Prove the following statement using the potential outcomes framework: If the stable unit treatment value assumption holds and selection bias is equal to zero then the difference in means between treatment and control groups is an unbiased estimate of the average treatment effect on the treated?
The objective of program evaluation is to evaluate the net difference, which appears with respect to the difference between treated and non-treated groups should be attributed to the intervention. Notably in this condition, other variables are constant (Benjamin, 2012).
Considering the study of treatment effect over the heterogeneity, the coefficient for any indicator variable is obvious not to reflect the complexity existing in the treatment effects. The most important consideration in this regard is the treatment assignment assumption (Accetturo & De Blasio, 2012). For example in the case of the analytical approach, the results are not dependent on the assumptions. Thus, they depend on the distributional and functional forms. On the contrary, when rebalancing approach is being used, the condition becomes more alike to the results that are generated by randomization.To resolve such challenges, counterfactual statistical methods such as Heckman’s sampling, propensity scoring-matching, score sub-classification, score weighting and matching estimators models are useful.
The hypothesis is holding a stable unit of treatment value and zero value for selection bias, relates with instrumental variable estimator. For a linear population model,
y = β0 + β1x1 + β2x2 + … βkxk + ε
E(s) = 0, Cov(xf, ε) = 0, Cov(xk, ε) ≠ 0, j=1, …. , K-1.
Here xk is correlated with ε and that xk is potentially endogenous. Here ε is believed to contain one omitted variable and is not correlated with any explanatory variables except xk. To resolve this, an observed variable z is essential to incorporate, such that
z is not correlated with ε and
z is not correlated with xk
In other case, the model considered as
y = xβ + ε
Where, it contains the constants, x = (1, x2, …, xk) and the exogenous variables is denoted with z = (1, x2, …, xk-1, z)
In practical measures, finding an exact value for the instrumental variable z is challenging. This z does not correlate with the regression model that links with endogeneity but it is correlated with independent variables which causes endogeneity.
This was resolved by Angrist, in their study (Angrist & Imbens, 1995), which focus over effect of education on wages. In this case, the residual for the regression was correlated with education, as it contains omitted ability. The authors used dichotomous variable, where the concers subjects were categorized as either born in first quarter of birth year (=1) or in remaining quarters (=0). Here they argued that compulsory school attendance was induced over people to attend school and thus, there exists a relationship between education and quarter of birth. Notably the quarter is a random variable and this not correlated with omitted variables of regression model.
Hence they suggest that non-ignorability should be avoided in the treatment received, for any indicator variable. Rather a precise and straightforward casual interpretation is needed. Such interpretation can avoid the constant effect imposed over all the units and hence it delineates the critical assumptions. Thus the focus is more over average casual effect on outcome.
This is also proved with (local average treatment effect) (Angrist & Pischke, 2010):
EE ≠ ITT (ATE) ≠ TT ≠ Naïve ATE
Where EE is efficacy effect, ITT is intent-to-treat, ATE is average treatment effect, TT is the average treatment effect for treated, and naïve is opposite to ignorable variable.
Benjamin, D., Brandt, L., McCaig, B., & Le Hoa, N. (2012). Evaluating the impact of a targeted land distribution program: Evidence from Vietnam (No. tecipa-461).
Accetturo, A., & De Blasio, G. (2012). Policies for local development: An evaluation of Italy's “Patti Territoriali”. Regional Science and Urban Economics, 42(1), 15-26.
Angrist, J., & Imbens, G. (1995). Identification and estimation of local average treatment effects.
Angrist, J., & Pischke, J. S. (2010). The credibility revolution in empirical economics: How better research design is taking the con out of econometrics (No. w15794). National Bureau of Economic Research.