using copulas to deal with endogeneity - an application to

IntroductionMethodology

ResultsConclusionReferences

Using copulas to deal with endogeneityAn application to development economics

Sanne Blauw and Philip Hans Franses

Summer School in Development EconomicsAlba di Canazei, July 16 2013

Sanne Blauw and Philip Hans Franses Using copulas to deal with endogeneity



Overview

Introduction

MethodologyEndogeneityCopulasEstimation

ResultsSimulationsTelephone use in Uganda

Conclusion




Motivation

I Endogeneity is a common problem in (development)economics

I Simultaneous causalityI Omitted variablesI Measurement error

I E.g. the impact of telephone use on development

I We need exogenous and relevant instruments

I What if we cannot find exogenous instruments?




EndogeneityCopulasEstimation

Endogeneity

I Regression modely = Xβ + ε (1)

I Endogeneity impliesE (X ′ε) 6= 0 (2)

β is biased and inconsistent.

Key idea: Let’s model the correlation between X and ε anduse this information in the likelihood function to obtainconsistent estimates.





Copulas

I Park, S. and Gupta, S. (2012). Handling endogenousregressors by joint estimation using copulas. MarketingScience, 31(4):567-586.

I Use copulas to model correlation between X and ε.

I A copula is a function that maps multiple CDFs to their jointCDF.

Sklar’s theorem





I Regression model

y = x1β1 + X2β2 + ε, (3)

where x1 is endogenous and X2 is exogenous.

I Joint CDF (using Gaussian copula)

G (x1, ε) = N(Φ−1(Fx(x1)),Φ−1(Fε(ε))), (4)

where Φ denotes the standard normal CDF, N is the bivariatestandard normal CDF with correlation coefficient ρ, Fx and Fε

are the marginal CDFs of x1 and ε.

I Joint PDF

g(x1, ε) =δδG (x1, ε)

δx1δεfx fε, (5)

where fx and fε are the marginal PDFs of x1 and ε.





Two methods

I Copula method 1: MLEI Assumption: Linearity

I Copula method 2: Including generated regressorx∗1 = Φ−1(Fx(x1)) in OLS Proof

I Assumptions: Linearity, Gaussian copula and ε ∈ N(0, σ2ε )

I We use F̂x , the empirical CDF of x1: x̂∗1 = Φ−1(F̂x(x1)).




SimulationsTelephone use in Uganda

Simulations

Data generating process

Variable Mean Std.Er.true beta -1

beta ols #200 0.692 0.215beta ols #400 0.685 0.151beta ols #1000 0.690 0.096

beta iv #200 -1.031 0.316beta iv #400 -1.005 0.22beta iv #1000 -1.005 0.14

beta copula #200 -0.885 1.015beta copula #400 -0.906 0.711beta copula #1000 -0.968 0.448

Replications 1000





Multicollinearity

I Endogenous and generated regressor can be highly correlated,implying multicollinearity.

I Multicollinearity is an efficiency problem.I Indicators of multicollinearity

I High correlation between endogenous and generated regressorI Joint significance, but separately insignificantI Inflated standard errorsI Variance inflation factor (VIF) > 10

Table 1: Simulation results VIF for copula method

Variable Mean Std. Dev.vif #200 22.622 4.306vif #400 22.42 2.975vif #1000 22.311 1.777Replications 1000





Multicollinearity


I Multicollinearity is an efficiency problem.

I Indicators of multicollinearity








Multicollinearity


I Multicollinearity is an efficiency problem.I Indicators of multicollinearity








Empirical data

I What is the impact of telephone use on economicdevelopment of households? Literature

I Data from Uganda (N=196), collected March-April 2010I Economic development: Progress out of Poverty Index (PPI)

Scorecard

I Telephone use

I Proportion of mobile phone users in householdI Log years mobile phone ownership (HoH)I Log mobile phone calls per week (HoH)I Log public phone calls per week (HoH)

I Estimation

I Heteroskedasticity robust standard errorsI Nonparametric bootstrap for standard errors of generated

regressor





Empirical data



Scorecard

I Telephone useI Proportion of mobile phone users in householdI Log years mobile phone ownership (HoH)I Log mobile phone calls per week (HoH)I Log public phone calls per week (HoH)

I Estimation

I Heteroskedasticity robust standard errorsI Nonparametric bootstrap for standard errors of generated

regressor





Empirical data



Scorecard

I Telephone useI Proportion of mobile phone users in householdI Log years mobile phone ownership (HoH)I Log mobile phone calls per week (HoH)I Log public phone calls per week (HoH)

I EstimationI Heteroskedasticity robust standard errorsI Nonparametric bootstrap for standard errors of generated

regressor





ols copulaConstant 3.129***

(0.19)Proportion mobile phone users in household 0.843***

(0.13)Education, head of household (years) 0.098**

(0.05)Farmer -0.134**

(0.06)Household size 0.071

(0.06)Area 1 -0.028

(0.07)Area 2 -0.189***

(0.07)Generated regressor 1

Observations 193R-squared 0.354Normality of endogenous variable (p-value)Joint sign. generated and endogenous (p-value)Correlation generated and endogenousVIF





ols copulaConstant 3.129*** 3.293***

(0.19) (0.20)Proportion mobile phone users in household 0.843*** 0.349

(0.13) (0.37)Education, head of household (years) 0.098** 0.091*

(0.05) (0.05)Farmer -0.134** -0.123*

(0.06) (0.06)Household size 0.071 0.046

(0.06) (0.06)Area 1 -0.028 -0.023

(0.07) (0.07)Area 2 -0.189*** -0.175**

(0.07) (0.07)Generated regressor 1 0.132

(0.10)Observations 193 193R-squared 0.354 0.358Normality of endogenous variable (p-value)Joint sign. generated and endogenous (p-value)Correlation generated and endogenousVIF






(0.19) (0.20)Proportion mobile phone users in household 0.843*** 0.349

(0.13) (0.37)Education, head of household (years) 0.098** 0.091*

(0.05) (0.05)Farmer -0.134** -0.123*

(0.06) (0.06)Household size 0.071 0.046

(0.06) (0.06)Area 1 -0.028 -0.023

(0.07) (0.07)Area 2 -0.189*** -0.175**


(0.10)Observations 193 193R-squared 0.354 0.358Normality of endogenous variable (p-value) 0.000Joint sign. generated and endogenous (p-value) 0.000Correlation generated and endogenous 0.958VIF 15.980






(0.16) (0.25)Log years mobile phone ownership 0.181*** 0.456**

(0.04) (0.21)Education, head of household (years) 0.081 0.093*

(0.05) (0.05)Farmer -0.170*** -0.183***

(0.06) (0.06)Household size -0.126** -0.124**

(0.05) (0.05)Area 1 -0.022 -0.011

(0.08) (0.08)Area 2 -0.140* -0.132*

(0.07) (0.07)Generated regressor 1 -0.279







(0.17) (0.37)Log mobile phone calls per week 0.122*** -0.306

(0.04) (0.27)Education, head of household (years) 0.096* 0.098*

(0.05) (0.05)Farmer -0.180*** -0.185***

(0.07) (0.07)Household size -0.134** -0.126**

(0.05) (0.05)Area 1 -0.063 -0.050

(0.08) (0.08)Area 2 -0.143* -0.121








(0.17) (0.18)Log public phone calls per week -0.066 0.271*

(0.06) (0.16)Education, head of household (years) 0.152*** 0.157***

(0.06) (0.06)Farmer -0.239*** -0.227***

(0.06) (0.06)Household size -0.114** -0.122**

(0.05) (0.05)Area 1 -0.101 -0.097

(0.08) (0.08)Area 2 -0.234*** -0.216***

(0.07) (0.07)Generated regressor 1 -0.249**






Summary of results

Significant Multicollinearity

Proportion of mobile phone users in household No Yes

Log years mobile phone ownership Yes Yes

Log mobile phone calls per week No Yes

Log public phone calls per week Yes No

Size of impact




Conclusion

I Exploration of copula method, an instrument-free method tohandle endogeneity.

I Mobile and public phone use has a positive causal effect oneconomic development.

I However, multicollinearity poses problems in some cases.

I This method is not the holy grail. It seems like you have to belucky with the distribution of the endogenous regressor and/orthe size of the impact!




THANK YOU!


Appendix

Chen, S., Schreiner, M., and Woller, G. (2008). Progress out ofPoverty IndexTM: A Simple Poverty Scorecard for Kenya.Technical report, Grameen Foundation.

Park, S. and Gupta, S. (2012). Handling endogenous regressors byjoint estimation using copulas. Marketing Science,31(4):567–586.


Appendix

Copulas

Sklar’s theoremLet H be a joint distribution function with margins F and G. Thenthere exists a copula C such that for all x,y in R,

H(x , y) = C (F (x),G (y)), (6)

Conversely, if C is a copula and F and G are distribution functions,then the function H defined by (6) is a joint distribution functionwith margins F and G.

Jump back


Appendix

Proof copula method 2

Jump back

I x∗1 = Φ−1(Fx(x1))

I ε∗ = Φ−1(Fε(ε))I [x∗1 ε

∗]′ follows bivariate standard normal distribution(Assumption 1: Gaussian copula). Gaussian copula

I (x∗1ε∗

)=

(1 0

ρ√

1− ρ2

)(ν1

ν2

), (7)

where ν1 and ν2 are independent random variables drawnfrom a standard normal distribution.

I Or:ε∗ = ρν1 +

√1− ρ2ν2 = ρx∗1 +

√1− ρ2ν2. (8)


Appendix


Jump back

I x∗1 = Φ−1(Fx(x1))I ε∗ = Φ−1(Fε(ε))

I [x∗1 ε∗]′ follows bivariate standard normal distribution

(Assumption 1: Gaussian copula). Gaussian copula

I (x∗1ε∗

)=

(1 0

ρ√

1− ρ2

)(ν1

ν2

), (7)



√1− ρ2ν2 = ρx∗1 +

√1− ρ2ν2. (8)


Appendix


Jump back

I x∗1 = Φ−1(Fx(x1))I ε∗ = Φ−1(Fε(ε))I [x∗1 ε

∗]′ follows bivariate standard normal distribution(Assumption 1: Gaussian copula). Gaussian copula

I (x∗1ε∗

)=

(1 0

ρ√

1− ρ2

)(ν1

ν2

), (7)



√1− ρ2ν2 = ρx∗1 +

√1− ρ2ν2. (8)


Appendix


Jump back

I ε∗ = ρx∗1 +√

1− ρ2ν2.

I Remember: ε∗ = Φ−1(Fε(ε)).

I By Assumption 2 (normally distributed structural error):

ε = F−1ε (Φ(ε∗)) = Φ−1

σ2ε

(Φ(ε∗)) = σεε∗. (9)

I Including (9) in the regression model:

y = x1β1 + X2β2 + ε (10)

= x1β1 + X2β2 + σε(ρx∗1 + (√

1− ρ2)ν2).


Appendix


Jump back

I Structural error: ε = σε(ρx∗1 + (√

1− ρ2)ν2).

I (1) σερx∗1 (correlated with x1)I (2) σε(

√1− ρ2)ν2 (uncorrelated with x1)

I New regression model:

y = x1β1 + X2β2 + σερx∗1 + σε(√

1− ρ2)ν2. (11)

I Key result: New structural error is not correlated with x1.


Appendix


Jump back

I Structural error: ε = σε(ρx∗1 + (√

1− ρ2)ν2).I (1) σερx∗1 (correlated with x1)I (2) σε(

√1− ρ2)ν2 (uncorrelated with x1)

I New regression model:

y = x1β1 + X2β2 + σερx∗1 + σε(√

1− ρ2)ν2. (11)

I Key result: New structural error is not correlated with x1.


Appendix

Data generating process

Jump back

I From Park and Gupta (2012): ε∗

x∗

z∗

∼ N

000

, 1 0.5 0

0.5 1 0.80 0.8 1

(12)

I ε = F−1ε (Φ(ε∗)) = Φ−1(Φ(ε∗)) = ε∗ [error, standard normal]

I x = F−1x (Φ(x∗1 )) = Φ(x∗) [endogenous, uniform]

I z = Φ(z∗) [instrument, uniform]

I Dependent variable

y = β · x + ε = −1 · x + ε (13)


Appendix

Distributions of endogenous regressor

Jump back

Jump to table with parameters


Appendix

Correlation between endogenous and generated regressor

Jump back


Appendix

Table 2: Parameters for simulated distributions

parametersuniform a = 0, b = 1

normal µ = 0, σ2 = 1bimodal P[N(0, 1)] = 0.5, P[N(5, 1)] = 0.5qmodal P[N(0, 1)] = 0.25, P[N(5, 1)] = 0.25, P[N(10, 1)] = 0.25, P[N(15, 1)] = 0.25chi2 df = 2beta1 α = β = 0.5beta2 α = 5, β = 1bernouilli P[X = 0] = 0.5, P[X = 1] = 0.5discrete P[X = 0] = 0.2, P[X = 1] = 0.2, P[X = 2] = 0.2, P[X = 3] = 0.2, P[X = 4] = 0.2poisson λ = 4nbinomial r = 4, p = 0.5

Jump back


Appendix

Literature

I Economic growth (Kathuria et al., 2009; Waverman et al.,2005)

I [M]ay be twice as large in developing countries compared todeveloped countries. (Waverman et al., 2005)

I PricesI 20% reduction in grain prices across Nigerian markets (Aker,

2008)I 5-7% increase in price of onions of farmers in Philippines (Lee

and Bellemere, 2012)I [N]ear-perfect adherence to the Law of One Price in the

South-Indian fisheries sector (Jensen, 2007)I Market participation

I Increase in market participation for farmers in Uganda growingperishable crops in remote areas (Muto and Yamano, 2009)

Jump back


Appendix

Table 3: Economic development scorecard

Question Answer Points

1. How many household members A: 3 or more 0are aged 25 or younger? B: 0, 1 or 2 82. How many household members A: Not all 0aged 6 to 17 are currently attending school? B: All 8

C: No children aged 6 to 17 213 What is the material of the walls of A: Mud/cow dung/grass/sticks 0the house? B: Other 54. What kind of toilet facility does A: Other 0your household use? B: Flush to sewer; flush to septic tank; 2

pan/bucket; covered pit latrine;or ventilation improved pit latrine

5. Does the household own a TV? A: No 0B: Yes 16

6. Does the household own a sofa? A: No 0B: Yes 14

7. Does the household own a stove? A: No 0B: Yes 12

8. Does the household own a radio? A: No 0B: Yes 8

9. Does the household own a bicycle? A: No 0B: Yes 5

10. How many head of cattle are A: None or unknown 0owned by the household currently? B: 1 or more 9

Note: The scorecard is a reproduction of the scorecard in Chen et al. (2008).

Jump backSanne Blauw and Philip Hans Franses Using copulas to deal with endogeneity

Appendix

Table 4: Descriptive statistics of economic development for the fullsample and for the three geographic areas

Total Area 1 Area 2 Area 3

Mean 37.14 40.38 31.69 40.92Median 37.00 37.00 27.00 42.00Maximum 86.00 86.00 67.00 67.00Minimum 9.00 9.00 10.00 10.00Std. Dev. 15.87 17.83 14.40 13.96Observations 196 56 77 63

Average poverty likelihood (%) 35.98 35.62 44.61 35.13County poverty level (%) 32.31 22.60 29.00 45.00

Jump back


Appendix

Jump backSanne Blauw and Philip Hans Franses Using copulas to deal with endogeneity

Appendix

Size of impact of years of ownership

Jump back

Roughly...

I Increasing the time of ownership from 2.5 (the average) to 3.5years (40%) increases the PPI score with 20% (coefficient is0.5).

I On an average PPI of 37, this means an increase from 37 to44.

I This corresponds with a drop in poverty likelihood of 2.6%(from 35.4% to 32.8%, see Chen et al., 2008).


using copulas to deal with endogeneity - an application to

Documents