A Course in Applied Econometrics 1. Introduction Lecture 5 2. - PowerPoint PPT Presentation

Outline “A Course in Applied Econometrics” 1. Introduction Lecture 5 2. Basics 3. Local Average Treatment Effects Instrumental Variables with Treatment Effect 4. Extrapolation to the Population Heterogeneity: Local Average Treatment Effects 5. Covariates Guido Imbens 6. Multivalued Instruments IRP Lectures, UW Madison, August 2008 7. Multivalued Endogenous Regressors 1 1. Introduction 2. Basics Linear IV with Constant Coefficients. Standard set up: 1. Instrumental variables estimate average treatment effects, with the average depending on the instruments. Y i = β 0 + β 1 · W i + ε i . There is concern that the regressor W i is endogenous, corre- 2. Population averages are only estimable under unrealistically lated with ε i . Suppose that we have an instrument Z i that is strong assumptions (“identification at infinity”, or under both uncorrelated with ε i and correlated with W i . the constant effect). In the single instrument / single endogenous regressor, we end 3. Compliers (for whom we can identify effects) are not nec- up with the ratio of covariances essarily the subpopulations that are ex ante the most inter- � N 1 i =1 ( Y i − ¯ Y ) · ( Z i − ¯ esting subpopulations, but need extrapolation for others. Z ) β IV ˆ N = . � N 1 1 i =1 ( W i − ¯ W ) · ( Z i − ¯ Z ) N 4. The set up here allows the researcher to sharply separate Using a central limit theorem for all the moments and the the extrapolation to the (sub-)population of interest from delta method we can infer the large sample distribution without exploration of the information in the data. additional assumptions. 2 3

Potential Outcome Set Up Let Y i (0) and Y i (1) be two potential outcomes for unit i , one 3. Local Average Treatment Effects for each value of the endogenous regressor or treatment. Let W i be the realized value of the endogenous regressor, equal to The key instrumental variables assumption is zero or one. We observe W i and � Y i (1) if W i = 1 Assumption 1 (Independence) Y i = Y i ( W i ) = Y i (0) if W i = 0 . Z i ⊥ ⊥ ( Y i (0) , Y i (1) , W i (0) , W i (1)) . Define two potential outcomes W i (0) and W i (1), representing the value of the endogenous regressor given the two values for the instrument Z i . The actual or realized value of the It requires that the instrument is as good as randomly assigned, endogenous variable is and that it does not directly affect the outcome. The assump- � tion is formulated in a nonparametric way, without definitions W i (1) if Z i = 1 W i = W i ( Z i ) = of residuals that are tied to functional forms. W i (0) if Z i = 0 . So we observe the triple Z i , W i = W i ( Z i ) and Y i = Y i ( W i ( Z i )). 4 5 Assumptions (ctd) Alternatively, we separate the assumption by postulating the existence of four potential outcomes, Y i ( z, w ), corresponding Compliance Types to the outcome that would be observed if the instrument was Z i = z and the treatment was W i = w . It is useful for our approach to think about the compliance Assumption 2 (Random Assignment) behavior of the different units Z i ⊥ ⊥ ( Y i (0 , 0) , Y i (0 , 1) , Y i (1 , 0) , Y i (1 , 1) , W i (0) , W i (1)) . W i (0) 0 1 and Assumption 3 (Exclusion Restriction) 0 never-taker defier W i (1) Y i ( z, w ) = Y i ( z ′ , w ) , for all z, z ′ , w. 1 complier always-taker The first of these two assumptions is implied by random assign- ment of Z i , but the second is substantive, and randomization has no bearing on it. 6 7

Monotonicity We cannot directly establish the type of a unit based on what we observe for them since we only see the pair ( Z i , W i ), not Assumption 4 (Monotonicity/No-Defiers) the pair ( W i (0) , W i (1)). Nevertheless, we can rule out some possibilities. W i (1) ≥ W i (0) . Z i This assumption makes sense in a lot of applications. It is 0 1 implied directly by many (constant coefficient) latent index models of the type: 0 complier/never-taker never-taker/defier W i W i ( z ) = 1 { π 0 + π 1 · z + ε i > 0 } , 1 always-taker/defier complier/always-taker but it is much weaker than that. 8 9 Implications for Compliance types: Distribution of Compliance Types Z i Under random assignment and monotonicity we can estimate 0 1 the distribution of compliance types: 0 complier/never-taker never-taker π a = Pr( W i (0) = W i (1) = 1) = E [ W i | Z i = 0] W i 1 always-taker complier/always-taker π c = Pr( W i (0) = 0 , W i (1) = 1) = E [ W i | Z i = 1] − E [ W i | Z i = 0] For individuals with ( Z i = 0 , W i = 1) and for ( Z i = 1 , W i = 0) π n = Pr( W i (0) = W i (1) = 0) = 1 − E [ W i | Z i = 1] we can now infer the compliance type. 10 11

Now consider average outcomes by instrument and treatment: E [ Y i | W i = 0 , Z i = 0] = π c π n · E [ Y i (0) | complier] + · E [ Y i (0) | never − taker] , π c + π n π c + π n Local Average Treatment Effect Hence the instrumental variables estimand, the ratio of these two reduced form esti- E [ Y i | W i = 0 , Z i = 1] = E [ Y i (0) | never − taker] , mands, is equal to the local average treatment effect E [ Y i | W i = 1 , Z i = 0] = E [ Y i (1) | always − taker] , β IV = E [ Y i | Z i = 1] − E [ Y i | Z i = 0] E [ W i | Z i = 1] − E [ W i | Z i = 0] E [ Y i | W i = 1 , Z i = 1] = π c π a = E [ Y i (1) − Y i (0) | complier] . · E [ Y i (1) | complier] + · E [ Y i (1) | always − taker] . π c + π a π c + π a From this we can infer the average outcome for compliers, E [ Y i (0) | complier] , and E [ Y i (1) | complier] , 12 13 4. Extrapolating to the Full Population 5. Covariates We can estimate Traditionally the TSLS set up is used with the covariates en- E [ Y i (0) | never − taker] , and E [ Y i (1) | always − taker] tering in the outcome equation linearly and additively, as We can learn from these averages whether there is any evi- Y i = β 0 + β 1 · W i + β ′ 2 X i + ε i , dence of heterogeneity in outcomes by compliance status, by comparing the pair of average outcomes of Y i (0); with the covariates added to the set of instruments. Given E [ Y i (0) | never − taker] , and E [ Y i (0) | complier] , the potential outcome set up with general heterogeneity in the effects of the treatment, one may also wish to allow for more and the pair of average outcomes of Y i (1): heterogeneity in the correlations between treatment effects and E [ Y i (1) | always − taker] , and E [ Y i (1) | complier] . covariates. If compliers, never-takers and always-takers are found to be Here we describe a general way of doing so. Unlike TSLS type substantially different in levels, then it appears much less plau- approaches, this involves modelling both the dependence of the sible that the average effect for compliers is indicative of aver- outcome and the treatment on the covariates. age effects for other compliance types. 14 15

Heckman Selection Model Flexible Alternative Model A traditional parametric model with a dummy endogenous vari- Specify ables might have the form (translated to the potential outcome set up used here): f Y ( w ) | X,T ( y | x, t ) = f ( y | x ; θ wt ) , W i ( z ) = 1 { π 0 + π 1 · z + π ′ 2 X i + η i ≥ 0 } , for ( w, t ) = (0 , n ) , (0 , c ) , (1 , c ) , (1 , a ). A natural model for the Y i ( w ) = β 0 + β 1 · w + β ′ 2 X i + ε i , distribution of type is a trinomial logit model: 1 with ( η i , ε i ) jointly normally distributed (e.g., Heckman, 1978). Pr( T i = complier | X i ) = a X i ) , Such a model impose restrictions on the relation between com- 1 + exp( π ′ n X i ) + exp( π ′ pliance types, covariates and outcomes: exp( π ′ ⎧ n X i ) η i < − π 0 − π 1 − π ′ ⎪ never − taker if 2 X i Pr( T i = never − taker | X i ) = a X i ) , ⎨ 1 + exp( π ′ n X i ) + exp( π ′ − π 0 − π 1 − π ′ 2 X i ≤ η i < − π 0 − π 1 − π ′ i is a complier if 2 X ⎪ ⎩ − π 0 − π ′ always − taker if 2 X i ≤ η i , Pr( T i = always − taker | X i ) = which imposes strong restrictions, e.g., if E [ Y i (0) | n, X i ] < E [ Y i (0) | c, X i ], then E [ Y i (1) | c, X i ] < E [ Y i (1) | a, X i ] 1 − Pr( T i = complier | X i ) − Pr( T i = never − taker | X i ) . 16 17 The log likelihood function is then, factored in terms of the Application: Angrist (1990) effect of military service contribution by observed ( W i , Z i ) values: The simple ols regression leads to: L ( π n , π a , θ 0 n , θ 0 c , θ 1 c , θ 1 a ) = � � log(earnings) i = 5 . 4364 − 0 . 0205 · veteran i exp( π ′ (0079) (0 . 0167) � n X i ) × a X i ) · f ( Y i | X i ; θ 0 n ) 1 + exp( π ′ n X i ) + exp( π ′ i | W i =0 ,Z i =1 In Table we present population sizes of the four treatmen/instrument � exp( π ′ � n X i ) 1 samples. For example, with a low lottery number 5,948 indi- × n X i ) · f ( Y i | X i ; θ 0 n ) + n X i ) · f ( Y viduals do not, and 1,372 individuals do serve in the military. 1 + exp( π ′ 1 + exp( π ′ i | W i =0 ,Z i =0 Z i � exp( π ′ � a X i ) 1 0 1 × a X i ) · f ( Y i | X i ; θ 1 a ) + a X i ) · f ( Y 1 + exp( π ′ 1 + exp( π ′ i | W i =1 ,Z i =1 0 5,948 1,915 exp( π ′ � a X i ) × a X i ) · f ( Y i | X i ; θ 1 a ) . W i 1 + exp( π ′ n X i ) + exp( π ′ 1 1,372 865 i | W i =1 ,Z i =0 18 19