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.