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8.6 Summary

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The least squares estimator $\hat{\beta}=(X^{\prime}X)^{-1}X^{\prime}Y$ is a random variable, since it is a function of the random variable $Y$ .
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To obtain a sampling distribution for $\hat{\beta}$ we write

$\hat{\beta}=AY$

where $A=(X^{\prime}X)^{-1}X^{\prime}$ and so $\hat{\beta}$ is a linear combination of $n$ normal random variables $Y_{1},\ldots,Y_{n}$ .
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From this it follows that the sampling distribution is

$\hat{\beta}\sim\operatorname{Normal}(\beta,\sigma^{2}(X^{\prime}X)^{-1}).$
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We can use this distribution to calculate confidence intervals or conduct hypothesis tests for the regression coefficients.
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The most frequent hypothesis test is to see whether or not a covariate has a ‘significant effect’ on the response variable. We can test this by testing

$H_{0}:\beta_{j}=0$

vs.

$H_{1}:\beta_{j}\neq 0.$

A one-sided alternative can be used if there is some prior belief about whether the relationship should be positive or negative.
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In a similar way, a sampling distribution can be derived for linear combinations of the regression coefficients $a^{\prime}\hat{\beta}$ :

$a^{\prime}\hat{\beta}\sim\operatorname{Normal}(a^{\prime}\beta,\sigma^{2}a^{% \prime}(X^{\prime}X)^{-1}a).$