厦门大学：《概率论与数理统计 Probability and Statistics for Economists》课程教学资源（课件讲稿）Chapter 09 Hypothesis Testing

0院究研清银南受王季大門厦餐 WI8目 /图YsmE Hypothesis Testing Professor Yongmiao Hong Cornell University Juy7,2020

Hypothesis Testing Professor Yongmiao Hong Cornell University July 7, 2020

CONTENTS 9.1 Introduction to Hypothesis Testing 9.2 Neyman-Pearson Lemma 9.3 Wald Test 9.4 Lagrangian Multiplier (LM)Test 9.5 Likelihood Ratio Test 9.6 lllustrative Examples 9.7 Conclusion Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 2/110

Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 2/110 9.1 Introduction to Hypothesis Testing 9.2 Neyman-Pearson Lemma 9.3 Wald Test 9.4 Lagrangian Multiplier (LM) Test 9.5 Likelihood Ratio Test 9.6 Illustrative Examples 9.7 Conclusion CONTENTS

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Question:Given a realization x"of a random sample X" from the population distribution f(x,0),we would also like to know whether the true parameter value 0o belongs to some specific subsetΘof the parameter spaceΘ. Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 3/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 3/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 1(9.1)[Return to Education] Let 0 measure the change in hourly wage given another year of education,holding all other factors fixed.Labor economists are interested in testing whether the return to education,con- trolling other factors,is zero.That is,whether or not 0 equal to zero. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 4/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 4/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 1 (9.1) [Return to Education]

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 1(9.1)[Hypothesis] A hypothesis is a statement about the population or some attributes of the population distribution.The two comple- mentary hypotheses in a hypothesis testing problem are called the null hypothesis and the alternative hypothesis,denoted by Ho and HA respectively Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 5/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 5/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Definition 1 (9.1) [Hypothesis]

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: .A null hypothesis is a statement about the population or some at- tributes of the population.The alternative hypothesis is that the statement in the null hypothesis is false. The goal of hypothesis testing is to decide,based on an observed data set x"generated from a population,which of two complementary hy- potheses is true. Suppose a random sample Xm is generated from a population distri- bution f(x,0)with some unknown value of parameter 0 O,where is a known finite-dimensional parameter space.In hypothesis test- ing,the parameter space is divided into two mutually exclusive and collectively exhaustive subsetsΘoandΘA,namelyΘo∩A=gand Θ0UΘA=Θ. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 6/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 6/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks:

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing The problem is to determine to which of these two subsets the true value of 0 belongs.That is,based upon an observed data set x",one is trying to choose between the two hypotheses H0:0∈Θ0 versus HA:0∈ΘA. The first hypothesis Ho is called the null hypothesis,and the second HA,is called the alternative hypothesis. ●Ho is called the“null”hypothesis because it is often stated as“no effects”or“no relationship”.One example,isHo:f=0 versus lA:0≠0，as is the case of Example9.1. Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 71110

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Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 2(9.2)[Constant Return to Scale Hypothesis] A production function Y=F(L,K) tells how much output Y to produce using inputs of labor L and capital k.A production technology is said to display a constant return to scale if the output increases by the same proportion as the inputs increase;that is,for all A >0, λF(L,K)=F(λL,入K). Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 8/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 8/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 2 (9.2) [Constant Return to Scale Hypothesis]

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 3(9.3)[Constant Return to Scale Hypothesis] Suppose a production function is given by Y=AKOLB, where Y is the output,K and L are the capital and labor inputs,A is a constant,and 0 (a,B)is a parameter vector. Then the constant return to scale hypothesis can be stated as H0:a+B=1. The alternative hypothesis Ho a+B 1 consists of two cases:a+B>1 and a+B <1,which imply an increasing return to scale and a decreasing return to scale respectively. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 9/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 9/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 3 (9.3) [Constant Return to Scale Hypothesis]

Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: The hypotheses can be divided into two basic categories- simple hypotheses and composite hypotheses. Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 10/110

Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 10/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: