### 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, ${\mu }_{t}≔{𝔼}_{t-1}\left[{v}_{t}\right]$ is the conditional mean of ${v}_{t}$.

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

$μt = ω + αvt-1 + βμt-1$

#### 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). Anyway, 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 $\mathrm{MEM}$ model 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 $t$. However, if $\alpha +\beta <1$, the variance proxy itself is mean reverting, and it fluctuates around:

$v ≔ ω 1-α-β$

the unconditional variance proxy implied by the $\mathrm{MEM}$ model.

Usual restrictions on the parameters are $\omega ,\alpha ,\beta >0$. Though, it is possible to have $\omega =0$ and $\alpha +\beta =1$; the conditional variance proxy is then an integrated process (shocks to the variance are persistent).

#### Prediction

Let ${v}_{T}$ be the last observation in the sample, and let $\stackrel{^}{\omega }$, $\stackrel{^}{\alpha }$ and $\stackrel{^}{\beta }$ be the QML estimators of the parameters $\omega$, $\alpha$ and $\beta$ respectively. The $\mathrm{MEM}$ model implies that the forecast of the conditional expectation of the variance proxy (realized variance or high-low range) at time $T+h$ is:

$μ^ T+h = ω^ + α^ + β^ μ^ T+h-1$

and so, by applying the above formula iteratively, we can forecast the conditional expectation of the variance proxy for any horizon $h$. Then, the forecast of the proxy for the compound volatility at time $T+h$ is:

$σ^ T+1:T+h μ = ∑ i=1 h μ^ T+i$

Notice that, for large $h$, this forecast of the proxy for the compound volatility converges to:

$h ω^ 1-α^-β^$

scaling over the horizon with the well known square-root law, times the square root of the estimate of the unconditional expectation of the variance proxy implied by the $\mathrm{MEM}$ model.

#### Asymmetric MEM Model

The Asymmetric $\mathrm{MEM}$ ($\mathrm{Asy-MEM}$), as the $\mathrm{GJR-GARCH}$, is able to capture the stylized fact that negative shocks at time $t-1$ have a stronger impact in the variance proxy at time $t$, than positive shocks. This asymmetry used to be called leverage effect because the increase in risk was believed to come from the increased leverage induced by the negative shocks, but nowadays we know that this channel is just too small. The $\mathrm{Asy-MEM}$ specification is:

$μt = ω + α+γIt-1vt-1 + βμt-1$

where $\gamma >0$, and:

where ${r}_{t}$ is the return at time $t$. So, the effective coefficient associated with a negative shock is $\alpha +\gamma$. In financial time series, we generally find that $\gamma$ is positive and statistically significant. Notice that the $\mathrm{MEM}$ is a restricted version of the $\mathrm{Asy-MEM}$, with $\gamma =0$.

#### MEM(p,q)

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

$μ t = ω + ∑ i=1 p αi v t-i + ∑ j=1 q βj μ t-j$

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.

An analogous generalization could be done with the $\mathrm{Asy-MEM}$ model.

#### Bibliography

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

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