multivariate probit
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
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Multivariate Probit
An Analysis of Access to Amenities in Delhi’s Slums
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Example: Coalmining and Respiratory Symptoms Ashford and Sowden, Biometrics, 1970 Is there a relationship? Best model? Standard approach: two probit equations
1. Wheezing and years in mine (age)
2. Breathlessness and years in mine (age)
3. Does this approach overlook anything?
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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)
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Multivariate Probit (MVP)
Extension of univariate probit (UVP) model Allows for:
1. Simultaneous estimation of multiple probit equations
2. Correlated disturbances across equations
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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
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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
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Where we’re going…
I. Univariate Probit
II. Bivariate Probit (i = 2)
III. Multivariate Probit (i = T)
IV. Delhi Slum Dwellers’ Access to Amenities
V. Published Applications and Extensions
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Univariate Probit
Dichotomous dependent variable Takes on a value of either 0 or 1
Estimate with OLS?
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Linear Probability Model
),()|1Pr( xx FY
),(1)|0Pr( xx FY
'),( xx F
'xY
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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?
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Normal Cumulative Distribution FunctionProperties Bounded by 0 and 1 Nonlinear relationship between P and X
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Univariate Probit Model
)'()()|1Pr()|0*Pr('
xxxx
dttYy
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
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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
)()|1Pr( XXY
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UVP: Mechanics
1. Group the data by RHS variable (income)
2. Calculate Phati for each i grouping of income
3. For each Phati, use the standard normal cdf to find Ii
4. Add 5 to each Ii
5. Use OLS to estimate β1, β2 in:
iii uXI 21
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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
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Bivariate Probit
0*1 y
, y 1 = 1 if , 0 otherwise
22'2
*2 xy , y 2 = 1 if , 0 otherwise
0],|[],|[ 212211 xxxx EE
1],|[],|[ 212211 xxxx VarVar
0,],|,[ 2121 xxCov
0*1 y11
'1
*1 xy
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BVP: Estimation Maximum Likelihood Bivariate Normal cdf:
212122211 ,),,(),Pr( dzdzzzxXxX
where φ2 represents the bivariate normal pdf and:
2/12
)1/()2)(2/1(
212 )1(2),,(
22
22
21
xxxxe
xx
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BVP: Estimation
Probabilities that enter the likelihood function:
21* iiiqq
),,(),|,Pr( *212212211 iiiii wwyYyY xx
Φ2: bivariate normal cdf
2,1,' jqw jijijij x12,12 2211 iiii yqyq
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BVP: Estimation
Function to be maximized:
n
iiii wwL
1212 ),,(lnlog *
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BVP: Estimation in Practice
Simulated Maximum Likelihood
1. Markov chain Monte Carlo
2. GHK simulator Geweke-Hajivassiliou-Keane smooth recursive
conditioning simulator Greene (2003) discusses this in Appendix E Cappellari and Jenkins (2003)
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BVP: Is it Necessary? H0: ρ = 0 (estimate independent probit
equations separately) Test statistic (Kiefer 1982):
n
iiiii
ii
n
iii
iiii
wwww
ww
ww
wwqq
LM
12211
221
2
121
2121
)()()()(
)()(
)()(
)()(
LM ~ χ2 with d.f. = (T)(T-1)/2 where T = # of equations
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BVP: More Test Statistics
z-statistic:
Likelihood ratio:
j = number of restrictions
)(~ˆ 2
ˆ
js
z
)(~)ˆlnˆ(ln2 2 jLLLR UR
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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)
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BVP: Measure of Goodness of FitMcFadden’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
0ln/ln1 LLLRI
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Multivariate Probit
iiiy '* x
yi = 1 if yi* > 0, 0 otherwise, i = 1,…,T
,0~',,1 MVNT
where
TTT
T
1
111
and ρii = 1, ρij = ρji
for i, j = 1, …, T
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MVP Application: Delhi SlumsDelhi slum dwellers’ access to amenities
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Delhi Slums
Tiebout sorting (Charles Tiebout 1956) Individuals sort themselves into
communities based on preferences of provisions of public goods
Assumptions1. Unlimited mobility
2. Unrestricted number of communities Implication: Heterogeneous preferences
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Delhi Slums
Heterogeneity in community composition Impact on economic outcomes
1. Reduced participation to secure community grants in US (Vigdor 2004)
2. Decreased maintenance of infrastructure projects in Pakistan (Khwaja 2001)
3. Less spending on education, sewers, roads in US (Alesina et al 1999)
4. Slower growth in Sub-Saharan Africa (Easterly and Levine 2003)
Channels?
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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:
)1/(1* ˆ1
milg
mil̂
mil̂
1/
*ˆ1
1ˆm
imi
ll
g
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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
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Delhi Slums: Data
Variable name Mean Median Min. Max. Std. dev.
Religious fractionalization 0.22 0.24 0 0.43 0.11Caste fractionalization 0.68 0.69 0.44 0.73 0.04Access to medical facility (DV) 0.55 1 0 1 0.50Access to sanitation services (DV) 0.70 1 0 1 0.46Access to drainage system (DV) 0.79 1 0 1 0.41Household PC income (monthly in Rupees) 841 686 186 5947 616Participation in political process (DV) 0.78 1 0 1 0.42Years in community of head of HH 16 16 0 84 0.42Mean to median income ratio 1.14 1.14 0.98 1.24 0.08
TABLE ISummary Statistics of Slum Data
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Delhi Slums: Public Goods
Provision of public goods is a latent variable Proxy with access to public goods
1. Medical facilities (MED)
2. Sanitation services (SAN)
3. Drainage (DRA)
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Delhi Slums: FractionalizationProxies for Fractionalization Religion
1. Hindu
2. Muslim
3. Sikh Caste
1. Backward castes and tribes
2. Scheduled castes and tribes
3. General Hindu
4. Muslim, Sikh, other
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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
aaay '* X DSMa ,,
1aiy 0* aiy
*aiy
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Delhi Slums: Econometric ModelAssumption:
where
UVP:
,0~',, MVNDSM
1
1
1
SDMD
SDMS
MDMS
0 SDMDMS
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Delhi Slums: Econometric ModelX 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
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RHS variables MVP UVP MVP UVP MVP UVP
REL -1.795 -1.712 -5.637 -5.03 -1.596 -1.464(1.38) (1.28) (3.45)** (3.11)** (1.15) (0.96)
CASTE -5.615 -5.546 16.345 15.702 9.837 9.104(1.87) (1.80) (3.49)** (3.36)** (3.36)** (2.81)**
PCINC 0 0 0 0 0 0(0.34) (0.67) (0.11) (0.16) (0.33) (0.20)
EDU1 0.206 0.238 0.225 0.278 0.186 0.158(0.89) (1.02) (0.97) (1.19) (0.81) (0.65)
EDU2 -0.044 -0.034 0.835 0.891 0.818 0.707(0.15) (0.12) (2.38)* (2.50)* (2.19)* (1.86)
EDU3 0.221 0.256 0.273 0.2 3.66(0.43) (0.49) (0.51) (0.37) -0.03
MMINRAT 4.487 4.545 2.558 2.456 0.297 0.487(3.47)** (3.47)** (1.65) (1.57) (0.20) (0.31)
POOR -0.246 -0.241 0.322 0.336 0.563 0.453(1.13) (1.09) (1.34) (1.36) (2.10)* (1.65)
POLITICAL 0.537 0.532 0.258 0.219 0.186 0.317(2.30)* (2.26)* (1.10) (0.93) (0.77) (1.27)
YRINCOMM 0.011 0.016 0.01 0.011 0.004 -0.001(1.04) (1.41) (1.01) (1.07) (0.35) (0.07)
Constant -1.391 -1.69 -13.068 -12.607 -6.416 -6.167(0.55) (0.64) (3.19)** (3.09)** (2.37)* (2.09)*
Coeff. z-stat.ρ MS 0.42 (3.81)**ρ MD -0.25 (1.94)ρ SD 0.52 (4.92)**
Absolute value of z-statistics in parentheses* significant at 5%; ** significant at 1%tedu3 dropped b/c it predicted success perfectly
Correlation Coefficients
TABLE II
n/at
Medical Facilities Sanitation Services Drainage
Dependent Variables are Access to the Specified Amenity
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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: 137.033.350
32.3021
LRI
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Future Research
Dependent variable More direct measure for spendingPanel data Changes in income v changes in
fractionalizationSemiparametric and nonparametric techniques Horowitz and Savin (2001)
1. Single-index modeling2. Median regression approach
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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.
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ReferencesAlesina, 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.
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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.
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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.