Elsevier

Finance Research Letters

Volume 23, November 2017, Pages 39-49
Finance Research Letters

The effect of non-trading days on volatility forecasts in equity markets

https://doi.org/10.1016/j.frl.2017.07.002Get rights and content

Abstract

Weekends and holidays lead to gaps in daily financial data. Standard models ignore these irregularities. Because this issue is particularly important for persistent time series, we focus on volatility modelling, specifically modelling of realized volatility. We suggest a simple way of adjusting volatility models, which we illustrate on an AR(1) model and the HAR model of Corsi (2009). We investigate daily series of realized volatilities for 21 equity indices around the world, covering more than 15 years, and we find that our extension improves the volatility models—both in sample and out of sample. For HAR models and for consecutive trading days, the mean squared error decreased by 2.34% in average and for the QLIKE loss function by 1.41%.

Introduction

Daily time series are probably the most commonly used data in empirical finance. The main advantage of these data is that they usually represent the highest frequency freely available for various datasets. In addition, for many purposes, daily time series can be considered regularly spaced data, and standard time series techniques can be applied to them. However, most financial data exist only for trading days. As we show in this paper, considering this trading gap can significantly improve the performance of models in some cases.

Many time series data exhibit autoregressive properties, meaning that a variable's value today is related to its value tomorrow. If we can more precisely capture this relationship, our models will be more precise. However, in case of daily financial series, data usually exist for weekdays but not for weekends or holidays. Weekends can be considered a break in the data or days with missing data. In either case, the dependence between two consecutive trading days, for example, Wednesday and Thursday, can be intuitively expected to be stronger than the dependence between two trading days separated by a weekend or holiday, for example, Monday and Friday. If we ignore these differences and assume the same dependence throughout the week, we will likely underestimate the dependence between consecutive weekdays and overestimate the dependence between Friday and Monday. We suggest that allowing the autoregressive coefficient to be dependent on whether a weekend separates the two observations sufficiently addresses this issue.

Obviously, this effect does not matter when very little dependence is reflected in the data. In the literature, volatility is understood to have a long memory (Bollerslev and Mikkelsen, 1996). In the empirical part of our paper, we thus focus on volatility. Volatility is not directly observable, but the concept of realized volatility (RV) calculated from high-frequency data (Andersen and Bollerslev, 1998) makes it observable for practical purposes. Models based on RV have been applied to stock markets (e.g., Christoffersen et al., 2010, Bugge et al., 2016), exchange rates (e.g., Andersen et al., 2001, Lyócsa et al., 2016), and commodities (Haugom et al., 2014, Birkelund et al., 2015, Lyócsa and Molnár, 2016). We thus focus on RV.

Long memory and the persistence of market volatility have led previous research to use fractionally integrated time series models (e.g., Bollerslev and Mikkelsen, 1996). However, the most popular model is the heterogeneous autoregressive (HAR) model of Corsi (2009), which captures the long-memory property as accurately as competing models and is easy to implement. We suggest a simple extension of this model, the NT-HAR model (non-trading days HAR), which allows the autoregressive coefficient to be dependent on whether a non-trading period occurs between two observations.

The data that we use in the empirical evaluation of our model are RV series for 21 equity indices around the globe, including equity indices for Brazil, Canada, China, France, Germany, Greece, India, Italy, Japan, Mexico, Singapore, South Korea, Spain, Switzerland, and the U.K.; three indices for U.S.; and one index for the Eurozone. We find that, in most cases, the NT-HAR model outperforms the benchmark HAR model, both in sample and out of sample.

The rest of the paper is organized as follows. Section 2 intuitively explains our model using a simulation example. Section 3 describes the data and the methodology; Section 4 presents the results; and Section 5 concludes.

Section snippets

Illustrative example

Our motivation can be intuitively explained as follows: given the measure of daily market volatility RVi,t, the lagged market volatility RVi,t-1 does not always precede the predicted volatility at time t by one calendar day. On average, in more than one-fifth of the cases (18.56%), the calendar-day difference between two consecutive days on equity markets is equal to or more than 3 (e.g., see column ``nC” in Table 2). One consequence of non-equidistant observations could be that the lagged

Data and volatility estimators

The predictive regression models used in this study are based on modelling-realized measures of daily volatility RVi,t for the ith stock market index for day t. The standard approach in the literature is to employ the RV given byj=1Nri,t,j2where ri,t,j denotes the jth intraday return. N denotes the number of intraday returns given the length of the trading hours and the sampling frequency.

The literature has recently provided several classes of realized estimators of market volatility, which

Results

Table 2 presents the summary statistics of the returns and RVs of the studied equity indices. In general, summary statistics reveal that these time series behave as expected. Autocorrelation in returns is rather low. By contrast, autocorrelation in RV is high. We can also observe that the days with the highest volatilities and those with the highest and lowest returns typically occur during financial crises or other significant market events.

The last column of Table 2 deserves special

Conclusion

Daily financial time series are usually irregularly spaced due to weekends, holidays and other non-trading days. However, standard time series models ignore this issue, which is most pronounced for highly persistent time series. Volatility usually exhibit long memory, and we therefore study this topic in the context of volatility modelling and forecasting. We suggest a simple way to improve volatility models by incorporating information that some observations occurred after weekends or

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    Lyócsa appreciates the support by the Slovak Research and Development Agency under contract No. APVV-14-0357 and by the Slovak Grant Agency under Grant No. 1/0406/17 and Grant No. 1/0257/18.

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