MISSPECIFICATION in terms of Regressors, Functional Forms, and Measurement

Dr. C. Ertuna

MISSPECIFICATIONin terms of

Regressors, Functional Forms, and Measurement

Dr. C. Ertuna

DEFINITION

Misspecification means that either functional form, or regressors, or measured data in an equation is incorrect.

In other words:• Omitting relevant variable(s)• Including irrelevant variable(s)• Incorrect functional form• Measurement errors

Dr. C. Ertuna

CONSEQUENCES of Omitting Relevant Variables

• E(u) ≠ 0• If omitted variable is correlated with the

regressors in the equation, then coefficients are biased and inconsistent.

• If a variable is omitted because it cannot be observed (measured) then we need to use a proxy variable to reduce effects of omission.

Dr. C. Ertuna

CONSEQUENCES of Including Irrelevant Variables

OLS estimators will be:• Unbiased,• Consistent,• However, not fully efficient.

Dr. C. Ertuna

Dropping Irrelevant Variables

If a variable is irrelevant, then after dropping it from the equation:• RSS will remain more or less unchanged, • will fall,• No sign change for coefficients,• No (appreciable) magnitude change for coefficients,• t-Statistics of remaining variables will not be seriously

affected.

Dr. C. Ertuna

CONSEQUENCES of Incorrect Functional Form

• Coefficients will be incorrect estimators of true population parameters.

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DIAGNOSIS of Misspecification

1. Observe regression residuals (including checking for Normality)

2. RESET test (Regression Specification Error Test)

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Normality Test

Analyze / Descriptive Statistics / Explore > Residuals (or variable under analysis) into

“Dependent List” > Plots: check “Normality plots with Tests”

Continue / Okay

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RESET Tests for MisspecificationF-Form

Step-1: Run OLS regression (After determining which variables should be in the model)

and save predicted values Step-2: Run an augmented regression where or and are added as

regressors (ENTER).

Step-3: Set Ho: Model correctly specified.

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RESET Tests for MisspecificationF-Form

Step-3: Set Ho: Model correctly specified.Step-4: Consider the auxiliary regression as unrestricted model and the

original model as restricted one.Step-5: Get RSS and k of Restricted Model (the model with fewer

variables) and get RSS and k and N of Unrestricted Model (the model with more variables)

Step-6: Apply the F-form of the Likelihood Ratio test (get organized and use Excel for computation).

F(, N - ) = p-value = FDIST(F-value; ; N - )

Definition of Test Parameters

= Residual Sum of Squares for Restricted Model (Model with fewer variables)

= Residual Sum of Squares for Unrestricted Model (Model with fewer variables)

= Number of parameters (included the intercept) in the Unrestricted model.

= Number of parameters (included the intercept) in the Unrestricted model.

N = Number of observations of the unrestricted model.

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Step-7: If p-value < alpha then conclude that there is significant evidence that the original model is misspecified.

Note: Since RESET test is an omnibus test we don’t know the source or type of misspecification.

RESET Tests for MisspecificationF-Form

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RESET Tests for MisspecificationLM Form

Step-1: Run OLS regression (After determining which variables should be in the model)

and save residuals and predicted values Step-2: Run an auxiliary regression where the dependent variable is and

independent variables are the original regressors plus or and (ENTER).

Step-3: Set Ho: Model correctly specified.

Dr. C. Ertuna

RESET Tests for MisspecificationLM Form

Step-3: Set Ho: Model correctly specified.Step-4: Compute LM = n* , where n = number of observations in auxiliary

regression = coefficient of determination of the auxiliary regression.

Step-5: Compute p-value by p-value = CHIDIST(LM;df) where df = h, h = number of additional regressors in the auxiliary regression.

Step-6: If p-value < alpha then conclude that there is significant evidence that the original model is misspecified.

Note: Since RESET test is an omnibus test we don’t know the source or type of misspecification.

Dr. C. Ertuna

Box-Cox Procedure for Functional Form Test

• When comparing the linear with the log-linear (or log-log) forms, we cannot compare the R2

because R2 is the ratio of explained variance to the total variance and the variances of y and log y are different.

• One solution to this problem is to consider a more general model of which both the linear and log-linear forms are special cases.

Dr. C. Ertuna

Box-Cox Procedure for Functional Form Test

• Consider the regression model with Box-Cox transformation:

• For this is a log-linear model and • For this is a linear model

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Box-Cox Procedure for Functional Form Test

Consider two regression models:

Ln

Step-1: Compute Geometric Mean of → Step-2: Transform by

Analyze / Reports / Case Summaries → Geometric Mean

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Box-Cox Procedure for Functional Form Test

Step-3: Run regressions:

Ln Step-4: Save and Step-5: Compute Box-Cox Test Statistics

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Box-Cox Procedure for Functional Form Test

Step-6: Set Ho: One functional form is not superior over the other.

Step-7: If p-value < alpha then conclude that there is significant evidence that one functional form is better than the other. Choose the functional form with lowest RSS as the better one.

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END

Next, Chapter 10: Dummy Variables

Dr. C. Ertuna

RESET Tests for Misspecification

Step-3: Set Ho: No significant difference between two functional form.Step-4: Compute LM = n* , where n = number of observations in auxiliary

regression = coefficient of determination of the auxiliary regression.

Step-5: Compute p-value by p-value = CHIDIST(LM;df) where df = k-1, k = number of parameters in the auxiliary regression.

Step-6: If p-value < alpha than conclude that there is significant evidence that the functional forms differ. Choose the functional form with lowest RSS.