这是一份 Imperial帝国理工大学 MATH97233作业代写的成功案例
\begin{aligned}
&y_{t}=\alpha+\sum_{i=0}^{\infty} \beta_{i} x_{t-i}+\varepsilon_{t} \
&y_{t}=f\left(x_{t}, x_{t-1}, x_{t-2}, \cdots\right)
\end{aligned}
showed that translates into the following linear econometric equation:
$$
y_{t}=\alpha+\sum_{i=1}^{p} \phi_{i} y_{t-i}-\sum_{j=1}^{q} \theta_{j} x_{i-j}+\sum_{i=1}^{m} \sum_{j=1}^{s} \beta_{i j} y_{t-i} x_{t-j}+\varepsilon_{t}
$$
where $p, q, m$, and $s$ are nonnegative integers.
MATH97233 COURSE NOTES ：
across-time averages of both sides of and utilizing the fact that $E[\varepsilon]=0$ by assumption:
$$
E[y]=f(x)+E[\varepsilon]=f(x)
$$
or, equivalently,
$$
f(x)=\frac{1}{T} \sum_{t=1}^{T} y_{t}
$$
where $T$ is the size of the sample.
To make sure that the estimation of $f(x)$ considers only the values around $x$ and not the values of the entire time series, the values of $y_{t}$ can be weighted by a weight function, $w_{t}(x)$. The weight function is determined by another function, known as a “kernel function, ” $K_{h}(x)$ :
$$
w_{t}(x)=\frac{K_{h}\left(x-x_{t}\right)}{\sum_{t=1}^{T} K_{h}\left(x-x_{t}\right)}
$$