Instrumental approach

Instrumental approach and two SLS/three SLS approach.

Instrumental approach or you can say variable is the part of the endogeneity problem. Here we will discuss the endogeneity problem.

Endogeneity problem means – when we have three types of situation (1) omitted variable bias, (2) measurement error, (3) Reverse causality. Here we will discuss the reverse causality.

Reverse causality arises when we have a function

Y= α + βX

Here Y variable is caused by the X variable. That means we are not sure the direction of causality. The direction of causality makes no sense which variable is dependent variable and which variable is independent variable. This cause will generate the biased parameter of interest.  We are interested to get the unbiased parameter of interest (β). To get unbiased parameter of interest, we take the help of instrumental variable approach. How instrumental variable approach is helping to get the parameter of interest (β). Let us discuss with an example.

Let you have data on bypass-surgery outcomes. You are interested to study the effect of outcome on experience. Here experience is measured through the number of surgeries performed by the surgeon in the previous year. Outcome is measured the success rate of performed surgeries in the previous year.

Outcome = β0 + β1Experience + X_hY_h + X_iY_i + ε -------- (eq-1)

Here X_{h =} represents the medical characteristics like whether medical is teaching hospital, X_i represents the individual characteristics presence of diabetics, age etc. Y_h   and Y_i are two vectors of coefficient of two variables. If we run the regression, we will get the positive coefficient β1. That means experience does have positive effect on outcome. If a doctor is more experienced, then success rate of surgeries will be more. Other variable are in the model are called control variables. Before concluding the above equation (eq-1), In another way we can say that more number of surgeries by doctor lead the superior outcome. That relation relationship leads the equation is as follow. That we will remind the reverse causality, here there is reverse causality between outcome and experience. That means we are sure whether experience does have cause on outcome or outcome does have cause on experience. To know the actual fact we will proceed to next equation.

Experience = α0 + α1Outcome + other stuff + ε -------- (eq-2)

Eq-2 will valid if doctor has superior medical surgeries, then more number of patients will come. That means outcome variable has positive effect on experience variable. Here we are confused whether outcome causes the experience or experience causes the outcome. In (eq-2) we add another variable that is other stuff. What is the importance of this variable we will discuss in detail in subsequent section?

To avoid our confuse or you can say to get the unbiased parameter of interest, we add another variable (other stuff) is called instrument variable in eq-2.  The importance of variable is more in this model for getting the unbiased parameter of interest. When we add an another variable called the instrumental variable, should know the features or characteristics it should have in this model.

(1)   The instrumental variable (other stuff) should not be directly related with variable “outcome” in any manner. It should be related with variable outcome not directly but through variable experience. The direction of relation is depicted here as follow.

             Other stuff                         Experience                             Outcome

(2)   The variable (other stuff) should not be related with error term of the model.  It should be uncorrelated with error term (ε).

(3)   Third characteristic- it (Instrumental variable – Other stuff) should be related with endogenous variable “Experience” in this model.

Here we can judge the impact of instrumental variable “other stuff” on variable “outcome” through the endogenous variable “experience”. This is the main and essential condition of instrumental variable approach.

To execute the strategy, we have an example- IIT professor came up with good instrument for surgeon experience. He noted that the caseloads of surgeons who retires or leave a hospital get distributed among the remaining surgeons. Assuming the caseload of an existing colleague is not associated with any other determinants of surgery outcomes, it can serve as an instrument for surgeon experience in the model represented by equation (1). In other words, existing colleagues’ volume is the “other stuff” required for equation (2).

To execute the strategy, professor first confirmed that there is a relationship between experience (the endogenous variable, call it “Experience”) and existing surgeons’ volume (the instrument, call it “other stuff”). This is called first stage of instrumental variable approach of second least square regression (IV/2SLS).

First stage –      Experience = α0 + λ1existing surgeon (stuff) + X_hY_h + X_iY_i + ε -------- (eq-3)

Where X_h and Xi are the vectors of coefficients on the various hospital and individual characteristics, respectively. For technical reason, it is essential to include all the independent repressors from equation (1) in the first stage, plus instrument variable (s). Nest, professor used the coefficients from the first stage to obtain predicted values for experience (call this variable experience –hat).

Then we run the second-stage regression. in which the endogenous variable (Experience ) is replaced by Experience –hat ; in all other respects it corresponds to the original model (equation -1).

Second stage-    Outcome = ψ0 + β1Experience -hat+ X_hY_h + X_iY_i + ε -------- (eq-4)

Why does this fix the problem? Because we have replaced Experience (endogenous variable), which is partially determined by outcome, with experience-hat, which is estimated using only exogenous variables. The coefficient on Experience-hat is an unbiased estimate of β1. The standard error must be corrected for the fact that Experience-hat is estimated by another regression and thus contains some noise.

How you will test your instrumental variable

Here we have to follow three tests that should perform when using instruments.  First, it should ensure that there is indeed a correlation between your proposed instrument and endogenous variable for which it is instrumenting. It should be checked that instrument variable should be statistically and economically significant coefficient in the first stage regression.

Second when you are sure that your first stage regression statistically significant, then go for the Housman test for endogeneity:

(1)   Regress Y =  a +bx +dx-hat res + other predictors (where X-hat is the residual from the first stage regression)

(2)   Test the significance of the d using the t statistics. If d is significant, then x is indeed endogenous and you should instrument it.

(3)   Third, you should design your own tests to confirm that your instrument truly does not belong as a predictor variable in the model ( i.e., test the assumption that “other stuff” is uncorrelated with the error term).

<----Joy hind---->