MATH38032 SECTION A Answer ALL four questions A1. Akaike’s information criterion is often used to choose between competing models fitted to an observed time series of length n. The AIC statistic can be defined by the formula AIC = ln σˆ2 + 2k n , where σˆ2 is an optimal (maximum likelihood or equivalent) estimate of σ2 and k is the number of estimated parameters (apart from σ2). a) (3 marks) Explain concisely the general behaviour of the two terms of the AIC statistic as the number of parameters increases. b) (2 marks) The table below gives the value of the AIC statistic for several models fitted to an observed time series of length n = 10. Model AIC Xt = c+ εt + θ1εt−1 17 Xt = c+ εt + θ1εt−1 + θ2εt−2 16.5 Xt = c+ εt + θ1εt−1 + θ2εt−2 + θ3εt−3 18.5 Use the AIC criterion to select a model for this time series. c) (3 marks) A modification, AIC corrected (AICC), of the AIC statistic is given by the following formula AICC = ln σˆ2 + 2k n− k − 1 . Calculate AICC for the models in part (b) above and select a model according to AICC. [8 marks] A2. Let xt, t = 1, . . . , n be an observed time series. The Box-Ljung test is based on the statistic QLB(h) = n(n+ 2) h∑ k=1 ρˆ2k n− k , where n is the length of the observed trajectory and ρˆk are sample autocorrelations of {xt}. Under the null hypothesis of interest the distribution of QLB(h) is approximately χ 2 h. a) (3 marks) State the null and alternative hypotheses in the Ljung-Box test. b) (3 marks) What is the distribution of the test statistic, under the same null hypothesis, when xt, t = 1, . . . , n represent the residuals from a fitted ARMA(p,q) model with intercept? [6 marks] 2 of 16 P.T.O. MATH38032 A3. Let {Xt} be an MA(q) time series with representation Xt = q∑ i=0 θiεt−i, where θ0 = 1 and {εt} is white noise with variance σ2. a) (3 marks) Show that EXtεt−k = θkσ2, for k = 0, 1, . . . , q. b) (7 marks) Let {Xt} be a stationary time series with mean EXt = 0. Assume further that the autocovariances, γk, of {Xt} are as follows: γk =  2 for k = 0, 0.8 for k = 12, 0 for k 6= 0,±12. Identify a model for {Xt} and compute its parameters. If more than one model is possible, choose an invertible one. [10 marks] A4. Consider the following model for a stationary process: Xt = − 310Xt−1 + 410Xt−2 + εt − 12εt−1, where {εt} ∼WN(0, σ2). a) (3 marks) Write down the model in operator form. b) (5 marks) Explain why this model is of higher order than necessary and give its reduced form. [8 marks] 3 of 16 P.T.O. MATH38032 SECTION B Answer 2 of the 3 questions B5. a) (12 marks) A stationary time series {Xt} follows the AR(2) model Xt = 0.01 + 0.2Xt−2 + εt, (1) where {εt} ∼WN(0, 0.02). i) What property of an AR model is referred to as causality property? ii) Find the mean of {Xt}. iii) Compute the first two autocorrelations, ρ1 and ρ2, and the variance, γ0, of the process {Xt}. iv) Assume that the values x100 = −0.01 and x99 = 0.02 of the series are observed at times t = 100 and t = 99. Compute xˆ101|100,99,... and xˆ102|100,99,..., the 1- and 2-step ahead forecasts at the forecast origin t = 100. v) Give the standard deviations of the forecast errors associated with the predictors in (iv). b) (12 marks) Let {Xt} be an AR(2) process with representation Xt = φ1Xt−1 + φ2Xt−2 + εt, εt ∼WN(0, σ2). (2) Let {Yt} be the “reversed” process defined by Yt = X−t. Show that {Yt} is also an AR(2) process with the same parameters as that of the model for {Xt}. [24 marks] B6. a) (8 marks) Let {Vt} be an I(1) process and {Zt} be a stationary process. i) Show that Vt +Zt is not stationary. (Hint: You may use the fact that a linear combination of stationary processes cannot be an I(1) process.) ii) Assume further that the time series {Vt} and {Zt} are independent of each other. Show that Vt + Zt is an I(1) process. b) (10 marks) Consider a random walk {ut} and a stationary AR(1) time series {et}, which evolve according to the following equations: u0 = 0, ut = ut−1 + ε1,t, for t ≥ 1, et = ρet−1 + ε2,t, for t ≥ 1, where |ρ| < 1 and {ε1,t} and {ε2,t} are white noise sequences, independent of each other (so, E ε1,tε2,s = 0 for all t, s). Consider also processes {Xt} and {Yt} which obey the following equations: Xt + βYt = ut Xt + αYt = et, where α and β are some constants, α 6= β, and {ut} and {et} are the processes defined above. 4 of 16 P.T.O. MATH38032 i) Show that Xt = α α−βut − βα−βet Yt = −1 α−βut + 1 α−βet. ii) Identify the processes {ut}, {et}, {Xt} and {Yt} as I(d) processes (e.g. I(1), I(0)). Give your reasons concisely. iii) For each of the following pairs of processes state, giving your reasons, if they are cointegrated of order one and, if so, give a cointegrating vector. • {ut} and {et}, • {Xt} and {Yt}. c) (6 marks) An analysis of two time series, {Xt} and {Yt}, established that they are well described by ARIMA(0,1,0) models. The Johansen test for cointegration was performed with the following results: ###################### # Johansen-Procedure # ###################### Test type: maximal eigenvalue statistic (lambda max), with linear trend Eigenvalues (lambda): [1] 0.31366569 0.04020297 Values of test statistic and critical values of test: test 10pct 5pct 1pct r <= 1 | 4.02 6.50 8.18 11.65 r = 0 | 36.89 12.91 14.90 19.19 Eigenvectors, normalised to first column: (These are the cointegration relations) x y x 1.0000000 1.000000 y -0.9989006 1.487965 ################################################################## Are the two time series cointegrated? If so, give a cointegrating vector and a cointegrating relation. Explain your decision. [24 marks] 5 of 16 P.T.O. MATH38032 B7. The number of overseas visitors to New Zealand is recorded for each month over the period 1977–1995. Let xt be the number of overseas visitors in time period t (in months) and zt = lnxt. a) (3 marks) Comment on the main features of the plots of {xt} and {zt} (see Figure 7.1). Why one might wish to build a model for {zt} rather than directly for {xt}? b) (3 marks) Comment on the main features of the sample autocorrelation function of {zt} (see Figure 7.1). c) (4 marks) An ARIMA(1,1,0) model was fitted to {zt}. Comment on the quality of the fit, using the numerical and graphical information in Figure 7.2. d) (4 marks) An ARIMA(1, 1, 0)(0, 1, 0)12 model was fitted to {zt}, see Figure 7.3. Why was this particular modification of the model in part (c) chosen? Did it achieve the desired effect? e) (2 marks) Identify the model in Figure 7.6 as a seasonal ARIMA(p, d, q)(ps, ds, qs)s model. Write it in operator form. f) (4 marks) Choose the best fitting model among those given in Figures 7.2–7.10. Explain the reasons for your choice. g) (4 marks) Comment on the quality of the fit of the model selected in part (f) above. Note: Write verbal answers concisely. You normally need one sentence or phrase for each worthy thought. Credit will not be given for irrelevant material. 6 of 16 P.T.O. MATH38032 Time X4 81 76 1980 1985 1990 1995 50 00 0 Time X4 81 76 1980 1985 1990 1995 10 .0 11 .5 0 1 2 3 4 5 0. 0 0. 4 0. 8 Lag AC F Figure 7.1: Visitors to New Zealand: the time series, {xt} (top), the logged time series, {zt} (middle), and the sample autocorrelation function of {zt} (bottom). 7 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 0)) Coefficients: ar1 0.0255 s.e. 0.0667 sigma^2 estimated as 0.04985: log likelihood = 18.17, aic = -32.34 Standardized Residuals Time 1980 1985 1990 1995 − 2 0 1 2 0.0 0.5 1.0 1.5 − 0. 5 0. 5 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.2: Visitors to New Zealand: Model 1 8 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 0), seasonal = list(order = c(0, 1, 0))) Coefficients: ar1 -0.4076 s.e. 0.0628 sigma^2 estimated as 0.005184: log likelihood = 259.3, aic = -514.6 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 − 0. 2 0. 4 1. 0 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.3: Visitors to New Zealand: Model 2 9 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 0), seasonal = list(order = c(1, 1, 0))) Coefficients: ar1 sar1 -0.3911 -0.2779 s.e. 0.0633 0.0672 sigma^2 estimated as 0.004784: log likelihood = 267.44, aic = -528.87 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 − 0. 2 0. 4 1. 0 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.4: Visitors to New Zealand: Model 3 10 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(0, 1, 1), seasonal = list(order = c(0, 1, 1))) Coefficients: ma1 sma1 -0.6727 -0.3706 s.e. 0.0726 0.0725 sigma^2 estimated as 0.004050: log likelihood = 284.64, aic = -563.27 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 − 0. 2 0. 4 1. 0 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.5: Visitors to New Zealand: Model 4 11 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 0), seasonal = list(order = c(0, 1, 1))) Coefficients: ar1 sma1 -0.3893 -0.4125 s.e. 0.0632 0.0706 sigma^2 estimated as 0.004556: log likelihood = 272.01, aic = -538.02 Standardized Residuals Time 1980 1985 1990 1995 − 2 0 2 0.0 0.5 1.0 1.5 − 0. 2 0. 4 1. 0 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.6: Visitors to New Zealand: Model 5 12 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(0, 1, 1), seasonal = list(order = c(1, 1, 0))) Coefficients: ma1 sar1 -0.6883 -0.2713 s.e. 0.0685 0.0674 sigma^2 estimated as 0.004182: log likelihood = 281.6, aic = -557.19 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 − 0. 2 0. 4 1. 0 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.7: Visitors to New Zealand: Model 6 13 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 1), seasonal = list(order = c(1, 1, 1))) Coefficients: ar1 ma1 sar1 sma1 0.2546 -0.845 0.2297 -0.5866 s.e. 0.0962 0.059 0.1618 0.1349 sigma^2 estimated as 0.003897: log likelihood = 288.35, aic = -566.71 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 − 0. 2 0. 4 0. 8 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.8: Visitors to New Zealand: Model 7 14 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 1), seasonal = list(order = c(1, 1, 0))) Coefficients: ar1 ma1 sar1 0.2396 -0.8409 -0.2868 s.e. 0.0972 0.0600 0.0669 sigma^2 estimated as 0.004077: log likelihood = 284.12, aic = -560.25 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 0. 0 0. 4 0. 8 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.9: Visitors to New Zealand: Model 8 15 of 16 P.T.O. MATH38032 Call: arima(x = visnzln, order = c(1, 1, 1), seasonal = list(order = c(0, 1, 1))) Coefficients: ar1 ma1 sma1 0.2527 -0.8393 -0.3909 s.e. 0.0948 0.0569 0.0717 sigma^2 estimated as 0.003935: log likelihood = 287.44, aic = -566.87 Standardized Residuals Time 1980 1985 1990 1995 − 3 − 1 1 3 0.0 0.5 1.0 1.5 0. 0 0. 4 0. 8 Lag AC F ACF of Residuals l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 60 0. 0 0. 4 0. 8 p values for Ljung−Box statistic lag p va lu e Figure 7.10: Visitors to New Zealand: Model 9 [24 marks] END OF EXAMINATION PAPER 16 of 16



























































































































































































































































































































































































































































































































































































































































































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