estimate of variance-covariance matrix for linear models with correlated observations

Question

For linear models $$y_{n \times1}=X_{n \times p}\beta_{p \times 1}+\epsilon_{n \times 1}, \text{ where }\epsilon \sim N(0,V)$$ If in a real life problem we have data as $(y_1,x_1),(y_2,x_2),...,(y_n,x_n)$ where $x_i$'s are row vectors in $X$, and we want to fit the data using the linear model above. Then how could we get a estimate of $V$?

I know usually, we are interested in estimating $\beta$ and the covariance of $\beta$. But here I want to find out the distribution of $y$, say $y \sim N_n(X\beta, V)$, then what is $V$? Is there an estimate of it?

For me, the vector $y$ is just one observation from the multinormal distribution $y \sim N_n(X\beta, V)$. Like in the univariate normal case, if we have an observation $z$ from $N(\mu, \sigma^2)$, we can estimate $\mu$ as $\hat{\mu}=\bar{z}=z$, but we can not estimate $\sigma^2$ since we only have one observation. So does that mean we cannot estimate $V$ here?

Moreover, if we have a linear mixed effect model $$y=X\beta+Z\alpha+\epsilon$$ where $\beta$ is the fixed effect and $\alpha$ is the random effects, and $$\epsilon \sim N(0, V) \text{ and }\alpha \sim N(0, \Omega)$$ Then is there a way to estimate $V$ and $\Omega$? Then we could find the covariance matrix of $y$.

Thanks!

Answer 1

In both models, the variance components (i.e., all parameters in the specification of the variance-covariance matrices $V$ and $\Omega$) are typically estimated using either the Restricted Maximum Likelihood or Maximum Likelihood approaches. These two approaches are, for example, implemented in function lmer() in the lme4 package, and functions gls() and lme() in the nlme package in R.