Exercise on impulse response functions for simple AR and MA error
processes (but not so simple that I did not screw up in class)
This is to figure out how impulse response functions differ for AR and
MA error processes
It is due 9/24
There will be more exercises
The MA1 error model with exogenous x and one lag of x may be denoted:
y_t = b x_t + d x_{t-1) + \nu_t + \mu \nu_{t-1}
whereas the corresponding model with AR1 error is
y_t = b x_t + d x_{t-1} + \epsilon_t
\epsilon_t = \phi \epsilon_{t-1} + \nu_t
Note that we always use \nu for an iid process, the \epsilon are not
iid.
Note that for the MA1 error process the stochastic difference equation
is already solved, since the only stuff on the RHS are forcing
variables.
Thus for the MA1 process you should convince yourselves that a change
in \nu only has an effect for two periods, same with a change in x.
Why is it different for the AR1 process. Obviously if x changes, y
only changes for two periods. What about \epsilon. This is not
obvious, but it becomes obvious when we write (1-\phi L)\epsilon = \nu
and then solve for \epsilon.
So convince yourselves that MA and AR error processes are
non-trivially different.
The mistake I made in class was talking about an AR process but
writing an MA process. Note that AR processes are used much more
often, but MA processes are the natural ones in terms of notation.