# 计量经济学 Econometrics ECON20110T

Consider a generic diffusive process governed by
$$d x=\mu(x) d t+\sigma(x) d w$$
for some vector-valued process $x$. We typically have a set of observations denoted by
$$x\left(t_{0}\right), x\left(t_{1}\right), \cdots, x\left(t_{N}\right)$$
for some set of $N$ times, usually equally spaced, e.g., $t_{i}=i h$ for some time step $h$. Following Yu (2014), we distinguish the following two limiting cases:
\begin{aligned} &N \rightarrow \infty, h \text { fixed } \ &h \rightarrow 0, N \text { fixed } \end{aligned}

## ECON20110T COURSE NOTES ：

for $i=1, \ldots, N$ and (instantaneous) covariance structure given by $d w_{i} d w_{j}=$ $\rho_{i j} \sigma_{i} \sigma_{j} d t \equiv X_{i j} d t$. Consider now the sample covariance of $T+1$ discretely observed realizations:
$$\bar{v}=\frac{1}{T} \sum_{i=1}^{T} \Delta z_{i} \Delta z_{i}^{T}-\bar{\mu} \bar{\mu}^{T}$$
where $\Delta z_{i} \equiv z_{i}-z_{i-1}$ and the sample mean $\bar{\mu}$ is given simply by
$$\bar{\mu}=\frac{1}{T} \sum_{i=1}^{T} \Delta z_{i}=\frac{1}{T}\left(z_{T}-z_{0}\right)^{71}$$