slides_10_asymptotic

# N is a smooth function of the estimator x n and so we

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̂ n is a smooth function of the estimator X ̄ n and so we can use a calculus approximation. 43

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Start with the general scalar case, and let g for Θ . Assume that Θ is an open set. Further, assume that g  is continuously differentiable on Θ , and denote its derivative by g 1 . We approximate the distribution of n ̂ n using linearization. Namely, we will approximate the distribution of n ̂ n using the asymptotic distribution of n ̂ n . 44
We need to apply the mean value theorem (MVT) along with the consistency and n -asymptotic normality of ̂ n . Because ̂ n p and Θ is an open set, ̂ n is in an open interval around with probability approaching one (wpa1). We will ignore that nicety and just act as if ̂ n is in the interval for n sufficiently large. 45

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We can apply the MVT as follows: g ̂ n g g 1 ̈ n    ̂ n where ̈ n is the mean value, which we know is on the line segment connecting and ̂ n . Because ̂ n p we also know ̈ n p (even though we do not generally know ̈ n ). 46
By Slutsky’s theorem, because g 1  is continuous, g 1 ̈ n p g 1 . Now we can use standard results from asymptotics: n g ̂ n g  g 1 ̈ n  n ̂ n  g 1  n ̂ n  g 1 ̈ n g 1  n ̂ n g 1  n ̂ n  o p 1 O p 1 g 1  n ̂ n  o p 1 47

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By the asymptotic equivalence lemma, n g ̂ n g  has the same asymptotic distribution as g 1  n ̂ n  . Let c be the asymptotic variance of n ̂ n , that is, n ̂ n d Normal 0, c  . It follows immediately that n ̂ n n g ̂ n g  d Normal 0, g 1  2 c  , that is Avar n ̂ n  dg d 2 c dg d 2 Avar n ̂ n  48
This approach to deriving the asymptotic variance of smooth functions of an estimator is called the delta method . It has widespread use an applied econometrics. The relationship between asymptotic variances is exactly as if we could compute the finite sample variances where ̂ n is a linear function of ̂ n : ̂ n a ̂ n a dg d 49

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Assuming c is also continuous on Θ , we can consistently estimate the asymptotic variance of n
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