Model Specification

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Model Specification Diagnostic Testing

• Lecture week 7
• Prepared by
• Dr. Zerihun Gudeta

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MODEL SPECIFICATION DIAGNOSTIC TESTING

• The 9th Assumption of the CLRM
• the regression model is correctly specified.
• Questions asked
• Criteria in choosing a model,
• Specification errors commonly encountered,
• Consequences of specification errors,
• Diagnostic tools commonly used to detect
  specification errors,
• Remedies to correct specification errors,
  performance of competing models.

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Model Selection Criteria

• Predictions made must be logically possible.
• Consistency with theory.
• Explanatory variables must be weakly exogenous
  (i.e. must not correlate with the error term).
• Parameter consistency.
• Data coherency (i.e. residuals from the model
  must be purely random.
• Should be encompassing (i.e. must include all
  rival models).

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Types of Specification Errors

• Omission of a relevant variable (s)
• Inclusion of an unnecessary variable (s)
• Adopting the wrong functional form
• Errors of measurement
• Incorrect specification of the stochastic error
  term

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Consequences of Model Specification Errors

• If a relevant variable is omitted
• The disturbance variance is incorrectly
  estimated.
• Standard errors of the parameters is incorrectly
  estimated.
• Confidence interval and hypothesis testing
  procedures give misleading conclusions.
• Forecast forecast interval is unreliable.

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Conseq. of Model Spec. Errors (Cont)

• If the model is over fitted
• OLS estimators are all unbiased and consistent.
• Error variance is correctly estimated.
• Confidence interval hypothesis-testing
  procedures remain valid.
• Estimated parameters are inefficient.

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Tests of Specification Errors

• (a) For the presence of unnecessary variable
• The t test , the F test
• Bottom-up approach or data mining not allowed
• (b) For the omission of variable (s)
• Look for model adequacy (R bar Square., t-ratios,
  signs of estimated coefficient, DW)
• If found not adequate, this may be attributed to
  one of the following omission of a relevant
  variable, the use of a wrong functional form,
  presence of serial correlation, etc .
• The Ramseys RESET Test
• F((R2new R2old)/number of new
  regressors)/((1-R2new)/(n-number of parameters in
  the new model))
• If significant, the model is mis-specified.
• The Lagrange Multiplier (LM) Test for Adding
  Variables
• ui a1 a2Xi akXk vi
• nR2 X2(number or restrictions)
• If significant, reject the restricted regression

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Nested versus non-nested models

• Example of Nested model
• Model A Yi ß1ß2X2i ß3X3i ß4X4i ß5X5iUi
• Model B Yi ß1ß2X2i ß3X3i Ui
• Model B is nested in Model A.
• If the hypothesis that ß4ß50 is not rejected
  (using either of the following the F-test,
  Lagrange Multiplier test and the Wald test),
  model A will reduce to model B.
• Two models are non-nested if one cannot be
  derived as a special case of the other.
• Model C Yi a1 a2X2i a3X3i Ui
• Model D Yi ?1 ?2Z2i ?3Z3i Ui

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Tests of Non-Nested Hypotheses

• Example of non-nested models
• Model C Yi a1 a2X2i a3X3i Ui
• Model D Yi ?1 ?2Z2i ?3Z3i Ui
• Encompassing F test
• Model E Yi ?1 ?2X2i ?3X3i ?4Z4i ?5Z5i vi
• Model E nests models C D but C D are
  non-nested models. If model C is correct ?4?50
  model D is correct if ?2?30 .
• Davidson-MacKinnon J Test
• Estimate model D include predicted values of D
  in model C test for the significance of the new
  variable in model C if not significant accept
  model C as the true model if significant accept
  model D. You may also start by estimating model C
  and follow the steps.

10
Model Selection Criteria

• Criteria used to choose a model among competing
  models R2, Adjusted R2, AIC, SIC, Mellows Cp,
  forecast chi-square.
• All aim at minimizing the RSS.
• Except R2, the others impose penalty for
  including an increasingly large number of
  regressors.
• Weakness of R2 as model selection criteria
  in-sample goodness of fit, dependent variables,
  irrelevant variables.
• AIC for both in-sample out-of-sample
  forecasting performance of a regression model.
  The model with the lowest value of AIC is
  preferred.
• SIC the lower the value of the SIC the better.

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Additional topic in Econometric Modeling

• Structural Stability
• When point of structural break is known
• The Chow Test Dummy Variable
• When point of structural break is not known.
• Recursive Least Squares (RELS)
• How the method works.
• Plot values of parameter estimates against each
  iteration.
• If structurally stable, changes in parameter
  values will be small and random. Otherwise if
  parameter values change significantly, it
  indicates structural break.

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Additional Topic in Econometric Modeling

• Chows Prediction Failure (stability) Test
• The predictive power of a regression model
• Stability of relationship between the dependent
  and independent variables beyond point of
  structural break.
• How the test is conducted F-test
• The null hypothesis no structural break.
• F ((S?t2-Set2)/n22)/ (S?t2/(n1-k))
• where n1 number of observation in the 1st
  period on which the initial regression is based,
  n2 number of observations in the 2nd or
  forecast period, Set2 RSS for (n1n2), Set2 RSS
  for n1, k of parameters estimated.
• Critical value obtained at n2 and n1 degrees of
  freedom.