{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Econ103fall10lec8

# Econ103fall10lec8 - Introduction Single Coecient Multiple...

This preview shows pages 1–10. Sign up to view the full content.

Introduction Single Coefficient Multiple Coefficients Example Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 8: Multiple Regression and Testing Sarolta Laczó October 19, 2010

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Introduction Single Coefficient Multiple Coefficients Example Introduction/1 General Multiple Regression Model: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ... + β k X ki + u i We have estimated the parameters β 0 , β 1 , ..., β 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 0 : β 2 = 0 or H 0 : β 3 = 1 Hypothesis tests regarding multiple coefficients For example H 0 : β 1 = β 2 = β 4 = 0 or H 0 : β 5 = 2 β 3 Example - the return to education

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ˆ β j - β j σ ˆ β j N (0 , 1) In practice, we have to estimate σ ˆ β j , that in turn depends on ˆ σ 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 σ 2 is ˆ σ 2 = 1 n - k - 1 X i ˆ 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 ˆ β j is a t with n - k - 1 degrees of freedom: ˆ β j - β j SE ( ˆ β j ) t n - k - 1 The t distribution converges to a normal when n is large. ˆ β j - β j SE ( ˆ β j ) N (0 , 1)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 : β j = 0 ) 2 Formulate the alternative hypothesis, either as two-sided ( H 1 : β j 6 = 0 ) or as one-sided ( H 1 : β j < 0 or H 1 : β j > 0 ) 3 Specify the level of significance α (e.g. α = 0 . 05 ) 4 Calculate the actual value of the t -statistic under the null. 5 Compute the critical value according to the significance level α . 6 Decide whether you can or cannot reject the null hypothesis.
Introduction Single Coefficient Multiple Coefficients Example Hypothesis Tests for a Single Coefficient/4

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 . 0 (8 . 73) - 1 . 10 (0 . 43) STR i - 0 . 65 (0 . 03) PctLE i We are interested in whether the STR affects test scores. 1 H 0 : β 1 = 0 2 H 1 : β 1 6 = 0 3 α = 5% 4 The actual value of the t -statistic is t act = ˆ β 1 - β 1 , 0 SE ˆ β 1 = - 1 . 10 - 0 0 . 43 = - 2 . 56 5 The critical values are z α/ 2 = - 1 . 96 and z 1 - α/ 2 = 1 . 96 6 We reject the null hypothesis. β 1 is negative and statistically significant. The STR has a negative and statistically significant effect on test scores.
Introduction

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 29

Econ103fall10lec8 - Introduction Single Coecient Multiple...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online