Impact Estimator for Panel Data: Difference-in-Differences

a) Difference-in-Differences estimators and assumptions

We now turn to the idea of using panel data to estimate the treatment effect under the

Difference-in-Differences approach, which is an increasingly popular method for identifying programme impact in the absence of purely experimental data (Ashenfelter & Card, 1985; Athey & Imbens, 2006). It takes sometime for a policy to be implemented or to take effect on the target group of participants. The observed outcomes over time may be attributed not only to the treatment but also the observed and unobserved factors such as economic conditions,

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other concurrent policies, individual motivation and ability, etc. It is necessary to exclude

such undesired attributes in order to assess the real effect of the policy. The DinD method

resolves the problem by comparing the control individuals, who share the same

characteristics, with the treated ones and then substracting the difference in the outcome variables of the treated ones over time by the difference of the control ones. In doing so, the desired effect can be expressed as follows:

, 1 , , 1 , ( - | 1) - ( - | 0) ATT DiD E Yi t Yi t D E Yi t Yi t D   _  _  (4.34) 1, 1 1, 0, 1 0, ( - ) - ( - ) ATT DiD Yi t Yi t Yi t Yi t   _{ } (4.35)

Equation (4.35) can be represented by the following regression equation (Angrist & Krueger, 1999; Meyer, 1995; Wooldridge, 2002): 0 0 1 1 it t i t i it Y   D D  D D (4.36) it

Y is the outcome of interest for individual i at period t. D_t is time dummy variable, taking

a value of zero for individual in pre-programme period and one for post-programme. D_i

denotes the treatment variable, which equals unity for an individual in a treatment group and

zero otherwise. Hence, D_i captures the differences in outcome between the treatment and

control groups due to policy change. The interaction term, D D_{i t} indicates the programme

participation and takes a value of one if the household i borrowed money and the observation

observed in the second period, and zero otherwise. The final term _it is the idiosyncratic

disturbance, which is assumed to be normally distributed with zero mean and constant variance. Under this assumption, equation (4.36) can be estimated by OLS from repeated

cross-sectional or panel data. The coefficent ₀ accounts for aggregate factors that affect the

outcome over time for both treatment group and control group. Coeficient ₁ captures the

mean of the potential time-invariant difference between the two groups. The coefficient ₁ is

the parameter of the programme impact since it measures the average effect of the treatment on the treated individuals, e.g., the average programme impact on the borrower group

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compared with the non-borrower group and its statistical significance can be tested by the conventional t-test.

However, for ₁ to be the consistent estimator of the average treatment effect on treated

individuals, two crucial assumptions have to be made (Blundell & Dias, 2000). The first assumption that, in the absence of the treatment, both the treatment and control individuals experience the same time-effect. Under this assumption, it is postulated that the unobserved factors, for example changes in economic conditions and other policy initiatives, may affect both groups in a similar manner. The second assumption is that there is no systematic changein the composition of individuals within each group. That is, before and after the treatment being implemented and takes effect, there is no big event that induces a majority of individuals in one group to move to another group.

Equation (4.36) provides an unbiased programme impact estimator under randomisation of programme participation, i.e., households are randomly selected in a microcredit programme. However, given non-random participation in microcredit programme, it is likely that

participants in a microcredit programme are selected based on pre-programme attributes. Consequently, the pattern of change in the outcome influenced by programme participation may vary systematically across the two groups if there are no credit programme. Ignoring these systematic differences may lead to biased estimates of the programme impact (Abadie, 2005; Islam, 2010).

To control for the pre-programme attributes, a vector of observable household characteristics and commune characteristics is included in equation (4.36). The programme impact estimator can be obtained from the following model:

0 0 1 1

it t i t i it i it

Y   D  D  D D X 

(4.37)

Equation (4.37) allows heterogeneity of the two groups in terms of the observable

characteristics such as individual demographic characteristics, geographical differences of residence, etc. That is, unbiasedness of the estimator is possible only when the treatment of

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is rarely satisfied in programme evaluation since programmes are normally designed to help groups with specific attributes. To capture the heterogeneity of individuals‟ attributes by

conditioning on covariates, X, such as demographic characteristics or housing

characteristics, the treatment effect model is specified as:

0 0 1 1 1

it t i t i it i t i it it

Y   D D  D D X D DX 

(4.38)

where _i captures the effects of the covariates on the outcome for each time period. The

parameter ₁ measures the treatment effects of X_it in case the treatment changes the slope of

the coefficients in equation (4.38). The coefficient δ₁ in equation (4.37) no longer explains the

full treatment effect as in equation (3.38) but, instead, δ + γ₁ X_{it 1} does (Pham, 2009).

Estimating equation (4.38) faces problems of endogeneity and selection bias (see also Athey & Imbens, 2006; Coleman, 1999; Heckman, 2001; Heckman & Vytlacil, 2005; Pitt &

Khandker, 1998). It isimportant to note that equation (4.38) is a non-linear model. Therefore, the treatment effects coefficient should be interpreted with care because in a non-linear

model, covariates X cannot be excluded by taking the differences across individual groups

and time periods. The treatment effect is conditional on both covariates and the functional form of the response functions (Ai & Norton, 2003). Hence, the magnitute of the effect and its

statistical significance may vary across X's values.

b) Estimation strategies

Recent studies have used panel data to tackle the major issues and shortcomings in

microcredit programmes. The fixed effects method using panel data is used to evaluate the impact of a single microcredit programme and the results confirm that microcredit is statistically significant in reducing poverty among poor borrowers and within the local economy in Bangladesh (Khandker, 2005). To estimate the impact estimator, the outcome equation is respecified as follows. First, the interaction terms in equation (4.37) are

101 0 0 0 1 it t i it i it it C   D  D X I  _(4.39) it it it I W  (4.40)

Next, the real term of borrowing is used as a dependent variable of the latent function; this is equivalent to the loan equation. Therefore, equations (4.39 & 4.40) can be re-written as a system of equations as follows:

0 0 0 1 it t i it i it it C   D  D X  FL  (4.41) it it it FL W  (4.42)

whereFL_it is the amount of loan if the household borrows in the post-period and zero

otherwise. C_itis consumption at the household level. W_itis a vector of household factors that

may be different from X_it. Again, two potential problems arise in estimating the model,

namely endogeneity and selection bias. The fixed effects model can be applied to solve potential biases using the maximum likelihood or two-step estimation methods(Angrist, 2004; Wooldridge, 2002). Equation (4.42) is estimated in the first step and equation (4.41) is

estimated in the second step. The coefficient, ₁, is now the impact of the microcredit

programme on the consumption outcome of the household.

To measure the impact of formal microcredit on household income, Coleman (2006)suggests using income as the dependent variable to credit constraint. The income model can be written as follows: 0 0 1 2 it t i it i it it I   D D X  FL  (4.43) it= itφ+εit FL W (4.44)

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where all notations in equations (4.43 and 4.44) are similar to equations (4.41 and 4.42) in the consumption model, except income is replaced as the outcome variable in equation (4.41). A similar estimation technique will be applied to estimate the system equations (4.43) and (4.44)

to obtain the coefficient microcredit programme impact, ₂, on the consumption outcome of

the household.

4.3

Chapter Summary

This chapter demonstrates a number of important issues in microcredit literature including asymmetric information, credit constraint and credit rationing, and consumption. Through the theoretical models, various economic relationships have been established for empirical estimation. Information asymmetry is the core of the lending principle that explains why lenders always select a certain borrower to grant a loan contract and hence there is always a mismatch between supply of and demand for credit in the formal rural credit market. This credit rationing creates a credit constraint on the rural household, particularly the rural poor. Subsequently, the household consumption model shows that credit helps reduce constraints to working capital in agricultural production or in non-farm income generating activities and hence, enhances the household‟s consumption growth.

Following the defined economic relationship, different empirical models are discussed in this study. First, the credit accessibility model is specified to determine factors affecting the household‟s decision to borrow from the formal credit sector under the conditions that the informal sector exists and interacts with the formal sector in the credit market. The model is expected to achieve consistent estimators for the determinants of the household‟s access to the formal and informal credit under the credit rationing assumption, selection bias and

interaction between informal and formal credit.

The Propensity Score Matching method and the Difference-in-Differences approach used for evaluating the impact of a microcredit programme on household consumption and income were also discussed. Their estimation strategies were proposed to obtain unbiased and consistent estimators, depending on the types of dependent variables as well as the nature of the dataset. The dataset and the estimation results for the empirical models will be presented and discussed in the following chapters.

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