Propensity Score Matching (PSM) Reference

2 Assumptions for treatment to be strongly ignorable (Rosenbaum Rubin 1983) thereby making PSM appropirate:
CIA (Conditional Independence Assump.): Conditional on observables (Xs) outcomes (Ys) are independent of treatment status (T=0 or 1).
 Common Support: For each value of observables (Xs) there is a positive probability of being treated and untreated. In other words observe all levels of X in both groups (treated and untreated).  

Pitfall--Curse of Dimensionality
The more observables the more difficult to find close matches across all Xs-hence need PS to gauge 'closeness'.

Matching Algorithms
Nearest Neighbor Matching
Radius Matching (calipers)
Kernel/local-linear matching (non-par that compares treated with w.avg. all all no treated; weighted by PS proximity.
Suggested test for robustness (Heinrich below)is to try all of them.(?)


Assumption and Spec Tests
For specification of the selection equations follow same rules as one would any other regression.
No great way of testing validity of CIA but can use to institutional knowledge to argue basis.
For common support one can test do an F-test (Hotelling test) on the joint X differences between treated and untreated.


Unobserved Heterogeneity and relaxing CIA
CIA can be somewhat relaxed if use DD on outcome comparisons.  Of course this requires the assumption that if there was selection on non-observables the non-observables be *time invariant*.  If that could strongly be argued then there is a strong case.  (requires panel data too).



PSM References: 
Heinrich, Maffioli, Vazquez, "A Primer for Applying Propensity-Score Matching", IDB Working paper 2010.
Angrist and Pirschke, "Mostly Harmless Econometrics", (Book) 2009.
Rosenbaum's "Observational Studies" (Book), 2002.

King, Gakidou, et al.-LANCET-2009

Title-"Public Policy for the poor? A randomized assessment of the Mexican Universal Health Insurance Programme"

Notes: Assesses Seguro Popular in Mexico.  Applies wedged experimental design (politically robust exp. design). This constituted a paired cluster rand. experiment where clusters pairs were selected from number of eligible clusters selected in accord to institutional constraints (needy clusters essentially).  Narrowed clusters based on closeness and likelihood of compliance (at hh level)-observables matched on did not include outcomes of interested. Used  baseline/endline survey data and main outcome of interest was effect of health expenditures on treated (where treatment were ads and increased public funding). In total 100 selected clusters (50 pairs). They estimate ITT as well as CACE (complier average causal effect:exclude always and never takers). CACE methodology is explained in Angrist, Imbs and Rubin 1996 (ref#18 in paper). ITT was estimated non-parametrically (reference to King 2009; (13)). CACE required assumption "only compliers benefited from [program]". Found no increase in participation but did find decreases in 'catastrophic expenditures (over 30% of hh capacity.  For most part other effects negligible-discuss it may be due to short assement period ~1 year.

Robertson-AER-2000

Title: Wage Shocks and North American Labor-Market Integration

Notes: Develops 3 sector model of international labor markets and bases empirical model from it.  Sees lagged wage in the US affecting labor demand in US. Begins with a linear labor demand equation and suppy equations for Mexico.Solves labor market equilibrium such that Mexican wages are based on lagged American wages. Using the final difference equilibrium equation they test two parameter estimates to check on hypotheses that American and US wages respond to same shocks and that they are converging. 

Robust vs. Clustered Errors

A constant problem is the homoscedasticity of error terms assumption is required to validate the Gauss-Markov Theorem (making straight foward OLS the BLUE). As the assumption can often be violated leading to funky standard error estimates, and therefore inference, the two most common corrections are estimating error variances and covariances (VCs) as a function of observables.  So now instead of estimating the error VCs unconditionally as in the basic OLS formulation an additional step is taken whereby the VCs are estimated with an additional preceding auxilliary regression whereby error components are a function of individual terms or cluster terms.  For example,the equation { error = individual (or group) effect + random component }. So while the error term was originally not homoscedastic the assumption here is that the random component is since the heterocedasticity was derived from the individual or group effects and these are controlled for.  Modeling the error structure as having an individual or group effect is what makes the error estimator as either robust (former) and clustered (the latter).  This method was introduced by White (1980), hence the name White estimator (among others), and is within the family of GLS methods.

Here are some helpful links:
http://www.stata.com/support/faqs/stat/cluster.html
http://sekhon.berkeley.edu/causalinf/sp2010/section/week7.pdf

Fafchamps, Shilpi-EJ-2005

Title: Cities and Specialization: Evidence from South Asia

Notes: No theoretical model. Uses non-parametric method called roughness penalty (Silverman 1982 referenced).  Specialization is basically measured by a Simpson Index-essentially Sum of squared number of hours in each occupation divided by squared number of total hours. City proximity is basically an integral of the product of urban households and distance to a specific household. They analyze how a household's and village specialization indexes are affected by city proximity. 

Emran, Shilpi-IIEPWP#11-undated(after 2008)

Title: The Extent of the Market and Stages of Agricultural Specialization

Notes: Applies gravity equations where specialization (in terms of a herfindal index of  concentration of crop-land use at the village-level) is regressed on a market size measure and controls. It uses Nepalese household survey data.

Key, Sadoulet, De Janvry-AJAE-2000

Title: Transactions Costs and Agricultural Household Supply Response

Notes: Build on an AHM of household supply to either market or home consumption and estimate it for only maize using Mexican data. It does allow for a simultaneous choice of production and consumption. They jointly estimate three regimes of (buyer, seller, autarky) by tacking on errors to the three theoretically derived equations.  They estimate the parameters by maximum likelyhood by assuming the errors in the 5 equations they end up with are ~ I.N.D. Funny to see them refer to identification in this paper as the ability to identify separate parameters within a model (as oppose to producing estimates of their product for example) while the now it seems identification is more associated with causation.  See Angrist and Pischke's book for a good discussion.

Their Abstract: "We develop and estimate a model of supply response when transactions costs create a situation where some producers buy, others sell, and others do not participate in markets. We present two rationales for why producing households may have different relationships to the market: proportional and fixed transactions costs. Using data on Mexican corn producers, we estimate an empirical model that allows for separate tests of the significance of both types of transactions costs, revealing that both fixed and proportional transactions costs matter for the estimation. The results provide consistent estimates of supply elasticity and measures of the relative importance of factors determining both proportional and fixed transactions costs."

Bowlus and Sicular-JDE-2003

Title: Moving toward markets? Labor allocation in rural China

Notes: Tests for separation property of AHM with Chinese Data uses Benjamin's (1992) approach.