Question

9. Consider the following MA(1) process: $y_t = e_t + heta_1 e_{t-1}$, where $e_t$ is a white noise process with zero mean and variance $sigma^2$. (a) Calculate the variance of $y_t$. (b) Calculate the autocovariance $gamma_s$ for $s = 1, 2$. (c) Calculate the autocorrelation $ ho_s$ for $s = 1, 2$. (d) Show that the partial autocorrelation, $eta_2$, is given by $eta_2 = frac{- heta_1^2}{1 + heta_1^2 + heta_1^4}$ [3 marks] [3 marks] [3 marks] [3 marks] [Total: 12 marks]

          9. Consider the following MA(1) process:
$y_t = e_t + 	heta_1 e_{t-1}$,
where $e_t$ is a white noise process with zero mean and variance $sigma^2$.
(a) Calculate the variance of $y_t$.
(b) Calculate the autocovariance $gamma_s$ for $s = 1, 2$.
(c) Calculate the autocorrelation $
ho_s$ for $s = 1, 2$.
(d) Show that the partial autocorrelation, $eta_2$, is given by
$eta_2 = frac{-	heta_1^2}{1 + 	heta_1^2 + 	heta_1^4}$
[3 marks]
[3 marks]
[3 marks]
[3 marks]
[Total: 12 marks]

$9. Consider the following MA(1) process: yt = et + heta1 et-1, where et is a white noise process with zero mean and variance sigma^2. (a) Calculate the variance of yt. (b) Calculate the autocovariance gammas for s = 1, 2. (c) Calculate the autocorrelation hos for s = 1, 2. (d) Show that the partial autocorrelation, eta2, is given by eta2 = frac- heta1^21 + heta1^2 + heta1^4 [3 marks] [3 marks] [3 marks] [3 marks] [Total: 12 marks]$

Added by Ronald S.

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