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 08 Parameter Estimation and Evaluation

海怨南曼王等大门餐 Parameter Estimation and Evaluation Professor Yongmiao Hong Cornell University April 21,2020

Parameter Estimation and Evaluation Professor Yongmiao Hong Cornell University April 21, 2020

CONTENTS 8.1 Population and Distribution Model 8.2 Maximum Likelihood Estimation 8.3 Asymptotic Properties of MLE 8.4 Method of Moments and Generalized Method of Moments 8.5 Asymptotic Properties of GMM 8.6 Mean Squared Error Criterion 8.7 Best Unbiased Estimators 8.8 Cramer-Rao Lower Bound 8.9 Conclusion Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 2/207

Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 2/207 8.1 Population and Distribution Model 8.2 Maximum Likelihood Estimation 8.3 Asymptotic Properties of MLE 8.4 Method of Moments and Generalized Method of Moments 8.5 Asymptotic Properties of GMM 8.6 Mean Squared Error Criterion 8.7 Best Unbiased Estimators 8.8 Cramer-Rao Lower Bound 8.9 Conclusion CONTENTS

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model A random sample XT=(X1,..,Xn)is from a popula- tion distribution fx(x). A realization x"of the random sample Xm constitutes a data set with sample size n. [Sampling Inference:The primary purpose of sta- tistical inference is to make inference of the population distribution fx(x)using an observed data x". Population Inference Sampling 光光子状天 量元米米子 Sample Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 3/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 3/207 Population and Distribution Model Population and Distribution Model Sampling Inference

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Parametric Approach]Consider a class of parametric candidate distributions F={f(,0):0∈Θ}， where f:2×O→R+is a known PMF/PDF function,2 is the support of Xi,is a parameter space that contains all plausible values for a p x 1 parameter vector 0,with p a fixed integer.Each value of 0e gives a distribution model for fx(x). Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 4/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 4/207 Population and Distribution Model Population and Distribution Model

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Correct Model Specification:Assume that the class of functions F contains the population distribution fx(x) that generates the observed data.That is,there exists some unkonwn parameter value 00∈曰such that fx(c)=f(x,0o）for all a∈2. If F contains fx(),we call that the class of models F is correctly specified for the population distribution fx(), and 00 is called the true value of 0. Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 5/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 5/207 Population and Distribution Model Population and Distribution Model

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model In contrast,F is said to be misspecified for fx()if there exists no value for 0ee such that fx(x)=f(x,0)for all xE1.This can occur when,e.g.,we specify a class of normal distribution models but the true population is a Gamma distribution. f红8 f fc.9,) fx@) fx) fc.,) Figure 8.1(a)Correct model specification Figure 8.1(b)Model misspecification Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 6/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 6/207 Population and Distribution Model Population and Distribution Model

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Example 1(8.1)[Discrete Choice Probit and Logit Models] The Probit and Logit models are popularly used when a de- pendent variable has binary outcomes,i.e.,there are two pos- sible outcomes 0 and 1.Examples include whether or not an individual is employed,whether or not a consumer makes a purchase,and whether or not a financial crisis (e.g.,default risk)occurs. A probit model assumes P(Y=1X)=Φ(01+02X), i=1,···,m, where (is the N(0,1)CDF,and Xi is an explanatory vari- able. 1 A logit model assumes P(Yi=1Xi)= 1+e-(01+02X) Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 71207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 7/207 Population and Distribution Model Population and Distribution Model Example 1 (8.1) [Discrete Choice Probit and Logit Models]

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Example 1(8.1)[Discrete Choice Probit and Logit Models] On 19 December 2018,$&P500 dropped 1.54%,the next day(20 December 2018),KLCI dropped 0.31%.The linear and logit regression model published on 14 December 2018(Read more here)predicted the KLCI would fall 0.37%and the chances of the drop are as high as 75%.This shows that the quantitative approach is indeed decent. Logit vs Profit Model(S&P500 Impact on KLCI) 09 Logit Probit 1 1 22 Probability=1+eo+) Probability e 2 dz 0.6 2πJ-0 0.5 0.2 0.2 15.0% 10.0% 50% 0.0% 50% 100w 15.w ◆一Logt一◆一升abt Chart shows the probability plot for both logit and probit models.Both models should give similar results. The slight difference is logit model has fatter tail. Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 8/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 8/207 Population and Distribution Model Population and Distribution Model Example 1 (8.1) [Discrete Choice Probit and Logit Models] On 19 December 2018, S&P500 dropped 1.54%, the next day (20 December 2018), KLCI dropped 0.31%. The linear and logit regression model published on 14 December 2018 (Read more here) predicted the KLCI would fall 0.37% and the chances of the drop are as high as 75%. This shows that the quantitative approach is indeed decent. Chart shows the probability plot for both logit and probit models. Both models should give similar results. The slight difference is logit model has fatter tail

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Example 2(8.2)[Survival/Duration Analysis] Suppose we are interested in the time length it takes for an unemployed person to find a job,the time length that elapses between two trades or two price changes,the length of a strike, the length before a cancer patient dies,and the length before a financial crisis (e.g.,credit default risk)comes out.Such analysis is called duration analysis or survival analysis. In practice,the main interest often lies in the question of how long a duration will continue,given that it has not finished yet.The hazard rate measures the chance that the duration will end now,given that it has not ended before.This hazard rate can be interpreted as the chance to find a job,to trade, to end a strike,etc. Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 91207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 9/207 Population and Distribution Model Population and Distribution Model Example 2 (8.2) [Survival/Duration Analysis]

Parameter Estimation and Evaluation Population and Distribution Model Population and Distribution Model Example 2(8.2)[Survival/Duration Analysis] Monthly Job-Finding Probability by Unemployment Duration 10 -Kaplan Meier Estimate O No controls -No controls,fitted Full controls 一Full controls.fted Job-finding probability 1.0 Survival Analysis 09 0.8 8%probability of surviving beyond 22 months 0.6 0 50%probability of surviving 0.4 . beyond 65 months 02 0 0 5 10 15 20 25 3 06 Unemployment duration(months) Source:Authors'calqulations from the Cumrent Population Survey. 30 40 50 60 Note:The job-finding probability is expressed relative to the job-finding probability of newly timeline unemployed workers. Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 10/207

Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 10/207 Population and Distribution Model Population and Distribution Model Example 2 (8.2) [Survival/Duration Analysis]