APPLICATION OF ARIMA MODEL FOR TESTING

“SERIAL INDEPENDENCE” OF STOCK PRICES AT THE HSEC

Cao Hao Thi – Pham Phu – Pham Ngoc Thuy
School of Industrial Management
HoChiMinh City University of Technology

ABSTRACT

The paper is an attempt to test the “serial independence” of stock prices at HoChiMinh City Stock Exchange Center (HSEC) in Vietnam by applying the ARIMA model for preliminary assessment in terms of its market efficiency. From findings derived, it appears to be that: (a) ARIMA model could be applied for testing the serial independence of stock prices at the HSEC; (b) It is failed to prove that the HSEC market is not a weak-form efficient one; and (c) the “sheep flock effect” psychology is a factor dominated at the HSEC during the past two years.

INTRODUCTION

The first stock exchange floor of Vietnam named “HoChiMinh City Stock Exchange Center” (HSEC) was officially opened in HCM City on July 20, 2000. After two and half years of operation, there are now 21 listed stocks and 41 bonds traded on the HSEC with a total capitalization of about VND 5,200 billion. This would be seen as a first success on the way of setting-up SEC in Vietnam.

VN-Index of the HSEC, however, has experienced a truly ups and downs movement and changed considerable during almost last two years. In the first section on July 28, 2000, VN-Index was 100 points and increased to a peak of 571.04 points in June 25, 2001 before sliding to lower 150 points in the first months of the year 2003. Therefore, the problem is: Why they fluctuated so much, lack of orientation, and whether or not they reflected to some extent the real health of the related stock companies?

From a common sense, some experts from the brokerage firms said that the stock prices at the HSEC has been fluctuated by “sheep flock effect” psychology of “naive speculators” (short time but lack of professional skills!) but not their estimates on the “intrinsic values” of stocks. However, in investment domain, this is probably the most controversial issue of “market efficiency” which has been and is likely to continue to be a topic of intensive debate in the investment community. Furthermore, the problem is much more difficult in the emerging markets like the case of the HSEC.

A market is efficient with respect to some particular information if that information is not useful in earning a positive excess return. “With respect to what information”, it could be defined three forms of market efficiency: weak, semistrong, and strong. For the case of the HESC, it seems to be to suspect that the market is a weak-form efficient one. In that case, the hypothesis states that “stock prices reflect historical price information and, therefore, an investor cannot “beat the market” by studying historical prices. There are many types of empirical tests for the above hypothesis and the first one is tests for “serial independence”. In other words, test of weak forms market efficiency would be failed if knowing how stock prices moved in the past cannot be translated into accurate predictions of future stock prices.

The available ARIMA model (Auto Regressive Integrated Moving Average) is an one using for forecasting future values of time series and then forecasted values depend on the own past values and the weighted success of current and lagged random disturbances. Therefore, it appears to be that the model could be seen as a tool for testing the serial independence in the issue of market efficiency.

From the above assessments, the problem of this study is that: (a) Would it be possible using the ARIMA model as a one among the tools for testing weakly efficient market hypothesis? And (b) Would it be possible providing an evidence (among others) on the weak-form efficiency of the HSEC’s market through applying ARIMA model for its historical records of stock prices?

ARIMA MODEL

• Stochastic process and its stationary

Any time series data can be thought of as being generated by a stochastic or random process. Broadly speaking, a stochastic process is said to be stationary if its mean and variance are constant over time and the value of covariance between two time periods depend only on the distance or lag between the two time periods and not on the actual time at which covariance is computed.

Using the graph of time series Yt = f(t), graph of autocorrelation function (Correlogram) or Dickey-Fuller test can identify the stationary of a time series.

If the time series is not stationary, difference it one or more times to achieve stationarity. If the original time series is Yt , then the first order difference is Wt = Yt – Yt-1 and the second order difference is Vt = Wt – Wt-1.

• Seasonality

Seasonality is just a cyclical behavior that occurs on a regular calendar basis. Often seasonal peaks and troughs are easy to spot by direct observation of the time series. However, if the time series fluctuates considerably, seasonal peaks and troughs might not be distinguishable from other fluctuations. Fortunately, the recognition of seasonality can be made easier with the help of autocorrelation function (SAC = f(t)). If the graph of SAC has peaks after m time period, then it is the sign of seasonality. A crude method of “deseasonalizing” data would be to a mth difference: Zt = Yt – Yt-m.

• Autoregressive Models

According to Box-Jenkins, any of stochastic processes can be modeled as and an ARIMA processes (AutoRegressive Integrated Moving Average). In which, the autoregressive process is an order of p if the current observation Yt is generated by a weighted average of past observations going back p periods, together with a random disturbance in the current period. This process is denoted as AR(p) and its equation is written as

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• Moving Average Models

In the moving average process of order q the current observation Yt is generated by a weighted average of random disturbances going back q periods, together with a random disturbance in the current period. This process is denoted as MA(q) and its equation is written as

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• AutoRegressive Integrated Moving Average Models

Many stochastic processes cannot be modeled as purely autoregressive or as purely moving average, since they have the qualities of both types of processes. The logical extension of the models is autoregressive integrated moving average process of order (p,q). This process is denoted as ARIMA(p, d, q) and its equation is written as

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• Specification of ARIMA models

Specification of ARIMA model is to choose the most appropriate values for p, d, q; in which p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms. This problem is partly resolved by examining both autocorrelation function (SAC) and the partial autocorrelation function (SPAC). SAC was defined in the previous part. SPAC is the correlation between Yt and Yt-k after removing the effect of intermediate Y’s.

• Model AR(p) is chosen if SPAC correlogram has significant spikes through lags p and cuts off after p; and SAC correlogram dies down.
• Model MA(q) is chosen if SAC correlogram has significant spikes through lags q and cuts off after p; and SPAC correlogram dies down. In summary,

|Type of model |Typical pattern of SAC = f(t) |Typical pattern of SPAC = f(t) |
|AR(p) |Decays exponentially with damped sine wave pattern or|Significant spikes through |
| |both |lags p |
|MA(q) |Significant spikes through lags q |Declines exponentially |
|ARMA(p, q) |Exponential decay |Exponential decay |

• Estimation of ARIMA models

Estimation of ARIMA model is to choose the most appropriate values for model parameters φi and θj. These parameters can be estimated by Ordinary Least Square Method:

where

Diagnostic checking

After determining of p, d, q and φi , θj , it is necessary to test whether residual terms εt of ARIMA model are white noise. This is a requirement of a good model. In theory, residual terms εt are generated by white noise process if:

The testing of white noise is based on SAC correlogram and SPAC correlogram of residual terms εt .

Forecasting

Based on the equation of ARIMA model, the values of point forecasting and confident interval will be determined

• Point Forecasting: [pic]
• Confident interval: [pic] For 95% confident level, k =2.

APPLICATION OF ARIMA MODEL IN FORECASTING THE STOCK PRICES AT THE HSEC.

• Stock price information

As mentioned above, up to March 2003, there are 21 stocks listed in the HSEC. However, in terms of time series of their prices, most of them are small sample sizes and are non-stationary. Therefore, in this research, only 5 stocks that have been listed and traded for a relatively long period including REE, SAM, HAP, TMS and LAF were selected for assessment. In addition, VN-Index, named MARKET PRICE, was also taken into account for comparison and assessing on the “sheep flock effect”. The main statistic parameters and the graph of these time series are as follows:

Table 1: THE STATISTIC PARAMETERS OF 5 COMPANY STOCK PRICES and MARKET PRICE

| | | | | | |MARKET |
| |REE_PRICE |SAM_PRICE |HAP_PRICE |TMS_PRICE |LAF_PRICE |PRICE |
| Mean | 36361.37 | 35800.00 | 59712.26 | 49498.83 | 34721.18 | 241.4512 |
| Median | 33100.00 | 30800.00 | 46800.00 | 40100.00 | 29700.00 | 210.5600 |
| Maximum | 95000.00 | 78500.00 | 146000.0 | 141000.0 | 82500.00 | 571.0400 |
| Minimum | 16000.00 | 17000.00 | 16000.00 | 17100.00 | 17000.00 | 100.0000 |
| Std. Dev. | 14701.51 | 12425.52 | 30103.32 | 23355.33 | 14717.06 | 93.01314 |
| Skewness | 1.435736 | 1.122369 | 0.618982 | 1.504105 | 1.433017 | 1.197553 |
| Kurtosis | 5.302573 | 3.908302 | 2.222365 | 5.069448 | 4.608104 | 4.506994 |
| Observations | 321 | 321 | 318 | 298 | 262 | 321 |

Source: Saigon Security Incorporation (July 2000 – June 2002)

GRAPHS OF 5 COMPANY STOCK PRICES and MARKET PRICE

• Applying ARIMA for time series of the REE’s stock prices

For illustration, the time series of REE_PRICE was selected to construct the ARIMA model. In this part, 320 past values of REE_PRICE were used to specify the equation of ARIMA model. The 321rst actual value of REE_PRICE is used to compare with the forecasted value that will be estimated from the equation of ARIMA model.

By using EViews software, the optimal model, which is purely autoregressive model ARIMA (2,0,0), was finally specified. The results of autoregressive model parameters φI is presented in following table:

Table 2: THE RESULT OF AUTOREGRESSIVE MODEL ARIMA(2,0,0) OF REE_PRICE

|Dependent Variable: REE_PRICE |
|Method: Least Squares |
|Date: 08/01/02 Time: 16:09 |
|Sample(adjusted): 3 321 |
|Included observations: 319 after adjusting endpoints |
|Convergence achieved after 8 iterations |
|Variable |Coefficient |Std. Error |t-Statistic |Prob. |
|C |39484.06 |8364.996 |4.720153 |0.0000 |
|AR(1) |1.426261 |0.050598 |28.18831 |0.0000 |
|AR(2) |-0.435760 |0.050449 |-8.637551 |0.0000 |
|R-squared |0.991086 | Mean dependent var |36488.09 |
|Adjusted R-squared |0.991030 | S.D. dependent var |14659.75 |
|S.E. of regression |1388.452 | Akaike info criterion |17.31913 |
|Sum squared resid |6.09E+08 | Schwarz criterion |17.35454 |
|Log likelihood |-2759.401 | F-statistic |17567.10 |
|Durbin-Watson stat |1.917363 | Prob(F-statistic) |0.000000 |
|Inverted AR Roots | .98 | .44 |

After determining of ARIMA model equation, it is necessary to conduct white noise testing of residual terms εt. Based on SAC correlogram of εt, the testing demonstrated residual terms εt are white noise. The results of testing is shown as follow:

Table 3: THE CORRELOGRAM OF REE_PRICE RESIDUALS

|Date: 08/01/02 Time: 16:11 |
|Sample: 3 321 |
|Included observations: 319 |
|Q-statistic probabilities | | | | | | |
|adjusted for 2 ARMA | | | | | | |
|term(s) | | | | | | |
|Autocorrelation |Partial Correlation | |AC | PAC | Q-Stat | Prob |
| .|. | | .|. | |1 |0.041 |0.041 |0.5496 | |
| .|. | | .|. | |2 |-0.055 |-0.056 |1.5109 | |
| *|. | | *|. | |3 |-0.137 |-0.133 |7.5749 |0.006 |
| .|. | | .|. | |4 |0.037 |0.045 |8.0134 |0.018 |
| .|* | | .|* | |5 |0.106 |0.090 |11.647 |0.009 |
| .|* | | .|* | |6 |0.189 |0.173 |23.386 |0.000 |
| .|. | | .|. | |7 |-0.049 |-0.044 |24.183 |0.000 |
| **|. | | *|. | |8 |-0.206 |-0.176 |38.175 |0.000 |
| .|* | | .|** | |9 |0.153 |0.214 |45.899 |0.000 |
| .|. | | .|. | |10 |0.058 |0.000 |47.032 |0.000 |
| .|* | | .|. | |11 |0.115 |0.053 |51.417 |0.000 |
| *|. | | *|. | |12 |-0.078 |-0.062 |53.442 |0.000 |
| *|. | | *|. | |13 |-0.117 |-0.080 |58.009 |0.000 |
| .|* | | .|** | |14 |0.111 |0.210 |62.154 |0.000 |
| .|** | | .|* | |15 |0.202 |0.088 |75.901 |0.000 |
| .|* | | .|* | |16 |0.143 |0.084 |82.797 |0.000 |
| *|. | | *|. | |17 |-0.133 |-0.075 |88.838 |0.000 |
| *|. | | *|. | |18 |-0.122 |-0.093 |93.879 |0.000 |
| .|. | | .|* | |19 |-0.011 |0.080 |93.920 |0.000 |
| .|* | | .|. | |20 |0.152 |-0.007 |101.83 |0.000 |

The results of forecasting model are stored in file name of REE_PRICEF. The graphs of REE_PRICE and REE_PRICEF are presented together as follow:

GRAPH OF ACTUAL REE_PRICE AND FORECASTED REE_PRICE

Based on the equation of forecasting model, which is ARIMA(2,0,0), the value of REE_PRICE at 321rst trading time (27 June 2002) is forecasted:

Forecasting point is [pic]= 32,962.75 VNÑ and 95% confident interval is [ 30,176.31 VNÑ, 35,749.19 VNÑ]

By comparing the actual value and forecasted value of REE_PRICE at 321rst trading time (27 June 2002), the forecasting error can be evaluated. The actual value is Yt = 32,900.00 VNÑ. This value falls within 95% confident interval. The forecasting error is ([pic]-Yt) / Yt *100 = (32,962.75 – 32,900.00) / 32,900.00* 100 = 0.19%

Applying ARIMA for time series of other stock prices

Similarly, this research also constructed the models for SAM_PRICE, HAP_PRICE, TMS_PRICE, LAF_PRICE and MARKET_PRICE. The main results of these time series models are shown in the following table; the graphs and the other results are presented more details in appendix:

Table 4: THE RESULTS OF TIME SERIES MODELS

| | | | | | |MARKET |
| |REE_PRICE |SAM_PRICE |HAP_PRICE |TMS_PRICE |LAF_PRICE |PRICE |
| | | | | | | |
|ARIMA (p,d,q) MODEL |ARIMA(2,0,0) |ARIMA(2,0,0) |ARIMA(2,0,0) |ARIMA(2,0,0) |ARIMA(2,0,0) |ARIMA(2,0,0) |
| | | | | | | |
|Actual Price Yt |32900.00 |28900.00 |38900.00 |37900.00 |21600.00 |200.50 |
|Forecasted Price [pic] | | | | | | |
| |32962.75 |28994.22 |39017.85 |37944.13 |21885.78 |200.14 |
|Forecasted Error | | | | | | |
|([pic]- Yt)/ Yt *100 |0.19% |0.33% |0.30% |0.12% |1.32% |- 0.18% |
|95% Confidence Interval | | | | | | |
|Maximum |35749.19 |31577.40 |45805.51 |41984.35 |24881.52 |216.02 |
|Minimum |30176.31 |26411.04 |32230.19 |33903.91 |18890.04 |184.26 | Relationship between stock prices and VN-Index

In addition, this research also introduced regression techniques to establish the relationship between 5 company stock prices (REE-PRICE, SAM_PRICE, HAP_PRICE, TMS_PRICE, LAF_PRICE) and MARKET_PRICE. The forecasted values of 5 company stock prices depend on the forecasted values of MARKET_PRICE, which are estimated from ARIMA model. The main results of these regression models are shown in the following table; the graphs and the other results are presented more details in appendix:

Table 5: THE RESULTS OF LINEAR REGRESSION MODELS

COMPANY STOCK PRICE = a + b* MARKET PRICE

| | | | | | |MARKET |
| |REE_PRICE |SAM_PRICE |HAP_PRICE |TMS_PRICE |LAF_PRICE |PRICE |
|Correlation Coefficient Between | | | | | | |
|Company Price |0.988974 |0.989002 |0.8714750 |0.976444 |0.944794 |1.000000 |
|And Market Price | | | | | | |
|Linear Regression Model | | | | | | |
|Intercept a |-1464.15 |3885.02 |-12056.36 |-15154.70 |-8599.74 | |
|Slope b | | | | | | |
| |156.62 |132.20 |295.73 |257.15 |163.07 | |
|Actual Price Yt | | | | | | |
| |32900.00 |28900.00 |38900.00 |37900.00 |21600.00 |200.50 |
|Forecasted Price [pic] | | | | | | |
| |29881.78 |30343.53 |47131.04 |36311.30 |24037.09 |200.14 |
|Forecasted Error | | | | | | |
|([pic]- Yt)/ Yt *100 |9.17% |4.99% |21.00% |-4.19% |11.28% |- 0.18% |

By comparing the results of time series models and regression models, the forecasting errors of time series are smaller than regression models.

IV. DISCUSSIONS AND CONCLUSIONS

The graphs of 5 forecasted time series of stock prices in the HSEC are fitted well with the graphs of 5 historical records of their “actual prices”. This is resulted from that the forecasting errors are very small and all of 5 actual prices at the 321st trading section (June 27, 2002) are fallen within 95% confident intervals. Moreover, the 5 ARIMA models of the 5 time series of stock prices are in the same form of ARIMA (2,0,0). It means that they are purely autoregressive, they are “not serial independence” and having known how stock prices moved in the past it could be translated into accurate predictions of future stock prices. From the above observations, it could be said that:

+ It would be possible using ARIMA model as an one among the tools for testing the serial independence of time series of stock prices; this is the first test for weakly efficient market hypothesis, at least in the case of the HSEC.

+ All of 5 stock prices have reflected historical price information. So it is failed to prove that the HSEC market is not a weak-form efficient one.

1) In addition, linear regression models between 5 stock prices and VN-Index have been formulated with correlation coefficients of about 0.94 to 0.99. It means that the movement of the particular price of a stock is followed the price of the other stocks and VN-Index but not the real health of the related company, or a good/ bad information disclosed from corresponding one. Consequently, it appears to be that the “sheep flock effect” psychology in securities trading is a factor dominated at the HSEC during the past two years. Moreover, with these highly significant serial correlations, a preferable trading rule might be developed.

2) It should be said that the above mentioned issues have been over simplified for preliminary assessments of the applying of ARIMA model in the HSEC and three year historical records of stock prices are probably not long enough to properly judge the outcomes but the basic point of the rational inference seems to be rather reasonable.

REFERENCES

Bowerman B.L., and O’Connell R.T., 1993. Forecasting and Time Series. 3rd ed., Wadsworth, Inc.

Charles J. Corrado and Bradford D. Jordan, 2002. Fundamentals of Investments – Valuation and Management. 2nd ed., McGraw-Hill.

Eviews, 2000. Quantitative Micro Software.

Gordon J. Alexander, William F. Sharpe and Jeffery V. Bailey, 2001. Fundamentals of Investments. 3rd ed., Prentice-Hall, Inc.

Jack Clark Francis and Richard W. Taylor, 2000. Theory and Problems of Investments. 2nd ed., McGraw-Hill.

Pindyck R.S., and Rubinfeld D.L., 1991. Econometric Models and Economic Forecast. 3rd ed., McGraw-Hill.

Ramanathan R., 2001. Introductory Econometrics with Applications. 5th ed., Harcourt College Publishers.

Thi Cao Haøo, vaø Caùc Coäng Söï, 1998. Baûn Dòch Kinh Teá Löôïng Cô Sôû (Basic Econometrics cuûa Gujarati D.N.). Chöông Trình Fulbright veà Giaûng Daïy Kinh Teá taïi Vieät Nam.

Contact : Mr. Cao Hao Thi

Address : School of Industrial Management, HoChiMinh City University of Technology

268 Ly Thuong Kiet Street, District 10, HoChiMinh City, Vietnam

Tel : 84 - 8 - 8650460

Fax : 84 - 8 – 8635058
Email : chthi@sim.hcmut.edu.vn
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