### Asymmetric Power Multiplicative Error Model

#### Introduction

Inferring variance dynamics from an appropriate variance proxy is an alternative to the classical approach. Popular examples of such proxies are the realized variance and the Parkinson daily range estimator (Parkinson, 1980).

The realized variance for a certain trading day is defined as the sum of the squared intraday returns. The intraday returns used by V-Lab are computed in five minutes intervals.

The Parkinson daily range estimator is defined as:

$vt ≔ 1 4ln2 phigh,t - plow,t 2$

where ${p}_{\mathrm{high},t}$ and ${p}_{\mathrm{low},t}$ are, respectively, the maximum and minimum log prices between the opening and the closing of day $t$. Under the assumption of constant variance and zero drift within the day, the range is an unbiased estimator of the latent variance of returns.

#### Pre-Processing

Both the realized variance and the range are estimators that only use information recorded between the opening and the closing of the trading day, so they do not take into account the overnight information. There are several solutions suggested in the literature that account for the overnight information. V-Lab corrects the variance proxy (the realized variance or the daily range) by multiplying the series by a time varying adjustment factor in order to ensure that the average of the variance proxy approximately matches the sample variance of the returns. The adjustment factor is obtained as follows.

Let ${V}_{t}$ be variance proxy at time $t$, i.e. ${V}_{t}$ is the realized variance or the high-low range at time $t$. And let ${V}_{t}^{e}$ be the exponentially weighted moving average of the variance proxy ${V}_{t}$. That is:

$Vte = 1-λ Vt-1 + λ Vt-1e$

for some $\lambda \in \left(0,1\right)$. Analogously, let ${r}_{t}^{2e}$ be the exponentially weighted moving average of the squared return at time $t$, so:

$rt2e = 1-λ rt-12 + λ rt-12e$

V-Lab uses $\lambda =0.05$ and, for the initial conditions of the recursion, V-Lab sets ${V}_{1}^{e}={V}_{1}$ and ${r}_{1}^{2e}={r}_{1}^{2}$.

Then, the adjustment factor ${c}_{t}$ is defined as:

$ct ≔ rt2e Vte$

and, finally, the adjusted variance proxy ${v}_{t}$ is defined as:

$vt ≔ ct Vt$

From now on, when we say variance proxy, we actually mean the adjusted variance proxy ${v}_{t}$, and that is the notation we use in the following subsections.

#### Definition

Consider a variance proxy (realized variance or high-low range, for instance) time series ${v}_{t}={\mu }_{t}{\epsilon }_{t}$, where and $D$ is any distribution with non-negative support with unit mean and variance ${\sigma }_{\epsilon }^{2}$. So, ${\mu }_{t}:={𝔼}_{t-1}\left[{v}_{t-1}\right]$ is the conditional mean of ${v}_{t}$.

Despite having serially uncorrelated error terms, the variance proxy time series, ${v}_{t}$, does not need to be serially independent. For instance, its conditional mean or powers of its conditional mean may depend on past information. The Asymmetric Power Multiplicative Error Model ($\mathrm{Asy. Power MEM}$) assumes a specific parametric form for this dependence. More specifically, we say that ${v}_{t}\sim \mathrm{Asy. Power MEM}$ if:

$μtδ = ω + α vt−1 − γvt−1st−1 δ + β μt−1δ$

where $-1<\gamma <1$, and ${s}_{t-1}=\mathrm{sign}\left({r}_{t-1}\right)$.

#### Estimation

V-Lab assumes that the distribution of the fundamental innovation ${\epsilon }_{t}$ is Chi-Square, i.e. $D\equiv {\chi }_{\left(1\right)}^{2}$. Notice that it does not imply that the variance proxy (realized variance or high-low range) ${v}_{t}$ is Chi-Squared distributed. Even though its conditional distribution is Chi-Square, its unconditional distribution may be far more complex. In fact, assuming that the conditional distribution is Chi-Square is not as restrictive as it seems, since the parameters can be estimated by Quasi-Maximum Likelihood (QML). Regardless, in a more general scenario, one could assume that the conditional distribution is Gamma, and its parameters could also be estimated by QML. This would be more general because the Gamma distribution nests both the Chi-Square and the Exponential distributions.

Notice that modeling the variance proxy (realized variance or high-low range) with the $\mathrm{Asy. Power MEM}$ captures a stylized fact in financial time series, variance (hence, volatility) clustering. The variance proxy is more likely to be high at time $t$ if it was also high at time $t-1$. Another way of seeing this is noting that a shock at time $t-1$ also impacts the variance proxy at time $t$. The $\mathrm{Asy. Power MEM}$ also captures in the variance proxy an idea analogous to the long-memory property of returns discussed in Ding, Granger, and Engle (1993). See the APARCH model description in V-Lab for a more detailed exposition. The $\mathrm{Asy. Power MEM}$ model, as the $\mathrm{Asy. MEM}$ model, additionally captures asymmetry in the variance proxy. That is, the variance proxy tends to increase more when returns are negative, as compared to positive returns of the same magnitude. Usual restrictions on $\mathrm{Asy. Power MEM}$ parameters are that $\delta ,\omega ,\alpha ,\beta >0$ and $-1<\gamma <1$.

#### Prediction

Let ${v}_{T}$ be the last observation in the sample and let $\left(\stackrel{ˆ}{\delta },\stackrel{ˆ}{\omega },\stackrel{ˆ}{\alpha },\stackrel{ˆ}{\gamma },\stackrel{ˆ}{\beta }\right)$ be the QML estimators of the parameters $\left(\delta ,\omega ,\alpha ,\gamma ,\beta \right)$. The $\mathrm{Asy. Power MEM}$ implies that the forecast of the conditional expectation of the variance proxy raised to the power $\delta$ at time $T+h$ is:

$μˆ T+h δˆ = ω + μˆ T+h-1 δˆ αˆ 𝔼T εT+h−1 − γˆ st+h−1 εT+h−1 δˆ + βˆ$

The above conditional expectation in the expression:

$𝔼T εT+h−1 − γˆ st+h−1 εT+h−1 δˆ$

is computed in-sample using the fitted values $\left(\stackrel{ˆ}{\gamma },\stackrel{ˆ}{\delta }\right)$ and using the historical residual series. This approach also avoids making a distributional assumption for ${\epsilon }_{t}$. And so, by applying the above formula iteratively, we can forecast the conditional expectation of realized variance (to the power $\stackrel{ˆ}{\delta }$) for any horizon $h$.

#### Asy. Power MEM(p,q)

The specific model just described can be generalized to account for more lags in the conditional volatility specification. An $\mathrm{Asy. Power MEM}\left(p,q\right)$ model assumes that:

$μtδ = ω + ∑ i=1 p αi vt−i − γi vt−i st−i δ + ∑ j=1 q βj μt−jδ$

The best model ($p$ and $q$) can be chosen, for instance, by Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC), or by Akaike Information Criterion (AIC). The former tends to be more parsimonious than the latter. V-Lab uses $p=1$ and $q=1$ though, because this is usually the option that best fits financial time series.

#### Bibliography

Bollerslev, T., 2008. Glosarry to ARCH (GARCH). CREATES Research Paper 2008-49.

Engle, R. F., 2002. New Frontiers for ARCH Models. Journal of Applied Econometrics 17: 425-446.

Ding, Z., Granger C.W., Engle R.F., 1991. A Long Memory Property of Stock Market Returns and a New Model. Journal of Empirical Finance 1 (1993) 83-106.

Parkinson, M., 1980. The Extreme Value Method for Estimating The Variance of The Rate of Return. Journal of Business 53: 61-65.