*
* Trend and difference models for Yen exchange rate, pp 348-360
*
cal(m) 1973
open data exch.dat
data(format=prn,org=columns) 1973:1 1996:7
*
set logyen = log(yen)
set dyen   = logyen-logyen{1}
graph(footer="Figure 12.7 Log Yen Exchange Rate",window="Figure 12.7")
# logyen
graph(footer="Figure 12.8 Change in Log Yen Exchange Rate",window="Figure 12.8")
# dyen
*
* The option DIFFS=1 causes BJIDENT to produce the autocorrelations for the
* series and its first difference. Examining these allows you to decide whether
* to difference the series or not. If you're interested in the possibility of
* seasonal differencing (item 3 in the Problems and Complements section on page
* 361), you can use the SDIFFS option as well. DIFFS=1,SDIFFS=1 will give you all
* combinations of 0 or 1 regular differences and 0 or 1 seasonal differences.
*
* METHOD=YULE uses the same method of calculation for the autocorrelations that
* is used in generating the graphs in the text. The default (METHOD=BURG)
* produces rather similar results for the first differenced series, but has quite
* different partial autocorrelation estimates for the undifferenced series. YULE
* works by estimating the autocorrelations directly, and the partial
* autocorrelations from them, while BURG estimates the partial autocorrelations
* directly, then "backs out" the autocorrelation estimates.
*
@bjident(number=12,diffs=1,method=yule) logyen
*
* The time trend is defined as t-1973:1 (which gives it a value of 0 in 1973:1)
* rather than just t, so the regressions will match up. This has no effect on the
* fit.
*
set time  = t-1973:1
*
* This does some fancy table building to do the AIC and BIC (SBC) for all
* combinations of ar and ma up to 3. This is similar to what the BJAutoFit
* procedure does, but you can't use that because of the addition of the trend
* variable.
*
* Since we're probably not all that interested in all the estimation output for
* 16 models (most of which will be rejected out of hand once we see the
* information criteria), the BOXJENK instruction in the loop uses the option
* NOPRINT. The information criteria are computed using the procedure REGCRITS.
* This also uses NOPRINT. All that are used are the values %aic and %sbc.
*
* Note that the information criteria are slightly different from those in the
* text for any model with MA terms. When there are MA terms, the residuals have
* to be generated recursively, and different programs use different ways to do
* that, yielding somewhat different estimates and different residuals.
*
dec rect aic(4,4) sic(4,4)
do mas=0,3
  do ars=0,3
     boxjenk(constant,ma=mas,ar=ars,regressors,maxl,noprint) logyen * 1994:12
     # time
     @regcrits(noprint)
     compute aic(ars+1,mas+1)=%aic
     compute sic(ars+1,mas+1)=%sbc
  end do ars
end do mas
*
* The headers are blank (which goes over the AR lag), 0, 1, 2, 3. The first
* column has 0, 1, 2 and 3. report(atrow=1,atcol=2) aic will put the 4x4 matrix
* in the four rows and four columns beginning at (1,2)
*
report(action=define,hlabels=||"","0","1","2","3"||)
report(atrow=1,atcol=1,fillby=columns) "0" "1" "2" "3"
report(atrow=1,atcol=2) aic
report(action=format,picture="*.###")
report(action=show,window="Table 12.1 AIC Values")
report(atrow=1,atcol=2) sic
report(action=format,picture="*.###")
report(action=show,window="Table 12.2 SIC Values")
*
* Pick the best model, estimate and forecast
*
disp "Chosen model for estimation in levels"
boxjenk(constant,ar=2,regressors,define=levelseq) logyen * 1994:12
# time
uforecast(equation=levelseq,stderrs=sefore) fyen 1995:1 1996:7
*
* Compute upper and lower 2 standard error bands
*
set upper = fyen+2.0*sefore
set lower = fyen-2.0*sefore
*
* Create the dummy variable for the forecast shading zone. Since we're going to
* need this to stretch out to 2010:12 later on, we'll take care of the whole
* range now.
*
set forezone * 2010:12 = t>=1995:1
graph(footer="Figure 12.11. Log Yen/Dollar Rate: History and Forecast",window="Figure 12.11",shading=forezone) 4
# logyen 1990:1 1994:12
# fyen 1995:1 1996:7
# upper 1995:1 1996:7 3
# lower 1995:1 1996:7 3
*
graph(footer="Figure 12.13. Log Yen/Dollar Rate: History, Forecast and Realization",window="Figure 12.13",shading=forezone) 4
# logyen 1990:1 1996:7
# fyen 1995:1 1996:7
# upper 1995:1 1996:7 3
# lower 1995:1 1996:7 3
*
* uForeErrors computes forecast error statistics on the comparison between the
* actual data (logyen) and the forecasts (fyen).
*
@uForeErrors logyen fyen
*
* Longer term forecasts. We need to extend the time trend variable out to the end
* of the new forecast sample.
*
set time * 2010:12 = (t-1973:1)
ufore(equation=levelseq,stderrs=sefore) fyen 1995:1 2010:12
set upper 1995:1 2010:12 = fyen+2.0*sefore
set lower 1995:1 2010:12 = fyen-2.0*sefore
graph(footer="Figure 12.12. Log Yen/Dollar Rate: History and Long-Horizon Forecast",window="Figure 12.12",shading=forezone) 4
# logyen 1990:1 1994:12
# fyen 1995:1 2010:12
# upper 1995:1 2010:12 3
# lower 1995:1 2010:12 3
*
* The Dickey-Fuller unit root test is done using the procedure DFUNIT. The main
* options for this are LAGS (number of added lags) and TREND, which puts a time
* trend into the regression.
*
@dfunit(lags=3,det=trend) logyen * 1994:12
*
* ARMA models for differenced series. Since we want forecasts of the
* undifferenced series, we use the DIFFS option on BOXJENK and use logyen as the
* dependent variable. The parameter estimates and any summary statistics based
* upon the residuals (such as the information criteria) will be identical to what
* you would get if you used the differenced series as the dependent variable and
* eliminated the DIFFS option. If you do the estimates both ways (and don't
* suppress the output), the one difference you will note will be with any summary
* statistics which use information about the dependent variable. With DIFFS=1 and
* the level variable as the dependent variable, the R**2 will use the sum of
* squared deviations of the level variable in the denominator, while if you use
* the differenced variable directly, it will be the sum of squared deviations of
* the differenced series. Since the former is generally MUCH higher than the
* latter, while the sum of squared residuals are the same, the R**2 will
* generally be quite high when the differencing is done by BOXJENK and much lower
* when you do it yourself. This doesn't mean that the former is "better" than the
* latter - the two models are the same information stated in different forms.
* It's just an important reminder that the R**2 isn't very useful in comparing
* estimates with different dependent variable.
*
* Again, we use some fancy programming to generate the table of information
* criteria. This time, we use the MEDIT instruction, which is specifically
* designed to display a single matrix of information.
*
do mas=0,3
  do ars=0,3
     boxjenk(constant,diffs=1,ma=mas,ar=ars,maxl,noprint) logyen 1973:5 1994:12
     @regcrits(noprint)
     compute aic(ars+1,mas+1)=%aic
     compute sic(ars+1,mas+1)=%sbc
  end do ars
end do mas
medit(hlabels=||"0","1","2","3"||,vlabels=||"0","1","2","3"||,picture="*.###",noedit) aic sic
*
disp "Chosen model for estimation in differences"
boxjenk(constant,diffs=1,ar=1,define=diffeq) logyen * 1994:12
ufore(equation=diffeq,stderrs=sefore) fyen 1995:1 1996:7
*
set upper 1995:1 1996:7 = fyen+2.0*sefore
set lower 1995:1 1996:7 = fyen-2.0*sefore
graph(footer="Figure 12.15. Log Yen/Dollar Rate: History and Forecast. Model in Differences",window="Figure 12.15",shading=forezone) 4
# logyen 1990:1 1994:12
# fyen 1995:1 1996:7
# upper 1995:1 1996:7 3
# lower 1995:1 1996:7 3
graph(footer="Figure 12.17. Log Yen/Dollar Rate: History, Forecast and Realization",window="Figure 12.17",shading=forezone) 4
# logyen 1990:1 1996:12
# fyen 1995:1 1996:7
# upper 1995:1 1996:7 3
# lower 1995:1 1996:7 3
@uForeErrors logyen fyen
*
* Longer horizon forecasts
*
ufore(equation=diffeq,stderrs=sefore) fyen 1995:1 2010:12
set upper 1995:1 2010:12 = fyen+2.0*sefore
set lower 1995:1 2010:12 = fyen-2.0*sefore
graph(footer="Figure 12.16. Log Yen/Dollar Rate: History and Long-Horizon Forecast",window="Figure 12.16",shading=forezone) 4
# logyen 1990:1 1994:12
# fyen 1995:1 2010:12
# upper 1995:1 2010:12 3
# lower 1995:1 2010:12 3