Multistep-Ahead Forecasts

[ac_1]

Hi Tom,

A few questions regarding multistep-ahead forecasts.

Convergence of multistep-ahead forecasts
Modelling a stationary variable with an AR(1) the OOS multistep-ahead forecasts generally converge 'horizontally' upon the long-term mean of the model as the forecast horizon increases, i.e. as h tends to infinity, the forecasts become the unconditional mean of the process.

For example:
An AR(1) for a stationary variable via LINREG in LEVELS,
CONSTANT = 0.1074520022, AR(1) = 0.6417228994, long-term mean = 0.1074520022/(1-0.6417228994) = 0.2999131176

And an AR(1) for a stationary variable via BOXJENK in LEVELS with DIFFS=0,
CONSTANT = 0.2999131177, AR(1) = 0.6417228994

Thus, the multistep-ahead forecasts tend towards 0.299913118. Indeed 0.28's-0.29's are approximately the values of the forecasts I get at longer horizons.
If there's no CONSTANT in the specification the multistep-ahead forecasts head towards zero.


Modelling non-stationary variables frequently result in 'non-horizontal' multistep-ahead forecasts.

How do I determine the long-term mean of multistep-ahead forecasts of a non-stationary variable?

Let's say I model the LOG-OF-THE-LEVEL with an AR(1) via BOXJENK, accounting for the non-stationarity with DIFFS=1, and get the following,
CONSTANT = 0.0039181489, AR{1} = 0.0924907759

What long-term average value do the multistep-ahead forecasts converge to, for a specification with and without a CONSTANT?


Prediction intervals and standard errors
How best to depict prediction intervals? Fan charts are typically displayed with a forecast and prediction intervals with high probabilities e.g. 0.80 to 0.95.
Displaying prediction intervals with low probabilities i.e. less than 50% intervals initially doesn't make much sense to a layman as that says there is less than a 50/50 chance of the actual being inside the prediction interval, so why forecast? However, a forecast is a conditional mean which has a standard error (SE). Is it worth displaying prediction intervals less than 50%, or should they begin from 50% and higher?


thanks,
Amarjit

[TomDoan]

<<snip>>
Modelling non-stationary variables frequently result in 'non-horizontal' multistep-ahead forecasts.

How do I determine the long-term mean of multistep-ahead forecasts of a non-stationary variable?

Let's say I model the LOG-OF-THE-LEVEL with an AR(1) via BOXJENK, accounting for the non-stationarity with DIFFS=1, and get the following,
CONSTANT = 0.0039181489, AR{1} = 0.0924907759

What long-term average value do the multistep-ahead forecasts converge to, for a specification with and without a CONSTANT?

It has no long-term average value. A simple RW would be forecast at the last observed level. A RW with drift would be forecast as a linear trend starting off the last observed value, with a trend rate equal to the drift. If you start putting in stationary components in addition, a non-drifting process will eventually converge on some level and a drifting process on a linear trend, but not based solely on the last observed value.

Prediction intervals and standard errors
How best to depict prediction intervals? Fan charts are typically displayed with a forecast and prediction intervals with high probabilities e.g. 0.80 to 0.95.
Displaying prediction intervals with low probabilities i.e. less than 50% intervals initially doesn't make much sense to a layman as that says there is less than a 50/50 chance of the actual being inside the prediction interval, so why forecast? However, a forecast is a conditional mean which has a standard error (SE). Is it worth displaying prediction intervals less than 50%, or should they begin from 50% and higher?

Fan charts often start as low as 20% (10 each way).

[ac_1]

Many Thanks!