You are interested in estimating how the number of days of work missed affects t

Question

You are interested in estimating how the number of days of work missed affects the size of an employee's annual raise. You collect data on the size of the raise ("raise") and the number of days of work missed ("missed") for 5 employees at a Santa Cruz convenience store. You also collect data on the distance that each employee lives from the convenience store. a) Regress the size of the raise on the number of days missed. That is, estimate the coefficients B_0 and B_1 in the regression below using ordinary least squares. Raise_i = beta_0 + beta_1missed_i + u_i b) You are concerned that the estimated effect of days missed on the size of the raise may be biased by omitted variables. Suppose that being motivated" is an omitted variable. Carefully explain how omitting "motivated" from the regression is likely to bias the estimate of B_1. c) You intend to use the distance an employee lives from the store as an instrumental variable for days of work missed. Compute the instrumental variables estimate for B_1 using the data above. Show the equation you use to get your answer. d) Compare the instrumental variables estimate B_1 above to the ordinary least squares estimate of B_1 in part a). Do they differ in the way we predicted in part b) of this question? Explain.

Explanation / Answer

a) Regression the size of raise on the number of days missed

coefficients 1 = [nxy - xy]/[n(x2 - (x)2] = [ 5* 60 - 27*17] / [ 5*211 - 272]

= -159/326 = -0.48773

0 = Mean(y) -1 * Mean (x) = (17/5) + 0.48773 * (27/5) = 6.0337

so y = -0.4877x + 6.0337 + ui

(b) Lets say being " motivated is an ommitted variable. The variable can be summarized as either motivated ( xm = 1) and demotivated ( xm = 0) . So employees can be either motivated or can be demotivated

Lets say y = a + b1x + b2 xm ; so an employeed can either be motivated or demotivated which will the error part in the question. Like for employees 2 and 5 , here are positve error ( real raise - estimated raise by regression), so these employees can be said motivated and on same pattern, employees 1, 3 and 4 have negative error , so it can be said that they are demotivated employees. Here when we have ommitted thae motivated variable that create the beiases in estimation of B1. It may have more different kind of relationship between raise and number of missed days. Here if b2 > 0 , then it will be positive biased and vice-versa.

(c) W e have regressed now, the number of days missed based on the distance of residence of the employee from the store.

coefficients 1 = Cov [ distance, raise] / Cov[ distance, missed days] = -2.8/6.4 = -0.4375

formula for Cov (xi , yi)= (1/n-1) (x -xi)(y - yi)

Here Cov [ distance, raise]  and COv[ distance, meissed days] will be calculted by formula for Cov (xi , yi)= (1/n-1) (x -xi)(y - yi)

(d) Here we have seen that B1 from part a is abou -0.4877 and by in part c, is - 0.4375. Yes they differ . and in part(b) we predicted that coefficent will change on external facotrs or may be becuase of some dummy variables We can say that, the correlation coefficient get reduced and it may be because of some ommitted variable.

Employee missed(x) raise(y) XY X^2 Y^2 2 1 5 5 1 25 5 2 6 12 4 36 3 5 3 15 25 9 1 9 2 18 81 4 4 10 1 10 100 1 Sum 27 17 60 211 75

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