LECTURE 4: THE SIMPLE REGRESSION MODEL
Regression Analysis
: The study of the relationship between one variable
(
dependent variable
) and one or more other variables (
independent,
or
explanatory, variables
) using a (typically linear) regression model.
What do we use regression analysis for?
•
To estimate the mean or average value of the dependent variable, given the
values of the independent variables.
o
What is the average income for people with a high school diploma?
o
What is the average income for people with a college degree?
•
To test a hypothesis implied by economic theory
o
If we increase the price will the quantity demanded fall?
•
To predict, or forecast, the mean value of the dependent variable given the
independent variables.
o
What will happen to GDP if we change the interest rate?
THE POPULATION REGRESSION FUNCTION
DETERMINISTIC POPULATION REGRESSION FUNCTION
•
Population Regression Function
, E(YX), is the conditional mean of the
dependent variable (Y) given any value of the independent variable (X).
•
In general, E(YX) can have any shape as a function of X.
•
We typically choose to think that the population regression function is a
linear function of the conditioning variable(s), that is, we specify a
linear
regression model
.
•
The
bivariate linear regression model
takes the form
E(YX) =
β
0
+
β
1
X
o
β
0
and
β
1
are the
unknown
population regression parameters
or
regression coefficients
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o
β
0
is the
intercept
. It measures E(YX = 0).
o
β
1
is the
slope
. It measures the average marginal change in Y given a
small change in X.
β
1
= E
(
dX
dY
)
o
β
0
and
β
1
are unknown and they are the primary objects of interest.
STOCHASTIC POPULATION REGRESSION FUNCTION
•
Equivalently the linear regression model can be written as
Y =
β
0
+
β
1
X +
ε
where
ε
is an unobservable
stochastic,
or
random, error term (
or
disturbance
).
o
ε
is a random variable, i.e. it has some distribution
o
ε
is nothing but the deviation of any realization of Y from its
conditional mean, E(YX), i.e.
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 Winter '07
 SandraBlack
 Regression Analysis, Yi, population regression function

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