R help - Aug 2022 - Fitted values from AR model

The model that you are fitting to the data is an AR(2) model, which means

y(t) = a0 + a1 * y(t-1) + a2 * y(t-2) + eps(t)                 (1)

The fitting procedure estimates the coefficients a0, a1, a2 (and the
variance of eps).

After the coefficients have been estimated, the fitted values can be
calculated using equation (1) (setting eps(t) = 0)
using
fitted(t) = a0 + a1 * y(t-1) + a2 * y(t-2)

Assuming the series is given for t=1,2,...,20, there is no problem to
apply equation (1) to get the
fitted values for t=3,4,...,20. But there is a problem for t=1 and
t=2. For example, for t=1, equation (1)
says

fitted(1) = a0 + a1 * y(0) + a2 * y(-1)

But there are no values given for y(0) and y(-1). So, either no fitted
values should be given for t=1,2, or some
other method is being used. Apparently, the arima functions in R and
python use different methodology
to generate these two fitted points. (For all the other values, the
fits are extremely close.)

HTH,
Eric

On Thu, Aug 11, 2022 at 9:53 PM bogus christofer
<bogus.christofer at gmail.com> wrote:
>
> Hi,
>
> I have below AR model and fitted values from the forecast package
>
> library(forecast)
> dta = c(5.0, 11, 16, 23, 36, 58, 29, 20, 10, 8, 3, 0, 0, 2, 11, 27, 47, 63,
> 60, 39)
> fit <- arima(dta, order=c(2,0,0))
> fitted(fit)
>
> This gives fitted values as
>
> Time Series:
> Start = 1
> End = 20
> Frequency = 1
>  [1] 13.461017  9.073427 18.022166 20.689420 26.352282 38.165635 57.502926
> 9.812106 15.335303  8.298995 11.543320  6.606999  5.800820  7.502621
> 9.930962 19.723966 34.045298 49.252447 57.333846 44.615067
>
>
> However when I compare this result with Python, I see significant
> difference particularly in the first few values as below
>
> from statsmodels.tsa.arima.model import ARIMA
> dta = [5.0, 11, 16, 23, 36, 58, 29, 20, 10, 8, 3, 0, 0, 2, 11, 27, 47, 63,
> 60, 39]
> fit = ARIMA(dta, order=(2, 0, 0)).fit()
> fit.predict()
>
> array([21.24816788, 8.66048306, 18.02197059, 20.68931006,
> 26.35225759,38.16574655, 57.503278 , 9.81253693, 15.33539514,
> 8.29894655,11.54316056, 6.60679489, 5.80055038, 7.50232004,
> 9.93067155,19.72374025, 34.04524337, 49.25265365, 57.3343347 , 44.6157026
])
>
> Any idea why there are such difference between R and Python results will be
> very helpful.
>
> Thanks,
>
>         [[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

Thanks Eric.

Is it possible to know what R uses as the initial values? Are these
configurable?

On Fri, 12 Aug 2022 at 13:44, Eric Berger <ericjberger at gmail.com>
wrote:
> The model that you are fitting to the data is an AR(2) model, which means
>
> y(t) = a0 + a1 * y(t-1) + a2 * y(t-2) + eps(t)                 (1)
>
> The fitting procedure estimates the coefficients a0, a1, a2 (and the
> variance of eps).
>
> After the coefficients have been estimated, the fitted values can be
> calculated using equation (1) (setting eps(t) = 0)
> using
> fitted(t) = a0 + a1 * y(t-1) + a2 * y(t-2)
>
> Assuming the series is given for t=1,2,...,20, there is no problem to
> apply equation (1) to get the
> fitted values for t=3,4,...,20. But there is a problem for t=1 and
> t=2. For example, for t=1, equation (1)
> says
>
> fitted(1) = a0 + a1 * y(0) + a2 * y(-1)
>
> But there are no values given for y(0) and y(-1). So, either no fitted
> values should be given for t=1,2, or some
> other method is being used. Apparently, the arima functions in R and
> python use different methodology
> to generate these two fitted points. (For all the other values, the
> fits are extremely close.)
>
> HTH,
> Eric
>
> On Thu, Aug 11, 2022 at 9:53 PM bogus christofer
> <bogus.christofer at gmail.com> wrote:
> >
> > Hi,
> >
> > I have below AR model and fitted values from the forecast package
> >
> > library(forecast)
> > dta = c(5.0, 11, 16, 23, 36, 58, 29, 20, 10, 8, 3, 0, 0, 2, 11, 27,
47,
> 63,
> > 60, 39)
> > fit <- arima(dta, order=c(2,0,0))
> > fitted(fit)
> >
> > This gives fitted values as
> >
> > Time Series:
> > Start = 1
> > End = 20
> > Frequency = 1
> >  [1] 13.461017  9.073427 18.022166 20.689420 26.352282 38.165635
> 57.502926
> > 9.812106 15.335303  8.298995 11.543320  6.606999  5.800820  7.502621
> > 9.930962 19.723966 34.045298 49.252447 57.333846 44.615067
> >
> >
> > However when I compare this result with Python, I see significant
> > difference particularly in the first few values as below
> >
> > from statsmodels.tsa.arima.model import ARIMA
> > dta = [5.0, 11, 16, 23, 36, 58, 29, 20, 10, 8, 3, 0, 0, 2, 11, 27, 47,
> 63,
> > 60, 39]
> > fit = ARIMA(dta, order=(2, 0, 0)).fit()
> > fit.predict()
> >
> > array([21.24816788, 8.66048306, 18.02197059, 20.68931006,
> > 26.35225759,38.16574655, 57.503278 , 9.81253693, 15.33539514,
> > 8.29894655,11.54316056, 6.60679489, 5.80055038, 7.50232004,
> > 9.93067155,19.72374025, 34.04524337, 49.25265365, 57.3343347 ,
> 44.6157026 ])
> >
> > Any idea why there are such difference between R and Python results
will
> be
> > very helpful.
> >
> > Thanks,
> >
> >         [[alternative HTML version deleted]]
> >
> > ______________________________________________
> > R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> > and provide commented, minimal, self-contained, reproducible code.
>
	[[alternative HTML version deleted]]