The very first thing she did was to look whether there 
was a fire in the fireplace, and she was quite pleased to 
find that there was a real one, and blazing away as 
brightly as the one she had left behind.  Through the 
Looking Glass,  Lewis Carroll.

9.  A KALMAN FILTER ESTIMATED MODEL OF ENERGY DEMAND; ONE 
DEPENDENT AND MULTIPLE EXPLANATORY VARIABLES 


The importance of energy has been underlined by the 
"energy crises" (very large price rises) of 1974 and 
1979, and these price rises have increased the 
significance of the question of how much does energy 
price affect energy demand.  

The drastic change since 1974 in energy prices also 
raises the possibility of a structural change; i.e. the 
price elasticity or income elasticity after 1973 may be 
rather different from the price elasticity or income 
elasticity before 1973.  The Kalman filter makes it easy 
to estimate a model with changing parameters, and this 
chapter will describe the estimation of such a model. 

The same model will be estimated for various values of V 
and W;  this is to determine the sensitivity of the fit 
to V and W, and to discover whether the choice of V and W 
is critically important.


9.1   THE MODEL 


Experience with energy models (for example Nordhaus, 1977 
and Bohi, 1981) leads to the expectation that the 
following variables are likely to influence energy 
demand. 

       - the price of energy 
       - the level of economic activity (e.g. industrial 
         production) 
       - the state of the economy (is it in a boom? a 
         slump?) 
       - the severity of winter (especially in the 
         residential sector) 

We would also expect that the adjustment to new 
conditions (for example higher energy prices) would not 
be accomplished instantaneously;  we would expect the 
movement of demand back to long term equilibrium to take 
some time. So we would want to build in some lag 
mechanism. 

There are many lag structures suggested in the 
literature. Maddala (1977) devotes an entire chapter to 
distributed lag models (as do other textbooks).  In the 
energy field, where dynamic models are used (often static 
models are used, for example Pindyck, 1979, because of 
the difficulty of specifying a dynamic version of the 
otherwise very flexible translog model) they tend to be 
either flow adjustment or stock adjustment models.  
Murota, 1982, says that generally demand for energy has 
been studied using the flow adjustment model (such as 
9.1.9 below).  Murota states that stock adjustment models 
are superior, and that the main reason for their 
infrequent use is lack of data on stocks of energy-using 


appliances.  The stock adjustment model is as follows:

Log E = Log K + Log U       	       	       equation 9.1.1

Log Kdes = a + b Log Y + c Log Pk + d Log pe

          + e X	       	       	       equation 9.1.2

Log K = Log Kdes + f(Log K-1 - Log Kdes) eq. 9.1.3

Log U = g + h Log Pe + j Log Y + k Z       equation 9.1.4

Where       E is energy consumption

       K is the stock of energy-using appliances

       Kdes is the desired stock of energy-using 

       appliances

       U is the utilization rate of energy-using

       appliances

       Y is income or output, as appropriate

       Pk is the price of energy-using appliances

       Pe is the price of energy

       X, Z are the residuals

       The subcript -1 denotes a one-period lag.



From 9.1.3,

Log Kdes = (Log K - f Log K-1) / (1 - f)

Substituting this in 9.1.2,

Log K = f Log K-1 + (1-f)(a + b Log Y + c Log Pk

        + d Log Pe + eX)       equation 9.1.5

The model can be estimated from 9.1.4 and 9.1.5 if data 
on stocks of energy-using appliances are available.  
Unfortunately, this is rarely the case, so it is 
necessary to eliminate K from the equations.

Lagging 9.1.1 and rearranging, 

Log K-1 = Log E-1 - Log U-1       	       equation 9.1.6

Substituting 9.1.5 and 9.1.6 in 9.1.1,

Log E = f(Log E-1 - Log U-1) + (1-f)(a + b Log Y + 

       c Log Pk + d Log Pe + eX) + Log U  equation 9.1.7 

Lagging 9.1.4 and substituting in 9.1.7,

Log E = f Log E-1 - fg - fh Log Pe-1 - fj Log Y-1 ) 

       fk Z-1 + (1-f)(a + b Log Y + c Log Pk

        + d Log Pe + eX) + g + h Log Pe + j Log Y + k Z



Rearranging this gives :

Log E = (1-f)(a+g) + f Log E-1 -fh Log Pe-1 - fj Log Y-1

       +(b-bf+j) Log Y + (c-cf) Log Pk + (d-df+h) Log Pe

       + fk Z-1 + k Z + (e-ef) X       equation 9.1.8

The problem in estimating this equation is that there is 
likely to be considerable collinearity between Log Y and 
Log Y-1, and between Log Pe and Log Pe-1, especially when 
there is only a limited amount of data.

This equation was estimated using OLS and omitting the 
variable Pk (for which no data were available), and the 
following results were obtained.

In the Domestic sector, none of the estimated parameters 
were statistically significant at the 95% confidence 
level.  The R2 was 0.61.  The correlation between Log Pe 
and Log Pe-1 was 0.97, and the correlation between Log Y 
and Log Y-1 was 0.99.  Klein's rule (Klein, 1962) would 
suggest that there is, as anticipated above, severe 
collinearity, as the correlations between the explanatory 
variables are much greater than the multiple correlation.

In the Industrial sector, only one of the parameters 
(that on Log Y) was statistically significant. The R2 was 
0.90.  The correlation between Log Pe and Log Pe-1 was 
0.95, and the correlation between Log Y and Log Y-1 was 
0.97. Klein's rule would again suggest that collinearity 
is a problem.

These results bear out the statements of Murota;  data on 
stocks of energy-using appliances are needed to estimate 
the stock-adjustment model.  In common with other 


researches (some examples are listed below) a flow 
adjustment model will now be tried.

The desired level of energy demand is related to the 
weather, the price of energy, the level of economic 
activity and the state of the economy (all defined in 
9.2) by : 

LogQdes,t = A + B logPE,t + C log Yt + D WEATHERt + E BSt
       	       	       equation 9.1.9 

Where       Qdes,t is the desired level of energy demand in 
       year t 
       PE,t is the price of energy in year t 
       Yt is the level of economic activity in year t 
       BSt is the state of the economy in year t 
       WEATHERt is the severity of winter in year t 

The second equation describes the lagged adjustment 
process of energy demand to desired energy demand.  A 
partial adjustment (Koyck) process is used, in which the 
size of the move towards equilibrium is proportional to 
the degree of disequilibrium. 

       LogQt = logQdes,t + G ( logQt-1 - logQdes,t )       	
       	       	       equation 9.1.10 

Rearranging 9.1.2 gives 

       Log Qt = G log Qt-1 + (1-G) log Qdes,t 

Substituting for log Qdes,t using equation 9.1.1 gives : 

       Log Qt = G log Qt-1 + (1-G)(A + B log PE,t + 

       	       	        C log Yt + D WEATHERt + E BSt) 



And rearranging this gives : 

Log Qt = G log Qt-1 + (1-G) B logPE,t + (1-G) C log Yt 

       	       	        + (1-G) D WEATHERt + (1-G) E BSt	 
       	       	       equation 9.1.11 

This reduced form of the structural equations 9.1.9 and 
9.1.10 is the equation that will be used in estimating 
the model of energy demand. 

This model has been used in various studies of energy 
demand, such as the Federal Energy Administration (1976), 
Taylor, Blattenburger and Verleger (1976) and the model 
described in chapter 13 of Nordhaus (1977).

Hall (1981) states that this model is probably that most 
often used in energy modelling, as it has the advantage 
of simplicity;  he then goes on to use the model.  He 
models demand for petroleum products, gas, coal and 
electricity separatly.  He reports for the UK domestic 
sector, short-run own-price elasticities in the range 
-0.15 to -0.56 for the four fuels, and short-run income 
elasticities in the range 0.18 to 1.7.  In the industrial 
sector, short-run own-price elasticities are between 
-0.12 and -0.54, and short-run income elasticities 
between 1.0 and 2.0.

Kraft and Kraft (1980) estimate this model for the USA 
industrial sector.  They report a short-run price 
elasticity of -0.1;  long run of -0.5.  Berndt, Fuss and 
Waverman (1978) estimate this model for the USA 
industrial sector and report a long run elasticity of 
-0.7.



In the UK, Wigley and Vernon (1982) use this lag 
structure, giving as a reason the limited data available 
(1955 to 1979) and the simplicity of this model.  They 
report a short-run income elasticity of 0.6 and a 
short-run price elasticity of -0.09 for the industrial 
sector; 0.53 and -0.26 for the domestic sector.

Common (1981) estimates a partial adjustment model (as in 
9.1.11).  He reports a long-run price elasticity of the 
combined domestic and industrial sectors of -0.2.

Hawdon and Tomlinson (1982) review various energy demand 
models, and state that most dynamic models have been flow 
adjustment models rather than stock adjustment models.  
They again give as the problem the lack of data on 
appliance stocks.

Two sectors will be examined;  the residential (i.e. 
private households) sector and the industrial sector. 
This division is made because we would, a priori, expect 
the parameters of the model to be different in the 
different sectors. 


9.2  THE DATA


The consumption of energy is measured in million therms 
(a therm is 100,000 British thermal units;  a British 
thermal unit is the amount of heat needed to raise the 
temperature of a pound of water from 60 to 61 degrees 
Fahrenheit).  The source of the data is various issues of 
"Digest of UK Energy Statistics" published by the UK 
Department of Energy. 

The level of economic activity is measured by industrial 


production (in the industrial sector model) and by 
consumer expenditure deflated by the consumer price index 
in the residential  sector model.  The source of the data 
is the International Monetary Fund's International 
Financial Statistics. 

The state of the economy is defined as the difference 
between the rate of growth of economic activity and the 
average rate of growth of economic activity over the 
period 1955 to 1980.  For example, if the average growth 
rate of industrial production is 2.2%, and the rate of 
growth in 1976 was 6%, then the state of the economy 
variable has the value 3.8.  If the rate of growth was 
1%, then the state of the economy variable has the value 
-1.2. The data are derived from the economic activity 
data above.  

The severity of winter is measured by the number of 
degree days in the heating season (October to April) 
minus the number of degree days in the average winter.  
Degree days are measured as the number of days that the 
mean daily temperature is below 60 degrees Fahrenheit, 
multiplied by the number of degrees below 60.  Thus a day 
whose mean temperature was 40 degrees Fahrenheit would 
contribute 20 degree days to the year's total.  The 
degree day data come from various issues of "Platts 
Oilgram News", published by McGraw-Hill Inc.
 
These figures were all then divided by 10000, to scale 
them so as to give a model parameter of the order of one. 

The price of energy is not simple.  It is, however, 
possible to collect data on the price of solid fuels 
(coal, coke etc.), liquid fuels (heating oil, kerosene), 
gas (producer gas, natural gas) and electricity, and the 
source of these figures is again "Digest of UK Energy 


Statistics".  The price of each fuel is first deflated by 
the consumer price index, to disentangle the effect of 
changing fuel prices from the general fall in the value 
of money.  To combine these price series into a single 
energy price series, it is necessary to weight the fuels 
according to their importance.  This will be done by 
using the fuel shares as weights, and a geometric mean 
was calculated using these weights.  Data were collected 
for 1958 to 1980. 


9.3   ESTIMATING THE MODEL USING ORDINARY LEAST SQUARES 



When the model was estimated, some parameters were found 
to be statistically insignificant. In particular, in the 
residential sector, the state of the economy variable was 
not a significant influence on energy demand.  This is 
not too surprising; one would not expect residential 
energy demand to over-respond to short-term fluctuations 
in the economy.  In the industrial sector, the weather 
variable was not significant, which probably reflects the 
lower proportion of energy used by the industrial sector 
for space heating compared with the residential sector. 



The parameter estimates are as follows: 

Table 15
OLS estimation  
(Over the period 1958 to 1980, 23 observations)

       	       RESIDENTIAL       	       INDUSTRIAL 
    parameter standard t-value parameter standard t-value 
                 error       	       	          error 

Lag 
G	-.02	.15	 0.1	   0.34	     0.31     1.1 
Constant 
(1-G)A	8.66	1.36	 6.4	   5.39	     2.63     2.1 
Price 
(1-G)B	-0.35	0.09	 3.8	   -0.18	     0.06     2.9 
Income 
(1-G)C	0.36	0.07	 5.1	   0.38	     0.16     2.4 
Weather 
(1-G)D	-0.72	0.17	 4.2	   	     -         -       - 
State of economy 
(1-G)E   -      -       - 	   0.64	     0.33     2.0 

Standard error of estimate   0.021 
R2               .80                          .89 
F - statistic    17.4                         26.5 
Durbin h 
statistic         1.4                          0.4 
DSSE             .00627                        .01226 

The Durbin h statistic is used because one of the 
explanatory variables is the lagged dependent variable;  
the hypothesis of zero autocorrelation in the residuals 
cannot be rejected (a test at the 5% level would take h 
greater than 1.645 as the critical region).  The dynamic 
sum of squared errors (DSSE) was again calculated on the 


last four observations.

The parameters are all correctly signed, and are all 
statistically significant at the 95% confidence level, 
except for G. The F statistics are both significant at 
the 99% confidence level, and the Durbin - Watson 
statistic is fairly satisfactory (the hypothesis of zero 
autocorrelation cannot be rejected with 99% confidence in 
both sectors). 

The long - run price elasticity ( = B / (1-G), see 
Maddala, 1977, p 143) is -.27 in the industrial sector, a 
bit less than the -0.35 of the residential sector.  This 
perhaps reflects the fact that residential consumers have 
considerable flexibility in the temperature that they 
choose to maintain their living accomodation at ( most 
energy consumed in the residential sector is for space 
heating).  In the industrial sector, there is less 
flexibility, as it is not so easy to reduce the energy 
input as process heat, or as work. 

The zero value of G in the residential sector indicates 
that the reaction to changed conditions is immediate;  
the mean lag is G/(1-G) (Johnston, 1972, p 299).  This is 
consistent with most of the reaction to changed 
conditions being short - term (for example turning down 
the house thermostat).  

The mean lag in the industrial sector is not 
significantly different from zero, but it is correctly 
signed.



9.4 ESTIMATING THE MODEL WITH THE KALMAN FILTER


In all the estimations, the prior information on the 
parameters was of great prior uncertainty;  M0 = 0, C0 = 
100;  this is done so that comparisons with OLS will be 
on an equitable basis.


9.4.1  W = 0 


The model was first estimated with W (controlling the 
degradation of the certainty attached to the parameter 
estimates) set to zero, to get an estimation as simolar 
as possible to OLS.  Four estimations were done, with V 
(the observation error) set to 0, 0.0005, 0.001 and to 
0.002.  This corresponds to a standard deviation on the 
observation model of 0, 0.022, 0.031 and 0.045.  As the 
observation model is logarithmic in form, these values 
correspond approximately to standard deviations on the 
observations of energy demand of 0, 2%, 3% and 4.5%.  
This probably spans the likely range of data precision.

When the model was estimated using OLS, the standard 
error of the estimates (calculated from the residual 
variance) was 0.021 in the residential sector and 0.027 
in the industrial sector, which indicates that the two 
middle V's are reasonable values.  The larger V of 0.002 
and the zero V are included to test the sensitivity of 
the Kalman filter.


First, we shall examine the results with zero V.  Telling 
the Kalman filter that V = 0 is equivalent to telling it 
that the observations are totally accurate, which means 


that the Kalman filter will be able to make very strong 
inferences about the parameters with a very small amount 
of data;  these inferences will of course be unwarranted 
unless the observation error is indeed zero, which is 
most unlikely.

The parameter estimates are as follows (standard errors 
in brackets).


Table 16

KALMAN FILTER ESTIMATION 1, V = 0, W = 0,0,0,0,0

       	       	           RESIDENTIAL	       	       INDUSTRIAL

G      (LAG)        	       -0.23       (0.0)       1.04       (0.0)
(1-G)A (CONSTANT)       10.26       (0.0)       -1.62       (0.0)
(1-G)B (PRICE)       	       -0.59       (0.0)       -0.23       (0.0)
(1-G)C (INCOME)	       0.56       (0.0)       0.40       (0.0)
(1-G)D (WEATHER)       -1.14       (0.0)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.85       (0.0)       

DSSE       	       	       0.448       	       0.0132


The standard errors are all zero because of the 
unrealistic value of V (this is not caused by the zero W, 
as will be seen below).  The values of G are implausible 
(we should expect G to lie between zero (instantaneous 
adjustment) and 1 (very slow adjustment)).  The other 
parameters are, however, plausible in both sign and 
magnitude. The dynamic sum of squared errors in the 
industrial sector is comparable, though larger, than in 
the OLS estimation, but the dynamic sum of squared errors 
in the residential sector is very large indeed.




The successive estimates of the parameters as the Kalman 
filter iterates through the data are all over the place;  
the parameter on the constant, A, changes by over 50 in 
one iteration at iterations 18 and 22 of the residential 
estimation.  This is in a logarithmic model, where such a 
change in the log energy demand means a change of e50 = 5 
. 1021.in energy demand, which is plainly absurd.


The Kalman filter is clearly putting a great deal of 
faith into each observation;  it is behaving as if it is 
being given very strong information at each iteration (as 
is the case).  The Kalman filter is being told that the 
observation error is zero.  This behaviour can often be 
seen in humans (which is why this paper has been 
anthropomorphising the Kalman filter), who will often 
have a great deal of faith in something on what other 
people would regard as very flimsy evidence.  Clearly the 
people holding these beliefs do not regard the evidence 
as flimsy, but as very strong.

If this is indeed a misjudgement of the strength of the 
evidence (in Kalman filter terms, the value of V), then 
when fresh evidence is presented that contradicts the old 
(in Kalman filter terms, a new observation which leads to 
very different parameters being calculated), and if they 
have great faith in this new evidence (V very small) then 
they will completely and suddenly change their beliefs 
(the parameters will change violently).  If this happens 
to them frequently, then they are probably 
over-estimating the strength of each piece of evidence (V 
is too small).



We can conclude from theoretical considerations, backed 
up by the results above, that if W is zero, then it is 
dangerous to set V to zero also.  Later on, we shall see 
that if W is sufficiently large, the value of V is less 
critical.

There is a parallel in everyday life.  If one never 
changes one's mind (W=0) then it is dangerous to regard 
anything as certain (V=0).  But if one is willing to 
shift one's views with new evidence (W greater than 0) 
then one will be able to adapt to changing circumstances.  
Or, if one never changes one's mind (W=0), then one will 
still retain flexibility as long as one does not regard 
anything as certain (V greater than 0) and so never fully 
fixes in one position.

So what happens if we have non-zero V? Parameter 
estimates are as follows (standard errors in brackets).


Table 17

KALMAN FILTER ESTIMATION 2, V = 0.0005, W = 0,0,0,0,0

                        RESIDENTIAL     INDUSTRIAL

G      (LAG)        	       -0.06       (0.16)       0.29       (0.25)
(1-G)A (CONSTANT)       8.84       (1.45)       5.64       (2.12)
(1-G)B (PRICE)       	       -0.30       (0.10)       -0.20       (0.05)
(1-G)C (INCOME)	       0.37       (0.08)       0.45       (0.12)
(1-G)D (WEATHER)       -0.79       (0.19)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.57       (0.26)       

DSSE       	       	       0.0062       	       0.0121




The parameters are not very different from the parameters 
estimated using ordinary least squares (see 9.3) and the 
standard errors attached to the parameters are very 
similar to the OLS estimation.  This is not of course 
surprising as the close relationship between the Kalman 
filter and OLS has been demonstrated in earlier parts of 
this paper.  The differences come from the differing 
values of V used by the two methods.  OLS, in effect, 
estimates V as part of the estimation process, and the 
Kalman filter will give answers fully identical to OLS 
only when we choose W=0 and V equal to the value 
estimated by OLS.

The parameter estimates above are all correctly signed, 
plausible, and statistically significant at the 95% 
confidence level, except (as in ordinary least squares) 
for the lag parameter G.  The dynamic sum of squared 
errors are about 15% worse than in OLS.  The violent 
changes observed in the parameter estimates in the 
previous run were not observed in this run.



The third run used a higher value for V.

Table 18

KALMAN FILTER ESTIMATION 3, V = 0.001, W = 0,0,0,0,0

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.006       (0.23)       0.37       (0.35)
(1-G)A (CONSTANT)       8.40       (2.04)       5.09       (2.95)
(1-G)B (PRICE)       	       -0.33       (0.14)       -0.18       (0.07)
(1-G)C (INCOME)	       0.36       (0.11)       0.39       (0.18)
(1-G)D (WEATHER)       -0.73       (0.27)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.66       (0.37)       

DSSE       	       	       0.0055       	       0.0121

The parameter estimates are almost the same as in run 2 
(and these estimated by OLS) and the standard errors are 
all rather higher (the higher standard errors are caused 
by the assumption of less precision in the observations).  
The dynamic sum of squared errors (DSSE) in the 
industrial sector is unchanged from run 2, but the DSSE 
in the residential sector shows an improvement;  it is 
11% smaller than run 2, and is 12% smaller than in the 
ordinary least squares estimation.  Thus, recognition of 
data imprecision leads to larger standard errors of 
estimates, but to better forecasts.

For the fourth run, V was doubled again.




Table 19

KALMAN FILTER ESTIMATION 4, V = 0.002, W = 0,0,0,0,0

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	        0.06       (0.32)       0.43       (0.48)
(1-G)A (CONSTANT)       8.00       (2.83)       4.61       (4.00)
(1-G)B (PRICE)       	       -0.33       (0.20)       -0.17       (0.10)
(1-G)C (INCOME)	       0.35       (0.15)       0.35       (0.24)
(1-G)D (WEATHER)       -0.71       (0.38)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.71       (0.51)       

DSSE       	       	       0.0048       	       0.0121
The parameter estimates are still very similar to those 
of run 2, and the standard errors are yet higher.  The 
dynamic sum of squared errors has fallen by further 13% 
in the residential sector, and is unchanged in the 
industrial sector.

Some general conclusions can be drawn from the four runs 
above.  Clearly it is possible to reduce the dynamic sum 
of squared errors (and therefore the expected forecasting 
errors) below the level offered by OLS by using the 
Kalman filter with W = 0, but the DSSE is not always 
reduced, even by a good choice of V.

Secondly, the value of V has some influence on the 
parameter estimates, but much more influence on the 
estimated standard errors of the estimates.  Comparing 
runs 2 and 4, we see that the parameter estimates differ 
by only a few per cent, but the standard errors of the 
estimates are (roughly) doubled in run 4 compared to run 
2.  This is not surprising, we have quadrupled V (the 
variance attached to the observations) so it seems very 
plausible that the variance attached to the parameter 


estimates should also quadruple.

9.4.2 W NON-ZERO

9.4.2.1  W = 10-_6_, 10-_6_, 10-_6_, 10-_6_, 10-_6_ 

We begin by introducing a very small, but non-zero W.  
W is the increase in parameter variance with passing 
time;  it may be easier to look at standard error (square 
root of variance).  A W of 10-6 means that the standard 
error increases by .001 each year, and this is small 
compared with the standard errors (estimated by OLS) 
which are at least 0.06.  Again, four runs were done with 
V = 0, .0005, .001, .002; to avoid confusion these will 
be called estimations 5 to 8.

Table 20

KALMAN FILTER ESTIMATION 5, V = 0, ALL W = 10-6

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.015       (0.09)       0.24       (0.12)
(1-G)A (CONSTANT)       11.06       (0.81)       6.22       (0.87)
(1-G)B (PRICE)       	       -0.27       (0.08)       -0.21       (0.03)
(1-G)C (INCOME)	       -0.08       (0.09)       0.44       (0.12)
(1-G)D (WEATHER)       -0.93       (0.10)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.52       (0.13)       

DSSE       	       	       0.0033       	       0.0177





Table 21

KALMAN FILTER ESTIMATION 6, V = 0.0005, ALL W = 10-6

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.15       (0.19)       0.22       (0.28)
(1-G)A (CONSTANT)       9.66       (1.80)       6.16       (2.38)
(1-G)B (PRICE)       	       -0.23       (0.15)       -0.21       (0.07)
(1-G)C (INCOME)	       0.34       (0.13)       0.50       (0.16)
(1-G)D (WEATHER)       -0.77       (0.22)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.50       (0.30)       

DSSE       	       	       0.0047       	       0.0125

It is interesting to compare run 6 with run 2.  The 
values of V are the same, only the values of W differ.

The effect of these differing W's is to increase the 
uncertainty attached to the parameter estimates, the 
standard error is around 0.05 higher (for the parameters 
other than the parameter on the constant term A).  The 
DSSE is very much better (24%) on run 6 than on run 2 in 
the residential sector, and a bit worse (4%) in the 
industrial sector.




Table 22

KALMAN FILTER ESTIMATION 7, V = 0.001, ALL W = 10-6

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.06       (0.25)       0.31       (0.37)
(1-G)A (CONSTANT)       9.04       (2.32)       5.54       (3.14)
(1-G)B (PRICE)       	       -0.27       (0.19)       -0.20       (0.09)
(1-G)C (INCOME)	       0.33       (0.16)       0.42       (0.21)
(1-G)D (WEATHER)       -0.75       (0.30)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.60       (0.40)       

DSSE       	       	       0.0045       	       0.0122

Increasing V has the usual effect of increasing the 
standard errors (compared to run 6).  In the both 
sectors, DSSE has fallen.

This run may also be compared to run 3, where V is the 
same, but W is zero.  The parameter estimates are 
slightly different, and the standard errors are slightly 
worse.



Table 23

KALMAN FILTER ESTIMATION 8, V = 0.002, ALL W = 10-6

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.02       (0.34)       0.40       (0.49)
(1-G)A (CONSTANT)       8.36       (3.04)       4.87       (4.13)
(1-G)B (PRICE)       	       -0.29       (0.24)       -0.18       (0.11)
(1-G)C (INCOME)	       0.32       (0.20)       0.37       (0.27)
(1-G)D (WEATHER)       -0.73       (0.40)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.69       (0.53)       

DSSE       	       	       0.0043       	       0.0121



Again the main effect of the increase in V has been to 
increase the standard errors of the estimates, but there 
are also small changes in the parameter estimates.  The 
DSSE in the residential sector, at .0043 is now 31% less 
than the DSSE in the OLS estimation but the DSSE in the 
industrial sector is only 2% less than OLS.

So what can be concluded from runs 5 to 8?

The main effect of setting W to 10-6 has been a small 
increase in the standard error of the estimates.  A 
second effect has been to mitigate the effects of setting 
V to 0; the results were not as disastrous as in run 1.  
There has also been a small change in the parameter 
estimates.

A very small value of W, then, has some effects, but not 
very large effects.  Let us now examine the effects of a 
W ten times larger.



9.4.2.2  W = 10-_5_, 10-_5_, 10-_5_, 10-_5_, 10-_5_


With W = 10-5, the standard error of the estimates 
increases by about .003 each year, which is still small 
compared with the standard errors estimated by ordinary 
least squares, of at least .06.  Four runs were done with 
V = 0, .0005, .001, .002 (as before) which will be called 
runs 9 to 12.


Table 24

KALMAN FILTER ESTIMATION 9, V = 0, ALL W = 10-5

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.26       (0.22)       0.0       (0.31)
(1-G)A (CONSTANT)       10.84       (2.49)       6.73       (2.59)
(1-G)B (PRICE)       	       -0.13       (0.25)       -0.19       (0.10)
(1-G)C (INCOME)	       0.27       (0.23)       0.81       (0.28)
(1-G)D (WEATHER)       -0.80       (0.30)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.26       (0.36)       

DSSE       	       	       0.0049       	       0.0152

In run 1 the effect of setting V to zero was to give very 
large dynamic sum of squared errors (DSSE).  In run 9, 
the non-zero W have fully mitigated the effect of zero V, 
as the DSSE's are small.  So a non-zero W can make the 
estimation more robust.



Table 25

KALMAN FILTER ESTIMATION 10, V = 0.0005, ALL W = 10-5

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.17       (0.29)       0.16       (0.41)
(1-G)A (CONSTANT)       9.91       (3.04)       6.13       (3.42)
(1-G)B (PRICE)       	       -0.16       (0.30)       -0.21       (0.13)
(1-G)C (INCOME)	       0.30       (0.27)       0.62       (0.33)
(1-G)D (WEATHER)       -0.79       (0.37)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.42       (0.42)       

DSSE       	       	       0.0040       	       0.0138

The standard errors of the estimates are about .12 higher 
than in run 6 (where V was as in run 10, but W was 10-6), 
apart from the constant (A).

Table 26

KALMAN FILTER ESTIMATION 11, V = 0.001, ALL W = 10-5

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.12       (0.34)       0.24       (0.48)
(1-G)A (CONSTANT)       9.40       (3.42)       5.72       (3.98)
(1-G)B (PRICE)       	       -0.17       (0.33)       -0.20       (0.14)
(1-G)C (INCOME)	       0.30       (0.30)       0.54       (0.37)
(1-G)D (WEATHER)       -0.77       (0.43)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.51       (0.53)       

DSSE       	       	       0.0037       	       0.0133



The standard errors of the estimates are greater than 
those of run 7 by about the same amount by which those of 
run 10 exceed run 6.

Table 27

KALMAN FILTER ESTIMATION 12, V = 0.002, ALL W = 10-5

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       -0.04       (0.41)       0.34       (0.57)
(1-G)A (CONSTANT)       8.70       (3.96)       5.12       (4.77)
(1-G)B (PRICE)       	       -0.19       (0.38)       -0.19       (0.16)
(1-G)C (INCOME)	       0.31       (0.33)       0.45       (0.41)
(1-G)D (WEATHER)       -0.75       (0.52)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.61       (0.64)       

DSSE       	       	       0.0035       	       0.0129

Again, the standard errors of the estimates are greater 
than those of run 8 by about the same amount by which 
those of run 10 exceed run 6.

So what conclusions can be drawn from runs 9 to 12?

The DSSE's in the residential sector are stable with 
respect to changes in V, and are all at least 22% better 
than OLS.  In the industrial sector the DSSE's are not as 
good as OLS, but only by 6 to 12% for non-zero V.  The 
higher W leads to higher standard errors of the parameter 
estimates, as would be expected.



9.4.2.3  W = 10-_4_, 10-_4_, 10-_4_, 10-_4_, 10-_4_



Since the increase in W from 10-6 to 10-5 was accompanied 
by a small improvement in DSSE it is natural to want to 
see what happens when W is increased by another factor of 
ten.


Table 28

KALMAN FILTER ESTIMATION 13, V = 0, ALL W = 10-4

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.04       (0.63)       0.31       (0.83)
(1-G)A (CONSTANT)       6.93       (6.31)       4.14       (6.46)
(1-G)B (PRICE)       	       -0.06       (0.79)       -0.17       (0.32)
(1-G)C (INCOME)	       0.38       (0.73)       0.70       (0.91)
(1-G)D (WEATHER)       -0.84       (0.93)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.51       (1.06)       

DSSE       	       	       0.0068       	       0.0160









Table 29

KALMAN FILTER ESTIMATION 14, V = 0.0005, ALL W = 10-4

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.06       (0.65)       0.34       (0.85)
(1-G)A (CONSTANT)       6.71       (6.42)       3.97       (6.62)
(1-G)B (PRICE)       	       -0.06       (0.81)       -0.17       (0.33)
(1-G)C (INCOME)	       0.39       (0.75)       0.66       (0.93)
(1-G)D (WEATHER)       -0.83       (0.96)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.54       (1.09)       

DSSE       	       	       0.0065       	       0.0158





Table 30

KALMAN FILTER ESTIMATION 15, V = 0.001, ALL W = 10-4

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.08       (0.67)       0.37       (0.87)
(1-G)A (CONSTANT)       6.50       (6.52)       3.83       (6.75)
(1-G)B (PRICE)       	       -0.06       (0.82)       -0.17       (0.34)
(1-G)C (INCOME)	       0.39       (0.76)       0.64       (0.94)
(1-G)D (WEATHER)       -0.83       (0.99)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.57       (1.11)       

DSSE       	       	       0.0064       	       0.0156






Table 31

KALMAN FILTER ESTIMATION 16, V = 0.002, ALL W = 10-4

       	       	       	       RESIDENTIAL       	       INDUSTRIAL

G      (LAG)        	       0.11       (0.70)       0.42       (0.90)
(1-G)A (CONSTANT)       6.16       (6.70)       3.57       (6.99)
(1-G)B (PRICE)       	       -0.07       (0.85)       -0.17       (0.35)
(1-G)C (INCOME)	       0.40       (0.79)       0.59       (0.97)
(1-G)D (WEATHER)       -0.81       (1.04)       -	       -
(1-G)E (BOOM/SLUMP)       -	       -	       0.62       (1.16)       

DSSE       	       	       0.0061       	       0.0153

In all four of these runs, the parameter estimates and 
the standard errors are almost unaffected by the value of 
V.  The parameter estimates are a bit different from the 
estimates of runs 9 to 12, but their standard errors 
are much larger (about twice the size).  The DSSE's are 
distinctly worse than in runs 9 to 12.

It would appear that we have used a W which is too large.  
The effect of a too-large W is that the estimation 
process "forgets" what is has learned too readily, and so 
is not able to forecast as well as it might.

9.4.2.4 CONCLUSIONS FROM NON-ZERO W


There do seem to be benefits from choosing W non-zero.  
In the residential sector, when W is 10-5 or 10-6 the 
DSSE is smaller than with ordinary least squares or with 
Kalman filtering with W = 0.  In the industrial sector, 
there is not much difference between the DSSE with W = 0 


and the DSSE with W = 10-5 or 10-6.

In both sectors, the DSSE is quite a lot larger when W is 
as large as 10-4, so it is important not to have W too 
large, if forecasting error is to be minimised.

The effects of zero or too small V can be mitigated by 
non-zero W;  this can be very useful, as the effects of 
zero V can be so devastating.

It is possible to have W too large; then the Kalman 
filter forgets what it has learnt from analysing past 
data too fast, and so produces inferior forecasts.



9.5 CONCLUSIONS ABOUT THE MODEL PARAMETERS AND THE 
STANDARD ERRORS ATTACHED TO THEM


Adjustment to new conditions is instantaneous in the 
residential sector;  none of the estimations produced a 
lag.  An explanation of this may lie in the importance of 
the weather variable, which was an important factor in 
all the estimations.  We would expect a cold winter to 
increase fuel consumption in the residential sector in 
that year, and not subsequently.  There is, however, a 
lag in the industrial sector, and most of the estimations 
agree that it is between 0.3 and 0.4, implying a mean lag 
of a year or less.  But this is not statistically 
significant;  we cannot reject the hypothesis that here 
also adjustment is instantaneous.

There are statistically significant price effects in both 
sectors.  This is confirmed by all the sensible 
estimations, although the statistical significance is 
less for the Kalman filter estimations when W is large, 
as would be expected.  In most estimations, the price 
elasticity is greater in the residential sector than in 
the industrial sector, and the standard errors attached 
to the estimate is also greater.  Reasons why this may be 
so are given in 9.3 above.

The income elasticities in the two sectors are almost 
equal in most of the estimations, but the standard error 
attached to the income elasticity is smaller in the 
residential sector.  The income elasticities are 
surprisingly low;  a priori one would have expected a 
figure not too far from 1.  This may be because the data 
used (heat supplied in therms) do not take into account 
the different efficiencies of the different fuels, and 
the changing efficiency of burning fuel caused by 


technological advance.  For example, electricity is 
"burnt" with 90% efficiency, whereas gas is burnt with 
50% efficiency.  Also an open coal fire can run at as 
little as 5% efficiency, while a modern central heating 
boiler can run at several times that efficiency.  Were it 
not for the technological improvements in fuel use over 
the last 20 years, much higher fuel consumption would be 
needed now, and the model would have estimated a much 
higher income elasticity.

It could be suggested that the model have a variable 
added to represent technological improvement.  
Unfortunately, there are no published data on efficiency 
of fuel-using equipment over the years, and the usual 
practice of using a time trend as a proxy would lead to 
multicollinearity with the other variables.

The weather parameter (used in the residential sector 
only) is highly significant in all the sensible 
estimations, and is about -0.7 to -0.8.  This means that 
a winter 1000 degree days colder than average causes an 
increase in fuel use of 7% to 8%.  Appendix F displays 
degree-day data;  a winter 300 degree-days colder than 
average could be regarded as a severe winter.  The 
notorious winter of 1963 was 619 degree-days colder than 
average.

The state of the economy variable (used in the industrial 
sector only) is generally significant, and has a value of 
around 0.5 to 0.6.  This means that when industrial 
production grows 1% faster than average, fuel consumption 
is 0.5% higher than would be expected if only the income 
elasticity is taken into consideration.  Thus, energy use 
fluctuates around its trend more violently than does 
industrial production.



9.6  CONCLUSIONS ABOUT THE KALMAN FILTER ESTIMATION

Table 32

       	       	       DYNAMIC SUM OF SQUARED ERRORS
       	       	       RESIDENTIAL       	       	       INDUSTRIAL

OLS       	                 0.00627       	                 0.01226

V =      0.0  .0005  .001  .002    0.0  .0005  .001  .002

W=0     .4481 .0062 .0055 .0048   .0132 .0121 .0121 .0121
W=10-6  .0033 .0047 .0045 .0043   .0177 .0125 .0122 .0121
W=10-5  .0049 .0040 .0037 .0035   .0152 .0138 .0133 .0129
W=10-4  .0068 .0065 .0064 .0061   .0160 .0158 .0156 .0153

In the residential sector, it is possible to improve on 
the OLS estimated model's forecasting ability with a 
suitable choice of W (W =/ 0), with the DSSE dropping from 
0.0063 to as little as 0.0033.  Further, this improvement 
is not sensitive to the value of W;  W can be increased 
or decreased by a factor of ten without changing the DSSE 
very much.  The choice of V also seems to be not too 
critical, provided that V is non-zero.

In the industrial sector, the Kalman filter estimations 
are about the same as OLS;  the best DSSE is around 
0.0120 compared with 0.0123 for OLS.  The same comments 
about the insensitivity of this result to the choice of V 
and W apply as in the residential sector.  It may be that 
in the industrial sector, the parameters are fairly 
stable, so there is very little benefit in Kalman 
filtering.

It is important not to have a too large W, as this causes 
the Kalman filter to "forget" what it has learned from 


previous data too fast, and so it is less able to 
forecast.