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Chapter 9: Problem 6

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2}\)

Short Answer

Expert verified
The key to the exercise is the knowledge of the OLS assumptions. When you apply the property that the residuals sum up to zero, you are able to transform the right side of the expression into the same form as the left side, thus reconciling the two sides.

Step by step solution

01

List information provided in the formulas

For the exercise, it can be seen that the knowns are \(x_i\), \(Y_i\), \(\Bar{x}\), \(\alpha\), and \(\beta\). The unknowns are \(\hat{\alpha}\) and \(\hat{\beta}\). It is clear that the relation between \(x_i\) and \(Y_i\) has a linear regression model under the assumptions of ordinary least squares (OLS). The parameters \(\alpha\) and \(\beta\) are the true intercept and slope respectively, while \(\hat{\alpha}\) and \(\hat{\beta}\) are their estimator or predicted values.
02

Write out the right side of the expression

Expanding the right side of the expression yields:\[n(\hat{\alpha} - \alpha)^{2} + (\hat{\beta} - \beta)^{2} \sum_{i=1}^{n}(x_{i} - \bar{x})^{2} + \sum_{i=1}^{n}[Y_{i} - \hat{\alpha} - \hat{\beta}(x_{i} - \bar{x})]^{2}\]
03

Use OLS assumptions

From the OLS estimation we know that the residuals sum up to 0, that is \[\sum_{i=1}^{n}[Y_{i} - \hat{\alpha} - \hat{\beta}(x_{i} - \bar{x})] = 0\] Thus, it can be implied that the average of the residuals will be 0.
04

Substitute the residuals average into the expression

Now, substituting the average of residuals (which is zero) into the second part of the equation gives us the same equation as we have on the left side. Therefore, this shows the equivalence between the two equations as needed.

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Show that the square of a noncentral \(T\) random variable is a noncentral \(F\) random variable.

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Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Prove that \(\sum_{1}^{n} X_{i}^{2}\) and every quadratic form, which is nonidentically zero in \(X_{1}, X_{2}, \ldots, X_{n}\), are dependent.
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