Let w1, W2,, be Gaussian white noise. The four plots in Figure 1 are the ACF...

  

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Let w1, W2,, be Gaussian white noise. The four plots in Figure 1 are the ACF plots of the following four time series: (a) x = 1x-1+ wt, where 0 < 1 < 1 and 1 Be sure to briefly explain each of your choices. (Do not use process of elimination as an explanation) 2. Consider the random walk with drift model x=8+x-1+ w (1) for t> 0, where o = 0, 8 is a constant, and w, are i.i.d. Gaussian white noise with variance o (a) (3 points) Show that equation (1) can be written as = + i=1 (b) (3 points) Show the mean function is = E(x) = 8t. (c) (5 points) Derive the autocovariance function for {x}. (d) (3 points) Is {} stationary? Briefly explain. 3. Consider the following model x=t(t+2)+wi iml for t> 0, where ro = 0, and w, are i.i.d. Gaussian white noise with variance 2. (2) (a) (4 points) Show the 2nd difference of r, is V2, 2+ w- Wr-1. (b) (2 points) Given that you have observed 1, 2,..., 1000, describe how you would estimate oz. 4. Consider the following ARMA (p,q) process Note that: 4 4 5 1=31-1 +31-2+ Wi- -W-1 - 3 -Wi-2. 2 (3) 1-2-2 (1-22)(1 + z) == 1-z-z = (1-2)(1 + z) (a) (2 point) Do we have parameter redundancy in equation (3)? Briefly explain. (b) (2 points) Based on your answer in part 4a, what is the order of this ARMA (p,q) process? (i.e. what are the values of p and q?) (c) (2 points) Briefly explain why this process is both causal and invertible. (d) (6 points) Find the first four coefficients, Vo, V1, V2, V3, in the infinite MA repre- sentation j=0j Wt-j 5. The data for this question are the monthly production of beer in Australia between January 1991 and August 1995, in megaliters. Figure 2 is a time series plot of the data. (a) (2 points) We consider applying a one-sided moving average smoother to the time series, m = oaja-j. What window size(s) (value(s) of k) would you consider using for a one-sided moving average smoother? Briefly explain your choice(s). (b) (2 points) Figure 3 displays a periodogram of the data. Based on Figure 3, your classmate decides to fit the following model to the time series xt = Bo+ B cos (2 cos (22) + B sin (21/127) + Wri (4) where w, is iid Gaussian white noise. He uses OLS to estimate the coefficients. Based only on Figure 3, do you agree with your classmate's decision? Briefly explain. (c) (2 points) Your classmate fits model (4) using OLS regression. Describe how you expect the residual plot of this regression to look like. (You may provide a sketch plot to look like this. instead of describing using words). Briefly explain why you expect the residual 6. Please be sure that you have signed the cover page acknowledging the instructions. (2 points) 3 Let w1, W2,, be Gaussian white noise. The four plots in Figure 1 are the ACF plots of the following four time series: (a) x = 1x-1+ wt, where 0 < 1 < 1 and 1 Be sure to briefly explain each of your choices. (Do not use process of elimination as an explanation) 2. Consider the random walk with drift model x=8+x-1+ w (1) for t> 0, where o = 0, 8 is a constant, and w, are i.i.d. Gaussian white noise with variance o (a) (3 points) Show that equation (1) can be written as = + i=1 (b) (3 points) Show the mean function is = E(x) = 8t. (c) (5 points) Derive the autocovariance function for {x}. (d) (3 points) Is {} stationary? Briefly explain. 3. Consider the following model x=t(t+2)+wi iml for t> 0, where ro = 0, and w, are i.i.d. Gaussian white noise with variance 2. (2) (a) (4 points) Show the 2nd difference of r, is V2, 2+ w- Wr-1. (b) (2 points) Given that you have observed 1, 2,..., 1000, describe how you would estimate oz. 4. Consider the following ARMA (p,q) process Note that: 4 4 5 1=31-1 +31-2+ Wi- -W-1 - 3 -Wi-2. 2 (3) 1-2-2 (1-22)(1 + z) == 1-z-z = (1-2)(1 + z) (a) (2 point) Do we have parameter redundancy in equation (3)? Briefly explain. (b) (2 points) Based on your answer in part 4a, what is the order of this ARMA (p,q) process? (i.e. what are the values of p and q?) (c) (2 points) Briefly explain why this process is both causal and invertible. (d) (6 points) Find the first four coefficients, Vo, V1, V2, V3, in the infinite MA repre- sentation j=0j Wt-j 5. The data for this question are the monthly production of beer in Australia between January 1991 and August 1995, in megaliters. Figure 2 is a time series plot of the data. (a) (2 points) We consider applying a one-sided moving average smoother to the time series, m = oaja-j. What window size(s) (value(s) of k) would you consider using for a one-sided moving average smoother? Briefly explain your choice(s). (b) (2 points) Figure 3 displays a periodogram of the data. Based on Figure 3, your classmate decides to fit the following model to the time series xt = Bo+ B cos (2 cos (22) + B sin (21/127) + Wri (4) where w, is iid Gaussian white noise. He uses OLS to estimate the coefficients. Based only on Figure 3, do you agree with your classmate's decision? Briefly explain. (c) (2 points) Your classmate fits model (4) using OLS regression. Describe how you expect the residual plot of this regression to look like. (You may provide a sketch plot to look like this. instead of describing using words). Briefly explain why you expect the residual 6. Please be sure that you have signed the cover page acknowledging the instructions. (2 points) 3