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Scalable Regularized Vector Multiplicative Error Models for Positive-valued Financial Time Series

Rohan Hemant Chhatre, Chiranjit Dutta, Nalini Ravishanker, Sumanta Basu

arXiv:2610.08443v1stat.COstat.APstat.ML

TL;DR

The paper studies modeling and forecasting multivariate positive-valued financial time series, which exhibit nonnegativity, persistence, and clustering patterns. The paper regularizes log-vMEMs with hierarchical lag structures and group penalties, uses multivariate gamma errors, and optimizes likelihoods with blockwise coordinate descent and GPU-accelerated integration. 3× speed up for the model fitting procedure; the Own-Other HLag structure with Adaptive Group Lasso performs best in out-of-sample forecasting measures.

  • Problem

    The paper studies modeling and forecasting multivariate positive-valued financial time series, which exhibit nonnegativity, persistence, and clustering patterns.

    Source: §1 · p. 1

  • Method

    The paper regularizes log-vMEMs with hierarchical lag structures and group penalties, uses multivariate gamma errors, and optimizes likelihoods with blockwise coordinate descent and GPU-accelerated integration.

    Source: §1 · p. 3, §6 · p. 22

  • Results

    3× speed up for the model fitting procedure; the Own-Other HLag structure with Adaptive Group Lasso performs best in out-of-sample forecasting measures.

    Source: §6 · p. 22

  • Takeaways & Limitations

    GPU acceleration supports holistic comparison of competing models through rolling-window validation and averaged forecast errors.

    Source: §6 · p. 22

    For the five Microsoft volatility measures, the model has difficulty fitting peaks and residuals remain heavy-tailed; truncating observations at the 95th quantile yields a good fit in the reported experiment.

    Source: §6 · p. 22

Abstract

from arXiv · show

The logarithmic multiplicative error model (log-vMEM) has been useful in modeling and forecasting multivariate positive-valued financial time series. The number of parameters grow rapidly with the dimension of the system and the lag order, making estimation computationally demanding in high-dimensional settings. This paper describes regularized estimation via hierarchical lag structures (Nicholson et al., 2020) for log-vMEM models with multivariate gamma error distribution of Tsionas (2004). The parameter estimation is performed using a blockwise coordinate descent algorithm with a Gauss-Seidel-style update scheme (Wright, 2015). This enables an efficient computation strategy compared to traditional penalized maximum likelihood approaches. The competing models are juxtaposed against each other by combining three hierarchical lag structures (componentwise, elementwise, own-other) and four penalties(group-lasso, adaptive group-lasso, group-mcp, and group-scad). Extensive simulation runs have been performed to test the parameter recovery for both the unpenalized and the penalized models. We apply the proposed methods to model the joint dynamics of robust intraday realized volatility measures for Microsoft (NASDAQ: MSFT) for the competing models. The numerical integration step of the log-likelihood is identified to be the principal computational bottleneck. We address this issue by using GPU-accelerated quadrature integration thus improving computational scalability of the proposed models.

1 Introduction

The paper addresses modeling and forecasting multivariate positive-valued financial series, whose dynamics include persistence and clustering. It extends log-vMEM with multivariate gamma errors, hierarchical lag regularization, blockwise optimization, and GPU-accelerated likelihood integration.

1 Introduction

  • The paper motivates these models by the prevalence of positive-valued financial series and describes MEM as a product of conditional expectation and unit-mean errors.The conditional expectation is modeled using its own past values and lagged observations.Source: §1 · pp. 1–2
  • The log-vMEM models the logarithm of the conditional mean using lagged log observations and lagged log conditional means.Its parameters are matrices, and the paper situates it among prior log-vMEM specifications with different error distributions.Source: §1 · p. 2
  • The multivariate gamma error specification models dependence among the components of positive-valued observations in the log-vMEM framework.The paper compares this specification with several convex and non-convex penalty structures.Source: §1 · p. 3
  • Hierarchical lag structures combined with group penalties support parsimonious, interpretable estimation and structured variable and lag selection in high-dimensional log-vMEMs.They allow maximal lag orders to vary across components rather than imposing one common order.Source: §1 · p. 3
  • A Gauss-Seidel block coordinate descent algorithm optimizes unpenalized and penalized likelihoods by updating parameter blocks conditionally on the remaining parameters.The authors describe this as a faster approach than optimizing the resulting penalized likelihood jointly.Source: §1 · p. 3
  • GPU-accelerated adaptive numerical integration reduces the computational burden of evaluating the multivariate gamma likelihood and improves scalability to higher dimensions.The approach is applied to Microsoft intraday volatility measures, with model performance evaluated over 27 possible rolling origins.Source: §1 · p. 3

2 Logarithmic Vector Multiplicative Error Models

The log-vMEM models positive-valued multivariate observations through conditional means and multivariate gamma errors, with likelihood estimation using blockwise updates. In simulations with d = 5, p = 10, and T = 1500, estimates closely approximate the true parameters with low average bias and RMSE.

2 Logarithmic Vector Multiplicative Error Models

  • The log-vMEM models positive-valued observations through conditional means, while multivariate gamma errors provide a joint model for their dependence.The gamma parameters are set to ensure unit error means, with an equicorrelation covariance matrix.Source: §2 · pp. 3–4

2.1 Log-likelihood function

  • Assuming stationarity allows the intercept to be expressed approximately as a function of β, simplifying the likelihood to depend on β and the autoregressive matrices.The stationarity condition requires the moduli of the roots of the stated matrix polynomial to exceed one.Source: §2.1 · pp. 5–6

2.2 Block Coordinate Descent Maximization of the Log-likelihood Function

  • Because the likelihood is non-convex and direct joint maximization is computationally arduous, alternating block coordinate descent updates the autoregressive matrix and gamma shape parameter.Rows of the matrix are updated sequentially using derivative-free optimization; the likelihood is repeatedly evaluated within each block update.Source: §2.2 · pp. 6–8

2.3 Results from a Simulation Study

  • Across 100 replications with d = 5, p = 10, and T = 1500, estimates closely approximated true parameters with low average bias and RMSE.Table 2 reports the true value, average estimate, bias, and RMSE for each parameter.Source: Table 2 · p. 9, §2.3 · p. 10

3 Penalized Maximum Likelihood Estimation

The section regularizes log-vMEM coefficient matrices using hierarchical lag structures and group penalties, with estimation and penalty selection implemented through rowwise optimization. Simulations examine recovery of active structures under three lag schemes.

3 Penalized Maximum Likelihood Estimation

  • The section formulates penalized maximum-likelihood estimation of log-vMEM parameters under the reviewed sparsity-inducing penalties.The objective penalizes the coefficient matrix A while retaining the unpenalized MLE of β.Source: §3 · p. 10, §3.2 · p. 12

3.1 Hierarchical lag (HLag) based penalty structures

  • Hierarchical lag penalties combine componentwise, own-other, or elementwise structures with group lasso, SCAD, or MCP penalties to impose structured sparsity.Componentwise shares a maximum lag across predictors within each equation; own-other adds a hierarchy favoring longer own-series than cross-series lags, while elementwise has no predefined relationships among maximum lags.Source: §3.1 · pp. 10–11

3.2 Implementation of penalized estimation

  • Penalized estimation optimizes one coefficient-matrix row at a time, with the penalty parameter selected by the smallest BIC.The procedure initializes each row from its unregularized MLE and leaves the estimate of β equal to its unpenalized MLE.Source: §3.2 · pp. 12–13

3.3 Simulated Data Analysis under HLag Penalties

  • Simulations use 100 replicates with d = 5, p = 10, and T = 1,500 under each of the three hierarchical lag structures.The study reports parameter recovery for group lasso, group MCP, and group SCAD estimates against the true simulated sparsity structures.Source: §3.3 · p. 13

4 Analysis of Realized Volatility Measures

The realized-volatility analysis fits competing models to five MSFT measures and evaluates their forecasts and fit. The Own-Other HLag model with an Adaptive Group Lasso penalty has the least forecast error among the competitors.

4 Analysis of Realized Volatility Measures

  • The five volatility measures are temporally dependent, positively skewed, heavy-tailed, and moderately to highly positively correlated.Kendall correlations range from 0.52 between Xt1 and Xt5 to 0.88 between Xt1 and Xt3.Source: §4 · p. 17
  • The Own-Other HLag model with Adaptive Group Lasso has the least forecast error among the competing models.The models use p = 10, are fit on 1,445 training observations, and forecast the first five test observations; penalty settings are selected by smallest in-sample BIC.Source: §4 · pp. 17–18
  • The selected model provides an adequate fit to all five series, with 95% prediction-band coverage from 84.0% to 94.6%.Residual autocorrelation is substantially reduced but remains, and Gamma QQ plots fit the distribution reasonably in its central part, with upper-tail departures.Source: §4 · p. 18

5 Computational Acceleration Using GPU-Based Quadrature Integration

GPU-based quadrature reduces the cost of repeated multivariate Gamma likelihood evaluations, with gains that increase as system dimension grows. The approach is also used for rolling-origin comparison of the forecasting models.

5 Computational Acceleration Using GPU-Based Quadrature Integration

  • Repeated likelihood evaluations make one-dimensional integration costly during block coordinate descent; fixed-node quadrature approximates the integral.The BOBYQA optimizer repeatedly evaluates the likelihood at trial parameter values.Source: §5 · p. 19
  • Calculations across times and quadrature nodes are independent conditional on parameters, so they can run simultaneously on GPU while data and quadrature quantities remain resident across evaluations.Source: §5 · pp. 19–20
  • In-sample fit bands generally cover actual values except for outliers, while residual Gamma QQ agreement is reasonable centrally but departs more in the upper tail.Source: Figure 6 · p. 20, Figure 7 · p. 21
  • At d = 50, 100, 200, and 500, GPU evaluation is approximately 3, 4.15, 8.4, and 17.1 times faster than CPU, respectively.At d = 5, CPU evaluation remains faster because the workload does not offset GPU execution overhead.Source: §5 · p. 20, Table 5 · p. 20
  • Across 27 rolling origins, Own-Other HLag with Adaptive Group Lasso performs best on out-of-sample forecasting measures; GPU validation took 158.79 hours and was 3× faster than CPU.The measures are averaged over the five realized volatility series at forecast horizons h = 1, . . . , 5.Source: Table 4 · p. 19, §5 · p. 21
  • Simulation runs use the CPU implementation for reproducibility on standard computing environments, while the GPU version is used for rolling-origin validation.Source: §5 · p. 21

6 Conclusion

The paper presents sparse log-vMEMs with multivariate Gamma errors, hierarchical lag regularization, and blockwise coordinate descent, reporting parameter recovery in simulations. In the MSFT application, GPU acceleration speeds fitting and one lag-penalty combination performs best, while heavy residual tails remain a limitation.

6 Conclusion

  • The proposed log-vMEM combines multivariate Gamma errors, blockwise coordinate descent, HLag regularization, and group SCAD and MCP penalties; simulations report successful parameter recovery.Source: §6 · p. 22
  • In five MSFT volatility series, in-sample fit misses peaks and residuals have heavy tails; truncating each series at its 95th quantile removes upper-tail QQ departures.The authors identify a mixture error distribution as a possible way to model both the heavy tail and the bulk.Source: §6 · p. 22
  • The Own-Other HLag with Adaptive Group Lasso performs best on out-of-sample forecasting measures, and GPU quadrature yields a 3× speedup for model fitting.Source: §6 · p. 22
  • The authors identify spatio-temporal applications and Bayesian shrinkage priors as directions for future research.Source: §6 · p. 22
  • Residual autocorrelation is substantially reduced, although statistically significant autocorrelation remains at several lags.Source: Figure 8 · p. 23
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