Problem Set 2 - Problem Set 2 Solutions Econ 120C Winter...

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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 chi-squared 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 chi-squared 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
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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 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) / 2 /n d N (0 , 1) (5) 2
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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) / 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) . (b) Let W n = n i =1 Z 2 i , (7) where Z 1 , Z 2 , . . . , Z n is an i.i.d. sample from the N (0 , 1) distri- bution, and let Z be another N (0 , 1) random variable which is independent of Z i , for i = 1 , . . . , n .
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