TS Module 7 Stationary mixed processes

(The attached PDF file has better formatting.)

Time series ARMA processes practice problems

*Question 7.1: ARMA(1,1)

An ARMA(1,1) process has ñ₁ = 0. Which of the following is true?

A.   ö₁ = è₁ or ö₁ × è₁ = 1

B.   ö₁ = è₁ and ö₁ × è₁ = 1

C.   ö₁ = –è₁ or ö₁ × è₁ = –1

D.   ö₁ = –è₁ and ö₁ × è₁ = –1

E.   ö₁ = è₁ and ö₁ × è₁ = –1

Answer 7.1: A

For an ARMA(1,1) process:

 for k ≥ 1

Intuition: If the residual in period t increases one unit:

        Moving average: è₁ causes the forecast to decrease è₁ units

        Autoregressive: ö₁ causes the forecast to increase ö₁ units

If ö₁ = è₁, these two effects offset each other. A change in the period t value does not affect the period t+1 value, so ñ₁ = 0.

è₁ and 1/è₁ produce the same autocorrelation function.

ö₁ = è₁ has the same effect as ö₁ = 1/è₁ or ö₁ × è₁ = 1.

*Question 7.2: Exponential decay

For which of the following processes do the autocorrelations have exponential decay?

A.   White noise

B.   MA(1)

C.   MA(2)

D.   ARMA(1,1)

E.   ARIMA(0,1,1)

Answer 7.2: D

See Cryer and Chan page 78: The ARMA(1,1) autocorrelation function decays exponentially as the lag k increases. The damping factor is ö, but the decay starts from initial value ñ₁, which also depends on è. This is in contrast to AR(1) autocorrelation, which also decays with damping factor ö but always from initial value ñ_o = 1.

*Question 7.3: Exponential decay

The autocorrelation function of an ARMA(1,1) process has exponential decay with a damping factor of 0.4. Which of the following is true?

A.   ö₁ – è₁ = 0.4

B.   ö₁ × è₁ = 0.4

C.   ö₁ × è₁ = 2.5

D.   ö₁ = 0.4

E.   è₁ = –0.4

Answer 7.3: D

*Question 7.4: Moving average and autoregressive processes

The moving average process Y_t = e_t + 0.5 e_t-1 + 0.5² e_t-2 + 0.5³ e_t-3 + … is equivalent to what autoregressive process?