a Find the autocovariance and autocorrelation functions for this process when \u03b8

# A find the autocovariance and autocorrelation

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a. Find the autocovariance and autocorrelation functions for this process when θ . 8. b. Compute the variance of the sample mean (X 1 + X 2 + X 3 + X 4 )/ 4 when θ . 8. c. Repeat (b) when θ . 8 and compare your answer with the result obtained in (b). 1.6. Let { X t } be the AR(1) process defined in Example 1.4.5. a. Compute the variance of the sample mean (X 1 + X 2 + X 3 + X 4 )/ 4 when φ . 9 and σ 2 1. b. Repeat (a) when φ . 9 and compare your answer with the result obtained in (a). 1.7. If { X t } and { Y t } are uncorrelated stationary sequences, i.e., if X r and Y s are uncorrelated for every r and s , show that { X t + Y t } is stationary with autoco- variance function equal to the sum of the autocovariance functions of { X t } and { Y t } .
42 Chapter 1 Introduction 1.8. Let { Z t } be IID N ( 0 , 1 ) noise and define X t Z t , if t is even , (Z 2 t 1 1 )/ 2 , if t is odd . a. Show that { X t } is WN ( 0 , 1 ) but not iid ( 0 , 1 ) noise. b. Find E(X n + 1 | X 1 , . . . , X n ) for n odd and n even and compare the results. 1.9. Let { x 1 , . . . , x n } be observed values of a time series at times 1 , . . . , n , and let ˆ ρ(h) be the sample ACF at lag h as in Definition 1.4.4. a. If x t a + bt , where a and b are constants and b 0, show that for each fixed h 1, ˆ ρ(h) 1 as n → ∞ . b. If x t c cos (ωt) , where c and ω are constants ( c 0 and ω ( π, π ]), show that for each fixed h , ˆ ρ(h) cos (ωh) as n → ∞ . 1.10. If m t p k 0 c k t k , t 0 , ± 1 , . . . , show that m t is a polynomial of degree p 1 in t and hence that p + 1 m t 0. 1.11. Consider the simple moving-average filter with weights a j ( 2 q + 1 ) 1 , q j q . a. If m t c 0 + c 1 t , show that q j q a j m t j m t . b. If Z t , t 0 , ± 1 , ± 2 , . . . , are independent random variables with mean 0 and variance σ 2 , show that the moving average A t q j q a j Z t j is “small” for large q in the sense that EA t 0 and Var (A t ) σ 2 /( 2 q + 1 ) . 1.12. a. Show that a linear filter { a j } passes an arbitrary polynomial of degree k without distortion, i.e., that m t j a j m t j for all k th-degree polynomials m t c 0 + c 1 t + · · · + c k t k , if and only if j a j 1 and j j r a j 0 , for r 1 , . . . , k. b. Deduce that the Spencer 15-point moving-average filter { a j } defined by (1.5.6) passes arbitrary third-degree polynomial trends without distortion.
Problems 43 1.13. Find a filter of the form 1 + αB + βB 2 + γ B 3 (i.e., find α , β , and γ ) that passes linear trends without distortion and that eliminates arbitrary seasonal components of period 2. 1.14. Show that the filter with coefficients [ a 2 , a 1 , a 0 , a 1 , a 2 ] 1 9 [ 1 , 4 , 3 , 4 , 1] passes third-degree polynomials and eliminates seasonal components with pe- riod 3. 1.15. Let { Y t } be a stationary process with mean zero and let a and b be constants. a. If X t a + bt + s t + Y t , where s t is a seasonal component with period 12, show that ∇∇ 12 X t ( 1 B)( 1 B 12 )X t is stationary and express its autocovariance function in terms of that of { Y t } .

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