Using copulas to deal with endogeneity

Size: px

Start display at page:

Download "Using copulas to deal with endogeneity"

Darlene Fitzgerald
5 years ago
Views:

1 An application to development economics Summer School in Development Economics Alba di Canazei, July

2 Overview Endogeneity Copulas Estimation Simulations Telephone use in Uganda

4 Motivation Endogeneity is a common problem in (development) economics Simultaneous causality Omitted variables Measurement error E.g. the impact of telephone use on development

5 Motivation Endogeneity is a common problem in (development) economics Simultaneous causality Omitted variables Measurement error E.g. the impact of telephone use on development We need exogenous and relevant instruments

6 Motivation Endogeneity is a common problem in (development) economics Simultaneous causality Omitted variables Measurement error E.g. the impact of telephone use on development We need exogenous and relevant instruments What if we cannot find exogenous instruments?

7 Endogeneity Endogeneity Copulas Estimation Regression model y = X β + ɛ (1)

8 Endogeneity Endogeneity Copulas Estimation Regression model y = X β + ɛ (1) Endogeneity implies E(X ɛ) 0 (2) β is biased and inconsistent.

9 Endogeneity Endogeneity Copulas Estimation Regression model y = X β + ɛ (1) Endogeneity implies E(X ɛ) 0 (2) β is biased and inconsistent. Key idea: Let s model the correlation between X and ɛ and use this information in the likelihood function to obtain consistent estimates.

10 Copulas Endogeneity Copulas Estimation Park, S. and Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4): Sklar s theorem

11 Endogeneity Copulas Estimation Copulas Park, S. and Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4): Use copulas to model correlation between X and ɛ. Sklar s theorem

12 Endogeneity Copulas Estimation Copulas Park, S. and Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4): Use copulas to model correlation between X and ɛ. A copula is a function that maps multiple CDFs to their joint CDF. Sklar s theorem

13 Endogeneity Copulas Estimation

14 Endogeneity Copulas Estimation

15 Endogeneity Copulas Estimation

16 Endogeneity Copulas Estimation Regression model y = x 1 β 1 + X 2 β 2 + ɛ, (3) where x 1 is endogenous and X 2 is exogenous.

17 Endogeneity Copulas Estimation Regression model y = x 1 β 1 + X 2 β 2 + ɛ, (3) where x 1 is endogenous and X 2 is exogenous. Joint CDF (using Gaussian copula) G(x 1, ɛ) = N(Φ 1 (F x (x 1 )), Φ 1 (F ɛ (ɛ))), (4) where Φ denotes the standard normal CDF, N is the bivariate standard normal CDF with correlation coefficient ρ, F x and F ɛ are the marginal CDFs of x 1 and ɛ.

18 Endogeneity Copulas Estimation Regression model y = x 1 β 1 + X 2 β 2 + ɛ, (3) where x 1 is endogenous and X 2 is exogenous. Joint CDF (using Gaussian copula) G(x 1, ɛ) = N(Φ 1 (F x (x 1 )), Φ 1 (F ɛ (ɛ))), (4) where Φ denotes the standard normal CDF, N is the bivariate standard normal CDF with correlation coefficient ρ, F x and F ɛ are the marginal CDFs of x 1 and ɛ. Joint PDF g(x 1, ɛ) = δδg(x 1, ɛ) f x f ɛ, (5) δx 1 δɛ where f x and f ɛ are the marginal PDFs of x 1 and ɛ.

19 Two methods Endogeneity Copulas Estimation Copula method 1: MLE Assumption: Linearity

20 Two methods Endogeneity Copulas Estimation Copula method 1: MLE Assumption: Linearity Copula method 2: Including generated regressor x1 = Φ 1 (F x (x 1 )) in OLS Proof

21 Two methods Endogeneity Copulas Estimation Copula method 1: MLE Assumption: Linearity Copula method 2: Including generated regressor x1 = Φ 1 (F x (x 1 )) in OLS Proof Assumptions: Linearity, Gaussian copula and ɛ N(0, σ 2 ɛ )

22 Two methods Endogeneity Copulas Estimation Copula method 1: MLE Assumption: Linearity Copula method 2: Including generated regressor x1 = Φ 1 (F x (x 1 )) in OLS Proof Assumptions: Linearity, Gaussian copula and ɛ N(0, σ 2 ɛ ) We use ˆF x, the empirical CDF of x 1 : xˆ 1 = Φ 1 ( ˆF x (x 1 )).

23 Simulations Simulations Telephone use in Uganda Data generating process Variable Mean Std.Er. true beta -1 beta ols # beta ols # beta ols # beta iv # beta iv # beta iv # beta copula # beta copula # beta copula # Replications 1000

24 Multicollinearity Simulations Telephone use in Uganda Endogenous and generated regressor can be highly correlated, implying multicollinearity.

25 Multicollinearity Simulations Telephone use in Uganda Endogenous and generated regressor can be highly correlated, implying multicollinearity. Multicollinearity is an efficiency problem.

26 Multicollinearity Simulations Telephone use in Uganda Endogenous and generated regressor can be highly correlated, implying multicollinearity. Multicollinearity is an efficiency problem. Indicators of multicollinearity High correlation between endogenous and generated regressor Joint significance, but separately insignificant Inflated standard errors Variance inflation factor (VIF) > 10

27 Multicollinearity Simulations Telephone use in Uganda Endogenous and generated regressor can be highly correlated, implying multicollinearity. Multicollinearity is an efficiency problem. Indicators of multicollinearity High correlation between endogenous and generated regressor Joint significance, but separately insignificant Inflated standard errors Variance inflation factor (VIF) > 10 Table 1: Simulation results VIF for copula method Variable Mean Std. Dev. vif # vif # vif # Replications 1000

28 Empirical data Simulations Telephone use in Uganda What is the impact of telephone use on economic development of households? Literature

29 Empirical data Simulations Telephone use in Uganda What is the impact of telephone use on economic development of households? Literature Data from Uganda (N=196), collected March-April 2010 Economic development: Progress out of Poverty Index (PPI) Scorecard Telephone use Proportion of mobile phone users in household Log years mobile phone ownership (HoH) Log mobile phone calls per week (HoH) Log public phone calls per week (HoH)

30 Empirical data Simulations Telephone use in Uganda What is the impact of telephone use on economic development of households? Literature Data from Uganda (N=196), collected March-April 2010 Economic development: Progress out of Poverty Index (PPI) Scorecard Telephone use Proportion of mobile phone users in household Log years mobile phone ownership (HoH) Log mobile phone calls per week (HoH) Log public phone calls per week (HoH) Estimation Heteroskedasticity robust standard errors Nonparametric bootstrap for standard errors of generated regressor

31 Simulations Telephone use in Uganda ols Constant 3.129*** (0.19) Proportion mobile phone users in household 0.843*** (0.13) Education, head of household (years) 0.098** (0.05) Farmer ** (0.06) Household size (0.06) Area (0.07) Area *** (0.07) Generated regressor 1 copula Observations 193 R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF

32 Simulations Telephone use in Uganda ols copula Constant 3.129*** 3.293*** (0.19) (0.20) Proportion mobile phone users in household 0.843*** (0.13) (0.37) Education, head of household (years) 0.098** 0.091* (0.05) (0.05) Farmer ** * (0.06) (0.06) Household size (0.06) (0.06) Area (0.07) (0.07) Area *** ** (0.07) (0.07) Generated regressor (0.10) Observations R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF

33 Simulations Telephone use in Uganda ols copula Constant 3.129*** 3.293*** (0.19) (0.20) Proportion mobile phone users in household 0.843*** (0.13) (0.37) Education, head of household (years) 0.098** 0.091* (0.05) (0.05) Farmer ** * (0.06) (0.06) Household size (0.06) (0.06) Area (0.07) (0.07) Area *** ** (0.07) (0.07) Generated regressor (0.10) Observations R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF

34 Simulations Telephone use in Uganda ols copula Constant 3.555*** 3.296*** (0.16) (0.25) Log years mobile phone ownership 0.181*** 0.456** (0.04) (0.21) Education, head of household (years) * (0.05) (0.05) Farmer *** *** (0.06) (0.06) Household size ** ** (0.05) (0.05) Area (0.08) (0.08) Area * * (0.07) (0.07) Generated regressor (0.20) Observations R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF

35 Simulations Telephone use in Uganda ols copula Constant 3.568*** 4.064*** (0.17) (0.37) Log mobile phone calls per week 0.122*** (0.04) (0.27) Education, head of household (years) 0.096* 0.098* (0.05) (0.05) Farmer *** *** (0.07) (0.07) Household size ** ** (0.05) (0.05) Area (0.08) (0.08) Area * (0.08) (0.08) Generated regressor (0.35) Observations R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF

36 Simulations Telephone use in Uganda ols copula Constant 3.660*** 3.552*** (0.17) (0.18) Log public phone calls per week * (0.06) (0.16) Education, head of household (years) 0.152*** 0.157*** (0.06) (0.06) Farmer *** *** (0.06) (0.06) Household size ** ** (0.05) (0.05) Area (0.08) (0.08) Area *** *** (0.07) (0.07) Generated regressor ** (0.11) Observations R-squared Normality of endogenous variable (p-value) Joint sign. generated and endogenous (p-value) Correlation generated and endogenous VIF 8.716

37 Summary of results Simulations Telephone use in Uganda 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

38 Summary of results Simulations Telephone use in Uganda 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

39 Exploration of copula method, an instrument-free method to handle endogeneity. Mobile and public phone use has a positive causal effect on economic development. However, multicollinearity poses problems in some cases. This method is not the holy grail. It seems like you have to be lucky with the distribution of the endogenous regressor and/or the size of the impact!

40 THANK YOU!

41 Appendix Chen, S., Schreiner, M., and Woller, G. (2008). Progress out of Poverty Index TM : A Simple Poverty Scorecard for Kenya. Technical report, Grameen Foundation. Park, S. and Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4):

42 Appendix Copulas Sklar s theorem Let H be a joint distribution function with margins F and G. Then there 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 function with margins F and G. Jump back

43 Appendix Proof copula method 2 Jump back x 1 = Φ 1 (F x (x 1 ))

44 Appendix Proof copula method 2 Jump back x 1 = Φ 1 (F x (x 1 )) ɛ = Φ 1 (F ɛ (ɛ))

45 Appendix Proof copula method 2 Jump back x 1 = Φ 1 (F x (x 1 )) ɛ = Φ 1 (F ɛ (ɛ)) [x1 ɛ ] follows bivariate standard normal distribution (Assumption 1: Gaussian copula). Gaussian copula

46 Appendix Proof copula method 2 Jump back x 1 = Φ 1 (F x (x 1 )) ɛ = Φ 1 (F ɛ (ɛ)) [x1 ɛ ] follows bivariate standard normal distribution (Assumption 1: Gaussian copula). Gaussian copula ( x 1 ɛ ) = ( 1 0 ρ 1 ρ 2 ) ( ν1 ν 2 ), (7) where ν 1 and ν 2 are independent random variables drawn from a standard normal distribution.

47 Appendix Proof copula method 2 Jump back x 1 = Φ 1 (F x (x 1 )) ɛ = Φ 1 (F ɛ (ɛ)) [x1 ɛ ] follows bivariate standard normal distribution (Assumption 1: Gaussian copula). Gaussian copula ( x 1 ɛ ) = ( 1 0 ρ 1 ρ 2 ) ( ν1 ν 2 ), (7) where ν 1 and ν 2 are independent random variables drawn from a standard normal distribution. Or: ɛ = ρν ρ 2 ν 2 = ρx ρ 2 ν 2. (8)

48 Appendix Proof copula method 2 Jump back ɛ = ρx ρ 2 ν 2.

49 Appendix Proof copula method 2 Jump back ɛ = ρx ρ 2 ν 2. Remember: ɛ = Φ 1 (F ɛ (ɛ)).

50 Appendix Proof copula method 2 Jump back ɛ = ρx ρ 2 ν 2. Remember: ɛ = Φ 1 (F ɛ (ɛ)). By Assumption 2 (normally distributed structural error): ɛ = F 1 ɛ (Φ(ɛ )) = Φ 1 (Φ(ɛ )) = σ ɛ ɛ. (9) σ 2 ɛ

51 Appendix Proof copula method 2 Jump back ɛ = ρx ρ 2 ν 2. Remember: ɛ = Φ 1 (F ɛ (ɛ)). By Assumption 2 (normally distributed structural error): ɛ = F 1 ɛ Including (9) in the regression model: (Φ(ɛ )) = Φ 1 (Φ(ɛ )) = σ ɛ ɛ. (9) σ 2 ɛ y = x 1 β 1 + X 2 β 2 + ɛ (10) = x 1 β 1 + X 2 β 2 + σ ɛ (ρx 1 + ( 1 ρ 2 )ν 2 ).

52 Appendix Proof copula method 2 Jump back Structural error: ɛ = σ ɛ (ρx 1 + ( 1 ρ 2 )ν 2 ).

53 Appendix Proof copula method 2 Jump back Structural error: ɛ = σ ɛ (ρx 1 + ( 1 ρ 2 )ν 2 ). (1) σ ɛ ρx 1 (correlated with x 1) (2) σɛ ( 1 ρ 2 )ν 2 (uncorrelated with x 1 )

54 Appendix Proof copula method 2 Jump back Structural error: ɛ = σ ɛ (ρx 1 + ( 1 ρ 2 )ν 2 ). (1) σ ɛ ρx 1 (correlated with x 1) (2) σɛ ( 1 ρ 2 )ν 2 (uncorrelated with x 1 ) New regression model: y = x 1 β 1 + X 2 β 2 + σ ɛ ρx 1 + σ ɛ ( 1 ρ 2 )ν 2. (11)

55 Appendix Proof copula method 2 Jump back Structural error: ɛ = σ ɛ (ρx 1 + ( 1 ρ 2 )ν 2 ). (1) σ ɛ ρx 1 (correlated with x 1) (2) σɛ ( 1 ρ 2 )ν 2 (uncorrelated with x 1 ) New regression model: y = x 1 β 1 + X 2 β 2 + σ ɛ ρx 1 + σ ɛ ( 1 ρ 2 )ν 2. (11) Key result: New structural error is not correlated with x 1.

56 Appendix Data generating process Jump back

57 Appendix Data generating process Jump back From Park and Gupta (2012): ɛ x N 0, z (12)

58 Appendix Data generating process Jump back From Park and Gupta (2012): ɛ x N 0, z (12) ɛ = Fɛ 1 (Φ(ɛ )) = Φ 1 (Φ(ɛ )) = ɛ [error, standard normal] x = Fx 1 (Φ(x1 )) = Φ(x ) [endogenous, uniform] z = Φ(z ) [instrument, uniform]

59 Appendix Data generating process Jump back From Park and Gupta (2012): ɛ x N 0, z (12) ɛ = Fɛ 1 (Φ(ɛ )) = Φ 1 (Φ(ɛ )) = ɛ [error, standard normal] x = Fx 1 (Φ(x1 )) = Φ(x ) [endogenous, uniform] z = Φ(z ) [instrument, uniform] Dependent variable y = β x + ɛ = 1 x + ɛ (13)

60 Appendix Distributions of endogenous regressor Jump back Jump to table with parameters

61 Appendix Correlation between endogenous and generated regressor Jump back

62 Appendix Table 2: Parameters for simulated distributions parameters uniform a = 0, b = 1 normal µ = 0, σ 2 = 1 bimodal P[N(0, 1)] = 0.5, P[N(5, 1)] = 0.5 qmodal P[N(0, 1)] = 0.25, P[N(5, 1)] = 0.25, P[N(10, 1)] = 0.25, P[N(15, 1)] = 0.25 chi2 df = 2 beta1 α = β = 0.5 beta2 α = 5, β = 1 bernouilli P[X = 0] = 0.5, P[X = 1] = 0.5 discrete P[X = 0] = 0.2, P[X = 1] = 0.2, P[X = 2] = 0.2, P[X = 3] = 0.2, P[X = 4] = 0.2 poisson λ = 4 nbinomial r = 4, p = 0.5 Jump back

63 Appendix Literature Economic growth (Kathuria et al., 2009; Waverman et al., 2005) [M]ay be twice as large in developing countries compared to developed countries. (Waverman et al., 2005) Prices 20% reduction in grain prices across Nigerian markets (Aker, 2008) 5-7% increase in price of onions of farmers in Philippines (Lee and Bellemere, 2012) [N]ear-perfect adherence to the Law of One Price in the South-Indian fisheries sector (Jensen, 2007) Market participation Increase in market participation for farmers in Uganda growing perishable crops in remote areas (Muto and Yamano, 2009) Jump back

64 Appendix Table 3: Economic development scorecard Question Answer Points 1. How many household members A: 3 or more 0 are aged 25 or younger? B: 0, 1 or How many household members A: Not all 0 aged 6 to 17 are currently attending school? B: All 8 C: No children aged 6 to What is the material of the walls of A: Mud/cow dung/grass/sticks 0 the house? B: Other 5 4. What kind of toilet facility does A: Other 0 your 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 0 B: Yes Does the household own a sofa? A: No 0 B: Yes Does the household own a stove? A: No 0 B: Yes Does the household own a radio? A: No 0 B: Yes 8 9. Does the household own a bicycle? A: No 0 B: Yes How many head of cattle are A: None or unknown 0 owned by the household currently? B: 1 or more 9 Note: The scorecard is a reproduction of the scorecard in Chen et al. (2008). Jump back

65 Appendix Table 4: Descriptive statistics of economic development for the full sample and for the three geographic areas Total Area 1 Area 2 Area 3 Mean Median Maximum Minimum Std. Dev Observations Average poverty likelihood (%) County poverty level (%) Jump back

66 Appendix

67 Appendix Size of impact of years of ownership Jump back Roughly... Increasing the time of ownership from 2.5 (the average) to 3.5 years (40%) increases the PPI score with 20% (coefficient is 0.5). On an average PPI of 37, this means an increase from 37 to 44. This corresponds with a drop in poverty likelihood of 2.6% (from 35.4% to 32.8%, see Chen et al., 2008).

Applied Health Economics (for B.Sc.)

Applied Health Economics (for B.Sc.) Helmut Farbmacher Department of Economics University of Mannheim Autumn Semester 2017 Outlook 1 Linear models (OLS, Omitted variables, 2SLS) 2 Limited and qualitative