TS Module 7 Stationary mixed processes

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Time series practice problems residuals

*Question 7.1: Expected Value

An ARMA(2,4) process has moving average parameters θ_j = 0.5^j for j = 1 to 4 and autoregressive parameters φ_k = 0.6^k 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 θ_j = 0.5^j for j = 1 to 4 and autoregressive parameters φ_k = 0.366^k for k = 1 to 2.  The parameter δ = θ₂ = 5, and σ² = 1.

Let ε_j be the residual for the j^th 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

ε_j is the residual for the j^th value in an AR(1) process with φ = 0.5.  If the expected variance of ε_j is 2 and the actual value of ε_j is 4, what is the expected value of ε_j+1?