Sructural Time Series using JDemetra+

Jean Palate

2025-05-13

Introduction

Data

We illustrate the various methods using two sets of time series:

• The retail data set contains monthly figures over US retail activity of various categories of goods and services from 1992 to 2010.
• The ABS data set contains long monthly series over Australian retail trade.

Overview of the supported state space forms

The general linear gaussian state-space model can be written in many different ways. The form considered in JD+ 3.0 is presented below.

$$y_t = Z_t \alpha_t + \epsilon_t,\quad \epsilon_t \sim N\left(0, \sigma^2 H_t\right),\quad t \gt 0$$

$$\alpha_{t+1} = T_t \alpha_t + \mu_t, \quad \mu_t \sim N \left(0, \sigma^2 V_t \right),\quad t \ge 0$$

$y_t$ is the observation at period t, $\alpha_t$ is the state vector. $\epsilon_t, \mu_t$ are assumed to be serially independent at all leads and lags and independent from each other.
In the case of multi-variate models, $y_t$ is a vector of observations. However, in most cases, we will use the univariate approach by considering the observations one by one (univariate handling of multi-variate models).

The innovations of the state equation will be modelled as

$$\mu_t = S_t \xi_t, \quad \xi_t \sim N\left( 0, \sigma^2 I\right)$$

In other words, $V_t=S_t S_t'$

The initial ($\equiv t=0$) conditions of the filter are defined as follows:

$$\alpha_{0} = a_{0} + B\delta + \mu_{0}, \quad \delta \sim N\left(0, \kappa I \right),\: \mu_{0} \sim N\left(0, P_*\right)$$

where $\kappa$ is arbitrary large. $P_*$ is the variance of the stationary part of the initial state vector and $B$ models the diffuse part. We write $BB'=P_\infty$.

The definition used in JD+ is quasi-identical to that of Durbin and Koopman[1].

In summary, the model is completely defined by the following quantities (possible default values are indicated in brackets):

$$\mathbf{Z_t}, \mathbf{H_t} [=0]$$

$$\mathbf{T_t}, \mathbf{V_t} [=S_t S_t'], \mathbf{S_t} [=Cholesky(V)]$$

$$\mathbf{a_{0}}[=0], \mathbf{P_*} [=0], \mathbf{B} [=0], \mathbf{P_\infty} [=BB']$$

Description of the state blocks

ar

Introduction

The auto-regressive block is defined by

$$\Phi\left(B\right)y_t=\epsilon_t$$

where:

$$\Phi\left(B\right)=1+\varphi_1 B + \cdots + \varphi_p B^p$$

is an auto-regressive polynomial.

Let $\gamma_i$ be the autocovariances of the process

Using those notations, the state-space block can be written as follows :

State block:

$$\alpha_t= \begin{pmatrix} y_t \\ y_{t-1} \\ \vdots \\ y_{t-p+1} \end{pmatrix}$$
The state block can be extended with additional lags. That can be useful in complex (multi-variate) models

Dynamics

$$T_t = \begin{pmatrix}-\varphi_1 & \cdots & \cdots & -\varphi_p \\ 1 & \cdots & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & 1 & 0 \end{pmatrix}$$

$$S_t = \sigma_{ar} \begin{pmatrix} 1 \\ 0 \\ \vdots\\ 0 \end{pmatrix}$$

$$V_t = S S'$$

Default loading

$$Z_t = \begin{pmatrix} 1 & 0 & \cdots & 0\end{pmatrix}$$

Initialization

$$\alpha_{-1} = \begin{pmatrix}0 \\ 0 \\ \vdots\\ 0 \end{pmatrix}$$

$$P_{*} = \Omega$$$\Omega$ is the unconditional covariance of the state array; it is computed by means of the auto-covariance function of the model

$$\Omega_t = \begin{pmatrix}\gamma_0 & \gamma_1 & \cdots & \gamma_p \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \\ \vdots & \ddots & \ddots & \vdots\\ \gamma_p & \cdots & \gamma_1 & \gamma_0 \end{pmatrix}$$

R code

The “ar” block is defined by specifying the coefficients $\phi_i$ of the ar polynomial and the innovation variance. More exactly, they correspond to the equation

$$y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t$$

The coefficients and/or the variance can be fixed

b_ar<-ar("ar", c(.7,-.4, .2), nlags=5, variance=1)

cat("T\n")
#> T
knit_print(block_t(b_ar))
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]  0.7 -0.4  0.2    0    0
#> [2,]  1.0  0.0  0.0    0    0
#> [3,]  0.0  1.0  0.0    0    0
#> [4,]  0.0  0.0  1.0    0    0
#> [5,]  0.0  0.0  0.0    1    0
cat("\nP0\n")
#> 
#> P0
knit_print(block_p0(b_ar))
#>            [,1]       [,2]       [,3]       [,4]       [,5]
#> [1,] 1.51552795 0.77018634 0.08695652 0.05590062 0.15838509
#> [2,] 0.77018634 1.51552795 0.77018634 0.08695652 0.05590062
#> [3,] 0.08695652 0.77018634 1.51552795 0.77018634 0.08695652
#> [4,] 0.05590062 0.08695652 0.77018634 1.51552795 0.77018634
#> [5,] 0.15838509 0.05590062 0.08695652 0.77018634 1.51552795

ar2

Introduction

An alternative representation of the auto-regressive block will be very useful for the purposes of reflecting expectations. The process is defined as above:

$$\Phi\left(B\right)y_t=\epsilon_t$$

where:

$$\Phi\left(B\right)=1+\varphi_1 B + \cdots + \varphi_p B^p$$

is an auto-regressive polynomial. However, modeling data that refers to expectations may require including conditional expectations in the state vector. Thus, the same type of representation that is used for the ARMA model will be considered here.

Let $\gamma_i$ be the autocovariances of the model. We also define the size of our state vector as $r0=max(p,h+1)$, where $h$ is the forecast horizon desired by the user. If the user needs to use $nlags$ lagged values, whose default value is zero. Then the size of the state vector will be $r=r0+nlags$

Using those notations, the state-space model can be written as follows :

State block:

$$\alpha_t= \begin{pmatrix} y_{t-nlags} \\ \vdots \\ y_{t-1} \\ \hline y_{t} \\ y_{t+1|t} \\ \vdots \\ y_{t+h|t} \end{pmatrix}$$

where $y_{t+i|t}$ is the orthogonal projection of $y_{t+i}$ on the subspace generated by ${y\left(s\right):s \leq t}$. Thus, it is the forecast function with respect to the semi-infinite sample. We also have that $y_{t+i|t} = \sum_{j=i}^\infty {\psi_j \epsilon_{t+i-j}}$

Dynamics

$$T_t = \begin{pmatrix} 0 &1 & 0 & \cdots & 0 \\0& 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1\\ -\varphi_r & \cdots & \cdots & \cdots &-\varphi_1 \end{pmatrix}$$

with $\varphi_{j}=0$ for $j>p$

$$S_t = \sigma_{ar} \begin{pmatrix} 0 \\ \vdots \\ 0\\ \hline 1 \\ \psi_1 \\ \vdots\\ \psi_s \end{pmatrix}$$

$$V_t = S S'$$

Default loading

$$Z_t = \begin{pmatrix} 0 & \cdots &0 & | & 1 & 0 & \cdots & 0\end{pmatrix}$$

Initialization

$$\alpha_{-1} = \begin{pmatrix} 0 \\ \vdots \\ 0\\ \hline 0 \\ 0 \\ \vdots\\ 0 \end{pmatrix}$$

$$P_{*} = \Omega$$

$\Omega$ is the unconditional covariance of the state array; it can be easily derived using the MA representation. We have:

$$\Omega\left(i,0\right) = \gamma_i$$

$$\Omega\left(i,j\right) = \Omega\left(i-1,j-1\right)-\psi_i \psi_j$$

b_ar2<-ar2("ar2", c(-.2, .4, -.1), nlags=3, nfcasts=2)
knit_print(block_t(b_ar2))
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]    0    1    0  0.0  0.0  0.0
#> [2,]    0    0    1  0.0  0.0  0.0
#> [3,]    0    0    0  1.0  0.0  0.0
#> [4,]    0    0    0  0.0  1.0  0.0
#> [5,]    0    0    0  0.0  0.0  1.0
#> [6,]    0    0    0 -0.1  0.4 -0.2

arma

Introduction

The arma block is defined by

$$\Phi\left(B\right)y_t=\Theta\left(B\right)\epsilon_t$$

where:

$$\Phi\left(B\right)=1+\varphi_1 B + \cdots + \varphi_p B^p$$

$$\Theta\left(B\right)=1+\theta_1 B + \cdots + \theta_q B^q$$

are the auto-regressive and the moving average polynomials.

The MA representation of the process is $y_t=\sum_{i=0}^\infty {\psi_i \epsilon_{t-i}}$. Let $\gamma_i$ be the autocovariances of the model. We also define: $r=\max\left(p, q+1\right), \quad s=r-1$.

Using those notations, the state-space block can be written as follows :

State block:

$$\alpha_t= \begin{pmatrix} y_t \\ y_{t+1|t} \\ \vdots \\ y_{t+s|t} \end{pmatrix}$$

where $y_{t+i|t}$ is the orthogonal projection of $y_{t+i}$ on the subspace generated by ${y\left(s\right):s \leq t}$.Thus, it is the forecast function with respect to the semi-infinite sample. We also have that $y_{t+i|t} = \sum_{j=i}^\infty {\psi_j \epsilon_{t+i-j}}$

Dynamics

$$T_t = \begin{pmatrix}0 &1 & 0 & \cdots & 0 \\0& 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1\\ -\varphi_r & \cdots & \cdots & \cdots &-\varphi_1 \end{pmatrix}$$

with $\varphi_{j}=0$ for $j>p$

$$S_t = \begin{pmatrix}1 \\ \psi_1 \\ \vdots\\ \psi_s \end{pmatrix}$$

$$V_t = S S'$$

Default loading

$$Z_t = \begin{pmatrix} 1 & 0 & \cdots & 0\end{pmatrix}$$

Initialization

$$\alpha_{-1} = \begin{pmatrix}0 \\ 0 \\ \vdots\\ 0 \end{pmatrix}$$

$$P_{*} = \Omega$$

$\Omega$ is the unconditional covariance of the state array; it can be easily derived using the MA representation. We have:

$$\Omega\left(i,0\right) = \gamma_i$$

$$\Omega\left(i,j\right) = \Omega\left(i-1,j-1\right)-\psi_i \psi_j$$

b_arma<-arma("arma", ar=c(-.2, .4, -.1), ma=c(.3, .6))
knit_print(block_t(b_arma))
#>      [,1] [,2] [,3]
#> [1,]  0.0  1.0  0.0
#> [2,]  0.0  0.0  1.0
#> [3,]  0.1 -0.4  0.2
knit_print(block_p0(b_arma))
#>           [,1]      [,2]      [,3]
#> [1,] 1.3501359 0.6394319 0.2517752
#> [2,] 0.6394319 0.3501359 0.1394319
#> [3,] 0.2517752 0.1394319 0.1001359

arima

Examples

Basic structural models

Time varying trading days

Regular period cubic splines

Modeling Nile riverflow

Time series with a sampling error