s2 is an unbiased estimator of sigma2

Problem

• Data: http://media.nek.lu.se/data/Estimating_sigma2.xlsx
• In this problem we will investigate if \(s^2\) is an unbiased estimator of \(σ^2\) .
• Our x-data is simulated in the D-column from a normal where you can set the expected value in B2 and the standard deviation in B3. Check the formula in cell D2.
• Error terms are simulated in the E-column with expected value 0 and you can set \(σ\) in B4
• Y-data is simulated in the F-column from the LRM where you can set \(β_1,β_2\) in B5 and B6. Check the formula in cell F2 and make sure it makes sense to you.
• \(β_1,β_2\) are estimated in A11 and A12 (check the formula)
• We have fitted values in the G column. Check the formula in cell G2 and make sure it makes sense to you.
• We have residuals in the H column. Check the formula in cell H2 and make sure it makes sense to you. Also, explain to yourself the difference between the error terms in the E column and the residuals in the H column.
• Finally, we have \(s^2\) in B22. Does “=SUMSQ(H2:H21)/18” make sense to you?
• Now here is the main point. \(s^2\) is an estimator of \(σ^2\) (which is in B21). By pressing F9, you can redo the simulation and get a new estimate \(s^2\) . If \(s^2\) is an unbiased estimator of \(σ^2\) then it should sometimes be bigger that \(σ^2\) and sometimes smaller, but on average equal to \(σ^2\) . This should be true no matter how we set the parameters in B2-B6, in particularly, no matter how we set \(σ\) . Try it!