Abstract: For the volatility of financial asset returns, as a key variable in the process of financial risk management, how to construct a suitable model to estimate and predict the volatility more accurately has important theoretical and practical significance. Zero returns often result from liquidity problems, price dispersion or rounding errors, data problems, market closures and other market specific characteristics of financial markets. In order to estimate volatility more accurately, we think it necessary to construct a reasonable model to model data containing zero returns, for studying the impact of zero returns on volatility estimation.
The causes of the emergence of zero returns have been examined in the literature. The first type of the literature argues that zero returns arise in a continuous time frame because the underlying price transformation process is not observed. The second one argues that zero returns arise naturally because of the discrete nature of price changes. The third one argues that price changes beyond zero return are continuous. The fourth one argues that as long as the residuals of the GARCH model can be zero, the GARCH model can be used to study the zero returns problem. On the basis of the fourth one this paper studies the problem of modelling financial data containing non-stationary zero return processes.
While there is literature on the causes and modelling of zero return generation, little attention has been paid to the fact that the zero returns process is a non-stationary case. Zero returns processes in practice are usually non-stationary, so the probability of zero returns may be time-varying or periodic. For inter-day data, a downward (upward) trend in the zero probability may be due to an upward (downward) trend in liquidity or an upward (downward) trend in the level of stock prices. For intra-day data, the zero probability tends to be non-stationary and cyclical: it will be lower when liquidity is low and higher when liquidity is high. It is therefore necessary to extend the existing GARCH family of models for financial data containing non-stationary zero returns processes.
This paper proposes a zero-inflated GARCH model to model financial data containing a non-stationary zero return process, where the zero probability may be trending or cyclical, or both. Then, an improved QMLE method, i.e. 0-adj QMLE method, is proposed for parameter estimation. Then, six major global exchange rate markets are selected as the research object, and the daily returns data of the exchange rate market and the 5-minute high-frequency returns data of the US dollar to offshore RMB exchange rate are modelled using the zero-inflated GARCH model in this paper, and the two methods of 0-adj QMLE and standard QMLE are applied to estimate the parameters and make a comparative analysis of the estimation results; meanwhile, the impact of zero returns on the volatility of the exchange rate market is discussed.
The main findings are: (1)The zero probability is trending in the daily data and cyclical in the high frequency data. (2)The zero-inflated GARCH model is a better fit to exchange rate data containing a non-stationary zero return process. (3)In the daily data, the previous day’s zero returns have a significant negative impact on the volatility of the next day in the USD/CNY market, and the previous day’s zero returns have a significant positive impact on the volatility of the next day in the USD/CNH market. (4)The previous day’s zero returns have a significant positive impact on the next day’s volatility in the USD/CNH market. In the USD/CNH market, the previous day’s zero returns have a significant positive impact on the next day’s volatility. For high frequency data, zero returns in the first five minutes tend to significantly increase volatility in the second five minutes.

Key words: volatility, zero return, zero expansion GARCH model, high-frequency data

摘要： 金融资产收益波动率作为金融风险管理等过程中的关键变量,如何构建合适模型来更加准确地估计和预测波动率具有重要的理论和现实意义。由于金融市场中流动性问题等原因,收益率常常为零。为了深入探讨零收益率对波动率估计的影响,需要构建合理模型对含有零收益率的数据进行建模。本文首先提出一种零膨胀GARCH模型来对含非平稳零收益率过程的金融数据进行建模,并提出一种改进QMLE方法,即0-adj QMLE方法来进行参数估计。然后以全球6个主要汇率市场为研究对象,使用本文零膨胀GARCH模型对日收益率数据和高频收益率数据进行建模,运用0-adj QMLE方法对模型进行估计,并深入讨论零收益率对后续时期波动率的影响。研究结果表明:就日度数据而言,USD/CNY市场中前一天零收益率对后一天波动率产生显著负向影响,USD/CNH市场中前一天零收益率对后一天波动率产生显著正向影响。就高频数据而言,前5分钟的零收益率往往显著提高后5分钟的波动率。

关键词: 波动率, 零收益率, 零膨胀GARCH模型, 高频数据