## Ritme Informatique

The following product is developed by Ritme Informatique, a third party company for use with GAUSS. Technical support is provided directly through the developer.

### TSM v1.2 Time Series and Wavelets for Finance

TSM is a GAUSS library for time series modeling in both time domain and frequency domain and works in conjunction with the GAUSS Application – Optimization. It is primarily designed for the analysis and estimation of ARMA, VARX processes, state space models, fractional processes and structural models. To study these models, special tools have been developed like procedures for simulation, spectral analysis, Hankel matrices, etc. Estimation is based on the Maximum Likelihood principle and linear restrictions may be easily imposed.

TSM deals with vector ARMA(p,q) processes defined in the following form:

Following LÜTKEPOHL [1991], several procedures enable one to get the VAR(1) representation, roots of the reverse characteristic polynomial, the pure AR and MA representations, the matrices of the response forecast errors and the orthogonal impulses (and those of the corresponding dynamic multipliers) and the forecast error variance decomposition matrices. Two types of estimation can be performed: Conditional Maximum Likelihood (based on REINSEL,[1993] and Exact Maximum Likelihood (based on ANSLEY and KOHN [1983]. Let q be the vector of parameters. Constrained maximum likelihood is obtained by imposing implicit linear restrictions in the form:

Related to ARMA processes (and to state space models), Hankel matrices may be computed. You can also determine the McMillan degree of an ARMA process (see Aoki [1987]).

### State Space Models

Analysis and Estimation of state space models (SSM) are included in TSM. The SSM form corresponds to the one presented in HARVEY [1990]. Filtering, (fixed-interval) smoothing and maximum likelihood (with implicit linear restrictions) may be easily undertaken. For time invariant SSM, three additional procedures permit computing initial conditions, forecasting processes and solving the algebraic Riccati equation. Note that for structural models (local level, local linear trend, basic structural and cycle models), maximum likelihood can be performed in the frequency domain.

### Spectral Analysis

TSM also contains spectral analysis procedures for the estimation of periodograms, cross-periodograms and coherencies, cross-amplitude spectra and phase spectra. Data windowing can be done in the frequency domain. The user has the choice between different lag window generators (rectangular, Hartlett, Daniell, Tukey, Parzen and Bartlett-Priestley) and may define his own generator. Note that there also exists a procedure for smoothing in the time domain, based on the Savitzky-Golay filter.

General maximum likelihood estimation can be undertaken. For ML estimation in the frequency domain (Whittle likelihood), special procedures are available. Linear restrictions may be imposed in this implicit, form-Jacobian, gradient and Hessian matrices (and information matrix in the frequency domain) allow one to easily perform Lagrange multiplier tests.

TSM also contains procedures for resampling and simulation, like bootstrap, surrogate data technique and kernel estimation.

### New in Version 1.2

Version 1.2 of TSM contains 48 supplementary procedures which concern tools for state space models, special time series regression and Time-Frequency analysis of 1-D signal. New time series regression methods are implemented in TSM: Recursive Least Squares (Brown, Durbin and Evans, Journal of the Royal Statistical Society, 1975), Flexible Least Squares (Kalaba and Tesfatsion, *Computers & Mathematics with Applications*, 1989) and Generalized Flexible Least Squares (Lütkepohl and Herwartz, *Journal of Econometrics*, 1996). FLS and GFLS are methods for estimating the paths of time-varying coefficients. TSM contains also the GFLS filter and smoother for approximately linear systems (Kalaba and Tesfatsion, *IEEE Transactions on Systems, Man and Cybernetics*, 1990):

where **y***t* is a *m*-dimension time series and at is the *n*-dimension state vector. The Generalized Method of Moments with implicit linear restrictions is now included.

TSM contains new tools to analyze state space models, for example impulse analysis, forecast error variance decomposition or theoretical Hankel matrix. We can now estimate parameters of multivariate model by maximum likelihood in the frequency domain, because TSM computes the multivariate periodogram and the spectral generating function of SSM. The algorithm for bootstrapping state space models (Stoffer and Wall, *JASA*, 1991) is implemented. We can now compute the gain matrices to obtain the innovations form representation:

This form is very useful to analyze the learning convergence.

### Time-Frequency Analysis

Time-Frequency analysis (wavelet analysis and wavelet packet analysis) can be now performed with TSM. Different quadrature mirror filters are available: Coiflet, Daubechies, Haar and Pollen. Wavelet procedures concern the discrete wavelet transform (DWT), the inverse wavelet transform (IWT), wavelet decomposition coefficients subband tools (extraction, insertion, selection and split), the scalogram of the wavelet coefficients and the wavelet decomposition coefficients plot. Wavelet packet analysis is composed with nine procedures. It includes the wavelet packet transform (to generate packet tables), the inverse wavelet packet transform, the basis selection, best basis (the tree prunning algorithm of Coifman and Wickerhauser) and best level selections. Different information cost functions are considered: Shannon entropy, log energy and *lp* norm. And the user can define its own additive cost functions.

TSM also contains tools for signal denoising based on thresholding techniques: Soft, Hard and Semi-Soft wavelet shrinkages, quantile thresholding, etc. Denoised time series are easily obtained by signal reconstruction with the inverse wavelet transform or the inverse wavelet packet transform.

Several domains are concerned by *Time-Frequency* analysis: time series forecasting, density estimation, outlier testing, power spectrum estimation (Moulin, *IEEE Transactions on Signal Processing*, 1994), fractal signals (Wornell and Oppenheim, *IEEE Transactions on Signal Processing*, 1992), fractional processes, etc.

### TSM 1.2 includes more than 95 procedures for:

- ARMA processes
- VARX processes
- Spectral analysis
- Maximum Likelihood Estimation, including: Time Domain Estimation, Frequency Domain Estimation for Univariate Processes
- Univariate Models
- State space models and the Kalman filter
- Resampling and Simulation.
- Estimation tools for time series analysis
- Time-Frequency Analysis including: Quadrature mirror filters
- Wavelet Analysis, with Periodic discrete wavelet transform, Wavelet Tools, Wavelet packet analysis with transform and basis functions, and Thresholding methods
- Matrix operators

**Extensively Illustrated and Documented**

The package is extensively documented with over 230 pages in 2 volumes. More than 100 examples illustrate TSM routines. These examples are not just applications, but should be viewed as extensions of the library. They concern, for example, the optimal order of VAR models, the Kolmogorov-Smirnov statistic in the frequency domain, CUSUM and CUSUMsq tests or normality test for probit models.

TSM is written by Thierry Roncalli and published by Ritme Informatique.

**Platforms:** Windows

**Requires:** GAUSS Mathematical & Statistical Systems v3.2 and above AND GAUSS Application “Optimization v3.1.

**Ritme Informatique**

34, boulevard Haussmann

6eme etage

75009 Paris – France

Phone: +33 (0)1 42 46 00 42

Fax: +33 (0)1 42 46 00 33

Email: info@ritme.com