econometrics layout.pdf

7/28/2019 Econometrics layout.pdf

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Chapter III:Multiple Regression Analysis

y =0+1x1+2x2+ . . . kxk+ u

( )

( )

( ) 0...2

...

0...2

0...2

1

,,33,2211

2

,2

1

,,33,221

2

1

2

1

,,33,221

1

1

2

==

==

==

=

=

=

=

=

=

ki

n

i

iKKiii

k

n

i

i

i

n

i

iKKiii

n

i

i

n

i

iKKiii

n

i

i

XXXXY

e

XXXXY

e

XXXY

e


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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

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