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40052 - predictable manner iii A plot of the residuals...

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STATISTICS 4005 ASSIGNMENT 2 Due date: February 12, 2004 1. Consider the model Z t = α 0 + α 1 t 2 + α 2 t 4 + a t , t = 1 , . . . , 5 where a t ’s are independently, identically distributed as N (0 , σ 2 a ). Write the model in matrix form. 2. Let { Y t } be a stationary process with V ar ( Y t ) = 1 and autocorrelation function ρ k = 0 . 5 k for k = 0 , 1 , 2 , ... . Find V ar 1 3 3 t =1 Y t . 3. Let X 1 , X 2 , ..., X 200 denote 200 consecutive observations of a stationary time series with mean μ and autocovariances γ k ’s. Let ¯ X = 200 i =1 X i / 200. (a) 4 marks Show that ¯ X is an unbiased estimator of μ . (b) 4 marks Calculate V ar ( ¯ X ) in terms of γ k ’s. 4. Fit the data set “dataset1.xls” by the following two models: (a) A model Z t = μ t + X t where E ( X t ) = 0, μ t = β 0 + β 1 cos( 2 πt 12 ) + β 2 sin( 2 πt 12 ) + β 3 cos( 2 πt 6 ) + β 4 sin( 2 πt 6 ) (b) A seasonal mean model (one mean for each month) - remember to omit the constant term from the fit. With each fit, include a residual analysis, i.e. i. A multiple time series plot of the data and fitted values. ii. A time series plot of the residuals.
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Unformatted text preview: predictable manner? iii. A plot of the residuals against the fitted values. Does the plot show non-constancy of error variance? iv. A histogram of residuals (with Options: Lower Bound = -10, Upper Bound = 14 and Interval Width = 2 in Splus) v. A plot of the residuals against the quantiles of Standard Normal. Ac-cording to (iii) and (iv), does the residuals look like normal? vi. An autocorrelation plot of the residuals, with Maximum Lag 36. (c) For each model, calculate by hand the fitted value and the actual residual for t = 18. (d) Which would you prefer, in practice? Justify your choice? NOTE: The data set “dataset1.xls” can be downloaded from http://www.sta.cuhk.edu.hk/khwu/courses/sta4005/ under the subsection Time Series Data used for Assignments....
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