《计量经济学》ch_04_wooldridge_5e_ppt
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Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (2/5)
Assumption MLR.6 (Normality of error terms)
independently of
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators 4.2 Testing Hypotheses about a Single Population Parameter: The t Test 4.3 Confidence Intervals 4.4 Testing Hypotheses about a Single Linear Combination of the Parameters 4.5 Testing Multiple Linear Restrictions: The F Test 4.6 An application— estimation of the weights of CPI components in China
Sampling distributions of the OLS estimators The OLS estimators are random variables We already know their expected values and their variances However, for hypothesis tests we need to know their distribution In order to derive their distribution we need additional assumptions Assumption about distribution of errors: normal distribution
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (5/5)
Terminology
„Gauss-Markov assumptions“ „Classical linear model (CLM) assumptions“
Multiple Regression Analysis: Inference
Chapter 4
Wooldridge: Introductory Econometrics: A Modern Approach, 5e Instructed by professor Yuan, Huiping
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.2 Testing Hypotheses about a Single Population Parameter: The t Test 4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators 4.2.2 Testing against One-Sided Alternatives 4.2.3 Two-Sided Alternativssion Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (4/5)
Discussion of the normality assumption (cont.) Examples where normality cannot hold: • Wages (nonnegative; also: minimum wage) • Number of arrests (takes on a small number of integer values) • Unemployment (indicator variable, takes on only 1 or 0) In some cases, normality can be achieved through transformations of the dependent variable (e.g. use log(wage) instead of wage) Important: For the purposes of statistical inference, the assumption of normality can be replaced by a large sample size
Assignments: Promblems 1, 2, 4, 5, 7, 8, 10
Computer Exercises C1, C2, C3, C8, C9 C8: smpl if marr=1 and fsize=2 (401ksubs.wf1)
The End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
The standardized estimators follow a standard normal distribution
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (3/5)
Discussion of the normality assumption The error term is the sum of „many“ different unobserved factors Sums of independent factors are normally distributed (CLT) Problems: • How many different factors? Number large enough? • Possibly very heterogenuous distributions of individual factors • How independent are the different factors? The normality of the error term is an empirical question At least the error distribution should be „close“ to normal
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Theorem 4.1 (Normal sampling distributions) Under assumptions MLR.1 – MLR.6:
The estimators are normally distributed around the true parameters with the variance that was derived earlier
It is assumed that the unobserved factors are normally distributed around the population regression function. The form and the variance of the distribution does not depend on any of the explanatory variables. It follows that:
Chapter 4 Multiple Regression Analysis: Inference
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (1/5)
Statistical inference in the regression model Hypothesis tests about population parameters Construction of confidence intervals
4.2.4 Testing Other Hypotheses about bj
4.2.5 Computing p-Values for t Tests 4.2.6 A Reminder on the Language of Classical Hypothesis Testing 4.2.7 Economic, or Practical, versus Statistical Significance
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (2/5)
Assumption MLR.6 (Normality of error terms)
independently of
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators 4.2 Testing Hypotheses about a Single Population Parameter: The t Test 4.3 Confidence Intervals 4.4 Testing Hypotheses about a Single Linear Combination of the Parameters 4.5 Testing Multiple Linear Restrictions: The F Test 4.6 An application— estimation of the weights of CPI components in China
Sampling distributions of the OLS estimators The OLS estimators are random variables We already know their expected values and their variances However, for hypothesis tests we need to know their distribution In order to derive their distribution we need additional assumptions Assumption about distribution of errors: normal distribution
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (5/5)
Terminology
„Gauss-Markov assumptions“ „Classical linear model (CLM) assumptions“
Multiple Regression Analysis: Inference
Chapter 4
Wooldridge: Introductory Econometrics: A Modern Approach, 5e Instructed by professor Yuan, Huiping
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.2 Testing Hypotheses about a Single Population Parameter: The t Test 4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators 4.2.2 Testing against One-Sided Alternatives 4.2.3 Two-Sided Alternativssion Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (4/5)
Discussion of the normality assumption (cont.) Examples where normality cannot hold: • Wages (nonnegative; also: minimum wage) • Number of arrests (takes on a small number of integer values) • Unemployment (indicator variable, takes on only 1 or 0) In some cases, normality can be achieved through transformations of the dependent variable (e.g. use log(wage) instead of wage) Important: For the purposes of statistical inference, the assumption of normality can be replaced by a large sample size
Assignments: Promblems 1, 2, 4, 5, 7, 8, 10
Computer Exercises C1, C2, C3, C8, C9 C8: smpl if marr=1 and fsize=2 (401ksubs.wf1)
The End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
The standardized estimators follow a standard normal distribution
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (3/5)
Discussion of the normality assumption The error term is the sum of „many“ different unobserved factors Sums of independent factors are normally distributed (CLT) Problems: • How many different factors? Number large enough? • Possibly very heterogenuous distributions of individual factors • How independent are the different factors? The normality of the error term is an empirical question At least the error distribution should be „close“ to normal
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Theorem 4.1 (Normal sampling distributions) Under assumptions MLR.1 – MLR.6:
The estimators are normally distributed around the true parameters with the variance that was derived earlier
It is assumed that the unobserved factors are normally distributed around the population regression function. The form and the variance of the distribution does not depend on any of the explanatory variables. It follows that:
Chapter 4 Multiple Regression Analysis: Inference
Chapter End © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 4 Multiple Regression Analysis: Inference
4.1 Sampling Distributions of the OLS Estimators (1/5)
Statistical inference in the regression model Hypothesis tests about population parameters Construction of confidence intervals
4.2.4 Testing Other Hypotheses about bj
4.2.5 Computing p-Values for t Tests 4.2.6 A Reminder on the Language of Classical Hypothesis Testing 4.2.7 Economic, or Practical, versus Statistical Significance