By Rainer Winkelmann; Stefan Boes
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Extra info for Analysis of microdata : with 38 figures and 41 tables
Second, if X and Y are independent, then any two functions h(X) and g(Y ) are independent as well. As mentioned already, the basic sampling model in microdata applications is the random sampling approach, which implies that observations are independent. 6. Consider the following joint probability function X=0 X=1 Y =0 Y =1 2/6 1/4 1/4 1/6 • Derive the conditional expectation function E(y|x) = y yf (y|x). • Are X and Y independent? The deﬁnitions of joint, conditional and marginal density functions in the continuous case are largely analogous.
3 Properties of the Maximum Likelihood Estimator 55 where the last equality applies the chain rule for taking derivatives of logarithmic functions. In this expression, θ0 is understood to be the true, albeit unknown, parameter of the model. We see that if the expected score is evaluated at the true parameter θ0 , the f (yi ; θ0 )-expressions cancel and we obtain ∂f (yi ; θ) dyi ∂θ E[s(θ0 ; yi )] = =0 θ=θ0 The expected score for each single observation, if evaluated at the true parameter, is zero, which was to be shown.
Finally, we make the parameters of the selected probability function dependent on the explanatory variables. By doing so, we obtain a conditional probability model. This approach is best illustrated with a few examples. 2. Binary Responses Suppose that Y is a binary variable. Obviously, the underlying distribution function is a Bernoulli distribution, which is fully determined by the success probability π. In this case, the counter-probability is 1−π, and the probability function can be written compactly as f (y; π) = π y (1 − π)1−y y = 0, 1 In a conditional probability model, π is speciﬁed as a function of X.
Analysis of microdata : with 38 figures and 41 tables by Rainer Winkelmann; Stefan Boes