计量经济学讲义11.

Specification
(1) y X ; (2a) i ~ iid (0, 2 ), i 1,2,...,n; (2b) i ~ iidN (0, ), i 1,2,...,n;
2
(3) E ( X ij i ) 0, j 1,2,...,k ; i 1,2,...,n; (4) X is nonstochas tic with full column rank (k 1).
• In an estimated equation, parameters are assumed to be constant over all observations in the sample: “Structural stability”. • 1.Tests of Structural Change • The estimated equation has relevance for data outside the sample data used in the estimation, which is embodied in the notion of parameter constancy: “Predictive accuracy” or “Out-of-sample prediction”. • 2.Tests of Predictive Accuracy
Tests of Structural Change
T estsof StructuralChange T heunrestrict ed modelis specifiedas y1 X 1 0 1 1 y 0 X , n n1 n2 2 2 2 2 H 0 : No structuralbreak H 0 : 1 2 T he test of H 0 is based on thetest of linear restrictio ns in termsof a restrictedand an unrestrict ed regression: y1 X 1 y1 X 1 0 b1 e1 y X b e with RSS r e ' e ; y 0 X b e Байду номын сангаас with RSS u e1 ' e1 e2 ' e2 2 2 2 2 2 2 [e ' e (e1 ' e1 e2 ' e2 )] /(k 1) F ~ F k 1, n 2(k 1) (e1 ' e1 e2 ' e2 ) /[n 2(k 1)] Large valuesof F - stat reject the null hypothesis of no structuralbreak.
2 2 ˆ2 y ˆ3 y ˆ4 , y ˆ Xb, y ˆ2 y ˆ12 y ˆ2 ˆn Z y y ˆ is excluded because its inclusion makes X Z haveless thanfull rank. Note that y




Tests of Parameter Constancy
Specification Error
• Problems with X
(1) Exclusionof relevant v ariables; (1) Inclusionof irrelevantvariables ; (1) Incorrectfunct ional form Nonlinearform,ie. quadratic form; (3) E ( X ij i ) 0, j 1,2,...,k ; i 1,2,...,n Simultaneo us equat ions,Endogeneit y; (4) T heX matrixhas less thanfull column rank Multicolli nearityor Collineari ty; Here notethatperfectcollineari ty means thatest imatesof b are not unique, but tosome degree collineari ty onlyincreasesstandarderrors.
Specification Error
• Problems with Functional Form
(1) Linearitydoes not hold Nonlineari ty between y and X. Loglineari ty or otherforms
Specification Error
• Problems with b
(1) Constancyof does not hold Structuralbreak
Specification Error
• Problems with u
(2a) E ( i2 ) i2 2 Heterosced asticity; (2a) E ( i j ) 0 Autocorrel ationor Serial correlatio n; (2b) Normality does not hold OLSis BLUE, but theinferenceis only asymptotic ally valid ;
Econometric Theory
Lecturer: Dr.Jingtao Yi Room 715 Business School, RUC
Lecture 11: k-Variable Linear Regression VIIII
• 1. Specification Error;
• 2. Tests of Parameter Constancy; • 3. Tests of Structural Change.
Test of Specification Error
• The Ramsey RESET Test
Ramsey argues that vari ous specificat ion errors (omittedvariables , incorrectfunctional form,correlatio n between X and ) give rise to a nonzero vector. H 0 : ~ N (0, 2 I ) H1 : ~ N (u , 2 I ), u 0 T he test of H 0 is based on an augmentedregression: y X Z T he test for specificat ion erroris 0. Z containspowers of thepredictedvaluesof thedependentvariable.