Specification Test

8/3/2019 Specification Test

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Specification test

Vid Adrison


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Outline

Redundant Variable

Omitted Variable

Functional Specification

Selection Criteria


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Redundant Variable Consequences

On the unbiasedness: remain unbiased Review the concept of unbiased estimator

On the variance: increases variance Proof:

Create a simulated demand function Simulation is useful as we know the true value of the parameter

Steps in conducting simulation; Assume that Qx is only a function of Px and Income Generate 200 data of Px, Py, INC, and Error via random draw

In excel the syntax is =rand() Generate log(Qx)= 0.5-0.5*log(Px)+0.5*log(INC)+Error Run log(Qx)=f[log(Px), log(INC)]

The parameter will be closer to the assigned values, as thenumber of draws increase

Repeating the above procedure for N times and get the averagevalues of the parameter Monte Carlo Simulation

As the comparison, run log(Qx)=f[log(Px), log(Py), log(INC)], seehow the parameter changes


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Redundant Variable

Test procedure in EVIEWS:

View | Coefficient Test | Omitted Variables | (WriteVariables | OK

H0: Variables do not belong to the model

H1: Variables belong to the model

This procedure is the same as omitted variabletest, thus, the hypotheses remain the same

Basically, omitted variable/redundant variable testare performed by comparing the likelihood ratiobetween restricted and unrestricted model


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Correct Specification RegressionDependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:44Sample: 1 60Included observations: 60

Variable Coefficient Std. Error t-Statistic Prob.

LOG(PX) -0.525034 0.035679 -14.71562 0.0000LOG(INC) 0.514221 0.045908 11.20119 0.0000

C 0.970042 0.095809 10.12477 0.0000

R-squared 0.828189 Mean dependent var 1.237024Adjusted R-squared 0.822161 S.D. dependent var 0.723513S.E. of regression 0.305112 Akaike info criterion 0.512434Sum squared resid 5.306335 Schwarz criterion 0.617151Log likelihood -12.37302 F-statistic 137.3802Durbin-Watson stat 2.276588 Prob(F-statistic) 0.000000

Redundant Variable caseDependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:44Sample: 1 60Included observations: 60


LOG(PX) -0.521201 0.035292 -14.76838 0.0000LOG(INC) 0.528201 0.046149 11.44567 0.0000LOG(PY) 0.070505 0.044328 1.590528 0.1173

C 0.890289 0.107022 8.318742 0.0000



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Omitted Variable Consequences

On the unbiasedness: more serious than redundantvariable case Omitted variable may be due to ignorance or data

unavailability

Example:

Dropping INC from the previous regression Excluding ability in wage offer function

For two variable-model, the sign of bias depends on thecorrelation between excluded variable and includedvariable

The direction of bias can be more complicated if we havethree or more regressors

See Wooldridge Chapter 3 for detail derivation

Corr (X1, X2) > 0 Corr(X1, X2) 0 Positive Bias Negative Bias

B2 < 0 Negative Bias Positive Bias


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Omitted Variable case

Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 17:45Sample: 1 60Included observations: 60


LOG(PX) -0.420876 0.061096 -6.888789 0.0000C 1.800707 0.107595 16.73598 0.0000


Omitted Variable Test

Omitted Variables: LOG(INC)

F-statistic 125.4667 Probability 0.000000Log likelihood ratio 69.81100 Probability 0.000000

Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 23:52

Sample: 1 60Included observations: 60


C 0.970042 0.095809 10.12477 0.0000LOG(PX) -0.525034 0.035679 -14.71562 0.0000LOG(INC) 0.514221 0.045908 11.20119 0.0000

R-squared 0.828189 Mean dependent var 1.237024Adjusted R-squared 0.822161 S.D. dependent var 0.723513S.E. of regression 0.305112 Akaike info criterion 0.512434Sum squared resid 5.306335 Schwarz criterion 0.617151

Log likelihood -12.37302 F-statistic 137.3802Durbin-Watson stat 2.276588 Prob(F-statistic) 0.000000


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Regression through Origin

Recall the interpretation of intercept

For Keynesian consumption function, it reflectsautonomous consumption; the amount of consumptionone will have if his/her income is zero

Some have no (logical) economic interpretation: I.e., production function (K=0, L=0 will definitely lead to

Y=0, demand function (price should be in the positivedomain)

In the absence of economic interpretation, one is

tempted to drop intercept It is essentially dropping vector of ONE in the matrix

notation

Is it the correct treatment ???


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Regression through Origin Note that an intercept does not have to have economic

interpretation One of several role of an intercept is to ensure zero conditional

mean on error Example of violation;

True Consumption = B0 + B1*Income + error If consumption is measured incorrectly, such as, understatement;

such that Observed consumption = True consumption understatement The regression would be;

Observed Consumption = B0 + B1*Income + error understatement

If we dont include B0, then E (error understatement) is differentfrom zero Bias in B1

If we include B0, B1 is not biased

Cost of using intercept if B0 is truly zero None Cost of deleting intercept if B0 is not zero Biased in slope

parameter


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Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/23/10 Time: 18:18

Sample: 1 60Included observations: 60


LOG(PX) -0.429977 0.057083 -7.532469 0.0000LOG(INC) 0.873992 0.048203 18.13144 0.0000

R-squared 0.519198 Mean dependent var 1.237024Adjusted R-squared 0.510909 S.D. dependent var 0.723513S.E. of regression 0.505989 Akaike info criterion 1.508162Sum squared resid 14.84945 Schwarz criterion 1.577973Log likelihood -43.24485 Durbin-Watson stat 1.730641


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Functional Specification What to choose:

A: ln(Qx)=f(Px, INC), B: ln(Qx)=f(Px, Py, INC), C: ln(Qx)=f(ln(Px),ln(INC)) D: ln(Qx)=f(ln(Px),ln(Py), ln(INC))??

Nested Model: A Vs B, or C Vs D Ramsey RESET

Basically add the polynomial of expected value as theregressor, as the proxy for unaccounted variable

If adding this proxy variable leads to significant increase inadjusted R square, the regression contains misspecification

Steps in Eviews: View | Stability Test | Ramsey RESETtest | (Include number of polynomial variable) | OK

H0: No misspecification error H1: Model contains specification error


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Ramsey RESET Test:


Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:33Sample: 1 60

Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.

C 1.088893 0.165015 6.598762 0.0000LOG(PX) -0.601195 0.093144 -6.454500 0.0000LOG(INC) 0.550684 0.061735 8.920113 0.0000FITTED^2 -0.043772 0.049433 -0.885480 0.3797

R-squared 0.830562 Mean dependent var 1.237024Adjusted R-squared 0.821485 S.D. dependent var 0.723513S.E. of regression 0.305692 Akaike info criterion 0.531863Sum squared resid 5.233065 Schwarz criterion 0.671486Log likelihood -11.95589 F-statistic 91.50122

Durbin-Watson stat 2.255349 Prob(F-statistic) 0.000000

Ramsey RESET Test:


Test Equation:Dependent Variable: LOG(QX)Method: Least SquaresDate: 02/24/10 Time: 00:34Sample: 1 60

Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.

C 0.502144 0.492582 1.019413 0.3124PX 0.005691 0.080800 0.070429 0.9441INC -0.004399 0.042664 -0.103115 0.9182

FITTED^2 0.413044 0.202524 2.039483 0.0461



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Non Nested Model: A Vs C (In theprevious slides)

Mizon and Richard (1986)

Ln(Qx) =B0 + B1*Px +B2*INC+B3*ln(Px)+B4*ln(INC)+e

Test using Wald

B1=B2=0 if null is rejected, then specification A ispreferred

B3=B4=0 if null is rejected, then specification C ispreferred


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Variable Coefficient Std. Error t-Statistic Prob.C 1.047136 0.119279 8.778901 0.0000

LOG(PX) -0.490712 0.064609 -7.595069 0.0000LOG(INC) 0.604477 0.083606 7.230085 0.0000

PX -0.017596 0.025031 -0.702977 0.4850INC -0.024155 0.018644 -1.295620 0.2005

R-squared 0.834092 Mean dependent var 1.237024Adjusted R-squared 0.822026 S.D. dependent var 0.723513S.E. of regression 0.305228 Akaike info criterion 0.544139Sum squared resid 5.124023 Schwarz criterion 0.718668

Log likelihood -11.32417 F-statistic 69.12739Durbin-Watson stat 2.314130 Prob(F-statistic) 0.000000

Wald Test:Equation: Untitled

Null Hypothesis: C(4)=0C(5)=0

F-statistic 0.978447 Probability 0.382341Chi-square 1.956895 Probability 0.375894

Wald Test:Equation: Untitled

Null Hypothesis: C(2)=0C(3)=0

F-statistic 53.70023 Probability 0.000000Chi-square 107.4005 Probability 0.000000


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Davidson-MacKinnon (1981) Use the similar principle as Ramsey, but different

predicted values

Recall Spec A: ln(Qx)=f(Px, INC)

Spec C: ln(Qx)=f(ln(Px), ln(INC))

Steps: to test if Spec A is correct: Run Spec C, get predicted value, say Z1

Run Spec A by adding Z1 into the equation

If Z1 is insignificant, then A is correctly specified

We can also perform the test in the other direction; Run Spec A, get predicted value, say Z2

Run Spec C by adding Z2 into the equation

If Z2 is insignificant, then C is correctly specified


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C 0.038918 0.176320 0.220726 0.8261PX 0.000279 0.020148 0.013860 0.9890INC -0.008157 0.013027 -0.626168 0.5337Z1 1.021596 0.099618 10.25511 0.0000




C 1.001958 0.176638 5.672389 0.0000LOG(PX) -0.534508 0.056757 -9.417454 0.0000

LOG(INC) 0.522164 0.059141 8.829176 0.0000Z2 -0.027657 0.128139 -0.215839 0.8299



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Selection Criteria

According to Hendry and Richard (1983), a modelchosen for empirical analysis should satisfy thefollowing criteria: Admissible (prediction made from the model must be

logically possible)

Consistent with theory: Make economic good sense Have weakly exogenous explanatory variables:

Regressors are uncorrelated with the error terms Constancy: The values of the parameters should be

stable. In other word, the parameter values obtainedusing within sample observation should not deviate

significantly from outside sample observation. Coherency: Residuals estimated from the model must be

purely random Encompassing: No other model explains better


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Selection Criteria

Evaluation of Competing Models

Three statistics for model evaluation criteriaavailable in most econometric software are; Adjusted R-Squared Choose model that generates

the highest Adjusted R squared

Akaike Information Criterion Choose model that

generates the smallest AIC

Schwarz Information Criterion Choose model that

generates the smallest SIC

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Specification Test