The dataset for this question contains quarterly beer production. Figure 4 displays the time series plot of the data, and Figure 5 displays the periodograms of the data with di?erent levels of smoothing (details provided in the captions). (a) What does Figure 5 indicate about the oscillatory behavior of this time series? (5 points) (b) Why do we usually want to apply some level of smoothing to a periodogram? (3 points)
1. Consider the following process xt = 4 5 xt�1 + wt � 5 6 wt�1 + 1 6 wt�2, (1) where wt is i.i.d. Gaussian white noise with variance 1. Note: 1 � 56z + 1 6z 2 = (1� 12z)(1� 1 3z). (a) Show that the spectral density of the process given in (1) is f(!) = 62 36 � 50 36 cos(2⇡!) + 1 3 cos(4⇡!) 41 25 � 8 5 cos(2⇡!) , You may use any method from class. (7 points) (b) Figure 1 displays the spectral density of the process given in (1). Briefly describe what Figure 1 is informing you about the behavior of this process. (2 points) (c) Figure 2 displays the spectral density of another process. Figure 3 displays the time series plots that were simulated from the two processes from Figures 1 and 2. Which of the two plots, Plot 1 or Plot 2, from Figure 3 is more likely to belong to the process given in (1)? Briefly explain. (2 points) (d) Why is the plot of any spectral density plotted for 0 ! 0.5? Briefly explain. (2 points) 2. The dataset for this question contains quarterly beer production. Figure 4 displays the time series plot of the data, and Figure 5 displays the periodograms of the data with di↵erent levels of smoothing (details provided in the captions). (a) What does Figure 5 indicate about the oscillatory behavior of this time series? (5 points) (b) Why do we usually want to apply some level of smoothing to a periodogram? (3 points) 3. For this question, we have monthly data that measure the amount of precipitation and inflow (the amount of water entering a body of water) at Shasta Lake, California. We consider fitting a model for the transformed (square root) inflow (yt) on the transformed (log) precipitation (xt). We consider the following three lagged regression models: • Model 1: lagged regression with ARMA(3, 2)⇥ (1, 0)12 errors; • Model 2: lagged regression with ARMA(3, 3)⇥ (1, 0)12 errors; • Model 3: lagged regression with ARMA(3, 4)⇥ (1, 0)12 errors. Recall in lagged regression we seek to a fit a model of the form yt = rX k=1 !kyt�k + sX k=0 �kxt�d�k + ut. (2) 3 where ut is a stationary noise process. The three models listed above can be viewed as specific cases of (2). Figure 6 displays the plot of the sample cross correlation (CCF) of yt and xt (with prewhitening). (a) Why do we prewhiten the predictor variable prior to examining a cross correlation plot? (2 points) (b) Based on Figure 6, your classmate suggests performing a (lagged) regression of yt against yt�1 and xt? Do you agree? Briefly explain. If you disagree, suggest an alternative (lagged) regression and briefly explain your choice. (3 points) (c) The three lagged regression models, Model 1, Model 2, and Model 3, are fitted. Their diagnostic plots and their estimated coe�cients of the error terms are dis- played in Figures 7 to 12. The three models and their corresponding figures are shown in the table below: Model Diagnostic Plots Estimated Coe�cients of Error Terms Model 1 Figure 7 Figure 8 Model 2 Figure 9 Figure 10 Model 3 Figure 11 Figure 12 Based only on Figures 7 to 12, which model will you select? Be sure to explain your selection, and why you did not select the others. You may assume the regression coe�cients, !k and �k, are significant in all the models. (8 points) 4. For the following statements, state whether they are true or false. If false, briefly explain why. (a) One of the assumptions in fitting a linear regression model is that the observations are all independent. With time series data, data are by their nature dependent on previous observations. Hence a linear regression model is not suited for time series data. (2 points) (b) One of the assumptions in fitting a linear regression model is that the observations are all independent. A consequence of this assumption not being met is that the estimated coe�cients, �̂, will be biased. (2 points) (c) When using the Cochrane-Orcutt method in building the regression model with AR errors (assuming one predictor), we transform the response and predictor variables (denoted by yt and xt respectively) using the AR operator, to obtain y⇤t = �(B)yt and x ⇤ t = �(B)xt. (2 points) (d) Let yt and xt denote the response and predictor variables respectively. The cross- covariance function (CCF), is defined as �xy(h) = E [(xt+h � µx)(yt � µy)]. We would expect the sample CCF to be significant when h > 0 and to be insignificant when h 0. Assume conditions to reliably interpret the values of the sample CCF to be met. (2 points) (e) The variance of xt can be found by integrating its spectral density, f(!), over �0.5 ! 0.5, i.e. V ar(xt) = R 1/2 �1/2 f(!)d!. (2 points) 4 " *⇐-BE⇒E☒¥*⇐TBafggBpG±••fB③oBBoa*••• 0.0 0.1 0.2 0.3 0.4 0.5 5 10 15 Spectral Density of ARMA(1,2) frequency f(om ega ) Figure 1: Spectral Density of ARMA(1,2) from Question 1. 0.0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 Spectral Density of Another Process frequency f(om ega ) Figure 2: Spectral Density of “Other Process” from Question 1. 6 Plot 1 Time x1 0 10 20 30 40 50 −6 −2 2 6 Plot 2 Time x2 0 10 20 30 40 50 −20 0 10 20 Figure 3: Time Series Plot of the Two Processes from Question 1. Top: Plot 1, Bottom: Plot 2 Time Series Plot of Beer Time data 0 10 20 30 40 50 60 70 250 300 350 400 450 500 Figure 4: Time Series Plot of Quarterly Beer Data 7 0.0 0.1 0.2 0.3 0.4 0.5 5 50 500 500 0 frequency spe ctru m Series: data Raw Periodogram bandwidth = 0.00401 0.0 0.1 0.2 0.3 0.4 0.5 50 200 100 0 500 0 frequency spe ctru m Series: data Smoothed Periodogram bandwidth = 0.0145 0.0 0.1 0.2 0.3 0.4 0.5 50 200 100 0 500 0 frequency spe ctru m Series: data Smoothed Periodogram bandwidth = 0.0244 0.0 0.1 0.2 0.3 0.4 0.5 50 200 100 0 500 0 frequency spe ctru m Series: data Smoothed Periodogram bandwidth = 0.0352 Figure 5: Periodograms of Beer Data, with no smoothing (top left), modified Daniell kernell applied twice each with m = 1 (top right), modified Daniell kernell applied twice each with m = 2 (bottom left), and modified Daniell kernell applied twice each with m = 3 (bottom right). 8 −20 −10 0 10 20 −0 .1 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 Lag C C F Figure 6: CCF of Transformed Inflow and Transformed Precipitation for Lake Shasta Data 9 Standardized Residuals Time 0 100 200 300 400 −3 −2 −1 0 1 2 3 4 Model: (3,0,2) (1,0,0) [12] 0 5 10 15 20 25 30 35 −0 .1 0.1 0.3 0.5 ACF of Residuals LAG AC F ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 4 Normal Q−Q Plot of Std Residuals Theoretical Quantiles Sa mp le Qu an tile s ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 p values for Ljung−Box statistic LAG (H) p v alu e Figure 7: Diagnostic Plots from Lagged Regression with ARMA(3, 2)⇥(1, 0)12 errors (Model 1) on Lake Shasta Data. Figure 8: Estimated Coe�cients of Error Terms from Model 1 on Lake Shasta Data. 10 Standardized Residuals Time 0 100 200 300 400 −3 −2 −1 0 1 2 3 4 Model: (3,0,3) (1,0,0) [12] 0 5 10 15 20 25 30 35 −0 .1 0.1 0.2 0.3 0.4 0.5 ACF of Residuals LAG AC F ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●