Multiple Regression Model
•
General Multiple Regression Model
.
•
k
regressor variables:
X
1
i
, ……,
X
ki
•
k
“slope” coefficients (parameters):
β
1
,….…,
β
k
•
Each “slope” coefficient
β
j
measures the effect of a one
unit change in the corresponding regressor
X
ji
,
holding
all else (e.g. the other regressors) constant
.
•
u
i
– still omitted variables (but hopefully there are less
in here since we are including more regressors!)
0
1
1
2
2
=
......
i
i
i
k
ki
i
Y
X
X
X
u
β
β
β
β
+
+
+
+
+
Multiple Regression Assumptions  I
•
As in the simple regression model, we need to make some
assumptions in order to estimate the coefficients
β
0
,
β
1
,….…,
β
k
.
The first 3 are very similar to our previous set of assumptions.
•
A1)
Cov
(
u
i
,
X
ji
) = 0 for every
j
(i.e.
u
i
is uncorrelated with each
of the
k
regressors)
or
A1b) E[
u
i

X
1i
= c
1
,
X
2i
= c
2
,
.....
,
X
ki
= c
k
] = E[
u
i

X
1i
,
X
2i
,….,
X
ki
] = 0
(i.e. the expectation of
u
i
is zero regardless of the values of the
k
regressors.)
(Note the minor notational change in Assumption 1b) )
Multiple Regression Assumptions II
•
A2) (
X
1i
,
X
2i
,….,
X
ki
,
Y
i
) are i.i.d.
(again, this is true with random
sampling)
•
A3) (
X
1i
,
X
2i
,….,
X
ki
,
Y
i
) have finite fourth moments
(again, this
is generally true in economic data).
•
We also need a fourth assumption in the multiple regression
model.
This fourth assumption addresses how the various
X
ji
’s
are related to each other.
Multiple Regression Assumptions III
•
A4) The regressors (
X
1i
,
X
2i
,….,
X
ki
) are not
perfectly
multicollinear.
This means that none of the regressors can
be written as a
perfect
linear function of
only
the other
regressors.
For example:
– If
X
2i
= 11 + 7
X
5i
+
X
4i
 3
X
9i
+ 5.5
X
3i
, then A4) is violated
– If
X
2i
= 11 + 7
X
5i
+
X
4i
 3
X
9i
+ 5.5
X
3i
+ W
i
(where W
i
is some other
variable that is not one of the
X
ji
’s), then A4) is not violated
•
Assumption 4) is rarely violated in practice, and when it is, it
is typically by accident. We will discuss A4) in more detail
momentarily.
Estimation  I
•
Under A1) – A4), the OLS estimators (
β
0
,
β
1
,….…,
β
k
), which
minimize:
are
unbiased
and
consistent
estimators of the parameters
(
β
0
,
β
1
,….…,
β
k
).
Moreover, the CLT implies that for each
j
,
•
The formulas for the OLS estimators (and their standard errors)
are too complicated to write down (unless one uses matrix
notation
☺
), but in STATA the estimates (and their SEs) can be
computed with the command (e.g. with 3 regressors):
“regress y x1 x2 x3”
or
“regress y x1 x2 x3, robust”
(
)
2
0
1
1
1
ˆ
ˆ
ˆ
.......
n
i
i
k
ki
i
Y
X
X
β
β
β
=
⎡
⎤
−
+
+
+
⎣
⎦
∑
(
)
(
)
(
)
(
)
ˆ
ˆ
ˆ
,Var
and
0,1
ˆ
j
j
j
j
j
j
N
N
SE
β
β
β
β
β
β
−
∼
∼
Estimation  II
•
Predicted (expected) values and residuals are the same
as before, i.e.
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 Spring '07
 SandraBlack
 Econometrics, Regression Analysis, Yi, dummy variables

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