The transition from traditional energy to cleaner energy sources has raised concerns from companies and investors regarding, among other things, the impact on financial downside risk. This article implements backtesting techniques to estimate and validate the value-at-risk (VaR) and expected shortfall (ES) in order to compare their performance among four renewable energy stocks and four traditional energy stocks from the WilderHill New Energy Global Innovation and the Bloomberg World Energy for the period 2005-2016. The models used to estimate VaR and ES are AR(1)-GARCH(1,1), AR(1)-EGARCH(1,1), and AR(1)-APARCH(1,1), all of them under either normal, skew-normal, Student’s t, skewed-t, Generalized Error or Skew-Generalized Error distributed innovations. Backtesting performance is tested through traditional Kupiec and Christoffersen tests for VaR, but also through recent backtesting ES techniques. The paper extends these tests to the skewed-t, skew-normal and Skew-Generalized Error distributions and applies it for the first time in traditional and renewable energy markets showing that the skewed-t and the Generalized Error distribution are an accurate tool for risk management in those markets. Our findings have important implications for portfolio managers and regulators in terms of capital allocation in renewable and traditional energy stocks, mainly to reduce the impact of possible extreme loss events.

The mixed causal‐non‐causal invertible‐non‐invertible autoregressive moving‐average (MARMA) models have the advantage of incorporating roots inside the unit circle, thus adjusting the dynamics of financial returns that depend on future expectations. This article introduces new techniques for estimating, identifying and simulating MARMA models. Although the estimation of the parameters is done using second‐order moments, the identification relies on the existence of high‐order dynamics, captured in the high‐order spectral densities and the correlation of the squared residuals. A comprehensive Monte Carlo study demonstrated the robust performance of our estimation and identification methods. We propose an empirical application to 24 portfolios from emerging markets based on the factors: size, book‐to‐market, profitability, investment and momentum. All portfolios exhibited forward‐looking behavior, showing significant non‐causal and non‐invertible dynamics. Moreover, we found the residuals to be uncorrelated and independent, with no trace of conditional volatility.

Financial assets often exhibit explosive price surges followed by abrupt collapses, alongside persistent volatility clustering. Motivated by these features, we introduce a mixed causal–noncausal invertible–noninvertible autoregressive moving average generalized autoregressive conditional heteroskedasticity (MARMA–GARCH) model. Unlike standard ARMA processes, our model admits roots inside the unit disk, capturing bubble-like episodes and speculative feedback, while the GARCH component explains time-varying volatility. We propose two estimation approaches: (i) Whittle-based frequency-domain methods, which are asymptotically equivalent to Gaussian likelihood under stationarity and finite variance, and (ii) time-domain maximum likelihood, which proves to be more robust to heavy tails and skewness—common in financial returns. To identify causal vs. noncausal structures, we develop a higher-order diagnostics procedure using spectral densities and residual-based tests. Simulation results reveal that overlooking noncausality biases GARCH parameters, downplaying short-run volatility reactions to news ($\alpha$) while overstating volatility persistence ($\beta$). Our empirical application to Bitcoin and Ethereum enhances these insights: we find significant noncausal dynamics in the mean, paired with pronounced GARCH effects in the variance. Imposing a purely causal ARMA specification leads to systematically misspecified volatility estimates, potentially underestimating market risks. Our results emphasize the importance of relaxing the usual causality and invertibility assumption for assets prone to extreme price movements, ultimately improving risk metrics and expanding our understanding of financial market dynamics.

This paper investigates new ways of estimating and identifying causal, noncausal, and mixed causal-noncausal autoregressive models driven by a non-Gaussian error sequence. We do not assume any parametric distribution function for the innovations. Instead, we use the information of higher-order cumulants, combining the spectrum and the bispectrum in a minimum distance estimation. We show how to circumvent the nonlinearity of the parameters and the multimodality in the noncausal and mixed models by selecting the appropriate initial values in the estimation. In addition, we propose a method of identification using a simple comparison criterion based on the global minimum of the estimation function. By means of a Monte Carlo study, we find unbiased estimated parameters and a correct identification as the data depart from normality. We propose an empirical application on eight monthly commodity prices, finding noncausal and mixed causal-noncausal dynamics.