ECON Program Evaluation, Binary Dependent Variable, Misc.

Transcription

1 ECON Program Evaluation, Binary Dependent Variable, Misc. Maggie Jones () 1 / 17

2 Readings Chapter 13: Section 13.2 on difference in differences Chapter 7: Section on binary dependent variables Chapter 6: Scaling, beta coefficients, log-level regressions, adjusted R 2 () 2 / 17

3 Program Evaluation With program or policy evaluation we are interested in the effect of a particular treatment Treatment could be any policy, e.g. providing rewards for student performance, increasing the minimum wage, implementing a basic guaranteed income, etc. To understand the effect of these policies, economists rely on a number of techniques to account for selection and non-randomness If we have a truly random allocation of people to treatment and control groups, then we can run the following regression: y = β 0 + β 1 treatment + u And we can interpret β 1 as the causal effect of treatment E.g., see the Roland Fryer paper from assignment 4 () 3 / 17

4 Program Evaluation Unless economists run experiments known as randomized control trials where they are able to randomly allocate people to treatment and control groups, then they must rely on alternative techniques to measure the effect of treatment One common methodology is known as difference-in-differences The basic idea is to obtain data on the outcomes of two groups (one group that is affected by treatment or a policy change and one that is not) for two time periods (before the treatment occurred and after the treatment occurred) Then we compare the change in the outcome variable of the control group before and after the policy change to the change in the outcome variable of the treatment group before and after the policy change () 4 / 17

5 Difference-in-Differences This methodology has the advantage of: i removes biases in 2nd period comparisons between treatment and control that result from permanent differences between these two groups ii removes biases in comparisons of the treatment group over time that could result from underlying trends in the outcome variable () 5 / 17

6 Source: () 6 / 17

7 Difference-in-Differences Treatment Control T-C Before E[y t = 1, a = 0] E[y t = 0, a = 0] E[y 1, 0] E[y 0, 0] After E[y t = 1, a = 1] E[y t = 0, a = 1] E[y 1, 1] E[y 0, 1] A - B E[y 1, 0] E[y 1, 1] E[y 0, 0] E[y 0, 1] (E[y 1, 1] E[y 0, 1]) The difference-in-differences estimate is the quantity (E[y 1, 1] E[y 0, 1]) (E[y 1, 0] E[y 0, 0]) (E[y 1, 0] E[y 0, 0]) () 7 / 17

8 Difference-in-Differences We can also obtain the difference-in-differences estimate by running the following regression: y = β 0 + β 1 treatment + β 2 after + β 3 treatment after + u Where treatment = 1 if the individual belongs to the treatment group after = 1 if the time period is after the policy change treatment after = 1 if the individual belong to the treatment group AND the time period is after the policy change ˆβ3 is thus the estimate of the effect of treatment () 8 / 17

9 Difference-in-Differences Two key assumptions: i The model is linear: we assume that the outcome is changing linearly over time ii Parallel trends assumption: in the absence of treatment, the treatment and control groups would have followed parallel trends () 9 / 17

10 Binary Dependent Variables We have already discussed the interpretation of a binary independent variable in our regression model We may also want to include a binary dependent variable in our regression model E.g. Does someone have a high school degree or not? E.g. Is an individual in the top one percent of the income distribution? E.g. Does the household have electricity or not? E.g. Does the household have access to clean drinking water? This type of qualitative information can be incorporated into the regression model as a left hand side variable in the linear probability model (one of several models for binary dependent variables) () 10 / 17

11 Linear Probability Model When a variable y is binary (i.e. 0 or 1) the expected value of the variable is the probability of success : E(y x) = Pr(y = 1 x) Then the coefficients ˆβ j from our regression function: E(y x) = β 0 + β 1 x β k x k = Pr(y = 1 x) represent the effect of a one unit change in x j on the probability of y Exact interpretation is: a 1 unit increase in x j changes y by ˆβ j 100 percentage points, holding all other variables fixed () 11 / 17

12 Linear Probability Model Two problems with the linear probability model: i You can obtain predicted probabilities that lie outside the [0, 1] interval ii Probabilities are forced to be linearly related to the xs, which might not always be the case Note: when y is binary, Var(y x) = p(x)(1 p(x)), where p(x) = β 0 + β 1 x β k x k, which violates the assumption of homoskedasticity Result: ˆβj is not biased, but Var( ˆβ j ) not correct and t and F statistics will not have t and F distributions () 12 / 17

13 Scaling Rescaling the dependent variable by some scalar, γ will result in a rescaling of all of the ˆβ j s by γ Example: regress y on several x variables Rescale y to be y/γ: y = ˆβ 0 + ˆβ 1 x ˆβ k x k y/γ = ˆα 0 + ˆα 1 x ˆα k x k ˆα 0 = ˆβ 0 /γ, ˆα 1 = ˆβ 1 /γ,..., ˆα k = ˆβ k /γ Note that all standard errors will also be rescaled (se(α k ) = se(β k )/γ), so that t and F statistics remain the same and inference is unchanged () 13 / 17

14 Beta Coefficients Sometimes key variables are measured in scales that are difficult to interpret (e.g. test scores), or we wish to have all x variables on the same scale so that we can compare the coefficients across x variables One method economists use in these situations is to standardize all of the variables so that they are all in units of standard deviations Consider the standard regression model where each y i is given by: y i = ˆβ 0 + ˆβ 1 x 1i + + ˆβ k x ki + û i () 14 / 17

15 Beta Coefficients Standardize all variables: y i ȳ = ˆβ ( ) 1ˆσ x1 x1i x 1 ˆσ y ˆσ y ˆσ x1 z y = ˆb 1 z ˆb k z k + ê + + ˆβ kˆσ xk ˆσ y ( ) xki x k ˆσ xk + ûi ˆσ y ˆbj are known as the standardized beta coefficients, all t and F statistics remain the same Interpretation: a 1 standard deviation increase in z 1 increases z y by ˆb 1 standard deviations holding all other variables constant () 15 / 17

16 Log-level regressions Suppose we have the regression: log y = β 0 + β 1 x β k x k + u Previously we said that a one unit increase in x j changes y by 100 ˆβ j % holding all other variables constant It turns out this is just an approximation that holds for ˆβ j ( 0.1, 0.1) The true interpretation is that, holding all other variables constant, a one unit change in x j leads to a 100 (e ˆβ j 1)% change in y () 16 / 17

17 Adjusted R 2 As we saw in the previous chapters, adding more variables to the regression model can only increase the R 2 This leads to a problem in which researchers may add irrelevant variables to the model in order to increase the R 2 to claim a better fit The Adjusted R 2 imposes a penalty for adding additional independent variables: R 2 = 1 SSR SST = 1 σ2 u σ 2 y The adjusted R 2 replaces σu 2 and σy 2 with their unbiased estimators: R 2 SSR/(n k 1) = 1 SST/(n 1) () 17 / 17