Chapter 8 6 Model Robustness check Modified Version

Chapter 8 6 Model: Robustness check Modified Version by Woraphon Yamaka © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. © kentoh/Shutterstock.

Robustness check: Heteroscedasticity ● Unconditional error variance is unaffected by heteroskedasticity (which refers to the conditional error variance) © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 3

Testing for heteroskedasticity ● 1. Bartlett test ● 2. Breusch Pagan test ● 3. Goldfeld Quandt test ● 4. Glesjer test ● 5. Test based on Spearman’s rank correlation coefficient ● 6. White test ● 7. Ramsey test ● 8. Harvey Phillips test ● 9. Szroeter test ● 10. Peak test (nonparametric) test © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 4

Testing Heteroskedasticity 1) Breusch-Pagan test for heteroskedasticity คาเฉลยนของ u 2 ตองนงเมอคา x 2, …, x k เปลยนไป © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. x 1, 5

Testing Heteroskedasticity 1) Breusch-Pagan test for heteroskedasticity (Procedures) ●Step 1: Run regression model Step 3: F-test �� LM-test ยง R-squared สง F สง จะมโอกาสปฎเสธ มากขน หรออาจใช (Lagrange multiplier statistic, LM). ยง R-squared สง LM สง จะมโอกาสปฎเสธ มากขน. © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. H 0 6

Testing Heteroskedasticity ● �� : Heteroskedasticity in housing price equations Heteroskedasticity © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 7

Testing Heteroskedasticity 2) The White test for heteroskedasticity Step 1: Run regression model Step 3: LM-test ● ขอเสยของ White test ● วธนตองประมาณ พารามเตอรเยอะ ซงนำไปสการประมาณทซบซอนเกนไปและเกดปญหา © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 8

Testing Heteroskedasticity ● �� White test �� © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 9

Solving Heteroskedasticity 1) log-log Model ใน Form ใหมนเราจะเหนวาแบบจำลองนไมมปญหา Heteroscedasticity 2) OLS robust estimation 3) Generalized Least Squares (GLS)This is an extension of OLS. 3. 1 Weighted least squares: Heteroskedasticity is known 3. 2 Feasible least squares: Heteroskedasticity is unknown The functional form of the heteroskedasticity is either known or unknown © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 10

Solving Heteroskedasticity by OLS robust estimation ● © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 11

Solving Heteroskedasticity by OLS robust estimation ● Example: Hourly wage equation Note: () �� OLS S. E. �� [] �� robust OLS S. E. Heteroskedasticity robust standard errors may be larger or smaller than their nonrobust counterparts The differences are often small in practice. F statistics are also often not too different. If there is strong heteroskedasticity, differences may be larger. To be on the safe side, it is advisable to always compute robust standard errors. © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 12

Solving Heteroskedasticity by Weighted least squares: Heteroskedasticity is known Original model Step 1: Transformed model Step 2: Run Transformed model using OLS © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 13

Solving Heteroskedasticity by Weighted least squares: Heteroskedasticity is known ● OLS in the transformed model is weighted least squares (WLS) ● Why is WLS more efficient than OLS in the original model? • �� weight �� weight �� Variance �� 14 © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.

Solving Heteroskedasticity by Weighted least squares: Heteroskedasticity is known ● Example: Savings and income Note that this regression model has no intercept ● The transformed model is homoskedastic ● ถา ขอสมมตฐานอนๆ การประมาณแบบจำลองท นน BEST นะครบ ใน Gauss-Markov ยงอย Transform น ดวยวธ OLS ยงถอวา © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. OLS 15

Example results ● Example: Financial wealth equation Net financial wealth Assumed form of heteroskedasticity �� WLS �� se(B) �� OLS Participation in 401 K pension plan © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 16

Important special case of heteroskedasticity • �� : �� / �� weight �� / �� (m i) Average contribution to pension plan in firm i Average earnings and age in firm i Percentage firm contributes to plan Heteroskedastic error term Error variance if errors are homoskedastic at the individual-level If errors are homoskedastic at the individual-level, WLS with weights equal to firm size mi should be used. If the assumption of homoskedasticity at the individual-level is not exactly right, one can calculate robust standard errors after WLS (i. e. for the transformed model). © 2016 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a ® license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 17

Solving Heteroskedasticity by Feasible least squares: Heteroskedasticity is unknown ● When heteroskedasticity function is unknown, we use OLS. Feasible GLS �� Step 1 : Run linear regression �� expo function �� weight �� weight �� Multiplicative error (�� X �� ) © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 18

Solving Heteroskedasticity by Feasible least squares: Heteroskedasticity is unknown Step 4: Run Transformed model using OLS © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 19

Solving Heteroskedasticity by Feasible least squares: Heteroskedasticity is unknown ● Example: Demand for cigarettes ● Estimation by OLS Cigarettes smoked per day Smoking restrictions in restaurants Logged income and cigarette price Reject homoskedasticity © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 20

Solving Heteroskedasticity by Feasible least squares: Heteroskedasticity is unknown ● Estimation by FGLS Now statistically significant ● Discussion • �� log(income) �� © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 21

Problem of Feasible least squares ● จะเปนอยางไรถาเรากำหนด heteroskedasticity function ผด ? • �� heteroskedasticity function �� , WLS �� consistent �� MLR. 1 – MLR. 4, �� robust standard errors �� • WLS �� consistent �� MLR. 4 �� • �� OLS �� WLS �� 4 �� • �� heteroskedasticity function �� WLS �� OLS © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 22

GLS in the linear probability model ● GLS in the linear probability model (LPM) : Logit and Probit �� Heteroscedasticity function �� Logit �� Probit �� Weighted MLE �� ● Discussion • �� Y �� 0 -1 �� • �� Robust MLE estimation © 2016 Cengage Learning ®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use. 23