Time Series Analysis in Valuation Models
Model 1
The Value-at-Risk (VaR) Model generalizes the ARCH model and is used for time series analysis to detect heteroskedasticity. $$ (1 - \beta{1} B - \cdots - \beta_{p} B^p) \sigma_{t | t-1}^2 = \omega + (\alpha_{1} B + \cdots + \alpha_{q} B^q) r_{t}^{2} $$
Derivation
Let’s start with the simplest $ARCH(1)$ model.
2 Given time series data $\left\{ p_{t} \right\}$ and its returns $\left\{ r_{t} \right\}$, the data exhibiting an ARCH effect with lag $1$, i.e., autoregressive conditional heteroskedasticity, can be represented as follows: $$ r_{t} = \sigma_{t | t-1} \varepsilon_{t} $$
$$ \begin{align} \sigma_{t | t-1}^2 = \omega + \alpha r_{t-1}^{2} \end{align} $$ Here, $\alpha$ and $\omega$ are coefficients yet unknown, and $\varepsilon_{t}$ is an innovation processed as iid with a mean of $0$ and variance of $1$, not necessarily assumed to be a white noise. $\sigma_{t | t-1}^2$ is called the Conditional Volatility, and according to the following mathematical development, the square of the return $r_{t}^2$ is an unbiased estimator of $\sigma_{t | t-1}^2$. $$ \begin{align*} E \left( r_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) =& E \left( \sigma_{t | t-1}^2 \varepsilon_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) \\ =& \sigma_{t | t-1}^2 E \left( \varepsilon_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) \\ =& \sigma_{t | t-1}^2 E \left( \varepsilon_{t}^2 \right) \\ =& \sigma_{t | t-1}^2 \end{align*} $$ That $r_{t}^{2}$ is an unbiased estimator of $\sigma_{t | t-1}^2$ means that by setting $\eta_{t} := r_{t}^{2} - \sigma_{t | t-1}^2$ and substituting $\eta_{t}$ into $(1)$, one can derive an autocorrelation model $AR(1)$ for $\left\{ r_{t}^{2} \right\}$. $$ \left( r_{t}^{2} \right) = \omega + \alpha \left( r_{t-1}^{2} \right) + \eta_{t} $$ Assuming that $r_{t}$ has a certain constant population variance $\sigma^2$ and taking the expected value of both sides yields: $$ \begin{align} \sigma^2 = \omega + \alpha \sigma^2 \end{align} $$ If the appearance of $\sigma$ is confusing, it’s important to note that this unindexed $\sigma$ represents the population variance of the return $r_{t}$, not the original time series data $p_{t}$. Since $E (r_{t}) = E (\sigma_{t | t-1} \varepsilon_{t} ) = 0$, it results in: $$ \begin{align*} \sigma =& \text{var} (r_{t}) \\ =& E(r_{t}^{2}) - E(r_{t})^2 \\ =& E(r_{t}^{2}) \end{align*} $$ It is evident in the $ARCH(1)$ model that the $p_{t}$ exhibits heteroskedasticity. According to $(2)$, since $\displaystyle \omega = \sigma^2 \left( 1 - \alpha \right)$, elucidating $(1)$ about $\sigma_{t | t-1}^2$ would mean: $$ \begin{align*} \sigma_{t | t-1}^2 =& \omega + \sigma r_{t}^{2} \\ =& (1 - \alpha) \sigma^2 + \alpha r_{t}^{2} \end{align*} $$ Thus, $\sigma_{t | t-1}^2$ appears as a weighted average of $\sigma^2$ and $r_{t}$, and if $\alpha$ is closer to $1$, it suggests a higher influence of the previous return $r_{t}$ and negligible ARCH effect as it nears $0$. Analysis then fundamentally shifts to estimating coefficient $\sigma$ in model $AR(1)$ without focusing on $\omega$. Simply put, it’s recreating an ARMA model.
Generalization
The generalization of the ARCH model can be done in the same steps. $$ \sigma_{t | t-1}^2 = \omega + \beta_{1} \sigma_{t-1| t-2}^2 + \cdots + \beta_{p}\sigma_{t -p | t-p-1}^2 $$ The above would be the autoregressive model $AR(p)$ of $\sigma_{t | t-1}^2$, $$ \sigma_{t | t-1}^2 = \omega + \alpha_{1} r_{t-1}^2 + \cdots + \alpha_{q} r_{t-q}^2 $$ and the above, the moving average model $MA(q)$ of $\sigma_{t | t-1}^2$. This way, the ARMA model for $\sigma_{t | t-1}^2$ could be referred to as a Generalized ARCH Model $GARCH(p,q)$. Using the backshift operator $B$ simplifies the equation as follows. $$ (1 - \beta{1} B - \cdots - \beta_{p} B^p) \sigma_{t | t-1}^2 = \omega + (\alpha_{1} B + \cdots + \alpha_{q} B^q) r_{t}^{2} $$ Conceptually, if the VaR model is akin to the ARMA model, finding its order $p,q$ shouldn’t differ, similarly employing the EACF method.
Furthermore, volatility clustering can now be defined more sophisticatedly. Instead of a vague description of ‘variance increasing and decreasing,’ it can be stated as ’there is volatility clustering when the data follows a higher order of the VaR model.’