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Instrument Variables and AB Testing – Part 1by@varunnakra1
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Instrument Variables and AB Testing – Part 1

by Varun NakraMay 9th, 2024
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This article explores the Mathematical details of least squares estimator in an unbiased and biased settings due to model specification errors.
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In my previous article, I delved into the statistical methods to perform causal analysis. One of the methodologies involved using “instrument variables”. We will delve deep into the math of that methodology in this article series. In the first part, we will cover the basics of model specification errors and confounding; and in the second part, we will cover the math behind instrument variables. Regression model specification errors – There are two prominent kinds of regression model specification errors:


  1. Specifying the model incorrectly by exclusion of relevant variables
  2. Specifying the model incorrectly by inclusion of irrelevant variables

Of course we will have to investigate how these regression model specification errors impact the regression. Before we do that, let’s cover the derivation of a basic result.

This proves that the least squares estimator is unbiased. Now, we go back to our earlier discussion on the regression model specification errors. Our objective will be to investigate the bias in the least squares error estimator because of those errors.

Now that we have covered the mathematical background of confounding, we are ready to delve deep into the math of instrument variables in the part 2 of this article.