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ps1 - X ∼ N μ x Σ what is the distribution of AX b...

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1. For a and x both p × 1 vectors, show by example that a 0 x x = a . 2. For A a p × p matrix and x a p × 1 vector, show by example that x 0 Ax x = A + A 0 . 3. Prove that the split sample estimator used to illustrate the Gauss-Markov theorem is unbi- ased. 4. Calculate the OLS estimates of the Nerlove model using Octave and GRETL, and provide printouts of the results. Interpret the results. 5. Do an analysis of whether or not there are influential observations for OLS estimation of the Nerlove model. Discuss. 6. Using GRETL, examine the residuals after OLS estimation and tell me whether or not you believe that the assumption of independent identically distributed normal errors is war- ranted. No need to do formal tests, just look at the plots. Print out any that you think are relevant, and interpret them as carefully and well as you are able to. 7. For a random vector
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Unformatted text preview: X ∼ N ( μ x , Σ ) , what is the distribution of AX + b , where A and b are conformable matrices of contants? 8. For a random vector X ∼ N ( μ x , Σ ) , what is the distribution of ( X-μ x ) Σ-1 ( X-μ x ) ? 9. Using Octave, write a little program that veri²es that Tr ( AB ) = Tr ( BA ) for A and B con-formable matrices of random numbers. Note: there is an Octave function trace. Note that checking equality using the Octave operation ” == ” will indicate that they are not equal (though in fact they are). Why is that? 10. For the model with a constant and a single regressor, y t = β 1 + β 2 x t + e t , which satis²es the classical assumptions, prove that the variance of the OLS estimator declines to zero as the sample size increases. 1...
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