Midterm

# Midterm - Midterm STAT 443 Spring 2009 1 If X_t and Y_t are...

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Midterm STAT 443 Spring 2009 1. If X_t and Y_t are uncorrelated (weakly) stationary sequences, i.e., if X_r ans Y_s are uncorrelated for every r and s, show that X_t+Y_t is (weakly) stationary with autocovariance function equal to the sum of the autocovariance functions of X_t and Y_t. 2. Let {Xt} satisfy the equation Xt =Zt + bZt-3, (1) where Zt is white noise with zero mean (E(Zt)=0) and unit variance (Var(Zt)=1). Such an equation (1) is called a moving-average model of order 3, or an MA(3) model. a) Find the autocovariance functions of Xt when b=-0.6; b) Find the autocorrelation functions of Xt when b=-0.6. 3. DESCRIPTIVE ABSTRACT: Monthly number of unemployed persons in Australia. Feb 1978 - Aug 1995. YOUR TASK: To forecast unemployment rate for the first eight months of 1995, i.e. from January to August 1995. You should a) Construct a linear regression model of the unemployment rate vs. time using either dummy variables or trigonometric functions; Your overall goal is to construct a model with high predictive power and residuals satisfying OLS assumptions. You can use any method to reach this goal (dummy variables or trigonometric functions). You can apply variance stabilizing transformations if you think that such transformations are appropriate. b) provide diagnostics of the assumptions (white noise + normality) for your best model, i.e. verify that residuals are homoscedastic (residual plots), uncorrelated (acf plot, the Ljung-Box test/plot, the Durbin-Watson test, the runs and Bartels tests) and normally distributed (Shapiro-Wilk test, QQ plot). c) construct 90% and 95% predictive intervals for the first eight months of 1995, i.e. from January to August 1995, using your best regression model; does the observed data fall into the constructed predictive intervals? d) compare the constructed 90% and 95% predictive intervals from your best regression model with the 90% and 95% predictive intervals from your best Holt-Winters model which you obtained in the 1

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homework 2; compare the residual diagnostics from the regression and Holt-Winters models. e) write your conclusion on the aptness of the obtained models; which model is better and why? Justify all your conclusions. 428800 424800 403400 398400 393500 380500 398300 387300 370400 372800 444600 449900 458100 424800 420600 400100 393000 387100 377500 400400 391400 363600 431000 441700 448500 415600 408000 416600 409300 387600 394500 407600 378500 359600 435700 433800 427700 413300 379500 379300 353700 378200 380600 394000 374000 375000 437600 443900 2
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Midterm - Midterm STAT 443 Spring 2009 1 If X_t and Y_t are...

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