Consider the model y = ?x+?, where the intercept of the
Chapter 7, Problem 19SE(choose chapter or problem)
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
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