# Types of Specification Errors 1. Omitted Variables. 2. Including an irrelevant variable. 3. Incorrect functional form. 2

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1 Notes on Model Specification To go with Gujarati, Basic Econometrics, Chapter 13 Copyright - Jonathan Nagler; April 4, 1999 Attributes of a Good Econometric Model A.C. Harvey: (in Gujarati, p ) ffl Parsimony: Explain alot with a little. ffl Identifiability: For a set of data, unique values exist for the parameters. ffl Goodness of Fit ffl Theoretical Consistency ffl Predictive Power 1. Outside the sample. 2. The model is based on quantities observed prior to the time the prediction is needed. 1

2 Types of Specification Errors 1. Omitted Variables. 2. Including an irrelevant variable. 3. Incorrect functional form. 2

3 Consequences of Omitted Variable: ffl Everything is wrong. Consequences of Including an Irrelevant Variable: 1. Estimates of the parameters are unbiased and consistent. 2. ff 2 is correctly estimated. 3. Hypotheses tests are valid. 4. The estimated standard errors are generally inefficient (i.e., the variances are larger than those of ^fi). ffl This violates the BLUE condition: the estimated parameters are not the best linear unbiased estimators. 3

4 Tests of Specification Errors Detecting the Presence of an Unnecesary Variable Note: Gujarati says to use a t-test or F-test. This is not a recomended procedure. We know that ff ^fij increases as Var(x j ) decreases. We do not want to infer that x j does not influence y simply because we don't have good observations on x j. Detecting Omitted Variables and Incorrect Functional Form Examine the residuals: specification errors show up as patterns in the plots. Suggested Plots to look at: 1. Y versus ^Y 2. e versus X j for some independent variable of interest. 3. Y versus ^Y versus X j for some independent variable of interest. 4

5 Ramsey RESET Test: 1. Get ^Y. 2. Add variants of ^Y ( ^Y 2, ^Y 3, etc) to the RHS of the model. 3. Estimate the new model. 4. If the ^Y terms are significant (use an F-test to test for joint-significance of all the ^Y terms), then you have specification error. From this, we do not get information on alternatives or corrections. But, it is a good test because we do not need to know what Xs are causing the problem. Other Tests: 1. Likelihood Ratio Test 2. Wald Test 3. Lagrange Multiplier Test 4. Hausman Test 5

6 Lagrange Multiplier Test for Adding a Variable ffl Estimate a restricted regression (some set of variables is omitted), compute the residuals e i. ffl Regress e i on all the regressors. ffl The quantity nr 2 will be distributed as a χ 2, where m m is the number of restrictions. ffl If nr 2 exceeds the critical value, we can reject the restricted model. This test is similar to the Ramsey RESET test. But, this test could be accomplished with a standard F-test. 6

7 Measurement Error Error in Measuring Y Say the truth is: Y Λ i = fi 0 + fi 1 X i + u i We measure Y Λ with error: Y i = Y Λ i + ffl i Now we estimate the model based on observed data: (Y i + ffl i ) = fi 0 + fi 1 X i + u i Y i = fi 0 + fi 1 X i + (u i ffl i ) Y i = fi 0 + fi 1 X i + fl i If we assume that ffl i is very stochastic and unrelated to u i : E[u i ] = E[ffl i ] = 0 cov(x i ;u i ) = cov(ffl i ;X i ) = 0 cov(ffl; u i ) = 0 Then we will have unbiased estimates. But, the variance will be increased. 7

8 With no measurement error: Var( ^fi) = ff 2 u ±(X i μ X) 2 (1) With measurement error: Var( ^fi) = ff 2 fl ±(X i X) μ = ff 2 ±(X i X) μ (2) 2 2 u + ff2 ffl 8

9 Measurement Error in X Say the truth is: Y i = fi 0 + fi 1 X Λ i + u i We measure X Λ with error: X i = X Λ i + ffl i Now we estimate the model based on observed data: Y i = fi 0 + fi 1 (X Λ i + ffl i ) + u i Y i = fi 0 + fi 1 X Λ i + (u i + fi 1 ffl i ) Y i = fi 0 + fi 1 X Λ i + fl i Now we cannot assume that cor(x i ;fl i ) = 0. So, our OLS estimates will be biased. 9

10 Leamer on Model Selection Reasons for Model specification tests (From Leamer (1978, as adapted in Gujarati)): 1. Hypothesis Testing: Choose a true model. 2. Interpretive: Interpret data involving several correlated variables. 3. Simplification: construct a fruitful" model. 4. Proxy: Choose between measures that purport to measure the same variable. 5. Data Selection: Select the appropriate data for estimation and prediction. 6. Post Data Model Construction: Improve an existing model. 10

11 Hendry on Model Selection From Hendry and Richard 1983, as reported in Gujarati: 1. Be data admissable. The predictions made from the model must be logically possible. 2. Be consistent with theory. 3. Have weakly exogenous regressors. 4. Exhibit parameter constancy. 5. Exhibit data coherency. The residuals must be stochastic. 6. Be encompassing. 11

12 Nested models: tests are straightforward via F-tests. Non-Nested Models: 1) Goodness of fit comparisons. 2) Estimate a model encompassing the two competing models. Model A: Y i = ff 10 + fi 11 X i + ffl i Model B: Y i = ff 20 + fi 21 Z i + v i Create an encompassing model which the competing models are nested within: Model C: Y i = ff 30 + fi 31 X i + fi 32 Z i + fl i If fi 31 is significant, but fi 32 is not significant, then we would prefer model A over model B. But, we cannot actually reject model B - we have no statistical test to do this. 12

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