Question

Discuss with an example one of the problems with standard regression analysis that can be solved by using instrumental variable(s). What are the conditions that the instrumental variable needs to satisfy? Discuss (you can refer to your example).

Answer 1

The problem with standard regression analysis is the auto-correlation, which is the endogeneity of x, meaning that changes in x are associated not only with changes in y but also changes in the error.

Consider the regression of earnings (y) on years of schooling (x). The error term u embodies all factors other than schooling that determine earnings, such as the ability. Suppose a person has a high level of u, due to high (unobserved) ability. This increases earnings, since y = x + u. But it may also lead to higher levels of x since schooling is likely to be higher for those with high ability.

The consequences of this correlation between x and u are that higher levels of x have two effects on y. There is both a direct effect via x and an indirect effect via u affecting x which in turn affects y. The goal of regression is to estimate only the rst effect, yielding an estimate of . The OLS estimate will instead combine these two effects, giving > b in this example where both effects are positive.

The OLS estimates the total effect + du=dx rather than alone. The OLS estimator is therefore biased and inconsistent for , unless there is no association between x and u.

This problem is solved by using instrumental variable(s).

Let the instrument variable is z

The instrumental variable(s) z has the property that changes in z are associated with changes in x but do not lead to change in y. The z and y will be correlated, but the only source of such correlation is the indirect path of z being correlated with x which in turn determines y. The more direct path of z being a regressor in the model for y is ruled out.

The conditions that the instrumental variable needs to satisfy:

1) z is uncorrelated with the error u

2) z is correlated with the regressor

3) z affects the outcome variable y only through x

For regression with scalar regressor x and scalar instrument z, the instrumental variables (IV) estimator is defined as

bIV = (z 'x)^(-1) z'y

wherein the scalar regressor case z, x and y are N*1 vectors.

This estimator provides a consistent estimator for the slope coefficient in the linear model y = x + u if z is correlated with x and uncorrelated with the error term.

bIV = (dy/dz) / (dx=dz)

bIV = ((z 'z)^(-1) z'y) / ((z 'z)^(-1) z'x

=(z'x)^(-1)z'y

This shows that the instrument variable resolves the problem of the auto-correlation, which is the endogeneity of x, meaning that changes in x are associated not only with changes in y but also changes in the error.