chapter-2 electricity demand projectionsshodhganga.inflibnet.ac.in/bitstream/10603/8179/9/09_chapter...

16

Chapter-2

ELECTRICITY DEMAND PROJECTIONS

2.1 Introduction

India faces formidable challenges in meeting its energy needs and providing adequate

energy of desired quality in various forms for different sectors of economy in a sustainable

manner and at reasonable costs. If one looks at the pattern of electricity supply and demand

scenario, the extent of power shortage varies up to 25.4% with all India average of

11.7%.Similarly energy shortage is up to 20% with all India average of 7.4% (Fig.2.1).

In order to fulfill this gap between demand and supply and to ensure sustainable energy in

future it is essential to project future electricity demand in the various sectors of economy.

These projections can then be used as inputs for optimizing available energy sources for

market allocations.

0

100

200

300

400

500

600

1981 1985 1990 1995 2000 2005

Year

Electricity Requirement and Availability(Billion kWh)

Requirement

Availability

Fig.2.1 Requirement and Availability of Electricity (Utilities) (CMIE, 2007)

Various studies for projecting energy demand have been conducted in the last two

decades. A survey of statistical methods to evaluate urban energy needs has been presented by

17 Balocoo and Grazzini[1].Determinants of energy demand in the literature include degree

days (DD) temperature by Al-Zayer and Al-Ibrahim[2] and Energy demand projections based

on the GDP has been reported by Dincer and Dost [3]. Connor [4] used neural network

models to simulate the impact of various physical and economical variables on electricity and

energy production levels. Galli [5] estimated the relationship between energy intensity and

income levels by forecasting long term energy demand in Asian emerging countries.Erdogan

and Dahl [6] investigated the impact of income, price and population on the aggregate,

industrial, manufacture and mining sectors of energy in Turkey. Eltony and Hosque [7]

presented a co-integrating relationship for the demand of electricity in Kuwait. Hondroyiannis

[8] estimating residential demand for electricity in Greece. Hunt et al [9] presented UK energy

demand for various sectors underlying trends and seasonality. Crompton and Wu [10]

presented energy consumption in China: past trends and future directions. Mirasgedis et al

[11] presented a model for mid-term electricity demand forecasting incorporating weather

influences. Shiu and Lam [12] have studied electricity consumption and economic growth in

china. Yemane [13] have estimated electricity consumption and economic growth: a time

series experience for 17 African countries. Yoo [14] studied electricity consumption and

economic growth: evidence from Korea. Nasr, et al. [15] studied econometric modeling for

electricity consumption in post-war Lebanon.

In India too, a number of forecasting studies have been made. Ghosh [16] presented

electricity consumption patterns related to economic growth in India. Population growth and

oil prices are projected by Majumdar and Parikh [17].In a report by planning commission of

India [18],future electricity demand has been projected on the basis of elasticity with respect

to GDP.TERI [19] has assumed various GDP growth scenario and used regression model to

project future energy and electricity demand.

The aim of the present study is to forecast electricity demand projections for India by

considering sectoral GDPs of the Indian economy. For that, the time series has been set up to

estimate future sectoral GDPs, number of consumers in various sectors and price indices of

electricity.

This data have been used for electricity demand forecast using econometric model. The

econometric model is based on logarithmic linearity and the various terms correspond to

industrial, agricultural and service sector GDPs, price indices and number of consumers.

18

2.2 Methodology

2.2.1 Time Series Projections

Mathematically a time series can be represented as follows

et= �1et-1 + �2et-2+…+ �pet-p+�t (1)

Where et is the error term for the year t expressed in the form of backward error terms up to

the autoregressive order p and �’s are the coefficients to reduce the corresponding error terms

to zero and to minimize the noise terms to �t ,which is a white noise .This is known as

autoregressive model .

The sectoral GDPs, number of consumers and price indices for the year t can be represented

as

yt = a + bt + et (2)

Where yt is the particular GDPs, number of consumers and price indices and therefore a

dependent variable, t is the explanatory or independent variable, a and b are constants and et

is the error term given by

et= ρet-1+�t (3)

Where ρ is another coefficient with a value less than unity. The variable yt for the subsequent

future years can be written in the following form

yt+1= a + b (t+1) + ρet + �t+1 (4)

yt+2= a + b (t+1) + ρ2et + ρet+1+ �t+2 (5)

:

yt+h= a + b (t+h) + ρhet + ρh -1et+1 +…+ �t+h (6)

To predict future sectoral growth ahead, the best predictor in terms of the current error at the

time t is ρhet. All other predictors ρh -1et+1, ρh -2et+2…,ρ et+h-1 should not be taken into account

since the error terms et+1, et+2... et+h-1 are unknown. As h increases, the amount of error

incrementally added to the forecast exponentially attenuates until an asymptote is

approximated [20].

19 2.2.2 Forecast of sectoral electricity: Econometric models

In order to avoid nonlinearities of an econometric model, we represent the electricity demand

and supply in year t by the following logarithmic equation

ln Eti = ai + bi ln Ati+ci ln Lti+di lnMti + pi ln Pt+ eti (7)

Where Eti is the electricity demand and consumption (GWh) of sector i in year t. The Ati

represents sectoral GDPs (Rs.crore), i.e. agriculture, industry, service and GDP per capita

(Rs.) for domestic, commercial and transport sectors. Ati is therefore the independent variable

in the time series equation (2). Lti represents sectoral electricity demand and consumption in

the year t (taken lag 1) for agricultural and industrial sectors. Mti is number of consumers for

domestic, commercial and other sectors. Pt represents price indices of electricity. eti is the

corresponding error or residual term and ai, bi, ci, di and pi are constants have to be estimated

for respective sectors by regression models. For Mathematical details refer to Apendix 2.1.

2.3 Result and Discussions

For calculating the coefficients in equation (7), SPSS 14.0 (Statistical Package for Social

Sciences) software has been used. The data for GDP, number of consumers and corresponding

electricity consumption has been taken from CMIE, 2007 [21].

The time series estimate of the various sector of GDP, number of consumers in various

sectors have been tested by using autoregressive time series methods and the result was

compared with the original time series. The comparison shows good agreement between

predicted and original time series. The statistics of residual i.e. autocorrelation function and

partial autocorrelation function have given good results. While a number of statistics are

reported, we have focused on two i.e. MAPE (mean absolute percentage error) and MaxAPE

(maximum absolute percentage error) [22]. Absolute percentage error is a measure of how

much a dependent series varies from its model-predicted level and provides an indication of

the uncertainty in the predictions. We have tested various time series for the MAPE and

MaxAPE and the result shows good agreement between actual data (past) and the generated

data.

20 Sectoral GDP forecast up to year 2045 has been given in Fig.2.2. The results of the

developed time series for the present and past data matched well. One can therefore

confidence inn the accuracy of the future forecast. From the figure it can be seen that the

service sector has largest value in 2006 as compared to other two sectors but after the year

2035 the industry sector becomes the largest component and shows faster growth as compared

to the other two sectors. It may be possible therefore, that service sector experience recession

after 2035.

0

200

400

600

800

1000

1200

1400

1600

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045

Year

Rs

.Tri

llion

GDP by Agriculture

GDP by Industry

GDP by Service

Fig 2.2 Time series forecasted value for various components of GDP from base year 2006 to 2045. (1 USD= 42.38 Indian Rs.in year 2005)

The time series forecasting of the number of consumers is shown in Fig.2.3. From the figure

it is clear that the number of consumers in domestic sector increases rapidly as compared to

commercial and other sectors. The number of commercial consumer curve has lowest slope.

So in future, number of commercial consumer may be stagnated.

Fig.2.4 shows the time series projection of GDP per capita and the price indices of electricity.

Price indices and GDP per capita increase with significant growth rate from year 2006 to

2045,where as GDP growth rate is relatively much less.

21

0

500

1000

1500

2000

2500

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045Year

Nu

mbe

r(in

Millio

n)

Domestic Consumers

Commercial Consumer

Other Consumers

Fig 2.3 Time series forecasted value of number of consumers in various sectors from base year 2006 to 2045

0

2000

4000

6000

8000

10000

12000

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045Year

GDP per Capita(Rs.Thousands)

Price Indices(1993-94=100)

Fig 2.4 Time series forecasted value of GDP per capita and price electricity from base year

2006 to 2045

The correctness of the predicted data can be examined from the values of statistical

parameters. In Table2.1, R2 and adjusted R2 value for all the sectors show very high predictive

power of the developed models. The Durbin-Watson (D-W) statistics, which is widely used

for testing the serial correlation is estimated and shows very small positive autocorrelation for

some of the sectors and in some sectors almost absence of autocorrelation.

22

Table2.1: Values of coefficients of econometric models together with statistical results for

electricity supply based on data for the period 1971 to 2005 in individual sectors.

Variables Coefficients Standard

Errors t-

Statistics

Statistics

Industrial sector Constant 1.947 0.831 2.344 R2 0.993

Ati 0.274 0.118 2.327 Adjusted R2 0.992

Pt -0.231 0.118 -1.962 Std. error of estimate 0.040

Lti 0.630 0.145 4.347 Durbin-Watson 1.740

Agricultural sector Constant -8.972 2.340 -3.834 R2 0.973

Pt -1.046 0.310 -3.379 Adjusted R2 0.970

Ati 2.064 0.306 6.740 Std. error of estimate 0.170

Lti 0.485 0.125 -2.476 Durbin-Watson 1.910

Domestic sector Constant -11.077 0.936 -11.839 R2 0.964

Pt -0.336 0.084 -4.007 Adjusted R2 0.952


Mti 1.003 0.075 13.410 Durbin-Watson 1.400

Commercial sector Constant -0.410 1.453 -0.282 R2 0.996

Pt 0.183 0.117 1.560 Adjusted R2 0.995

Mti 0.372 0.123 3.014 Std error of estimate 0.049

Ati 0.348 0.122 2.857 Durbin-Watson 1.620

Transport sector Constant 5.400 0.478 11.405 R2 0.991

Pt 0.517 0.145 3.566 Adjusted R2 0.989

Ati 0.131 0.125 1.044 Std error of estimate 0.061

Durbin-Watson 1.350

Other sectors Constant 2.372 0.101 23.552 R2 0.994

Mti -0.015 0.015 -1.016 Adjusted R2 0.993


Durbin-Watson 1.760

The value of D-W statistics in agricultural sector, industrial sector and other sectors are 1.91,

1.74 and 1.76 respectively which gives conclusive result for the absence of autocorrelation.

The D-W statistics in commercial sector, domestic sector and transport sectors are 1.62, 1.4

and 1.4 respectively which gives a moderate absence of autocorrelation. According to

econometric regression theory, if the residuals are not independent (or in other words the

23 errors are serially correlated), the use of the F-and t-tests and confidence intervals is not

strictly valid and the estimate of the coefficients may be unstable [23].The t-statistics also

shows almost satisfactory result for each variable in all sectors.

Since a logarithmic linear equation (7) has been used to forecast sectoral electricity demand,

the coefficients directly measure the elasticity. The coefficients in forecasting industrial sector

demand are very small to unity. So these are inelastic in nature. For agricultural sector the

long term price elasticity is -1.04 and elasticity with respect to GDP contribution by

agriculture is 2.06 which show a strong elastic behavior. All other sectors show inelastic

behavior except number of consumer in domestic sector where the long term elasticity is

1.003.

The forecasted electricity consumption using econometric model (Eqn.7) for various sectors

is shown in Fig.2.5.From the figure it is clear that the electricity demand by industrial sector is

largest over the years and will be the dominating sector in the electricity consumption. The

time series of sectoral consumption of electricity agriculture sector was dominating before

2004 as compared to the domestic sector but after that, the domestic sector electric energy

consumption increases with high growth rate. The commercial sector electricity consumption

also shows substantial growth over time but less than in absolute term with respect to

industrial, agricultural and domestic sector consumption. The transport sector and the other

sectors show very slow growth in the forecasting horizon. The industrial sector electricity

consumption, in the base year 2005, was 138 billion kWh which increases to 588 billion kWh

with an average growth rate of about 8% in the year 2045.The Agricultural and domestic

sector electric consumptions were 89 and 96 billion kWh respectively, in year 2005, and

increase up to 287 and 397 billion kWhs in the year 2045 with an average growth of 6% and

8% respectively.

24

0

100

200

300

400

500

600

700

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045

Year

Ele

ctr

icity

Consum

ptio

n(B

illio

n k

Wh)

IndustrialAgricultural

DomesticCommercial

TransportOthers

Fig.2.5 Sectoral electricity supply projection from base year 2006 to 2045 using econometric models.

In Table 2.2 below, R2 and adjusted R2 value for all electricity requirement sectors like

end use electric consuming sectors show very high predictive power of the developed models.

The Durbin-Watson statistics, in agricultural, industrial and domestic sectors are 2.15, 2.07

and 1.63 respectively which gives a conclusive result for the absence of autocorrelation. The

D-W statistics in commercial, transport and others sectors are 1.43, 1.4 and 1.29 respectively

which gives a moderate absence of autocorrelation. .The t-statistics also shows almost

satisfactory results for each variable in all sectors.

The coefficients in forecasting industrial sector demand are very small to unity. So these are

inelastic in nature. All other sectors also show inelastic behavior because the values of

coefficients are smaller than unity. In forecasting the sectoral electricity consumption and

requirement by multiple econometric models the problem of multicollinearity has been tested

and satisfactory results are obtained except for some sectors where multicollinearity is slightly

problematic. The variance of inflation factor (VIF) [24] is found less than five in most of the

sectors except for the agricultural sector in both the cases and transport sector in consumption

projection and other sectors in electricity requirement projection. Due to the limitation on data

availability in some sectors the VIF factor is more than five but less than ten. So the model

can be considered slightly multicollinear in these sectors.

25 Table 2.2: Values of coefficients of econometric models together with statistical results for electricity requirement based on data for the period 1971 to 2005 in individual sectors.

Variables Coefficients Standard

Errors t-

Statistics Statistics

Industrial sector Constant 2.121 0.918 2.309 R2 0.994

Ati 0.174 0.101 1.734 Adjusted R2 0.992

Pti -0.091 0.095 -0.965 Std. error of estimate 0.038

Lti 0.677 0.142 4.756 Durwin-Watson 2.070

Agricultural sector Constant 0.518 0.672 0.771 R2 0.998

Ati -0.150 0.132 -1.119 Adjusted R2 0.970

Pti -0.016 0.111 -0.134 Std. error of estimate 0.056

Lti 1.129 0.070 16.190 Durwin-Watson 2.150

Domestic sector Constant -9.392 0.998 -9.412 R2 0.984

Pt -0.085 0.090 -0.954 Adjusted R2 0.972


Mti 0.930 0.080 11.650 Durbin-Watson 1.630

Commercial sector Constant 0.670 1.370 0.489 R2 0.998

Pt 0.433 0.110 3.908 Adjusted R2 0.997

Mti 0.348 0.117 2.980 Std error of estimate 0.047

Ati 0.180 0.115 1.610 Durbin-Watson 1.43

Transport sector Constant 4.588 0.488 9.393 R2 0.980

Pt 0.265 0.150 1.773 Adjusted R2 0.970

Ati 0.306 0.129 2.370 Std error of estimate 0.063

Durbin-Watson 1.400

Other sectors Constant 3.323 0.821 4.045 R2 0.992

Pt 0.278 0.212 1.312 Adjusted R2 0.991


Durbin-Watson 1.294

As stated, the present electricity scenario is characterized by shortage of supply .Scenarios

have therefore been developed considering the patterns of supply-demand gap since the year

1971 and taking 2005 as the base year. The forecasted electricity demand using econometric

model (Eqn.7) for various sectors is shown in Fig.2.6. From the figure it is clear that the

electricity demand in the industrial sector is highest over the years. It increases to 7.5 times in

comparison to the base year 2005 in the year 2045. It is also seen that the electricity

consumption in the agriculture sector was dominating before 2004 as compared to the

domestic sector. But after that, the domestic sector electric energy demand increases rapidly

26 and the agricultural electricity requirement remains almost constant up to the forecasted

period. The commercial sector electricity demand also shows substantial growth over time but

less than in absolute term with respect to the industrial and domestic sectors demand. The

agricultural sector electricity demand is higher as compared to the commercial sector up to the

year 2020, but after that there is a rapid increase in the electricity requirement for this sector

with respect to the base year 2005. The transport sector shows very slow growth during the

forecasting period. The other sectors electric requirement shows significant growth rate over

the forecasting period and becomes about 11.5 times with respect to base year 2005.The

industrial sector electricity demand in base year 2005 was 211 billion kWh which increases to

1537 billion kWh with an average growth rate of about 7.4%.The domestic and commercial

sector electric requirement were 146 and 48 billion kWh in year 2005 and increases up to

1219 and 1099 billion kWh with an average growth of 5.5% and 8.2% respectively. The other

sector electric requirement grows with an average growth rate 8.3%.

The total electricity supply and demand by summation of all electrical energy consuming

sectors is shown by Fig.2.7. The total electricity end use consumption increases with an

average growth rate of about 7% and becomes 1.52 PWh in 2045 which is four fold from base

year 2005.The total electric requirement increases exponentially and becomes 5.06PWh with

6.9% average annual growth rate. The big gap between the total electricity end use

consumption and over all electricity requirements is due to the fact that the electricity

requirement data contains the electric deficit and transmission and distribution(T&D) losses

over the years and these considerations also included in the model prediction. The energy

deficit is 7.4% and the T&D losses are 31.5% of the total electricity production in the base

year 2005.

27

0

200

400

600

800

1000

1200

1400

1600

1800

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045

Year

Ele

ctr

icity R

equirem

ent(

Bill

ion k

Wh)

Industrial

Agricultural

DomesticCommercial

Transport

Other

Fig.2.6 Sectoral electricity requirement projection from base year 2006 to 2045 using econometric model

0

1000

2000

3000

4000

5000

6000

2006 2009 2012 2015 2018 2021 2024 2027 2030 2033 2036 2039 2042 2045

Year

Billio

n k

Wh

Electricity Consumption

Electricity Requirement

Fig.2.7 The total electricity requirement and end use consumption projections from base year 2005-2045.

2.4 Conclusions

Long-term electricity demand forecasting in power systems is a complicated task

because it is affected directly or indirectly by various factors primarily associated with the

economy and the population. In this work, two approaches have been applied, first is the time

28 series and second is multiple regression model. It is common to use a combination of

econometric and time series models to achieve greater precision in the forecasts. This has the

advantage of establishing causal relationships as in an econometric model along with the

dependency relationship. The model is shown to provide high accuracy forecasts up to the

year 2045. This is very useful in planning fuel procurement, scheduling unit maintenance, and

imports.

The forecast presented in this paper suggests that significant growth in electricity

demand can be expected in India until 2045.The sectoral electricity demand shows the

different growth rate over time. The forecast faster growth in electricity consumption is

consistent with the anticipated, relatively moderate rate of economic growth in India in the

coming decades. In addition, the faster growth in electricity consumption also reflects the fact

that there will be further structural changes in the Indian economy and that subsequently some

energy-intensive sectors in the economy are expected to grow. Concomitant with the faster

growth in electricity demand will be a continuation of the change in the market shares, with

oil, natural gas, and hydroelectricity becoming increasingly important energy sources at the

expense of coal, reflecting government policies towards the use of cleaner energy in India. It

is also observed that in the business as usual scenario, the supply–demand gap may increase

up to 7%, which is a highly unsustainable scenario. Policy interventions at this point like

reducing demand and alternatives like nuclear need to be considered urgently.

References:

[1] Balocco, C. and Grazzini, G. (1997) ’A statistical method to evaluate urban

energy needs’, Int.J.Energy Res., Vol.21, No.14, pp. 1321-1330.

[2] Al-Zayer, J. and Al-Ibrahim, A. (1996) ‘Modeling the impact of temperature

on electricity consumption in the eastern province of Saudi Arabia’,

J.Forecast.Vol.15, No.2, pp.97-106.

[3] Dincer, I. and Dost, S. (1997)’Energy and GDP’, Int.J.Energy Res., Vol.21,

No.2, pp.153-167.

[4] Connor, J.T. (1996) ‘A robust neural network filter for electricity demand

prediction’, J.Forecast Vol.15, No.6, pp 437-458.

29 [5] Galli, R. ( 1998) ‘The relationship between energy intensity and income

levels: forecasting long term energy demand in Asian emerging countries’, The

Energy Journal ,Vol.19, No.4,pp. 85-105.

[6] Erdogam, M.and Dalh, C. (1997)’Energy demand in Turkey’, J.Energy Devel.

Vol.21, No.2, pp. 173-187.

[7] Eltony, M.N.and Hosque, A. (1997) ‘A co integrating relationship in the

demand for energy: the case of electricity in Kuwait’, J.Energy Devel.Vol.21,

No.2, pp.293-301.

[8] Hondroyiannis, G. (2004)’Estimating residential demand for electricity in

Greece’, Energy Economics, Vol.26, No.3, pp.319-334.

[9] Hunt et al.(2003) ‘Underlying trends and seasonality in UK energy demand: a

sectoral analysis’, Energy Economics,Vol.25,pp-93-118.

[10] Crompton, P. and Wu, Y. (2005) ‘Energy consumption in China: past trends

and future directions’, Energy Economics, Vol.27, No.1, pp.195-208.

[11] Mirasgedis, et al. (2006)’Models for mid-term electricity demand forecasting

incorporating weather influences’, Energy, Vol.31, pp.208-227.

[12] Shiu, A. and Lam, P. (2004) ‘Electricity consumption and economic growth in

china’, Energy Policy, Vol. 32, No.1, pp.47-54.

[13] Yemane, Wolde-Rufael, (2006)’Electricity consumption and economic growth:

a time series experience for 17 African countries’, Energy Policy, Vol.34,

No10, pp.1106-1114.

[14] Yoo, Seung-Hoon (2005)’Electricity consumption and economic growth:

evidence from Korea’, Energy Policy, Vol.33, No.12, pp.1627-1632.

[15] Nasr, et al. (2000)’Econometric modeling for electricity consumption in post-

war Lebanon’, Energy Economics, Vol.22, No.6, pp.627-640.

[16] Ghosh, S. (2002) ‘Electricity consumption and economic growth in India’

Energy Policy, Vol. 30, No.2, pp.125-129.

[17] Majumdar, S.and Parikh, J. (1996)’Energy demand forecasts with investment

constraints’, J.Forecast. Vol.15, No.6, pp. 459-476.

[18] PC, (2006) Integrated energy policy: Report of the Expert Committee,

Planning Commission, New Delhi, India.

30 [19] TERI (2005) ‘TERI Energy Data Directory and Yearbook2004-

05(TEDDY)’, The Energy and Resource Institute, New Delhi.

[20] Yaffee, R. and McGee, M. (2000)’ Introduction to time series analysis and

forecasting with applications of SAS and SPSS’ Academic Press, New York.

[21] CMIE (2007) ‘India’s energy sector’, Centre for Monitoring the Indian

Economy, Mumbai, India.

[22] Gujarati, D.N. (2005) ‘Basic Econometrics’, Tata McGraw-Hill Publishing

Company Limited, New Delhi.

[23] Makridakis S., et al (1998) ‘Forecasting: Methods and Applications (3rd

edition)’, John Wiley & Sons, Inc, New York.

[24] Levine, et al. (2004) ‘Business Statistics: A First Course’, Pearson Education,

New Delhi.

Appendix 2.1

Fundamentals of Econometric Equations

The Simple Two variable Models

The simple two variable linear equation can be written as

eXY ++= βα (1)

Where

Y = dependent variable

X = independent variable

� ,� = parameters to be estimated

e = a random error term.

31 The least square criterion can be seen to be the determination of the set of values of � and �

that minimizes the expression

2

11

2)( i

n

i

i

n

i

i xye βα −−=��==

(2)

Differentiating eqn (2) with respect to � and � one obtains

( ) )(22

iii xye βαα

−−−=∂

∂�� (3)

( ) )(22

iiii xyxe βαβ

−−−=∂

∂�� (4)

Which when set equal to zero, yields

��∧∧

+= ii xny βα (5)

��∧∧

+= 2

iiii xxyx βα (6)

where ∧

α and ∧

β denotes the least square estimator of � and �. These two equations can be

solved for ∧

α and ∧

β to yield

( )� �

� � �−

−=

∧

22

ii

iiii

xxn

yxyxnβ (7)

32

( )� �

� � � �−

−=

∧

22

2

ii

iiiii

xxn

yxxyxα (8)

It should be noted that we hypothesize the residual to be characterized by a probability

distribution of zero mean and some unknown variance 2

uσ ,i.e.

0=ieE (9)

Where the notation

E denotes expected value of , and

2

0

u

iieeEσ

=

Extensions to the Multivariable Case:

The n equations of linear model is written as

ikikiiii eXXXY +++++= ββββ ...3322 (11)

can be written more compactly in the matrix notation as

)1()1()()1( nxkxnxknx

eXY += β (12)

Where

��

�

�

��

�

�

=

nY

Y

Y

Y.

2

1

��

�

�

��

�

�

=

knnn

k

k

XXX

XXX

XXX

X

.1

.....

.1

.1

32

23222

13121

��

�

�

��

�

�

=

nβ

β

β

β.

2

1

��

�

�

��

�

�

=

ne

e

e

e.

2

1

for i � j

for i = j (10)

33

There is k-1 explanatory or independent variables. Now let ∧

β denote the least squares

estimate of �.Then we may write

eXY +=∧

β (13)

The sum of squares is given by

eeeT

n

i

i =�=1

2 (14)

�

��

−�

��

−=

∧∧

ββ XYXY

T

∧∧∧

+−= βββ XXYXYYT

T

T

T

T 2 (15)

Differentiating eqn (15) to obtain the value of � that minimizes the sum of squares

( )∧

+−=∂

∂β

βXXYXee

TTT 22 (16)

Which when set equal to zero and solving for ∧

β ,yields

∧

+−= βXXXX TT 220 (17)

∧

= βXXXXTT (18)

By multiplying both sides by ( ) 1−XX

T ,we obtain

34

( ) ( )∧−−

= βXXXXYXXXTTTT 11

(19)

But a matrix multiplied by its inverse yields the identity matrix and any vector multiplied by

the identity matrix of comfortable dimension yields again the vector, thus

( ) YXXXTT 1−∧

=β (20)

To establish the mean and variance of ∧

β ,let us substitute eqn(12) in to eqn(20),to yield

( ) ( )eXXXXTT +=

−∧

ββ1

(21)

( ) eXXXTT 1−

+= β (22)

Taking expectation of both sides

( ) { }eEXXXETT 1−∧

+=��

��

ββ (23)

from which it follows that ββ =��

�� ∧

E if the expected value of the residuals is zero.

The variance of ∧

β follows from the definition

{ }��

��

��

��

�

��

−�

��

−=

∧∧ T

EVar βββββ (24)

Since from eqn (21)

35

( ) ( )eXXXXTT +−=−

−∧

ββββ1

( ) ( ) ( ) eXXXXXXXTTTT 11 −−

−−= ββ

( ) eXXXTT 1−

−= (25)

then inserting eqn (25) in to eqn (24) ,one obtains

( ) ( ){ }11 −−∧

=��

��

XXXeeXXXEVarTTTTβ

( ) { } ( ) 11 −−= XXXeeEXXX

TTTT (26)

Thus if { } IeeET 2σ= ,an assumption noted previously,

( ) 12 −∧

=��

��

XXVarTσβ (27)

It can be shown that

{ } 2)( σkneeET −= (28)

from which follows that our estimate of 2σ , say S2 is given by

kn

eeS

T

−=2 (29)

36 but from eqn (15)

∧∧∧

+−= βββ XXYXYYeeTTTTT 2

YXYYTT

∧

−= β (30)

chapter-2 electricity demand projectionsshodhganga.inflibnet.ac.in/bitstream/10603/8179/9/09_chapter...

Documents