1. Econometric methods of analysis and forecasting of financial markets Lecture 6. Models of restricted dependent variables

2. This lecture helps to understand: Different types of restricted dependent variables Censored and truncated dependent variables Appropriate model choice Probit and logit estimations Heckman 2-step procedure Tobit model Probit model in estimation of default probability

3. Contents Binary dependent variable Linear probability model Limited dependent variables Logit and Probit Some examples 2-step Heckman procedure Censored and truncated dependent variables Tobit Probit model in estimation of default probability

4. Binary dependent variable

5. Linear probability model

6. Limited dependent variables General case of binary dependent variable is limited dependent variable, when its range of values is somehow restricted: only positive (negative) values Count variables

7. Some examples some stocks pay dividends and others do not which factors affect whether countries default on their sovereign debt some firms are engaged in stock splits and others do not

8. Logit and Probit

9. The Logit model

10. The Probit model

11. Logit and Probit estimation

12. Probit or logit?

13. Heckman 2-step procedure Multinomial linear dependent variables: more alternatives for dependent variable (for example, investor choice between different mutual funds or credit ratings). In case of credit ratings, dependent variable is ordered. First case: multinomial logit/probit, second case: ordered logit/probit. In case when firms would have received lower credit ratings (because of weak financials) they elect not to solicit a rating => sample selection bias => inconsistent coefficients => 2-step Heckman procedure

14. Heckman 2-step procedure

15. Censored and Truncated Variables the range of dependent variables values is limited for some reason (not necessary dummies) Example: charitable donations by individuals When many values of dependent variable are zero, OLS gives biased and inconsistent estimates => use maximum likelihood method Censored: dependent variable is censored so that values above or below this point are not observable. Truncated: observations are missing when dependent variable is above or below some point.

16. Tobit model

17. Probit model in estimation of default probability Default modelling helps to identify factors that affect the ability of firms to pay back borrowed money. This ability is called the financial health of the firm. The model:  Find out firm determinants of financial health. Take into account ratios of the firm: profitability ratios, liquidity ratios, leverage ratios...  Identify the weights given to these factors to construct a single measure of financial health of the firm.  This single measure of financial health is then turned into a probability of default.

18. Conclusions We’ve covered different types of restricted dependent variables, censored and truncated dependent variables We’ve learned how to make appropriate model choice How to perform probit and logit estimations What is Heckman 2-step procedure what is Tobit model How to use probit model in estimation of default probability

19. References Brooks C. Introductory Econometrics for Finance. Cambridge University Press. 2008. Cuthbertson K., Nitzsche D. Quantitative Financial Economics. Wiley. 2004. Tsay R.S. Analysis of Financial Time Series, Wiley, 2005. Y. Ait-Sahalia, L. P. Hansen. Handbook of Financial Econometrics: Tools and Techniques. Vol. 1, 1st Edition. 2010. Alexander C. Market Models: A Guide to Financial Data Analysis. Wiley. 2001. Cameron A. and Trivedi P.. Microeconometrics. Methods and Applications. 2005. Lai T. L., Xing H. Statistical Models and Methods for Financial Markets. Springer. 2008. Poon S-H. A practical guide for forecasting financial market volatility. Wiley, 2005. Rachev S.T. et al. Financial Econometrics: From Basics to Advanced Modeling Techniques, Wiley, 2007.