Econometric Applications of Maximum Likelihood Methods by Jan Salomon Cramer

By Jan Salomon Cramer

The arrival of digital computing allows the empirical research of monetary types of a long way larger subtlety and rigour than prior to, while many attention-grabbing rules weren't up as the calculations concerned made this impracticable. The estimation and checking out of those extra complicated types is generally according to the tactic of utmost chance, that is a well-established department of mathematical statistics. Its use in econometrics has ended in the advance of a few certain innovations; the explicit stipulations of econometric learn in addition call for definite alterations within the interpretation of the elemental argument. This booklet is a self-contained advent to this box. It includes 3 components. the 1st bargains with normal positive factors of utmost chance tools; the second one with linear and nonlinear regression; and the 3rd with discrete selection and similar micro-economic versions. Readers may still already be acquainted with simple statistical conception, with utilized econometric learn papers, or with the literature at the mathematical foundation of utmost probability concept. they could additionally attempt their hand at a few complex econometric study in their personal.

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28 2. Likelihood Berndt et al. (1974). Both (53) and (54) are consistent estimators, for they contain sums of independent variates which_(upon suitable scaling by n) will converge to the mean expected value H of (43). Hence V2 and V$ converge to H~l. In particular applications it may turn out that two (or possibly even all three) of these estimators coincide. When they differ, the choice that must be made among them is often influenced by considerations of computational convenience. 1 the covariance matrix of the asymptotic distribution offiis - p)(p -0)T^ = a2(XTX/n)-1 and this is asymptotically efficient in the sense that it meets the Cramer-Rao bound.

Consider the hypothesis that a single regression coefficient is zero and derive the Wald test statistic, making use of the familiar OLS regression formula for Vfi. Compare with the standard Mest. j 4- fi2 = 0. CHAPTER 4 The use of likelihood in econometrics This chapter is concerned with the justification of likelihood methods in the specific context of econometric analyses. This question prompts responses at various levels. In discussing it we cannot entirely avoid philosophical and ideological issues on which readers must form their own opinions.

So far we have, however, treated the observed {#, y) as a single amorphous whole, and apart from one or two asides we have not mentioned sample size at all. For the asymptotic argument we must change our ways. 7 Independent observations We assume that the observed sample {x, y) can be partitioned into n elements {x{ 9yt}t and that the y{ can be treated as realizations of independent random variables yt. Xi) (39) where w* has the same dimensions asy { . As before y{ may represent a vector or a matrix, and the elements of this array need not be stochastically independent.

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