Structural breaks in Box-Cox transforms of realized volatility: a model selection perspective

Abstract

Autoregressive (AR) models such as the heterogeneous autoregressive (HAR) model capture the linear footprint inherent in realized volatility. We draw upon the fact that the HAR model is a constrained AR model and cast the problem of estimating structural breaks in the autoregressive volatility dynamics as a model selection problem. A two-step Lasso-type procedure is used to consistently estimate the unknown number and timing of structural breaks. Empirically, we find the number of breaks to be heavily influenced by Box-Cox transformations applied to realized volatility series of eight stock market indices: For example, while we find breaks in the original series, no breaks are found in log-realized volatility, a measure often used in applied research, across a wide range of lag lengths. These Box-Cox transformations lead to different volatility processes with distinct autoregressive dynamics and affect the estimation of structural breaks. Importantly, the log-transformation considerably reduces the number of price jumps which might otherwise be selected as structural breaks.

Keywords:

• Realized volatility
• Box-Cox transformation
• Structural breaks
• Group Lasso

JEL codes:

• C22
• C52
• G15

Acknowledgments

Funding by the German Research Foundation (Grant PE 2370/2-1) is greatly appreciated. I am grateful to Karsten Schweikert, Francesco Audrino, Daniele Ballinari, Alexander Schmidt, and two anonymous referees for valuable comments and suggestions that helped to substantially improve the quality of the paper. Moreover, I am grateful to Francesco Audrino, Fabio Sigrist, and Daniele Ballinari for sharing their realized volatility data on individual-level stocks. All views expressed in the paper are my own and do not necessarily represent the views of d-fine GmbH.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Note that this naive estimator is generally biased in the presence of microstructure noise or if $p_t$ evolves as a jump-diffusion of the general form $d{p}_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t+d{J}_t$, where $J_t$ denotes a finite activity jump process (e.g. Audrino and Knaus 2016).

2 See https://realized.oxford-man.ox.ac.uk/data.

3 Both the kernel density plots and the autocorrelation function plots are available upon request.

4 For some realized volatility series, the group-LARS algorithm fails to converge for p>20, which is denoted in the tables by ‘– ’. To be more precise, in these cases, a nearly singular matrix is encountered when computing the decent direction in the group-LARS algorithm.

5 Besides the realized volatility of stock market indices, we have also considered the (transformed) realized volatility based on the MedRV estimator for a sample of 19 liquid stocks. Considering individual-level stocks constitutes an interesting empirical exercise in its own right since the idiosyncratic features of these stocks or specific trading days may have an impact on the persistence of the realized volatility series. However, also considering individual-level stocks, we find that results are in line with the reported results for the (transformed) realized volatility of stock market indices. These additional results are available upon request.

6 See also Casini and Perron (2019a) and Casini and Perron (2019b) for recent additions to the literature covering methods for the detection of structural breaks in continuous time.

7 The estimation can easily be implemented along the lines of Bai and Perron 2003 with the strucchange package in the statistical software environment R. See https://CRAN.R-project.org/package=strucchange and also Zeileis et al. (2003) for further comments. The parameter h, which governs the maximum number of structural breaks m that are allowed, is set to h = 0.1 when considering a model with breaks only in $c_j$ (e.g. Jung and Maderitsch 2014) and we allow for a larger m by setting h = 0.02 when considering breaks in $c_j$ and the dynamic effects ${\text{β}}_j$. The latter choice is based on the results reported for the two-step Lasso procedure where we find that structural breaks do not occur in quick succession. The Bayesian Information Criterion (BIC) is used to select the number of structural breaks. Note that the procedure of Chan et al. (2014) is computationally more efficient compared to the procedure outlined in Bai and Perron (2003), which is especially evident when considering long time series and several specifications. Thus, we have not considered a smaller number for h in our empirical application.

Additional information

Funding

This work was supported by Deutsche Forschungsgemeinschaft [grant number PE 2370/2-1].