TS Module 6 Stationary autoregressive processes

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TS Module 6 Stationary autoregressive processes

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By NEAS - 8/31/2013 1:23:23 PM

TS Module 6 Stationary autoregressive processes

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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

₁ + ö₂ < 1

₂ – ö₁ < 1

ö₂| < 1

(See Cryer and Chan page 72, equation 4.3.11)

Part B:

ö₁ is given, the conditions are

₂ < 1 – ö₁

₂ < 1 + ö₁

ö₂| < 1

Illustration:

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?

(x) = 1 + ö₁(x) + ö₂(x²)

(x) = 1 + ö₂(x) + ö₁(x²)

(x) = 1 – ö₁(x) – ö₂(x²)

(x) = 1 – ö₂(x) + ö₁(x²)

(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

₁ = ö₁ ã₀ + ö₂ ã₁

Divide this equation by

ã₀ to get

₁ = ö₁ ñ₀ + ö₂ ñ₁

_o is the correlation of a random variable with itself, which is always 1, so we derive that

_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

₂ = ö₁ ã₁ + ö₂ ã₀

Divide by

ã₀ to get

₂ = ö₁ ñ₁ + ö₂ ñ₀ = ö₂ + ö₁² / (1 – ö₂)

Cryer and Chan write this as

₂ = [ ö₂ (1 – ö₂) + ö₁² ] / (1 – ö₂)