5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1)

Transcription

1 5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1) Assumption #A1: Our regression model does not lack of any further relevant exogenous variables beyond x 1i, x 2i,..., x Ki and the included exogenous variables are not irrelevant Two types of violations: Omitting relevant exogenous variables Including irrelevant exogenous variables 208

2 Example: Analysis of the wage structure in an enterprise with 20 workers What are the factors driving the wage y i (in euros)? Education x 1i (years after school) Age x 2i (in years) Time spent in the enterprise x 3i (in years) 209

3 i y i x 1i x 2i x 3i

4 Alternative specifications: Specification #1: Specification #2: Specification #3: y i = α + β 1 x 1i + u i y i = α + β 1 x 1i + β 2 x 2i + u i y i = α + β 1 x 1i + β 2 x 2i + β 3 x 3i + u i Assumption: Specification #2 satisfies all A, B, C assumptions (i.e. Specification #2 is the data-generating specification) 211

5 Regression output for Specification #1 Dependent Variable: WAGE Method: Least Squares Date: 10/06/04 Time: 18:57 Sample: 1 20 Included observations: 20 Variable Coefficient Std. Error t-statistic Prob. C EDUCATION R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

6 Regression output for Specifications #2 and #3 Dependent Variable: WAGE Method: Least Squares Date: 10/06/04 Time: 18:58 Sample: 1 20 Included observations: 20 Variable Coefficient Std. Error t-statistic Prob. C EDUCATION AGE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: WAGE Method: Least Squares Date: 10/06/04 Time: 18:58 Sample: 1 20 Included observations: 20 Variable Coefficient Std. Error t-statistic Prob. C EDUCATION AGE TIME_IN_FIRM R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

7 Results: β 1 (education) significantly positive in all specifications β 2 (age) positive in #2 and #3, but only significant in #2 β 3 (time spent in firm) negative and insignificant in #3 214

8 5.1 Omitted-Variable Bias Obviously: Substantial differences in the estimates from #1 and #2 Question: What are the impacts of omitting relevant variables on OLS estimators hypothesis tests of the remaining model parameters? 215

9 To this end: Consider the linear regression model in matrix notation y = Xβ + u with the (N 1) vector y (N [K + 1]) matrix X ([K + 1] 1) vector β (N 1) vector u 216

10 Now: Partitioning of X and β into so that X = [ X 1 X 2 ] and β = [ β1 β 2 ], X 1 contains the column of 1 s and K 1 exogenous variables X 2 contains the remaining exogenous variables β 1 contains the parameters α, β 1,... β K1 β 2 contains the parameters β K1 +1,... β K 217

11 Explicitly: X 1 = 1 x 11 x K1 1 1 x 12 x K x 1N x K1 N and X 2 = β 1 = x (K1 +1)1 x (K1 +2)1 x K1 x (K1 +1)2 x (K1 +2)2 x K2... x (K1 +1)N x (K 1 +2)N x KN α β 1. β K1, β 2 = β K1 +1 β K1 +2. β K 218

12 Regression model can be written as y = X 1 β 1 + X 2 β 2 + u 1. Impact on the OLS estimators: Consider the following scenario: Let the true data-generating model be given by y = X 1 β 1 + X 2 β 2 + u Let the misspecified model be given by y = X 1 β 1 + ũ (i.e. the variables x K1 +1,..., x K are omitted) 219

13 OLS estimator of the misspecified (incomplete) specification: β 1 = (X 1 X 1) 1 X 1 y = (X 1 X 1) 1 X 1 (X 1β 1 + X 2 β 2 + u) = (X 1 X 1) 1 X 1 X 1β 1 + (X 1 X 1) 1 X 1 X 2β 2 + (X 1 X 1) 1 X 1 u = β 1 + (X 1 X 1) 1 X 1 X 2β 2 + (X 1 X 1) 1 X 1 u 220

14 The expectation vector is given by E ( β 1 ) = β 1 + (X 1 X 1) 1 X 1 X 2β 2 β 1 the OLS estimator β 1 is biased Definition 5.1: (Omitted-variable bias) We call the bias (X 1 X 1) 1 X 1 X 2β 2, caused by omission of the relevant exogenous variables in the linear regression model, omittedvariable-bias. 221

15 Remarks: (I) The omitted-variable bias is zero, if β 2 = 0 (K K1 ) 1 (justified omission) X 1 and X 2 are orthogonal, i.e. if X 1 X 2 = 0 (K1 +1) (K K 1 ) (very unlikely in real-world situations) 222

16 Remarks: (II) In general, the omitted-variable bias does not vanish for N the OLS estimator β 1 is inconsistent The absolute value of the omitted-variable bias can be substantial (cf. the estimates ˆβ 1 in the Specifications #1 and #2) 223

17 2. Impact on hypothesis tests: Setting: Consider t-tests, F -tests under the A, B, C assumptions (cf. Section 4.3) Forms of the test statistics: r T = β q ˆσ 2 r (X X) 1 r (t-test, cf. Slide 186) [ R β q ] [ R(X X) 1 R ] 1 [ R β q ] F = ˆσ 2 (F -test, cf. Slide 200) 224

18 Consider now: the misspecified incomplete model y = X 1 β 1 + ũ We already know that E ( β 1 ) = E [ (X 1 X 1) 1 X 1 y ] = β 1 + (X 1 X 1) 1 X 1 X 2β 2 β 1 OLS estimator is biased (cf. Slide 221) 225

19 Similarly, we can show that E (ˆσ 2) = E [ ũ ũ/(n K1 1)] = E [ ( ) ) y X 1 β 1 (y X 1 β 1 /(N K 1 1) ] = σ 2 + β 2 X 2 [ IN X 1 (X 1 X 1) 1 X 1] X2 β 2 /(N K 1 1) (see class) σ 2 estimator ˆσ 2 is also biased 226

20 Summary: The misspecified incomplete model yields biased estimators of β and σ 2 test statistics T and F no longer have the desired statistical properties results of the t- and F -tests are highly unreliable Theorem 5.2: (Summary) The omission of relevant variables entails 1. biased OLS estimators, 2. unreliable conclusions from t- and F -tests. 227

21 5.2 Using Irrelevant Variables Now: Consider the regression model y = X 1 β 1 + X 2 β 2 + u, where the variables in X 2 are assumed to be irrelevant to y Question: What is the impact of including irrelevant variable in the matrix X 2 on the OLS estimator and hypothesis tests? 228

22 1. Impact on the OLS estimators Line of argument: Under the A, B, C assumptions (without #B4) the OLS estimator β = (X X) 1 X y is an unbiased estimator of β (cf. Theorem 4.3, Slide 170) The proof does not depend on whether some of the parameters in β are equal to zero or on whether X contains an irrelevant block matrix X 2 unbiasedness of the OLS estimator remains valid The same result holds for the estimator ˆσ 2 = û û/(n K 1) unbiasedness of estimator ˆσ 2 remains valid 229

23 However: OLS estimators of the overloaded model are inefficient Sketch of proof: (I) (see class) Calculate the covariance matrix Cov specified model ( β 1 ) of the correctly y = X 1 β 1 + ũ Result: Cov ( β 1 ) = σ 2 [X 1 X 1] 1 230

24 Sketch of proof: (II) Calculate the block matrix Cov ( β 1 ) of the overloaded model y = X 1 β 1 + X 2 β 2 + u Result: Cov ( β 1 ) = σ 2 [ X 1 ( IN X 2 (X 2 X ) ] X 2 X 1 Show that the matrix is positively definite Cov ( β 1 ) Cov ( β 1 ) relative efficiency of the OLS estimator from the correctly specified model 231

25 Illustration: Standard errors in the Specifications #2 and #3 (cf. Slide 213) 232

26 2. Impact on hypothesis tests Line of argument: t- and F -test statistics rest on the OLS estimators β and ˆσ 2 Both estimators are unbiased tests are based on correct conditions test results are principally valid However: Owing to the inefficiency of the estimators, the tests from the overloaded model have smaller power than the tests from the correctly specified model 233

27 Theorem 5.3: (Summary) The inclusion of irrelevant variables entails 1. unbiased, but inefficient OLS estimators, 2. usable, but unnecessarily imprecise t- and F -tests. Remark: The impact of including irrelevant variables is far less severe than that resulting from the omission of relevant variables 234

28 5.3 Diagnostics and Re-Specification Problem: Selection of the correct exogenous variables Trade-off: Risk of biased estimation versus inefficient estimation Important criterion: Economic theory 235

29 Statistical criteria: (I) Choosing between bias and inefficiency Adjusted R 2 : R 2 adj = 1 (1 N 1 R2 ) N K 1 Akaike s information criterion: AIC = ln [ ] Sûû N + 2(K + 1) N Schwarz-criterion: SC = ln [ ] Sûû N + K ln(n) N 236

30 Statistical criteria: (II) t-tests for testing significance of single variables F -tests for testing significance of groups of variables Example: Regression output for the Specifications #1, #2, #3 on Slides 212,

31 Specification strategies: Backward selection: Start with an overloaded model and reduce number of variables by applying t- and F -tests Forward selection: Start with a small number of variables and add additional variables by applying t- and F -tests (not recommended due to omitted-variable bias) Problem with both strategies: Final specification depends on the order of the selection process 238