TS Module 7 Stationary mixed processes
(The attached PDF file has better formatting.)
Time series practice problems residuals
*Question 7.1: Expected Value
An ARMA(2,4) process has moving average parameters θj = 0.5j for j = 1 to 4 and autoregressive parameters φk = 0.6k for k = 1 to 2. The parameter δ = θ0 = 0, and σ2 = 1. All values for t < 12 were the mean, and all residuals for t < 12 were zero.
If ε12 = 1, what is the expected value of (ε13 – ε14) × (ε13 + ε14)?
A. –0.5
B. –0.25
C. 0
D. +0.11
E. +0.42
Answer 7.1: C
(ε13 – ε14) × (ε13 + ε14) = ε132 – ε142
The expected value of the squared residual is constant.
*Question 7.2: Error Terms
A correctly specified ARMA(p,q) process has moving average parameters θj = 0.5j for j = 1 to 4 and autoregressive parameters φk = 0.366k for k = 1 to 2. The parameter δ = θ2 = 5, and σ2 = 1.
Let εj be the residual for the jth value. What is the expected value of ρ(ε1,ε2)?
A. –1/6
B. –1/12
C. 0
D. 1/12
E. 1/6
Answer 7.2: C
If the time series process is correctly specified, any serial correlation is eliminated by the moving average parameters.
The expected value of the correlation of the residuals depends on the serial correlation of the time series. If the time series is correctly specified, this value is zero. In practice, many time series do not eliminate all the serial correlation; many items can cause serial correlation.
Illustration: Sales figures are affected by inflation, population growth, changes in income, changes in taste, introduction of new products, and a host of other factors. Most of these are serially correlated, and they cause serial correlation in the time series of sales figures. We can not always eliminate the serial correlation with a moving average or autoregressive model.
*Question 7.3: Error Terms
εj is the residual for the jth value in an AR(1) process with φ = 0.5. If the expected variance of εj is 2 and the actual value of εj is 4, what is the expected value of εj+1?
A. 0
B. 1
C. 2
D. 3
E. 4
Answer 7.3: A
The expected value of the residual is zero, regardless of the past values, as long as the process is correctly specified.