TimeSeriesBook.pdf

Compute the variance of x 1 x 2 x 3 x 4 4 exercise

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Compute the variance of ( X 1 + X 2 + X 3 + X 4 ) / 4 . Exercise 2.5.2. Check whether the following stochastic difference equations possess a stationary solution. If yes, is the solution causal and/or invertible with respect to Z t WN(0 , σ 2 ) ? (i) X t = Z t + 2 Z t - 1 (ii) X t = 1 . 3 X t - 1 + Z t (iii) X t = 1 . 3 X t - 1 - 0 . 4 X t - 2 + Z t (iv) X t = 1 . 3 X t - 1 - 0 . 4 X t - 2 + Z t - 0 . 3 Z t - 1 (v) X t = 0 . 2 X t - 1 + 0 . 8 X t - 2 + Z t (vi) X t = 0 . 2 X t - 1 + 0 . 8 X t - 2 + Z t - 1 . 5 Z t - 1 + 0 . 5 Z t - 2 Exercise 2.5.3. Compute the causal representation with respect Z t WN(0 , σ 2 ) of the following ARMA processes: (i) X t = 1 . 3 X t - 1 - 0 . 4 X t - 2 + Z t (ii) X t = 1 . 3 X t - 1 - 0 . 4 X t - 2 + Z t - 0 . 2 Z t - 1 (iii) X t = φX t - 1 + Z t + θZ t - 1 with | φ | < 1 Exercise 2.5.4. Compute the autocovariance function of the following ARMA processes: (i) X t = 0 . 5 X t - 1 + 0 . 36 X t - 2 + Z t (ii) X t = 0 . 5 X t - 1 + 0 . 36 X t - 2 + Z t + 0 . 5 Z t - 1 Thereby Z t WN(0 , σ 2 ) .
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46 CHAPTER 2. ARMA MODELS
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Chapter 3 Forecasting Stationary Processes An important goal of time series analysis is forecasting. In the following we will consider the problem of forecasting X T + h , h > 0, given { X T , . . . , X 1 } where { X t } is a stationary stochastic process with known mean μ and known autocovariance function γ ( h ). In practical applications μ and γ are unknown so that we must replace these entities by their estimates. These estimates can be obtained directly from the data as explained in Section 4.2 or indirectly by first estimating an appropriate ARMA model (see Chapter 5) and then inferring the corresponding autocovariance function using one of the methods explained in Section 2.4. Thus the forecasting problem is inherently linked to the problem of identifying an appropriate ARMA model from the data (see Deistler and Neusser, 2012). 3.1 The theory of linear least-squares fore- casts We restrict our discussion to linear forecast functions , also called linear pre- dictors , P T X T + h . Given observation from period 1 up to period T , these predictors have the form: P T X T + h = a 0 + a 1 X T + . . . + a T X 1 = a 0 + T X i =1 a i X T +1 - i with unknown coefficients a 0 , a 1 , a 2 , . . . , a T . In principle, we should index these coefficients by T because they may change with every new observations. 47
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48 CHAPTER 3. FORECASTING STATIONARY PROCESSES See the example of the MA(1) process in Section 3.1.2. In order not to overload the notation, we will omit this additional index. In the Hilbert space of random variables with finite second moments the optimal forecast in the mean squared error sense is given by the conditional expectation E ( X T + h | c, X 1 , X 2 , . . . , X T ). However, having practical applica- tions in mind, we restrict ourself to linear predictors for the following rea- sons: 1 (i) Linear predictors are easy to compute. (ii) The coefficients of the optimal (in the sense of means squared errors) linear forecasting function depend only on the first two moments of the time series, i.e. on E X t and γ ( j ), j = 0 , 1 , . . . , h + T - 1.
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