In the lecture entitled Maximum likelihood we have demonstrated that, under certain assumptions, the distribution of the maximum likelihood estimator of a vector of parameters can be approximated by a multivariate normal distribution with mean and covariance matrixwhere is the log-likelihood of one observation from the sample, evaluated at the true parameter , and the gradient is the vector of first derivatives of the log-likelihood.

Because is unknown, also this covariance matrix is unknown. Here we discuss methods to consistently estimate it.

We make the same assumptions made in the aforementioned lecture. So, for example, where are the realizations of the first terms of an IID sequence . Under these assumptions we also have that the information equality holds, so thatwhere the Hessian matrix is the matrix of second-order partial derivatives of the log-likelihood function.

The first estimate of the asymptotic covariance matrixis called outer product of gradients (OPG) estimate and it is computed asIt takes its name from the fact that the gradient is a column vector, its transpose is a row vector, and the product between a column and a row is called outer product.

Provided some regularity conditions are satisfied, the OPG estimator is a consistent estimator of , that is, it converges in probability to .

Proof

We provide only a sketch of the proof and we refer the reader to Newey and McFadden (1994) for a more rigorous exposition. Provided some regularity conditions are satisfied (see the source just cited), we have the following equality between probability limits:where has been replaced by because, being a consistent estimator, it converges in probability to . Because the sample is IID, by the Law of Large Numbers we have thatNow, the formula for the covariance matrix (see the lecture entitled Covariance matrix) yieldsBut the expected value of the gradient evaluated at is , so thatThus,Because matrix inversion is continuous, by the Continuous Mapping theorem we have which is exactly the result we needed to prove.

The second estimate of the asymptotic covariance matrixis called Hessian estimate and it is computed as

Under some regularity conditions, the Hessian estimator is also a consistent estimator of .

Proof

Again, we do not provide an entirely rigorous proof (for which you can see Newey and McFadden - 1994) and we only a sketch the main steps. First of all, under some regularity conditions, we have thatwhere has been replaced by because, being a consistent estimator, it converges in probability to . Now, since the sample is IID, by the Law of Large Numbers we have thatBy the information equality, we haveTherefore,Because matrix inversion is continuous, by the Continuous Mapping theorem we have which is what we needed to prove.

The third estimate of the asymptotic covariance matrixis called Sandwich estimate and it is computed aswhere is the OPG estimate and is the Hessian estimate.

Also the Sandwich estimator is a consistent estimator of .

Proof

This is again a consequence of the Continuous Mapping theorem:where the last equality follows from the consistency of the OPG and Hessian estimators.

Newey, W. K. and D. McFadden (1994) "Chapter 35: Large sample estimation and hypothesis testing", in Handbook of Econometrics, Elsevier.

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