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Technical range forecastingIn Chapter 4 we demonstrated that closing price and technical range are cointegrated. In this chapter, we will show how this property can be used to improve technical range forecasting. 8.1 Introduction Technical range, defined as the difference between the high and low prices, is an important component of the candlestick, and gauges the variability of the price movement during a specific time period. The larger the technical range, the higher risk investors are confronted with. Therefore, the question of how to describe the dynamics of technical range is of great interest to investors. Voluminous literature is available investigating highlow price range. Early application of range in the field of finance can be traced to Mandelbrot (1971), and academic work on the rangebased volatility estimator started from the early 1980s. Many authors, such as Parkinson (1980), Garman and Klass (1980), developed from range several volatility estimators which are found to be far more efficient than the returnbased volatility estimators. Also, Alizadeh et al. (2002) found that the conditional distribution of the log range is approximately Gaussian, which facilitates the maximum likelihood estimation of stochastic volatility models. Moreover, as pointed by Alizadeh et al. (2002), and Brandt and Diebold (2006), the rangebased volatility estimator is robust to microstructure noise such as bidask bounce. Through Monte Garlo simulation, Shu and Zhang (2006) found that range estimators are fairly robust toward microstructure effects, which is consistent with the finding of Alizadeh et al. (2002). Using a proper dynamic structure for the conditional expectation of range, Chou (2005) proposed the conditional autoregressive range (CARR) model to describe the dynamics of range. Empirical studies performed on the S&P500 index, both insample and outofsample, show that the CARR model does provide a more accurate volatility estimator compared with the GARCH model. Brandt and Jones (2006) formulated a model that is analogous to Nelson’s (1991) EGARCH model, but uses the square root of the intraday price range in place of the absolute return and find that the rangebased volatility estimators oiler a significant improvement over their returnbased counterparts. Using the dynamic conditional correlation (DCC) model proposed by Engle (2002), Chou et al. (2007) extended CARR to a multivariate context and find that this range based DCC model performs better than other returnbased volatility models in forecasting covariances. Other models concerning rangebased volatility include the ACARR (asymmetric CARR) of Chou (2006), the FACARR (feedback ACARR) of Xie (2019), the extended CARR of Xie and Wu (2019). However, all the available academic literature concerning range is on the Parkinson range. No academic research, to the best of our knowledge, is available investigating the technical range. This chapter, based on the theoretical properties of technical range, proposes a vector error correction model (VECM) to describe and forecast the dynamics of technical range. The chapter is organized as follows: Section 2 presents the econometric methodologies with some discussions. We present the empirical results in Section 3. The summary is presented in Section 4.
The first benchmark model is a simple Moving Average (MA) model which is widely used in technical analysis. A ALA model of order q is given by where x, is /«('I'll,). Another benchmark model is the ARMA model. The ARMA model is used for the reason that it is the most commonly used technique in univariate time series modeling and forecasting. An ARMA model of order (p, q) is presented as follows:
where x_{r} is A/w(Til,). We don’t model ln(TR_{t}) because it is a unit root process. Both MA and ARMA models are univariate time series models, they don’t take into consideration the cointegration between the closing price and the technical range. To see if this cointegration can be used to improve technical range forecasting, we propose to use a vector error correction model (VECM). The VECM model is presented as follows
where It is a lag structure, T, is a p x 1 vector of variables and is integrated of order one, 1(1), p is a p x 1 vector of constants, and e_{f} is a p x 1 vector of white noise error terms. Гу is a p x p matrix that represents shortterm adjustments among variables across p equations at the /th lag. Ylj tTAT", ■ and ajl‘ T_{:}_ i are the vector autoregressive (VAR) components in first differences and errorcorrection components respectively. [I is a p x r matrix of cointegrating vectors, and a is a p x r matrix of speed of adjustment parameters. The cointegrating vector fi shows the longterm equilibrium relationship between the concerned variables while the adjustment factor a shows the speed of adjustment towards equilibrium in case there is any deviation. A larger a suggests a faster convergence toward longrun equilibrium in cases of shortrun deviations from the longrun equilibrium. In this chapter, T_{t} = (ln(C_{t}), ln{TR,))^{T}, where C, and 111, are respectively closing price and technical range. 8.2.2 Outofsample forecast evaluation To evaluate the outofsample forecasting accuracy, both mean absolute error (MAE) and root mean squared error (RAISE) are used
where X_{t} and XF,(Mj) are, respectively, the observations and forecasts reported by model i. Models that report smaller MAE (RAISE) are said to have better forecasts. To see if there is significant difference between two competing models for out ofsample forecasting, the DAI statistic (Diebold and Alariano, 1995) is used. Let the forecasting error of model i be
We test the superiority of model j over model i with a rtest of /Jjj coefficient in
where a positive estimate of j indicates support for model j. To further gain insight into the difference between two competing models, we follow the approach of Mincer and Zarnowitz (1969) in running the following regression:
A test of the unbiasedness of the predicted volatility can be performed by a joint test of a = 0 and b = 1. To determine the relative information content of two competing volatility models, we also run a forecast encompassing regression:
If model i dominates model j, then it is expected that b is statistically significant while c is not.
We collected the monthly data of Standard and Poors 500 (S&P500) stock index for the sample period from 1950.01 to 2014.12 with 780 observations. For each month, four pieces of price information, opening, high, low and closing prices were collected. The data set was downloaded from website http://finance.yahoo.com. Table 8.1 presents the summary statistics of log closing price, stock return, log technical range, and differenced log technical range. The JarqueBera statistics reject the null hypothesis of normal distribution for both closing price and technical range. However, the null hypothesis of normal distribution for differenced log technical range can not be rejected. The ADF statistics show that the unit root hypothesis for closing price and technical range can not be rejected. The unit root hypothesis is rejected for both return and differenced log technical range. 8.3.2 Insample estimation For the ARMA model, the SIC criteria prefers the ARMA (0, 1) model. The estimation result is presented as follows
The Rsquarc shows that the ARMA(0, 1) model can explain respectively 37.3% variation of the total variance of differenced log technical range. Table 8.1 Summary statistics of closing price and technical range
Note: C, is the closing price, TR, is the technical range, and r, is the stock return. We use ***, **, and * to mean significance at the level of 1%, 5%, 10%. Tabic 8.2 Vector error correction estimates on S&P500 stock index
The VECM model estimates are reported in Table 8.2. We select the lag k = 5 by the SIC criteria. The cointegration vector is given by/? = (1.000, 0.933, 3.033)^{7}. The cointegration relationship between closing price and technical range is presented as:
The speed of adjustment factor is given by a = (0.007, 0.267). The fact that the adjustment factor a = 0.007 is small and insignificant in the closing price suggests that closing price is exogenous to the changes in technical range. The large and significant adjustment factor a = 0.267 in the technical range means that technical range responds quickly to the changes in closing price. For technical range the Яsquare statistic is reported to be 0.447, which is much larger than 0.373 of the ARMA model. This result indicates that VECM reports better in sample technical range forecasts than ARMA does. 8.3.3 Outofsample forecast For practical purposes, the more important thing is the outofsample forecasting performance. For outofsample predicting, the whole T data observations are divided into an insample portion composed of the first m observations and an Technical range forecasting 67 Table 8.3 Outofsample MAE and RAISE for VECM, ARMA and MA
Table 8.4 Outofsample forecasting evaluation: DM test
We use ***, **, and * to mean significance at the level of 1%, 5%, 10%. Table 8.5 Outofsample forecasting evaluation: regression and encompassing regression
outofsample portion composed of the last q observations. A static forecasting procedure is used. To be specific, we use the first m observations to obtain the estimates of the parameters. Then these estimates are kept fixed for outof sample forecasting period. For the simple MA model, we set q= 12. We used the data over 2001.012014.12 as the outofsample evaluation period. Table 8.3 reports the outofsample forecasting results. It is clear both MAE and RMSE show that VECM dominates ARMA, and ARMA dominates the simple MA model. Table 8.4 reports the results of the DM test. The DM statistics show that VECM significantly outperforms the ARMA, and ARMA significantly dominates the MA model. Table 8.5 reports the regression [ Kq. (8.8)J and encompassing regression  Eq. (8.9)J results. The regression results show that VECM reports the least biased forecasts and has the largest Rsquare. The encompassing regression results show that VECM has the most informative forecasts. Once the VECM forecasts are included, the coefficient on the forecasts reported by ARMA (MA) becomes negative or insignificant. Figure 8.1 presents the time series plots of technical range forecasts reported by different models. It is clear from Figure 8.1 that Figure 8.1 Outofsample technical range forecasting the forecasts given by the VECM model are more flexible and adaptive to evolution of real technical range. 8.4 Summary Using the cointegration relation between the closing price and the technical range, a VECM model is proposed for technical range forecasting. An empirical study is performed on the monthly S&P500 stock index. The results show that the VECM dominates both ARMA and MA for both insample and outof sample forecasting. The results obtained in this chapter are interesting and important as they indicate that the statistical properties of the technical range can be used for improving technical range forecasting. 
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