what is the likelihood function $p(y|a,tau)$ of simple linear regression model?

Question

For a simple linear regression model without intercept, that is
$$y_i=ax_i+varepsilon_i$$
where $varepsilon_isim_{iid} N(0, tau^2), i=1,2,dots, n$ and $x_i$ is a fixed covariate.
My question is what is the likelihood function $p(y|a,tau)$?
My idea is
$$y_i sim N(ax_i, tau^2)$$(not sure right?)
Then
$$p(y|a,tau)=prod_{i=1}^n p(y_i|a,tau)proptoprod_{i=1}^nexp(-frac{(y_i-ax_i)^2}{2tau^2})$$
Is it right? Thanks.

Camille Gontier · Accepted Answer

That's indeed correct. As mentioned in the comment, it is usually easier to work with the log-likelihood:
$$
log p(y|a,tau) = sum_{i} log p(y_i|a,tau)
$$
You can then easily compute the derivative of the log likelihood with respect to $a$ and $tau$ to derive the Maximum Likelihood Estimators of these parameters.

what is the likelihood function $p(y|a,tau)$ of simple linear regression model?

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