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m06 - UNIVERSITY OF TORONTO Faculty of Arts and Science...

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Unformatted text preview: UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL - MAY 2006 EXAMINATIONS STA457H1 S/STA2202HlS Duration - 3 Hours No Aids Allowed Instructions: 0 Please answer all 4 questions in the booklets provided. 0 Marks for each question are in parentheses following the question number. 1. (12) Consider the AR(p) process 2. (9) Xt :¢1 Xl—l +¢2Xt—2 +”'+¢pXt—p +Zt where t = 0,14,... ;cov(Z,,Xs) = 0 for each 3 <1 and {z,}~ WN(0,0'2). a) What conditions on the coefﬁcients (If, ,¢2 ,. ..,¢p are required to ensure that {X} is a stationary process? b) What is the power transfer function for an AR(p) process? Use the Filtering Theorem to derive the spectral density for the stationary AR(p) process {Xr }. 0) Suppose p =1 and consider the AR(l) process X! = ¢1Xt—l +Zl ' Sketch both the autocorrelation function (ACF) and the spectral density for the case where (151 < 0. Brieﬂy outline what information the spectral density conveys about an AR(l) process with {1), < 0 as compared to an AR(l) process with {151 > 0. (1) Brieﬂy describe how to determine the partial autocorrelation ﬁmction (PACF) of an AR(p) process. What properties of this PACF make it so useﬁil for ﬁtting AR(p) models to observed data? Suppose a plot of a time series x1 ,. . .xn reveals a deterministic seasonal or periodic component of d time units but there does not appear to be any polynomial trend in the data. page 1 a) How would you expect the sample ACF corresponding to x1,...xn to appear? What about the periodogram? b) You decide to ﬁt the following model directly to the data x1,...xn : X, =st+Zt where s, = acosm t) +bsin(/1 t) , ,1 2 Zn / d and {2,} is a stationary process (not necessarily VWV(0,0'2)). Note that a and b are ﬁxed unknown parameters. How would one determine suitable values for a and b ? Is there another way to estimate the trend s, without assuming the parametric form 3, = a cos(/l t) + bsin(/l t) ? 0) Alternatively, suppose you decide to difference the data x1 ,. . .x,l using the seasonal difference operator V dx, = x, — x,_d . Brieﬂy describe what effect this would have on the data by refen‘ing to the power transfer function of the linear ﬁlter V dx, = x, — xH, . After differencing, brieﬂy outline how you would then proceed to model the data. 3. (15) Consider the following observed time series data: x1,...xn . A plot of the sample 4. (9) ACF indicates that an ARMA(p,q) model may be appropriate for modelling the data: X, =¢1X,_1+¢2X,_2 +---+¢pX,_p +Z, +9xZ,_1 +022,_2 +...+¢9qZ,_q where {z,}~ WN(0,0'2). a) What feature of the sample ACF may have led to this conclusion? b) Derive the spectral density of an ARMA(p,q) process. c) Brieﬂy describe how to determine suitable values for p and q. (1) Brieﬂy describe one method for estimating the parameters for this model. e) Brieﬂy describe how to assess the model's goodness-of-ﬁt. Consider the ARCH (1) process X =0'Z l t I where {z,}~IID N(0,1) and of =0;0 +011X,2_1,oz0 > 0, (1120. a) For what type of data might this model be appropriate? b) Calculate E (X ,) and var(Xt) . c) Show that E(X,X,+h)=0 for h >0. Total Marks = (45) Total Pages = (2) page 2 ...
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m06 - UNIVERSITY OF TORONTO Faculty of Arts and Science...

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