Introduction to econometric theory by Kuan C.-M.

By Kuan C.-M.

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T , the likelihood function of θ is L(θ) := f (ζ 1 , . . , ζ T ; θ). In what follows, we will use L and f interchangeably. Note, however, that a postulated density function need not be the true density function that generates the random variables. When f is differentiable and non-zero with probability one, the gradient vector of log L(θ), ∇θ log L(θ) = 1 ∇ L(θ), L(θ) θ c Chung-Ming Kuan, 2001 28 CHAPTER 2. STATISTICAL CONCEPTS is known as the score vector, denoted as s(ζ 1 , . . , ζ T ; θ).

Under [A2](ii), yt , t = 1, . . , T , have the constant variance σo2 and are pairwise uncorrelated (but not necessarily independent). Although conditions [A2] and [A3] impose the same structures for the mean and variance of y, the latter is much stronger because it also specifies the distribution of y. 3 that uncorrelated normal random variables are also independent. Therefore, yt , t = 1, . . d. (independently and identically distributed) normal random variables under [A3]. 1) with [A1] and [A3] is also known as the classical normal linear model.

A test may yield incorrect inferences. A test is said to commit the type I error if it rejects the null hypothesis when the null hypothesis is in fact true; a test is said to commit the type II error if it accepts the null hypothesis when the alternative hypothesis is true. Suppose that we are interested in testing H0 : θ o = a against H1 : θ o = b. Let IP0 be the probability when θ o = a and IP1 the probability when θ o = b. The probability of the type I error is then α = IP0 ((z 1 , . . , z T ) ∈ Γ) = Γ f0 (ζ 1 , .

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