is the coefficient of the intercept an unbiased estimate of the average student performance across all the schools of India

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You want to study the student performance in the schools of India. You use random data from a sample of 350 schools in India for 2010, where the variables are: SP – Student performance (average for class XII students in that school) (%); CS – Class size, i.e. total students in class XII divided by number of sections in class XII); EXP END – Expenditure per student (Rs.); INCOME – average parental Income (Rs.); and RES – proportion of students in the reserved category (%). You can assume that the total number of schools is very large. Hence, you can think of the data as random 350 draws from all the schools with replacement. Therefore, each observation is an independent and identically distributed draw from the population distribution.

1. You regress SP on just an intercept 1.

(a) Show that the value of the coefficient of the intercept is the sample mean of SP. [Hint. You can either solve the minimization problem of choosing b such that P350 i=1(SPi − b) 2 is minimized. Alternatively, directly use P350 i=1 SPi ∗ 1 P350 i=1 1 2 as the formula for the coefficient when there is only a single regresssor which is a constant 1 in this case.]

(b) Is the coefficient of the intercept an Unbiased Estimate of the Average Student performance across all the schools of India? Can you come up with a better Unbiased Estimate? Give reasons for each. [Hint. Note that the Gauss-Markov assumptions are satisfied and hence the coefficient has the properties corresponding to the Gauss-Markov theorem: – SP = P350 i=1 SPi 350 ∗ 1 + SP − P350 i=1 SPi 350 ! ; – The expectation of the error term equals 0. Since there is no variability in the setup, the conditional expectation also equals 0 – There is no multicollinearity – Since there is no variability in the regressors, it is a homoscedastic setup. Further, independence implies an absence of autocorrelation.

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