Identification in a nutshell Start by comparing expenditures for households marginally close to S=0; since Y=Yo+rB we have that E{|S=0}=E{S=0}+E{RB|S=0 E{y|S=0}=E{lS=0}+ES=0 Consider the difference around eligibility E{yS=0}-E{y|s=0}=E{S=0-}-E{xS=0} +ERBIS'=0*)
Identification in a nutshell Start by comparing expenditures for households marginally close to S *=0; since Y = Y0+Rβ we have that Consider the difference around eligibility: * * * 0 * * * 0 | 0 | 0 | 0 | 0 | 0 | 0 E Y S E Y S E R S E Y S E Y S E R S + + + − − − = = = + = = = = + = * * * * 0 0 * | 0 | 0 | 0 | 0 | 0 E Y S E Y S E Y S E Y S E R S + − + − + = − = = = − = + =
Identification in a nutshell Key identifying restriction(the mean consumption profile under the no-retirement alternative is smooth enough at zero) E{x|S=0}=E{|S=0 The result rests upon a weak regularity condition: if none of the heads were to retire no discontinuity in household consumption would take place at the time they become eligible (i.e. at S=0) see Hahn et al. (2001)and Battistin and Rettore( 2006) This amounts to assuming that any idiosyncratic shocks relevant to the retirement choice and correlated with Yo(e.g. health shocks) do not occur selectively at either side of the eligibility threshold
Key identifying restriction (the mean consumption profile under the no-retirement alternative is smooth enough at zero): The result rests upon a weak regularity condition: if none of the heads were to retire no discontinuity in household consumption would take place at the time they become eligible (i.e. at S*=0) – see Hahn et al. (2001) and Battistin and Rettore (2006). This amounts to assuming that any idiosyncratic shocks relevant to the retirement choice and correlated with Y0 (e.g. health shocks) do not occur selectively at either side of the eligibility threshold. * * 0 0 E Y S E Y S | 0 | 0 + − = = = Identification in a nutshell
Identification in a nutshell By using simple algebra we have EBR=1,S=0 EYIS=0-E(YIS PrR=1S=0 Estimators of the causal effect of retirement on consumption are analogue estimators obtained by replacing the quantities in the last expression by their empirical counterparts Following Imbens and angrist(1994) and hanh et al (2001), it can be shown that this expression coincides with the iv estimator obtained by instrumenting the endogenous variable r with the eligibility status defined from s
By using simple algebra we have: • Estimators of the causal effect of retirement on consumption are analogue estimators obtained by replacing the quantities in the last expression by their empirical counterparts. • Following Imbens and Angrist (1994) and Hanh et al. (2001), it can be shown that this expression coincides with the IV estimator obtained by instrumenting the endogenous variable R with the eligibility status defined from S* . * * * * | 0 | 0 | 1, 0 Pr 1| 0 E Y S E Y S E R S R S + − + + = − = = = = = = Identification in a nutshell
Endogeneity of s The s variable may be the outcome of individual choices( time to enter the labour market, temporary exits etc). This might casts doubts that our identification strategy is marred by an endogeneity problem Consider the regression we use to get the numerator of the iv estimate(the reduced form) Y=80+81S*+62S2+831(S>0)+8 The mean of y conditional on s is E{YS}=8+81S+82S2+631(S>0)+E{eS} where the last term does not vanish if s is endogenous
Endogeneity of S * • The S* variable may be the outcome of individual choices (time to enter the labour market, temporary exits, etc). This might casts doubts that our identification strategy is marred by an endogeneity problem. • Consider the regression we use to get the numerator of the IV estimate (the reduced form): Y= δ0 + δ1 S * + δ2 S *2 + δ3 1(S*>0) +ε The mean of Y conditional on S* is: E{Y|S*} = δ0 + δ1 S * + δ2 S *2 + δ3 1(S*>0) + E{ε|S*} where the last term does not vanish if S* is endogenous