The Simple Regression Model
y = b0 + b1x + u
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Some Terminology
In the simple linear regression model,
where y = b0 + b1x + u, we typically refer
to y as the



Dependent Variable, or
Left-Hand Side Variable, or
Explained Variable, or
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Some Terminology
we typically refer to x as the





Independent Variable, or
Right-Hand Side Variable, or
Explanatory Variable, or
Regressor, or
Control Variables
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A Simple Assumption
The average value of u, the error term, in
the population is 0. That is,
E(u) = 0
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We also assume
E(u|x) = 0
E(y|x) = b0 + b1x
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E(y|x) as a linear function of x, where for any x
the distribution of y is centered about E(y|x)
y
f(y)
. E(y|x) = b + b x
0
.
x1
1
x2
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Ordinary Least Squares (OLS)
Let {(xi,yi): i=1, …,n} denote a random
sample of size n from the population
For each observation in this sample, it will
be the case that
yi = b0 + b1xi + ui
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Population regression line, sample data points
and the associated error terms
E(y|x) = b0 + b1x
.{
u4
y
y4
y3
y2
y1
.} u3
u2 {.
} u1
.
x1
x2
x3
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x
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Basic idea of regression is to estimate the
population parameters from a sample
Intuitively, OLS is fitting a line through the
sample points such that the sum of squared
residuals is as small as possible.
The residual, û, is an estimate of the error
term, u, and is the difference between the
fitted line (sample regression function) and
the sample point
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Sample regression line, sample data points
and the associated estimated error terms (residuals)
y
.
y4
û4 {
yˆ  bˆ0  bˆ1 x
y3
y2
y1
.} û3
û2 { .
û1
}
.
x1
x2
x3
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x
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One approach to estimate
coefficients
Given the intuitive idea of fitting a line, we can
set up a formal minimization problem
That is, we want to choose our parameters such
that we minimize the following:
n
 uˆi 
i 1
2
n

  yi  bˆ0  bˆ1 xi
i 1
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2
11
It could be shown that estimated
coefficient is
n
bˆ1 
  x  x  y  y 
i 1
i
i
n
x  x 
i 1
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Summary of OLS slope estimate
The slope estimate is the sample covariance
between x and y divided by the sample
variance of x
If x and y are positively correlated, the
slope will be positive
If x and y are negatively correlated, the
slope will be negative
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Algebraic Properties of OLS: in
English
The sum of the OLS residuals is zero
Thus, the sample average of the OLS
residuals is zero as well
The sample covariance between the
regressors and the OLS residuals is zero
The OLS regression line always goes
through the mean of the sample
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Algebraic Properties of OLS: In
mathematics:
n
n
 uˆ
i 1
n
 uˆi  0 and thus,
n
 x uˆ
i 1
i
i
i 1
i
0
0
y  bˆ0  bˆ1 x
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More terminology
We can think of each observatio n as being made
up of an explained part, and an unexplaine d part,
yi  yˆ i  uˆi We then define the following :
  y  y  is the total sum of squares (SST)
  yˆ  y  is the explained sum of squares (SSE)
 uˆ is the residual sum of squares (SSR)
2
i
2
i
2
i
Then SST  SSE  SSR
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Notations Alert
The notation SSR (Sum of Squared
Residuals) in this handout and my other
lecture slides= ESS (Error Sum of Squares)
in our textbook.
The notation SSE (Sum of Squared
Explained) in this handout and my other
lecture slides = RSS (Regressed Sum of
Squares) in our textbook.
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Goodness-of-Fit
How do we think about how well our
sample regression line fits our sample data?
Can compute the fraction of the total sum
of squares (SST) that is explained by the
model, call this the R-squared of regression
R2 = SSE/SST = 1 – SSR/SST
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OLS regressions
Now that we’ve derived the formula for
calculating the OLS estimates of our
parameters, you’ll be happy to know you
don’t have to compute them by hand
Regressions in GRETL are very simple.
Have you installed the software yet?
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Under some conditions, OLS esimated
coefficients are unbiased.
Define
S    xi  x  . It could be shown that :
2
2
x
bˆ1  b1
 x  x u


i
S
i
2
x
 
 1 
ˆ
E b1  b1  
2    xi  x  * E  ui   b1
 Sx 
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Unbiasedness Summary
The OLS estimates of b1 and b0 are
unbiased
Remember unbiasedness is a description of
the estimator – in a given sample we may be
“near” or “far” from the true parameter
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Variance of the OLS Estimators
Now we know that the sampling
distribution of our estimated coefficient is
centered around the true parameter
Want to think about how spread out this
distribution is
Assume Var(u|x) =Var(u) = s2
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Variance of OLS estimators
s2 is called the error variance
s, the square root of the error variance is
called the standard deviation of the error
E(y|x)=b0 + b1x and Var(y|x) = s2
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Variance of OLS estimator
 
Var bˆ1  s
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S
2
x
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Variance of OLS Summary
The larger the error variance, s2, the larger
the variance of the slope estimate
The larger the variability in the xi, the
smaller the variance of the slope estimate
Problem that the error variance is unknown
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Estimating the Error Variance
We don’t know what the error variance, s2,
is, because we don’t observe the errors, ui
What we observe are the residuals, ûi
We can use the residuals to form an
estimate of the error variance
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Estimating the Error Variance
uˆi  yi  bˆ0  bˆ1 xi
an unbiased estimator of s is
2
1
2
sˆ 
uˆi  SSR /  n  2 

 n  2
2
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Estimating Standard Error of
coefficients Estimate
 
se bˆ1  sˆ /
 x  x  
2
1
2
i
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