 '15 at 20:27
• @Hirek good job! By the way, I really appreciate your questions here. I wish we had more people ask questions about econometrics and yours are good ones as well. – Andy Mar 5 '15 at 10:29

You have the following NLS model:

$$y=x(\beta_0) + u \\ E(u)=0 \\ E(uu^T)=\Omega$$

The proofs are for the asymptotic variance matrix of $$n^{1/2}(\hat \beta -\beta_0),$$ but if we were to actually calculate this with software for the purpose of testing some hypotheses, we don't need the $\sqrt n$, so we leave them out. In other words, we have shown that $$\sqrt n (\hat \beta -\beta_0)$$ has variance matrix $B_{0}^{-1}M_0 B_{0}'^{-1},$ where $B$ is the "bread" and $M$ is the "meat" of the sandwich. It follows that $\hat \beta$ has an asymptotic variance matrix $n^{-1}B_{0}^{-1}M_0 B_{0}'^{-1},$ so all the $n$s would cancel. Another difference is that software would typically use some sort of finite sample correction, though that is more of an aside.

I have the newer edition, where $\hat X$ is written as $X_0$, which is the matrix of derivatives of $x(\beta)$ with respect to $\beta$, evaluated at the true value $\beta_0$. It's defined on p. 551.

• Thanks so much @Dimitriy V. Masterov but why again can we leave out the factors of n? I.e. is there a compelling reason to do so other than saying we don't need them? – Hirek Mar 3 '15 at 23:49
• Practitioners care about $V(\hat \beta)$, not $V(f(\hat \beta))$. – Dimitriy V. Masterov Mar 4 '15 at 0:14
• Right, I see that but still why can we leave out n? Isn't it an important normalization to achieve some type of convergence? – Hirek Mar 4 '15 at 0:17
• I tried to clarify in the body of the answer. Maybe it would be helpful to consider a linear constant-only model, so your only parameter is the intercept, or $\hat \beta = \bar y$. You want to test the hypothesis that $\bar y = 10,$ so you need $V(\bar y)$. – Dimitriy V. Masterov Mar 4 '15 at 0:32
• Yes, I understand that I need a variance estimate to test the null hypothesis you suggest to compute a t statistic that can be judged against a standard normal but my question here is about the mathematical intuition: Why is it that I write down an estimator 1/n {matrix operations} and conclude its plim is the desired parameter and then, when using it in C or MATLAB I exclude the factor 1/n? Like am I not programming up a different thing then? – Hirek Mar 4 '15 at 0:38