Section 5

# Section 5 - 5 Section Econ I40 GSI Edson Severnini 4.3 The...

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Section 5 - Econ I40 GSI: Edson Severnini APPENDIX 4.3 The Multiple Regression Model in Matrix Form ESTIMATING 02, t TESTS To calculate the variance-covariance matrix of the estimated parameters, we need to determine an estimate for the scalar o'. A natural choice is .2_ J- N-k (A4.2o) It is tedious, but not difficult, to prove that s2 provides an unbiased estimator of o2.It follows that r2(X'X)-t yields an unbiased estimator of Var (p). we rely on the use of the / test when s2 is used to approximate o2. To do so, we use the following statistical results: l. G'€lo2 is distributed as chi square with N - k degrees of freedom. 2. (N - k)s2lo2 is distributed as chi square with N - k degrees of freedom. ,. @, - Bil, for i : L, 2, . ., k, isnormally distributedwith mean 0 and variance o2V;, where I/; is the lth diagonal element of (X'X)- t. 4. (N - k)s2lo2 and p; - Biare independently distributed. It follows that (/.4.2r) 6',6 rN- ft Fi- Fi sYVi is r-distributed with N - k degrees of freedom. This allows us to construct confidence intervals for individual regression parameters in a manner analo- gous to the procedure described in Chapter 2. To test a hypothesis about a particular value of Ft, we substitute that value into Eq. (A4.2I). If the / value is great enough in absolute value, we reject the null hypothesis at the appro- priately chosen level of confidence. A 95 percent confidence interval for Bi is given by p, t t,1s{vS (1'4.22) where /, is the critical value of the r distribution associated with a 5 percent

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Section 5 - 5 Section Econ I40 GSI Edson Severnini 4.3 The...

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