Maximum likelihood estimator for model without the predictor variable

Question

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?

Answer 1

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.