The Probability Distribution of Future Demand: The Case of Hydro Quebec
Author(s): Jean-Thomas Bernard and Michael R. Veall
Source: Journal of Business & Economic Statistics, Vol. 5, No. 3 (Jul., 1987), pp. 417-424
Published by: American Statistical Association
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Journalof Business&EconomicStatistics,July1987,Vol.5, No.3
? 1987 American Statistical Association
The
Future
Probability Distribution of
Demand
The Case of HydroQuebec
Jean-Thomas Bernard
of Economics,UniversiteLaval,Ste. Foy,QuebecG1K7P4, Canada
Department
Michael R. Veall
of Economics,McMaster
OntarioL8S4M4,Canada
Hamilton,
Department
University,
Thebootstrapping
distribution
of
techniqueof Efron(1979)is usedto estimatethe probability
futureelectricity
demandforHydroQuebec.Theapplication
followsthe regressionapproach
of
andPeters(1984a,b)butalso allowsforseriallycorrelated
Freedman
disturbances
anduncervariableforecasts.
taintyinthe independent
KEYWORDS:Bootstrapping;
intensivestatistics;Electricity
demand;ForeComputationally
cast uncertainty;
Serialcorrelation.
1. INTRODUCTION
time andlow fixed-cost,highvariable-costcapacitysuch
as thermal generation and rather less high lead time
and high fixed-cost, low variable-costcapacitysuch as
nuclear (Ellis 1980; Veall 1983). Therefore, the planning process needs more than simple point projections
of future demand; it needs an estimate of the entire
probabilitydistributionof future peak demand, conditionalon currentinformation.
In this articlea regressionmodel is employedin the
estimationof the probabilitydistributionof futureelectricity demand for Hydro Quebec, one of the largest
utilities in North America. The model has been kept
very simple to ensure the tractabilityof the approach
and to keep the emphasisof the articleon the second
moment ratherthan the first moment of the forecast.
The technique used is bootstrappingas proposed by
Efron (1979, 1982) and describedin general terms by
Efron and Gong (1983). The particularapplicationto
forecastingfollows the Freedmanand Peters (1984a,b)
one-equation approach, modified to accommodatea
recursivesystem with autocorrelateddisturbancesand
stochasticforecastsof the independentvariables.Veall
(1987a)appliesa somewhatsimilarapproachto Ontario
Hydrodata, althoughin thatcase the recursivestructure
is simplerand there is no need to treat the serial correlation, which representsa significantcomplication.
Computationallyintensive techniquessuch as bootstrappingare becomingincreasinglyimportantas tools
of applied statisticsand econometrics.They appearto
be particularlyvaluable in cases in which standarderrors or confidenceintervalshave either no known formulas or the analytic expressionsavailable are based
on tenuous asymptoticand parametricassumptions.
A common situationof this sort is the predictionof
a futurevalue of a time series given some set of current
information.If, for example,a regressionmodelis used,
it is straightforwardto forecast the future value of a
dependentvariableby takingthe innerproductof forecastsof the independentvariableswiththe least-squares
coefficient estimates from the observed sample. But
although there is a standardformula (e.g., Johnston
1984, p. 195) for confidenceintervalsof such forecasts,
it requiresthat the dependentvariablebe normallydistributed(an assumptionthat cannotbe relaxedfor large
samples) and that the predicted values of the independent variablesbe somehow known with certainty.
This latter assumptionespeciallyis unlikelyto be even
approximatelytrue in most practicalsituations.
Confidence-interval-typeinformationis likely to be
particularlyimportantin the predictionof future electricitydemand. First, most utilities are less concerned
about expected future demandthan about the amount
of capacityrequiredto ensuresome (presumablylarge)
probabilitythat there will be no shortage. Second, the
degree of forecastuncertaintyitself is importantto the
utility'sdecisions. All other things being equal, as the
degree of demand uncertaintyincreases, the optimal
mix of capacity will include relatively more low lead
2.
THE MODEL
Many utilities forecast peak demand by forecasting
averagedemandfor the relevantperiod and assuming
that averagedemandis some given constantproportion
(calleda "loadfactor")of peak demand.Our approach
instead employs a regressionlinear in logarithms.To
illustrate, we use annual data for Hydro Quebec for
417
418
Journalof Business & EconomicStatistics,July 1987
the period 1965-1983 and estimate the ordinaryleast
squares(OLS) regression
log(peakt) = -.652696
+ 1.126434 log(AMW,),
(.017234)
(.153962)
R2
=
.9960, DW = 2.0403,
(1)
where peaktis the annualpeak demandin megawatts
(Mw), AMW,is the averagedemandover the year also
in Mw, standarderrors are in parentheses, and DW
representsthe Durbin-Watsonstatistic. The constant
load factor assumption would imply that the
log(AMW,) coefficient is unity, a hypothesisthat can
be easily rejected at the 5% level of significance.
The next step is to estimate a forecastingequation
for AMW,. Again emphasizingsimplicity,we estimate
with OLS and the same sample period as in (1):
log(AMW,) = -2.849834
+ .066910 log(pricet)
(2.307198) (.130148)
+ .983279 log(GPP,)
(.308234)
+ .026412 time,,
(.010444)
R2
=
.9942, DW = 1.3464,
(2)
where pricetis the real price of electricitycalculated
somewhatcrudelyby dividingaverageHydro Quebec
revenuesper Mw by the grossprovincialproduct(GPP)
deflator for Quebec, GPP, is the real gross provincial
product, time, is a linear time trend equal to the last
two digits of the year, and standarderrors are in parentheses.Two things should be noted about these estimates. First, the log(price,)coefficientdoes not have
a negative sign, as would be expected theoretically.
Second, the value of the Durbin-Watson statistic is
fairlylow, indicatingthe possibilityof seriallycorrelated
disturbances.As the test statistic is in the indecisive
region of the Durbin-Watsontables, the exact probability value of the null hypothesiswas calculatedusing
the SHAZAM package'suse of the routinesof Imhof
(1961). As the value was only .0109, we reject the null
hypothesisof no serial correlation.
We therefore estimate a new model for AMW, by
takingthe residualsfrom (2) and estimatingtheir firstorderautoregressivecoefficientas .3030.Wethentransform Equation (2) using the noniteratedprocedureof
Cochraneand Orcutt(1949)with the PraisandWinsten
(1954) weighting of the first observation by (1
-
p2)1/2.
The result is
log(AMW,) =
1.695002 -
.15127 log(price,)
(2.481269) (.141747)
+ .830044 log(GPP,) + .031422 time,,
(.011210)
(.331061)
p = .303009, R2 (on transformed variables) = .9970,
DW (on transformedvariables) = 1.8800, (3)
wherestandarderrorsare in parentheses.The DurbinWatsonstatisticnow cannot reject the null hypothesis
of no autocorrelationat the 5% level, and the price
coefficienthas a negativesign, althoughits value is not
significantlydifferentfrom0 at the 5% level. The coefficientsof log(GPP)andtime seem to be estimatedwith
acceptable precision, although the trend coefficient's
estimate of a 3.1% per annum growth in demand for
givenpriceandincomemay seem somewhathigh. (This
may be partlydue to increaseduse of electric heating
in both the residentialand commercialsectors.)
Equations(1) and (3) are then taken as the working
model and subjected to a series of diagnostic tests.
Monte Carloresearchby Whiteand MacDonald(1980)
suggests that a suitable test for normalityin the disturbances is the variation on the Shapiro and Wilk
(1965)test suggestedby ShapiroandFrancia(1972)and
appliedto the OLS residuals.The test values are .9080
and .9821for Equations(1) and (3), respectively,where
the 95% acceptance region is [.9010, 1.0000], so the
null hypothesisof normalitycannot be rejected at the
5% level, although the value of the test on (1) does
indicatemore than a 90% probabilitythat the disturbances are nonnormal.Similarconclusionsare yielded
by the ordinaryShapiro-Wilktest statistic.The Breusch
and Pagan (1979) test (TR2 version; see Judge, Griffiths, Hill, Liitkepohl, and Lee 1985, p. 447) cannot
rejectthe null hypothesisof homoscedasticityfor either
equation, with statisticsof .0169 and 1.5942 compared
with respectivechi-squaredcriticalvaluesof 3.8415and
7.8147. The Engle (1982) test againstan alternativeof
first-orderautoregressiveconditionalheteroscedasticity
(ARCH) effects giveschi-squaredstatisticsof .3392and
2.1591 compared with a critical value of 3.8415 and
hence cannot reject the null hypothesis.
Finally, as a test againstthe potential correlationof
log(AMW,) and the disturbancein (1), a specification
test was conducted as describedby Hausman (1978).
The test value was a t statisticof -.5876 and hence is
unable to reject the null hypothesisof no correlation.
3. THE BOOTSTRAP
Now that a simple model has been developed, it can
be used to forecast and the probabilitydistributionof
the forecast errorscan then be estimated. First, however, some discussionof the techniqueof bootstrapping
is required.This will be brief because, as noted, there
are both formaland informaldescriptionsof the bootstrap available elsewhere and in particularFreedman
and Peters (1984a,c) discussedthe bootstrapin a forecastingcontext.
Much of the intuition behind bootstrappingcan be
given by a simple example, such as the estimationof
the median of a sample of independentobservations
drawn from the same, perhaps unknown, underlying
distribution.The probabilitydistributionof the estimated median is typically complex, except for a few
casessuchas one in whichthe observationsare normally
419
Bernardand Veall:FutureDemandof HydroQuebec
distributed.The bootstrap,however, replacesthis calculationby creatinga largenumberof artificialsamples
by drawingrandomlyand with replacementfrom the
originalsample. The probabilitydistributionof the estimated median can then be approximatedby the histogramof the mediansof the artificialsamples.
We shall now jump directly to the specifics of our
context. Firstwrite the model of Equations(1) and (3)
as
Y,= [1 X]a + e,,
,
t=
1,
. . ., T,
(4)
are mutually independent, Yt is
log(peakt), Xt is log(AMWt), and Zt is a 1 x 4 row
vector correspondingto the explanatoryvariables in
Equation (3). The advantageof a two-equationmodel
(ratherthan solvingfor Xt and makingYta functionof
Z,) is that the recursive system imposes structure on
the dynamicsas well as on the way in whichthe variables
Zt affect Yt. These restrictions(whichare not rejected
for this example by the Hausmanspecificationtest in
Sec. 2) can improvethe forecastif true. As the tests in
Section 2 indicate, this structurealso appearsto correspondto disturbancesthat are homoscedasticand serially uncorrelated.Moreover, althoughnot discussed
in this articlefor purposesof brevity, analysisof the X,
forecast is also an importantpotential output.
The forecast process consideredhere consists of es-
where fGLSis the original Prais-Winsten estimate. The
sample can then be completed by using
= &o+ &a1X
Y
t +
t=
t,
1,...,
and forecast Xt for the target period T*:
XT* = ZT fGLS
(5)
for a given set of values for ZT*(GLS representsgeneralized least squares). Note that T* is assumedto be
far enoughin the futurethat the contributionof the last
sample disturbance ATto the optimal forecast for T* is
assumednegligible (which seems reasonablegiven the
smallestimateof p andthe fact that the forecasthorizon
is seven years). The forecastXT*is then used to predict
YT*as
YT* = [1 XT*]aOLS-
(6)
To bootstrap this process, take the Prais-Winsten
estimators (p,
fGLS)
and construct B artificial samples.
For each artificialsample i, this is done by drawingT
times randomlywith replacementfrom the elements of
I to create an artificial residual set 0'. This can be used
to create an artificialfi by taking the first element of
Af and dividingit by (1 - /2)1/2 to reversethe effect of
the Prais-Winsten transformationon the first observation, thus yieldingAfi. The rest of the vector can then
be constructedby the iterativeprocedure
Ai = P;At-i + At,
T,
(9)
where &Oand &aare the original OLS estimates and
E are from artificialresidualssampled randomlyand
same exogenous data Zt (t = 1, .. ., T), but each
differingin the stochasticcomponent.
The originalestimationprocesscan then be repeated
on all of the artificialsamples. The forecastfor period
T* can then be bootstrappedas the distributionsof
XT*
= ZT*
flGLS,
yi'. = [1 xT.]i,
t = 2,...,
T.
(10)
where /GLSand a' are the estimates on artificial sample
i. But, because the distributionof the forecasterroris
more interestingthan the distributionof the forecast,
we also bootstrapwhat Freedmanand Peters (1984a,c)
called "simulatedactuals."First, we randomlysample
B more residuals independently from
i = 1, ...,
a E, call these 'iT- and Ei*, respectively, and
each of and
again transformthe former into fii- by dividing by
(1
p2)1/2. Then the simulatedactualsare
XiT
=
ZT* AGLS +
iT*,
Y'i.
= [1 X'iT]
+ E'T*,
(11)
timating fJOLSand then using a first-order autoregression
coefficient on the OLS residualsto estimate a f. This
is then used in a Prais-Winstentransformto estimate
iGLS
(8)
represent B artificialsamples, all consistent with the
underlyingmodel assumedin (1) and (3), all havingthe
At = P_t-1 + qt, qt iid, t = 1, . . ., T,
rt
Xt = ZtfGLS + pt,
with replacement from e. The sets (Yi, Xi, Zt) then
iid
xt = z,t + P,
where Et and
The artificialsample is then developed as
that is, the forecastvaluesplus a bootstrappedresidual.
The simulatedforecasterrorsui andeiare the difference
between (11) and the bootstrapprediction:
UT*
=
XT*
-
ZT*1GLS,
e.*
Y'.
-
[1 XT.]&';
(12)
therefore, the distributionof XT*and YT*,conditional
on the forecast information,can be estimated as the
empiricaldistributionof these forecasterrorscentered
on the actual forecast (XT*, YT*).
Before discussingthe results, it mightbe worthwhile
to discuss the reasons for using bootstrappingrather
than some analytic approach.The simplest answer is
that there are no standardanalyticformulasavailable,
even for large samples. For example, even if the disturbances in (4) are normal, the forecast error e,T will
be a complexmixtureof a seriesof productsof normally
distributedvariates. (This is complicatedfurtherwhen
ZT*is allowed to be randomas in Sec. 4.) Recall also
that XT* and YT*are logarithms, but what is of more
interestis a forecast of these exponents, which are actual average and peak demand. If (XT*, YT*) are unbiased forecasts of (XT*, YT*), (exp(XT*), exp(YiT))
are not unbiased forecasts of (exp(XT.), exp(YTr)) [see
Goldberger(1968) for a discussionof the parametric
normalcase]. To deal with this, we alter the procedure
420
Joural of Business&Economic
Statistics,July1987
1974-1983 trend is projected to 1990. Given that real
pricesfell until 1973and then grewsteadilyafterwards,
a similar1974-1983 trendwas projectedfor real price.
More formally, the following OLS results were obtained for 1965-1983:
slightlyby setting the forecast errorsas the difference
in the exponentsof the simulatedactuals (11) and the
exponentsof the bootstrappingforecasts(10) andcenter
these on (exp(XTr), exp(YTr)). The result is, therefore,
a bootstrapestimate of the probabilitydistributionof
peak demandusing the linear as opposed to the logarithmicscale. A bootstrapestimate of the bias due to
takingexponents is the mean of this probabilitydistribution estimate less the original forecast (exp(XT*),
exp(YT.)), which is just the average of the forecast
errorson the linear scale (see Efron 1982, p. 33).
Although all of these considerationssuggest that
there can be no realistic analytic alternative to the
bootstrap,there is also some theoreticalsupportfor the
effectivenessof the bootstrapin small samples(Beran
1982;Hall 1986). Moreover,limitedMonte Carloanalysis in related situations(Freedmanand Peters 1984b;
Veall 1986, 1987b)suggeststhatthe performanceshould
be acceptable.
log(GPP,) = 6.785629 + 2.191371Dt + .043964T1l
(.264790)
(.190046)
+ .014966T2, +
t,,
(.002347)
R2 = .9898, DW = 1.2259,
log(pricet) = 1.012691 - 4.838032Dt - .015200Tt1
(.005647)
(.389925) (.543280)
+ .047675T2 + t,,
(.004816)
R2 = .8806, DW = 1.8111, (14)
where Dt is 0 for 1965-1973 and 1 thereafter, T1 =
(1 - Dt) * time, and T2, = D * time,.
4.1 Forecasts of the Exogenous Variables
To use the bootstraptechniquedescribedin Section
3, forecastsof the exogenousvariableswill be needed.
One approachthat could be used is to take these forecasts from other sources. The referee correctlyemphasizes, however, the advantageof explicit models at all
stages, and we follow this advice, althoughthe equations used are somewhatsimplistic.
Therefore,we examinedthe time series of both GPP,
and P,, as in Figures1 and 2. GPPtgrew fairlyrapidly
up untilthe 1973oil shock, afterwhichgrowthwasmuch
slower. GPP in 1982 was substantiallybelow even this
slower trend with the 1983 value also below trend but
by a smallermargin.A researcherexaminingthese data
in 1984mightreasonablyhave adopteda modelin which
the 1982-1983recessionwas transitorybut in whichthe
lower growth since 1974 would continue. Hence the
There is an indicationof serialcorrelationby the low
Durbin-Watsonvalue for (13) that correspondsto an
exact probabilityvalue of .003 for the null hypotheses
of no serial correlationusing the Imhof procedure.It
is, therefore, assumedthat
/Vt= Pi t-1 + Ot
with
ct(
iid. Using the same GLS procedure used to
log(GPP,) = 6.711738 + 2.231869Dt
(.229394) (.346317)
+ .045077Tt, + .015349T2t,
(.003319)
(.002884)
p1 = .374714, R2 (on transformed variables) = .9992,
DW (on transformedvariables) = 1.6770, (16)
4+
+
+
+
+
4.
+
+
4+
+
+**
0LL
+-
0
o
4.*
9.8+
4
*
4.
U- -
LII. 1
65
?
70
(15)
obtain (3), the resultsare
10.2
10.0
(13)
and
4. RESULTS
a.
(.002752)
75
80
Figure 1. Plot of the Log of Gross Provincial Product, Quebec, 1965-1983.
85
Bernardand Veall:FutureDemandof HydroQuebec
421
0.1
u
x
-LJ
I-
+
0.0
- -
65
-
+
T
t
+
+
I
75
+
I
85
80
+
0
-0.21
-i
+
-0.3
Figure2. Plot of the Real Average Price of Electricity,Quebec, 1965-1983.
with almost the same coefficient estimates as (13).
Equations(14) and (16) each pass at the 5% level the
Breusch-Pagan(1979) and Engle (1982)/ARCH diagnostictests as used previouslyin (1) and (3). Of interest
for Section 4.5, the null hypotheses of normalityalso
cannot be rejected using the same tests as applied to
(1) and (3).
Using (16), the forecast for 1990 GPP (ignoringthe
time serialcorrelationeffect fromthe 1983disturbance)
is 30,484.9million1971dollars.Using (14), the forecast
for real price is 1.59264 or about 60% above its 1971
level. [The regressionsand calculationswere all done
on a VAX 11/780 in double precision.The readerwho
checksthese forecastsusing (14) and (16) will note very
small discrepanciesin the last digit.]
4.2
The Simple Boolstrap
This will be the firstof three sets of resultspresented
to illustratethe outputfromthis kindof approach.First,
we try a simplerapproachthanthatdescribedin Section
3 and assume 1990 average demand will be with certainty 16,254 Mw, the value predictedby putting the
preceding forecasts of real GPP and real price into
Equation (1). (This is somewhathigher than the comparableforecastof 15,526Mw in Hydro Quebec 1985.)
Under this assumptiononly Equation (1) needs to be
bootstrapped;the resultingestimate of the probability
distributionfor peak demand is graphedin Figure 3.
The OLS forecast for peak demandis 28,834 Mw and
the bootstrapestimate of the bias, as describedin Section 3, is only 15 Mw. (As the bias estimatesare always
similarlysmall, they will not be reportedin the following.) The amount of capacityto be 95% sure that demand will be met is 29,954 Mw.
In this simple case we can also easily calculate the
95% point analyticallyunder the assumptionof normalityusing the method of Salkever(1976). This value
is 29,666 Mw, somewhat less than the nonparametric
bootstrapvalue.
4.3
Recursive Boolstiap With Certain
Exogenous Variables
Here we proceedexactlyas in Section3, allowingfor
uncertaintyin average demand by recursivelybootstrappingEquations(1) and (3) but assumingthat the
1990 real GPP and real price forecastsare knownwith
certainty. This does not change forecast average demand or peak demand, but the added uncertaintyincreasesthe 95%point by over 1,500Mw to 31,556Mw,
as in Figure 3.
4.4
Bootstiapping
the Entire Model
We now relaxthe unrealisticassumptionused in Section 3 and so far in Section4 that 1990priceand income
are knownwith certainty.One methodwould be to use
subjectiveprobabilitydistributionsfor these forecasts,
but we instead bootstrap the entire model, including
Equations (14) and (16). It should be noted that this
extra layer of bootstrappingincreases the computational burdenconsiderably.
The bootstrappingprocedureof Sections 3 and 4.3
is modified in four ways. First, bootstrapsamples are
also created for log(Pi) and log(GPPi) in the usual
way using (14) and (16), with the serial correlationfor
the latter handled just as for log(AMW,) as described
in (7) and (8). Second, in bootstrappinglog(AMW,)
in (8), log(P,) and log(GPP,) in Z, are replaced
by log(Pi) and log(GPPi), the correspondingbootstrap values. "Simulated actuals" for log(Pl99) and
log(GPP1990)are then calculatedas the forecastsof those
values plus bootstrapresidualsfrom these corresponding equations, and these values are put in ZiT along
with a 1 (for the constant) and timeTr = timel990= 90.
The bootstrap method then proceeds as before, with
the thirdmodificationbeingthat for (11), the simulated
actualsformulafor averagedemand,ZT. is replacedby
ZIT..To emphasize,this means that each bootstraprun
uses a differentvalue for the 1990exogenousvariables
422
Journalof Business&Economic
Statistics,July1987
=A.&---------
24000
26000
28000
30000
32000
I
34000
-
nW
n%
II
Figure3. Plots of BootstrapEstimatesof the ProbabilityDensityof 1990 Peak ElectricityDemand, Quebec. Theinnercurve is the estimate
conditionalon an average demand of 16,254 Mw and employs Equation(1). The next curve is the recursivebootstrapestimate employing
average demand Equation(3), as well as Equation(1), and is conditionalon estimates of GPP of 30,484.9 million1971 dollarsand a real
price of 1.59264, where the 1971 real price is 1.0. The outermostcurve represents a recursive bootstrapdescribed in Section 4.4, where
GPP and real price are not assumed exogenous but are also bootstrapped,this time fromtime trendregressions (14) and (16), which feed
into (3), which in turnfeeds into (1). Densityestimates have been normalizedto approximatelythe same height with verticaldashed lines
indicatingboundariesof 5% upper tails.
to reflectthe uncertaintyof
and log(GPP1990)
log(P1990)
these forecastsgiven currentinformation.
There is one final modificationthat is not strictly
necessary but that seems appropriate.As noted, the
growthincreaseis much less smooth after 1973;this is
reflectedby a sample standarddeviationof the (transformed) residualsof (16) of .0221 for 1974-1983 comparedwith only .0099for 1965-1973.Althoughbecause
of the small sample this difference is not statistically
significantusing a X2test, it still seems appropriateto
constructartificialsamplesfor 1965-1973log(GPP) using only 1965-1973 residuals and for 1974-1983
log(GPP) using only 1974-1983 residuals.For the constructionof the simulatedactualsof log(GPP99o),only
1974-1983 residualsare used.
All of these modificationsdo not change forecast
peak, but they do increase the uncertainty,as shown
in Figure3. The 95% point is 31,841 Mw.
4.5
Parametric and Smoothed Bootstraps
For each of (1), (3), (14), and (16), tests on residuals
have been unable to reject the null hypothesesof normality. Since these equationsare in logs, we might expect more skewness after taking exponents than is
apparentin Figure 3, but this is not inconsistentgiven
the relative tightnessof fit of all of the equationsand
the lack of power of normalitytests in such small samples. But the small samples cause anotherproblemin
that the tails of the probabilitydistributionsof disturbancesare likely to be poorlyestimatedby the frequency
of residuals.In some situationsa researchermay feel
thatit is worthwhileto sacrificethe nonparametriccharacter of the bootstrap because of this; therefore, we
experimentwith a parametricbootstrapusing the normal distribution.This is exactlyas describedin the previous sections except for every instance in which a
bootstrapresidual is drawn, instead a pseudorandom
variateis generatedwith mean 0 and varianceequal to
the sample variance of the correspondingresiduals.
When this is applied to the "complete"bootstrapof
Section 4.4 the tails do become more importantwith,
for example, a 95% point of 32,139 Mw (not shown in
Fig. 3). An intermediatepositionis the smoothedbootstrap (Efron 1982, p. 30), which instead uses a convolution of the empirical residual distributionand a
constant s multipliedby the normal distributionwith
the same variance,scaledby dividingby (1 + s2)1/2.We
use s = .1 and find the results almost identicalto the
nonsmoothedbootstrapwith, for example, a 95%point
of 31,816 Mw.
Bernardand Veall:FutureDemandof HydroQuebec
TableA.1. ElectricityDemand and Its Determinants,Quebec,
1965-1983
Peak
demand
Average
megawatts
Real
price
GPP
Time
6,450.0
7,050.0
7,800.0
8,000.0
8,650.0
9,300.0
9,450.0
10,250.0
11,450.0
11,950.0
13,350.0
14,800.0
15,800.0
17,050.0
17,600.0
19,400.0
19,700.0
18,400.0
19,800.0
4,224.0
4,589.0
4,954.0
5,214.0
5,639.0
6,027.0
6,141.6
6,626.0
7,123.0
7,900.0
8,105.0
9,073.0
9,452.0
10,217.0
10,434.0
11,282.0
11,256.0
11,119.0
11,416.0
1.01312
.94621
1.00383
1.00225
1.00774
.97959
1.00000
.89634
.84642
.77242
.79034
.76675
.83040
.90802
.97764
.99728
.99405
1.11771
1.14199
15,526.47794
16,301.18064
16,799.53669
17,421.83723
18,330.99493
18,869.11237
19,923.23555
21,023.36693
22,331.32845
23,593.38093
23,768.58888
24,738.42980
25,198.21432
25,756.75781
26,752.87497
26,942.16301
27,224.75647
25,873.76142
26,832.19945
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
NOTE: Peak demandand averagemegawattsare given in thousandsof megawattsand
are fromHydroQuebec (1985). Priceis calculatedon average revenueper kilowatthour
(source: HydroQuebec, "DemandReports,"1965-1983) dividedby the GPP deflator
(source:ConferenceBoardof Canada)and normalizedso that the 1971 value = 1.00.
GPP is gross provincialproductin millionsof 1971 dollars(source:ConferenceBoardof
Canada).
5.
CONCLUSIONS
This article has illustrated the case of bootstrapping
to estimate the probability distribution of future peak
demand for Hydro Quebec, conditional on current information. The results illustrate that reasonable estimates of demand uncertainty are typically very large
and hence such measures should likely play an important role in the planning process. If anything, these large
uncertainty estimates from the nonparametric bootstrap
are likely to be conservative because, as shown, parametric bootstrap estimates are even larger. Moreover,
we have largely employed deterministic time trends as
opposed to more volatile autoregressive integrated
moving average processes, and it has been assumed
throughout that there will be no structural shifts before
the target period of the forecast.
ACKNOWLEDGMENTS
Thanks are due to W. Cheng, J.-F. Dion, D. Fretz,
and G. Green for research assistance and to B. Efron,
R. Ellis, I. McLeod, R. Tibshirani, an associate editor,
and an anonymous referee for valuable advice. Related
work was presented by Veall at the World Congress of
the Econometric Society, Cambridge, Massachusetts,
in August 1985. The assistance of Formation des Chercheurs et Aide a la Recherche, the Department of Education, the Government of the Province of Quebec,
the Centre for the Study of International Economic
Relations at the University of Western Ontario, and the
Social Sciences and Humanities Research Council of
Canada is acknowledged.
423
APPENDIX: THE DATA
When needed in the bootstrapping procedure, random numbers are generated using the routine of Wickman and Hill (1982). When a normal random number
generator is needed, this routine is supplemented with
that of Beasley and Springer (1977). The observed data
are presented in Table A.1.
[ReceivedOctober1985. RevisedJuly 1986.]
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