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# 金融代写|金融数学FINANCIAL MATHEMATICS代写|MAT3330 Time Series Models for Aggregated Data: Modeling the Variance

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## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Time Series Models for Aggregated Data: Modeling the Variance

In the $\operatorname{ARMA}(p, q)$ model $\phi(B) Y_{t}=\delta+\theta(B) \varepsilon_{t}$ for series $\left{Y_{t}\right}$, when the errors $\varepsilon_{t}$ are independent r.v.’s (with the usual assumptions of zero mean and constant variance $\left.\sigma^{2}\right)$, an implication is that the conditional variance of $\varepsilon_{t}$ given the past information is a constant not depending on the past. This, in turn, implies the same feature for $l$-step ahead forecast errors $e_{t}(l)=\sum_{i=0}^{l-1} \psi_{i} \varepsilon_{t+l-i}$. However, in some settings, particularly for financial data, the variability of errors may exhibit dependence on the past variability. Modeling the variance, which is essential for studying the risk-return relationship is an important topic in finance.

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Autoregressive Conditional Heteroscedastic (ARCH) Models

The autoregressive conditional heteroscedastic (ARCH) models were originally proposed by Engle (1982) [120], Engle and Kraft (1983) [124], to allow for the conditional error variance in an ARMA process to depend on the past squared innovations, rather than be a constant as in ARMA models with independent errors. For example, an $\operatorname{AR}(p)$ process with $\operatorname{ARCH}(q)$ model errors is given as $Y_{t}=\sum_{i=1}^{p} \phi_{i} Y_{t-i}+\delta+\varepsilon_{t}$, where, $\mathrm{E}\left(\varepsilon_{t} \mid \varepsilon_{t-1}, \varepsilon_{t-2}, \ldots\right)=0$ and
$$h_{t}=\operatorname{Var}\left(\varepsilon_{t} \mid \varepsilon_{t-1}, \varepsilon_{t-2}, \ldots\right)=\omega_{0}+\sum_{i=1}^{q} \omega_{i} \varepsilon_{t-i}^{2}$$
with $\omega_{0}>0$ and $\omega_{i} \geq 0$ for $i=1, \ldots, q$. In the generalized ARCH (GARCH) model, introduced by Bollerslev (1986) [46], it is assumed that
$$h_{t}=\omega_{0}+\sum_{i=1}^{q} \omega_{i} \varepsilon_{t-i}^{2}+\sum_{i=1}^{k} \beta_{i} h_{t-i}$$
where $\beta_{i} \geq 0$ for all $i=1, \ldots, k$. Much subsequent research on and applications of the ARCH and GARCH models have occurred since the publication of their research papers. It is not possible to cover the range of applications resulted from this research here, but these results are widely available to the curious reader.

Let us briefly discuss some basic results and implications of the ARCH and GARCH models. The errors $\varepsilon_{t}$ in the model have zero mean, since $\mathrm{E}\left(\varepsilon_{t}\right)=$ $\mathrm{E}\left[\mathrm{E}\left(\varepsilon_{t} \mid \varepsilon_{t-1} \ldots\right)\right]=0$, and they are serially uncorrelated; that is, $\mathrm{E}\left(\varepsilon_{t} \varepsilon_{t-j}\right)=0$ for $j \neq 0$ (since for $j>0$, for example, $\mathrm{E}\left(\varepsilon_{t} \varepsilon_{t-j}\right)=\mathrm{E}\left[\mathrm{E}\left(\varepsilon_{t} \varepsilon_{t-j} \mid \varepsilon_{t-1} \ldots\right)\right]=$ $\mathrm{E}\left[\varepsilon_{t-j} \mathrm{E}\left(\varepsilon_{t} \mid \varepsilon_{t-1} \ldots\right)\right]=0$ ). But the $\varepsilon_{t}$ are not mutually independent r.v.’s since they are inter-related through their conditional variances (i.e., their second moment). We will also assume that the $\varepsilon_{t}$ have equal unconditional variances, $\operatorname{Var}\left(\varepsilon_{t}\right)=\sigma^{2}$, for all $t$, so they are weakly stationary. Consider then, the simple case of the first-order $\mathrm{ARCH}$ or $\mathrm{ARCH}(1)$ model,
$$h_{t}=\omega_{0}+\omega_{1} \varepsilon_{t-1}^{2} .$$

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Autoregressive Conditional Heteroscedastic (ARCH) Models

$$h_{t}=\operatorname{Var}\left(\varepsilon_{t} \mid \varepsilon_{t-1}, \varepsilon_{t-2}, \ldots\right)=\omega_{0}+\sum_{i=1}^{q} \omega_{i} \varepsilon_{t-i}^{2}$$

$$h_{t}=\omega_{0}+\sum_{i=1}^{q} \omega_{i} \varepsilon_{t-i}^{2}+\sum_{i=1}^{k} \beta_{i} h_{t-i}$$

(因此 $j>0$ ， 例如， $\mathrm{E}\left(\varepsilon_{t} \varepsilon_{t-j}\right)=\mathrm{E}\left[\mathrm{E}\left(\varepsilon_{t} \varepsilon_{t-j} \mid \varepsilon_{t-1} \ldots\right)\right]=\mathrm{E}\left[\varepsilon_{t-j} \mathrm{E}\left(\varepsilon_{t} \mid \varepsilon_{t-1} \ldots\right)\right]=0$ )。但是 $\varepsilon_{t}$ 不是相互独立的 $r v$ ，因为它们䓰过们伯条件方差（即它们的二阶矩）相互关联。我们还将假设
$\varepsilon_{t}$ 具有相等的无条件方差， $\operatorname{Var}\left(\varepsilon_{t}\right)=\sigma^{2}$ ，对所有人 $t$ ，所以它们是弱静止的。那么考虑一下一阶 的简单情况ARCH或者 $\mathrm{ARCH}(1)$ 模型，
$$h_{t}=\omega_{0}+\omega_{1} \varepsilon_{t-1}^{2} .$$

## MATLAB代写

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