Re: [Gretl-users] New plot for arma models
On Mon, 27 Mar 2017, Sven Schreiber wrote:
Am 27.03.2017 um 08:47 schrieb Riccardo (Jack) Lucchetti:
> On Mon, 27 Mar 2017, Sven Schreiber wrote:
>> When you say "sample periodogram", I guess you're using the
smoothed
>> periodogram? (As the raw periodogram isn't a consistent estimator.)
>
> I thought that the sample periodogram is better for the purpose I had in
> mind, which is to provide a basic visual "mis-specification test". Maybe
> I'm wrong, but the whole idea was to provide a sort-of
> "actual-vs-fitted" plot. If the model is correctly specified, you should
> see the spectrum nicely going through the periodogram points.
I think it depends on how you define "through".
Intuitively I'd say that if you are using the correct model (an estimated
version of the DGP), then asymptotically the parametrically estimated
spectrum and the non-parametrically estimated one (for example, using the
smoothed periodogram) will converge on each other.
Yeah, that's correct. But then, you'd be missing another nice feature of
this plot, that is its ability to spot overparametrisation graphically.
What is known as the "common factor" problem in arma estimation is
essentially, in the frequency domain, a problem of overfitting, with the
model trying to drive the spectrum through random peaks in the
periodogram.
Try this and you'll see what I mean:
<hansl>
nulldata 256
set seed 101010
setobs 1 1 --special-time-series
#true model = ma(1)
e = normal()
y = e + 0.5*e(-1)
# overparametrised model
arima 3 4 ; y --window
</hansl>
I don't think that the same result would hold for the raw sample
periodogram, because it is not a consistent estimator of the spectrum.
In that sense the periodogram ordinates are not "actuals" like
datapoints are, I think.
Well, they are. They're the squares of the moduli of the FFT of the
residuals (phew!), and the FFT is a linear, invertible transformation of
the actual data. The new plot, in spirit, is not very different from what
you'd get if you plotted the estimated variance from a GARCH model
through the squared residuals.
Graphically I would expect the sample periodogram to
"wiggle" around, and the
parametric spectrum estimate will be comparably smooth and go "through" the
periodogram in a common-sense way. I think that would basically mean that the
user is doing the smoothing of the periodogram herself mentally, in her
imagination.
Let's be creative then! :-)
(Smoothing is of course not the only way to obtain a consistent
estimator.)
Of course not. But in order to get a consistent estimator some work has to
be done anyway; and my point is: why should we, if consistency is not
needed?
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Riccardo (Jack) Lucchetti
Dipartimento di Scienze Economiche e Sociali (DiSES)
Università Politecnica delle Marche
(formerly known as Università di Ancona)
r.lucchetti(a)univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti
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