Here’s a note that clarifies two notions in econometric that I found confusing somehow. And mid-term exam is on the way so here’s a note.
playground
Let ${Y_1, Y_2, \ldots, Y_n}$ be a collection of independent and identically sampled random variables out of $f(y; \theta)$, where $\theta \in \Theta$ is a (possibly unknown) parameter associated with the distribution. In general, we want to estimate $\theta$ using our samples $Y$. An estimation of $\theta$ - random variable $W$ can be expressed as $$ W = h(Y_1, \ldots, Y_n) $$
unbiasedness
Definition If $\forall \theta\in \Theta$, $\mathbb E[W] = \theta$, $W$ is an unbiased estimator.
Remark small sample property: the ‘unbiasedness’ property is defined for any serve of sample size–doesn’t matter how large $n$ is it shall holds anyway.
consistency
Definition Let $W_n$ be an estimator of $\theta$ given sample ${Y_1, Y_2, \ldots, Y_n}$. If $n \to \infty$, for any $\epsilon > 0$: $$ P[|W_n - \theta| > \epsilon] \to 0 $$ we call such $W_n$ a consistent estimator. We’d also call $\theta$ the probability limit of $W_n$, denoted as $\text{plim}(W_n) = \theta$.
Properties
continuity Let $g(\cdot)$ be some continuous function, define another parameter $\gamma = g(\theta)$. Assume $\text{plim}(W_n) = \theta$. Then if we take an estimation of $\gamma$ as $G_n = g(W_n)$, we’d also have $\text{plim}(G_n) = \gamma$.
e.g. Sample variance $S_n^2 = \frac1{n-1} | \mathbf Y - \bar Y|_2^2$ is an unbiased (hence consistent) estimator of $\sigma^2$. Then, $S_n = \sqrt{S_n^2}$, despite being not biased (i.e. $\mathbb E[S_n] \ne \sigma$), it’s consistent ($\text{plim}S_n = \sigma$).
composition If $\text{plim}T_n = \alpha$, $\text{plim}U_n = \beta$:
(i) $\text{plim}(T_n + U_n) = \alpha + \beta$
(ii) $\text{plim}(T_nU_n) = \alpha\beta$
(iii) $\text{plim}(T_n/U_n) = \alpha/\beta$ (for $\beta\ne 0$).
Actually, there might be that for general $\text{plim}T_n^k = \alpha^k, k = 1, …, K$ and continuous function $f: \mathbb R^K \to \R$ $$ \text{plim}f(\mathbf T) = f(\alpha) $$