Econometrics – lecture 7 – model specifications

The omission of a relevant explanatory variable causes the regression coefficients to be biased and the standard errors to be invalid.. OMISSION OF A RELEVANT VARIABLE If we estimate th

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MODEL SPECIFICATIONS

Dr Tu Thuy Anh Faculty of International Economics

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 Model misspecification:

 Omitting relevant variables

 Including irrelevant variables

 Function specification: Ramsey test

 Structural change: the Chow test

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Consequences of Variable Misspecification

X

Y  b1  b2 2  b3 3 

u X

Y  b1  b2 2 

3 3

2 2 1

ˆ

X b

X b b

ˆ b b X

Y  

To keep the analysis simple, we will assume that there are only two

possibilities Either Y depends only on X2, or it depends on both X2 and X3

OMISSION OF A RELEVANT VARIABLE

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X

Y  b1  b2 2  b3 3 

u X

Y  b1  b2 2 

3 3

2 2 1

ˆ

X b

X b b

assumptions are valid

Likewise we will not encounter any problems if Y depends on both X2 and X3and we fit the multiple regression

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X

Y  b1  b2 2  b3 3 

u X

Y  b1  b2 2 

3 3

2 2 1

ˆ

X b

X b b

ˆ b b X

Y   Correct specification, no problems

Correct specification,

no problems

In this sequence we will examine the consequences of fitting a simple

regression when the true model is multiple The omission of a relevant

explanatory variable causes the regression coefficients to be biased and the standard errors to be invalid

Coefficients are biased (in general) Standard errors are invalid.

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u X

3 3

2 2

3 2

2)

(

X X

X

X b

E

i

i i

b b

b

2

b

3

The strength of the proxy effect depends on two factors: the strength of the

effect of X3 on Y, which is given by b3, and the ability of X2 to mimic X3

The ability of X2 to mimic X3 is determined by the slope coefficient obtained

when X3 is regressed on X2, the term highlighted in yellow

Y Y

X

X b

i

i i

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 If we estimate the restricted (bad) model

instead of the right (unrestricted) one, then

 the estimate of parameter B2 will be biased,

and the bias will depend on:

 The magnitude of the omitted parameter (B3)

 The correlation between the included and the omitted variables (X2 and X3)

 R2 is affected

will be affected more

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Example: Teaching Ratings

Correlation Coefficients, using the observations 1 - 463

5% critical value (two-tailed) = 0,0911 for n = 463

minority age female onecredit beauty

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Model 1: OLS, using observations 1-463

Dependent variable: course_eval

coefficient std error t-ratio p-value -

-Mean dependent var 3,998272 S.D dependent var 0,554866

Sum squared resid 120,0996 S.E of regression 0,513766

R-squared 0,155647 Adjusted R-squared 0,142657

F(7, 455) 11,98202 P-value(F) 4,67e-14

Log-likelihood -344,5811 Akaike criterion 705,1623

Schwarz criterion 738,2641 Hannan-Quinn 718,1936 10

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-Mean dependent var 3,998272 S.D dependent var 0,554866

F(6, 456) 9,354823 P-value(F) 1,09e-09

Schwarz criterion 756,7121 Hannan-Quinn 739,1504

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X

Y  b1  b2 2  b3 3 

u X

Y  b1  b2 2 

3 3

2 2 1

ˆ

X b

X b b

INCLUSION OF AN IRRELEVANT VARIABLE

Including irrelevant variables: The effects are different from those of omitted variable misspecification In this case the coefficients in general remain

unbiased, but they are inefficient The standard errors remain valid, but are needlessly large

Coefficients are unbiased (in general), but inefficient.

Standard errors are valid (in general)

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u X

Y  b1  b2 2 

3 3 2

2 1

Y   

u X

X

Y  b1  b2 2  0 3 

INCLUSION OF AN IRRELEVANT VARIABLE

Rewrite the true model adding X3 as an explanatory variable, with a

coefficient of 0 Now the true model and the fitted model coincide Hence b2

will be an unbiased estimator of b2 and b3 will be an unbiased estimator of 0

However, the variance of b2 will be larger than it would have been if the

correct simple regression had been run because it includes the factor 1 / (1 –

r2), where r is the correlation between X2 and X3

The standard errors remain valid, but they will tend to be larger than those obtained in a simple regression, reflecting the loss of efficiency

,

2 2 2

2 2

3 2

2

1

X X i

u b

r X

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Mean dependent var 3,998272 S.D dependent var 0,554866

F(6, 456) 14,00557 P-value(F) 1,19e-14

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F(6, 456) 14,00557 P-value(F) 1,19e-14

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F(7, 455) 11,98202 P-value(F) 4,67e-14

Schwarz criterion 738,2641 Hannan-Quinn 718,1936 16

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Function specification

 Why the model has to be linear?

 The effects of a bad specification for our

functional form would as serious as the omitted Variables problem: biased estimates

 Ramsey (1969) proposes a test with the

following hypothesis:

 H0: Right functional form

 Ha: Mistaken functional form

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Ramsey RESET test

 Original model

 We may suspect that some variables should be

introduced non-linearly (e.g in quadratic or

cubic terms) To detect it the RESET test

proposes using powers of the adjusted

endogenous variables from the original

equation into an auxiliary regression:

 Hypothesis H0: 1=0 versus Ha: 10

 If more terms involved, possible F tests

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RESET test for specification (squares and cubes) Test statistic: F = 1,821350,

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Structural change

 What if parameters are NOT constant over the

sample  structural change

 Male vs female

 ASEAN vs non-ASEAN membership

 HQ class vs regular class

 If we do not control for structural changes, our

predictions will not be reliable, our estimation will be inefficient and biased for the not

controlled parameter.

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The Chow test

 We assume that we may have two subsamples:

 Compute the sum of squares of the residuals

for every model (SSRT, SSR1 and SSR2)

 Finally we compute the following test:

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coefficient std error t-ratio p-value

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