In Example 6.7 we tested the joint null hypothesis

in the model

By substituting the restrictions into the model and rearranging variables, show how the model can be written in a form where least squares estimation will yield restricted least squares estimates.

Data From Example 6.7:-

In this example, we consider a joint test of two of Big Andy's conjectures. In addition to proposing that the optimal level of monthly advertising expenditure is \(\$ 1900\), Big Andy is planning staffing and purchasing of inputs on the assumption that when PRICE \(=\$ 6\) and \(A D V E R T=1.9\), sales revenue will be \(\$ 80,000\) on average. In the context of our model, and in terms of the regression coefficients \(\beta_{k}\), the conjecture is

Are the conjectures about sales and optimal advertising compatible with the evidence contained in the sample of data? We formulate the joint null hypothesis

The alternative is that at least one of these restrictions is not true. Because there are \(J=2\) restrictions to test jointly, we use an \(F\)-test. A \(t\)-test is not suitable. Note also that this is an example of a test with two restrictions that are more general than simply omitting variables. Constructing the restricted model requires substituting both of these restrictions into our extended model, which is left as an exercise. Using instead computer output obtained by supplying the two hypotheses directly to the software, we obtain a computed value for the \(F\)-statistic of 5.74 and a corresponding \(p\)-value of 0.0049 . At a 5\% significance level, the joint null hypothesis is rejected. As another exercise, use the least squares estimates to predict sales revenue for \(P R I C E=6\) and \(A D V E R T=1.9\). Has Andy been too optimistic about the level of sales, or too pessimistic?

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Ho B3 +3.84 = 1, B +62 +1.93 + 3.614 = 80