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On the stability of electric utility fossil fuel demand
Electric Power Systems Research, 5 (1982) 53 - 62
53
On the Stability of Electric Utility Fossil Fuel Demand NOEL D URI*
Department of Economzcs and Business Management, The Cathohc Unwers~ty of America, Washington, DC 20064 (U S A ) (Received July 20, 1981)
SUMMARY
The w e w that the regzonal demand for fossd fuels by electric utlhtzes in the United States zs characterized by stable relationships ~s subjected to an objectwe statzst~cal test The test utd~zes a statzsttc whzch equals the ratio o f the sum o f squared residuals o f one period predzctzon from the k + 1 perzod to the rth period to the sum o f squared reszduals o f one period predzctzon from the k + 1 to the Tth period, where k ts the n u m b e r o f estzmated coeffzclents m the model and T ~s the sample szze The results suggest, for the period 1973 1978, that the fosszl fuel demand functlons m Regzon H and Region VIII and the coal demand function zn Regzon X became unstable around 1977 For the other regions, the results mdzcate no appreczable (stattstzcally slgmfzcant) change ~n the relatwe zmportance o f the underlying determinants o f the demand for fossd fuels
INTRODUCTION
In a recent paper, Un [1] develops and Implements a regmnal short-term forecasting structure for the demand for fossil fuel by electric utflltms in the United States. The methodology m based on a translog price possibility frontmr and exphcltly takes into account the existence of lnterfuel substitution in the generation of electrical energy. Using regional data for ten federal regions, the results suggest that the responsiveness of the
*The author IS an econommt wlth the U S Department of Energy The vlews expressed are those of the author and do not necessarily represent the policies of the Department of Energy or the views of other Department of Energy staff members 0378-7796/82/0000-0000/$02 75
demand for coal, residual fuel oil, distillate fuel o11, and natural gas by electric utilltms to relative price changes is significant In a forecasting setting, the demand model performs remarkably well when actual and forecast values for 1979 are compared. The essence of the model is to produce high frequency (quarterly) forecasts that will provide the planner and pohcy maker with some mslght mto fossil fuel consumption by type Uneqmvocally, the model is adequate for this purpose over a one year horizon. Is It appropriate over a longer period9 This comes down to the question of whether the observed structural relationships have been stable over time. (Stability is defined m the statistical sense of the estimated coefflcmnts of the explanatory varmbles remmnmg constant over time.) Thin Is questmnable. There is no reason to believe a p n o n that m the dynamic environment of the energy arena over the past decade the relative importance of the factors influencing the demand for fossil fuels m the generatmn of electrical energy has remained unchanged One reason for the absence of an empmcal mvestlgatmn of this issue rests on the fact that the usual procedures for testing whether a regression relationship is constant over time in the statlstmal sense, namely the use of the F-test [2] or d u m m y vamables [3], reqmre prior knowledge of the point m time when the change in the estimated coefflcmnts is suspected Except for obvious dramatic changes such as a war, the knowledge may not be readily available The questmn of the stability of the functional relationships underlying fossil fuel demand is of cntmal importance. Forecasts are made on the basis of past behavior. If this has been subject to change, then any references drawn from the observed value, especl© Elsevier Sequola/Printed m The Netherlands
54 ally as the forecast horizon is ext ended, will be at least in part unsatisfactory One must be reasonably certain that the demand for fossil fuels has n o t changed structurally The purpose of this paper, then, is to determine whether stable demand for fossil fuels used in the generation of electrical energy existed over the period 1973 - 1978 This is accomphshed by utlhzlng a statlstmal test developed by Brown e t al [4] that is designed to d e t e c t shifts in a regression relationship This approach is a d o p t e d in preference to those previously m e n t i o n e d because it does n ot require prior knowledge of the shifts but rather tests for the presence of such occurrences over the sample pe r m d (Note t ha t there appears to have been little previous use of this test because o f the complexltms of applying it One ap p h cat m n is that by Kahn [5] lookmg at the stablhty of the dem a nd for m o n e y ) Before this is done, however, it is useful to briefly review the previous exposition of the model
A RECAPITULATION
It was h y p o t h e s i z e d that there exists a twice dlfferentlable p r o d u c t i o n frontier relating the generatton of electrmal energy to the inputs of capital, labor, and energy inputs coal, residual fuel oil, distillate fuel oil, and natural gas Corresponding t o the p r o d u c t i o n posslblhty frontmr is a price possibility frontier where the cost of generating electrmal energy is a f u n c t m n of the factor input prmes and a vector of exogenous factors (e g , weather, natural gas curtailments, etc ) N e x t it was assumed that the fossil fuels constitute a separable and h o m o g e n e o u s energy aggregate In specifying an exact functmnal form having the minimal a p n o r t restrictions, the translog price posslblhty frontmr was selected It is expressed as
log PE
= ~ a t l o g Pt + 1
+ . ~ - Z Z T t , l ogp, logPj + t
I
+ ~ ~k log Z k + k
+ g1 ~ k
CkllogZ~ l o g Z t + !
+ ~8,h t
logP, l o g Z k
(1)
k
where PE denotes the prme of energy, P,, t = c, r, d, g, denotes the price of coal, residual fuel oil, distillate fuel oil, and natural gas, respectwely, Zk, k = 1, 2, 3, 4, denotes the total fossil fuel generatmn (for k = 1), the n u m b e r of heating degree days (for k = 2), the n u m b e r of cooling degree days (for k = 3), gas curtailments (for k = 4), and at, %j, ~3k, Ckt, and 8 tk are parameters In deriving a forecasting equation, use was made of Shephard's lemma that perm i t t ed the derwatlon of cost share equations, S,, for each of the fossil fuels of the form S, = at + ~, "r,, log Pj + ~ 1
7,k log Zk
(2)
k
for i = c, r, d, g and k = 1, 2, 3, 4 These four relationships serve as the basis for forecasting the demand for fossil fuels Consequently, t h e y are the ones t hat must be subjected to the test of stablhty Before domg so, it is instructive to discuss the test because it is n o t y e t a c o m m o n element m the researcher's tool k~t
TEST
OF STABILITY
As previously noted, statlstmal tests made to ascertain w het her a regression relationship over the sample horizon is stable have been hmlted either to the use of d u m m y variables when a change in the relationship is suspected or to sphttmg the sample period at t hat p o m t and performing a Chow test Both of these techniques reqmre a przorz knowledge of the p o m t m time when the function shifts and this reformation might n o t be available A m e t h o d of d e t e r m m m g w het her a regression relationship is constant over a gaven period based on an examination of the residuals of the relationship has been developed by Brown e t al [4] This procedure will be used to test the stability of the investment function previously developed To gave an appreciation of the test, it is brmfly discussed here Essentially, thin approach necessitates the computation of one-period prediction residuals, whmh are obtained by applying the regression estimated with r -- 1 observations to predict the rth observation using k expl anat ory variables (mcludmg the constant) The m e t h o d is based
55
on a test statistic S(r), which equals the ratio o f the sum o f squared residuals of one period prediction from the (k + 1)th period to the rth period to the sum of squared residuals of one period prediction from the (k + 1)th to the Tth period, where T denotes the sample size The null hypothesis that the regression relationship is constant over time implies that the ex p ected value of the test statistic S(r), E [ S ( r ) ] , will lie along (in a statistical sense) its mean value line Consider the basic regression model
Yt =x't 8t + ~t
t = 1, 2,
,T
(3)
where Yt is a vector of observations on the d e p e n d e n t variable, x t is a column vector of observations on k regressors, 0t is a vector of regression coefficients (the subscript t implylng that the 0s may n o t be constant over tune) and ~t is a vector o f normally and indep en d en tly distributed variables with mean zero and variables 0 2, t = 1, 2, , T. The first element in xt will be taken to equal unity for all values of t since the model contams a constant The hypothesis of constancy over time, which is d e n o t e d by H0, is 01 ---- 0 2 ----
0-2
=
G2
-----
---- 0 T = 0 _-- 0 . 2 _--
0.2
The test is mo r e concerned with detecting differences among t he 0s than among the as Let br be the least squares estimate from the sample o f the first r observations and let F Yr -- xrbr- 1 Wr =
[1 + X ' r ( X ' _ l X r _ l ) - l x r ] 1/2
r=k+
l,k+
2....
T
(4)
where X' r - 1 = [ x l , x 2 . . . . . Xr-1] It is shown by Brown et al t hat these transformed residuals, Wk- I, W~+2. . . . , are normally distributed i ndependent variables with zero means and finite variances. Equation (4) is just a generalization of the Helmert orthogonal transformation The variables Wr can be obtained w i t h out repeated matrix Inversion by the relationships br
= br_l
+
(X'rXr)-lx'r~Y --x'rbr-1)
and (/'r)l'r) - 1 = ( / ' r _ i X r _ l ) -1 +
(5)
(X'r_ 1 X r _ 1) ?
-I
, , Xrxr(Xr-lXr-1) -1
I
1 + xr(Xr-,X,-1)-lXr
(6)
If 0t is constant up to period t = to and different from then on, t he Wr will have mean zero up to to and non-zero mean from then on The suggestion is made to plot the followmg variable
/=k+l S(r) =
T
Z
r = k + 1, k + 2,
,T
w?
/=k+l
The value of S(r) will lie between zero and one S ( r ) = 0 i f r = k + l , S ( r ) = I I f r = T . T h e expected value of S(r) is E[S(r)] = (r -- })/ (T -- }) and, given the null hypothesis that the 8s are constant, the plot of S(r) should be along this mean-valueline Two-sided slgnlheance tests are performed by drawing a pmr of lines S(r) = +Co + (r -- })/ (T -- }) parallel to the mean-valuellne. If the sample path of S(r) crosses either of these slgnlhcance lines, the null hypothesis of the constancy of the regression coefficients is rejected The statistic Co is distributed as Pike's modified K o l m o g o r o v - S m l r n o v statistic and values for various levels of significance are provided in D urbm [6] This test of stability is approxi m at e and tends to be optimistic. It IS suggested, however, that the approxi m at i on is good when T is large, as is the case with the problem currently under investigation, and when the residuals ~t are serially uncorrelated The n e x t step is to Implement the test Before doing so, recall that the data originally used to estimate the share equations consisted of quarterly data for ten D e p a r t m e n t of Energy regions*. Fossil fuel c o n s u m p t i o n and *Reglon I Connectmut, Massachusetts,Marne, N e w Hampshlre, Rhode Island and Vermont, Reglon II N e w Jersey and N e w York, Region Ill Dlstnct of Columbla, Delaware, Maryland, Pennsylvania, Vlrglma and West Vlrglma, Region IV Alabama, Florida, Georgia, Kentucky, Mmsmslppl, North Carohna, South Carolina, and Tennessee, Reglon V llhnom, Indiana, Michigan, Minnesota, Ohlo, and Wmconsm, Reglon VI Arkansas, Loumlana, N e w Mexico, Oklahoma, and Texas, Region VII Iowa, Kansas, Mmsourl,and Nebraska, Reglon VIII Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming, Reglon IX Arizona, Cahfornla, and Nevada, Reglon X Idaho, Oregon, and Washington
56 price data were obtained from the Federal Energy Regulatory Commission as were generation data Weather data came from the National Oceanic and Atmospheric Administration while estimates of natural gas curtailments came from a D e p a r t m e n t of Energy survey
EMPIRICAL RESULTS The fuel share equations were estimated, as alluded to previously, with serial correlation being corrected for via the C o c h r a n e - O r c u t t [7] lteratlve t e c h m q u e Because of the stability test requirements, no e f f o r t was made to impose the s y m m e t r y contramts, %j = 7j, The sample plots of the statistic S(r) for each of four fossil fuels (coal, residual fuel oil, distillate fuel oil, and natural gas) for each of the ten regions are given in Figs 1 - 10 Since there are nine parameters estimated for each share equation, the test is made for the period 1975 Q2 {quarter two) through 1978 Q4
(quarter 4) On each Figure ,s drawn a set of confidence lines corresponding to a significance level of 10% These values were obtained as previously indicated From the Figures it can be observed t hat the sample path of S(r) travels outside the 10% confidence bands for all fuels in Region II and Regmn VIII and for coal in Regmn X In each instance, the S(r) crosses the 10% significance line sometime during the 1977 period In the c o n t e x t of stability, this implies a structural shift m the demand for the specific fossil fuels during this period In the case of Regmn II, the result can be traced to the precipitous increase in the world price of crude oil In 1978, for example, 78% of the total generation was by residual fuel oil and distillate fuel oil of which slightly m excess of 50% came from crude oil which was i m port ed The effects are cumulative so that while the price increase in crude oil did n o t immediately have an unsettling effect, by 1977, however, the changes brought about by such an increase had the effect of upsetting
(a) COAL
(b) RESIDUAL FUEL OIL
100
100
075
0 75
0S0
"~ 0OO (n
025
000
/jj
fJ"
O25
000 1975
1976
1977
1975
1978
(c) DISTILLATE FUEL OIL lOO
o.
y l /
o,,
_
I~f
•
.
i
.
1976
Fig 1 Region I
~
1977 YEAR
1977
1978
(d) NATURAL GAS ~ ~
/
026 OOO ~ . , 1975
1976 YEAR
YEAR
i
o
.
.
1978
~
~o"
0O0 1975
1976 YEAR
1977
1978
57
(a) COAL
(b) RESIOUAL
lO0
100
O76
o 76
S
OSO
FUEL
OIL
" 060 025
025
ooo
1975
I/~'~i 1976
",
. • 1977
I
• . , I 1978
• i J', , I 1975 1976
0oo
YEAR
.
, , i 1977
,
, , 1978
i
,
,
I
* * , 1978
I
YEAR
(C) DISTILLATE FUEL OIL
(d) NATURAL
GAS
tO0
/ °7"I"
¢'~" ~,,,~"
f ,,
I /
O76
'= c4
:: ..L':,./.-',:-.., ..., 197S
1976
1977
O6O
O26
• .
000
1976
1976
, i
,
1976
YEAR
,
, I
,
1977
1976
YEAR
Fig 2 Reglon II (8) COAL
.=
(b) RESIDUAL
.o.//
o.~
076
//
000
~,~-
g , I.~, . . 1976 1976
i
,
, , 1977
I
, , , i 1976
O00V',
""r
_
I
.
. * I 1977
YEAR FUEL
OIL
(d) NATURAL
I oo
,.,,,/,4 I/
o-I-
J ~/' "~
, I #, . I 1976 1976
YEAR (C) D I S T I L L A T E
OIL
f/
f
! . V°,'7 o001-
06O
025
FUEL
,p ..V
~
GAS
C ~
,
~
076
//~+" "~
{4
..,TJ
OW
~
o. 0 OO I,P'; , I X . . .I . . . 1976 1976 1977 YEAR
Fig 3 Region III
I
i i 19711
|
000
"~" 1976
I /~1)1
-'
1976
" " I 1977
YEAR
" " " 1978
I
58
(b) RESIDUAL FUEL OIL
(a) COAL 100
,oor
o~,/f
O75
O5O
,~
,,,,o~
"9"
0 25
0OO
,.,,-'~'-, I / . . . 1975 1976
I
~ , 1977
I
, , ,
i
0 00 ~
1978
~ ~ 1975
J
JJ ~ m 1976
J
m d 1977
m ~ 1978
YEAR
YEAR
(d) NATURAL GAS
(c) DISTILLATE FUEL OIL 100~
.=
,,.,,~j /
o~
0 75
O5O
0 25
?: ......
000 1975
1976
1977
000
,
I I ,
1975
1975
,
1
,
1976
,
,
I
,
,
1977
.
i
1978
YEAR
YEAR
Fig 4 Region IV
(a) COAL
(b) RESIDUAL FUEL OIL
"r
100
~
075
.~°oo~ + o ~ j _./,,0 25
ooo
~o,,i-
,//. ~7/
19"/ ~"
4 ~"
000 1975
1976
1977
M"
1975
1978
, I 1976
,
. , I 1977
, , , 1978
i
YEAR
YEAR
(d) N A T U R A L GAS
(c) DISTILLATE FUEL OIL 100
O75
075
O5O
.,¢o
ooo
, ~ 1 ,1975 / I . . . , . 1977 ..,
1975
YEAR
Fig 5 Regmn V
000 1975
1975
, 1 , , , I 1976 1977 YEAR
,
, , 1978
J
59 (a) COAL
0
(b) RESIDUAL FUEL OIL
J/ .+:Y / /
100
75
/~"
+V
O50
-
100
0 75
--~ 050 (n
0 25
y/
~,o"
0 25
000
1975
1976
1977
,
0 O0
1978
•
I
1975
YEAR
4/
,~/
0
75
. , i I , , i I 1976 1977 1978 YEAR
1975
.
,
.
I
,
1977
n
I
1978
~
I'Y
= o.
000
I
(d) NATURAL GAS
~,.~,~
000
•
1976 YEAR
(c) DISTILLATE FUEL OIL
0 75
/ .
I~"
0
1975
1976 1977 YEAR
1978
Fig 6 Region VI
(a) COAL
(b) RESIDUAL FUEL OIL
~,>~///
loo
1-F
0 75
.J
"~ 0 50 ~ ~
025
O25
000V,
,
1975
I/,
,
,
I
,
1976
•
•
I
*
1977
*
,
I
1978
O00
~/ . I . . .I 1976 1977
19~
YEAR
loo
(C)
DISTILLATE
, , , ! 1978
YEAR FUEL
OIL
,,,,~./
000
(d) NATURAL GAS
~0,00~ ~
~
02S 000
"
'
1975
'
I
'
'
"
1976 1977 YEAR
Fig 7 Region VII
I
I
*
|
1978
J
000 1975
1976 1977 YEAR
1978
I
60
(b) RESIDUAL
(a) COAL
lOO"
FUEL OIL
100
f
0 75
.~ r,/)
/
oGo
/ ./~" o 25
O25
00o
~
iL
1975
.
.
i
~
1976
•
I
.
.
1977
.
000
I
1979
.,~<'/.,~,..., ..., 1975
1976
YEAR (C) D I S T I L L A T E
(d) NATURAL
FUEL OIL
.....//.
,oor o75
1977
1978
YEAR
lOO~
1~
GAS
.,+/
075
r#)
O60
025
0001¢0; , I X • , 1976 1975
I
,
, , 1977
t
,
, , 1975
i
000
% i
II.
1975
• I • • , I 1975
8
Region
lOO
~
o.
(b) RESIDUAL ,
~
/
o76
~.y/ /
~o, /,~@@"
¢ i 1975
J /
O25
, . . 1979
197/
I
, , , t
000
, I , , . i 1975
1978
1976
-
'=V
~C-'4
,~y
/
o.l-.
.
o,,/ / ~ j,/.,', 1975
1976
000~1 1977
YEAR
Fig 9 Region IX
(d) NATURAL
FUEL OIL
100'
0 76
1977
, . , i 1978
YEAR
YEAR (C) D I S T I L L A T E
FUEL OIL
lOO
.,.o~//
000
1970
VIII
(a) COAL
025
• . . I
YEAR
YEAR Fig
1977
1978
GAS
o+/.~. ~,y
/
i I .X". • I 1975 1975
M o''
,
YEAR
, , I 1977
•
, • 1978
I
61
(a) COAL tO0
100
O76
o 79
# O6O
-
(b) RESIDUAL FUEL OIL /
,j
~ooo
/
O26
026
/,e" •
0OO
I
,
,
1976
1976
•
I
•
1977
•
•
I
OOO
1978
1975
100
100
O76
076
o~
1978
O6O (4
026
026
0OO
1977
(d) NATURAL GAS
(c) DISTILLATE FUEL OIL
~
1976 YEAR
YEAR
.
.
1976
.
.
I
.
19711
, 1977
,
I
•
•
,
|
1978
0OO 1976
1971
1977
1978
YEAR
YEAR
Fig i0 Regmn X
the hmtormal mterrelatmnshlps between the factors mfluencmg the demand for fossil fuels. In the case of the demand for fossil fuels m Region VIII, the instability m a reflectmn of the very rapid economm growth m the region (Per capita income grew over the 1973 - 1978 p e n o d at more than 8% per year ) In partmular, total generatmn m 1978 was 97% higher (14% per year) than m 1973 with all of thin coming from mcreased use of coal. It is not surprising that the mcreased use of coal resulted since coal is indigenous to the region while the other fossil fuels must be (for the most part) imported The instability m the use of coal m Region X arose m 1977 due to the hydroelectric shortage over the January 1977 - O c t o b e r 1977 period. A slgmfmant portion of the defmlt m generatmn due to the water shortage was made up by increased nuclear generation [8]. The remmnder of the shortfall was satinfled by increased use of coal destabillzmg this partmular functmnal relationship It m interesting to note that the other fossil fuels con-
tmued to respond d u n n g this period as they did over the entire historical permd.
IMPLICATIONS
Major attention has focused on the demand for fossil fuel mputs m the generatxon of electrmal energy. Of slgmfmant concern m the context of drawing meaningful references over the hmtormal period as well as over the forecast horizon is whether the observed relationships (e.g., the pnce elasticmes) are stable. The lmphcatlons of the stability test are clear Events over the years 1973 - 1978 have had the effect of destabilizing the demand for fossil fuels m Region II and Region VIII and for coal m Region X, while leaving vtrtually unchanged the demand for coal, residual fuel oil, distillate fuel o11, and natural gas That is, for the majority of regions, the relative Importance of the pnce of coal, the price of residual fuel oil, the price of dmtillate fuel oil and the price of natural gas together with weather, generation and natural gas curtail-
62 m e n t s in Influencing the share o f total expenditures (and h e n c e d e m a n d ) has r e m a i n e d constant One m u s t be careful, however, to avoid inferring t h a t t h e q u a n t i t y d e m a n d e d o f coal, residual fuel oil, distillate fuel o11, and natural gas r e m a i n e d u n c h a n g e d in the stable regions The e s t i m a t i o n results clearly s h o w t h e price o f all of the fossil fuels influencing each e x p e n d i t u r e share Thus, an increase in the price o f coal does lead to an increase m the q u a n t i t y o f natural gas used In the g e n e r a t i o n process The m a g n i t u d e o f this response for each o f the fossil fuel inputs for m o s t regions r e m a i n e d u n a l t e r e d over the sample p e r i o d A n o t h e r w a y o f expressing this is t h a t the e x p e n d i t u r e share elasticities for the fossil fuels did n o t vary F o r the regions d e m o n s t r a t i n g m s t a b l h t y , on the o t h e r h a n d , such a c o n c l u s i o n is n o t f o r t h c o m i n g T h e r e is an Identifiable shift in the relative i m p o r t a n c e o f the factors Influencing the d e m a n d for fossil fuels U n f o r t u nately, the p a u c i t y o f data precludes a precise m e a s u r e m e n t o f the e x t e n t o f this shift (i e , n o t e n o u g h degrees o f f r e e d o m exist to empirically estimate t h e share e q u a t i o n s post1 9 7 7 ) T h e n a t u r e o f the change, h o w e v e r , was m a n i f e s t in r e o p t l m l z l n g the fossil fuel mix m such a w a y as t o take a c c o u n t o f fuel availability and relatively larger price differentials In a forecasting setting the l n s t a b I h t y presents quite a p r o b l e m for longer-run considerations As can be seen f r o m the s h o r t - r u n forecasts, the stability or lack t h e r e o f pres e n t e d little p r o b l e m with regard t o f o r e c a s t a c c u r a c y In the larger setting (i e , t h e longerrun), the e s t i m a t e d coefficients in R e g i o n II,
R e g i o n V I I I and R e g i o n X are n o t a d e q u a t e reflections o f the w a y electric utlhtles actually altered their fuel e x p e n d i t u r e s for s o m e o f the sample p e r i o d w h i c h gives rise t o a c o n c e r n t h a t f u t u r e behavior will m o r e closely e m u l a t e the responsiveness over the unstable p e r i o d t h a n over the stable p e r i o d leading t o unacceptable forecasts and h e n c e p o o r p o h c y based on t h o s e forecasts There is n o easy r e s o l u t i o n t o this d i l e m m a N o t e n o u g h d a t a or u n d e r s t a n d i n g o f the institutional anomalies exist to i m p r o v e on the situation F o r the time being, it is simply a fact o f hfe t h a t m u s t be a c c e p t e d
REFERENCES 1 N D Urn, Forecasting fossil fuel demand, Electric Power Systems Research, 4 (1981) 247 - 254 2 G Chow, Tests of equahty between two sets of coefhclents m two linear regressions, Econometr~ca, 28 (1960) 591 - 605 3 J Kmenta, Elements o f Econometrics, MacMillan, New York, 1971 4 R L Brown, J Durbm and J M Evans, Techtuques for testing the constancy of regression relationships over time, J R Star S o c , 37 (1975) 149 - 163 5 M S Kahn, The stability of the demand for money function m the U S 1901 - 1965,J Pohtzcal E c o n , 82 (1974) 1205 - 1219 6 J Durbm, Tests for serial correlation m regression analysis based on the penodogram of least-squares residuals, Bmmetrtka, 56 (1969) 1 - 15 7 D Cochrane and G H Orcutt, Apphcatlons of least squares regressions to relationships containing autocorrelated error terms, J A m Star A s s o c , 44 (1949) 32- 61 8 Edlson Electric Institute, Statistical Yearbook, Edlson Electric Institute, New York, annual
53
On the Stability of Electric Utility Fossil Fuel Demand NOEL D URI*
Department of Economzcs and Business Management, The Cathohc Unwers~ty of America, Washington, DC 20064 (U S A ) (Received July 20, 1981)
SUMMARY
The w e w that the regzonal demand for fossd fuels by electric utlhtzes in the United States zs characterized by stable relationships ~s subjected to an objectwe statzst~cal test The test utd~zes a statzsttc whzch equals the ratio o f the sum o f squared residuals o f one period predzctzon from the k + 1 perzod to the rth period to the sum o f squared reszduals o f one period predzctzon from the k + 1 to the Tth period, where k ts the n u m b e r o f estzmated coeffzclents m the model and T ~s the sample szze The results suggest, for the period 1973 1978, that the fosszl fuel demand functlons m Regzon H and Region VIII and the coal demand function zn Regzon X became unstable around 1977 For the other regions, the results mdzcate no appreczable (stattstzcally slgmfzcant) change ~n the relatwe zmportance o f the underlying determinants o f the demand for fossd fuels
INTRODUCTION
In a recent paper, Un [1] develops and Implements a regmnal short-term forecasting structure for the demand for fossil fuel by electric utflltms in the United States. The methodology m based on a translog price possibility frontmr and exphcltly takes into account the existence of lnterfuel substitution in the generation of electrical energy. Using regional data for ten federal regions, the results suggest that the responsiveness of the
*The author IS an econommt wlth the U S Department of Energy The vlews expressed are those of the author and do not necessarily represent the policies of the Department of Energy or the views of other Department of Energy staff members 0378-7796/82/0000-0000/$02 75
demand for coal, residual fuel oil, distillate fuel o11, and natural gas by electric utilltms to relative price changes is significant In a forecasting setting, the demand model performs remarkably well when actual and forecast values for 1979 are compared. The essence of the model is to produce high frequency (quarterly) forecasts that will provide the planner and pohcy maker with some mslght mto fossil fuel consumption by type Uneqmvocally, the model is adequate for this purpose over a one year horizon. Is It appropriate over a longer period9 This comes down to the question of whether the observed structural relationships have been stable over time. (Stability is defined m the statistical sense of the estimated coefflcmnts of the explanatory varmbles remmnmg constant over time.) Thin Is questmnable. There is no reason to believe a p n o n that m the dynamic environment of the energy arena over the past decade the relative importance of the factors influencing the demand for fossil fuels m the generatmn of electrical energy has remained unchanged One reason for the absence of an empmcal mvestlgatmn of this issue rests on the fact that the usual procedures for testing whether a regression relationship is constant over time in the statlstmal sense, namely the use of the F-test [2] or d u m m y vamables [3], reqmre prior knowledge of the point m time when the change in the estimated coefflcmnts is suspected Except for obvious dramatic changes such as a war, the knowledge may not be readily available The questmn of the stability of the functional relationships underlying fossil fuel demand is of cntmal importance. Forecasts are made on the basis of past behavior. If this has been subject to change, then any references drawn from the observed value, especl© Elsevier Sequola/Printed m The Netherlands
54 ally as the forecast horizon is ext ended, will be at least in part unsatisfactory One must be reasonably certain that the demand for fossil fuels has n o t changed structurally The purpose of this paper, then, is to determine whether stable demand for fossil fuels used in the generation of electrical energy existed over the period 1973 - 1978 This is accomphshed by utlhzlng a statlstmal test developed by Brown e t al [4] that is designed to d e t e c t shifts in a regression relationship This approach is a d o p t e d in preference to those previously m e n t i o n e d because it does n ot require prior knowledge of the shifts but rather tests for the presence of such occurrences over the sample pe r m d (Note t ha t there appears to have been little previous use of this test because o f the complexltms of applying it One ap p h cat m n is that by Kahn [5] lookmg at the stablhty of the dem a nd for m o n e y ) Before this is done, however, it is useful to briefly review the previous exposition of the model
A RECAPITULATION
It was h y p o t h e s i z e d that there exists a twice dlfferentlable p r o d u c t i o n frontier relating the generatton of electrmal energy to the inputs of capital, labor, and energy inputs coal, residual fuel oil, distillate fuel oil, and natural gas Corresponding t o the p r o d u c t i o n posslblhty frontmr is a price possibility frontier where the cost of generating electrmal energy is a f u n c t m n of the factor input prmes and a vector of exogenous factors (e g , weather, natural gas curtailments, etc ) N e x t it was assumed that the fossil fuels constitute a separable and h o m o g e n e o u s energy aggregate In specifying an exact functmnal form having the minimal a p n o r t restrictions, the translog price posslblhty frontmr was selected It is expressed as
log PE
= ~ a t l o g Pt + 1
+ . ~ - Z Z T t , l ogp, logPj + t
I
+ ~ ~k log Z k + k
+ g1 ~ k
CkllogZ~ l o g Z t + !
+ ~8,h t
logP, l o g Z k
(1)
k
where PE denotes the prme of energy, P,, t = c, r, d, g, denotes the price of coal, residual fuel oil, distillate fuel oil, and natural gas, respectwely, Zk, k = 1, 2, 3, 4, denotes the total fossil fuel generatmn (for k = 1), the n u m b e r of heating degree days (for k = 2), the n u m b e r of cooling degree days (for k = 3), gas curtailments (for k = 4), and at, %j, ~3k, Ckt, and 8 tk are parameters In deriving a forecasting equation, use was made of Shephard's lemma that perm i t t ed the derwatlon of cost share equations, S,, for each of the fossil fuels of the form S, = at + ~, "r,, log Pj + ~ 1
7,k log Zk
(2)
k
for i = c, r, d, g and k = 1, 2, 3, 4 These four relationships serve as the basis for forecasting the demand for fossil fuels Consequently, t h e y are the ones t hat must be subjected to the test of stablhty Before domg so, it is instructive to discuss the test because it is n o t y e t a c o m m o n element m the researcher's tool k~t
TEST
OF STABILITY
As previously noted, statlstmal tests made to ascertain w het her a regression relationship over the sample horizon is stable have been hmlted either to the use of d u m m y variables when a change in the relationship is suspected or to sphttmg the sample period at t hat p o m t and performing a Chow test Both of these techniques reqmre a przorz knowledge of the p o m t m time when the function shifts and this reformation might n o t be available A m e t h o d of d e t e r m m m g w het her a regression relationship is constant over a gaven period based on an examination of the residuals of the relationship has been developed by Brown e t al [4] This procedure will be used to test the stability of the investment function previously developed To gave an appreciation of the test, it is brmfly discussed here Essentially, thin approach necessitates the computation of one-period prediction residuals, whmh are obtained by applying the regression estimated with r -- 1 observations to predict the rth observation using k expl anat ory variables (mcludmg the constant) The m e t h o d is based
55
on a test statistic S(r), which equals the ratio o f the sum o f squared residuals of one period prediction from the (k + 1)th period to the rth period to the sum of squared residuals of one period prediction from the (k + 1)th to the Tth period, where T denotes the sample size The null hypothesis that the regression relationship is constant over time implies that the ex p ected value of the test statistic S(r), E [ S ( r ) ] , will lie along (in a statistical sense) its mean value line Consider the basic regression model
Yt =x't 8t + ~t
t = 1, 2,
,T
(3)
where Yt is a vector of observations on the d e p e n d e n t variable, x t is a column vector of observations on k regressors, 0t is a vector of regression coefficients (the subscript t implylng that the 0s may n o t be constant over tune) and ~t is a vector o f normally and indep en d en tly distributed variables with mean zero and variables 0 2, t = 1, 2, , T. The first element in xt will be taken to equal unity for all values of t since the model contams a constant The hypothesis of constancy over time, which is d e n o t e d by H0, is 01 ---- 0 2 ----
0-2
=
G2
-----
---- 0 T = 0 _-- 0 . 2 _--
0.2
The test is mo r e concerned with detecting differences among t he 0s than among the as Let br be the least squares estimate from the sample o f the first r observations and let F Yr -- xrbr- 1 Wr =
[1 + X ' r ( X ' _ l X r _ l ) - l x r ] 1/2
r=k+
l,k+
2....
T
(4)
where X' r - 1 = [ x l , x 2 . . . . . Xr-1] It is shown by Brown et al t hat these transformed residuals, Wk- I, W~+2. . . . , are normally distributed i ndependent variables with zero means and finite variances. Equation (4) is just a generalization of the Helmert orthogonal transformation The variables Wr can be obtained w i t h out repeated matrix Inversion by the relationships br
= br_l
+
(X'rXr)-lx'r~Y --x'rbr-1)
and (/'r)l'r) - 1 = ( / ' r _ i X r _ l ) -1 +
(5)
(X'r_ 1 X r _ 1) ?
-I
, , Xrxr(Xr-lXr-1) -1
I
1 + xr(Xr-,X,-1)-lXr
(6)
If 0t is constant up to period t = to and different from then on, t he Wr will have mean zero up to to and non-zero mean from then on The suggestion is made to plot the followmg variable
/=k+l S(r) =
T
Z
r = k + 1, k + 2,
,T
w?
/=k+l
The value of S(r) will lie between zero and one S ( r ) = 0 i f r = k + l , S ( r ) = I I f r = T . T h e expected value of S(r) is E[S(r)] = (r -- })/ (T -- }) and, given the null hypothesis that the 8s are constant, the plot of S(r) should be along this mean-valueline Two-sided slgnlheance tests are performed by drawing a pmr of lines S(r) = +Co + (r -- })/ (T -- }) parallel to the mean-valuellne. If the sample path of S(r) crosses either of these slgnlhcance lines, the null hypothesis of the constancy of the regression coefficients is rejected The statistic Co is distributed as Pike's modified K o l m o g o r o v - S m l r n o v statistic and values for various levels of significance are provided in D urbm [6] This test of stability is approxi m at e and tends to be optimistic. It IS suggested, however, that the approxi m at i on is good when T is large, as is the case with the problem currently under investigation, and when the residuals ~t are serially uncorrelated The n e x t step is to Implement the test Before doing so, recall that the data originally used to estimate the share equations consisted of quarterly data for ten D e p a r t m e n t of Energy regions*. Fossil fuel c o n s u m p t i o n and *Reglon I Connectmut, Massachusetts,Marne, N e w Hampshlre, Rhode Island and Vermont, Reglon II N e w Jersey and N e w York, Region Ill Dlstnct of Columbla, Delaware, Maryland, Pennsylvania, Vlrglma and West Vlrglma, Region IV Alabama, Florida, Georgia, Kentucky, Mmsmslppl, North Carohna, South Carolina, and Tennessee, Reglon V llhnom, Indiana, Michigan, Minnesota, Ohlo, and Wmconsm, Reglon VI Arkansas, Loumlana, N e w Mexico, Oklahoma, and Texas, Region VII Iowa, Kansas, Mmsourl,and Nebraska, Reglon VIII Colorado, Montana, North Dakota, South Dakota, Utah, and Wyoming, Reglon IX Arizona, Cahfornla, and Nevada, Reglon X Idaho, Oregon, and Washington
56 price data were obtained from the Federal Energy Regulatory Commission as were generation data Weather data came from the National Oceanic and Atmospheric Administration while estimates of natural gas curtailments came from a D e p a r t m e n t of Energy survey
EMPIRICAL RESULTS The fuel share equations were estimated, as alluded to previously, with serial correlation being corrected for via the C o c h r a n e - O r c u t t [7] lteratlve t e c h m q u e Because of the stability test requirements, no e f f o r t was made to impose the s y m m e t r y contramts, %j = 7j, The sample plots of the statistic S(r) for each of four fossil fuels (coal, residual fuel oil, distillate fuel oil, and natural gas) for each of the ten regions are given in Figs 1 - 10 Since there are nine parameters estimated for each share equation, the test is made for the period 1975 Q2 {quarter two) through 1978 Q4
(quarter 4) On each Figure ,s drawn a set of confidence lines corresponding to a significance level of 10% These values were obtained as previously indicated From the Figures it can be observed t hat the sample path of S(r) travels outside the 10% confidence bands for all fuels in Region II and Regmn VIII and for coal in Regmn X In each instance, the S(r) crosses the 10% significance line sometime during the 1977 period In the c o n t e x t of stability, this implies a structural shift m the demand for the specific fossil fuels during this period In the case of Regmn II, the result can be traced to the precipitous increase in the world price of crude oil In 1978, for example, 78% of the total generation was by residual fuel oil and distillate fuel oil of which slightly m excess of 50% came from crude oil which was i m port ed The effects are cumulative so that while the price increase in crude oil did n o t immediately have an unsettling effect, by 1977, however, the changes brought about by such an increase had the effect of upsetting
(a) COAL
(b) RESIDUAL FUEL OIL
100
100
075
0 75
0S0
"~ 0OO (n
025
000
/jj
fJ"
O25
000 1975
1976
1977
1975
1978
(c) DISTILLATE FUEL OIL lOO
o.
y l /
o,,
_
I~f
•
.
i
.
1976
Fig 1 Region I
~
1977 YEAR
1977
1978
(d) NATURAL GAS ~ ~
/
026 OOO ~ . , 1975
1976 YEAR
YEAR
i
o
.
.
1978
~
~o"
0O0 1975
1976 YEAR
1977
1978
57
(a) COAL
(b) RESIOUAL
lO0
100
O76
o 76
S
OSO
FUEL
OIL
" 060 025
025
ooo
1975
I/~'~i 1976
",
. • 1977
I
• . , I 1978
• i J', , I 1975 1976
0oo
YEAR
.
, , i 1977
,
, , 1978
i
,
,
I
* * , 1978
I
YEAR
(C) DISTILLATE FUEL OIL
(d) NATURAL
GAS
tO0
/ °7"I"
¢'~" ~,,,~"
f ,,
I /
O76
'= c4
:: ..L':,./.-',:-.., ..., 197S
1976
1977
O6O
O26
• .
000
1976
1976
, i
,
1976
YEAR
,
, I
,
1977
1976
YEAR
Fig 2 Reglon II (8) COAL
.=
(b) RESIDUAL
.o.//
o.~
076
//
000
~,~-
g , I.~, . . 1976 1976
i
,
, , 1977
I
, , , i 1976
O00V',
""r
_
I
.
. * I 1977
YEAR FUEL
OIL
(d) NATURAL
I oo
,.,,,/,4 I/
o-I-
J ~/' "~
, I #, . I 1976 1976
YEAR (C) D I S T I L L A T E
OIL
f/
f
! . V°,'7 o001-
06O
025
FUEL
,p ..V
~
GAS
C ~
,
~
076
//~+" "~
{4
..,TJ
OW
~
o. 0 OO I,P'; , I X . . .I . . . 1976 1976 1977 YEAR
Fig 3 Region III
I
i i 19711
|
000
"~" 1976
I /~1)1
-'
1976
" " I 1977
YEAR
" " " 1978
I
58
(b) RESIDUAL FUEL OIL
(a) COAL 100
,oor
o~,/f
O75
O5O
,~
,,,,o~
"9"
0 25
0OO
,.,,-'~'-, I / . . . 1975 1976
I
~ , 1977
I
, , ,
i
0 00 ~
1978
~ ~ 1975
J
JJ ~ m 1976
J
m d 1977
m ~ 1978
YEAR
YEAR
(d) NATURAL GAS
(c) DISTILLATE FUEL OIL 100~
.=
,,.,,~j /
o~
0 75
O5O
0 25
?: ......
000 1975
1976
1977
000
,
I I ,
1975
1975
,
1
,
1976
,
,
I
,
,
1977
.
i
1978
YEAR
YEAR
Fig 4 Region IV
(a) COAL
(b) RESIDUAL FUEL OIL
"r
100
~
075
.~°oo~ + o ~ j _./,,0 25
ooo
~o,,i-
,//. ~7/
19"/ ~"
4 ~"
000 1975
1976
1977
M"
1975
1978
, I 1976
,
. , I 1977
, , , 1978
i
YEAR
YEAR
(d) N A T U R A L GAS
(c) DISTILLATE FUEL OIL 100
O75
075
O5O
.,¢o
ooo
, ~ 1 ,1975 / I . . . , . 1977 ..,
1975
YEAR
Fig 5 Regmn V
000 1975
1975
, 1 , , , I 1976 1977 YEAR
,
, , 1978
J
59 (a) COAL
0
(b) RESIDUAL FUEL OIL
J/ .+:Y / /
100
75
/~"
+V
O50
-
100
0 75
--~ 050 (n
0 25
y/
~,o"
0 25
000
1975
1976
1977
,
0 O0
1978
•
I
1975
YEAR
4/
,~/
0
75
. , i I , , i I 1976 1977 1978 YEAR
1975
.
,
.
I
,
1977
n
I
1978
~
I'Y
= o.
000
I
(d) NATURAL GAS
~,.~,~
000
•
1976 YEAR
(c) DISTILLATE FUEL OIL
0 75
/ .
I~"
0
1975
1976 1977 YEAR
1978
Fig 6 Region VI
(a) COAL
(b) RESIDUAL FUEL OIL
~,>~///
loo
1-F
0 75
.J
"~ 0 50 ~ ~
025
O25
000V,
,
1975
I/,
,
,
I
,
1976
•
•
I
*
1977
*
,
I
1978
O00
~/ . I . . .I 1976 1977
19~
YEAR
loo
(C)
DISTILLATE
, , , ! 1978
YEAR FUEL
OIL
,,,,~./
000
(d) NATURAL GAS
~0,00~ ~
~
02S 000
"
'
1975
'
I
'
'
"
1976 1977 YEAR
Fig 7 Region VII
I
I
*
|
1978
J
000 1975
1976 1977 YEAR
1978
I
60
(b) RESIDUAL
(a) COAL
lOO"
FUEL OIL
100
f
0 75
.~ r,/)
/
oGo
/ ./~" o 25
O25
00o
~
iL
1975
.
.
i
~
1976
•
I
.
.
1977
.
000
I
1979
.,~<'/.,~,..., ..., 1975
1976
YEAR (C) D I S T I L L A T E
(d) NATURAL
FUEL OIL
.....//.
,oor o75
1977
1978
YEAR
lOO~
1~
GAS
.,+/
075
r#)
O60
025
0001¢0; , I X • , 1976 1975
I
,
, , 1977
t
,
, , 1975
i
000
% i
II.
1975
• I • • , I 1975
8
Region
lOO
~
o.
(b) RESIDUAL ,
~
/
o76
~.y/ /
~o, /,~@@"
¢ i 1975
J /
O25
, . . 1979
197/
I
, , , t
000
, I , , . i 1975
1978
1976
-
'=V
~C-'4
,~y
/
o.l-.
.
o,,/ / ~ j,/.,', 1975
1976
000~1 1977
YEAR
Fig 9 Region IX
(d) NATURAL
FUEL OIL
100'
0 76
1977
, . , i 1978
YEAR
YEAR (C) D I S T I L L A T E
FUEL OIL
lOO
.,.o~//
000
1970
VIII
(a) COAL
025
• . . I
YEAR
YEAR Fig
1977
1978
GAS
o+/.~. ~,y
/
i I .X". • I 1975 1975
M o''
,
YEAR
, , I 1977
•
, • 1978
I
61
(a) COAL tO0
100
O76
o 79
# O6O
-
(b) RESIDUAL FUEL OIL /
,j
~ooo
/
O26
026
/,e" •
0OO
I
,
,
1976
1976
•
I
•
1977
•
•
I
OOO
1978
1975
100
100
O76
076
o~
1978
O6O (4
026
026
0OO
1977
(d) NATURAL GAS
(c) DISTILLATE FUEL OIL
~
1976 YEAR
YEAR
.
.
1976
.
.
I
.
19711
, 1977
,
I
•
•
,
|
1978
0OO 1976
1971
1977
1978
YEAR
YEAR
Fig i0 Regmn X
the hmtormal mterrelatmnshlps between the factors mfluencmg the demand for fossil fuels. In the case of the demand for fossil fuels m Region VIII, the instability m a reflectmn of the very rapid economm growth m the region (Per capita income grew over the 1973 - 1978 p e n o d at more than 8% per year ) In partmular, total generatmn m 1978 was 97% higher (14% per year) than m 1973 with all of thin coming from mcreased use of coal. It is not surprising that the mcreased use of coal resulted since coal is indigenous to the region while the other fossil fuels must be (for the most part) imported The instability m the use of coal m Region X arose m 1977 due to the hydroelectric shortage over the January 1977 - O c t o b e r 1977 period. A slgmfmant portion of the defmlt m generatmn due to the water shortage was made up by increased nuclear generation [8]. The remmnder of the shortfall was satinfled by increased use of coal destabillzmg this partmular functmnal relationship It m interesting to note that the other fossil fuels con-
tmued to respond d u n n g this period as they did over the entire historical permd.
IMPLICATIONS
Major attention has focused on the demand for fossil fuel mputs m the generatxon of electrmal energy. Of slgmfmant concern m the context of drawing meaningful references over the hmtormal period as well as over the forecast horizon is whether the observed relationships (e.g., the pnce elasticmes) are stable. The lmphcatlons of the stability test are clear Events over the years 1973 - 1978 have had the effect of destabilizing the demand for fossil fuels m Region II and Region VIII and for coal m Region X, while leaving vtrtually unchanged the demand for coal, residual fuel oil, distillate fuel o11, and natural gas That is, for the majority of regions, the relative Importance of the pnce of coal, the price of residual fuel oil, the price of dmtillate fuel oil and the price of natural gas together with weather, generation and natural gas curtail-
62 m e n t s in Influencing the share o f total expenditures (and h e n c e d e m a n d ) has r e m a i n e d constant One m u s t be careful, however, to avoid inferring t h a t t h e q u a n t i t y d e m a n d e d o f coal, residual fuel oil, distillate fuel o11, and natural gas r e m a i n e d u n c h a n g e d in the stable regions The e s t i m a t i o n results clearly s h o w t h e price o f all of the fossil fuels influencing each e x p e n d i t u r e share Thus, an increase in the price o f coal does lead to an increase m the q u a n t i t y o f natural gas used In the g e n e r a t i o n process The m a g n i t u d e o f this response for each o f the fossil fuel inputs for m o s t regions r e m a i n e d u n a l t e r e d over the sample p e r i o d A n o t h e r w a y o f expressing this is t h a t the e x p e n d i t u r e share elasticities for the fossil fuels did n o t vary F o r the regions d e m o n s t r a t i n g m s t a b l h t y , on the o t h e r h a n d , such a c o n c l u s i o n is n o t f o r t h c o m i n g T h e r e is an Identifiable shift in the relative i m p o r t a n c e o f the factors Influencing the d e m a n d for fossil fuels U n f o r t u nately, the p a u c i t y o f data precludes a precise m e a s u r e m e n t o f the e x t e n t o f this shift (i e , n o t e n o u g h degrees o f f r e e d o m exist to empirically estimate t h e share e q u a t i o n s post1 9 7 7 ) T h e n a t u r e o f the change, h o w e v e r , was m a n i f e s t in r e o p t l m l z l n g the fossil fuel mix m such a w a y as t o take a c c o u n t o f fuel availability and relatively larger price differentials In a forecasting setting the l n s t a b I h t y presents quite a p r o b l e m for longer-run considerations As can be seen f r o m the s h o r t - r u n forecasts, the stability or lack t h e r e o f pres e n t e d little p r o b l e m with regard t o f o r e c a s t a c c u r a c y In the larger setting (i e , t h e longerrun), the e s t i m a t e d coefficients in R e g i o n II,
R e g i o n V I I I and R e g i o n X are n o t a d e q u a t e reflections o f the w a y electric utlhtles actually altered their fuel e x p e n d i t u r e s for s o m e o f the sample p e r i o d w h i c h gives rise t o a c o n c e r n t h a t f u t u r e behavior will m o r e closely e m u l a t e the responsiveness over the unstable p e r i o d t h a n over the stable p e r i o d leading t o unacceptable forecasts and h e n c e p o o r p o h c y based on t h o s e forecasts There is n o easy r e s o l u t i o n t o this d i l e m m a N o t e n o u g h d a t a or u n d e r s t a n d i n g o f the institutional anomalies exist to i m p r o v e on the situation F o r the time being, it is simply a fact o f hfe t h a t m u s t be a c c e p t e d
REFERENCES 1 N D Urn, Forecasting fossil fuel demand, Electric Power Systems Research, 4 (1981) 247 - 254 2 G Chow, Tests of equahty between two sets of coefhclents m two linear regressions, Econometr~ca, 28 (1960) 591 - 605 3 J Kmenta, Elements o f Econometrics, MacMillan, New York, 1971 4 R L Brown, J Durbm and J M Evans, Techtuques for testing the constancy of regression relationships over time, J R Star S o c , 37 (1975) 149 - 163 5 M S Kahn, The stability of the demand for money function m the U S 1901 - 1965,J Pohtzcal E c o n , 82 (1974) 1205 - 1219 6 J Durbm, Tests for serial correlation m regression analysis based on the penodogram of least-squares residuals, Bmmetrtka, 56 (1969) 1 - 15 7 D Cochrane and G H Orcutt, Apphcatlons of least squares regressions to relationships containing autocorrelated error terms, J A m Star A s s o c , 44 (1949) 32- 61 8 Edlson Electric Institute, Statistical Yearbook, Edlson Electric Institute, New York, annual