Category A COMPANION TO Theoretical Econometrics

Hypothesis Testing with Artificial Regressions

Artificial regressions like the GNR are probably employed most frequently for hypothesis testing. Suppose we wish to test a set of r equality restrictions on 9. Without loss of generality, we can assume that these are zero restrictions. This allows us to partition 9 into two subvectors, 91 of length k – r, and 92 of length r, the restrictions being that 92 = 0. If the estimator 0 is not only root-n consistent but also asymptotically normal, an appropriate statistic for testing these restric­tions is

0ї(^г (02))-102, (1.19)

which will be asymptotically distributed as x2(r) under the null if ^«(02) is a suit­able estimate of the covariance matrix of 02.

Suppose that r(9) and R(9) define an artificial regression for the estimator 0...

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The OPG Regression

By no means all interesting econometric models are regression models. It is there­fore useful to see if artificial regressions other than the GNR exist for wide classes of models. One of these is the outer-product-of-the-gradient regression, or OPG
regression, a particularly simple artificial regression that can be used with most models that are estimated by maximum likelihood. Suppose we are interested in a model of which the loglikelihood function can be written as

n

€(9) = (0), (1.27)

t = 1

where €t() denotes the contribution to the loglikelihood function associated with observation t. This is the log of the density of the dependent variable(s) for observation t, conditional on observations 1,…, t – 1. Thus lags of the dependent variable(s) are allowed. The key feature of (1...

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An Artificial Regression for GMM Estimation

Another useful artificial regression, much less well known than the OPG regres­sion, is available for a class of models estimated by the generalized method of moments (GMM). Many such models can be formulated in terms of functions f(0) of the model parameters and the data, such that, when they are evaluated at the true 0, their expectations conditional on corresponding information sets, Qt, vanish. The Qt usually contain all information available prior to the time of observation, and so, as with the GNR and the OPG regression, lags of depend­ent variables are allowed.

Let the n x l matrix W denote the instruments used to obtain the GMM estim­ates. The tth row of W, denoted Wt, must contain variables in Qt only...

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Artificial Regressions and HETEROsKEDASTiciTy

Covariance matrices and test statistics calculated via the GNR (1.7), or via arti­ficial regressions such as (1.35) and (1.36), are not asymptotically valid when the assumption that the error terms are iid is violated. Consider a modified version of the nonlinear regression model (1.3), in which E(uuT) = Q, where Q is an n x n diagonal matrix with tth diagonal element ю2 Let V denote an n x n diagonal matrix with the squared residual й] as the tth diagonal element. It has been known since the work of White (1980) that the matrix

(XTX)-1XTVX(XTX)-1 (1.37)

provides an estimator of var(S), which can be used in place of the usual estim­ator, s2(XTX)-1...

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Double-Length Regressions

Up to this point, the number of observations for all the artificial regressions we have studied has been equal to n, the number of observations in the data. In some cases, however, artificial regressions may have 2n or even 3n observations. This can happen whenever each observation makes two or more contributions to the criterion function.

The first double-length artificial regression, or DLR, was proposed by Davidson and MacKinnon (1984a). We will refer to it as the DLR, even though it is no longer the only artificial regression with 2n observations. The class of models to which the DLR applies is a subclass of the one used for GMM estimation. Such models may be written as

f (y, 0) = є t, t = 1,…, n, £t ~ NID(0, 1), (1.47)

where, as before, each ft () is a smooth function that depends on...

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