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Multiple Regression Analysis
y = b0 + b1x1 + b2x2 + . . . bkxk + u
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Parallels with Simple Regression
b0 is still the intercept
b1 to bk all called slope parameters
u is still the error term (or disturbance)
Still need to make a zero conditional mean
assumption, so now assume that
E(u|x1,x2, …,xk) = 0
Still minimizing the sum of squared
residuals, so have k+1 first order conditions
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Interpreting Multiple Regression
yˆ  bˆ0  bˆ1 x1  bˆ 2 x2  ...  bˆ k xk , so
yˆ  bˆ x  bˆ x  ...  bˆ x ,
1 1
2 2
k
k
so holding x2 ,..., xk fixed implies that
yˆ  bˆ1 x1 , that is each b has
a ceteris pa ribus interpreta tion
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A “Partialling Out” Interpretation
Consider t he case where k  2, i.e.
yˆ  bˆ  bˆ x  bˆ x , then
0
1 1
bˆ1   rˆi1 yi 
2 2
 rˆ
2
i1
, where rˆi1 are
the residuals from the estimated
regression xˆ1  ˆ0  ˆ2 xˆ 2
Economics 20 - Prof. Anderson
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“Partialling Out” continued
Previous equation implies that regressing y
on x1 and x2 gives same effect of x1 as
regressing y on residuals from a regression
of x1 on x2
This means only the part of xi1 that is
uncorrelated with xi2 is being related to yi so
we’re estimating the effect of x1 on y after
x2 has been “partialled out”
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Simple vs Multiple Reg Estimate
~ ~
~
Compare the simple regression y  b 0  b1 x1
with the multiple regression yˆ  bˆ0  bˆ1 x1  bˆ 2 x2
~
Generally, b1  bˆ1 unless :
bˆ  0 (i.e. no partial effect of x ) OR
2
2
x1 and x2 are uncorrelat ed in the sample
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Goodness-of-Fit
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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Goodness-of-Fit (continued)
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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Goodness-of-Fit (continued)
We can also think of R 2 as being equal to
the squared correlatio n coefficien t between
the actual yi and the values yˆ i
 y  y yˆ  yˆ 



  y  y   yˆ  yˆ  
2
R
2
i
i
2
2
i
i
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More about R-squared
R2 can never decrease when another
independent variable is added to a
regression, and usually will increase
Because R2 will usually increase with the
number of independent variables, it is not a
good way to compare models
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Assumptions for Unbiasedness
Population model is linear in parameters:
y = b0 + b1x1 + b2x2 +…+ bkxk + u
We can use a random sample of size n,
{(xi1, xi2,…, xik, yi): i=1, 2, …, n}, from the
population model, so that the sample model
is yi = b0 + b1xi1 + b2xi2 +…+ bkxik + ui
E(u|x1, x2,… xk) = 0, implying that all of the
explanatory variables are exogenous
None of the x’s is constant, and there are no
exact linear relationships among them
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Too Many or Too Few Variables
What happens if we include variables in
our specification that don’t belong?
There is no effect on our parameter
estimate, and OLS remains unbiased
What if we exclude a variable from our
specification that does belong?
OLS will usually be biased
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Omitted Variable Bias
Suppose the true model is given as
y  b 0  b1 x1  b 2 x2  u , but we
~ ~
~
estimate y  b  b x  u , then
~
b1
x


 x
i1
i1
0
 x1  yi
1 1
 x1 
2
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Omitted Variable Bias (cont)
Recall the true model, so that
yi  b 0  b1 xi1  b 2 xi 2  ui , so the
numerator becomes
 x  x b
b  x  x 
i1
1
0
2
1
i1
1
 b1 xi1  b 2 xi 2  ui  
 b 2  xi1  x1 xi 2    xi1  x1 ui
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Omitted Variable Bias (cont)
~
b  b1  b 2
 x  x x   x
 x  x    x
i1
1
i1
i2
2
i1
1
i1
 x1 ui
 x1 
2

since E( ui )  0, taking expectatio ns we have
 
~
E b1  b1  b 2
 x  x x
 x  x  
i1
i1
1
i2
2
1
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Omitted Variable Bias (cont)
Consider t he regression of x2 on x1
~
~
~
~
x2   0   1 x1 then  1
 
~
x  x x


 x  x  
i1
i1
~
1
i2
2
1
so E b1  b1  b 2 1
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Summary of Direction of Bias
Corr(x1, x2) > 0 Corr(x1, x2) < 0
b2 > 0
Positive bias
Negative bias
b2 < 0
Negative bias
Positive bias
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Omitted Variable Bias Summary
Two cases where bias is equal to zero

b2 = 0, that is x2 doesn’t really belong in model

x1 and x2 are uncorrelated in the sample
If correlation between x2 , x1 and x2 , y is
the same direction, bias will be positive
If correlation between x2 , x1 and x2 , y is
the opposite direction, bias will be negative
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The More General Case
Technically, can only sign the bias for the
more general case if all of the included x’s
are uncorrelated
Typically, then, we work through the bias
assuming the x’s are uncorrelated, as a
useful guide even if this assumption is not
strictly true
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Variance of the OLS Estimators
Now we know that the sampling
distribution of our estimate is centered
around the true parameter
Want to think about how spread out this
distribution is
Much easier to think about this variance
under an additional assumption, so
Assume Var(u|x1, x2,…, xk) = s2
(Homoskedasticity)
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Variance of OLS (cont)
Let x stand for (x1, x2,…xk)
Assuming that Var(u|x) = s2 also implies
that Var(y| x) = s2
The 4 assumptions for unbiasedness, plus
this homoskedasticity assumption are
known as the Gauss-Markov assumptions
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Variance of OLS (cont)
Given the Gauss - Markov Assumption s
 
Var bˆ j 
s
2

SST j 1  R
2
j

, where
SST j   xij  x j  and R is the R
2
2
j
2
from regressing x j on all other x' s
Economics 20 - Prof. Anderson
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Components of OLS Variances
The error variance: a larger s2 implies a
larger variance for the OLS estimators
The total sample variation: a larger SSTj
implies a smaller variance for the estimators
Linear relationships among the independent
variables: a larger Rj2 implies a larger
variance for the estimators
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Misspecified Models
Consider again the misspecifi ed model
 
s
~ ~
~
~
y  b 0  b1 x1 , so that Var b1 
SST1
~
Thus, Var b  Var bˆ unless x and
 
1
 
1
2
1
x2 are uncorrelat ed, then they ' re the same
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Misspecified Models (cont)
While the variance of the estimator is
smaller for the misspecified model, unless
b2 = 0 the misspecified model is biased
As the sample size grows, the variance of
each estimator shrinks to zero, making the
variance difference less important
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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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Error Variance Estimate (cont)
sˆ   uˆ
 n  k 1  SSR df
thus, sebˆ   sˆ SST 1  R 
2
2
i
j
j
2 12
j
df = n – (k + 1), or df = n – k – 1
df (i.e. degrees of freedom) is the (number
of observations) – (number of estimated
parameters)
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The Gauss-Markov Theorem
Given our 5 Gauss-Markov Assumptions it
can be shown that OLS is “BLUE”
Best
Linear
Unbiased
Estimator
Thus, if the assumptions hold, use OLS
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