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ps6 - Problem Set 6 Introduction to Econometrics Fall 2006...

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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 < σ 2 u < . The demand of bikes is given by: ln ( Q ) = γ 0 + v where E ( v ) = 0 , var ( v ) = σ 2 v , 0 < σ 2 v < and cov ( u, v ) = 0 . All the variables in the supply and in the demand have nonzero fi 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 fi cient β 1 . Is the OLS coef fi 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 . You 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 fi rst three assumption for IV regression hold, that the instruments are not weak and that your sample is large.
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