Metoda uparivanja po vjerojatnosti sklonosti

master's thesis

• Author: Sanja Kovačević
• Mentor: Sonja Radas (mentor)
• Committee member: Sonja Radas (predsjednik povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu koncentrirali smo se na procjenu prosječnog efekta tretmana koristeći različite pretpostavke i ovisno o njima regresijske metode, metode bazirane na vjerojatnosti sklonosti ili metode instrumentalnih varijabli. Varijable \(y_1\) i \(y_0\) označavale su ishod sa, odnosno bez tretmana, dok je binarna varijabla \(w\) označavala tretman, a \(\textbf{x}\) vektor opaženih kovarijabli. Ono što smo htjeli procijeniti bila je razlika između ishoda sa i bez tretmana, \(y_1-y_0\). Budući da se pojedinac ne može istovremeno nalaziti u oba stanja, tu razliku nije moguće direktno izračunati. Stoga smo promatrali sljedeća dva očekivanja: prosječan efekt tretmana (average treatment effect - ATE) koji je definiran kao: \(AT E \equiv E( y_1-y_0 )\) i prosječan efekt tretmana tretiranih jedinki (average treatment effect on the treated \(AT E_1\)) definiran sa: \(AT E_1 \equiv E( y_1-y_0 | w = 1)\). Također, promatrali smo i prosječan efekt tretmana uvjetno na \(\textbf{x}\) i prosječan efekt tretmana tretiranih uvjetno na \(\textbf{x}\) : \(AT E_1 (\textbf{x}) = E( y_1-y_0 | \textbf{x} , w = 1)\) i \(E( y_1-y_0 | \textbf{x}) = AT E(\textbf{x})\). Ključna pretpostavka bila je nezavisnost o tretmanu (eng. ignorability of a treatment) dana uvjetima: ATE.1: Uvjetno na \(\textbf{x}\), \(w\) i (\( y_0, y_1\) ) su nezavisni. ATE.1’: (a) \(E(y_0 | \textbf{x}, w) = E(y_0 | \textbf{x}\)); (b) \(E(y_1 | \textbf{x}, w) = E(y_1 | \textbf{x})\). Propensity score ili vjerojatnost sklonosti \(p(\textbf{x})\) označava vjerojatnost da jedinka primi tretman ako nam je poznat vektor kovarijabli \(\textbf{x}\). Važna pretpostavka pri korištenju metoda baziranih na vjerojatnosti sklonosti bila je jaka nezavisnost o tretmanu (eng. strong ignorability of a treatment) (uz danu kovarijablu \(\textbf{x}\)) koja se sastoji od pretpostavke ATE.1 i uvjeta \(0 < p(\textbf{x}) < 1, \forall \textbf{x}\). Koristeći te metode objasnili smo ideju uparivanja na temelju vjerojatnosti sklonosti, koju su prvi put predložili Rosenbaum i Rubin 1983. godine. Metoda instrumentalnih varijabli (IV) koristi se za procjenu \(AT E\) i \(AT E_1\) kada nismo sigurni vrijedi li nezavisnost o tretmanu, ali i za procjenu lokalnog prosječnog efekta tretmana (LATE). Promatrali smo najjednostavniji slučaj - kada je instrumentalna varijabla z binarna - te smo LATE definirali na subpopulaciji sastavljenoj od jedinki kod kojih varijabla z određuje hoće li te jedinke sudjelovati u tretmanu. U drugom dijelu rada pokazali smo korištenje uparivanja na temelju vjerojatnosti sklonosti u praksi. Procjenjivali smo utjecaj IWT subvencija na razvoj R&D-a u kompanijama u Flandriji i kako se taj utjecaj mijenja ovisno o određenim parametrima.

Abstract (english)

In this paper focus was on estimating average treatment effects. We used different assumptions and different methods- regression methods, methods based on the propensity score and instrumental variables methods. Variable \(y_1\) denoted the outcome with treatment and \(y_0\) without the treatment, while \(\textbf{x}\) denoted a vector of observed covariates and w denoted receiving the treatment. To estimate the difference \(y_1-y_0\), we observed average treatment effect (ATE) defined as \(AT E \equiv E( y_1-y_0 )\) and average treatment effect on the treated (\(AT E_1\) ) defined as \(AT E_1 \equiv E( y_1-y_0 | w = 1)\). Key assumption was assumption called ignorability of treatment (given observed covariates \(\textbf{x}\)): ATE.1: Conditional on \(\textbf{x}\), \(w\) and ( \( y_0, y_1\)) are independent. ATE.1’: (a) \(E(y_0 | \textbf{x}, w) = E(y_0 | \textbf{x}\)); (b) \(E(y_1 | \textbf{x}, w) = E(y_1 | \textbf{x})\). Propensity score \(p(\textbf{x})\) is the probability of receiving the treatment given the covariates. For using methods based on the propensity score we needed strong ignorability of a treatment assumption. We described matching algorithm based on the propensity score. For estimating the local average treatment effect (LATE), we used the instrumental variables method in the simplest scenario- when an instrumental variable \(z\) is binary. Finally, we showed how to use the propensity score matching in practice. We estimated the impact of IWT subsidies on firms’ R&D intensity and employment in the Flemish region.

• Keywords: procjena prosječnog efekta tretmana; regresijske metode; metode bazirane na vjerojatnosti sklonosti; metode instrumentalnih varijabli
• Keywords (english): estimating average treatment effects; regression mathods; methods based on the propensity score; instrumental variables method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:037779
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-19 11:09:18