VARX-L: Structured regularization for large vector autoregressions with exogenous variables

Abstract

The vector autoregression (VAR) has long proven to be an effective method for modeling the joint dynamics of macroeconomic time series as well as forecasting. A major shortcoming of the VAR that has hindered its applicability is its heavy parameterization: the parameter space grows quadratically with the number of series included, quickly exhausting the available degrees of freedom. Consequently, forecasting using VARs is intractable for low-frequency, high-dimensional macroeconomic data. However, empirical evidence suggests that VARs that incorporate more component series tend to result in more accurate forecasts. Conventional methods that allow for the estimation of large VARs either tend to require ad hoc subjective specifications or are computationally infeasible. Moreover, as global economies become more intricately intertwined, there has been substantial interest in incorporating the impact of stochastic, unmodeled exogenous variables. Vector autoregression with exogenous variables (VARX) extends the VAR to allow for the inclusion of unmodeled variables, but it similarly faces dimensionality challenges.We introduce the VARX-L framework, a structured family of VARX models, and provide methodology that allows for both efficient estimation and accurate forecasting in high-dimensional analysis. VARX-L adapts several prominent scalar regression regularization techniques to a vector time series context in order to greatly reduce the parameter space of VAR and VARX models. We also highlight a compelling extension that allows for shrinking toward reference models, such as a vector random walk. We demonstrate the efficacy of VARX-L in both lowand high-dimensional macroeconomic forecasting applications and simulated data examples. Our methodology is easily reproducible in a publicly available R package.

LASSO vector autoregression structures for very short‐term wind power forecasting

“…Let ∥.∥ r represents both vector and matrix L r norms. The standard LASSO‐VAR (sLV) loss function is expressed as where λ > 0 is a scalar regularization (or penalty) parameter controlling the amount of shrinkage.…”

“…As can be easily seen, the lLV and the sLV are obtained considering α = 0 and α = 1, respectively. Here, as proposed in Nicholson et al ,. the wihin‐group sparsity is estimated based on the number of time series/variables, and set as α = 1/( k + 1).…”

Bayesian structure selection for vector autoregression model

“…In Example 2, we compare the VAGSA with the structured regularization method proposed by Nicholson et al (2014) with respect to these three different grouping structures. In this example, the CGS is used for comparison purposes by ignoring the structure assumptions of the time series data, and the errors are set as i.i.d.…”

“…In particular, Lasso-type methods have been used to reduce the coefficient dimensionality in VAR-for example, Lasso-VAR proposed by Hsu, Hung, and Chang (2008). Nicholson, Matteson, and Bien (2014) generalized their works to cover more types of VAR structures, and in addition to Lasso they also proposed penalized least squares approaches for group selection and sparse group selection. Nicholson, Matteson, and Bien (2014) generalized their works to cover more types of VAR structures, and in addition to Lasso they also proposed penalized least squares approaches for group selection and sparse group selection.…”

Penalised inference for lagged dependent regression in the presence of autocorrelated residuals

“…In [1,6,15,20], no exogenous variables are included in the model. In contrast to this, [12] studied the asymptotic properties of adaptive lasso in high dimensional time series models when the number of exogenous variables increases as a function of the number of observations and [17] considered the case of VAR models with exogenous variables. While both models cover a lagged regression in the presence of exogenous variables, they do not consider autocorrelated residuals.…”