May 2004 Prof. Himayatullah 1 Basic Econometrics Chapter 7 MULTIPLE REGRESSION ANALYSIS: The Problem of Estimation

May 2004 Prof. Himayatullah The three-Variable Model: Notation and Assumptions Y i = ß 1 + ß 2 X 2i + ß 3 X 3i + u i (7.1.1) ß 2, ß 3 are partial regression coefficients With the following assumptions: + Zero mean value of U i: : E(u i |X 2i,X 3i ) = 0.  i (7.1.2) + No serial correlation: Cov(u i,u j ) = 0,  i # j (7.1.3) + Homoscedasticity: Var(u i ) =  2 (7.1.4) + Cov(u i,X 2i ) = Cov(u i,X 3i ) = 0 (7.1.5) + No specification bias or model correct specified (7.1.6) + No exact collinearity between X variables (7.1.7) (no multicollinearity in the cases of more explanatory vars. If there is linear relationship exits, X vars. Are said to be linearly dependent) + Model is linear in parameters

May 2004 Prof. Himayatullah Interpretation of Multiple Regression E(Y i | X 2i,X 3i ) = ß 1 + ß 2 X 2i + ß 3 X 3i (7.2.1) (7.2.1) gives conditional mean or expected value of Y conditional upon the given or fixed value of the X 2 and X 3

May 2004 Prof. Himayatullah The meaning of partial regression coefficients Y i = ß 1 + ß 2 X 2i + ß 3 X 3 +….+ ß s X s + u i ß k measures the change in the mean value of Y per unit change in X k, holding the rest explanatory variables constant. It gives the “direct” effect of unit change in X k on the E(Y i ), net of X j (j # k) How to control the “true” effect of a unit change in X k on Y? (read pages )

May 2004 Prof. Himayatullah OLS and ML estimation of the partial regression coefficients This section (pages ) provides: 1. The OLS estimators in the case of three- variable regression Y i = ß 1 + ß 2 X 2i + ß 3 X 3 + u i 2. Variances and standard errors of OLS estimators 3.8 properties of OLS estimators (pp ) 4.Understanding on ML estimators

May 2004 Prof. Himayatullah The multiple coefficient of determination R 2 and the multiple coefficient of correlation R This section provides: 1.Definition of R 2 in the context of multiple regression like r 2 in the case of two-variable regression 2.R =  R 2 is the coefficient of multiple regression, it measures the degree of association between Y and all the explanatory variables jointly 3. Variance of a partial regression coefficient Var(ß^ k ) =  2 /  x 2 k (1/(1-R 2 k )) (7.5.6) Where ß^ k is the partial regression coefficient of regressor X k and R 2 k is the R 2 in the regression of X k on the rest regressors

May 2004 Prof. Himayatullah Example 7.1: The expectations-augmented Philips Curve for the US ( ) This section provides an illustration for the ideas introduced in the chapter Regression Model (7.6.1) Data set is in Table 7.1

May 2004 Prof. Himayatullah Simple regression in the context of multiple regression: Introduction to specification bias This section provides an understanding on “ Simple regression in the context of multiple regression”. It will cause the specification bias which will be discussed in Chapter 13

May 2004 Prof. Himayatullah R 2 and the Adjusted-R 2 R 2 is a non-decreasing function of the number of explanatory variables. An additional X variable will not decrease R 2 R 2 = ESS/TSS = 1- RSS/TSS = 1-  u^ 2 I /  y^ 2 i (7.8.1) This will make the wrong direction by adding more irrelevant variables into the regression and give an idea for an adjusted-R 2 (R bar ) by taking account of degree of freedom R 2 bar = 1- [  u^ 2 I /(n-k)] / [  y^ 2 i /(n-1) ], or (7.8.2) R 2 bar = 1-  ^ 2 / S 2 Y (S 2 Y is sample variance of Y) K= number of parameters including intercept term –By substituting (7.8.1) into (7.8.2) we get R 2 bar = 1- (1-R 2 ) (n-1)/(n- k) (7.8.4) –For k > 1, R 2 bar < R 2 thus when number of X variables increases R 2 bar increases less than R 2 and R 2 bar can be negative

May 2004 Prof. Himayatullah R 2 and the Adjusted-R 2 R 2 is a non-decreasing function of the number of explanatory variables. An additional X variable will not decrease R 2 R 2 = ESS/TSS = 1- RSS/TSS = 1-  u^ 2 I /  y^ 2 i (7.8.1) This will make the wrong direction by adding more irrelevant variables into the regression and give an idea for an adjusted-R 2 (R bar ) by taking account of degree of freedom R 2 bar = 1- [  u^ 2 I /(n-k)] / [  y^ 2 i /(n-1) ], or (7.8.2) R 2 bar = 1-  ^ 2 / S 2 Y (S 2 Y is sample variance of Y) K= number of parameters including intercept term –By substituting (7.8.1) into (7.8.2) we get R 2 bar = 1- (1-R 2 ) (n-1)/(n- k) (7.8.4) –For k > 1, R 2 bar < R 2 thus when number of X variables increases R 2 bar increases less than R 2 and R 2 bar can be negative

May 2004 Prof. Himayatullah R 2 and the Adjusted-R 2 Comparing Two R 2 Values: To compare, the size n and the dependent variable must be the same Example 7-2: Coffee Demand Function Revisited (page 210) The “game” of maximizing adjusted-R 2 : Choosing the model that gives the highest R 2 bar may be dangerous, for in regression our objective is not for that but for obtaining the dependable estimates of the true population regression coefficients and draw statistical inferences about them Should be more concerned about the logical or theoretical relevance of the explanatory variables to the dependent variable and their statistical significance

May 2004 Prof. HimayatullahProf. Himayatullah Partial Correlation Coefficients This section provides: 1. Explanation of simple and partial correlation coefficients 2.Interpretation of simple and partial correlation coefficients (pages )

May 2004 Prof. Himayatullah Example 7.3: The Cobb- Douglas Production function More on functional form Y i =  1 X  2 2i X  3 3i e U i (7.10.1) By log-transform of this model: lnY i = ln  1 +  2 ln X 2i +  3 ln X 3i + U i =  0 +  2 ln X 2i +  3 ln X 3i + U i (7.10.2) Data set is in Table 7.3 Report of results is in page 216

May 2004 Prof. Himayatullah Polynomial Regression Models Y i =  0 +  1 X i +  2 X 2 i +…+  k X k i + U i (7.11.3) Example 7.4: Estimating the Total Cost Function Data set is in Table 7.4 Empirical results is in page Summary and Conclusions (page 221)