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

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

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

8. Byron Gangnes 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. Byron Gangnes Properties of AR(1) Model Key properties are (see book, p. 146-147): 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. Why is this last result important? Does it look familiar?

10. Byron Gangnes 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?