Time-Series Models for Volatility Forecasts and Statistical Arbitrage

In Chapter 7, Linear Models – From Risk Factors to Asset Return Forecasts, we introduced linear models for inference and prediction, starting with static models for a contemporaneous relationship with cross-sectional inputs that have an immediate effect on the output. We presented the ordinary least squares (OLS) learning algorithm, and saw that it produces unbiased coefficients for a correctly specified model with residuals that are not correlated with the input variables. Adding the assumption that the residuals have constant variance guarantees that OLS produces the smallest mean squared prediction error among unbiased estimators.

We also encountered panel data that had both cross-sectional and time-series dimensions, when we learned how the Fama-Macbeth regressions estimate the value of risk factors over time and across assets. However, the relationship between returns across time is typically fairly...

Tools for diagnostics and feature extraction

A time series is a sequence of values separated by discrete intervals that are typically even spaced (except for missing values). A time series is often modeled as a stochastic process consisting of a collection of random variables, , with one variable for each point in time, . A univariate time series consists of a single value, y, at each point in time, whereas a multivariate time series consists of several observations that can be represented by a vector.

The number of periods, , between distinct points in time, t_i, t_j, is called lag, with T-1 distinct lags for each time series. Just as relationships between different variables at a given point in time is key for cross-sectional models, relationships between data points separated by a given lag are fundamental to analyzing and exploiting patterns in time series.

For cross-sectional models, we distinguished between input and output variables, or target and predictors...

How to diagnose and achieve stationarity

The statistical properties, such as the mean, variance, or autocorrelation, of a stationary time series are independent of the period—that is, they don't change over time. Thus, stationarity implies that a time series does not have a trend or seasonal effects. Furthermore, it requires that descriptive statistics, such as the mean or the standard deviation, when computed for different rolling windows, are constant or do not change significantly over time. A stationary time series reverts to its mean, and the deviations have a constant amplitude, while short-term movements are always alike in a statistical sense.

More formally, strict stationarity requires the joint distribution of any subset of time-series observations to be independent of time with respect to all moments. So, in addition to the mean and variance, higher moments such as skew and kurtosis also need to be constant, irrespective of the lag between different observations...

Univariate time-series models

Multiple linear-regression models expressed the variable of interest as a linear combination of the inputs, plus a random disturbance. In contrast, univariate time-series models relate the current value of the time series to a linear combination of lagged values of the series, current noise, and possibly past noise terms.

While exponential smoothing models are based on a description of the trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data. ARIMA(p, d, q) models require stationarity and leverage two building blocks:

• Autoregressive (AR) terms consisting of p lagged values of the time series
• Moving average (MA) terms that contain q lagged disturbances

The I stands for integrated because the model can account for unit-root non-stationarity by differentiating the series d times. The term autoregression underlines that ARIMA models imply a regression of the time series on its own values...

Multivariate time-series models

Multivariate time-series models are designed to capture the dynamic of multiple time series simultaneously and leverage dependencies across these series for more reliable predictions. The most comprehensive introduction to this subject is Lütkepohl (2005).

Systems of equations

Univariate time-series models, like the ARMA approach we just discussed, are limited to statistical relationships between a target variable and its lagged values or lagged disturbances and exogenous series, in the case of ARMAX. In contrast, multivariate time-series models also allow for lagged values of other time series to affect the target. This effect applies to all series, resulting in complex interactions, as illustrated in the following diagram:

Figure 9.9: Interactions in univariate and multivariate time-series models

In addition to potentially better forecasting, multivariate time series are also used to gain insights into cross-series dependencies...

Cointegration – time series with a shared trend

We briefly mentioned cointegration in the previous section on multivariate time-series models. Let's now explain this concept and how to diagnose its presence in more detail before leveraging it for a statistical arbitrage trading strategy.

We have seen how a time series can have a unit root that creates a stochastic trend and makes the time series highly persistent. When we use such an integrated time series in their original, rather than in differenced, form as a feature in a linear regression model, its relationship with the outcome will often appear statistically significant, even though it is not. This phenomenon is called spurious regression (for details, see Chapter 18, CNNs for Financial Time Series and Satellite Images, in Wooldridge, 2008). Therefore, the recommended solution is to difference the time series so they become stationary before using them in a model.

However, there is an exception when there...

Statistical arbitrage with cointegration

Statistical arbitrage refers to strategies that employ some statistical model or method to take advantage of what appears to be relative mispricing of assets, while maintaining a level of market neutrality.

Pairs trading is a conceptually straightforward strategy that has been employed by algorithmic traders since at least the mid-eighties (Gatev, Goetzmann, and Rouwenhorst 2006). The goal is to find two assets whose prices have historically moved together, track the spread (the difference between their prices), and, once the spread widens, buy the loser that has dropped below the common trend and short the winner. If the relationship persists, the long and/or the short leg will deliver profits as prices converge and the positions are closed.

This approach extends to a multivariate context by forming baskets from multiple securities and trading one asset against a basket of two baskets against each other.

In practice, the strategy...

Summary

In this chapter, we explored linear time-series models for the univariate case of individual series, as well as multivariate models for several interacting series. We encountered applications that predict macro fundamentals, models that forecast asset or portfolio volatility with widespread use in risk management, and multivariate VAR models that capture the dynamics of multiple macro series. We also looked at the concept of cointegration, which underpins the popular pair-trading strategy.

Similar to Chapter 7, Linear Models – From Risk Factors to Return Forecasts, we saw how linear models impose a lot of structure, that is, they make strong assumptions that potentially require transformations and extensive testing to verify that these assumptions are met. If they are, model-training and interpretation are straightforward, and the models provide a good baseline that more complex models may be able to improve on. In the next two chapters, we will see two examples...