Post on 03-Apr-2018
transcript
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Chapter III:Multiple Regression Analysis
y =0+1x1+2x2+ . . . kxk+ u
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( ) 0...2
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0 is still the intercept 1to kall called slope parameters
uis still the error term (or disturbance)
Still minimizing the sum of squaredresiduals
Sampling Distributions as n
1
n1
n2
n3n1 < n2 < n3
Goodness-of-Fit
how well our sample regression line fitsour sample data?
Can compute the fraction of the total sumof squares (TSS) that is explained bythe model, call this the R-squared ofregression
R2 = ESSE/TSS = 1 RSS/TSS
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Adjusted R-Squared
The adjusted R2
takes into account thenumber of variables in a model, andmay decrease
( )[ ]( )[ ]
( )[ ]1
1
1
11
2
2
=
nSST
nSST
knSSRR
kn
1n)R1(1R 22
=
v
)()1(
)1(
)1)(1(
)(
)(
)1(SS
2
2
2
2
kn
R
kR
Rk
Rkn
knRSS
kE
F
=
=
=
You can compare the fit of 2 models(with the same y) by comparing the
adj-R2
You cannot use the adj-R2 to comparemodels with different ys
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A special case of exclusion restrictions isto test H0: 1 = 2 == k= 0
Since the R2 from a model with only anintercept will be zero, the Fstatistic issimply
( ) ( )11 22
=
knR
kRF
Wald test
(U)
(R)
H0: m == k-1=0
H1: ?
uXXXXXYkkmmmm+++++= 111122110 .......
vXXXYmm
+++++= 1122110 ....
(U), k: number of parameters, RSS(U) df.(n-k)
(R), m: number of parameters, RSS(R) df. (n-m)
Wald test
),(2
22
~
)/()1(
)/()(
)/(
)/()(knmk
U
RU
U
UR
WF
knR
mkRR
knRSS
mkRSSRSSF
=
=
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Chapter 4: Dummy Variables
A dummy variable is a variable thattakes on the value 1 or 0
Examples: male (= 1 if are male, 0otherwise), south (= 1 if in the south, 0otherwise), etc.
Dummy variables are also called binaryvariables, for obvious reasons
y
x
y = 0+ 1x
y = (0+ 0) + (1 + 1)x
d = 0
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The Chow Test
If run the restricted model for groupone and get SSR1, then for group twoand get SSR2
Run the restricted model for all to getSSR
( )[ ] ( )[ ]1
12
21
21
+
+
+
+=
k
kn
SSRSSR
SSRSSRSSRF
Chapter 6: Heteroskedasticity
What is Heteroskedasticity
homoskedasticity implied that conditionalon the explanatory variables, the variance
of the unobserved error, u, was constant If this is not true, that is if the variance
ofuis different for different values of thexs, then the errors are heteroskedastic
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x1
f(y|x)
x3
Why Worry AboutHeteroskedasticity?
OLS is still unbiased and consistent, evenif we do not assume homoskedasticity
The standard errors of the estimates arebiased if we have heteroskedasticity
we can not use the usual tstatistics
Testing for Heteroskedasticity
H0: Var(u|x1, x2,, xk) = 2, which is
equivalent to H0: E(u2|x1, x2,, xk) =
E(u2) = 2
The White Test
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SSRur = SSR1 + SSR2 Note, k+ 1 restrictions (each of the
slope coefficients and the intercept)
Note the unrestricted model wouldestimate 2 different intercepts and 2different slope coefficients, so the df isn 2k 2
What happens if we include variables inour specification that dont belong?
There is no effect on our parameterestimate, and OLS remains unbiased
What if we exclude a variable from ourspecification that does belong?
OLS will usually be biased
Too Many or Too Few Variables
The homoskedastic normal distributionwith a single explanatory variable
E(y|x) =0+ 1x
Normal
distribution
s
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A Weaker Assumption
Without assumptions, OLS will bebiased and inconsistent
The error variance: a larger 2 impliesa larger variance for the OLS estimators
Chapter 7: Multicollinearity
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Multicollinearity is a statisticalphenomenon in which two or morepredictor variables in a multipleregression model are highly correlated
r223 =1
2X2 + 3X3 =0
X2 = X3
r223 < 1
2X2 + 3X3 + v =0
Effects
Model is not identified: we cannotestimate the separate influence of X1
and X2 on Y
=
)1(var
2
23
2
2
2^
2rx
i
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Detect
high R2
but none of the variables showsignificant effects
regress each of the Xs on all of theother Xs
Variance Inflation Factors (VIF)
Autocorrelation
Misspecification
Data Manipulation
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Checking for Autocorrelation
Test: Durbin-Watson statistic:
d =(e
i e
i1)2
ei
2, for n and K- 1 d.f.
Positive Zone of No Autocorrelation Zone of Negative
autocorrelation indecision indecision autocorrelation|_______________|__________________|_____________|_____________|__________________|___________________|
0 d-lower d-upper 2 4-d-upper 4-d-lower 4
Autocorrelation is clearly evident
Ambiguous cannot rule out autocorrelation
Autocorrelation in not evident
Dealing with autocorrelation
There are several approaches toresolving problems of autocorrelation.
Lagged dependent variables
Differencing the Dependent variable
Chapter 8: Choosing good model
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R2
T
F
Multicollinearity;
Heteroskedasticity;.
L, AIC
L =-n/2(1+log2N+log(RSS/n))
AIC=(RSS/n)e2k/n
SC=(RSS/n).nk/n