main-content

This book presents an introduction to linear univariate and multivariate time series analysis, providing brief theoretical insights into each topic, and from the beginning illustrating the theory with software examples. As such, it quickly introduces readers to the peculiarities of each subject from both theoretical and the practical points of view. It also includes numerous examples and real-world applications that demonstrate how to handle different types of time series data. The associated software package, SSMMATLAB, is written in MATLAB and also runs on the free OCTAVE platform.

The book focuses on linear time series models using a state space approach, with the Kalman filter and smoother as the main tools for model estimation, prediction and signal extraction. A chapter on state space models describes these tools and provides examples of their use with general state space models. Other topics discussed in the book include ARIMA; and transfer function and structural models; as well as signal extraction using the canonical decomposition in the univariate case, and VAR, VARMA, cointegrated VARMA, VARX, VARMAX, and multivariate structural models in the multivariate case. It also addresses spectral analysis, the use of fixed filters in a model-based approach, and automatic model identification procedures for ARIMA and transfer function models in the presence of outliers, interventions, complex seasonal patterns and other effects like Easter, trading day, etc.

This book is intended for both students and researchers in various fields dealing with time series. The software provides numerous automatic procedures to handle common practical situations, but at the same time, readers with programming skills can write their own programs to deal with specific problems. Although the theoretical introduction to each topic is kept to a minimum, readers can consult the companion book ‘Multivariate Time Series With Linear State Space Structure’, by the same author, if they require more details.

### Chapter 1. Quick Introduction to SSMMATLAB

Abstract
In this chapter, we will present some examples on how SSMMATLAB can easily handle some of the more popular univariate and multivariate time series models. In this way, the user can quickly familiarize himself/herself with this software tool.
Víctor Gómez

### Chapter 2. Stationarity, VARMA, and ARIMA Models

Abstract
Statistically speaking, a time seriesy is a finite set of values {y 1…, y n} taken by certain k-dimensional random vectors {Y 1…, Y n}. The proper framework in which to study time series is that of stochastic processes.
Víctor Gómez

### Chapter 3. VARMAX and Transfer Function Models

Abstract
The vector random process {Y t} is said to follow a vector autoregressive moving average model with exogenous variables or VARMAX model if it satisfies an equation of the form.
$$\displaystyle \Phi (B) Y_{t} = \Omega (B) Z_{t} + \Theta (B) A_{t},$$
where B is the backshift operator, BY t = Y t−1, Φ(B) = I + Φ1B + ⋯ + ΦpB p, Ω(B) = Ω0 + Ω 1B + ⋯ + ΩrB r, Θ(B) = I + Θ 1B + ⋯ + ΘqB q, {Z t} is a process of strongly exogenous inputs with respect to {Y t}, and {A t} is a multivariate white noise process. It is assumed that Z t and A v are orthogonal for all v ≤ t.
Víctor Gómez

### Chapter 4. Unobserved Components in Univariate Series

Abstract
As mentioned in Gómez and Maravall (Seasonal adjustment and signal extraction in economic time series. In: Peña D, Tiao GC, Tsay RS (eds), A course in time series analysis, (chap 8). Wiley, New York, 2001), there exist at present two approaches to the problem of specifying a model in which several unobserved components that follow ARIMA models are present. The first one begins by specifying directly the models for the components and is called the structural time series approach. The other approach, called the ARIMA model based (AMB) method, starts by identifying a model for the observed series and derives from it the appropriate models for the components.
Víctor Gómez

### Chapter 5. Spectral Analysis

Abstract
Spectral analysis is a discipline that provides a frequency domain approach to time series. See Gómez (Multivariate time series models with linear state space structure. Springer, New York, 2016, Sect. 3.4) for an introduction to the frequency domain.
Víctor Gómez

### Chapter 6. Computing Echelon Forms by Polynomial Methods

Abstract
Given a VARMA, VARMAX, or time invariant state space model, we will show in this chapter how to compute the echelon form by polynomial methods. More specifically, we will use the stable algorithm described in Gómez (Multivariate time series models with linear state space structure. Springer, New York, 2016, Appendix to Chapter 5) to pass from a right matrix fraction description (MFD) to a left coprime MFD in echelon form. In SSMMATLAB, function pecheform.m applies this algorithm to compute the echelon form of a VARMA or VARMAX model. The previous function can also be used with time invariant state space models once they are put into VARMA form. We will show how to do this later in Sect. 6.2. All the MATLAB instructions that appear in this chapter can be found in the script file varmax2echelon_d.m of SSMMATLAB.
Víctor Gómez

### Chapter 7. Multivariate Structural Models

Abstract
Multivariate structural models are defined in a way similar to that of univariate structural models, described in Sect. 4.​1. For example, let the stochastic vector Y t satisfy Y t = P t + S t + I t, where P t is the trend, S t is the seasonal, and I t is the irregular component.
Víctor Gómez

### Chapter 8. Cointegrated VARMA Models

Abstract
In this section, it is briefly shown how cointegrated VARMA models can be handled in SSMMATLAB. The user can consult Gómez (Multivariate time series models with linear state space structure. Springer, New York, 2016, Sect. 5.7) for more information on the subject. The VARMA models can be ordinary, multiplicative as in Sect. 2.​1.​5, or in echelon form as in Sect. 3.​1.​6. The following discussion is valid for all these types of models. Later, we will specify the different functions appropriate for each model.
Víctor Gómez

### Chapter 9. Simulation of Common Univariate and Multivariate Models

Abstract
In Chap. 1, we showed how to simulate series following several of the univariate or multivariate models more used in practice. These include ARIMA, structural and transfer function models, as well as VARMA, VARMAX, and innovations state space models.
Víctor Gómez

### Chapter 10. The State Space Model

Abstract
The state space model considered in SSMMATLAB is
$$\displaystyle \begin {array}{rcl} \alpha _{t+1} &=& W_t\beta + T_t\alpha _t + H_t\epsilon _t, \\ Y_t &=& X_t\beta + Z_t\alpha _t + G_t\epsilon _t, \qquad t=1,\ldots ,n, \end {array}$$
where {Y t} is a multivariate process with $$Y_{t}\in \mathbb {R}^{p}$$, W t, T t, H t, X t, Z t, and G t are time-varying deterministic matrices, $$\beta \in \mathbb {R}^{q}$$ is a constant bias vector, $$\alpha _{t}\in \mathbb {R}^{r}$$ is the state vector, and {𝜖 t} is a sequence of uncorrelated stochastic vectors, $$\epsilon _{t}\in \mathbb {R}^{s}$$, with zero mean and common covariance matrix σ 2I.
Víctor Gómez

### Chapter 11. SSMMATLAB Examples by Subject

Abstract
In this chapter, we present several examples of the use of SSMMATLAB to analyze models that can be put into state space form. The examples are classified by subject. All script files and the corresponding data sets are included in SSMMATLAB.
Víctor Gómez