17 Introduction
When making a draft of this book, I didn’t intend to devote an entire chapter to state-space (SS) models until I started using them a lot in my daily work and then realized they were too important to be neglected. In addition, the subject often goes unmentioned in both data science and econometrics textbooks despite its usefulness in many applications where alternative methods are certainly inferior.
After a brief introduction, the next two sections will focus on basic applications of SS models: 1. Estimating time-varying coefficients from regression models; and 2. Estimating a common underlying process when there are multiple sources of information. For those interested in delving into the details of the subject, I strongly recommend starting by reading the excellent Holmes, Scheuerell, and Ward (2021) – which will be a valuable reference for the following applications.
Let’s think about state-space models as a representation of an idea rather than a method. Suppose we need to measure air temperature in the city of Rio de Janeiro. We can’t observe it directly, but we have data collected by a sensor. This sensor performs very well on average, although it’s subject to measurement errors. The main goal is to accurately estimate the real air temperature from the somewhat noisy data we get from the sensor.
In the SS representation, this problem is summarized by two equations. The observation equation tells us the data we observe (\(y_t\)) is the real unobserved temperature (\(x_t\)) plus \(v_t\), which is a Gaussian error with zero mean and variance \(\sigma^2_v\). We could add terms for seasonality, exogenous regressors, or dummy variables as well. For now, we’ll stick to the simplest specification. In matrix form:
\[ \begin{align} y_t = Z_tx_{t} + v_t \end{align} \]
The state process (or transition) equation describes how the unobservable real air temperature (the hidden state, \(x_t\)) evolves over time. It’s usually characterized by an autoregressive process – often a random walk – in which \(w_t\) is also a Gaussian error term with zero mean and variance \(\sigma^2_w\). In matrix form:
\[ \begin{align} x_t = B_tx_{t-1} + w_t \end{align} \]
Finding the hidden state trajectory requires us to solve this linear stochastic dynamical system, which can be accomplished by several algorithms – the best-known are the Expectation-Maximization (EM) and the Kalman Filter and Smoother, the latter of which you may already have heard of. In the next two sections, we’ll see how to build and estimate such models with the MARSS package.