PS_4.pdf - Econometrics II Fall 2019 Problem Set 4 due by 10:56 Monday November 18 in the box labeled Econometria II next to office 20.141(For each

# PS_4.pdf - Econometrics II Fall 2019 Problem Set 4 due by...

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Econometrics II. Fall 2019 Problem Set 4 due by 10:56, Monday, November 18 in the box labeled Econometria II next to office 20.141 (For each person of your problem set group, please put their seminar group number next to their name.) Problem 1 . Suppose Y i are random variables with mean μ and variance σ 2 = 30. We have a sample of n = 10 observations, and would like to predict the value of an out-of-sample Y OOS . 1. Consider predictor Y / 2. Calculate its MSPE. [Hint: it may depend on μ .] 2. Consider predictor Y . Calculate its MSPE. 3. Suppose μ = 1. Calculate MSPE’s of both predictors. Which one is smaller? 4. Suppose μ = 10. Calculate MSPE’s of both predictors. Which one is smaller? Problem 2 Prove the following results about conditional means, forecasts, and forecast errors: 1. Let W be a random variable with mean μ w and variance σ 2 w . Let c be a constant and show that E[( W - c ) 2 ] = σ 2 w + ( μ w - c ) 2 2. Suppose we are interested in forecasting a random variable Y t and to do this we have random variables

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Unformatted text preview: Y t-1 , Y t-2 , . . . . Let f t-1 denote some forecast for Y t that is based on Y t-1 , Y t-2 , . . . . We judge the quality of the forecast by the conditional mean squared forecast error E[( Y t-f t-1 ) 2 | Y t-1 , Y t-2 , . . . ]. Show that taking f t-1 = E[ Y t | Y t-1 , Y t-2 , . . . ] mini-mizes the conditional mean squared forecast error. 3. Consider the AR(p) model under the assumption E[ u t | Y t-1 , . . . , Y t-p ] = 0. Show that this assumption implies that Cov[ u t , u t-j ] = 0 for all j 6 = 0. 1 Problem 3 Consider the stationary AR(1) model Y t = 2 . 7 + 0 . 7 Y t-1 + u t , where u t is iid with E[ u t ] = 0 and variance Var[ u t ] = 9. 1. Compute the mean and variance of Y t . 2. Compute the first two autocovariances of Y t . 3. Compute the first two autocorrelations of Y t . 4. Suppose Y T = 102 . 3.Compute Y T +1 | T = E[ Y T +1 | Y T , Y t-1 , . . . ]. 2...
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