Multivariate Time Series

The models we discussed in the previous chapter only depended on the previous values of the single variable of interest. Those models are appropriate when we only have a single variable in our time series. However, it is common to have multiple variables in time-series data. Often, these other variables in the series can improve forecasting of the variable of interest. We will discuss models for time series with multiple variables in this chapter. We will first discuss the correlation relationship between time-series variables, then discuss how we can model multivariate time series. While there are many models for multivariate time-series data, we will discuss two models that are both powerful and widely used: autoregressive integrated moving average with exogenous variables (ARIMAX) and vector autoregressive (VAR). Understanding these two models will extend the reader’s model toolbox and provide building blocks for the reader to learn more about multivariate...

Multivariate time series

In the previous chapter, we discussed models for univariate time series or a time series of one variable. However, in many modeling situations, it is common to have multiple time-varying variables that are measured together. A time series consisting of multiple time-varying variables is called a multivariate time series. Each variable in the time series is called a covariate. For example, a time series of weather data might include temperature, rain amount, wind speed, and relative humidity. Each of these variables, in the weather dataset, is a univariate time series, and together, a multivariate time series and each pair of variables are covariates.

Mathematically, we typically represent a multivariate time-series as a vector-valued series, as follows:

X = x 0,0 x 0,1 ⋮ , x 1,0 x 1,1 ⋮ , … , x t,0 x t,1 ⋮ 

Here, each X instance consists of multiple...

ARIMAX

In the previous chapter, we discussed the ARIMA family of models and demonstrated how to model univariate time-series data. However, as we mentioned in the previous section, many time series are multivariate, such as stock data, weather data, or economic data. In this section, we will discuss how we can incorporate information from covariate variables when modeling time-series data.

When we model a multivariate time series, we typically have a variable we are interested in forecasting. This variable is commonly called the endogenous variable. The other covariates in the multivariate time series are called exogenous variables. Recall from Chapter 11, ARIMA Models, the equation representing the ARIMA model:

y′ t = c + ϕ 1 y′ t−1 + … + ϕ p  y ′  t−p + ϵ t + θ 1 ϵ t−1 + … + ϕ q ϵ t−q

Here, y′ ...

VAR modeling

The AR(p), MA(q), ARMA(p,q), ARIMA(p,d,q)m, and SARIMA(p,d,q) models we looked at in the last chapter form the basis of multivariate VAR modeling. In this chapter, we have discussed ARIMA with exogenous variables (ARIMAX). We will now begin discussion on the VAR model. First, it is important to understand that while ARIMAX requires leading (future) values of the exogenous variables, no future values of these variables are required for the VAR model as they are all autoregressive to each other – hence the name vector autoregressive – and by definition not exogenous. To start, let us consider the two-variable, or bivariate, case. Consider a process y t that is the output of two different input variables, y t1 and y t2. Note that in matrix form, we are discussing the case of an nxm matrix (y n,m) where n corresponds to the point in time and m corresponds to the variables involved (variables 1,2, … , m). We exclude the comma from notation...

Summary

In this chapter, we provided an overview of multivariate time-series and how they differ from the univariate case. We then covered the math and intuition behind two popular approaches to solving problems using multivariate time-series models—ARIMAX and the VAR model framework. We walked through examples for each model using a step-by-step approach. This chapter concludes our discussions on time-series analysis and forecasting. At this point, you should be able to identify and assess the statistical properties of time series, transform them as needed, and construct models that are useful for fitting and forecasting both univariate and multivariate cases.

In the next chapter, we will begin our discussion on survival analysis with an introduction to time-to-event (TTE) variables.

References

[1] Liang, X., Zou, T., Guo, B., Li, S., Zhang, H., Zhang, S., Huang, H. and Chen, S. X. (2015). Assessing Beijing’s PM2.5 pollution: severity, weather impact, APEC and winter heating. Proceedings of the Royal Society A, 471, 20150257.

[2] Dua, D. and Graff, C. (2019). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science.