By Rainer Winkelmann; Stefan Boes
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Extra resources for Analysis of microdata : with 38 figures and 41 tables
Recall the examples of conditional probability models from Chapter 2. • yi |xi is Bernoulli distributed with parameter πi = exp(xi β)/[1+exp(xi β)]. • yi |xi is Poisson distributed with parameter λi = exp(xi β) • yi |xi is normally distributed with parameters µi = xi β and σ 2 . In order to accommodate such models within the previous framework, we have to extend the assumption of random sampling to pairs of observations (yi , xi ), requiring that the i-th draw is independent from all other draws i = i.
Yn , and is thus a function of the data, the estimate is the value taken by that function for a speciﬁc data set. The same distinction can be made for the likelihood function itself, or for any function of the likelihood function. For instance, for each point θp , L(θp ; y) is a random variable, as are log L(θp ; y) or ∂ log L(θp ; y)/∂θ, since all these functions depend on the random sample that has been drawn. Of course, in practice, a single sample is the only information we have. However, the derivation of general properties of the maximum likelihood estimator, such as consistency or asymptotic normality, require the analysis of the behavior of the estimator in repeated samples, which can be conducted based on the assumption that we know the true data generating process f (y; θ0 ).
In most applications, the ML estimator θˆ is a non-linear 54 3 Maximum Likelihood Estimation function of the dependent variable and it will be biased in small samples. 4, . . , 1). A common way to investigate the small sample properties of ML estimators is by means of Monte Carlo simulations. However, such simulations provide results for speciﬁc parameter values only, and one cannot prove general results in this way. For information about this issue see Gouri´eroux and Monfort (1996). 1 Expected Score A crucial property of the ML method is that E[s(θ; y)], the expected score, if evaluated at the true parameter θ0 , is equal to zero.