Learning outcomes: Describe Wold’s theorem. Define a general linear process. Relate rational distributed lags to Wold’s theorem
Questions:
508.1. Wold's representation theorem points to an appropriate model for a covariance stationary residual such that:
a. Any autoregressive process of (p) order can be expressed as a rational polynomial of lagged errors
b. Any purely nondeterministic covariance-stationary process is a linear regression of y(t) on a lagged conditional mean
c. Any purely nondeterministic covariance-stationary process is some linear combination of lagged values of a white noise process
d. Any autoregressive moving average model, ARMA(p,q), can be shown as the sum of autoregressive (AR) and moving average (MA) processes
508.2. Wold’s theorem tells us that when formulating forecasting models for covariance stationary time series we need only consider models according to the the general linear process. “General” refers to the fact that any covariance stationary series can be captured by the process. “Linear” reflects the fact that the Wold representation expresses the series as a linear function of its innovations. Under the general linear process, each of the following is true EXCEPT which is false?
a. Unconditional mean is constant
b. Unconditional variance is constant
c. Conditional mean moves over time in response to the information set
d. Conditional variance moves over time in response to the information set
508.3. In regard to rational distributed lag models, each of the following is true EXCEPT which is false?
a. It is possible to approximate the Wold representation using a rational distributed lag
b. ARMA and ARIMA forecasting models are simply rational approximations to the Wold representation
c. Rational distributed lags produce models that are parsimonious yet provide accurate approximations to the Wold representation
d. ARMA(1,1) is not practical for two reasons: it cannot be covariance stationary; and its unconditional mean is time-varying while its conditional mean is fixed
Answers here:
Questions:
508.1. Wold's representation theorem points to an appropriate model for a covariance stationary residual such that:
a. Any autoregressive process of (p) order can be expressed as a rational polynomial of lagged errors
b. Any purely nondeterministic covariance-stationary process is a linear regression of y(t) on a lagged conditional mean
c. Any purely nondeterministic covariance-stationary process is some linear combination of lagged values of a white noise process
d. Any autoregressive moving average model, ARMA(p,q), can be shown as the sum of autoregressive (AR) and moving average (MA) processes
508.2. Wold’s theorem tells us that when formulating forecasting models for covariance stationary time series we need only consider models according to the the general linear process. “General” refers to the fact that any covariance stationary series can be captured by the process. “Linear” reflects the fact that the Wold representation expresses the series as a linear function of its innovations. Under the general linear process, each of the following is true EXCEPT which is false?
a. Unconditional mean is constant
b. Unconditional variance is constant
c. Conditional mean moves over time in response to the information set
d. Conditional variance moves over time in response to the information set
508.3. In regard to rational distributed lag models, each of the following is true EXCEPT which is false?
a. It is possible to approximate the Wold representation using a rational distributed lag
b. ARMA and ARIMA forecasting models are simply rational approximations to the Wold representation
c. Rational distributed lags produce models that are parsimonious yet provide accurate approximations to the Wold representation
d. ARMA(1,1) is not practical for two reasons: it cannot be covariance stationary; and its unconditional mean is time-varying while its conditional mean is fixed
Answers here:
