gasmodel

Overview

A package for estimation, forecasting, and simulation of generalized autoregressive score (GAS) models of Creal et al. (2013) and Harvey (2013), also known as dynamic conditional score (DCS) models or score-driven (SD) models.

Model specification allows for various conditional distributions, different parametrizations, exogenous variables, higher score and autoregressive orders, custom and unconditional initial values of time-varying parameters, fixed and bounded values of coefficients, and missing values. Model estimation is performed by the maximum likelihood method and the Hessian matrix.

The package offers the following functions for working with GAS models:

• gas() estimates GAS models.
• gas_simulate() simulates GAS models.
• gas_forecast() forecasts GAS models.
• gas_filter() obtains filtered time-varying parameters of GAS models.
• gas_bootstrap() bootstraps coefficients of GAS models.

The package handles probability distributions by the following functions:

• distr() provides table of supported distributions.
• distr_density() computes the density of a given distribution.
• distr_mean() computes the mean of a given distribution.
• distr_var() computes the variance of a given distribution.
• distr_score() computes the score of a given distribution.
• distr_fisher() computes the Fisher information of a given distribution.
• distr_random() generates random observations from a given distribution.

In addition, the package provides the following datasets used in examples:

• bookshop_sales contains times of antiquarian bookshop sales.
• ice_hockey_championships contains the results of the Ice Hockey World Championships.
• sp500_daily contains daily S&P 500 prices.

Installation

You can install the development version of gasmodel from GitHub using package devtools as:

install_github("vladimirholy/gasmodel")

Example

As a simple example, let us model volatility of daily S&P 500 prices in 2021 in the fashion of GARCH models. We estimate the GAS model based on the Student’s t-distribution with time-varying volatility and plot the filtered time-varying parameters:

library(tidyverse)
library(gasmodel)

data <- sp500_daily %>%
mutate(return = log(close) - log(lag(close))) %>%
filter(format(sp500_daily$date, "%Y") == "2021") %>% select(date, return) summary(data) #> date return #> Min. :2021-01-04 Min. :-0.023512 #> 1st Qu.:2021-04-05 1st Qu.:-0.006242 #> Median :2021-07-04 Median :-0.001423 #> Mean :2021-07-03 Mean :-0.001029 #> 3rd Qu.:2021-10-01 3rd Qu.: 0.003219 #> Max. :2021-12-31 Max. : 0.026013 model_gas <- gas(y = data$return, distr = "t", par_static = c(TRUE, FALSE, TRUE))

model_gas
#> GAS Model: Student‘s t Distribution / Mean-Variance Parametrization / Unit Scaling
#>
#> Coefficients:
#>                    Estimate  Std. Error  Z-Test  Pr(>|Z|)
#> mean            -0.00145631  0.00042388 -3.4357 0.0005911 ***
#> log(var)_omega  -2.16158419  0.76650952 -2.8200 0.0048018 **
#> log(var)_alpha1  0.54442475  0.15216805  3.5778 0.0003465 ***
#> log(var)_phi1    0.78322463  0.07644654 10.2454 < 2.2e-16 ***
#> df              10.11802479  6.59431541  1.5344 0.1249422
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: 870.9712, AIC: -1731.942, BIC: -1714.295

ggplot(mapping = aes(x = data$date, y = model_gas$fit\$par_tv[, 2])) +
labs(title = "Filtered Time-Varying Volatility", x = "Date", y = "log(var)") +
geom_line(color = "#771468") +
theme_bw()

Case Studies

To further illustrate the usability of GAS models, the package includes the following case studies in the form of vignettes:

• case_durations analyzes the timing of online antiquarian bookshop orders.
• case_rankings analyzes the strength of national ice hockey teams using the annual Ice Hockey World Championships rankings.

Supported Distributions

Currently, there are 19 distributions available.

The list of supported distribution can be obtained by the distr() function:

print(distr(), right = FALSE, row.names = FALSE)
#>  distr_title                     param_title     distr     param    type        dim   orthog default
#>  Bernoulli                       Probabilistic   bernoulli prob     binary      uni    TRUE   TRUE
#>  Categorical                     Worth           cat       worth    categorical multi FALSE   TRUE
#>  Double Poisson                  Mean            dpois     mean     count       uni    TRUE   TRUE
#>  Exponential                     Rate            exp       rate     duration    uni    TRUE  FALSE
#>  Exponential                     Scale           exp       scale    duration    uni    TRUE   TRUE
#>  Gamma                           Rate            gamma     rate     duration    uni   FALSE  FALSE
#>  Gamma                           Scale           gamma     scale    duration    uni   FALSE   TRUE
#>  Generalized Gamma               Rate            gengamma  rate     duration    uni   FALSE  FALSE
#>  Generalized Gamma               Scale           gengamma  scale    duration    uni   FALSE   TRUE
#>  Geometric                       Mean            geom      mean     count       uni    TRUE   TRUE
#>  Geometric                       Probabilistic   geom      prob     count       uni    TRUE  FALSE
#>  Multivariate Normal             Mean-Variance   mvnorm    meanvar  real        multi FALSE   TRUE
#>  Multivariate Student‘s t        Mean-Variance   mvt       meanvar  real        multi FALSE   TRUE
#>  Negative Binomial               NB2             negbin    nb2      count       uni    TRUE   TRUE
#>  Negative Binomial               Probabilistic   negbin    prob     count       uni   FALSE  FALSE
#>  Normal                          Mean-Variance   norm      meanvar  real        uni    TRUE   TRUE
#>  Plackett-Luce                   Worth           pluce     worth    ranking     multi FALSE   TRUE
#>  Poisson                         Mean            pois      mean     count       uni    TRUE   TRUE
#>  Skellam                         Difference      skellam   diff     integer     uni   FALSE  FALSE
#>  Skellam                         Mean-Dispersion skellam   meandisp integer     uni   FALSE  FALSE
#>  Skellam                         Mean-Variance   skellam   meanvar  integer     uni   FALSE   TRUE
#>  Student‘s t                     Mean-Variance   t         meanvar  real        uni   FALSE   TRUE
#>  Weibull                         Rate            weibull   rate     duration    uni   FALSE  FALSE
#>  Weibull                         Scale           weibull   scale    duration    uni   FALSE   TRUE
#>  Zero-Inflated Geometric         Mean            zigeom    mean     count       uni   FALSE   TRUE
#>  Zero-Inflated Negative Binomial NB2             zinegbin  nb2      count       uni   FALSE   TRUE
#>  Zero-Inflated Poisson           Mean            zipois    mean     count       uni   FALSE   TRUE

Details of each distribution, including its density function, expected value, variance, score, and Fisher information, can be found in vignette distributions.

Generalized Autoregressive Score Models

The generalized autoregressive score (GAS) models of Creal et al. (2013) and Harvey (2013), also known as dynamic conditional score (DCS) models or score-driven (SD) models, have established themselves as a useful modern framework for time series modeling.

The GAS models are observation-driven models allowing for any underlying probability distribution $\dpi{110}&space;\bg_white&space;p(y_t|f_t)$ with any time-varying parameters $\dpi{110}&space;\bg_white&space;f_t$ for time series $\dpi{110}&space;\bg_white&space;y_t$. They capture the dynamics of time-varying parameters using the autoregressive term and the lagged score, i.e. the gradient of the log-likelihood function. Exogenous variables can also be included. Specifically, time-varying parameters $\dpi{110}&space;\bg_white&space;f_{t}$ follow the recursion

where $\dpi{110}&space;\bg_white&space;\omega$ is a vector of constants, $\dpi{110}&space;\bg_white&space;\beta_i$ are regression parameters, $\dpi{110}&space;\bg_white&space;\alpha_j$ are score parameters, $\dpi{110}&space;\bg_white&space;\varphi_k$ are autoregressive parameters, $\dpi{110}&space;\bg_white&space;x_{ti}$ are exogenous variables, $\dpi{110}&space;\bg_white&space;S(f_t)$ is a scaling function for the score, and $\dpi{110}&space;\bg_white&space;\nabla(y_t, f_t)$ is the score given by

Alternatively, a different model can be obtained by defining the recursion in the fashion of regression models with dynamic errors as

The GAS models can be straightforwardly estimated by the maximum likelihood method. For the asymptotic theory regarding the GAS models and maximum likelihood estimation, see Blasques et al. (2014), Blasques et al. (2018), and Blasques et al. (2022).

The use of the score for updating time-varying parameters is optimal in an information theoretic sense. For an investigation of the optimality properties of GAS models, see Blasques et al. (2015) and Blasques et al. (2021).

Generally, the GAS models perform quite well when compared to alternatives, including parameter-driven models. For a comparison of the GAS models to alternative models, see Koopman et al. (2016) and Blazsek and Licht (2020).

The GAS class includes many well-known econometric models, such as the generalized autoregressive conditional heteroskedasticity (GARCH) model of Bollerslev (1986), the autoregressive conditional duration (ACD) model of Engle and Russel (1998), and the Poisson count model of Davis et al. (2003). More recently, a variety of novel score-driven models has been proposed, such as the Beta-t-(E)GARCH model of Harvey and Chakravarty (2008), the discrete price changes model of Koopman et al. (2018), the directional model of Harvey (2019), the bivariate Poisson model of Koopman and Lit (2019), and the ranking model of Holý and Zouhar (2021). For an overview of various GAS models, see Harvey (2022).

The extensive GAS literature is listed on www.gasmodel.com.

References

Blasques, F., Gorgi, P., Koopman, S. J., and Wintenberger, O. (2018). Feasible Invertibility Conditions and Maximum Likelihood Estimation for Observation-Driven Models. Electronic Journal of Statistics, 12(1), 1019–1052. doi: 10.1214/18-ejs1416.

Blasques, F., Koopman, S. J., and Lucas, A. (2014). Stationarity and Ergodicity of Univariate Generalized Autoregressive Score Processes. Electronic Journal of Statistics, 8(1), 1088–1112. doi: 10.1214/14-ejs924.

Blasques, F., Koopman, S. J., and Lucas, A. (2015). Information-Theoretic Optimality of Observation-Driven Time Series Models for Continuous Responses. Biometrika, 102(2), 325–343. doi: 10.1093/biomet/asu076.

Blasques, F., Lucas, A., and van Vlodrop, A. C. (2021). Finite Sample Optimality of Score-Driven Volatility Models: Some Monte Carlo Evidence. Econometrics and Statistics, 19, 47–57. doi: 10.1016/j.ecosta.2020.03.010.

Blasques, F., van Brummelen, J., Koopman, S. J., and Lucas, A. (2022). Maximum Likelihood Estimation for Score-Driven Models. Journal of Econometrics, 227(2), 325–346. doi: 10.1016/j.jeconom.2021.06.003.

Blazsek, S. and Licht, A. (2020). Dynamic Conditional Score Models: A Review of Their Applications. Applied Economics, 52(11), 1181–1199. doi: 10.1080/00036846.2019.1659498.

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31(3), 307–327. doi: 10.1016/0304-4076(86)90063-1.

Creal, D., Koopman, S. J., and Lucas, A. (2013). Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics, 28(5), 777–795. doi: 10.1002/jae.1279.

Davis, R. A., Dunsmuir, W. T. M., and Street, S. B. (2003). Observation-Driven Models for Poisson Counts. Biometrika, 90(4), 777–790. doi: 10.1093/biomet/90.4.777.

Engle, R. F. and Russell, J. R. (1998). Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica, 66(5), 1127–1162. doi: 10.2307/2999632.

Harvey, A. C. (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press. doi: 10.1017/cbo9781139540933.

Harvey, A. C. (2022). Score-Driven Time Series Models. Annual Review of Statistics and Its Application, 9(1), 321–342. doi: 10.1146/annurev-statistics-040120-021023.

Harvey, A. C. and Chakravarty, T. (2008). Beta-t-(E)GARCH. Cambridge Working Papers in Economics, CWPE 0840. doi: 10.17863/cam.5286.

Harvey, A., Hurn, S., and Thiele, S. (2019). Modeling Directional (Circular) Time Series. Cambridge Working Papers in Economics, CWPE 1971. doi: 10.17863/cam.43915.

Holý, V. and Zouhar, J. (2021). Modelling Time-Varying Rankings with Autoregressive and Score-Driven Dynamics. Journal of the Royal Statistical Society: Series C (Applied Statistics). doi: 10.1111/rssc.12584.

Koopman, S. J. and Lit, R. (2019). Forecasting Football Match Results in National League Competitions Using Score-Driven Time Series Models. International Journal of Forecasting, 35(2), 797–809. doi: 10.1016/j.ijforecast.2018.10.011.

Koopman, S. J., Lit, R., Lucas, A., and Opschoor, A. (2018). Dynamic Discrete Copula Models for High-Frequency Stock Price Changes. Journal of Applied Econometrics, 33(7), 966–985. doi: 10.1002/jae.2645.

Koopman, S. J., Lucas, A., and Scharth, M. (2016). Predicting Time-Varying Parameters with Parameter-Driven and Observation-Driven Models. Review of Economics and Statistics, 98(1), 97–110. doi: 10.1162/rest_a_00533.