Econ 427 lecture 12 slides

1. Econ 427 lecture 12 slides MA (part 2) and Autoregressive Models

2. Moving Average (MA) models • Last time we looked at moving average models. • MA(1) • Past shocks (innovations) in the series feed into the succeeding period.

3. Properties of an MA(1) series • An MA(1) has a “short memory”—only last period’s shock matters for today • We saw this in the shape of the autocorrelation function: • There is one signif bar in the autocor graph

4. MA(q) series • Higher order MA processes involve additional lags of white noise: • What does the autocorrelation function for an MA(q) look like?

5. Autoregressive Models • Relates the current value of a series to its own past lags. An AR(1) is: How would I write that in lag operator form? We would like to know what its time-series properties are. How can we figure that out?

6. Properties of AR(1) Model • Transform it into an expression involving lags of epsilon by “backward substitution”:

7. Properties of AR(1) Model • So we can write • Or: As long as • The last step comes from the fact that the summation is a geometric series. See http://mathworld.wolfram.com/GeometricSeries.html

8. Autoregressive Models • Notice that you can use algebra on the original AR(1) expression in lag operator form to get this same result In lag operator form: Divide both sides by the expression in parentheses (the lag polynomial)

9. Properties of AR(1) Model • Key properties are (see book, p. 146-147): • Why is this last result important? Does it look familiar? • The variance will only be finite if |phi| < 1. Covariance stationarity requires this. Intuition, if phi = 1, the series can wander infinitely far away from its starting point, since any shock is permanent.

10. AR(p) series • Higher order AR processes involve additional lags of y: • What do the autocorrelation and partial autocorrelation functions for an AR(p) look like?