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Unformatted text preview: Introduction Single Coefficient Multiple Coefficients Example Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 8: Multiple Regression and Testing Sarolta Lacz October 19, 2010 Introduction Single Coefficient Multiple Coefficients Example Introduction/1 General Multiple Regression Model: Y i = + 1 X 1 i + 2 X 2 i + ... + k X ki + u i We have estimated the parameters , 1 ,..., k . Now, we will study whether the estimated coefficients are statistically significant. whether certain relationships between parameters hold. whether a group of parameters are jointly significant. Introduction Single Coefficient Multiple Coefficients Example Introduction/2 Outline: Hypothesis tests for a single coefficient For example H : 2 = 0 or H : 3 = 1 Hypothesis tests regarding multiple coefficients For example H : 1 = 2 = 4 = 0 or H : 5 = 2 3 Example  the return to education Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/1 Testing a hypothesis about a single coefficient is done in the same way as in the case of the simple regression. The CLT tells us that j j j N (0 , 1) In practice, we have to estimate j , that in turn depends on 2 , the estimated variance of the error term. We have n k 1 degrees of freedom to estimate this variance, thus an unbiased estimator of 2 is 2 = 1 n k 1 X i u 2 i Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/2 As a consequence of this, for a small n , the distribution of the standardized j is a t with n k 1 degrees of freedom: j j SE ( j ) t n k 1 The t distribution converges to a normal when n is large. j j SE ( j ) N (0 , 1) Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/3 Steps for hypothesis testing: 1 Formulate the null hypothesis (e.g. H : j = 0 ) 2 Formulate the alternative hypothesis, either as twosided ( H 1 : j 6 = 0 ) or as onesided ( H 1 : j < or H 1 : j > ) 3 Specify the level of significance (e.g. = 0 . 05 ) 4 Calculate the actual value of the tstatistic under the null. 5 Compute the critical value according to the significance level . 6 Decide whether you can or cannot reject the null hypothesis. Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/4 Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/5  Example The estimated regression line (with robust standard errors) is: \ Test Scores i = 686 ....
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This note was uploaded on 10/23/2010 for the course STAT 10 taught by Professor Davis during the Spring '10 term at UCLA.
 Spring '10
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