A dynamic model for correcting quarterly electricity consumption data

Thalassinos El

This study proposes a method of correcting quarterly electricity data in such a waj as to make them available to modellers in time series analysis. These models are quite d@erent in nature from econometric models since they no longer predict,fLture movements of a variable by relating it to a set of variables in a causal framework. Instead they base their prediction on the past behaviour of the variable and that variable alone. The proposed technique is based on a geometric distributed lag model betlceen the variables y, and 0,. Keywords: Quarterly data; Geometric distributed lag model

A plethora of different methods exist for forecasting a time series solely from its own current and past values. They range from the relatively simple exponen- tially weighted moving average of Holt 114) and Winters [25], to the complex techniques involved in the Bayesian approach detailed by Stevens and Harrison [22] and the methodology of Box and Jenkins [3].

In forecasting future energy and electricity demand in Greece the electricity supply industry (Public Power Corporation) distinguishes between the short term (up to two years), medium term (up to five years) and the long term (more than five years).

This study proposes a method of correcting avail- able quarterly electricity data in such a way as to make them usable to time series analysis modellers. Efficient use of this particular technique necessarily requires considerable experience of the model development process in order to avoid erroneous specifications. The idea of this study arises from the requirement for a time series model for a large number of observations (at least 50 and preferably 100 or more for ARIMA modelling). They should also be of high quality and, ideally, not have undergone any transformation because of season. weather or strike, which might distort their accuracy. In practice other considerations must be taken into account.

The author is with the University of Piraeus, 40 Karaoli

and Dimitriou. 18532 Piraeus, Greece.

Final manuscript received 20 September 1991.

The model and the suggested technique

A missing data technique can be indicated by a geometric distributed lag model between the flow variables y, and O’t of the form:

17, 4

J!r=U+ C hj~,_j+ C (Clj+djt)Qlj+“,

j=O j=l

t=1.2 T ,..., (1)

where

bj = bvj vj=(l -/?)A, O<A< I pj= 1

Using Klein’s suggestion model (1) can be written in a matrix form as:

y=X(A)w+u (2)

where

x(~)‘=[s;,.X;,x;, . ,.x:1 (3)

.~,=[lZ,~ZllQllQr*Qr3Qr4 QrlTR,Q,,TR,Q,3TR,Q,,TR,l (4)

no=E(yO)=(l-~) ~ ~j~,,_j

(initial value (5)

j=O parameter)

f-1

Ztz=(l -1.) c PDpj (6)

j=O

z,l =A’ (7)

N” = (ad) (8)

186 0140/9883/92/030186~06 (0 1992 Butterworth-Heinemann Ltd

Correcting quarterly electricity data: T. El

with respect to Y? and the unknown parameters w

taking into account the annual available information on yp formulated by the relation:

CYP = Y: (15)

where yf : (T,/4) annual available observations of y,. If the model is correctly specified, we obtain

consistent and asymptotically efficient estimators of the disaggregated (quarterly) observations yy and the parameters w of the under estimation (quarterly) specification (2) (see Sargan and Drettakis [20], p 39).

Matrix C is a (q1,4) x T1) aggregation ma&x of the

form (Gilbert [S], pp 223-225):

C=p!!iil I I f] (16)

Defining e4= (1.1 .l.l)’ and writing C= (ZT1,448ek,) we are able to prove the following aggregation relations.’

y1 = C’(CC’)_ lcy4 and Cyp = y: = Cy, (17)

where Y< is a (T,/4 x 1) vector giving the (T,/4) annual observations of y and y1 is a T1 vector of the quarterly averages of yp.

A Lagrangian procedure is used to minimize Equation (14) subject to Equation (15) (see Sargan and Drettakis [20], ~~42-44; Gilbert [8], pp 225-226).

F Lagr=(y~-x(EJw)‘(y”-X(~~)w)-2L’(Cyf!-y:)

(18)

where L is a (T,/4) vector of the Lagrangian multiplier. Differentiating Equation (18) with respect to yp,

setting it equal to zero and after some algebraic manipulations we obtain the constrained equation:

yp-Xxp(E.)W-C’L=O (19)

Premultiply Equation (19) by C and recalling that:

cyp=y:=c_$, (20)

we can solve Equation (19) for L to obtain:

L=(CC’)_‘C’(yvp-xp@)w) (21)

By substitution of Equation (21) into Equation (19) we obtain:

yp=xf!(~)w+c’(cc’)-‘cY~-c’(cc’)~’Cxq(~)w

(22) Using the time-aggregation relations

y1 = C’(CC’)_lC ~~ and x,(n)=C’(CC’)-‘CXF(A) ,Q

(22)

a’ = (c@lo) (9)

6’= (qlq2q3q&ld&4) (10)

Qc~=l, if t is a time point associated with the first quarter

= 0, otherwise (11)

u’= (UIUZU~, . . . ) u,) (12)

E(u) = 0

V(u) = ow,

E(x(n)‘u) = 0

u - NID(0, o%Z,)

and

Y,(Txl) vector of the dependent variable (electricity consumption) @,,(Txl) vector of the independent variable TR, = long-run trend, for t = 1,2,3,. . , T

Q, = dummy variables for seasonality

We further assume that in the first Tl quarters only annual observations are available on y,. We define T, = T - Tl and for algebraic convenience we assume that (T,/4) is an integer. We split the specification (2) as

[;]_[Z!!]w+[~;] (13)

where

y$? is a T,Xl vector giving the first T,(quarterly) observations on y etc

The suggested disaggregation approach is to estimate simultaneously the under estimation parameters w and the ‘missing-disaggregated’ quarterly observations y$? taking into account all available information (ie the existing by assumption ( Tl/4) of annual observations of y,). This approach was suggested by Anderson [l] and further extended by Sargan and Drettakis [20] and Gilbert [S]. This states that under normality in the residuals of Equation (2) we treat the y? ‘missing’ quarterly observations as unknown parameters which have to be estimated simultaneously with the other parameters w of the quarterly model (13) taking into account the available annual observations (if they are available) and the appropriate mathematical and stochastic specification between the dependent and the independent variable(s).

By minimizing the sum of squares function

F=(y”-X(k)w)‘(y”-X(E,)w) (14)

where

Y? y”= ___ il YP

ENERGY ECONOMICS July 1992

1 * denotes the Kronecker product. For details see Harvey ([lo], pp 358-359) and Gilbert ([S]. pp 224225).

187

Correcting quarter/j, electricity data: T. El

can be written as:

jp=y, +(x?(L)-R,(L)W) (23 I

Thus given 2. and conditional upon the estimated vector of coefficients w, we may estimate the (Ti) disaggregated quarterly observations ~4) by adding to the (T,) observed quarterly averages y1 the (T,) weighted (with the estimation vector w) deviations of the explanatory variables X?(L) from their quarterly averages X(1,). ie (X?(A) - R, (3.)).

Relation (23) suggests the following iterative pro- cedure. Applying Klein’s suggestion to estimate (2) for the subperiod T2 to obtain a starting value for ~1, say M’,., and using this starting value in the first step of the following iterative scheme:

()<j_< 1 ~~_._ ~~

_ r=1,2,3,...

9P,,+1=rl+(xp(n,-x,(~~,~v,,

VQ y&+1 =. l,r+l

Yf I

’ r+ 1 = [XQ(E.)'XQ(j")] ‘X$)y&+ 1

~ (until convergency)

We can avoid such an expensive computation pro- cedure by noting that Equation (23) can be written as:

$?-X~(~)M’=_~i -X$)u (24)

which substituted into the unconstrained sum of squares function (14) we obtain:

F=(_V, -x,(/l)w)‘(jl -xl(l)w)

+(~lf_xf(i)W)‘(~4~Xf(i”)~) (25)

which is immediately recognized as the function that, given Iti. is minimized by ordinary least squares estimation of Equation (13) after replacing both the disaggregated quarterly observations y? and the corresponding values of the exogeneous variables X? by their quarterly averages ri and X1(i_) respectively.

Therefore Equation (2) may be estimated directly using standard regression packages: for given values of i what is required is the replacement of both the missing quarterly data and the corresponding values of the exogenous variables by their annual averages. Of course degree of freedom correction will be necessary.

Correcting quarterly residential electricity consumption data

In this section we give an example of how the suggested correction technique can be used in correct-

ing quarterly electricity consumption data. We apply data for the period 1980: I-1987: IV and correct quarterly residential electricity consumption data for the period 1983: I-1985: IV.

These data are published without errors so we shall be able to judge for the ability of the suggested

correction technique, to correct the ‘erroneously’ published data of the period 1983: I-1985: IV. The specification used to model the residential electricity consumption was:

QFEOIK,=r+/l i (1 -i.)AjQGNIC,_j j=O

IL,

+y 1 (1 -k)k’QRPFOIK,_, j=O

4

+ GQPOZL, I + C (gj + djt )Q,j + 11, j=l

(26)

O<i< 1 O<ti<I

where

QFEOIK, = quarterly residential electricity consumption (KWh x 103)

QGNIC, = quarterly gross national income (drachma x lo”, 1970 prices)

QRPFOIK, = quarterly relative price index of residential electricity consumption (1982=100)

QPOIL, =unit value index of imports of the SITC3 section (1982= 100)

An estimate of Equation (26) is obtained using:

(i) all the available quarterly observations (Case A) (ii) only the correct published data (Case B)

(iii) all the quarterly and annual data using the correction technique (Case C)

The results are given in Table I. By evaluating the relation (23) we may obtain the

corrected quarterly residential electricity consumption data for the period 1983: I-1985: IV. These data are given in Table 2 along with the actual (without errors) published residential electricity consumption data.

In order to compare the disaggregated quarterly data with the actual we use some well known techniques. We first expressed the variables as relative (percentage) changes of the form:

where

pt = disaggregated relative percentage changes. a, = actual relative percentage changes. P, = pt -u, (error of relative percentage changes).

188 ENERGY ECONOMICS July 1992

Correcting quarterly electricity data: T. El

Table 1. Estimates of the parameters of Equation (26).

Parameters

d(

P(1 -A)

“J(1 -K)

6

42

q3

q4

R2 DW

Case A Case B

- 1484130 - 3343632

(4.3) (6.6) 49994.31(1-0.89) 66054.5(1-0.87)

(13.9) (11.9) - 1281449(1-0.7) -1118188(1-0.5)

(5.6) (5.4) -49.96 -70.7

(3.4) (3.4) -521801.8 -515450.2

(15.2) (11.3) - 1027377 ~ 1034133

(29.5) (21.9) -788122.5 ~ 803637.6

(22) (16.3)

0.97412 0.97446 1.87 2.21

Case C

- 782073

(2.7) 43235.07(1-0.9)

(14.8) ~ 1264327(1-0.7)

(6.07) -45.36

(3.4) -515505.8

(16.3) -988883.2

(30) - 734522.8

(22.5)

0.97715 1.68

No&: ( ) = t-statistics.

Table 2. Actual and corrected residential electricity consumption data.

QEOIK AEOIK FQEOIK FAEOIK

1983: 1 0.218527E+07 0.219047Ef07 1983: 2 O.l68315E+07 O.l69397E+07 1983: 3 0.127664E + 07 O.l26996E+07 1983: 4 O.l62396E+07 0.676902E+O7 0.161462E-tO7 0.676902E + 07 1984: 1 0.234673E + 07 0.229694E + 07 1984: 2 0.182964E + 07 O.l80056E+07 1984: 3 0.140534E + 07 O.l41156E+07 1984: 4 O.l65885E+07 0.724056E f07 O.l73150E+07 0.724056E + 07 1985: 1 0.256977E + 07 0.248870E + 07 1985: 2 O.l92714E+07 0,188743E+07 1985: 3 O.l41053E+07 0,147104E+07 1985: 4 O.l77758E+07 0.768502E + 07 O.l83785E+07 0.768502E +07

I.21 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ I 123412341234

1983 1984 1985

Figure 1. Actual and corrected residential electricity consumption.

A graphical presentation of the actual and the disaggregated quarterly observations as levels and as relative percentage changes are given in Figures 1 and 2 respectively.

In addition we can obtain the control chart in order to trace a possible tendency of the estimated percent-

ENERGY ECONOMICS July 1992

0.6

-0.4 ’ ’ ’ ’ ’ ’ ’ ’ ’ 1 ’ ’ 234123412341

1983 1984 1985

Figure 2. Actual and corrected residential electricity con- sumption as relative percentage change.

age relative changes to overestimate or to under- estimate the actual relative changes. The cumulative errors are plotted in Figure 3 where we observe that they all fall within the limits of the confidence bound. So we may conclude that there is no systematic

189

Correcting quarterly electricity data: T. El

IO Upper limit

CT b 6-

:

b : 2-

0 . /

f -2 - ‘0

5 E 2 -6-

Lower limit -10 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’

2 19383 412341234 1984 1985

Figure 3. Control chart.

a, = 100 ( y*t -p-I

yQt- I )

I . I

. . . . . . . .

I’

d p,=100 (

YQf -_YfJt-I YQ-I 1

Figure 4. Prediction realization diagram.

tendency for overestimation or underestimation of the actual percentage relative changes.

Another quantitative measure of the forecast error e, is the mean square error which can be mathematic- ally decomposed into three constituent elements each of which has a very distinct interpretation. In mathematical terms this decomposition can be written in the following way:

(l/(T-1)) i (~,-a,)’ t=2 1

= U”MSE + U”MSE + U’MSE =(~-ii)“+(Sp-sSn)Z+2(1-~)saSp (28)

where

U” = (j - ii)‘/MSE: bias proportion U” = (Sp-Sa)‘/MSE: variance proportion UC = 2( 1 -r)SaSp/MSE: covariance proportion

l=U”+U”+U’ (29)

In order to evaluate relation (29) we use the data from Table 1 to obtain:

1=0.0051106+0.011861+0.98303 (30)

Equation (30) indicates that 0.51% of the mean square error results from bias, 1.18% from errors of variation and 98.3% from less than perfect correlation of actual and predictive values. The small bias component tends

190

to confirm the conclusion reached earlier using the control chart that there is no persistence tendency to estimate incorrectly the magnitude percentage changes of the dependent variable.

Finally in relation to the above analysed criteria we give the prediction realization diagram in Figure 4. Examining the scatter diagram the obvious correlation between the observed and the forecasted points are shown. In all cases (19 quarterly observations) the direction of change of the dependent variable was correctly forecasted.

Using the iterative scheme given above with speci- fication (26) as basis and starting values for the non-linear least squares estimates for the subperiod T, = 1975. I,. . . , 1987.IV, we obtained the disaggre- gated observations of the residential electricity con- sumption for the subperiod TI = 1979.1,. . . , 1979.IV as shown in Table 3.

Finally using the time-aggregation matrix it is not difficult to see the consistency of the annually published and corrected electricity consumption data for the residential electricity consumption. A graphical presentation of the above results is given in Figures 5 and 6.

Conclusions

Forecasting procedures attempt to reach conclusions about future movements in data series by a systematic approach using minimum amount of information

Table 3. Residential electricity consumption data.

QEOIK FQEOIK (1) (2)

1979: 1 O.l07542E+07 O.l57316E+07 1979: 2 0.1418328+07 O.l57442E+O7 1979: 3 0,105994E+O7 O.l01071E+07 1979: 4 0.156735E307 O.l26272E+07

0.81 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 123412341234

1978 1979 1980

Figure 5. Quarterly residential electricity consumption data.

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- actual 2.6 - - - - corrected

0.61 ' ” ” ” ” ’ ” "'I "'I ” ” 1 1976 1978 1980 1982 1984 1986

Figure 6. Quarterly residential electricity consumption data.

concerning the environment generating previously observed outcomes. This study has shown how one technique involves a sophisticated way of weighting together past data values with missing or erroneously published data in a way to improve the series and make it usable in time series analysis. The success of

the method relies on both the quality and quantity of the available data as can be seen in the text above.

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Correcting quarterly electricity data: T. El

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