Transcript Review of Probability and Statistics - fu

Multiple Regression Analysis
y = b0 + b1x1 + b2x2 + . . . bkxk + u
1. Estimation
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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 , thatis each b has
a ceteris paribus interpretation
Prof. Dr. Rainer Stachuletz
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A “Partialling Out” Interpretation
Consider the 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 theestimated
regression xˆ1  ˆ0  ˆ2 xˆ2
Prof. Dr. Rainer Stachuletz
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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 are being related to yi
so we’re estimating the effect of x1 on y
after x2 has been “partialled out”
Prof. Dr. Rainer Stachuletz
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Simple vs Multiple Reg Estimate
~ ~
~
Comparethesimple regression y  b 0  b1 x1
wit h themultipleregression yˆ  bˆ0  bˆ1 x1  bˆ 2 x2
~
Generally,b1  bˆ1 unless :
bˆ  0 (i.e.no partialeffectof x ) OR
2
2
x1 and x2 are uncorrelated in thesample
Prof. Dr. Rainer Stachuletz
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Goodness-of-Fit
We can thinkof each observation as being made
up of an explainedpart,and an unexplained part,
yi  yˆ i  uˆi Wethendefine thefollowing:
  y  y  is the totalsum of squares (SST )
  yˆ  y  is theexplainedsum of squares (SSE)
 uˆ is theresidual sum of squares (SSR)
2
i
2
i
2
i
T henSST  SSE  SSR
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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Goodness-of-Fit (continued)
We can also thinkof R 2 as being equal to
thesquared correlation coefficient between
theactual yi and the values yˆ i
 y  y yˆ  yˆ 



  y  y   yˆ  yˆ  
2
R
2
i
i
2
2
i
i
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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Omitted Variable Bias
Suppose the true modelis given as
y  b 0  b1 x1  b 2 x2  u, but we
~ ~
~
estimatey  b  b x  u, then
0
~
b1
x  x  y


 x  x 
i1
1
1 1
i
2
i1
1
Prof. Dr. Rainer Stachuletz
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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
numeratorbecomes
 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
Prof. Dr. Rainer Stachuletz
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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, t akingexpect at ions we have
 
~
E b1  b1  b 2
 x  x x
 x  x  
i1
i1
1
i2
2
1
Prof. Dr. Rainer Stachuletz
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Omitted Variable Bias (cont)
Consider the regressionof 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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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)
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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Variance of OLS (cont)
Given theGauss - MarkovAssumptions
 
Var bˆ j 
s
2

SSTj 1  R
2
j

, where
SSTj   xij  x j  and R is theR
2
2
j
2
from regressing x j on all ot herx' s
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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Misspecified Models
Consider again themisspecified model
 
s
~ ~
~
~
y  b 0  b1 x1 , so thatVar b1 
SST1
~
T hus,Var b  Var bˆ unless x and
 
1
 
1
2
1
x2 are uncorrelated, then they
' re thesame
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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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)
Prof. Dr. Rainer Stachuletz
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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
Prof. Dr. Rainer Stachuletz
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