Technical range forecasting

In 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 high-low price range. Early application of range in the field of finance can be traced to Mandelbrot (1971), and academic work on the range-based 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 return-based 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 range-based volatility estimator is robust to microstructure noise such as bid-ask 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 in-sample and out-of-sample, 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 intra-day price range in place of the absolute return and find that the range-based volatility estimators oiler a significant improvement over their return-based 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 return-based volatility models in forecasting covariances. Other models concerning range-based 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.

  • 8.2 Econometric methods
  • 8.2.1 The model

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 xr is A/w(Til,). We don’t model ln(TRt) because it is a unit root process.

Both MA and ARMA models are univariate time series models, they don’t take into consideration the co-integration between the closing price and the technical range. To see if this co-integration 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 ef is a p x 1 vector of white noise error terms. Гу is a p x p matrix that represents short-term adjustments among variables across p equations at the /th lag. Ylj tT-AT", ■ and ajl‘ T:_ i are the vector autoregressive (VAR) components in first differences and error-correction components respectively. [I is a p x r matrix of co-integrating vectors, and a is a p x r matrix of speed of adjustment parameters. The cointegrating vector fi shows the long-term 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 long-run equilibrium in cases of short-run deviations from the long-run equilibrium. In this chapter, Tt = (ln(Ct), ln{TR,))T, where C, and 111, are respectively closing price and technical range.

8.2.2 Out-of-sample forecast evaluation

To evaluate the out-of-sample forecasting accuracy, both mean absolute error (MAE) and root mean squared error (RAISE) are used

where Xt 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- of-sample 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 r-test 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.

  • 8.3 An empirical study
  • 8.3.1 The data

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 Jarque-Bera 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 In-sample estimation

For the ARMA model, the SIC criteria prefers the ARMA (0, 1) model. The estimation result is presented as follows

The R-squarc 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

MQ)

ln( TR,)

n

Aln( TR,)

Mean

5.345

2.473

0.006

0.007

St.D.

1.328

1.494

0.042

0.469

Max.

7.634

5.791

0.151

1.561

Min.

2.836

-1.109

-0.245

-1.515

Kurt.

1.756

2.144

5.468

3.113

SKew

0.140

0.001

-0.670

0.035

Jarque-Bera

52.859***

23.790***

256.134***

0.569

ADF

-0.788

-1.473

26.501***

-19.178***

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

Vector Error Correction Estimates

f-statistics in [ ]

Со-integrating Eq:

CointEq 1

ln(Ct_ i):

1

ln(TR,_i):

-0.933 [-28.564]

C:

-3.033

Error Correction:

Д ln(Ct)

Aln{TRt)

CointEq 1

-0.007 [-1.497]

0.267 [ 6.392]

Д/я( C,_,)

0.045 [ 1.214]

-2.4187 [-7.776]

Mn(Ct_2)

-0.050 [-1.317]

-0.127 [-0.393]

д ln(C'_3)

0.067 [ 1.769]

0.370 [ 1.151]

ШС,_4)

0.055 [ 1.448]

0.385 [ 1.195]

Д/я(С,_5)

0.095 [ 2.517]

0.054 [ 0.169]

Aln(TR^)

-0.016 [-2.950]

-0.607 [-13.093]

ln{TR,_2)

-0.009 [-1.588]

-0.388 [-7.830]

Aln(TRt_3)

-0.007 [-1.140]

-0.217 [-4.485]

Д/и(ГЯ,_4)

-0.003 [-0.669]

-0.119 [-2.714]

Aln(TR^)

-0.003 [-0.619]

-0.077 [-2.238]

C

0.005 [ 3.204]

0.025 [ 1.877]

R-square

0.026

0.447

The VECM model estimates are reported in Table 8.2. We select the lag k = 5 by the SIC criteria. The co-integration vector is given by/? = (1.000, -0.933, -3.033)7. The co-integration 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 Я-square 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 Out-of-sample forecast

For practical purposes, the more important thing is the out-of-sample forecasting performance. For out-of-sample predicting, the whole T data observations are divided into an in-sample portion composed of the first m observations and an

Technical range forecasting 67

Table 8.3 Out-of-sample MAE and RAISE for VECM, ARMA and MA

VECM

ARMA

MA (12)

MAE

0.282

0.304

0.323

RMSE

0.351

0.381

0.401

Table 8.4 Out-of-sample forecasting evaluation: DM test

ARMA

VECM

ARMA

2.903***

MA( 12)

2.252**

3.207***

We use ***, **, and * to mean significance at the level of 1%, 5%, 10%.

Table 8.5 Out-of-sample forecasting evaluation: regression and encompassing regression

Cons.

MA(12)

ARMA

VECM

R2

1.112***

0.741***

0.258

0.745*

0.819***

0.327

-0.634

1.126***

0.433

0.858**

-0.371

1.162***

0.334

-0.610

-0.086

1.205***

0.444

-0.738*

-0.289

1.436***

0.450

-0.983**

0.336

-0.706*

1.574***

0.455

We use ***, **, ai

id * to mean significance

at the level of 1%, 5%.

, 10%.

out-of-sample 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 out-of- sample forecasting period. For the simple MA model, we set q= 12. We used the data over 2001.01-2014.12 as the out-of-sample evaluation period.

Table 8.3 reports the out-of-sample 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 R-square. 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

Out-of-sample technical range forecasting

Figure 8.1 Out-of-sample 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 co-integration 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 in-sample and out-of- 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.

 
Source
< Prev   CONTENTS   Source   Next >