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UI STAT 6560 - Applied time series analysis

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April, 25, 2008. Exam 2 for S156: Applied time series analysisName:1. (a) The following AR(2) model was fitted to a time series of size 30. Explain why the fittedmodel is stationary.ar1 ar2 interceptEstimate 0.51 0.07 0.47SE 0.18 0.18 0.29(b) Construct a 95% confidence interval for each of the three parameters. Which coefficient(s)are significant? Explain your answer to get credit.(c) Test the hypothesis that the stationary mean of the time series equals 1, at 5% significancelevel.(d) An AR(1) model was subsequently fitted to the data, resulting in the following fit:ar1 interceptEstimate 0.55 0.48SE 0.15 0.27The AIC of the fitted AR(1) model equals67.74, while that of the fitted AR(2) modelequals 69.6. Which model do you prefer?Name two reasons in support of your pref-erence.2. Let {et} and {Nt} be two independent time series, each of which is iid with zero mean andfinite variances σ2e> 0 and σ2N> 0.(a) Define the process Wt= Nt+ et− θ1et−1− θ2et−2, for all integer t, where θ1and θ26= 0 aretwo fixed parameters. Argue that {Wt} has finite memory of two lags.(b) What ARIMA(p,d,q) model can be used to model {Wt}, i.e. determine the orders, p, dand q?(c) Let {Xt} be a stationary AR(2) process defined by the equation (1 − φ1B − φ2B2)Xt= et,and Yt= Xt+ Ntwhere {Nt} and {et} are as in part (a). Compute (1 − φ1B − φ2B2)Yt.Hence, or otherwise, show that {Yt} is an ARMA(2,2) process.3. This question concerns whether or not thenonstationary time series shown in theright figure should be made stationary bytaking the first difference. It is found thatthe first differences of the series can be fit-ted by an AR(8) mo del, based on AIC. Afew ADF tests were carried out to assessthe need for differencing the data, whichare reported below.●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●TimeY0 20 40 60 80 1000 5 10 15 20 25 30ADF.test(Y,selectlags=list(Pmax=0),itsd=c(0,0,0))Estimate Std. Error t value Pr(>|t|)adf.reg 0.014 0.008 1.788 0.1ADF.test(Y,selectlags=list(Pmax=0),itsd=c(1,0,0))Estimate Std. Error t value Pr(>|t|)adf.reg -0.009 0.015 -0.642 0.1ADF.test(Y,selectlags=list(Pmax=0),itsd=c(1,1,0))Estimate Std. Error t value Pr(>|t|)adf.reg -0.528 0.09 -5.844 0.01ADF.test(Y,selectlags=list(select.lag=’aic’,Pmax=8),itsd=c(1,1,0))Estimate Std. Error t value Pr(>|t|)adf.reg -0.843 0.108 -7.813 0.01Lag orders: 1 3 8(a) Explain the meaning of the option itsd=c(1,1,0)(b) Among the above 4 ADF tests, which one is the most appropriate test for checking theneed for differencing? Explain your answer to get credits.(c) What is your recommendation on whether or not differencing is required? Explain youranswer. If differencing is not recommended, what do you recommend to do to deal withthe non-stationarity in the data.4. Identify an ARIMA(p,d,q) model for the time-series variable Y , given its ACF, PACF andEACF. Explain your reasoning to get credits.●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●Y0 50 100 150 200 250 3004 6 8 10 12 14 16 185 10 15 200.0 0.2 0.4 0.6ACFSeries Y5 10 15 20−0.4 −0.2 0.0 0.2 0.4 0.6Partial ACFSeries Y0 1 2 3 4 5 6 7 8 9 10 11 12 130 x x x x x o o o o o o o x x1 x x x o o o o o o o o o o o2 x x x x o o o o o o o o o o3 x x x x o o o o o o o o o o4 x x x o x o o o o o o o o o5 x o x o o o x o o o o o o o6 x x x x o x o o o o o o o o7 x x x o o x x o x o o o o


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