ACF and PACF of ARMA processes

Question 1

How is Autocovariance function denoted for an ARMA process?

Answer 1

ACVF

Question 2

How is the Autocorrelation function denoted for an ARMA process?

Answer 2

ACF

Question 3

Why do we only ever display the positive values of h in the ACVF function

Answer 3

As the covariance of minus h is the same as plus h

Question 4

What is significant about the MA process covariance and ACF

Answer 4

The ACVF/ACF of the MA(q) vanishes at lags greater than q so The ACF plot can be used to identify the order of a MA process
in practice.

Question 5

Why do we use the PACF

Answer 5

The partial ACF is a tool to mimic the behaviour of the ACF of
MA models for AR models - it will cut off for lags greater than p

Question 6

Describe the PACF in words

Answer 6

By stationarity, the PACF phi hh is the correlation between X t and X t−h with the linear dependence of {X h−1 , X h−2 , . . . , X 1 } removed on each

Question 7

How do we find the coefficients beta in the PACF formula

Answer 7

The linear combination minimises a certain expectation

Question 8

Why is no intercept included in the regression of X h on {X h−1 , X h−2 , . . . , X 1 }

Answer 8

The process has a zero mean

Question 9

Describe in words what Xh ^h-1 is

Answer 9

regression of X h on {X h−1 , X h−2 , . . . , X 1 }