Stats 461_Midterm 2_Fall 2015.pdf - Statistics 461 Midterm...

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Statistics 461, Midterm Exam Instructor: Xinyu Song, November 23, 2015 First Name: Last Name: Be sure to show all relevant work! 1. (30 points) Consider the time series y t = - 2 t + a t + 0 . 5 a t - 1 , where a t iid N (0 , σ 2 a ), t = 1 , · · · , T . (a) What are the mean function and autocovariance function of this time series? Is this time series weakly stationary? Justify your answer. (b) Consider the first differences of the time series above, that is, consider ω t = O y t = y t - y t - 1 . What are the mean function and autocovariance function of this time series? Is this time series weakly stationary? Justify your answer. 2. (25 points) Consider the daily log returns of Qualcomm stock from 1/03/2010 to 10/31/2015. Answer the following questions based on R output. Justify your answers. (a) Based on the summary statistics of log returns r t , does the log returns have heavy tails? Perform a test and draw the conclusion based on your test statistics. (b) A standard GARCH(1,1) model has been applied to the log return series and R output has been provided below. Write down the fitted model (include error distribution). 1
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(c) Is the model fitted in part (b) adequate based on the following residual plots and test results? If not, describe a method to improve the model fit. 2
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(d) A standard GARCH(1,1) model with Student t innovations has been applied to the log return series. Write down the fitted model (include error distribution). Why is the estimated shape parameter not an integer?
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