By George G. Judge
This booklet is meant to supply the reader with an organization conceptual and empirical knowing of uncomplicated information-theoretic econometric versions and strategies. simply because such a lot facts are observational, practitioners paintings with oblique noisy observations and ill-posed econometric versions within the type of stochastic inverse difficulties. for this reason, conventional econometric equipment in lots of situations are usually not acceptable for answering a few of the quantitative questions that analysts desire to ask. After preliminary chapters take care of parametric and semiparametric linear chance types, the focal point turns to fixing nonparametric stochastic inverse difficulties. In succeeding chapters, a relatives of energy divergence measure-likelihood features are brought for a number conventional and nontraditional econometric-model difficulties. eventually, inside of both an empirical greatest probability or loss context, Ron C. Mittelhammer and George G. pass judgement on recommend a foundation for selecting a member of the divergence relations
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Extra info for An Information Theoretic Approach to Econometrics
In this fully parametric model formulation, a large amount of information about the sampling process is specified but the parameters β and σ 2 nonetheless remain unknown and unobserved and are, indeed, unobservable. 10) and Because the ML estimator for β is a linear combination of the multivariate normally distributed vector Y, it follows that the ML estimator has a normal distribution N(β, σ 2 (x x)−1 ). It can also be shown that the ML estimator of n σ 2 can be bias-adjusted to define the unbiased estimator S2 = ( n−k )S2ML ∼ n−k 2σ2 Gamma ( 2 , n−k ).
The unbiasedness property follows from the condition E [ε |X] = 0, and E [βˆ ] = E x [E [βˆ |X]] = E x [β + (X X)−1 X E [ε|X]] = β. 10) given that E [(X X)−1 ] exists. 11) and, therefore, the estimator S 2 is a conditionally and unconditionally unbiased estimator of σ2 . 3 LS Estimator Properties: Asymptotics The asymptotic properties of the LS estimator when X is stochastic parallel those established in the case where x is fixed. 4 General Linear Model with Stochastic X 27 p p estimator, if n−1 X ε → 0 and n−1 X X → , then βˆ exists with probability p converging to 1 and βˆ → β when n → ∞.
This result is precisely the same as the case of nonstochastic x. The typical MLE asymptotics of consistency, asymptotic normality, and asymptotic efficiency also apply to these estimators. Furthermore, βˆ is conditionally n 2 normally distributed in finite samples, and S 2 = ( n−k )S ML is conditionally Chi-square(n-k, 0) distributed. 3. The asymptotic distributions of the test statistics are again Chi-square (W, LM, and pseudo-likelihood ratio [PLR]statistics) and normal (Z-statistic). 3. These assumptions are sufficient conditions.