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By Ian Birnbaum

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What we have done here, in effect, is to·apply constrained linear regression to the sample. The line obtained is taken as an estimate of the population line. Now, in the unconstrained case, by analogy, it seems we should choose the estimate by minimising the residuals for each set of values of X 1 , X 2 and X 3. Unfortunately, however, it is most unlikely that we will get the same values of the regression coefficients for each set of values of X 1 , X 2 and X 3. e4 2 for each set of X 1 , X 2 and X 3 values in the sample.

If this is the case we say that b4 1 is (conditionally) unbiased. 9 Hence we need to assume unconstrained regression to ensure unbiasedness; in the constrained case we normally have biased estimates (in the sense defined here). Secondly, we would like the variance of our estimates to be as small as The Theory of Causal Analysis 29 possible so that they tend to cluster close to the population value (assuming unbiasedness). To ensure this we need to assume the homoscedasticity of ~4 on x1, x2 and x3.

One, called the forward approach, first chooses that predictor variable which correlates best with the response variable; this is step 1. Now a second predictor variable is chosen which, together with the first, correlates best; this is step 2. The process continues until some criterion measure like the increase in correlation obtained becomes small enough (or a hypothesis test based on this idea can be used), when the variables chosen are retained as predictors and the rest discarded. The second method is the backward approach.

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