Q: We said in the lectures that it is not a good idea to over dierence a time series. The task in this exercise is to investigate the eect of over- dierencing via a simulation study. Consider the following stationary AR(1) model: Xt ?? Xt??1 = Zt where Zt is a white-noise process with variance 2Z= 0:5. Suppose that data (x1; :::; xn) are simulated from the above process and then these data are dierenced once giving rise to a series (y1; :::; yn) where: yt = Xt ?? Xt??1 Using the command ar in R t an autoregressive model to the simulated data (y1; :::; yn) and nd the order of the autoregressive model which ts the data best according to the AIC value. How far or how close are your answers from the truth with regards to the order selected using the AIC criterion. Comment on your results. Note that your conclusions may dier for various choices of the datasets size n and the value of and therefore your investigation should cover these aspects too. Your answer should be no longer than four pages. 1
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