Nonlinear co-integration is studied for score-driven models, using a new multivariate dynamic conditional score/generalized autoregressive score model. The model is named t-QVARMA (quasi-vector autoregressive moving average model), which is a location model for the multivariate t-distribution. In t-QVARMA, I(0) and co-integrated I(1) components of the dependent variables are included. For t-QVARMA, the conditions of the maximum likelihood estimator and impulse response functions (IRFs) are presented. A limiting special case of t-QVARMA, named Gaussian-QVARMA, is a Gaussian-VARMA specification with I(0) and I(1) components. As an empirical application, the US real gross domestic product growth, US inflation rate, and effective federal funds rate are studied for the period of 1954 Q3 to 2020 Q2. Statistical performance and predictive accuracy of t-QVARMA are superior to those of Gaussian-VAR. Estimates of the short-run IRF, long-run IRF, and total IRF impacts for the US data are reported.

A new class of multivariate nonlinear quasi-vector autoregressive (QVAR) models is introduced. It is a Markov switching score-driven model with stochastic seasonality for the multivariate t-distribution (MS-Seasonal-t-QVAR). As an extension, we allow for the possibility of having common-trends and nonlinear co-integration. Score-driven nonlinear updates of local level and seasonality are used, which are robust to outliers within each regime. We show that VAR integrated moving average (VARIMA) type filters are special cases of QVAR filters. Using exclusion, sign, and elasticity identification restrictions in MS-Seasonal-t-QVAR with common-trends, we provide short-run and long-run impulse response functions for the global crude oil market.

For policy decisions, capturing seasonal effects in impulse responses are important for the correct specification of dynamic models that measure interaction effects for policy-relevant macroeconomic variables. In this paper, a new multivariate method is suggested, which uses the score-driven quasi-vector autoregressive (QVAR) model, to capture seasonal effects in impulse response functions (IRFs). The nonlinear QVAR-based method is compared with the existing linear VAR-based method. The following technical aspects of the new method are presented: (i) mathematical formulation of QVAR; (ii) first-order representation and infinite vector moving average, VMA (∞), representation of QVAR; (iii) IRF of QVAR; (iv) statistical inference of QVAR and conditions of consistency and asymptotic normality of the estimates. Control data are used for the period of 1987:Q1 to 2013:Q2, from the following policy-relevant macroeconomic variables: crude oil real price, United States (US) inflation rate, and US real gross domestic product (GDP). A graphical representation of seasonal effects among variables is provided, by using the IRF. According to the estimation results, annual seasonal effects are almost undetected by using the existing linear VAR tool, but those effects are detected by using the new QVAR tool.

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We contribute to the literature on empirical macroeconomic models with time-varying conditional moments, by introducing a heteroskedastic score-driven model with Student’s t-distributed innovations, named the heteroskedastic score-driven $t$ -QVAR (quasi-vector autoregressive) model. The $t$ -QVAR model is a robust nonlinear extension of the VARMA (VAR moving average) model. As an illustration, we apply the heteroskedastic $t$ -QVAR model to a dynamic stochastic general equilibrium model, for which we estimate Gaussian-ABCD and $t$ -ABCD representations. We use data on economic output, inflation, interest rate, government spending, aggregate productivity, and consumption of the USA for the period of 1954 Q3 to 2022 Q1. Due to the robustness of the heteroskedastic $t$ -QVAR model, even including the period of the coronavirus disease of 2019 (COVID-19) pandemic and the start of the Russian invasion of Ukraine, we find a superior statistical performance, lower policy-relevant dynamic effects, and a higher estimation precision of the impulse response function for US gross domestic product growth and US inflation rate, for the heteroskedastic score-driven $t$ -ABCD representation rather than for the homoskedastic Gaussian-ABCD representation.