ps6 - Problem Set 6 Introduction to Econometrics - Fall...

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Problem Set 6 Introduction to Econometrics - Fall 2006 Due Date: Tuesday, December 12th, BEFORE CLASS. Each student is responsible for handing one handwritten solution. If working in groups, indicate the members of the group (to facilitate grading). PLEASE SHOW YOUR WORK . 1. The supply of bikes is given by the following equation: ln ( Q )= β 0 + β 1 ln ( P )+ u where u denotes factors other than price that determine supply. You know that E ( u )=0 and var ( u )= σ 2 u ,where 0 2 u < . The demand of bikes is given by: ln ( Q )= γ 0 + v where E ( v )=0 , var ( v )= σ 2 v , 0 2 v < and cov ( u, v )=0 . All the variables in the supply and in the demand have nonzero f nite fourth moments. a. Solve the two simultaneous equations and get ln ( Q ) and ln ( P ) as functions of u and v . b. Derive E [ln ( P )] and E [ln ( Q )] . c. Using the expression for ln ( P ) obtained in a. ,derive cov [ln ( P ) ,u ] . Are ln ( P ) and u correlated? d. Using the expression for ln ( P ) and ln ( Q ) obtained in a., derive var [ln ( P )] and cov [ln ( Q ) , ln ( P )] . e. You collect a random sample of 10000 observations ( Q i ,P i ) and compute (ln ( Q i ) , ln ( P i )) for each i =1 , ..., 10000 . You want to estimate the supply of bikes: ln ( Q i )= β 0 + β 1 ln ( P i )+ u i In particular, you are interested in the coef f cient β 1 . Is the OLS coef f cient b β 1 an unbiased estimator of β 1 in this application? Explain. f. Use the fact that: b β 1 = s ln( Q )ln( P ) s 2 ln( P ) p −→ cov [ln( P ) , ln( Q )] var [ln( P )] and your results in d. to show that the OLS estimator is inconsistent in this application. 2. Consider the following regression model: Y i = β 0 + β 1 X i + u i where corr ( X i ,u i ) 6 =0 . You want to use the variables Z 1 i and Z 2 i as instruments for X i .Y o u check for instrument exogeneity using the overidentifying restrictions test and the value of the heteroskedasticity-robust J -statistic you obtain is J =15 . 7 . Assume that the f rst three assumption for
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This note was uploaded on 07/28/2010 for the course ECON 410 taught by Professor Staff during the Fall '08 term at University of Wisconsin.

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ps6 - Problem Set 6 Introduction to Econometrics - Fall...

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