10.8  USING PRIOR INFORMATION


Prior information can be very useful in making forecasts.  
Fildes and Stevens (1978) show that in a situation where 
there is little or no data prior information can be 
important.  Section 10.8 of this paper will examine the 
effects of feeding prior information into the estimation 
process.  The sensitivity of the results will be examined 
by feeding in different priors.

The Kalman filter can be used to incorporate prior 
information into the estimation process.  This is done 
very easily.  In all the estimations done so far, the 
initial values of the mean and variance of the parameters 
were set to 0 and 100;  this corresponds to great prior 
uncertainty.  In this part of the paper, the prior mean 
and variance for the income, wool price and synthetic 
fibre price elasticity will be set to values that 
represent the introduction of prior information.

In all of the runs done in part 10.8, V was set to values 
for final and preliminary data.


10.8.1  THE PRIOR INFORMATION, CASE 1


A reasonable a priori value for income (i.e. consumer 
expenditure) elasticity would be 1.0;  obviously the 
average expenditure elasticity (averaged over all 
consumer goods and services) must be 1.0, and there is no 
strong prior reason why wool or total fibres should be 
very different from this average.  In addition, various 
work done in the area of fibre demand (reviewed in 
Newland, 1979) points to an elasticity of around 1.



A reasonable a priori value for the wool price elasticity 
is -0.5;  various other values (between, say, -0.1 and 
-1.0) might also be acceptable.  It also seems plausible 
that the cross elasticity with respect to synthetic fibre 
price is equal in magnitude (but opposite in sign) to the 
wool price elasticity;  i.e. 0.5.  Thus the prior mean 
for the three elasticities will be set to 1.0, -0.5, 0.5.

We specify the strength of this prior information by 
specifying the variance of the prior estimates.  It is 
easier, however, to think in terms of the standard 
deviation.  A standard deviation of 0.2 is large enough 
to give sufficient leeway for the Kalman filter to be 
able to modify the parameter estimates, yet not so large 
that the prior information will immediately be swamped by 
the data.  This means that the prior variance = 0.04.

In a real life situation, a consideration of the source 
and quality of the prior information would give a figure 
for the prior variance.




10.8.1.1  PRIOR DATA CASE 1, W = 10-3


The runs of 10.5.2 were repeated with the prior 
information of 10.8.1.  The results are given below.



Table 60

              C            D             E           DSSE    

Belgium   1.041 (.24)   -.247 (.12)   .507 (.21)    .0188
France    1.057 (.23)   -.307 (.12)   .469 (.20)    .0088
Germany   1.029 (.23)   -.292 (.12)   .421 (.20)    .0133
Holland   1.058 (.23)   -.228 (.13)   .343 (.21)    .0750
Italy      .824 (.16)   -.419 (.18)   .465 (.22)    .0693
Japan     1.040 (.17)   -.395 (.18)   .449 (.22)    .1143
UK        1.056 (.21)   -.236 (.08)   .408 (.15)    .0662
USA       1.023 (.22)   -.543 (.10)   .637 (.15)    .1991

AVERAGE   1.016         -.333         .462          .0706


The DSSE is 9% worse than the DSSE of 10.5.2, and the 
parameter estimates are very different.  Where the 
standard errors of the estimates of 10.5.2 are small 
(e.g. the UK wool price elasticity), the estimates of 
10.8.1.1 are very close to those of 10.5.2.  This is 
because a small standard error implies lots of relevant 
information in the data, and so the prior is largely 
ignored.  In the case of the UK wool price elasticity, 
the 10.5.2 estimate is -0.206 and the 10.8.1.1 estimate 
is -0.236.

When the standard errors of the estimates of 10.5.2 are 


large (for example, UK income elasticity), the estimates 
of 10.8.1.1 are close to the prior information;  the 
reason for this is exactly the same as above, but in 
reverse.


10.8.1.2  PRIOR DATA CASE 1, W = 10-2


The estimations of 10.8.1.1 were repeated, but with W = 
10-2.  This would give lower weight to older information, 
and so increase the speed at which the prior information 
became irrelevant.  In 10.5.1 above, W = 10-2 was tried 
without prior information (great prior uncertainty).  The 
results included a lot of wrongly signed parameters.




Table 61

              C            D             E           DSSE    

Belgium   1.103 (.34)   -.324 (.26)   .440 (.42)    .0219
France    1.078 (.28)   -.399 (.27)   .423 (.41)    .0098
Germany   1.031 (.27)   -.331 (.27)   .440 (.42)    .0200
Holland   1.091 (.27)   -.265 (.27)   .344 (.42)    .0858
Italy      .969 (.39)   -.483 (.34)   .468 (.39)    .0789
Japan     1.232 (.44)   -.486 (.34)   .449 (.39)    .1352
UK        1.053 (.24)   -.226 (.22)   .474 (.38)    .0584
USA       1.039 (.25)   -.497 (.23)   .546 (.39)    .1578

AVERAGE   1.074         -.376         .448          .0710


Many of the countries' DSSE have got worse.  But the 


USA's, DSSE is very good.  Although the USA elasticities 
are very close to the prior elasticities, this is not 
because the prior information is too strong.  For 
example, the estimate of the synthetic fibre price 
elasticity rises to 0.786 in 1974, before falling back to 
0.546.  So the improvement in the USA must come from the 
following effect.

In the 1960's, wool prices, income and synthetic fibre 
prices were all highly multicollinear (see Solomon, 
1980), so the elasticity estimates are imprecise.  This 
does not affect forecasting, however, as long as these 
variables remain collinear.  In the 1970's, this 
collinearity ended as wool prices rose very fast indeed, 
then fell.  If the parameter estimates are imprecise just 
before then, the forecast will be highly inaccurate and 
make a large contribution to DSSE.

If, however, some prior information has broken the 
multicollinearity (see Solomon, 1980) then the forecasts 
of the early 1960's are barely affected, and the 
forecasts of the 1970's are much improved.  This is what 
has happened in this case.




10.8.1.3  PRIOR DATA CASE 1, W = 10-4


The estimations were again repeated, but with W = 10-4, 
which would give more weight to older information.


Table 62

              C            D             E           DSSE    

Belgium    .879 (.16)   -.247 (.07)   .556 (.12)    .0238
France     .915 (.18)   -.256 (.05)   .512 (.11)    .0148
Germany   1.000 (.19)   -.295 (.07)   .332 (.11)    .0069
Holland    .878 (.19)   -.211 (.08)   .364 (.12)    .0650
Italy      .745 (.08)   -.303 (.10)   .398 (.15)    .0539
Japan      .818 (.07)   -.191 (.10)   .317 (.15)    .0739
UK        1.049 (.20)   -.317 (.03)   .322 (.07)    .0854
USA       1.047 (.20)   -.721 (.06)   .840 (.07)    .3550

AVERAGE    .916         -.318         .455          .0848


The average DSSE is rather worse than 10.8.1.1;  this 
overall view conceals the fact that there is a large 
deterioration in the USA DSSE.




10.8.1.4  PRIOR DATA CASE 1, W = 0.0


Just to see what would happen, a run was done for the 
prior information of 10.8.1 with W = 0.


Table 63

              C            D             E           DSSE    

Belgium    .787 (.09)   -.269 (.06)   .593 (.07)    .0261
France     .673 (.04)   -.248 (.03)   .716 (.05)    .0200
Germany    .867 (.07)   -.321 (.05)   .297 (.05)    .0056
Holland    .351 (.07)   -.214 (.06)   .436 (.06)    .0840
Italy      .670 (.06)   -.367 (.04)   .432 (.06)    .0787
Japan      .641 (.03)   -.253 (.02)   .262 (.03)    .1776
UK         .163 (.07)   -.425 (.02)   .300 (.04)    .1950
USA        .817 (.08)   -.905 (.04)   .955 (.04)    .7278

AVERAGE    .621         -.375         .499          .1644


The average DSSE is very much worse than when W was 10-2, 
10-3 or 10-4.  This is mostly because of the large 
increase in the DSSE for the USA.  This confirms the 
hypothesis of a parameter shift in the USA, as with a 
zero W, a shift in the parameters cannot be accomodated 
as well as with a non-zero W.




10.8.2  THE PRIOR INFORMATION, CASE 2


The prior information used in this part of the paper is 
the same as 10.8.1, except that the standard error is 
doubled (the variance is quadrupled).  Thus it will be 
possible to see what happens when less precise prior 
information is used.


10.8.2.1  PRIOR DATA CASE 2, W = 10-3



Table 64

              C            D             E           DSSE    

Belgium    .981 (.38)   -.188 (.13)   .504 (.26)    .0212
France    1.009 (.41)   -.252 (.13)   .466 (.26)    .0132
Germany   1.023 (.41)   -.253 (.13)   .342 (.25)    .0096
Holland    .968 (.42)   -.169 (.14)   .205 (.27)    .0698
Italy      .765 (.18)   -.331 (.24)   .423 (.34)    .0617
Japan     1.019 (.18)   -.268 (.25)   .397 (.36)    .0925
UK        1.042 (.41)   -.218 (.08)   .364 (.17)    .0701
USA       1.039 (.41)   -.507 (.11)   .658 (.16)    .2150

AVERAGE    .981         -.273         .420          .0691


The average DSSE is 2% lower than case 1 when this weaker 
prior information is used.  This rather interesting 
result shows that weaker prior information can result in 
better forecasting performance;  this would be expected 
when parameters are drifting, or when the prior 
information is wrong.




10.8.2.2  PRIOR DATA CASE 2, W = 10-2




Table 65

              C            D             E           DSSE    

Belgium   1.086 (.53)   -.261 (.30)   .473 (.51)    .0186
France    1.063 (.49)   -.344 (.31)   .446 (.51)    .0085
Germany   1.021 (.47)   -.289 (.31)   .429 (.51)    .0174
Holland   1.070 (.47)   -.219 (.30)   .283 (.52)    .0849
Italy      .953 (.44)   -.457 (.43)   .468 (.51)    .0766
Japan     1.228 (.50)   -.449 (.45)   .451 (.53)    .1289
UK        1.040 (.43)   -.212 (.23)   .452 (.41)    .0603
USA       1.021 (.45)   -.443 (.25)   .573 (.42)    .1644

AVERAGE   1.060         -.334         .447          .0700


Just as in 10.8.1, an increase in W gives a worsening of 
the DSSE (.0700 compared with .0691).  The DSSE for the 
USA, however, is much better, which reinforces the 
hypothesis that the parameters have changed during the 
estimation period.




10.8.2.3  PRIOR DATA CASE 2, W = 10-4



Table 66

              C            D             E           DSSE    

Belgium    .801 (.19)   -.219 (.07)   .548 (.13)    .0271
France     .798 (.24)   -.240 (.06)   .509 (.12)    .0192
Germany    .968 (.27)   -.276 (.07)   .289 (.12)    .0064
Holland    .659 (.27)   -.176 (.08)   .311 (.13)    .0688
Italy      .712 (.09)   -.240 (.11)   .304 (.20)    .0493
Japan      .807 (.07)   -.101 (.11)   .173 (.19)    .0545
UK        1.031 (.35)   -.310 (.04)   .301 (.07)    .0873
USA       1.119 (.32)   -.725 (.06)   .867 (.07)    .3686

AVERAGE    .862         -.286         .413          .0852



Just as in 10.8.1 when W = 10-4, the DSSE's are much 
worse than for W = 10-3, especially for the USA.


10.8.3  THE PRIOR INFORMATION, CASE 3



In this series of runs, the prior information was as in 
10.8.1, but the standard deviation is a half (the 
variance a quarter) of the 10.8.1 values.  Thus the prior 
information is much stronger.




10.8.3.1  PRIOR DATA CASE 3, W = 10-3




Table 67

              C            D             E           DSSE    

Belgium   1.074 (.15)   -.327 (.10)   .479 (.16)    .0191
France    1.075 (.13)   -.377 (.10)   .444 (.15)    .0084
Germany   1.031 (.12)   -.341 (.10)   .461 (.16)    .0165
Holland   1.085 (.12)   -.294 (.11)   .415 (.16)    .0750
Italy      .884 (.14)   -.468 (.12)   .471 (.15)    .0740
Japan     1.079 (.16)   -.466 (.13)   .452 (.15)    .1271
UK        1.055 (.11)   -.260 (.08)   .452 (.13)    .0641
USA       1.026 (.12)   -.601 (.09)   .592 (.14)    .1809

AVERAGE   1.039         -.392         .471          .0706


There is very little difference between these results and 
those with the prior information of case 1.




10.8.4  THE PRIOR INFORMATION, CASE 4


In this series of runs, the prior information was as in 
10.8.1, but the wool and synthetic fibre price 
elasticities were set to -0.2 and 0.2 (compared with -0.5 
and 0.5).  The prior variances are set to 0.04.


10.8.4.1  PRIOR DATA CASE 4, W = 10-3




Table 68

              C            D             E           DSSE    

Belgium   1.035 (.24)   -.185 (.12)   .296 (.21)    .0071
France    1.057 (.23)   -.238 (.12)   .253 (.20)    .0051
Germany   1.032 (.23)   -.228 (.12)   .212 (.20)    .0058
Holland   1.060 (.23)   -.157 (.13)   .124 (.21)    .0481
Italy      .822 (.16)   -.217 (.18)   .179 (.22)    .0198
Japan     1.039 (.17)   -.190 (.18)   .154 (.22)    .0529
UK        1.059 (.21)   -.201 (.08)   .307 (.15)    .0535
USA       1.027 (.22)   -.486 (.10)   .537 (.15)    .1643

AVERAGE   1.016         -.238         .258          .0446



The average DSSE is 37% lower than in 10.8.1.1;  a 
considerable improvement, which is shared by all the 
countries.  The prior information on the income 
elasticity has not been altered, so it is not surprising 
to find that the income elasticities have barely changed.  


The prior information on the wool and synthetic fibre 
price elasticities has been changed substantially, and 
(as we might expect) this has changed the final 
estimates.  But the average wool price elasticity has 
changed much less than the average synthetic fibre price 
elasticity, which would indicate that, in general, there 
is more information in the data about the effect of wool 
price changes than about the effect of synthetic fibre 
price changes.  This is probably because the price of 
wool fluctuates much more violently over this period than 
does the synthetic fibre price (see appendix F).


10.8.4.2  PRIOR DATA CASE 4, W = 10-2




Table 69

              C            D             E           DSSE    

Belgium   1.104 (.34)   -.220 (.26)   .186 (.42)    .0068
France    1.080 (.28)   -.268 (.27)   .153 (.41)    .0071
Germany   1.033 (.27)   -.225 (.27)   .185 (.42)    .0062
Holland   1.093 (.27)   -.158 (.27)   .089 (.42)    .0470
Italy      .967 (.39)   -.234 (.34)   .168 (.39)    .0208
Japan     1.232 (.44)   -.234 (.34)   .141 (.39)    .0614
UK        1.060 (.24)   -.163 (.22)   .307 (.38)    .0404
USA       1.046 (.25)   -.413 (.23)   .364 (.39)    .1059

AVERAGE   1.077         -.239         .199          .0370


When the older information is allowed to decay more 
quickly, slightly better forecasts are obtained than in 


10.8.4.1.  The wool price elasticity is unchanged, but 
the synthetic fibre price elasticity is lower.  This 
indicates that the synthetic fibre price elasticity is 
lower at the end of the time period than at the 
beginning.


10.8.4.3  PRIOR DATA CASE 4, W = 10-4



Table 70

              C            D             E           DSSE    

Belgium    .856 (.16)   -.213 (.07)   .457 (.12)    .0189
France     .909 (.18)   -.235 (.05)   .422 (.11)    .0096
Germany    .999 (.19)   -.261 (.07)   .251 (.11)    .0065
Holland    .871 (.19)   -.170 (.08)   .270 (.12)    .0576
Italy      .740 (.08)   -.206 (.10)   .255 (.15)    .0248
Japan      .815 (.07)   -.102 (.10)   .133 (.15)    .0444
UK        1.047 (.20)   -.304 (.03)   .286 (.07)    .0806
USA       1.046 (.20)   -.694 (.06)   .811 (.07)    .3484

AVERAGE    .910         -.273         .354          .0739



The average DSSE is much worse than when W is 10-3.  The 
higher average synthetic fibre price elasticity confirms 
the hypothesis that this elasticity is lower at the end 
of the time period;  the lower W allows the earlier data 
to have more weight, and that has pulled the elasticity 
up.




10.8.5  THE PRIOR INFORMATION, CASE 5


In this series of runs, the prior information was as in 
10.8.4, but with the prior variances set to 0.16;  the 
prior information is weaker.


10.8.5.1  PRIOR DATA CASE 5, W = 10-3



Table 71

              C            D             E           DSSE    

Belgium    .967 (.38)   -.164 (.13)   .396 (.26)    .0132
France    1.007 (.41)   -.225 (.13)   .356 (.26)    .0073
Germany   1.024 (.41)   -.227 (.13)   .237 (.25)    .0063
Holland    .966 (.42)   -.141 (.14)   .091 (.27)    .0546
Italy      .762 (.18)   -.203 (.24)   .195 (.34)    .0223
Japan     1.017 (.18)   -.133 (.25)   .153 (.34)    .0484
UK        1.042 (.41)   -.205 (.08)   .323 (.17)    .0640
USA       1.039 (.41)   -.486 (.11)   .620 (.16)    .2004

AVERAGE    .978         -.223         .296          .0521



Comparing this run with 10.8.4.1, we can see that the 
weakening of the prior information has made the DSSE 
worse, whereas (comparing 10.8.2.1 with 10.8.1.1) with 
the higher price elasticities, weakening the prior 
information improved the DSSE.  This leads us to the 
conclusion that this prior information is better than the 
prior information of 10.8.1 to 10.8.3.




10.8.5.2  PRIOR DATA CASE 5, W = 10-2





Table 72

              C            D             E           DSSE    

Belgium   1.086 (.53)   -.192 (.30)   .248 (.51)    .0059
France    1.065 (.49)   -.249 (.31)   .203 (.51)    .0058
Germany   1.024 (.47)   -.219 (.31)   .203 (.25)    .0063
Holland   1.073 (.47)   -.148 (.30)   .056 (.52)    .0500
Italy      .951 (.47)   -.228 (.43)   .170 (.51)    .0209
Japan     1.227 (.50)   -.216 (.45)   .147 (.53)    .0596
UK        1.045 (.43)   -.167 (.23)   .326 (.41)    .0453
USA       1.026 (.45)   -.384 (.25)   .442 (.42)    .1234

AVERAGE   1.062         -.225         .224          .0397



When the W is increased, the DSSE improves (mainly an 
improvement for the USA).  The average wool price 
elasticity is unchanged, but the synthetic fibre price 
elasticity has changed a little.



10.8.5.3  PRIOR DATA CASE 5, W = 10-4



Table 73

              C            D             E           DSSE    

Belgium    .788 (.19)   -.209 (.09)   .516 (.13)    .0252
France     .795 (.24)   -.234 (.06)   .481 (.12)    .0168
Germany    .967 (.27)   -.266 (.07)   .264 (.12)    .0063
Holland    .654 (.27)   -.164 (.08)   .281 (.13)    .0662
Italy      .709 (.09)   -.203 (.11)   .219 (.20)    .0338
Japan      .805 (.07)   -.068 (.11)   .096 (.19)    .0428
UK        1.029 (.35)   -.306 (.04)   .291 (.07)    .0859
USA       1.118 (.32)   -.717 (.06)   .858 (.07)    .3666

AVERAGE    .858         -.271         .376          .0805



Reducing the rate at which data become irrelevant has 
worsened the DSSE.  This has given more weight to the 
older data.  The USA is particularly hard hit by this, 
because of the large change in the model's parameters 
over the estimation period that has been demonstrated by 
the earlier runs.


10.9 CONCLUSIONS FROM THE ESTIMATIONS WITH PRIOR 
INFORMATION


Clearly prior information can be useful, but not greatly 
so when there is a lot of information in the data.  The 
best estimations with prior information had a DSSE of 
.037, compared to .038 for the best estimation without 
prior information.  Prior information is of greater value 
in forecasting at earlier time periods, as at earlier 
time there is less information in the data, and the prior 
information is correspondingly more useful.

Again, all of the Kalman filter estimations with prior 
information were much better (lower DSSE) than the OLS 
estimation.


10.10  CONCLUSIONS


1.  Recognising that the pre-1970 data were less precise 
than the post-1970 data led to improved forecasts.  This 
was true in all the runs that were repeated both with and 
without this assumption (see, for example, 10.4.2).  This 
is evidence for the hypothesis that it is better to 
recognize the variable precision of data, and take 
appropriate action.

2.  When the relevance of old data decays with time 
(non-zero W), the forecasts are much more accurate than 
when all data are assumed to have equal relevance (see 
10.5.2).  This lends weight to the suggestion that old 
data be given less weight than more current data.

3.  The Kalman filter is able to forecast the past much 
better than OLS for those countries with bad forecasts 
under OLS.  If the Kalman filter is supplied with prior 
information then the Kalman filter mean forecasting error 
was about the same (see the runs of chapter 10).

4.  When the DSSE from the Kalman filter is compared with 
the DSSE from OLS, we can see that:
    - All of the Kalman filter models show an average 
improvement in forecasting accuracy.
    - The OLS-estimated model has a DSSE more than five       
times greater than the best Kalman filter-estimated       
models.
    - This average change reflects the substantial 
improvement in the case of countries for which the OLS 
method performs badly because of drifting or unstable 
parameters.  Where there is greater stability, the use of 
the Kalman filter may produce slightly less forecasting 
accuracy than OLS, but any deterioration is very slight.



5.  The performance of the Kalman filter is not extremely 
sensitive to choice of W.  Order of magnitude increases 
or decreases in W do not affect the forecasting 
performance unduly (see 10.5.1).

6.  Combining two time series into a common estimation 
process with restrictions across the equations is done 
very simply, as is the incorporation of any available 
prior information into the estimation.

7.  When the lower precision of preliminary data is taken 
account of, we can expect an improvement in the model's 
forecasting ability (see 10.4).

8.  The Kalman filter copes successfully with shocks to 
the system that might be expected to affect the parameter 
values (see 10.6.10).