Topic 8 Lecture 1 Estimating Policy Effects in the Presence of. Endogeneity via the Linear Instrumental Variables (IV) Method

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1 Econometrics for Health Policy, Health Economics, and Outcomes Research Topic 8 Lecture 1 Estimating Policy Effects in the Presence of Endogeneity via the Linear Instrumental Variables (IV) Method Copyright Joseph V. Terza, Ph.D All Rights Reserved Recall that for cases in which (precluding the implementation of the SLRM). We turned to the Multiple Linear Regression Model (MLRM) and the Ordinary Least Squares (OLS) Estimation Method 1

2 Example: y = number of yearly visits to the doc x p = copay Person y 15 (x p* = $15) (x p = $15) (x p $15) A (y x p = $15) B (y x p $15)

3 In this case, = E[y x p = 15] E[y 15] = Why the un-equality? Confounders, e.g., income. Let s condition on income (x ) Person y 15 (x p* = $15) (x p = $15, x = $50K) (x p $15, x = $50K) A (y x p= $15, x = $50K) B (y x p $15, x = $50K) Now 2. = E[y x p = 15, x = $50K] E[y 15 x = $50K] = 2.8 and we have narrowed the difference.

4 By conditioning on more observable confounders (e.g. x 4 = age, x 5 = education, etc. ) we can further narrow the difference. If the vector of observable confounders x o = [ x, x 4,..., x K] is comprehensive, then which implies that. From this we obtain 4

5 In the Multiple Linear Regression Model we have that Therefore, as in the SLRM, PE = (x p2 - x p1)â p. OLS is used to obtain un unbiased estimate â p (along with the remainder of the regression parameters) 5

6 Now suppose that x o is not comprehensive, i.e.. In this case there remain unobservable confounders (x u). Assuming, of course, that x u encompasses all unobservable confounders, we have and. From this we obtain 6

7 In the linear case we may write Therefore, as in the SLRM and MLRM PE = (x p2 - x p1)â p. OLS, however, is not unbiased. 7

8 Formal Assumptions of the Model Assumption a: The policy variable can be any of the four possible types (binary, count, discrete, or continuous) and the relevant representative value of is its expected value. This implies that the policy effect of interest is defined as. (8-) 8

9 Assumption b (linearity in the parameters of the true [albeit counterfactual] regression model): The counterfactual value of the random variable representing the outcome (y), at a (counterfactually) fixed value of the policy variable, can be written as a linear combination of the parameters, i.e. (8-4) where â 1,..., â K, â u are parameters, and is a random error term defined such that E[ x, x 4,..., x K, x u] = 0. (8-5) 9

10 Note that (8-4) and (8-5) yield and (8-6) (8-7) The policy maker plans a change in x p from x p1 to x p2, and from (8-7) it follows that =. (8-8) 10

11 Assumption c (comprehensiveness of x u): There exist unobservable confounders (x u) and. (8-9) In conjunction with assumption (b), this implies that (8-10) Therefore, although Assumption (c) does not require that we know the joint distribution of (x p, x o, x u, y), it does require that the joint distribution of x p, x o, x u, and y be a member of the class of distributions for which. 11

12 (8.5) DEF: The pseudo sampling model corresponding to (8-4) and (8-10), is defined as (8-11) where x i = [1 x pi x oi x ui] and x oi = [x i,..., x Ki]. Note the important distinction between (8-4) and (8-11). In (8-4) x p is essentially controlled. This is the counterfactual world of policy analysis. Expression (8-11) represents the factual/observable world of sampling. Nothing is fixed or randomized. This is reason for the different representations of the error term ( vs. e). 12

13 The true sampling model is, however (8-12) where. Now OLS applied to (8-12) cannot be unbiased. 1

14 Recall that the unbiasedness of OLS in the MLRM depended on the fact that E[e x, x ] = 0. p o Specifically recall (7.8) THEOREM: OLS estimators in the MLRM can be written as linear combinations of the regression errors e,..., e n. Specifically, (7-15) where w ki is a function of only the observable x s (x p and x o). Now so E[e] = 0 only if E[e x, x ] = 0. p o Therefore from (7-15), = 0 only if E[e x, x ] = 0. p o 14

15 The analogous condition in (8-12) would be E[e* x, x ] = 0. (8-1) p o But (8-1) cannot be true because x and x are, by definition, p u correlated (see p. 14 of Chapter 7 Advanced Econometrics Course). Therefore, in this case OLS will be biased. 15

16 The Reduced Form Model To deal with this problem we add the following assumption to the list: Assumption c : There exist identifying instrumental variables (IV), i.e. variables that satisfy the following properties: 1) they are highly correlated with x p 2) they are neither included among the elements of x o nor correlated with e in (8-11) ) they are not correlated with (mean independent of) x u 16

17 The Instrumental Variables Estimator Given the existence of IVs it is reasonable to assume that x = wá + x (8-14) p u + where w is the set of variables that includes both x o and w (the set of identifying instrumental variables). Under these assumptions, the following two-stage estimator is feasible: Stage 1: Apply OLS to (8-14) and save where is the OLS estimate of á. Stage 2: Apply OLS to the following version of (8-11) (8-15) There are a number of alternative ways to compute the IV estimator. 17