Introduction to Linear
Regression
(SW Chapter 4)
The class size/test score policy
question:
 What is the effect on test scores of
reducing STR by one
student/class?
 Object of policy interest:
Test score
STR
 This is the slope of the line relating
test score and STR
This suggests that we want to draw a
line through the Test Score v. STR
scatterplot – but how?
Some Notation and Terminology
(Sections 4.1 and 4.2)
We assume that the conditional mean
of test scores, conditional on the
student-teacher ratio, is linear. That is:
E(TS │STR) = 0 + 1STR
where β0 and β1 are (unknown)
constants.
We call
TS = 0 + 1STR
the population regression line and
1 = slope of population regression
line
Test score
=
STR
= change in test score for a unit
change in STR
 Why are 0 and 1 “population”
parameters?
 We would like to know the
population value of 1.
 We don’t know 1, so must
estimate it using data.
How can we estimate 0 and 1 from
data?
Recall that Y was the least squares
estimator of Y: Y solves,
n
min m  (Yi  m) 2
i 1
By analogy, we will focus on the
least squares (“ordinary least
squares” or “OLS”) estimator of the
unknown parameters 0 and 1,
which solves,
n
min b0 ,b1 [Yi  (b0  b1 X i )]2
i 1
The OLS estimator minimizes the
average squared difference between
the actual values of Yi and the
prediction (predicted value) based on
the estimated line.
 This minimization problem can be
solved using calculus (App. 4.2).
 The result is the OLS estimators
of 0 and 1, ̂ 0 and ˆ1 ,
respectively.
Why use OLS, rather than some
other estimator?
 OLS is a generalization of the
sample average: if the “line” is
just an intercept (no X), then the
OLS estimator is just the sample
average of Y1,…Yn (Y ).
 Like Y , the OLS estimator has
some desirable properties: under
certain assumptions, it is unbiased
(that is, E( ˆ1 ) = 1), and it has a
tighter sampling distribution than
some other candidate estimators of
1 (more on this later)
 Importantly, this is what everyone
uses – the common “language” of
linear regression.
Application to the California Test
Score – Class Size data
Estimated slope = ˆ1 = – 2.28
Estimated intercept = ˆ0 = 698.9
Estimated regression line:
TS = 698.9 – 2.28*STR
This is also called the sample
regression line.
Interpretation of the estimated slope
and intercept
TS= 698.9 – 2.28*STR
 Districts with one more student per
teacher on average have test scores
that are 2.28 points lower.
Test score
 That is,
= –2.28
STR
 The intercept (taken literally)
means that, according to this
estimated line, districts with zero
students per teacher would have a
(predicted) test score of 698.9.
 This interpretation of the intercept
makes no sense – it extrapolates
the line outside the range of the
data – in this application, the
intercept is not itself economically
meaningful.
Predicted values & residuals:
One of the districts in the data set is
Antelope, CA, for which STR = 19.33
and TS = 657.8
predicted value:
YˆAntelope = 698.9 – 2.28*19.33 = 654.8
residual:
uˆ Antelope = 657.8 – 654.8 = 3.0
The OLS regression line is an
estimate, computed using our sample
of data; a different sample would have
given a different value of ˆ1 .
How can we:
 quantify the sampling uncertainty
associated with ˆ1 ?
 use ˆ1 to test hypotheses such as 1
= 0?
 construct a confidence interval for
1?
Like estimation of the mean, we
proceed in four steps:
1. The probability framework for
linear regression
2. Estimation
3. Hypothesis Testing
4. Confidence intervals