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?