Our basic model was:
We add an autoregressive component by adding a lagged observation.
AR(
A shock bumps up the output variable, which bumps up output variables forever, at a decreasing rate.
The Dickey-Fuller test tests if there is a unit root.
The AR(
We can rewrite this as:
We test if
If the coefficient on the last term is
If the last term is
If our model has no intercept it is:
If our model has a time trend it is:
We include more lagged variables.
If no unit root, can do normal OLS?
The standard AR(
The variance is:
Assuming the errors are IID we have:
This is independent of historic observations, which may not be desirable.
Consider the alternative formulation:
This allows for conditional heteroskedasticity.
We add previous error terms as input variables
MA(
Unlike AR models, the effects of any shocks wear off after
This is harder to fit the OLS, the error terms themselves are not observed.
We include both AR and MA
Estimted using Box-Jenkins
Uses differences to remove non statiority
Also estiamted with box-jenkins