ˆβk&βkσ'nj'1w2k,j-N[0,1](18)The error variance can be estimated similar to the case of the two-variable linearσ2regression model, namely using the sum of squared residualsSSR''nj'1ˆU2j,(19)whereˆUj'Yj&'ki'1ˆβiXi,j(20)is the OLS residual.It can be shown that under Assumptions 1-3, 'nj'1ˆU2jσ2-χ2n&k.(21)Since the expected value of a distributed random variable is n!k, the result (21) suggests toχ2n&kestimate byσ2ˆσ2'1n&kjnj'1ˆU2j.(22)Due to (21), this estimator is unbiased: Moreover, it can be shown that underE[ˆσ2]'σ2.Assumptions 1-3, is independent of the ‘s, hence it follows from (18) and (21) and'nj'1ˆU2jˆβithe definition of the tdistribution that under Assumptions 1 and 2,
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