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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
Ati 0.582 0.088 6.642 Std. error of estimate 0.035
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
Ati 0.555 0.013 43.344 Std. error of estimate 0.063
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
Ati 0.454 0.093 4.864 Std. error of estimate 0.038
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
Ati 0.395 0.141 2.809 Std. error of estimate 0.079
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.
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Company Limited, New Delhi.
[23] Makridakis S., et al (1998) ‘Forecasting: Methods and Applications (3rd
edition)’, John Wiley & Sons, Inc, New York.
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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)
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