1 / 43

Multivariate Probit

Multivariate Probit. An Analysis of Access to Amenities in Delhi’s Slums. Example: Coalmining and Respiratory Symptoms. Ashford and Sowden, Biometrics , 1970 Is there a relationship? Best model? Standard approach: two probit equations Wheezing and years in mine (age)

Download Presentation

Multivariate Probit

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Multivariate Probit An Analysis of Access to Amenities in Delhi’s Slums

  2. Example: Coalmining and Respiratory Symptoms • Ashford and Sowden, Biometrics, 1970 • Is there a relationship? • Best model? • Standard approach: two probit equations • Wheezing and years in mine (age) • Breathlessness and years in mine (age) • Does this approach overlook anything?

  3. Coalmining and Respiratory Symptom • Each physiological system has a certain tolerance: tolerance vector • Ashford and Sowden: ignoring important information if you estimate equations separately • Model in spirit of Seemingly Unrelated Regression (SUR)

  4. Multivariate Probit (MVP) • Extension of univariate probit (UVP) model • Allows for: • Simultaneous estimation of multiple probit equations • Correlated disturbances across equations

  5. MVP vs. UVP How is it better? • Does not ignore information across equations • Better prediction of conditional and joint probabilities • More consistent estimation • More efficient estimation

  6. Why MVP? • Access to amenities in Delhi slums • Correlation between access to sanitation services and access to drainage for a given household • Separate UVP estimation would ignore this

  7. Where we’re going… • Univariate Probit • Bivariate Probit (i = 2) • Multivariate Probit (i = T) • Delhi Slum Dwellers’ Access to Amenities • Published Applications and Extensions

  8. Univariate Probit Dichotomous dependent variable • Takes on a value of either 0 or 1 • Estimate with OLS?

  9. Linear Probability Model

  10. Linear Probability Model Shortcomings • Cannot constrain probabilities to the 0-1 interval • Negative variances • Heteroscedasticity of ε that depends on X • Logically not attractive Solutions?

  11. Normal Cumulative Distribution Function Properties • Bounded by 0 and 1 • Nonlinear relationship between P and X

  12. Univariate Probit Model y*: latent dependent variable Y: binary dependent variable x: vector of explanatory variables t: standardized normal variable φ: normal pdf β: measures impact of changes in x Φ: normal cdf

  13. UVP Example Y = 1: individual purchased refrigerator in last year Y = 0: individual has not purchased refrigerator in last year X = individual’s income per annum

  14. UVP: Mechanics • Group the data by RHS variable (income) • Calculate Phati for each i grouping of income • For each Phati, use the standard normal cdf to find Ii • Add 5 to each Ii • Use OLS to estimate β1, β2 in:

  15. Two probit equations Y1 = 1: individual purchased refrigerator in last year, 0 otherwise Y2 = 1: individual purchased dishwasher in last year, 0 otherwise X = individual’s income per annum

  16. Bivariate Probit , y 1 = 1 if , 0 otherwise , y 2 = 1 if , 0 otherwise

  17. BVP: Estimation • Maximum Likelihood • Bivariate Normal cdf: where φ2 represents the bivariate normal pdf and:

  18. BVP: Estimation • Probabilities that enter the likelihood function: Φ2: bivariate normal cdf

  19. BVP: Estimation • Function to be maximized:

  20. BVP: Estimation in Practice Simulated Maximum Likelihood • Markov chain Monte Carlo • GHK simulator • Geweke-Hajivassiliou-Keane smooth recursive conditioning simulator • Greene (2003) discusses this in Appendix E • Cappellari and Jenkins (2003)

  21. BVP: Is it Necessary? • H0: ρ = 0 (estimate independent probit equations separately) • Test statistic (Kiefer 1982): LM ~ χ2 with d.f. = (T)(T-1)/2 where T = # of equations

  22. BVP: More Test Statistics z-statistic: Likelihood ratio: j = number of restrictions

  23. BVP: Properties of the Estimator • Considers unobservable heterogeneity • Random components of one equation are allowed to be freely correlated with the random components of the other • Takes into account unobservable characteristics that might affect both dependent variables • More efficient and consistent than separate ML estimation of UVP models • UVP does not account for the correlation between error terms: assumes exogeneity of dep var covariates, so does not give consistent estimates of parameters (Maddala 1983)

  24. BVP: Measure of Goodness of Fit McFadden’s likelihood ratio index (LRI): • lnL: maximized value of log-likelihood function for specification at hand • lnL0: maximized value of log-likelihood function calculated with only a constant term • Bounded by 0, 1, increases as fit improves

  25. Multivariate Probit yi= 1 if yi* > 0, 0 otherwise, i = 1,…,T and ρii = 1, ρij= ρji for i, j = 1, …, T where

  26. MVP Application: Delhi Slums Delhi slum dwellers’ access to amenities

  27. Delhi Slums Tiebout sorting (Charles Tiebout 1956) • Individuals sort themselves into communities based on preferences of provisions of public goods • Assumptions • Unlimited mobility • Unrestricted number of communities • Implication: Heterogeneous preferences

  28. Delhi Slums Heterogeneity in community composition • Impact on economic outcomes • Reduced participation to secure community grants in US (Vigdor 2004) • Decreased maintenance of infrastructure projects in Pakistan (Khwaja 2001) • Less spending on education, sewers, roads in US (Alesina et al 1999) • Slower growth in Sub-Saharan Africa (Easterly and Levine 2003) • Channels?

  29. Delhi Slums: Model (Alesina et al 1999) Model: • g*: amount of public good provided in equilibrium • : median distance from the type of public good most preferred by the median voter • α: parameter from individual’s utility function (0<α<1) • Punchline: g* and are inversely related:

  30. Delhi Slums: H0 and H1 H0: Public goods provision is not affected by the degree to which preferences are polarized H1:Public goods provision is negatively affected by polarization of preferences

  31. Delhi Slums: Data

  32. Delhi Slums: Public Goods Provision of public goods is a latent variable • Proxy with access to public goods • Medical facilities (MED) • Sanitation services (SAN) • Drainage (DRA)

  33. Delhi Slums: Fractionalization Proxies for Fractionalization • Religion • Hindu • Muslim • Sikh • Caste • Backward castes and tribes • Scheduled castes and tribes • General Hindu • Muslim, Sikh, other

  34. Delhi Slums: Econometric Model • : amount of public good a (latent) accessible by slum dweller i • Map to the observed realizations: 1 represents access, 0 otherwise if , 0 otherwise • Lose information

  35. Delhi Slums: Econometric Model Assumption: where UVP:

  36. Delhi Slums: Econometric Model X vector: • Religious fractionalization (frd) • Caste fractionalization (fcd) • Per capita household income (pcinc) • Education dummies (edu1, edu2, edu3) • Mean-to-median income ratio (mminrat) • Poverty dummy (poor) • Political participation dummy (political) • Years in community (yrincomm) Proxies for lhatm

  37. Delhi Slums: MVP vs. UVP Is MVP necessary? • H0: ρMS = ρMD = ρSD = 0? • Stata reports the LR test statistic = 37.78 ~ χ2(3), so reject H0 • Yes, MVP is an improvement on UVP • Not ignore information contained in covariance matrix • goodness of fit:

  38. Future Research Dependent variable • More direct measure for spending Panel data • Changes in income v changes in fractionalization Semiparametric and nonparametric techniques • Horowitz and Savin (2001) • Single-index modeling • Median regression approach

  39. MVP in Practice Ashford and Sowden (1970) Zhao, X. and M. Harris (2004) “Demand for Marijuana, Alcohol and Tobacco: Participation, Levels of Consumption and Cross-equation Correlations,” The Economic Record, 80(251): 394-410. Greene, W. (1998) “Gender Economics Courses in Liberal Arts Colleges: Further Results,” The Journal of Economic Education, 29(4): 291-300. Christofides, L., T. Stengos, and R. Swidinsky (1997) “Welfare Participation and Labour Market Behavior in Canada,” The Canadian Journal of Economics, 30(3): 595-621.

  40. References Alesina, A., R. Baqir, and W. Easterly (1999) “Public Goods and Ethnic Divisions”, The Quarterly Journal of Economics, 114(4): 1243-1284. Ashford, J.R. and R.R. Sowden (1970) “Multi-variate Probit Analysis,” Biometrics, September: 535-546. Cappellari, L. and S.P. Jenkins (2003) “Multivariate probit regression using simulated maximum likelihood,” Stata Journal, 3(3): 221-235. Easterly, W., and R. Levine (1997) “Africa’s Growth Tragedy: Policies and Ethnic Divisions,” Quarterly Journal of Economics, 112(4), 1203-1250. Greene, W. (2003) Econometric Analysis (Fifth Edition), Delhi: Pearson Education.

  41. References (cont.) Horowitz, J. and N.E. Savin (2001) “Binary Response Models: Logits, Probits and Semiparametrics,” Journal of Economic Perspectives, 15(4): 43-56. Kiefer, N. (1982) “Testing for Dependence in Multivariate Probit Models,” Biometrika, 69(1): 161-166 Khwaja, A.I. (2001) “Can Good Projects Succeed in Bad Communities? Collective Action in the Himalayas,” John F. Kennedy School of Government Faculty Research Working Paper Series RWP01-043. URL: http://ssrn.com/abstract=295571 Maddala, G.S. (1983) Limited Dependent Variables in Econometrics, Cambridge: Cambridge University Press.

  42. References (cont.) Tiebout, C. (1956) “A Pure Theory of Local Expenditures,” Journal of Political Economy, 64(5): 416-424. Vigdor, J. (2004) “Community Composition and Collective Action: Analyzing Initial Mail Response to the 2000 Census,” The Review of Economics and Statistics. 86(1): 303-312.

More Related