Derivation of beta hat 1 from the simple linear regression equation

Asked Feb 10 '20 at 23:32

Active Feb 11 '20 at 00:00

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I have a no-intercept relationship:

$$y_{i} = \beta_{1}x_{i} + \varepsilon_{i}$$

where $\varepsilon_{i} \sim \text{ iid } \mathcal{N}(0, \sigma^{2})$, and $i = 1, \dots, n$.

How do I derive $\hat{\beta}_{1}$, the least-squares estimator of $\beta_{1}$?

edited Feb 10 '20 at 23:35

Alexis

asked Feb 10 '20 at 23:32

CC25

write the likelihood of your data – gunes Feb 11 '20 at 00:01
You don't need any assumptions about the $\varepsilon_i$ except that they have zero expectation. Thus, a likelihood-based derivation would not clearly reveal the underlying concepts, which are brought out most strongly in the algebraic formula of the "normal equations" and the geometric perspective. See https://stats.stackexchange.com/a/46508/919 (inter alia) for details and working code. – whuber Feb 11 '20 at 17:12

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