My question is about consistency of OLS parameter estimate for a linear regression model $y_t = x_t' b + \epsilon_t$ ($t = 1,2, \ldots, T)$. Here $x_t$ and $b$ are $1\times k$ vectors (there are $k$ explanatory variables).
It is known that OLS gives an unbiased estimate $\beta$ such that $E(\beta) = b$.
I know that consistency $plim_{T \to \infty} \beta = b$ holds when regressors $x_t$ are exogenous and errors $\epsilon_t$ are homoscedastic and uncorrelated. But it may hold under conditions where errors are serially correlated. I would like to see an example with exogeneity and homoscedasticity where the estimate is not consistent.
Consider the linear model $Y_i=X_i\theta + ϵ_i$ with $\theta \in \Theta$. The OLS estimator or MLE is $\hat{\theta}=\theta_0+(X′X)^{−1}X′ϵ$. If $lim_{n→\infty}n^{−1}X′X=Q$ where $Q$ is singular, or if the set $\Theta$ is not compact or if $ϵ_i$ is such that the objective function $\mathbb{Q}_0(\theta)$ is not continuous or does not have a unique maximum in $θ_0$, or if $\hat{\mathbb{Q}}_n(\theta)$ does not converge (uniformily) in probability to $\mathbb{Q}_0(\theta)$; then $plim \hat{\theta}\neq \theta_0$.