SISTEMAS DE CONTROL

The system of equation (4.26) is rewritten as follows:

3 3 3 3 3 3 3 3 3 3 =  +  + + + u IFL X H M G (4.27-1) * 2 2 2 2 2 2 2 2 2 2 = + + + + B  X H M G IFLu (4.27-2) * 1 0 B if B  0 B otherwise 3 3 3 3 3 3 3 3 3 1 =  +  + + + +B u FL X H M G (4.27-3)

Since IFL is an endogenous variable in the equation (4.27-2), this simultaneous equation

system is referred to as the probit model with continuous endogenous variable at the right

hand side (probit with continuous endogenous variable RHS), therefore, the standard probit or

OLS method can give biased estimates and joint estimation of this equation system is needed in order to obtain consistent estimates (Wooldridge, 2002). River and Vuong propose a useful two-stage conditional maximum likelihood method to estimate this system. Their method provides some advantages over the standard 2SLS since it is more efficient and it gives direct estimates of the parameters of interest and their correct asymptotic standard error (Pham & Izumida, 2002; Pham, 2009). The standard statistical test can be applied to test whether the

endogeneity problem exists by testing the correlation of the u₃and u₂in equation systems

(4.27-1 & 4.27-2). On the other hand, since Bis an endogenous variable in equation (4.27-3),

this binary endogenous variable in a simultaneous equation system (4.27-2 & 4.27-3) is

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i

 , _i,_i, _i (i=1,3),  and λ in the system of equations (4.26) can be estimated using the following steps:

Step 1. Regress IFL on exogenous variables to obtain the residual .

Step 2. Plug in to equation (4.27-2) as a regressor and run the standard probit to get the vectors of consistent coefficients ₂, ₂, ₂,₂, ₂, and .

This method also provides a simple test for the endogeneity of IFL by conducting the familiar

t-test for the coefficient  where the null hypothesis H :₀ 0. If H holds, this means ₀ u₁

and u₂ are uncorrelated, endogeneity is not a problem and IFL is an exogenous variable.

Once all the coefficients ₂, ₂, ₂,₂, ₂, and  in the probit model are well treated, we estimate the sample selection model using the following procedures:

Step 3. Estimate the probit model for the probability of borrowing from formal credit, then calculate the inverse Mills ratio of the probit model.

Step 4. Take into account the inverse Mills ratio in the loan amount, equation (4.27-3) and run the regression to obtain the coefficients and the coefficient,ˆλof the inverse Mills ratios.

Using the inverse Mills ratio to correct sample selection bias yields consistent coefficients for the determinants of access to the credit model. In addition, White‟s heteroskedasticity

consistent standard errors will also be used in the weighted least square regression to correct for asymptotic bias in ˆλ in step 4. Again, a t-test for the presence of sample selection is applied similar to the endogeneity problem in step 2.

Equation (4.27-3) is the main equation of interest in which the formal loan amount in log form is explained by a set of independent variables and the error term, u₁. The error term u₁is assumed to be normally distributed. Selection equation (4.27-2) is called the latent equation

because its dependent variable is unobservable. Only the choice variable B is observed when

the latent value is believed to be positive, i.e. a choice is made. This system of equations can be estimated by the standard sample selection Heckman model if the latent equation is well specified.

Since IFL is assumed to be endogenous, which influences the borrowing behaviour of the

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estimated. This problem forms another system of equations (4.27-1) and (4.27-2) to be estimated. Cameron and Trivedi (2009) suggest estimating the two equations (4.27-1) and

(4.27-2) simultaneously using instrumental variables for IFL, so called ivprobit model.

Equation (4.27-2) might be refered to as the “structural” equation in this system. This structural equation is the main interest and the other equation, called the reduced-form

equation, serves only as a source of identifying instruments for IFL. The reduced form

equation explains the variation in the endogeneous variable by including instrumental variables that are excluded from the structural equation.

In general, the model can be estimated by different methods depending on its distributional assumptions. Since access to formal credit is a binary variable and an informal loan is a continuous variable, their covariate error terms can be normalised to 1. Consistently, the model can be estimated by the conditional maximum likelihood method where the likelihood function is derived from the joint distribution of the error terms. Under a bivariate normal distribution, the conditional maximum likelihood method outperforms the ordinary 2SLS for two reasons. First, there is no requirement for the functional forms to be specified for the model as long as there are valid instrumental variables to control for the endogeneity and to achieve identification in the simultaneous equations. Second, the conditional maximum likelihood method fits the model by recursive algrorithms to obtain the coefficients with full observations in the sample hence more information will be included in the covariates in the

model for the ordinary 2SLS method18.

The bivariate normal distribution of the error terms in equations (4.27-1) and (4.27-2) can be specified following the standard procedures introduced by River and Vuong (1988) and

explained in detail in Wooldridge (2002). Since u₂and u₃ are correlated and form the

bivariate normality of ( ,u u_{2 3}) in a form u u₂| ₃ _{23 3}u e₁, and e₁ is normally distributed and

independent of u₂ and u₃, a test for exogeneity of the endogeneous variable, which is

equivalent to testing the null hypothesis H₀:₂₃ 0 of the error terms can be performed using an asymptotic t-test (Cameron & Trivedi, 2009; Wooldridge, 2002). This type of model

can be estimated using IV probit model. The valid instruments must significantly explain IFL

but not for the selection equation. To find the covariate that not only serves as the instrument

18_{Roodman (2009) introduces the cmp command which provides a flexible way to estimate different types of}

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but also helps determine the dependent endogenous IFL, we include more exogeneous

variables into the reduced equation then estimate the simultaneous equations by the conditional recursive mixed process (Roodman, 2009).

Under similar assumptions for the joint distribution of u₂and u₁, the correlation between u₂

and u₁ can be estimated following the Heckman (1979) twostep method. First, run a probit

model of selection equation (4.27-2) using all obsevations in the sample. The estimates of the probit model are then used to construct consistent estimates of the inverse Mills ratio term. Next, using OLS to estimate the outcome equation, which is the equation (4.27-3), with the

inverse Mills ratio term substituted for B to obtain consistent and asymtotically normal

estimators for the coefficients and



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 indicates the correlation between the unobserved factors in the selection equation and

outcome equation. A standard t-test of the null hypothesis that H₀:0, i.e. ₁₂ 0, can be carried out to test whether selection bias is present in the model (Vella, 1998; Wooldridge, 2002).