Question

Consider a regression of y on x1, x2 and x3. You are told that x1 and...

  1. Consider a regression of y on x1, x2 and x3. You are told that x1 and x3 are positively correlated but x2 is uncorrelated with the other two variables.
    1. [3] What, if anything, can you say about the relative magnitudes of the estimated coefficients on each of the three explanatory variables?
    1. [6] What, if anything, can you say about the precision with which we can estimate these coefficients?
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Answer #1

PART a

The magnitudes of the coefficients of the model depend first of all on the correlation that the independent variables have with the response variable "y".

But suppose that x1, x2 and x3  are strongly correlated with y. Since x1 and x3 are positively correlated, then the coefficients of both variables will have the same sign in the model, but only one of them will be significantly different from zero, while the coefficient of x2 will be significantly different from zero and with a sign opposite to that of the other two.

PART b

The precision with which we estimate the coefficients of a variable does not depend on the correlation that variable has with the others, it mainly depends of the variability of the variable itself, of the variability of the errors and of the size of the sample

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