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Times Series Analysis Midterm 1 1. Suppose Find . Compare your answer to what would have been obtained if Describe the effect that that autocorrelation in has on 2. Let be a zero mean white noise process. Suppose that the observed process is where is either or . (a) Find the theoretical autocorrelation function for both when and when (b) Show that the time series is second order stationary regardless of the value of . (c) Suppose that you can observe the series; and suppose that you can produce good estimates of the autocorrelations . Do you think that you could determine which value of is correct (3 or 1/3) based on the estimate of ? Why or why not? 3. Let be an process of the special form where follows a white noise process. (a) Find the range of values of for which the process is stationary. (b) Find the estimator of via Maximum Likelihood. Assume that and that is known. You may assume that there is no uncertainty with respect to the first two observations and . (c) Consider now the model , where and . Show that if the process cannot be stationary. (Hint: take variance of both sides) 4. Suppose that you have a data vector that represents your time series data in the R software. (a) Briefly mention two methods to smooth the time series . (b) How do you find the sample ACF and PACF of lag 10 for using the R software? Write the corresponding R commands. (c) How do you estimate an AR model of order 5 for using R? Write the R command. 5. Briefly comment on the behavior of the theoretical autocorrelation function (ACF) of a (a) MA(1) process; (b) AR(2) process with complex reciprocal roots, (c) ARMA(1,1) process. Assume that all these processes are causal and invertible. 6. Suppose that we have an process of the form (a) Show that this process is causal and invertible (b) Write the model as a general linear process and obtain the associated weights that give this representation. (c) Given data corresponding to this process, how would you predict ? 7. Let be white noise, where Let Determine the numerical values of (a) (b) The covariance between and . (c) Show that the correlation between and is 0.513.

Times Series Analysis Midterm 1 1. Suppose Find . Compare your answer to what would have been obtained if Describe the effect that that autocorrelation in has on 2. Let be a zero mean white noise process. Suppose that the observed process is where is either or . (a) Find the theoretical autocorrelation function for both when and when (b) Show that the time series is second order stationary regardless of the value of . (c) Suppose that you can observe the series; and suppose that you can produce good estimates of the autocorrelations . Do you think that you could determine which value of is correct (3 or 1/3) based on the estimate of ? Why or why not? 3. Let be an process of the special form where follows a white noise process. (a) Find the range of values of for which the process is stationary. (b) Find the estimator of via Maximum Likelihood. Assume that and that is known. You may assume that there is no uncertainty with respect to the first two observations and . (c) Consider now the model , where and . Show that if the process cannot be stationary. (Hint: take variance of both sides) 4. Suppose that you have a data vector that represents your time series data in the R software. (a) Briefly mention two methods to smooth the time series . (b) How do you find the sample ACF and PACF of lag 10 for using the R software? Write the corresponding R commands. (c) How do you estimate an AR model of order 5 for using R? Write the R command. 5. Briefly comment on the behavior of the theoretical autocorrelation function (ACF) of a (a) MA(1) process; (b) AR(2) process with complex reciprocal roots, (c) ARMA(1,1) process. Assume that all these processes are causal and invertible. 6. Suppose that we have an process of the form (a) Show that this process is causal and invertible (b) Write the model as a general linear process and obtain the associated weights that give this representation. (c) Given data corresponding to this process, how would you predict ? 7. Let be white noise, where Let Determine the numerical values of (a) (b) The covariance between and . (c) Show that the correlation between and is 0.513.

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