The Simple Linear Regression Model:
Specification and Estimation
Prepared by Vera Tabakova, East Carolina University
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2.1 An Economic Model
2.2 An Econometric Model
2.3 Estimating the Regression Parameters
2.4 Assessing the Least Squares Estimators
2.5 The Gauss-Markov Theorem
2.6 The Probability Distributions of the Least
Squares Estimators
2.7 Estimating the Variance of the Error Term
Principles of Econometrics, 3rd Edition
Slide 2-2
Figure 2.1a Probability distribution of food expenditure y given income x = $1000
Principles of Econometrics, 3rd Edition
Slide 2-3
Figure 2.1b Probability distributions of food expenditures y
given incomes x = $1000 and x = $2000
Principles of Econometrics, 3rd Edition
Slide 2-4

The simple regression function
E( y| x )   y| x  1   2 x
Principles of Econometrics, 3rd Edition
(2.1)
Slide 2-5
Figure 2.2 The economic model: a linear relationship between
average per person food expenditure and income
Principles of Econometrics, 3rd Edition
Slide 2-6

Slope of regression line
E( y| x ) dE( y| x )
2 

x
dx
(2.2)
“Δ” denotes “change in”
Principles of Econometrics, 3rd Edition
Slide 2-7
Figure 2.3 The probability density function for y at two levels of income
Principles of Econometrics, 3rd Edition
Slide 2-8
Assumptions of the Simple Linear Regression Model – I
The mean value of y, for each value of x, is given by the
linear regression
E  y | x   1  2 x
Principles of Econometrics, 3rd Edition
Slide 2-9
Assumptions of the Simple Linear Regression Model – I
For each value of x, the values of y are distributed about their
mean value, following probability distributions that all have
the same variance (homoscedasticity),
var  y | x   2
Principles of Econometrics, 3rd Edition
Slide 2-10
Assumptions of the Simple Linear Regression Model – I
The sample values of y are all uncorrelated (no autocorrelation), and have zero covariance, implying that there is
no linear association among them,
cov yi , y j  0


This assumption can be made stronger by assuming that the
values of y are all statistically independent.
Principles of Econometrics, 3rd Edition
Slide 2-11
Assumptions of the Simple Linear Regression Model – I
The variable x is not random, and must take at least two
different values.
Principles of Econometrics, 3rd Edition
Slide 2-12
Assumptions of the Simple Linear Regression Model – I
(optional) The values of y are normally distributed about their
mean for each value of x,
y
Principles of Econometrics, 3rd Edition
N 1  2 x,  2 
Slide 2-13
Assumptions of the Simple Linear Regression Model – I
•The mean value of y, for each value of x, is given by the linear regression
E ( y | x )  1  2 x
•For each value of x, the values of y are distributed about their mean value, following probability distributions
that all have the same variance,
2
var( y | x)  
•The sample values of y are all uncorrelated, and have zero covariance, implying that there is no linear
association among them,
cov  yi , y j   0
This assumption can be made stronger by assuming that the values of y are all statistically independent.
•The variable x is not random, and must take at least two different values.
•(optional) The values of y are normally distributed about their mean for each value of x,
y ~ N 1  2 x  , 2 
Principles of Econometrics, 3rd Edition
Slide 2-14

2.2.1 Introducing the Error Term
 The random error term is defined as
e  y  E ( y | x )  y  1   2 x
(2.3)
 Rearranging gives
y  1  2 x  e
(2.4)
y is dependent variable; x is independent variable
Principles of Econometrics, 3rd Edition
Slide 2-15
The expected value of the error term, given x, is
E  e | x   E  y | x  1 2 x  0
The mean value of the error term, given x, is zero.
Principles of Econometrics, 3rd Edition
Slide 2-16
Figure 2.4 Probability density functions for e and y
Principles of Econometrics, 3rd Edition
Slide 2-17
Assumptions of the Simple Linear Regression Model – II
SR1. The value of y, for each value of x, is
y  1  2 x  e
Principles of Econometrics, 3rd Edition
Slide 2-18
Assumptions of the Simple Linear Regression Model – II
SR2. The expected value of the random error e is
E (e)  0
Which is equivalent to assuming that
E ( y )  1  2 x
Principles of Econometrics, 3rd Edition
Slide 2-19
Assumptions of the Simple Linear Regression Model – II
SR3. The variance of the random error e is
var(e)    var( y)
2
The random variables y and e have the same variance
because they differ only by a constant.
Principles of Econometrics, 3rd Edition
Slide 2-20
Assumptions of the Simple Linear Regression Model – II
SR4. The covariance between any pair of random errors,
ei and ej is
cov(ei , e j )  cov( yi , y j )  0
The stronger version of this assumption is that the random
errors e are statistically independent, in which case the values
of the dependent variable y are also statistically independent.
Principles of Econometrics, 3rd Edition
Slide 2-21
Assumptions of the Simple Linear Regression Model – II
SR5. The variable x is not random, and must take at least two
different values.
Principles of Econometrics, 3rd Edition
Slide 2-22
Assumptions of the Simple Linear Regression Model – II
SR6. (optional) The values of e are normally distributed about
their mean
e

N 0, 2

if the values of y are normally distributed, and vice versa.
Principles of Econometrics, 3rd Edition
Slide 2-23
Assumptions of the Simple Linear Regression Model - II
•SR1.
y  1  2 x  e
•SR2. E ( e)  0  E ( y )  1   2 x
•SR3. var(e)  2  var( y)
•SR4. cov(ei , e j )  cov( yi , y j )  0
•SR5. The variable x is not random, and must take at least two different values.
•SR6. (optional) The values of e are normally distributed about their mean e ~ N(0, 2 )
Principles of Econometrics, 3rd Edition
Slide 2-24
Figure 2.5 The relationship among y, e and the true regression line
Principles of Econometrics, 3rd Edition
Slide 2-25
Principles of Econometrics, 3rd Edition
Slide 2-26
Figure 2.6 Data for food expenditure example
Principles of Econometrics, 3rd Edition
Slide 2-27

2.3.1 The Least Squares Principle
 The fitted regression line is
yˆi  b1  b2 xi
(2.5)
 The least squares residual
eˆi  yi  yˆi  yi  b1  b2 xi
Principles of Econometrics, 3rd Edition
(2.6)
Slide 2-28
Figure 2.7 The relationship among y, ê and the fitted regression line
Principles of Econometrics, 3rd Edition
Slide 2-29
 Any other fitted line
yˆi*  b1*  b2* xi
 Least squares line has smaller sum of squared residuals
N
N
i 1
i 1
if SSE =  eˆi2 and SSE * =  eˆi*2 then SSE < SSE *
Principles of Econometrics, 3rd Edition
Slide 2-30
 Least squares estimates for the unknown parameters β1
and β2 are obtained my minimizing the sum of squares
function
N
S  1 , 2    ( yi  1  2 xi )
2
i 1
Principles of Econometrics, 3rd Edition
Slide 2-31

The Least Squares Estimators
xi  x  yi  y 


b2 
2
  xi  x 
(2.7)
b1  y  b2 x
(2.8)
Principles of Econometrics, 3rd Edition
Slide 2-32

2.3.2 Estimates for the Food Expenditure Function
b2
 x  x  y  y  18671.2684



 10.2096
1828.7876
x  x 
i
i
2
i
b1  y  b2 x  283.5735  (10.2096)(19.6048)  83.4160
A convenient way to report the values for b1 and b2 is to write
out the estimated or fitted regression line:
yˆi  83.42  10.21xi
Principles of Econometrics, 3rd Edition
Slide 2-33
Figure 2.8 The fitted regression line
Principles of Econometrics, 3rd Edition
Slide 2-34

2.3.3 Interpreting the Estimates
 The value b2 = 10.21 is an estimate of 2, the amount by
which weekly expenditure on food per household increases
when household weekly income increases by $100. Thus, we
estimate that if income goes up by $100, expected weekly
expenditure on food will increase by approximately $10.21.
 Strictly speaking, the intercept estimate b1 = 83.42 is an
estimate of the weekly food expenditure on food for a
household with zero income.
Principles of Econometrics, 3rd Edition
Slide 2-35

2.3.3a Elasticities
 Income elasticity is a useful way to characterize the responsiveness
of consumer expenditure to changes in income. The elasticity of a
variable y with respect to another variable x is
percentage change in y y y y x



percentage change in x x x x y
 In the linear economic model given by (2.1) we have shown that
2 
Principles of Econometrics, 3rd Edition
E  y 
x
Slide 2-36
 The elasticity of mean expenditure with respect to income is
E ( y ) / E ( y ) E ( y ) x
x



 2 
x / x
x
E( y)
E( y)
(2.9)
 A frequently used alternative is to calculate the elasticity at the
“point of the means” because it is a representative point on the
regression line.
x
19.60
ˆ  b2  10.21
 .71
y
283.57
Principles of Econometrics, 3rd Edition
Slide 2-37

2.3.3b Prediction
 Suppose that we wanted to predict weekly food expenditure for a
household with a weekly income of $2000. This prediction is
carried out by substituting x = 20 into our estimated equation to
obtain
yˆi  83.42  10.21xi  83.42  10.21(20)  287.61
 We predict that a household with a weekly income of $2000 will
spend $287.61 per week on food.
Principles of Econometrics, 3rd Edition
Slide 2-38

2.3.3c Examining Computer Output
Figure 2.9 EViews Regression Output
Principles of Econometrics, 3rd Edition
Slide 2-39

2.3.4 Other Economic Models
 The “log-log” model
ln( y)  1  2 ln( x)
d [ln( y )] 1 dy
 
dx
y dx
d [1  2 ln( x)] 1
 2
dx
x
dy x
2  
dx y
Principles of Econometrics, 3rd Edition
Slide 2-40
 2.4.1 The estimator b2
N
b2   wi yi
(2.10)
xi  x
wi 
2
(
x

x
)
 i
(2.11)
b2  2   wi ei
(2.12)
i 1
Principles of Econometrics, 3rd Edition
Slide 2-41
 2.4.2 The Expected Values of b1 and b2
 We will show that if our model assumptions hold, then E  b    , which means
2
2
that the estimator is unbiased.
 We can find the expected value of b2 using the fact that the expected value of a sum
is the sum of expected values
E (b2 )  E  2   wi ei   E  2  w1e1  w2 e2 
 E  2   E  w1e1   E  w2 e2  
 E (2 )   E ( wi ei )
 wN eN 
 E  wN eN 
(2.13)
 2   wi E (ei )  2
using E  wi ei   wi E  ei  and E (ei )  0
Principles of Econometrics, 3rd Edition
Slide 2-42
2.4.3 Repeated Sampling
Principles of Econometrics, 3rd Edition
Slide 2-43

The variance of b2 is defined as var  b2   E b2  E  b2  
2
Figure 2.10 Two possible probability density functions for b2
Principles of Econometrics, 3rd Edition
Slide 2-44
 2.4.4 The Variances and Covariances of b1 and b2
 If the regression model assumptions SR1-SR5 are correct (assumption SR6 is not
required), then the variances and covariance of b1 and b2 are:


xi2

var(b1 )   
2 
N
(
x

x
)
  i

(2.14)
2
var(b2 ) 
 ( xi  x )2
(2.15)


x
cov(b1 , b2 )  2 
2 
(
x

x
)
 i

(2.16)
2
Principles of Econometrics, 3rd Edition
Slide 2-45
 2.4.4 The Variances and Covariances of b1 and b2
 The larger the variance term  2, the greater the uncertainty there is in the
statistical model, and the larger the variances and covariance of the least squares
estimators.
2
 The larger the sum of squares,  ( xi  x ) , the smaller the variances of the least
squares estimators and the more precisely we can estimate the unknown
parameters.
 The larger the sample size N, the smaller the variances and covariance of the
least squares estimators.
2
 The larger this term  xi is, the larger the variance of the least squares estimator
b1 .
 The absolute magnitude of the covariance increases the larger in magnitude is
the sample mean x , and the covariance has a sign opposite to that of x .
Principles of Econometrics, 3rd Edition
Slide 2-46

The variance of b2 is defined as var  b2   E b2  E  b2  
2
Figure 2.11 The influence of variation in the explanatory variable x on precision of estimation
(a) Low x variation, low precision (b) High x variation, high precision
Principles of Econometrics, 3rd Edition
Slide 2-47
Link: Gauss-Markov Theorem
Gauss-Markov Theorem: Under the assumptions
SR1-SR5 of the linear regression model, the estimators
b1 and b2 have the smallest variance of all linear and
unbiased estimators of b1 and b2. They are the Best
Linear Unbiased Estimators (BLUE) of b1 and b2
Principles of Econometrics, 3rd Edition
Slide 2-48
1.
The estimators b1 and b2 are “best” when compared to similar estimators, those
which are linear and unbiased. The Theorem does not say that b1 and b2 are the
best of all possible estimators.
2.
The estimators b1 and b2 are best within their class because they have the
minimum variance. When comparing two linear and unbiased estimators, we
always want to use the one with the smaller variance, since that estimation rule
gives us the higher probability of obtaining an estimate that is close to the true
parameter value.
3.
In order for the Gauss-Markov Theorem to hold, assumptions SR1-SR5 must
be true. If any of these assumptions are not true, then b1 and b2 are not the best
linear unbiased estimators of β1 and β2.
Principles of Econometrics, 3rd Edition
Slide 2-49
4.
The Gauss-Markov Theorem does not depend on the assumption of normality
(assumption SR6).
5.
In the simple linear regression model, if we want to use a linear and unbiased
estimator, then we have to do no more searching. The estimators b1 and b2 are
the ones to use. This explains why we are studying these estimators and why
they are so widely used in research, not only in economics but in all social and
physical sciences as well.
6.
The Gauss-Markov theorem applies to the least squares estimators. It does not
apply to the least squares estimates from a single sample. (In other words, you
can have a weird sample.)
Principles of Econometrics, 3rd Edition
Slide 2-50
 If we make the normality assumption (assumption SR6 about the error term) then
the least squares estimators are normally distributed

2  xi2 
b1 ~ N  1 ,
2 
N
(
x

x
)
 i


(2.17)


2
b2 ~ N  2 ,
2 
(
x

x
)
 i


(2.18)
A Central Limit Theorem: If assumptions SR1-SR5 hold, and if the sample
size N is sufficiently large, then the least squares estimators have a distribution
that approximates the normal distributions shown in (2.17) and (2.18).
Principles of Econometrics, 3rd Edition
Slide 2-51
The variance of the random error ei is
var(ei )  2  E[ei  E(ei )]2  E(ei2 )
if the assumption E(ei) = 0 is correct.
Since the “expectation” is an average value we might consider estimating σ2 as the
average of the squared errors,
2
e
i
ˆ 2 
N
Recall that the random errors are
ei  yi  1  2 xi
Principles of Econometrics, 3rd Edition
Slide 2-52
The least squares residuals are obtained by replacing the unknown parameters by their
least squares estimates,
eˆi  yi  yˆi  yi  b1  b2 xi
2
ˆ
e

i
ˆ 2 
N
There is a simple modification that produces an unbiased estimator, and that is
2
ˆ
e
i
ˆ 2 
N 2
(2.19)
ˆ 2 )  2
E(
Principles of Econometrics, 3rd Edition
Slide 2-53

Replace the unknown error variance  in (2.14)-(2.16) by
2
̂ 2 to obtain:
2


x

2
i
var  b1   ˆ 
2 
N
(
x

x
)
  i

(2.20)
ˆ 2
var  b2  
2
(
x

x
)
 i
(2.21)


x
cov  b1 , b2   ˆ 
2 
(
x

x
)
 i

(2.22)
2
Principles of Econometrics, 3rd Edition
Slide 2-54

The square roots of the estimated variances are the “standard errors” of b1
and b2.
se  b1   var  b1 
(2.23)
se  b2   var  b2 
(2.24)
Principles of Econometrics, 3rd Edition
Slide 2-55
2
ˆ
e
304505.2
ˆ 2   i 
 8013.29
N 2
38
Principles of Econometrics, 3rd Edition
Slide 2-56

The estimated variances and covariances for a regression are arrayed
in a rectangular array, or matrix, with variances on the diagonal and
covariances in the “off-diagonal” positions.
 var  b 
cov  b1 , b2  
1


cov  b1 , b2 
var  b2  

Principles of Econometrics, 3rd Edition
Slide 2-57

For the food expenditure data the estimated covariance matrix is:
C
INCOME
Principles of Econometrics, 3rd Edition
C
1884.442
-85.90316
INCOME
-85.90316
4.381752
Slide 2-58
var  b1   1884.442
var  b2   4.381752
cov  b1 , b2   85.90316
se  b1   var  b1   1884.442  43.410
se  b2   var  b2   4.381752  2.093
Principles of Econometrics, 3rd Edition
Slide 2-59












assumptions
asymptotic
B.L.U.E.
biased estimator
degrees of freedom
dependent variable
deviation from the
mean form
econometric model
economic model
elasticity
Gauss-Markov
Theorem
heteroskedastic
Principles of Econometrics, 3rd Edition












homoskedastic
independent variable
least squares
estimates
least squares
estimators
least squares principle
least squares residuals
linear estimator
prediction
random error term
regression model
regression parameters
repeated sampling






sampling precision
sampling properties
scatter diagram
simple linear
regression function
specification error
unbiased estimator
Slide 2-60
Principles of Econometrics, 3rd Edition
Slide 2-61
N
S (1 , 2 )   ( yi  1  2 xi ) 2
(2A.1)
i 1
S
 2 N 1  2 yi  2   xi  2
1
(2A.2)
S
 2   xi2  2  2 xi yi  2   xi  1
2
Principles of Econometrics, 3rd Edition
Slide 2-62
Figure 2A.1 The sum of squares function and the minimizing values b1 and b2
Principles of Econometrics, 3rd Edition
Slide 2-63
2   yi  Nb1    xi  b2   0
2   xi yi    xi  b1    xi2  b2   0
Nb1    xi  b2   yi
  xi  b1    xi2  b2   xi yi
b2 
Principles of Econometrics, 3rd Edition
N  xi yi   xi  yi
N  x    xi 
2
i
2
(2A.3)
(2A.4)
(2A.5)
Slide 2-64
 1

2
(
x

x
)

x

2
x
x

N
x

x

2
x
N
x

N
x
 i

 i

 i

N


2
2
i
2
2
i
(2B.1)
  xi2  2 N x 2  N x 2   xi2  N x 2
 ( xi  x )
2
  x  N x   x  x  xi   x
2
i
2
2
i
 ( xi  x )( yi  y )   xi yi  N x y   xi yi 
Principles of Econometrics, 3rd Edition
2
i
xi 



2
(2B.2)
N
 xi  yi
N
(2B.3)
Slide 2-65
We can rewrite b2 in deviation from the mean form as:
( xi  x )( yi  y )

b2 
 ( xi  x )2
Principles of Econometrics, 3rd Edition
Slide 2-66
  xi  x   0
( xi  x )( yi  y )  ( xi  x ) yi  y  ( xi  x )

b2 

2
 ( xi  x )
 ( xi  x )2
 ( xi  x ) 
( xi  x ) y i


 
y   wi yi
2
2  i
 ( xi  x )
  ( xi  x ) 
Principles of Econometrics, 3rd Edition
Slide 2-67
To obtain (2.12) replace yi in (2.11) by yi  1  2 xi  ei and simplify:
b2   wi yi   wi (1  2 xi  ei )
 1  wi  2  wi xi   wi ei
 2   wi ei
Principles of Econometrics, 3rd Edition
Slide 2-68
  xi  x  
1

 wi   
2
2
   xi  x     xi  x 
  xi  x   0
 wi xi  1
2  wi xi  2
( xi  x )  0
Principles of Econometrics, 3rd Edition
Slide 2-69
  xi  x 
2
   xi  x  xi  x 
   xi  x  xi  x   xi  x 
   xi  x  xi
xi  x  xi   xi  x  xi



 wi xi 
2
  xi  x    xi  x  xi
Principles of Econometrics, 3rd Edition
1
Slide 2-70
b2  2   wi ei
var  b2   E b2  E  b2  
Principles of Econometrics, 3rd Edition
2
Slide 2-71
var  b2   E 2   wi ei  2 
 E   wi ei 
2
2


 E   wi2 ei2  2 wi w j ei e j 
i j


[square of bracketed term]
  wi2 E  ei2   2 wi w j E  ei e j 
[because wi not random]
i j
 2  wi2

2
  xi  x 
Principles of Econometrics, 3rd Edition
2
Slide 2-72
2  var  ei   E ei  E  ei   E ei  0  E  ei2 
2

2

cov  ei , e j   E  ei  E  ei   e j  E  e j    E  ei e j   0




2
w

 i 

 xi  x 
2
x

x


 i 
2



2 

1
  xi  x  
2
2 2
  xi  x     xi  x 
2
var  aX  bY   a 2 var  X   b2 var Y   2ab cov  X , Y 
Principles of Econometrics, 3rd Edition
Slide 2-73
var  b2   var  2   wi ei 
[since 2 is a constant]
=  wi2 var  ei    wi w j cov  ei , e j 
[generalizing the variance rule]
i j
=  wi2 var  ei 
[using cov  ei , e j   0]
  2  wi2
[using var  ei    2 ]

2
  xi  x 
2
Principles of Econometrics, 3rd Edition
Slide 2-74

Let b2*   ki yi be any other linear estimator of β2.

Suppose that ki = wi + ci.
b2*   ki yi   ( wi  ci ) yi   ( wi  ci )(1  2 xi  ei )
  ( wi  ci ) 1   ( wi  ci ) 2 xi   ( wi  ci ) ei
(2F.1)
 1  wi  1  ci  2  wi xi  2  ci xi   ( wi  ci ) ei
 1  ci  2  2  ci xi   ( wi  ci ) ei
Principles of Econometrics, 3rd Edition
Slide 2-75
E (b2* )  1  ci  2  2  ci xi   (wi  ci )E (ei )
 1  ci  2  2  ci xi
ci  0 and ci xi  0
b2*   ki yi  2   (wi  ci ) ei
Principles of Econometrics, 3rd Edition
(2F.2)
(2F.3)
(2F.4)
Slide 2-76
 ci  xi  x  
1
c
w




i i 
2
2
   xi  x     xi  x 
 ci xi 
x
  xi  x 
2
 ci  0
var  b2*   var 
 2    wi  ci  ei     wi  ci  var  ei 
2
 2   wi  ci   2  wi2  2  ci2
2
 var  b2   2  ci2
 var  b2 
Principles of Econometrics, 3rd Edition
Slide 2-77