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Statistics 451 Spring 2013 Examination 1 Name You must show all of your work 1. Consider the following MA(1) time series model at ∼ nid(0, σa2 ) Zt = (1 − θ1 B)at , where θ1 = .5 and σa = 4. (a) Derive an expression for the variance of Zt . (b) Derive an expression for ρ2 , the true ACF value for this model at lag 2. 2. The following figure shows monthly data on the concentration of CO2 in the air at the Mauna Loa Observatory (on top of a volcano in Hawaii). 370 375 380 ppm 385 390 395 Monthly CO2 Concentration at Mauna Loa 2001−2012 2002 2004 2006 2008 2010 2012 Year The following models were fit to the data Model 1: Yt Model 2: Yt Model 3: Yt at = = = ∼ β0 + β1 Time + at β0 + β1 Time + β2 Aug + β3 Dec + · · · + β12 Sep + at β0 + β1 Time + β2 Aug + β3 Dec + · · · + β23 Year:MonthSep + at nid(0, σa2 ) where Yt is the CO2 concentration in ppm, Time is been defined as (2001.00, 2001.08333, . . . , 2012.917) and Aug is 1 in August and 0 otherwise, Dec is 1 in month December and 0 otherwise,. . . , Sep is 1 in month September and 0 otherwise (there is no dummy variable for April in this model). The results of the regression fit using ordinary least squares for Model 1 showed βb0 = −3635.04, βb1 = 2.002, σ ba = 2.183 (142 degrees of freedom) and for Model b b b 2, β0 = −3701.04, β1 = 2.036, β2 = −4.38, . . . βb12 = −5.93, σ ba = 0.348 (131 degrees of freeb b b dom). and for Model 3, β0 = −3693.34, β1 = 2.032, β2 = 3.126, . . . βb23 = 0.021, σ ba = 0.361 (120 degrees of freedom). 1 (a) What is the practical interpretation of the parameter β1 in Model 1? (That is, how would you explain the meaning to someone who did not know much statistics, using the appropriate units of the coefficient?) (b) What is the practical interpretation of the parameter β2 in Model 2? (c) What is the practical interpretation of the parameter σa in Model 2? (d) Explain why there really is no “practical” interpretation for β0 in these models, for this application. (e) Explain why the estimate of σa is larger in Model 3 than it is in Model 2. (f) Compare the results from fitting Models 1 and 2, assuming that at ∼ nid(0, σa2 ). Do a statistical test to see if there is evidence of a seasonal effect in the data. (g) Compare the results from fitting Models 2 and 3, assuming that at ∼ nid(0, σa2 ). Do a statistical test to see if there is evidence that Model 3 is better than Model 2. 2 (h) The following figure shows the ACF of the residuals from Model 2. ACF −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Series residuals(MaunaLoa.fit2) 0 5 10 15 20 Lag What does this plot tell us about our model assumptions? (i) What does this plot tell us about the PACF function of the residuals and why would this helpful? 3. Briefly explain why variability that is a percentage of the level of a time series will tend to cause a time series regression model to exhibit non-constant variance. 4. For a given time series realization Z1 , Z2 , . . . , Zn , the true autocorrelation function (ACF) ρ1 , ρ2 , . . . can be estimated by the sample autocorrelation function ρb1 , ρb2 , . . .. (a) What is the practical interpretation of the sample statistic ρb1 ? (b) What is the practical interpretation of the sample statistic ρb1 /Sρb1 , where Sρb1 is the standard error of ρb1 ? 3 5. Consider the following time series model Zt = µ + a t , at ∼ nid(0, σa2 ) Suppose that a realization of length n = 20 is available and that Z̄ and Sz are the sample mean and sample standard deviation, respectively. Then an approximate prediction interval for the next observation Z21 (one step ahead forecast) is Z̄ ± t(.975,19) Sz where t(.975,19) is the 0.975 quantile of the Student t distribution. (a) Why is the prediction interval approximate? What is missing? (b) Why do we use 19 degrees of freedom for the t quantile?. (c) Give an expression for an approximate prediction interval for Z22 , based on the same 20 observations, (two-step ahead forecast). 6. For each of the following differencing schemes, write down Wt as a function of past and present values of Zt . (a) Wt = (1 − B)2 Zt (b) Wt = (1 − B)(1 − B12 )1 Zt 7. The ARMA(1,1) model can be concisely written as φ1 (B)Zt = θ0 + θ1 (B)at . Show how to expand this shorthand version to the unscrambled form in which Zt is written as a function of past values of Z and a. 4