A Closer Look at High-Frequency Data and Volatility Forecasting in a HAR Framework 1

Transcription

1 A Closer Look at High-Frequency Data and Volatility Forecasting in a HAR Framework 1 Derek Song ECON 21FS Spring 29 1 This report was written in compliance with the Duke Community Standard

2 2 1. Introduction The volatility of asset returns is an area of vital importance for research in financial theory risk management and derivative valuation methods are all dependent on being able to accurately measure and forecast volatility. The recent availability of high-frequency price data has given rise to new models of volatility that have yielded significant improvements in the accuracy of volatility measurements and forecasting. The ability to accurately predict future volatility is particularly important in a practical sense because of its implications for asset management. Recent literature, such as Andersen, Bollerslev, Diebold, Labys (23), show that using high-frequency data, simple linear autoregressive regression models have better predictive capabilities than the more sophisticated ARCH/GARCH and stochastic volatility models. One such model is the heterogeneous autoregressive (HAR) model, a simple autoregressive model for realized volatility first proposed by Corsi (23). When setting up a regression model, researchers are afforded several degrees of freedom, including methods for calculating RV and regression methodology. In the current literature, comparisons of different regression models are often made using a particular choice for the sampling interval. This raises the question of whether or not those results would hold given different choices for model parameters, since we would like to ensure some level of consistency when comparing models. In this paper, we seek to add to the existing literature by empirically examining the sensitivity of several HAR model forecasts to various sampling and regression methods. In particular, we compare four models, HAR-RV, HAR-RAV, and both of the above models with implied volatility added in. The three factors that we will consider are (1) sampling interval, which runs

3 3 from 1 minute to 3 minutes, (2) sub-sampling, and (3) the effect of using robust regressions instead of OLS to control for outliers and leverage points. We compare the models by measuring both in-sample fit and out-of-sample performance on a synthetic portfolio constructed to mimic the S&P 1. We find that when the sampling interval is set at 5 minutes or higher, there is little variation in forecast accuracy for different intervals, and that any noisiness is eliminated by using sub-sampling. Furthermore, our results suggest that including implied volatility has a significant impact on forecasting accuracy. The rest of the paper proceeds as follows: theoretical and mathematical background of volatility and regression models (2), research methodologies (3), data preparation (4), empirical results (5), and a conclusion summarizing the paper and the most important results (6). All tables and figures are given in the end of the paper. 2. Theoretical Background 2.1 Stochastic Model of Asset Returns In this paper, we assume a widely used model of asset prices that includes jumps. We assume that the log-prices of a stock, denoted follow the stochastic differential equation given below: = + + (2.1.1) Here, is a time-varying drift component, represents a time-varying volatility component of the asset price, is a standard Wiener process, is the magnitude of the jump, and represents a counting process which is commonly assumed to be a Poisson process so that jumps are rare.

4 4 2.2 Market Microstructure Noise Stock prices are commonly assumed to have a theoretical fundamental price, calculated as the sum of all discounted future dividend payments. Market microstructure noise is defined as any short-term deviations of the spot price from the fundamental value of a stock, and is modeled by = + (2.2.1) Note that is the logarithm of the observed price, and therefore the error term is proportional to the observed price. Market microstructure noise arises due to various market frictions, including the bid-ask bounce. Because market microstructure noise distorts price data at high frequencies, it can become problematic for the estimation of realized volatility. Bandi, Russell (28) show that in the presence of noise, the RV estimator will diverge to infinity almost surely. However, Forsberg, Ghysels (27) argue that RAV is much more robust to sampling errors and jumps. In this paper, we discuss two ways of circumventing this problem, which we discuss in Sections 2.3 and Models of Volatility in Asset Returns We let, denote the logarithmic (geometric) return at some intra-day time, given by, =,,, where is the logarithm of the observed price. We will now define two different measures of volatility, Realized Variance (RV) and Realized Absolute Value (RAV). We define RV as the sum of the squared log-returns, and RAV as the sum of the absolute log-returns. As such, RV will be measured in variance units, while RAV is measured in standard deviation units. Letting be the number of times

5 5 we sample within each day (in our data, we have per-minute returns, yielding a maximum of 384 samples per day), we calculate the daily RV as follows: =, + (2.3.1) The next measure, RAV, is defined as: = 2, (2.3.2) There are several key points to note about these measures; the first is that because of the way in which they are defined, RV and RAV are not directly comparable. Secondly, as the sampling interval increases, we are throwing away more and more of the data. Zhang, Mykland, Aït-Sahalia (25) propose an alternative sampling methodology known as sub-sampling. Assuming we sample every minutes, we calculate our measure from each starting point =1,2,, and average those calculations, resulting in no discarded data points. Sub-sampling has two distinct advantages over the traditional sampling method: reduced bias from microstructure noise and the ability to use all of the available data points regardless of sampling interval. 2.4 HAR Regression Models In this paper, we rely on the Heterogeneous Autoregressive (HAR) models first introduced by Müller et al (27) and Corsi (23) to forecast volatility. Recent papers (see: Andersen, Bollerslev, Diebold, Labys (23) or Andersen, Bollerslev, Huang (27)) have shown empirically that simple linear models can often predict future volatility more accurately than more sophisticated models that can formally capture long memory processes and persistence. The HAR framework developed by Corsi is

6 6 attractive because it is easily estimated using OLS, and is significantly more parsimonious than the HARCH model of Müller et al (1997). The expected future variance over an h-day horizon is given by a linear combination of average historical RV s over different time scales, which can capture the persistence seen in time series data without making the sort of restrictive assumptions seen in ARFIMA and GARCH models. In order to calculate the model, we will define, as the average RV over a given time span, h. This is mathematically represented as follows:, = 1 h (2.4.1), is then defined analogously. In this paper, we want to calculate 22-day ahead forecasts, which correspond to the number of trading days within a calendar month. Thus, we can set up a HAR-RV regression as:, = +, +, +, + (2.4.2) The dependent variables correspond to daily, weekly, and monthly lagged regressors, which were chosen by Corsi in his paper. Also, it should be noted that Andersen, Bollerslev, Diebold (27) established that in general, the jump effects embedded in RV measures are not significant within the context of a HAR regression. Forsberg, Ghysels (27) extend the HAR-class models by using historical RAV to forecast future RV, which they find to be a significantly better predictor of RV than historical RV. The model is analogous to the one for HAR-RV, and is defined as:, = +, +, +, + (2.4.3) It should be noted that the physical interpretation of the HAR-RAV model is not identical to the HAR-RV model, since RAV and RV are in different units.

7 7 2.5 Hybrid HAR - Implied Volatility Regressions There is a large literature on the use of options and model-free implied volatility to forecast future volatility. Poon, Granger (25) and a literature review by Blair, Poon, Taylor (21) find that implied volatility is a better predictor of volatility than the commonly used time-series models. Mincer and Zarnowitz (1969) proposed a simple framework with which to evaluate the efficiency of implied volatility-based forecasting:, = + +, (2.5.1) If implied volatility were perfectly efficient, = and =1. However, numerous papers, including Becker, Clements, White (23) find that implied volatility is not a perfectly efficient estimator. Jiang, Tian (25) showed that model-free implied volatility is better than options-implied volatility at predicting future volatility and endorsed the new CBOE VIX methodology for its use of model-free implied volatility. Fradkin (28) found evidence that adding implied volatility to HAR models almost always improved model fit, which suggests that implied volatility contains information not present in historical realized volatility. We will define hybrid HAR-RV- IV and HAR-RAV-IV models identical to those used by Fradkin:, = +, +, +, + + (2.5.2), = +, +, +, + + (2.5.3) 3. Data Preparation The high-frequency stock price data used in this paper were obtained from an online vendor, price-data.com. For this paper, we follow Law (27) and select 4 of the largest MCAP stocks from the S&P 1 (OEX) and aggregate those stocks to form a

8 8 portfolio that we claim can proxy for the S&P 5 (SPX) for two reasons; the OEX is a subset of the SPX, and there exists a high degree of correlation between these two indices. In Figure 1a, we show a scatterplot of daily open-to-close returns for our synthetic proxy portfolio (SPP) versus daily open-to-close returns of the SPX. Our requirements for inclusion were that data for the stock be present from Jan. 3, 2 up through Dec. 31, 28; we also checked for inconsistencies in the data and adjusted the prices for stock splits. In creating the portfolio, we kept only the data for those days in which all 4 stocks traded, yielding a total of 224 days. We use an equal-weighting scheme to construct our portfolio by buying $25 of each stock at the initial price. In Section 2.2, we discussed the problem of market microstructure noise, and we now claim, citing Figures 1b, 2a, and 2b, that the process of aggregating stocks has averaged out most if not all of the microstructure noise. Our implied volatility data was taken from the CBOE website. We used the VIX, a model-free implied volatility index which uses options on the SPX to calculate the 1- month ahead implied volatility for reasons described in Section 2.5. Because intra-day data was not available, we use only the closing price of the VIX in our regressions. We transformed the data so that it is measured in the same units as RV. Also, we naturally include only those days for which the SPP exists. Our in-sample data runs for 7 years, from the beginning of 2 until the end of 26, yielding 1743 data points. Our out-of-sample data runs from the beginning of 27 until the end of 28, yielding 497 data points. We therefore have 24 independent month-long periods for the out-of-sample result, which should be sufficient to accurately gauge out-of-sample performance.

9 9 4. Regression Methodologies 4.1 Robust Regressions with Iterative Huber Weighting Poon, Granger (25) discusses the common problem of sample outliers to volatility estimation. These leverage points are problematic because they can unduly influence OLS estimators, especially when the regressions use only historical volatility. Because manually removing outliers in a data set this large is infeasible, we will deal with these leverage points by using robust regressions as a comparison for OLS regressions. We employ an iterative Huber weighting scheme over bisquare weighting because it converges significantly faster for our regressions. Our regressions are run in MATLAB, using the regress and robustfit commands to estimate the OLS and robust coefficients, respectively. 4.2 Evaluating Regression Performance There are a number of different methods for evaluation forecast accuracy. We will use Mean Absolute Percentage Error () because it is a measure of relative accuracy, allowing us to compare results when the RV measures we forecast vary due to sampling interval and sub-sampling. Letting be the residual, and be the actual value, we define as: = 1 (4.2.1) The main problem with is that the measure is not upper-bounded and so we must be careful of very small or zero values for. As Figure 3 shows, our RV is lower and upper-bounded by values that are within a reasonable range of each other.

10 1 5. Empirical Results 5.1 In-Sample Results The in-sample surface plots (Figures 4-6) show a marked increase in variation in fit when the sampling interval for either side of the regression is small ( <5 min). This effect is significantly more pronounced for OLS regressions than for the robust regressions. Above this threshold, the surface plot is relatively flat, suggesting that any choice of large sampling interval ( 5 min) has little bearing on fit. Sample fit increases when the sampling interval decreases in each of the models. For the HAR-RAV models, fit decreases when the sampling interval decreases. Adding implied volatility appears to curtail most of that variability, however. Sub-sampling eliminates noisiness in our regressions, producing a smooth surface plot; however, it does not improve fit uniformly across all sampling intervals. Therefore, although using sub-sampling is able to ensure some degree of consistency in our results, it does not play a major role in fit. Between models, we see that RAV produces a better fit than RV for OLS regressions over large sampling intervals. However, the addition of implied volatility provides the best fit, and there no longer appears to be a significant difference between RV-IV and RAV-IV. Furthermore, the variability seen at smaller intervals is diminished greatly by the inclusion of IV. With regards to the robust regressions, the differences between RV and RAV alone are not significant, and robust regressions have also decreased the variability at the lower sampling intervals. Adding IV improves fit, but the magnitude of the improvement is not as large as for OLS. Finally, the robust regressions appear to offer the best fit for each of the four regression models.

11 11 We report OLS coefficients for selected combinations for each model in Table 1 and robust coefficients in Table 2. The standard errors for the OLS coefficients are Newey-West standard errors with a lag of 44 days. We find that in general, the coefficients are significant at the =.5 level or better. The robust regression coefficients are, with few exceptions, highly coefficient ( <.1), however, this is very likely because the standard errors are not robust to serial correlations. 5.2 Out-of-sample Results From Figures 7-9, we see that the variability towards the small intervals is generally larger than in the in-sample results, particularly when using OLS regressions. Using robust regressions improves consistency when implied volatility is not included; however, when IV is included, the variation seen in at the small intervals is curtailed. However, with regards to HAR-RV and HAR-RAV, we see the same general pattern as in the in-sample data. Along the of the regression, decreasing sampling interval results in a small improvement in accuracy, while along the, we see a drastic decline in accuracy as the sampling interval decreases. For the out-of-sample comparisons, we see many of the same results discussed above. HAR-RV and HAR-RAV perform very similarly in the out-of-sample period, and the inclusion of implied volatility helps to improve performance. Robust estimation procedures appear to provide the most accurate forecasts across all models. We should note that the out-of-sample period used in this paper encompasses a period of unusually high volatility due to the recent economic turmoil, as seen in Figure 3. Fradkin (27) and Forsberg, Ghysels (27) both found clear evidence that HAR-RAV

12 12 offered the best predictions of future volatility; however, they used 25 and as their out-of-sample periods, respectively, which were both periods of relatively low volatility. This may imply that HAR-RAV offers a significant advantage over HAR-RV when the overall volatility is low and persistence effects are not as strong. However, further analysis of this topic is beyond the scope of this paper. 6. Conclusion In this paper, our goal was to examine the impact that different sampling and regression methodologies have on volatility forecasting in order to gain a better understanding of how the choices we make with regards to modeling and estimation can affect our results. To that end, we examined three factors: sampling interval, subsampling, and robust or OLS regressions. First, we found that forecast performance can vary greatly when the sampling interval falls below 5 min; for most of the models, decreasing the sampling interval on the of the regression improved accuracy, but decreasing the sampling interval on the hurt accuracy. Beyond 5 minutes, there is a high level of consistency in our results. Secondly, sub-sampling is able to reduce the noisiness in our regression results, but it does not yield any true improvements in overall forecast accuracy. Finally, our results show that using robust estimation procedures and implied volatility both improve forecasting performance over the base HAR-RV and HAR-RAV models, although the robustly estimated models fared the best out-of-sample.

13 13 7. Tables and Figures Figure 1: SPP Data Comparison of Daily Open-to-Close Log-returns.15 1 Intra-day Price Movements of SPP for 1 Day SPP Log-returns.5 Value of SPP S&P 5 Log-returns Time of Day Figures 1a and 1b: 1a shows a scatterplot showing SPX intra-day returns vs. SPP intra-day returns. 1b is a plot of the price movements in our portfolio SPP within an arbitrarily chosen day. Figure 2: Volatility Signature Plots.3 RV Volatility Signature.14 RAV Volatility Signature Annualized Units No Sub-sampling With Sub-sampling Sampling Interval (Minutes) Sampling Interval (Minutes) Figure 2: These are volatility signature plots, introduced in Andersen, Bollerslev, Diebold, Labys (1999). The fact that RV and RAV are decreasing as the sampling interval becomes smaller than 5 min suggests that market microstructure noise no longer biases either volatility measure. Annualized Units No Sub-sampling With Sub-sampling Figure 3: 1-month Ahead Mean RV and VIX Plots Annualized Volatility (%) Month Ahead RV VIX 1-Month Ahead Mean RV and VIX Date Figure 3: A plot showing annualized values for the VIX and the annualized monthly volatility for our synthetic portfolio (SPP).

14 14 Figure 4: OLS In-Sample Surface Plot w/out Sub-sampling In-Sample HAR-RV In-Sample HAR-RAV In-Sample HAR-RV-IV In-Sample HAR-RAV-IV Figures 4-9: These are all surface plots, with the sampling interval (from 1 min up to 3 min) of the lefthand side of the regression on the left axis, the sampling interval (also from 1 3 min) of the right-hand side of the regression on the right axis, and the for each combination on the vertical axis. Figure 5: OLS In-Sample Surface Plot w/ Sub-sampling In-Sample HAR-RV In-Sample HAR-RAV In-Sample HAR-RV-IV In-Sample HAR-RAV-IV

15 15 Figure 6: Robust In-Sample Surface Plot w/ Sub-sampling In-Sample HAR-RV In-Sample HAR-RAV In-Sample HAR-RV-IV In-Sample HAR-RAV-IV Figure 7: OLS Out-of-Sample Surface Plot w/out Sub-sampling Out-of-Sample HAR-RV Out-of-Sample HAR-RAV Out-of-Sample HAR-RV-IV Out-of-Sample HAR-RAV-IV

16 16 Figure 8: OLS Out-of-Sample Surface Plot w/ Sub-sampling Out-of-Sample HAR-RV Out-of-Sample HAR-RAV Out-of-Sample HAR-RV-IV Out-of-Sample HAR-RAV-IV Figure 9: Robust Out-of-Sample Surface Plot w/ Sub-sampling Out-of-Sample HAR-RV Out-of-Sample HAR-RAV Out-of-Sample HAR-RV-IV Out-of-Sample HAR-RAV-IV

17 17 Table 1: Coefficients for Select OLS Regressions (w/ Sub-sampling) HAR-RV HAR-RAV HAR-RV-IV HAR-RAV-IV (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (x1-5 ) 1.4*** 2.4*** -3.4*** -2.2*** 1.1*** 1.5*** -2.1** ***.6**.4***.2***.11**.3*.3**.1*.37***.15***.8**.1***.32**.12**.7**.4**.18.21***.2.3* **.14***.9*.9 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (x1-5 ) 1.6** 2.7*** -5.7*** -4.6*** 1.* 1.3** -3.* -3.6**.34***.11**.1***.4***.17*.6*.4*.3**.64**.27***.1**.1***.53**.22**.1*.9**.7.25** ***.22**.18**.1 Table 1: Coefficients reported with significance. (x,y) sampled at x min, sampled at y min Significance levels: * = p<.5 ** = p<.1 *** = p<.1 P-values obtained from Newey-West Standard Errors w/ Lag length of 44 Table 2: Coefficients for Select Robust Regressions (w/ Sub-sampling) HAR-RV HAR-RAV HAR-RV-IV HAR-RAV-IV (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (x1-5 ) 1.3*** 2.1*** -2.7*** -1.6*** 1.1*** 1.5*** 1.6*** -.9***.18***.8***.3***.2***.11***.4***.2***.1***.3***.15***.1***.4***.25***.1***.5***.3***.21***.17***.3***.3***.1***.8***.2***.2*** ***.11***.8***.7*** (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (x *** 1.9*** -4.2*** -3.3***.9*** 1.2*** -1.7*** -2.1*** ).3***.12***.5***.3***.12***.6***.2***.2***.37***.27***.1***.1***.28***.18***.1***.1***.27***.21***.4***.4*** -.1.5***.2.1*** ***.17***.17***.11*** Table 2: Same significance levels as in Table 1. P-values obtained from heteroskedasticity-robust SE s.

18 18 8. References 1. Andersen, T., T. Bollerslev, and F. Diebold, Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility. The Review of Economics and Statistics, (4): p Andersen, T., et al., Realized Volatility and Correlation. Working Paper, Northwestern University, Andersen, T., et al., Modelling and Forecasting Realized Volatility. Econometrica, (2): p Andersen, T., T. Bollerslev, and X. Huang, A Semiparametric Framework for Modelling and Forecasting Jumps and Volatility in Speculative Prices. Working Paper, Duke University, Bandi, F. and J. Russell, Microstructure Noise, Realized Variance, and Optimal Sampling. Review of Economic Studies, (2): p Becker, R., A. Clements, and S. White, On the Informational Efficiency of S&P5 Implied Volatility. North American Journal of Economics and Finance, (2): p Blair, B., S.-H. Poon, and S. Taylor, Forecasting S&P 1 Volatility: The Incremental Information Content of Implied Volatilities and High-Frequency Index Returns. Journal of Econometrics, (1): p Corsi, F., A Simple Long Memory of Realized Volatility. Unpublished Manuscript, University of Logano, Forsberg, L. and E. Ghysels, Why Do Absolute Returns Predict Volatility So Well? Journal of Financial Econometrics, 27. 5(1): p Fradkin, A., The Informational Content of Implied Volatility in Individual Stocks and the Market. Unpublished Manuscript, Duke University, Jiang, G. and Y. Tian, The Model-Free Implied Volatility and its Informational Content. Review of Financial Studies, (4): p Law, T.H., The Elusiveness of Systematic Jumps. Unpublished Manuscript, Duke University, Mincer, J. and V. Zarnowitz, The Evaluation of Economic Forecasts, in Economic Forecasts and Expectations, J. Mincer, Editor. 1969, NBER: New York. 14. Muller, U., et al., Volatilities of Different Time Resolutions - Analyzing the Dynamics of Market Components. Journal of Empirical Finance, (2-3): p Poon, S.-H. and C. Granger, Practical Issues in Forecasting Volatility. Financial Analysts Journal, (1): p Zhang, L., L. Mykland, and Y. Ait-Sahalia, A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data. Journal of the American Statistical Association, 25. 1: p