Computational Details

Chapter Contents

Time Series Analysis and Control Examples

Computational Details

Least Squares and Householder Transformation

Consider the univariate AR(p) process

$y_t = \alpha_0 + \sum_{i=1}^p \alpha_i y_{t-i} + \epsilon_t$

Define the design matrix X.

$X= [1 & y_p & ... & y_1 \ \vdots & \vdots & \ddots & \vdots \ 1 & y_{T-1} & ... & y_{T-p} ]$

Let y = (y_p+1, ... ,y_n)'. The least squares estimate, $\hat{a}=(X^'X)^{-1}X^'y$ ,is the approximation to the maximum likelihood estimate of $a=(\alpha_0,\alpha_1, ... ,\alpha_p)$ if $\epsilon_t$ is assumed to be Gaussian error disturbances. Combining X and y as

$Z= [X\,\vdots\,y]$

the Z matrix can be decomposed as

$Z= Q{U}= Q[R& w_1 \ 0 & w_2 ]$

where Q is an orthogonal matrix and R is an upper triangular matrix, w₁ = (w₁, ... ,w_p+1)', and w₂ = (w_p+2,0, ... ,0)'.

$Q^'y= [w_1 \ w_2 \ \vdots \ w_{T-p} ]$

The least squares estimate using Householder transformation is computed by solving the linear system

Ra = w₁

The unbiased residual variance estimate is

$\hat{\sigma}^2 = \frac{1}{T-p} \sum_{i=p+2}^{T-p} w_i^2 = \frac{w_{p+2}^2}{T-p}$

and

${\rm AIC}=(T-p)\log(\hat{\sigma}^2) + 2(p+1)$

In practice, least squares estimation does not require the orthogonal matrix Q. The TIMSAC subroutines compute the upper triangular matrix without computing the matrix Q.

Bayesian Constrained Least Squares

Consider the additive time series model

$y_t = T_t + S_t + \epsilon_t,\hspace*{.25in}\epsilon_t\sim N(0,\sigma^2)$

Practically, it is not possible to estimate parameters a = (T₁, ... ,T_T,S₁, ... ,S_T)', since the number of parameters exceeds the number of available observations. Let $\nabla_L^m$ denote the seasonal difference operator with L seasons and degree of m; that is, $\nabla_L^m = (1-B^L)^m$ .Suppose that T=L*n. Some constraints on the trend and seasonal components need to be imposed such that the sum of squares of $\nabla^k T_t$ , $\nabla_L^m S_t$ , and $(\sum_{i=0}^{L-1} S_{t-i})$ is small. The constrained least squares estimates are obtained by minimizing

$\sum_{t=1}^T \{(y_t-T_t-S_t)^2 + d^2[s^2(\nabla^k T_t)^2 + (\nabla_L^m S_t)^2 + z^2(S_t+ ... +S_{t-L+1})^2]\}$

Using matrix notation,

(y-Ma)'(y-Ma) + (a-a₀)'D'D(a-a₀)

where $M= [I_T\,\vdots\,I_T]$ , y = (y₁, ... ,y_T)', and a₀ is the initial guess of a. The matrix D is a 3T×2T control matrix in which structure varies according to the order of differencing in trend and season.

$D= d [E_m & 0 \ z{F}& 0 \ 0 & s{G}_k ]$

where

$E_m & = & C_m\otimes{I}_L,\hspace*{.25in} m=1,2,3 \ F& = & [1 & 0 & ... & 0 \ ... ...ddots & \ddots & \ddots & \ddots & 0 \ 0 & ... & 0 & -1 & 3 & -3 & 1 ]_{Tx T}$

The n×n matrix C_m has the same structure as the matrix G_m, and I_L is the L×L identity matrix. The solution of the constrained least squares method is equivalent to that of maximizing the following function

$L(a) = \exp\{-\frac{1}{2\sigma^2}(y-M{a})^'(y-M{a})\} \exp\{-\frac{1}{2\sigma^2}(a-a_0)^'D^'D(a-a_0)\}$

Therefore, the PDF of the data y is

$f(y|\sigma^2,a) = (\frac{1}{2\pi})^{T/2} (\frac{1}{\sigma})^T \exp\{-\frac{1}{2\sigma^2}(y-M{a})^'(y-M{a})\}$

The prior PDF of the parameter vector a is

$\pi(a|{D},\sigma^2,a_0) = (\frac{1}{2\pi})^T (\frac{1}{\sigma})^{2T}|{D}^'D| \exp\{-\frac{1}{2\sigma^2}(a-a_0)^'D^'D(a-a_0)\}$

When the constant d is known, the estimate $\hat{a}$ of a is the mean of the posterior distribution, where the posterior PDF of the parameter a is proportional to the function L(a). It is obvious that $\hat{a}$ is the minimizer of $\Vert{g}(a| d)\Vert^2 = (\tilde{y}-\tilde{D}a)^'(\tilde{y}-\tilde{D}a)$ ,where

$\tilde{y} = [y\ D{a}_0 ]$

$\tilde{D} = [M\ D ]$

The value of d is determined by the minimum ABIC procedure. The ABIC is defined as

${\rm ABIC} = T\log[\frac{1}T\Vert{g}(a| d)\Vert^2] + 2\{\log[\det(D^'D+M^'M)] - \log[\det(D^'D)]\}$

State Space and Kalman Filter Method

In this section, the mathematical formulas for state space modeling are introduced. The Kalman filter algorithms are derived from the state space model. As an example, the state space model of the TSDECOMP subroutine is formulated.

Define the following state space model:

$x_t & = & F{x}_{t-1} + G{w}_t \y_t & = & H_t{x}_t + \epsilon_t$

where $\epsilon_t\sim N(0,\sigma^2)$ and $w_t\sim N(0,Q)$ .If the observations, (y₁, ... ,y_T), and the initial conditions, $x_{0|}$ and $P_{0|}$ , are available, the one-step predictor $(x_{t| t-1})$ of the state vector x_t and its mean square error (MSE) matrix $P_{t| t-1}$ are written as

$x_{t| t-1} = F{x}_{t-1| t-1}$

$P_{t| t-1} = F{P}_{t-1| t-1}F^' + G{Q}G^'$

Using the current observation, the filtered value of x_t and its variance $P_{t| t}$ are updated.

$x_{t| t} = x_{t| t-1} + K_t e_t$

$P_{t| t} = (I- K_t H_t)P_{t| t-1}$

where $e_t = y_t - H_t{x}_{t| t-1}$ and $K_t = P_{t| t-1}H^'_t[H_t P_{t| t-1}H^'_t + \sigma^2{I}]^{-1}$ .The log-likelihood function is computed as

$\ell = -\frac{1}2\sum_{t=1}^T \log(2\pi v_{t| t-1}) - \sum_{t=1}^T \frac{e_t^2}{2v_{t| t-1}}$

where v_t|t-1 is the conditional variance of the one-step prediction error e_t.

Consider the additive time series decomposition

$y_t = T_t + S_t + T\!D_t + u_t + x^'_t\beta_t + \epsilon_t$

where x_t is a (K×1) regressor vector and $\beta_t$ is a (K×1) time-varying coefficient vector. Each component has the following constraints:

$\nabla^k T_t & = & w_{1t},\hspace*{.25in}w_{1t}\sim N(0,\tau_1^2) \ \nabla_L^m S... ...\sum_{i=1}^6 \gamma_{it}(T\!D_t(i)-T\!D_t(7)) \ \gamma_{it} & = & \gamma_{i,t-1}$

where $\nabla^k = (1-B)^k$ and $\nabla_L^m = (1-B^L)^m$ .The AR component u_t is assumed to be stationary. The trading day component TD_t(i) represents the number of the ith day of the week in time t. If k=3, p=3, m=1, and L=12 (monthly data),

$T_t & = & 3T_{t-1} - 3T_{t-2} + T_{t-3} + w_{1t} \ \sum_{i=0}^{11} S_{t-i} & = & w_{2t} \ u_t & = & \sum_{i=1}^3 \alpha_i u_{t-i} + w_{3t}$