Question

1.Consider a regression of y on two explanatory variables, x1 and x2, which are potentially correlated (though not perfectly). Say that x1 can take on any value between 1 and 100. A researcher draws a random sample of observations, with information on y, x1 and x2. She runs a regression on this sample, which we refer to as regression A. She then takes the subset of the data where x1 is restricted to only take values between 1 and 50, but there is no restriction on x2. She runs another regression, which we refer to as regression B. (a) [4] Do you expect the estimated coefficients to differ between regressions A and B? Explain. (b) [5] Do you expect any difference in the precision of the estimated coefficients between regressions A and B? Explain.

Answer 1

a)The estimated coefficients are expected to differ because the descriptive statistics of x1 and y have changed impacting the standard deviations and the mean of the data. The mean of x is expected to reduce and no restriction on x2 might not produce a similar result as in regression A.

b)As the data on x2 is not restricted, the model maynot produce a similar R^2. The new model can produce either a better R2 or a lower R^2 model. If one produces a model with a better R^2, the precision of the coefficients is expected to increase(lower SD) and similarly if the new model has a lower R^2, the precision is expected to reduce(higher SD).