# What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?

In a ARCH(m) model we have $$\sigma_n^2=\sum_{i=1}^{m} \alpha_i u_{n-i}^2$$ where $u_i$ is defined as the continuously compounded return during day $i$ (between the end of day $i-1$ and the end of day $i$), $\sigma_n^2$ is the variance rate and $\alpha_i$ is the amount of weight given to the observation $i$ days ago.

An extension is to assume that there is a long-run average variance rate $V_L$ and that this should be given some weight $\gamma$. This leads to the model that takes the form $$\sigma_n^2=\gamma V_L+\sum_{i=1}^{m} \alpha_i u_{n-i}^2.$$ What are the significant implications of this extension?
Why has Engle won the Nobel Prize for the development of this "simple" model?

• Could you please elaborate on $L$? I understand that $V_L$ represents some kind of long-run variance, but it can be helpful to introduce $L$ in the maths above in order to help you. – JejeBelfort Jun 20 '17 at 14:51
• Is your equation right? Or should it be $\sigma_n^2=\gamma V_L+(1-\gamma) \sum_{i=1}^{m} \alpha_i u_{n-i}^2$ – noob2 Jun 20 '17 at 16:23

The best answer to your question is probably given by the Nobel prize committee itself in "The Prize in Economic Sciences 2003 - Advanced Information" document. You should read it in full. Below is an excerpt.

According to the committee:

Financial economists have long since known that volatility in returns tends to cluster and that the marginal distributions of many asset returns are leptokurtic, which means that they have thicker tails than the density of the normal distribution with the same mean and variance. Even though the time clustering of returns was known to many researchers, returns were still modeled as independent and identically distributed over time. Examples include Mandelbrot (1963) and Mandelbrot and Taylor (1967) who used so-called stable Paretian distributions to characterize the distribution of returns. Robert Engle’s modelling of time-varying volatility by way of autoregressive conditional heteroskedasticity (ARCH) thus signiﬁed a genuine breakthrough.

Suppose at time $t$ we observe the stochastic vector $(u_t, \mathbb{x}_t^{'})$ where $u_t$ is a scalar and $\mathbb{x}_t$ may contain some lags of $u_t$. The predictive model for the variable $u_t$ is $$u_t = \mathbb{E}(u_t | \mathbb{x}_t) + \epsilon_t \tag{3.1} \label{one}$$

It is typically assumed that $\mathbb{E}(u_t | \mathbb{x}_t)$ has a parametric form and $\mathbb{E}\epsilon_t = 0$, $\mathbb{E}\epsilon_t \epsilon_s = 0, \forall t \not = s$

When estimating the parameters in $\mathbb{E}(u_t | \mathbb{x}_t)$, it was typically assumed that the unconditional variance of the error term $\epsilon_t$ is constant or time-varying in an unknown fashion. Engle considered the alternative assumption that, while the unconditional error variance − if it exists − is constant, the conditional error variance is time-varying.

This revolutionary notion made it possible to explain systematic features in the movements of variance over time and, a fortiori, to estimate the parameters of conditional variance jointly with the parameters of the conditional mean. The literature is wholly devoid of earlier work with a similar idea.

Engle parameterized the conditional variance of $\epsilon_t$ in the model $\eqref{one}$ such that large positive or negative errors $\epsilon_t$ were likely to be followed by another large error of either sign and small errors by a small error of either sign. Nowdays this is usually refered to as volatility clustering. He assumed that $\epsilon_t$ can be decomposed as $\epsilon_t = z_t \sigma_t^{\frac{1}{2}}$ where ${z_t}$ is a sequence of iid random variables with zero mean and unit variance and where $$\sigma_t = \operatorname{Var}(\epsilon_t | \mathcal{F}_t) = \alpha_0 + \sum_{j=1}^m \alpha_j \epsilon_{t-j}^{2} \tag{3.2}\label{two}$$

In $\eqref{two}$ $\epsilon_t = u_t - \mathbb{E}(u_t | \mathbb{x}_t)$, $\alpha_0 > 0$ and $\alpha_j \geq 0, j = 1, \cdots m$, the information set $\mathcal{F}_t = \sigma(\{\epsilon_{t - j}, j \geq 1\})$

Equation $\eqref{two}$ deﬁnes the ARCH model introduced in "Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inﬂation", where the conditional variance is a function of past values of squared errors.

In this classic paper, Engle developed the estimation theory for the ARCH model, gave conditions for the maximum likelihood estimators to be consistent and asymptotically normal, and presented a Lagrange multiplier test for the hypothesis of no ARCH (equal conditional and unconditional variance) in the errors $\epsilon_t$.

• I think you forgot to square $\sigma_t$ in (3.2) and perhaps two lines below. Also the title of the ARCH original paper is not separated from the text that follows. Other than that +1. – Richard Hardy Jun 20 '17 at 19:12
• @RichardHardy Thanks. $\sigma_t$ is a variance, not a standard deviation. That's why it is not squared. – zer0hedge Jun 20 '17 at 20:00
• Usual notation is for $\sigma$ to be a standard deviation. – Dom Jun 21 '17 at 5:02
• @Dom Agree, but I followed Nobel prize committee - see the document. – zer0hedge Jun 21 '17 at 6:31

$V_L$ is the long-run variance (or the unconditional variance) if and only if $\gamma=1-\sum_{i=1}^n \alpha_i$, because the long-run variance compatible with the model $$\sigma_n^2 = \gamma V_L + \sum_{i=1}^n \alpha_i u_{n-i}^2$$ is $$\sigma^2=\frac{\gamma V_L}{1-\sum_{i=1}^n \alpha_i}.$$

The presence of the intercept $\gamma V_L$ restricts $\sigma_n^2$ to never drop below $\gamma V_L$ (if $\alpha_i>0, \forall i$). You can say this is a fundamental (in the statistical, not the subject-matter sense) level of variance. I am not aware of any special role this term would play in financial theory, but then again I am not very well versed in financial theory either.

The ARCH model and its extensions have proven to be quite useful tools for modelling macroeconomic and financial data. The idea of modelling the conditional variance in this simple way now seems trivial, but someone had to propose it first. Interestingly, the simple model often yields a pretty good fit for the empirical data. The original paper by Engle has also been a starting point for many extensions and refinements of the model -- a sign there was something valuable about the original idea. Tens of thousands of citations (for ARCH and GARCH, and their extensions) later we see that the model has been widely accepted in the financial community, which is a very important measure when it comes to Nobel prizes. You cannot rule out something that has become a standard and a benchhmark in a sufficiently large field from the Nobel prize considerations, can you?