I am considering the linear regression model
\begin{equation} Y_{i} = A t_{i} + B + \varepsilon_{i} \tag{$1$} \end{equation}
where $A$ and $B$ are unknown real parameters and $(\varepsilon_{i})_{1 \leq i \leq n}$ are observed values of independent and identically distributed gaussian random variables with mean $0$ and variance $\sigma^{2}$ ($\sigma^{2}$ is unknown).
In matrix form, $(1)$ writes :
\begin{equation} Y=X\alpha + \varepsilon \tag{2} \end{equation}
where $X = \begin{bmatrix} t_{1} & 1 \\ \vdots & \vdots \\ t_{n} & 1 \end{bmatrix}$, $\alpha = \begin{bmatrix} A \\ B \end{bmatrix}$ and $\varepsilon \sim \mathcal{N}(0,\sigma^{2}\mathrm{I}_{n})$.
Using Matlab, I generated some data artificially : I took $n=100$, $A=3$, $B=-1$ and $\sigma^{2} = 0.64$ and the parameters estimates I obtain are : $\widehat{A}=3.0096$, $\widehat{B}=-0.9260$ and $\widehat{\sigma^{2}}=0.6337$.
The log-likelihood of $(2)$ is the function
$$ \ell(A,B,\sigma^{2}) = -n \log(\sigma) - \frac{1}{2\sigma^{2}} \Vert Y-X\alpha \Vert^{2} $$
This may be a silly question but I do not understand why the true values of the parameters do not maximize the log-liklihood in my case. Is it because the estimates I obtained using Matlab are observed values of random variables which asymptotically converge to the true values of the parameters and here, $n$ is not "big enough" ?
