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We have the following simple linear model without the predictor variable: $y_t = \beta_t + \epsilon_t \quad \text{ where } t = 1, 2, \ldots,n; \ \epsilon_t \sim N(0, \sigma^2)$

What is the maximum likelihood estimator for $\beta_t$?

Proceeding with the log-likelihood function and removing the constant term and the term with $\sigma^2$, we find that the problem reduces to the following minimization problem:

$\min_{\beta_t}\sum_{t=1}^{n}(y_t - \beta_t)^2$

The maximum likelihood estimator for $\beta$ is therefore the corresponding $y$. Is the interpretation correct?

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  • $\begingroup$ Is $\sigma^2$ known? $\endgroup$ – StubbornAtom Apr 10 '20 at 14:42
  • $\begingroup$ I think I have followed, and described, an MLE method, but agree that the final result for a single $\beta_{MLE}$ in our model, if single $\beta$ were the case, would be the same as the least square solution of single $\beta$. But it is the MLE interpretation about different $\beta$s that I am not fully certain yet about. $\endgroup$ – Mathophile Apr 10 '20 at 14:46
  • $\begingroup$ Yes, $\sigma^2$ is know. Say, it equals 1. $\endgroup$ – Mathophile Apr 10 '20 at 14:48
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If you're estimating a separate $\beta_t$ parameter for each data point, the maximum likelihood solution is to set $\epsilon = 0$, and so $\sigma = 0$. This model can perfectly fit your data, but is useless, since it has one parameter per data point.

A more useful model is $y_t = \beta + \epsilon_t$, with only one $\beta$ parameter. The MLE of $\beta$ is just the mean of the $y$ values. The MLE of $\sigma^2$ isn't exactly the same as the sample variance, but it's close.

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  • $\begingroup$ Thanks. I agree it's more interesting to have only one $\beta$ in the model for practical purpose. However, on a pure theoretical pursuit, if we have $\sigma \ne 0$ and hence $\epsilon = 0$ does not hold true, what can we say about the MLE of $\beta$? $\endgroup$ – Mathophile Apr 10 '20 at 14:18
  • $\begingroup$ $\sigma = 0, \beta_t = y_t$ is the MLE, and has a likelihood of 1, since all values of $y$ are exactly where the model predicts them to be. If you restrict your analysis to, for instance, $\sigma \geq .1$, then $\sigma = .1, \beta_t = y_t$ is the MLE. $\endgroup$ – Eoin Apr 14 '20 at 10:12

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