TS Module 7 Stationary mixed processes practice problems

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Time series practice problems residuals

*Question 7.1: Expected Value

An ARMA(2,4) process has moving average parameters è_j = 0.5^j for j = 1 to 4 and autoregressive parameters ö_k = 0.6^k for k = 1 to 2.  The parameter ä = è₀ = 0, and ó² = 1.  All values for t < 12 were the mean, and all residuals for t < 12 were zero.

If å₁₂ = 1, what is the expected value of (å₁₃ – å₁₄) × (å₁₃ + å₁₄)?

A.    –0.5

B.     –0.25

C.     0

D.    +0.11

E.     +0.42

Answer 7.1: C

(å₁₃ – å₁₄) × (å₁₃ + å₁₄) = å₁₃² – å₁₄²

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 è_j = 0.5^j for j = 1 to 4 and autoregressive parameters ö_k = 0.366^k for k = 1 to 2.  The parameter ä = è₂ = 5, and ó² = 1.

Let å_j be the residual for the j^th value.  What is the expected value of ñ(å₁,å₂)? 

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

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