Consider the two following linear projections in the population: L(y|1, X1, X2,

Question

Consider the two following linear projections in the population:

L(y|1, X1, X2, X3) = b0 + b1*X1 + b2*X2 + b3*X3

L(y|1, X1, X2) = a0 + a1*X1 + a2*X2

(a) Provide the formula relating a2 to b2 .
(b) Give an intuitive explanation for your answer to part (a).
(c) Suppose that X3 is uncorrelated with X1 and X2 . Does a2 = b2 ? Explain.
(d) Suppose that Cov(X2, X3) = 0, Cov(X1, X3) != 0, and Cov(X1, X2) != 0. Does a2 = b2? Explain.

Consider the two following linear projections in the population: L(y 1, X1, X2, X3) Bo X1 B2 X2 33 X3 (a) Provide the formula relating a2 to B2. (b) Give an intuitive explanation for your answer to part (a). (c) Suppose that X3 is uncorrelated with X1 and X2. Does a2 Ba? Explain. (d) Suppose that Co (X 2, X3) 30, Cov (X1, X3) 0, and Cov(X1, X2) 0. Does a 32? Explain.

Explanation / Answer

The first linear projections is nothing but adding another variable (X3) to the second equation.A parameter estimate in a regression model (e.g., b1, b2) will change if a variable, X1, X2, is added to the model that is:

An estimated beta will not change when a new variable is added, if either of the above are uncorrelated. Note that whether they are uncorrelated in the population (i.e., (Xi,Xj)=0(Xi,Xj)=0, or (Xj,Y)=0(Xj,Y)=0) is irrelevant. What matters is that both sample correlations are exactly 0. This will essentially never be the case in practice unless you are working with experimental data where the variables were manipulated such that they are uncorrelated by design.

Note also that the amount the parameters change may not be terribly meaningful (that depends, at least in part, on your theory). Moreover, the amount they can change is a function of the magnitudes of the two correlations above.

On a different note, it is not really correct to think of this phenomenon as "the coefficient of a given variable [being] influenced by the coefficient of another variable". It isn't the betas that are influencing each other. This phenomenon is a natural result of the algorithm that used to estimate the slope parameters. Imagine a situation where Y is caused by both X1 and X2 which in turn are correlated with each other. If only X1 is in the model, some of the variation in Y that is due to X2 will be inappropriately attributed to X1. This means that the value of b1 is biased; this is called the omitted variable bias.

(b), (c) ,(d) So a2=b2 when X3 is not correlated to X1,X2 and Y

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