TS Module 6 Stationary autoregressive processes practice problems

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NEAS	NEAS posted 14 Years Ago #10197 #
Supreme Being Group: Administrators Posts: 4.3K, Visits: 1.3K	TS Module 6 Stationary autoregressive processes (The attached PDF file has better formatting.) Time series autoregressive processes practice problems ** Exercise 6.1: Stationarity of AR(2) process What are the three conditions for an AR(2) process to be stationary? Given ö₁, what values of ö₂ create a stationary AR(2) process? Part A: The three conditions are ö ₁ + ö₂ < 1 ö ₂ – ö₁ < 1 \| ö₂\| < 1 (See Cryer and Chan page 72, equation 4.3.11) Part B: If ö₁ is given, the conditions are ö ₂ < 1 – ö₁ ö ₂ < 1 + ö₁ \| ö₂\| < 1 Illustration: If ö₁ = 0.9, then ö₂ must be less than 0.1 and more than –1. Question 6.2: Characteristic polynomial An AR(2) process is Y_t = ö₁ Y_t-1 + ö₂ Y_t-2 + e_t . What is the characteristic polynomial for this time series? ö (x) = 1 + ö₁(x) + ö₂(x²) ö (x) = 1 + ö₂(x) + ö₁(x²) ö (x) = 1 – ö₁(x) – ö₂(x²) ö (x) = 1 – ö₂(x) + ö₁(x²) ö (x) = 1 + ö₂(x) – ö₁(x²) Answer 6.2: C (See Cryer and Chan page 71, equation 4.3.9) The characteristic polynomial reverses the sign of the autoregressive coefficient. * Exercise 6.3: AR(2) autocorrelations An AR(2) process with mean zero is Y_t = ö₁ Y_t-1 + ö₂ Y_t-2 + å_t What is ã₀, the variance of Y_t? What is ã₁, the covariance of Y_t and Y_t-1? What is ã₂, the covariance of Y_t and Y_t-2? What is ñ₁, the autocorrelation of lag 1? What is ñ₂, the autocorrelation of lag 2? Part A: Multiply the expression for the AR(2) process by Y_t-1 and take expectations to get E(Y_t × Y_t-1) = ö₁ E(Y_t-1 × Y_t-1) + ö₂ E(Y_t-2 × Y_t-1) + E(å_t × Y_t-1) The time series observations are uncorrelated with the error terms in subsequent periods, so E( å_t, Y_t-1) = 0. E(Y_t, Y_t-1) = ã₁, E(Y_t-1, Y_t-1) = ã₀, and E(Y_t-2, Y_t-1) = ã₁, so ã ₁ = ö₁ ã₀ + ö₂ ã₁ Divide this equation by ã₀ to get ñ ₁ = ö₁ ñ₀ + ö₂ ñ₁ ñ _o is the correlation of a random variable with itself, which is always 1, so we derive that ñ _o = è₁ + è₂ ñ₁ ñ₁ = è₁ / (1 – è₂) Part B: Multiply the expression for the AR(2) process by Y_t-2 and take expectations to get E(Y_t × Y_t-2) = ö₁ E(Y_t-1 × Y_t-2) + ö₂ E(Y_t-2 × Y_t-2) + E(å_t × Y_t-2) The time series observations are uncorrelated with the error terms in subsequent periods, so E( å_t, Y_t-2) = 0. E(Y_t, Y_t-2) = ã₂, E(Y_t-1, Y_t-2) = ã₁, and E(Y_t-2, Y_t-2) = ã₀, so ã ₂ = ö₁ ã₁ + ö₂ ã₀ Divide by ã₀ to get ñ ₂ = ö₁ ñ₁ + ö₂ ñ₀ = ö₂ + ö₁² / (1 – ö₂) Cryer and Chan write this as ñ ₂ = [ ö₂ (1 – ö₂) + ö₁² ] / (1 – ö₂) Attachments TS Module 6 Stationary autoregressive processes more pps df.pdf (1.9K views, 44.00 KB) Edited 11 Years Ago by NEAS 0
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