Consider the model y = ?x+?, where the intercept of the

Chapter 7, Problem 19SE

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Consider the model \(y=\beta_{x}+\varepsilon\), where the intercept of the line is known to be zero. Assume that values \(\left(x_{1}, y_{1}\right)\), . . . , \(\left(x_{n}, y_{n}\right)\) are observed, and the least-squares estimate \(\hat{\beta}\) of \(\beta\) is to be computed.

a. Derive the least-squares estimate \(\hat{\beta}\) in terms of \(x_{i}\) and \(y_{i}\).

b. Let \(\sigma^{2}\) denote the variance of \(\varepsilon\) (which is also the variance of y). Derive the variance \(\sigma_{\widehat{\beta}}^{2}\) of the least-squares estimate, in terms of \(\sigma^{2}\) and the \(x_{i}\).

Equation Transcription:

Text Transcription:

y=beta{x}+varepsilon

(x_1,y_1)

(x_n,y_n)

hat{beta}

beta

hat{beta}

x_i

y_i

sigma^2

varepsilon

sigma_hat{beta}^2

sigma^2

x_i