instrumental variables

The technique of instrumental variables is used in regression analysis. A standard assumption of regression is that the explanatory variables are uncorrelated with the unexplained component. When that assumption fails, ordinary linear regression will not provide consistent estimates. However, if an instrument is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the regression, that is correlated with the explanatory variable, and that is uncorrelated with the error term. For the regression equation

y_i = x_i \beta + \epsilon_i,

the standard assumption is that

E x_i \epsilon_i = 0.

The method of moments estimating equation would then be

\sum_i x_i (y_i - x_i \beta) = 0.

This produces the estimator

\hat{\beta} = \frac{\sum_i x_i y_i}{\sum_i x_i^2} = \frac{\sum_i x_i (x_i \beta + \epsilon_i)}{\sum_i x_i^2} = \beta + \frac{\sum_i x_i \epsilon_i}{\sum_i x_i^2}.

When x and

\epsilon

are uncorrelated, the final term vanishes in the limit providing a consistent estimator. When the standard assumption fails, an instrument z known to be uncorrelated with the error term can be used. This provides the instrumental variables moment condition

\sum_i z_i (y_i - x_i \beta) = 0,

which produces consistent estimates. This produces the estimator

\hat{\beta} = \frac{\sum_i z_i y_i}{\sum_i z_i x_i} = \frac{\sum_i z_i (x_i \beta + \epsilon_i)}{\sum_i z_i x_i} = \beta + \frac{\sum_i z_i \epsilon_i}{\sum_i z_i x_i}.

When z and

\epsilon

are uncorrelated, the final term vanishes in the limit providing a consistent estimator. Classic examples of the technique include the errors in variables problem, and the recovery of structural parameters from simulataneous equations models such as supply and demand.

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