Time Series Analysis Autocorrelation and Partial Autocorrelation 3

Question 1

When both the ACF and PACF of a stationary series show gradual decay, which model class is most likely appropriate?

Accepted Answer

ARMA(p,q) with both p>0 and q>0

Answer

When both the ACF and PACF tail off gradually without cutting off, an ARMA model with both AR and MA components is typically indicated.

Question 2

The sample autocorrelation at lag k is computed as the ratio of:

Accepted Answer

The lag-k autocovariance to the lag-0 autocovariance (variance)

Answer

The sample autocorrelation at lag k is the lag-k autocovariance divided by the lag-0 autocovariance (i.e., the variance), normalizing it to lie in [-1, 1].

Question 3

For which of the following series would you expect the ACF to decay very slowly, remaining significantly positive for many lags?

Accepted Answer

A non-stationary series with a unit root

Answer

A unit root (non-stationary) series has autocorrelations that decay extremely slowly, often remaining close to 1 for many lags.

Question 4

In a seasonal ARIMA model, the seasonal ACF and PACF are examined at multiples of the seasonal period S. If S=12 and there is a spike only at lag 12 in the PACF (with the seasonal ACF tailing off), what seasonal component is suggested?

Accepted Answer

Seasonal AR(1)

Answer

A single spike at lag 12 in the seasonal PACF with a tailing-off seasonal ACF is the hallmark of a seasonal AR(1) component.

Question 5

The Yule-Walker equations are used to:

Accepted Answer

Relate the theoretical PACF of an AR(p) process to its parameters

Answer

The Yule-Walker equations express the autocovariances (and thus ACF) of an AR(p) process as a linear function of the AR parameters, enabling parameter estimation.

Time Series Analysis Practice Test

Time Series Analysis Practice Test

Time Series Analysis Autocorrelation and Partial Autocorrelation 3