Autoregressive Time-Series Models: Structure, Stationarity, Estimation, and Forecasting

All flashcards

What is an autoregressive model?

A time-series model in which the current value depends linearly on previous values of the same series, plus a random innovation.

What does AR(p) mean?

An autoregressive model of order $p$, using up to $p$ lagged values as predictors.

What is a white-noise innovation?

A random input term, commonly with zero mean and constant variance, representing new variation not explained by the model’s past values.

What does the backshift operator do?

It shifts a time series backward: $BX_t=X_{t-1}$, so $B^iX_t=X_{t-i}$.

How do AR, ARMA, ARIMA, and VAR relate?

AR is a basic model family; ARMA adds moving-average terms, ARIMA incorporates differencing, and VAR extends autoregression to multiple interrelated series.

Quiz questions

1. Under the usual assumptions, when is a zero-mean AR(1) process weak-sense stationary?Answer: When $|\phi|<1$.
2. In an AR(1) process, what is the contribution of a one-time innovation $\varepsilon_1$ to the value $k$ steps later?Answer: $\phi^k\varepsilon_1$.
3. For an AR($p$) model using $\Phi(z)=1-\sum_{i=1}^{p}\phi_i z^i$, what root condition gives weak-sense stationarity?Answer: Every root of $\Phi(z)=0$ lies outside the unit circle.
4. For a true AR($p$) process, what happens to the population partial autocorrelation at lags greater than $p$?Answer: It is zero at every lag greater than $p$.
5. In ordinary least squares estimation of an autoregressive model, what criterion determines the coefficient estimates?Answer: Minimizing squared prediction errors using lagged observations.