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Estimation and Comparison of Time Series Models

## Question 1

(a) A researcher estimated an MA(1) â€” GARCH(1,1) model for the daily percentage returns on the MX 200 Australian stock market index over the last five weeks of trading and obtained the results: r, = 0.34 + 1.05u1_ r + u, â€¢ = 0.27 + 0.0514_, + 0.93a? I The log-likelihood was 718.2Â
(i) Is the process for rt invertible in this model? Justify your answer. (0.5 marks) (ii) Is the conditional variance of r1 always positive in this model? Justify your answer. (0.5 marks) (iii) Will a large shock to returns in this model lead to forecasts of the conditional variance that are high and remain high for many periods into the future? Justify your answer. (2 marks)Â  (b) The researcher also estimated an ARMA(1,1) â€”GARCH(1,1) in Mean model and obtained the following results: n = 0.25 + 0.39r,_, + + 0.964 + u, â€¢ = 0.39 +0.08u1_1+ 0.9241 The log-likelihood was 720.SÂ  Is there a trade-off between risk and return in this model? Explain your answer fully. (2 marks)Â

(ii) Will the forecasts of the conditional variance of n converge to a finite number as the forecast horizon increases? Justify your answer. (1mark)

(iii) The log-likelihood here is larger (720.5 versus 718.2). Is this to be expected? Justify your answer. (1 mark)Â

(iv) Conduct a statistical test to determine which of the two models is better supported by the data. Be sure to state the null and alternative hypotheses, calculate the test statistic and report the 5% critical value and state your conclusion. (3 marks)Â

Question 2 (10 marks)

Â (a) Suppose that a researcher has observations on the log stock price indexes of Korea (k1), Japan (10 and Malaysia (m1). Each of these stock price index series are 1(1). The researcher also has observations on the yield spread between the 6-month and 3-month treasury bill rate for Australia. The yield spread, which is denoted YS,, is an 1(0) series

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(i) Suppose the researcher decides to estimate the following regression by OLS:Â  Mc e Pt) + ii?c UtÂ  and finds that the estimated residuals are white noise. What do you conclude? Explain your answer fully. (1 mark)

(ii) Suppose the researcher decides to estimate the following regression by OLS:Â  = 130 + 13111+ u, and finds that the estimated residuals are a random walk (without drift). What do you conclude? What regression equation would you recommend the researcher estimate next (if at all)? (2 marks)Â

(iii) The researcher estimates the following regression by OLS: j, = 130 + ft,YS, + u,Â  and finds the ACF of the estimated residuals. What pattern would you expect to see in the ACF of the estimated residuals? (1 mark).Â

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(b) Suppose a researcher has data on the stock price indexes for two countries. Let Pl, be the stock price index for country 1 and P2, the stock price Index for country 2. Both stock price indexes are 1(1) variables.

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01 If the two stock price indexes are not cointegrated with each other, how would you describe the movement of Pli and P21 over time relative to one another? Justify your answer. (2 marks)Â

(ii) The researcher estimates the regressionÂ  P 11 = Po + Pint + utÂ  and decides to perform an ADF test on the estimated residuals. In performing the ADF regression, should the researcher include a constant term and/or a trend term? Explain you answer fully. (1 mark)Â  (iii) Is the ADF test statistic [-distributed? Justify your answer. (1 mark)Â