Application of the Generalized Propensity Score. Evaluation of Public Contributions to Piedmont Enterprises Michela Bia Alessandra Mattei Dipartimento di Statistica “ G. Parenti ” Università di Firenze
Project’s goals Techniques based on the propensity score have long been used for causal inference in observational studies for reducing bias caused by non-random treatment assignment ( Rosenbaum and Rubin, 1983b ) The propensity score method is usually confined to binary treatment scenarios. In many cases of interest the treatment takes on more than two values ( e.g drug applied in different doses or a treatment applied over different time periods ) We implement an extension of the propensity score method , in a setting with a continuous treatment . The methodology is applied to the public contributions supplied to the Piedmont enterprises, during years 2001 - 2003 . Due to the variety of funds set by public policies, the treatment turns out to be a continuous variable . We are interested in the effect of different amounts of contribution on the occupational level.
Our empirical study: economic supports to Piedmont Industry This study covers all measures - basically grants and loans at special rates - of financial support in favour of enterprises in Piedmont between 2001 and 2003 (regional, given to regions, national and EU co-financed): � Economic supports to productive activities in depressed areas, economic supports to investments (488/92 Industry, 266/97, L. 140/97 , 341/95, 1329/65, 662/96) � Economic supports to investments for enterprises (DOCUP 2000- 2006 Ob.2 areas ) � Economic supports to the environment safeguard (R.L. 598/94) Research and development - Applied research (L.297/99 D.M.593/00 )
The Equivalent Gross Subsidy computation All data concerning loans at special rates are turned into Equivalent Gross Subsidy through a specific computation: − − p N mra sra mrent srent ∑ ∑ = + X fin + + t t ( 1 mra ) ( 1 mra ) = = + t 1 t p 1 where: � X is the EGS financing estimation (net benefit of enterprises); � mra is the market rate; � sra is the subsidized rate; � p is the pre-depreciation period; � fin is the total financed amount; � N is the financing term; � mrent is the financing rent with market rate; � srent is the financing depreciation rent with subsidized rate;
Basic framework We consider a sample of units i=1,2..,N and, for each unit, a set of potential unit-level outcomes Y i (t) for t ∈τ [SUTVA, (Rubin 1980a )] In the binary treatment τ = { 0,1 } In the continuous case τ ⊂ [ t 0, t 1 ] We are interested in the average treatment effects estimation, for example: µ (t) - µ (t+ ∆ t) =E [ Y i (t) ]- E [ Y i ( t+ ∆ t) ] Weak unconfoundedness assumption ( Imbens and Hirano, 2004) Y(t) ⊥ T | X for all t ∈ τ Generalized Propensity Score Let r(t,x) be the conditional density function of the treatment given r(t,x) = f T | X (t | x) the covariates :
Balancing property Balancing of pre-treatment variables given the generalized propensity score Within strata with the same value of r(t,X) , the probability that T = t does not depend on the value of X : X ⊥ 1 { T=t }| r(t,X) This definition does not require unconfoundedness. Weak unconfoundedness assumption given the generalized propensity score (Imbens and Hirano, 2004) Y(t) ⊥ T | r(t,X) for all t ∈ τ
Van Dik, Lu and Imbens: three different approaches for the GPS implementation Van Dyk - Imai (2003), Imbens - Hirano (2004) and Lu et al . (2001) develop methods that implement the generalized propensity score. Van Dik and Imai apply analysis techniques mostly based on sub- classification. They introduce the generalized propensity score through a propensity function : = f ( t x ) r ( T , X ) ψ ψ ( T X ) ψ where parameterizes this distribution. Dik and Imai assume that θ θ r (., X ) depends on X only trough a specific function ( X ) so that is ψ sufficient for T . ∧ ∧ They compute for each observation and sub-classify observations r ( t , X ) ψ with the same or similar values of gps into a number of sub-classes of equal size.
The average causal effect is a weighted average of the within sub- classes effects, with weights equal to the relative size of the sub- classes: ≈ ∑ ∧ S = E [ Y ( t )] E [ Y ( t ) T t , r ] W s s = s 1 � In contrast to sub-classification method , Lu et al . (2001) suggest matching pairs of units on r ^. They propose a distance measure that decreases when the propensity scores become similar and the received treatments become dissimilar. The treatment effect can be evaluated by examining the difference in response between the “high” and “low” treatment. In the continuous case matching procedures are more difficult than in binary treatment. This because the matched pairs should not only have similar r ^ , but also different treatment levels.
� Imbens and Hirano’s procedure (2004) for the dose-response estimation is mostly based on the regression on the propensity score technique . We will apply it in our empirical study. First the GPS is estimated through the conditional distribution of the treatment variable given the covariates ∼ β σ 2 T X N ( h ( X ; ), ) i i i with the estimated GPS equal to ∧ = φ gps ( T i X ; ) i To verify whether the specification is correct, one can verify if it balances the covariates.
Once the correct specification is obtained, the conditional expectation of Y given T and the GPS - E [ Y | T = t, R = r ] - is estimated: B(t,r)=E [ Y(t) | r(t,X) = r ] = E [ Y | T = t, R = r ] = B(t,r) i) Hence, the dose-response function µ (t) = E [ B(t,r(t,X) ] is obtained averaging the conditional expectation over the score r(t,X) evaluated at a certain level of the treatment t : µ (t) = E [ B(t,r(t,X) ] = E [ E [ Y(t) | r(t,X) ]] =E [ Y(t) ] ii)
The Program The program gpscore.ado estimates the generalized propensity score and tests the Balancing Hypothesis according to the following algorithm: 1. Assume a normal distribution for the treatment given the covariates: ∼ β σ 2 T X N ( h ( X ; ), ) i i i where β is the parameter vector, β is a known function of h ( X , ) i the covariates which depends on the parameters β and σ 2 Estimate β and σ 2 by maximum likelihood 2. 3. Estimate the gps applying the normal probability density function evaluated for all values of T and X : ∧ = φ gps ( T i X ; ) i
4 Test the Balancing Property 1) Split the treatment ’s range in k equally spaced intervals, where k is chosen by the user. 2) Calculate the mean or a percentile of the treatment and evaluate the gps at that specific level of T . Let t k,p be the chosen value of the treatment. 3) Split the estimated gps’ range in j equally spaced intervals, where j can be arbitrarily chosen. 4) Within each j-th interval of gps, for each covariate compute the differences between the mean for units with t i > t k,p and that for units with t i <= t k,p 5) Combine the differences in means, calculated in previous step, weighted by the number of observations in each group of gps i interval and then in each treatment interval 6) If the test fails , the Balancing Test is not satisfied and one or more of the following alternatives can be tried: a) Specify a different propensity score; b) Specify a different partition of the range of the estimated gps ; c) Specify a different sub-classification of the treatment.
Syntax gpscore is a regression-like command gpscore varlist [ if exp] [ in range] [fweight iweight pweight], gpscore ( string ) predict(string) sd(string) Cutpoints(varname numeric index ( string ) nq_gps ( numlist ) [ regression_type(string) Detail level(real 0.01) ]
The economic supports to Piedmont Enterprises The administrative data are collected by ASIA (1996-2003). The different types of funds assigned to the industries are supplied by Finpiemonte, Mediocredito Bank. The final database is obtained merging contributions archives relative to each type of measure with ASIA archive administrative data (2000- 2003) with Industry Census (2001) data, so that to have: - business name; - municipality and corporate address; - industrial activity field (Ateco 2002); - juridical classification; - employees (mean by year, permanent and temporary, 2001-2003); - grant concession and payment date (according to each law); - subsidized financing (based on E.G.S computation for loans); - company type (according to the number of employees and local unit localization) - craft or non-craft enterprise.
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