Time Series Analysis: Assignment Questions for STA4807

Define tomorrow. university of south africa Tutorial letter 013/0/2022 Time Series STA4807 Year module Department of Statistics ASSIGNMENT 03 QUESTIONS STA4807/013/0/2022

ASSIGNMENT 03 Unique Nr.: 771150 Fixed closing date: 20 July 2022 QUESTION 1 [16] (a) Consider the seasonal process ( 1 − Φ B 12 ) X t = ( 1 − Θ B 12 ) ϵ t , where ϵ t ∼ N ( 0 ,σ 2 ϵ ) . (i) Express the above model as an infinite moving average ( MA ( ∞ ) ) process, i.e. find the first 6 nonzero ψ weights and hence or otherwise, (3) (ii) find the autovariance function of the process, γ x k ; and (6) (iii) determine the variance of the l -step ahead forecast, V ar [ ̂ X n ( l )] ≡ V ar [ e n ( l )] based on the minimum mean square error ( MMSE ) l -step ahead forecast, ̂ X n ( l ) = E [ X n + l ∣ X n ,X n − 1 ,... ] forecast of X n + l at the origin n , where e n ( l ) = X n + l − ̂ X n ( l ) = l − 1 ∑ j = 0 ψ j a n + l − j . (3) (b) The π weights of the above process in (a) when it is expressed as an infinity autoregressive ( AR ( ∞ ) ) process are given by π j = { ( Φ − Θ ) Θ (( j / 12 ) − 1 ) , for j = 12 , 24 , 36 ,... 0 , otherwise. (i) Find the l -step ahead forecasts and describe how they are obtained. (3) (ii) Write down the difference equation that gives the eventual l -step ahead forecasts. (1) QUESTION 2 [27] The time series plot in Figure 1 shows the monthly hotel room occupancy in the USA from 1977 to 1990 (168 observations). SPSS was used for analysis. (a) Which of the two models: Y t = TN t × SN t × IR t , and Y t = TN t + SN t + IR t , where TN t , SN t ,IR t are trend, seasonal and irregular components, respectively, would you suggest adequately fits the hotel room occupancy series data? Give a reason(s) for your answer. (3) (b) Which operator(s) and/or transformation would you apply to this series to make it stationary? (3) 2

STA4807/013/0/2022 (c) The output in Figure 2 is from fitting the linear regression Model 1 to the natural log transformed series comprising linear trend and additive seasonality modelled using dummy variables i.e. , ln ( Y t ) = TN t + SN t + ε t = β 0 + β 1 t + β s 1 x s 1 ,t + β s 2 x s 2 ,t + ... + β s 10 x s 10 ,t + β s 11 x s 11 ,t + ε t , where sL are the seasons with L = 1 ,..., 12 and ε t is the error term. This multiple linear regression model (Model 1) in vector-matrix form is W = X β + ε , where W t = ln ( Y t ) and ε is a vector of error terms. Write down the design matrix X for Model 1. (4) (d) Figure 2 gives the estimation output from fitting Model 1 given in (c). (i) Write down the estimated model in Figure 2. (3) (ii) Taking into account both the seasonality and trend components in Model 1, find the forecast for June of 1991 ( Y n + 6 ), ̂ Y n ( 6 ) , given that V ar [ e n ( l )] = 0 . 2 . (4) (e) Discuss the adequacy of Model 1 fitted in Figure 2 suggesting what could be done to make this model adequate. (3) Figure 1: Monthly hotel room occupancy in the USA from Jan 1977 to Dec 1990 3

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Figure 2: Model 1. (f) The model given in (c) was improved upon by augmenting a SARIMA ( p,d,q ) × ( P,D,Q ) L model to it to describe the autocorrelated residuals to give Model 2. Figure 3 gives the output resulting from fitting Model 2. (i) Name and write down the SARIMA ( p,d,q ) × ( P,D,Q ) L model used to describe the autocorrelated residuals. (4) (ii) Discuss the adequacy of Model 2 suggesting what could be done to make this model adequate. (3) 4

STA4807/013/0/2022 Figure 3: Model 2. QUESTION 3 [29] (a) Let { a j } be the impulse response function (IRF) of a filter with transfer function (TF) A ( ω ) and suppose that ∑ ∣ a j ∣ < ∞ . Show that a k = 1 2 π ∫ π − π e iωk A ( ω ) dω, 5

where A ( ω ) = ∑ j a j e − iωj . (4) (b) Show that a symmetric linear filter with IRF { a j } q j = − q will pass through a linear trend µ t = a + bt without distortion if and only if ∑ a j = 1 and ∑ ja j = 0 . (4) (c) The TF of a filter is given by A ( ω ) = sin 2 ( ω / 2 ) , − π < ω < π. (i) Find the IRF of the filter. (4) (ii) Let { X t } 5 t = 0 = { 3 . 0 , 2 . 5 , 2 . 6 , 3 . 2 , 2 . 8 , 2 . 8 } be a segment (partial realization) of a time series. Suppose that the series is passed through the linear filter A ( ω ) . Obtain the output series. (2) (d) Let { X 0 ,X 1 ,...,X n − 1 } be a finite realization of a time series with fast fourier transform (FFT) { ˜ X ( ω 0 ) ,..., ˜ X ( ω n − 1 )} . Show that the inverse FFT is given by X t = 1 n n − 1 ∑ k = 0 ˜ X ( ω k ) e iω k t , ω k = 2 πk n and k = 0 , 1 ,...,n − 1 . (8) (e) Let { X 0 ,X 1 ,...,X n − 1 } be a finite realization of a time series and define the periodogram by I ( ω j ) = 1 n ∣ n − 1 ∑ t = 0 ( X t − ¯ X ) e i 2 πjt / n ∣ 2 j = 0 , 1 ,...,n − 1 . Prove that 1 n ∑ n − 1 j = 0 I ( ω j ) = 1 n n − 1 ∑ t = 0 ( X t − ¯ X ) 2 , i.e., an estimator of the variance of the series. (7) 6

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STA4807/013/0/2022 QUESTION 4 [28] (a) If X 1 ,X 2 ,...,X n are independent normal random variables, i.e., X t ∼ N ( µ,σ 2 ϵ ) : (i) find the respective distributions of a p = 2 n ∑ n t = 1 X t cos ( 2 πpt / n ) and b p = 2 n ∑ n t = 1 X t sin ( 2 πpt / n ) , (5) and (ii) show that a p and b p are independent, for p = 1 ,..., ( n / 2 ) − 1 . (3) (b) Show that the periodogram I ( ω ) = 1 πn ∣ ∑ n t = 1 X t e − itω ∣ 2 , is asymptotically unbiased, i.e., E [ I n ( ω )] = f ( ω ) as n → ∞ . (6) (c) The finite Fast Fourier Transform (FFT) ˜ X ( ω j ) = n − 1 ∑ t = 0 X t e − 2 πj n t of a realization of a time series is displayed in the following table: j ˜ X ( ω j ) 0 86 . 00 1 − 1 . 07 + 8 . 28 i 2 d 3 − 4 . 43 + 2 . 77 i 4 − 3 . 84 − 2 . 13 i 5 6 6 e 7 f 8 − 11 . 66 + 1 . 31 i 9 − 1 . 07 − 8 . 28 i (i) Supply the missing values d , e and f which have been obliterated from the output. (3) (ii) Calculate the periodogram { I ( ω j )} 9 1 = 1 nπ {∣ ˜ X ( ω j )∣ 2 } 9 1 . (8) (iii) Determine the mean and the variance of the series. (3) [Total marks=100] 7

STA4807_2022_TL_013_0_B

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