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Unformatted text preview: Problem Set 2 Solutions Econ 120C / Winter 2008 Due date: February 4, 2008 1 Exercises 1. Use the central limit theorem, Slutsky’s theorem and the continuous mapping theorem (as appropriate) to describe the asymptotic distrib utions in the following cases. (a) Let W 1 and W 2 be independent chisquared random variables, each with n degrees of freedom. Let F = W 1 W 2 (1) define another random variable. i. Describe the distribution of X in the language of the course. The Fdistribution with m and n degrees of freedom, denoted F m,n , is defined to be the distribution of the ratio of a chi squared random variable with degrees of freedom m, divided by m, to an independently distributed chisquared random vari able with degrees of freedom n, divided by n. In order to describe the distribution of X , first divide both numerator and denominator by n. Then X has an F distribu tion with ( n, n ) degrees of freedom according to the definition of the F distribution. ii. What is the probability limit of X as n tends to ∞ ? The law of large numbers says that if Y i , i = 1 , ..., n are in dependently and identically distributed with E ( Y i ) = μ y and var( Y i )= σ 2 y < ∞ , then ¯ Y p → μ y . 1 A Chi square random variable with n degrees of freedom can be expressed as the sum of the squared values of an i.i.d. sample of size n from a standard normal distribution, Z 1 i , Z 2 i , . . . , Z ni . Thus we can write W i /n = n X j =1 Z 2 ji /n, (2) which is the mean of a random sample of n draws from the distribution of Z 2 , the square of a standard normal random variable. By the law of large numbers, the probability limit of W i /n is 1 (since E ( Z 2 ji ) = 1 for all i, j ). Since W i /n converges to 1 in probability respectively, X degenerates to a point mass at 1 asymptotically according to the ratio version of Slutsky’s Theorem. i.e.W 1 /n p → 1 , W 2 /n p → 1 implies W 1 /n W 2 /n p → 1 / 1 = 1 (3) iii. Let V n = √ n ( W 1 n ) /W 2 . What is the asymptotic distribu tion of V n as n tends to ∞ ? In order to analyze convergence in distribution, divide both numerator and denominator by sample size n and change the V n to the following form. V n = √ n ( W 1 n ) /W 2 = √ n ( W 1 n ) /n W 2 /n = √ n ( W 1 /n 1) W 2 /n (4) From the previous problem, plim ( W 2 /n ) = 1 . Further, since W 1 follows chi square distribution, we can show that E ( W 1 /n ) = 1 and V ar ( W 1 /n ) = (2 n ) /n 2 = 2 /n. Then, we can see the as ymptotic distribution of W 1 /n by applying the Central Limit Theorem. ( W 1 /n 1) / q 2 /n d → N (0 , 1) (5) 2 Then, we can rearrange V n to use the previous results V n = √ n ( W 1 /n 1) W 2 /n = √ 2( W 1 /n 1) / q 2 /n W 2 /n (6) We can see that numerator converges to a distribution √ 2 N (0 , 1) = N (0 , 2) while denominator converges to 1 in probability. Fi nally,the version of Slutsky’s theorem for ratios implies V n converges to a distribution N (0 , 2) ....
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This note was uploaded on 04/08/2008 for the course ECON 120C taught by Professor Stohs during the Winter '08 term at UCSD.
 Winter '08
 Stohs
 Econometrics

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