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### test06ect

Course: SCRO 0560, Fall 2009
School: East Los Angeles College
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Word Count: 420

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Econometrics M.Phil. Test 2006 Answer two questions You have two hours 1. Suppose Y1 , . . . , Yn are independent identically Poisson [] distributed with density y exp () , fYi (y) = y! (i) Show E (Yi ) = and Var (Yi ) = . (ii) Calculate expectation and variance of the sample average. (iii) What is the maximum likelihood estimator of . (iv) Find the Cramr-Rao lower bound. Suppose Yi |Xi = xi are Poisson [exp (xi...

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Econometrics M.Phil. Test 2006 Answer two questions You have two hours 1. Suppose Y1 , . . . , Yn are independent identically Poisson [] distributed with density y exp () , fYi (y) = y! (i) Show E (Yi ) = and Var (Yi ) = . (ii) Calculate expectation and variance of the sample average. (iii) What is the maximum likelihood estimator of . (iv) Find the Cramr-Rao lower bound. Suppose Yi |Xi = xi are Poisson [exp (xi )] and the pairs (Yi , Xi ) are independent. (v) What properties can you show for the maximum likelihood estimator ? 2. Consider the model given by Y = X + u, where u Nn 0, 2 In , (1) for y N0 , > 0. and Y Rn , X Rnk , Rk and the regressors X are xed. (i) Derive the least squares estimator for . Now, consider the special case where k = 2 and Yi = 1 + 2 Xi + ui (ii) Derive the least squares estimator for 1 . (iii) Derive the distribution of 1 . Is 1 unbiased? (iv) How would you estimate the variance of 1 ? An econometrician runs the regression Yi = + vi . (2) (3) (v) Derive the least squares estimator arising from this regression. (vi) Derive the distribution of When is an unbiased estimator for 1 ? What are the . practical implications of this result? Illustrate with an example. Now, suppose X1 , . . . , Xn are independently, identically N[, 2 ] distributed. Argue (vii) that is the maximum likelihood estimator for 1 + 2 . (viii) Suppose you are interested in the regression (3), but runs the regression (2). Is the estimator 1 the an ecient estimate for ? What are the practical implications of this result? Illustrate with an example. 3. Consider the following demand and supply model: Demand: Supply: qi = ad,p pi + bd,x xi + ud,i , qi = as,p pi + us,i , where ud,i us,i IIN2 [0, ] , Here qi and pi denote log quantities and log prices and xi is strongly exogenous regressors. (i) Interpret this model. (ii) Derive the reduced form system. (iii) Discuss the identication of the structural equations. (iv) What is the maximum likelihood estimators of the identied structural mean parameter(s)? Sketch a derivation of this result. Co...

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