fan chart levels forecasts from returns forecasts

[ac_1]

Hi Tom,

I have calculated log returns from asset price levels and generated n-step ahead forecasts for the log returns. I have then calculated n-step ahead levels forecasts via:
levels_forecasts = exp(returns_forecasts) * price_level(fixed at end of estimation of regression date).

To create a fan chart for the returns_forecasts, the 95% prediction intervals are:

Code: Select all

set lower95 beginf endf = returns_forecasts+%invnormal(.025)*stderrs
set upper95 beginf endf = returns_forecasts+%invnormal(.975)*stderrs


Do I just exp(stderrs) & have levels_forecasts instead to calculate 95% prediction intervals for the levels_forecasts?

Amarjit

[TomDoan]

Sorry; in what form are you computing the standard errors? If they are the standard errors in the n step forecasts for the log returns, then you need

exp(forecast+x stderr)*price to get an estimate of the x standard error percentile.

[ac_1]

I'm puzzled. Attached is an example, g10xrate.xls with 3 graphs:
(1) LOG-RETURNS: forecasts and Prediction Intervals - the forecasts are correct, the prediction intervals are calculated using the formula from mwh3p366.rpf.
(2) LEVELS: forecasts - forecasts are correct and calculated from LOG-RETURNS.
(3) LEVELS: forecasts and Prediction Intervals - I am unclear as to the formula for the prediction intervals.

Code: Select all

*===============================
* read in data
OPEN DATA "/Users/Shared/RATS/Examples/g10xrate.xls"
DATA(FORMAT=XLS,NOLABELS,ORG=COLUMNS,TOP=2) 1 6237 USXJPN USXFRA USXSUI USXNLD USXUK $
USXBEL USXGER USXSWE USXCAN USXITA
*
* transform data into log form
log USXJPN / lUSXJPN
*
* compute diff-log, approx. % percentage change
set dlUSXJPN = 100.0*( lUSXJPN - lUSXJPN{1} )
*
*
*===============================
linreg(define=fxeq) dlUSXJPN 6000 6237
# constant dlUSXJPN{1 to 10}
uforecast(equation=fxeq,stderrs=stderrs) dl_fxfore 6238 6248
*
*
* (1) graph LOG-RETURNS: forecasts and Prediction Intervals
*
set lower95 6238 6248 = dl_fxfore+%invnormal(.025)*stderrs
set upper95 6238 6248 = dl_fxfore+%invnormal(.975)*stderrs
prin 6200 6248 dlUSXJPN dl_fxfore stderrs lower95 upper95
*
graph(footer="LOG-RETURNS: forecasts and Prediction Intervals",ovcount=2,overlay=fan,ovsame) 4
# dlUSXJPN 6200 6237
# dl_fxfore 6238 6248
# lower95 6238 6248
# upper95 6238 6248
*
*
* (2) graph LEVELS: forecasts
*
set level_fxfore 6238 6248 = exp(dl_fxfore)*USXJPN(6237)
prin 6200 6248 USXJPN dl_fxfore level_fxfore
*
graph(footer="LEVELS: forecasts") 2
# USXJPN 6000 6237
# level_fxfore 6238 6248
*
*
* (3) graph LEVELS: forecasts and Prediction Intervals
*
set level_lower95 6238 6259 = exp(level_fxfore+%invnormal(.025)*(stderrs))*USXJPN(6237)
set level_upper95 6238 6259 = exp(level_fxfore+%invnormal(.975)*(stderrs))*USXJPN(6237)
prin 6200 6248 USXJPN dl_fxfore level_fxfore level_lower95 level_upper95
*
graph(footer="LEVELS: forecasts and Prediction Intervals",ovcount=2,overlay=fan,ovsame) 4
# USXJPN 6200 6237
# level_fxfore 6238 6248
# level_lower95 6238 6248
# level_upper95 6238 6248


[TomDoan]

The MWH example is doing a completely different forecast---the differences and seasonal differences are automatically incorporated into the forecasting formulas by BOXJENK.

On the other hand, you are estimating a model in log returns and your UFORECAST is forecasting log returns. A forecast of the log return from t+8 to t+9 combined with the price at t is not going to give you a forecast of the price at t+9 since log(price(t+9))-log(price(t)) is the sum of all the intermediate steps, not just the final period's return. If your goal is to forecast log price, you would be better off letting BOXJENK take care of the differencing itself, the way that is done in the MWH example (except you wouldn't need the seasonal differences)---the model you are using is an ARIMA(10,1,0) model on log price.

[ac_1]

Thanks for the reply.

My goal is to forecast PRICE. The methodology I have used is to model LOG-RETURNS, generate forecasts for LOG-RETURNS with prediction intervals; then convert LOG-RETURNS forecasts to PRICE forecasts and calculate prediction intervals.

As per MWH example is the following code correct for LOG-PRICE with prediction intervals?

Code: Select all

*===============================
boxjenk(ar=10,diffs=1,define=fxeq) lUSXJPN
uforecast(equation=fxeq,stderrs=stderrs) l_fxfore 6238 6248
*
*
* graph LOG-PRICE: forecasts and Prediction Intervals
*
set lower95 6238 6248 = l_fxfore+%invnormal(.025)*stderrs
set upper95 6238 6248 = l_fxfore+%invnormal(.975)*stderrs
prin 6200 6248 USXJPN l_fxfore stderrs lower95 upper95
*
graph(footer="LOG-PRICE: forecasts and Prediction Intervals",ovcount=2,overlay=fan,ovsame) 4
# lUSXJPN 6200 6237
# l_fxfore 6238 6248
# lower95 6238 6248
# upper95 6238 6248
*

[TomDoan]

That looks correct.