# Background

### Structural

time series models

A structural time series model is one in which the trend, seasonal and error term, plus other relevant components, are modeled explicitly.
The components of the model are referred to as **time-varying components** and are often not visible by looking at a graph of the time series alone. They can however be extracted with the use of econometric time series methodology.
The algorithms of the Time Series Lab software are based on the **score-driven** methodology which was developed independently at VU University Amsterdam and Cambridge University.

### Academic

foundation

The time series methodology of the Time Series Lab software package is well-founded and has appeared in highly respected academic journals. For an overview of score-driven publications, we refer to the www.gasmodel.com website. The score-driven methodology was developed independently at VU University Amsterdam and Cambridge University. Currently, the knowledge and experience of both universities have been combined and Professor S.J. Koopman (VU Amsterdam) and Professor A.C. Harvey (Cambridge) are part of the Time Series Lab team.

The Time Series Lab team has a combined experience of over 80 years in the field of time series analysis, both in academia and the industry. Time series research of the Time Series Lab team has been published in top statistical and econometrical journals for decades and it continues to this day. All this combined knowledge accumulates in the Time Series Lab software products.

The close connection between academia and the industry is established by positions held at major universities and research institutes with the combination of consultancy work for hedge funds, (central) banks, and marketing firms.

### Score-driven

methodology

Score-driven models or Generalized Autoregressive Score models were proposed in their full generality in Creal, Koopman, and Lucas (2013) as developed at the time at VU University Amsterdam. Simultaneously, Harvey authored the book Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series (2013) in which the mechanism to update the parameters over time is the scaled score of the likelihood function. The score-driven model encompasses other well-known models such as the generalized autoregressive conditional heteroskedasticity (GARCH) model, autoregressive conditional duration (ACD) model, autoregressive conditional intensity (ACI) model, and Poisson count models with time-varying mean.

In the class of score-driven models, the latent time-varying parameter vector $\alpha_t$ is updated over time using an autoregressive updating function based on the score of the conditional observation probability density function, see Creal, Koopman, and Lucas (2013) and Harvey (2013). The updating function for $\alpha_t$ is given by \[ \alpha_{t+1} = \omega + \sum_{i=1} ^p A_i s_{t-i+1} + \sum_{j=1} ^q B_j \alpha_{t-j+1}, \] where $\omega$ is a vector of constants, $A$ and $B$ are fixed coefficient matrices and $s_t$ is the scaled score function which is the driving force behind the updating equation. The unknown coefficients $\omega$, $A$ and $B$ depend on the static parameter vector $\psi$. The definition of $s_t$ is \[ \qquad s_t = S_t \cdot \nabla_t, \qquad \nabla_t = \frac{\partial \, \log \, p(y_t | \alpha_t, \mathcal{F}_{t-1} ; \psi)}{\partial \alpha_t}, \] for $t = 1,\ldots,n,$ and where $\nabla_{t}$ is the score vector of the (predictive) density $p(y_t | \alpha_t, \mathcal{F}_{t-1} ; \psi)$ of the observed time series $y = \left(y_1',\ldots,y_n' \right)'$. The information set $\mathcal{F}_{t-1}$ usually consists of lagged variables of $\alpha_t$ and $y_t$ but can contain exogenous variables as well. To introduce further flexibility in the model, the score vector $\nabla_{t}$ can be scaled by a matrix $S_t$. Common choices for $S_t$ are unit scaling, the inverse of the Fisher information matrix, or the square root of the Fisher inverse information matrix. The latter has the advantage of giving $s_t$ a unit variance since the Fisher information matrix is the variance matrix of the score vector. In this framework and given past information, the time-varying parameter vector $\alpha_t$ is perfectly predictable one-step-ahead. The score-driven model has three main advantages:

- the 'filtered' estimates of the time-varying parameter are optimal in a Kullback-Leibler sense.
- since the score-driven models are observation driven, their likelihood is known in closed-form.
- the forecasting performance of these models is comparable to their parameter-driven counterparts.

The second point emphasizes that static parameters can be estimated in a straightforward way using maximum likelihood methods.

### Probability

distributions

The most well-known probability distribution is the Normal, or Gaussian, distribution. Despite its popularity, many time series require a different distribution than the Normal distribution. Time Series Lab offers a variety of probability distributions and we are working to extent the library of distributions further.

- Normal
- Student
*t* - Generalized Error Distribution
- Exponential Generalized Beta 2
- Exponential
- Weibull

- Poisson
- Negative Binomial
- Bernoulli
- Skellam

### Customizable

software

The Time Series Lab software is proprietary software that can be downloaded using this link. The software is licensed without cost, even for commercial purposes. Make sure to read the license agreement before installing the program. Some users / companies require more specialized features of the software. Therefore, Time Series Lab can be made available in a commercial package, fully customized to meet the clients' needs.

- Automatic distribution and component selection
- The ability to sequentially analyze univariate time series thereby eliminating the need to select one time series at a time
- Multivariate analysis of a panel of time series to uncover dynamics between time series
- Specific output requirements
- Extended graphical and forecasting capabilities

From the above, sequentially analyzing univariate time series and multivariate analysis are by far the most comprehensive when it comes to the correct implementation of time series methodology and coding effort. Customization of this software program is not free of cost. Please Contact us for more information about customized versions and pricing of the software.

# Bibliography

### References

Creal, D. D., S. J. Koopman, and A. Lucas (2013). Generalized autoregressive score
models with applications. *Journal of Applied Econometrics* 28, 777–795.

Harvey, A. C. (2013). *Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series.* Econometric Series Monographs. Cambridge:
Cambridge University Press.