will fully reflect any new information generated in the market; therefore, investors will not be able to gain profit from the market [1]. On the other hand, technical trading investor believe that some simple trading rules will be able to identify the trend change. One of the most successful technical trading rules is the Moving Average (MA) Trading Rule [2]. The Simple Moving Average Trading Rule claims that a Buy or Sell signal emits when short-term moving average price of length S, exceeds long-term moving average price of length L. Various moving average trading rules also specify the differences by a band of α%.That is, Buy if  otherwise Sell. (1)

We will denote the trading rules as MA(S, L, α). Popular moving average trading rules are MA(1, 5, 0), MA(1, 15, 0), MA(1, 50, 0), MA(1, 100, 0), MA(1, 150, 0), MA(1, 200, 0), MA(1, 5, 1), MA(1, 15, 1), MA(1, 50, 1), MA(1, 100, 1), MA(1, 150, 1), MA(1, 200, 1), etc. [3]. In this study, we will only show the result that Buy and Sell Days from MA (1, 15, 0) have no predictable capacity. Similar results can be found for other various simple moving average trading rules.

As a comparison to the study by about the market inefficiency of the S&P 500 price index from MA (1, 15, 0), a total 5037 dividend adjusted price indices from 10/6/1989 to 9/29/2009 for about twenty years is obtained from Yahoo Finance [4]. Day 15 to Day 5038 are classified as a Buy or a Sell Day according to Equation (1). For instance, the dividend adjusted closing price for Day 15 is 337.93, and the average of adjusted closing prices from Day 1 to Day 15 is 115ΣP15−i14i=0 = 347.52. Since 337.93 < 347.52, Day 15 is classified as a Sell Day. We can classify the rest Day 16 to Day 5037 the same way. Daily rate of return at time t, Rt, is defined as in Equation (2) as in Fama (1965).

For instance, the return for Day 15 is ln (337.93/342.5) = -0.0134. We can calculate the rest Day 16 to Day 5037 the same way. We then use a simple regression model to describe the relationship between the dependent variable, the daily rate of return, Rt, and the dummy independent variable, a Buy or a Sell day. That is, Rt = β0 + β1D + εt, where D is a dummy variable denoting a Buy Day (D = 1) or a Sell Day (D = 0), from t = 1 to 5023. The error term εt follows a normal distribution with mean E(εt) = 0 and variance E(εt) = an unknown constant.

The error terms, εt’s, are independent. We obtain an estimated regression equation = -0.0038 + 0.0069D by using the Regression function in the Data Analysis tool in Excel. 

Table 1 shows the Excel printout for the regression analysis. Based on the T-test with a p-value of 0.0000 for H0: β1 = 0 v.s. Ha: β1 ≠ 0, which is equivalent to H0: μs = μb v.s. Ha: μs ≠ μb where μs and μb are the means of daily returns for Buy and Sell Days, respectively, we may erroneously infer that the classification of Buy and Sell Days does have a predictive capacity on daily rate of returns and that the market is not efficient, as concluded from the study [5]. However, when we look at the Normality Plot in Figure 1 and the Residual Plot in Figure 2, we find that the residuals neither follow a normal probability distribution nor are independent with constant variance. 

That is, the model assumptions for the error terms of the regression are violated; therefore, the conclusion from the T-test in Table 1 is misleading. It indirectly disproves that the conclusion Ha: μs ≠ μb is not true as claimed by the T-test for Independent Two Samples in Pistole and Metghalchi (2010). We thus refute that the market inefficiency for the S&P 500 price index concluded by using the Student T-test for Independent Two Samples in Metghalchi, M., Gaza-Gomez, X., Chen, C., and and Monsef, S. (2005), and Pistole and Metghalchi (2010) is wrong. Nevertheless, the low coefficient of determinant R2 of 8.5% indicates that the dummy variable for Buy/Sell Days cannot be used to predict the daily returns from the S&P 500.

An Alternative Way to Define the Buy and Sell Days via Rt

In Equation (1), Buy/Sell Days are defined by comparing the short-term moving average price of length S, and the long-term moving average price of length L. In this session, instead of using the price, we use the rate of return, Rt, to classify days into Buy/Sell Days.

That is, Buy if otherwise sell. (3)

To be consistent, we let S = 1, L = 15, and α = 0 as in the last section. Since we cannot calculate the daily rate of return for Day 1, we will start to classify days into Buy and Sell Days from Day 16 to Day 5037. For instance, the daily rate of return for Day 16 is -0.0085, and the average of adjusted closing prices from Day 2 to Day 16 is 115ΣR16−i14i=0 = -0.0046. Since -0.0085 < -0.0046, Day 16 is classified as a Sell Day. We can classify the rest Day 17 to Day 5037 the same way. We re-run the regression analysis of daily returns defined in Equation (2) against the dummy variable for Buy or Sell Days using the criteria laid out in equation (3). An estimated regression equation ????????????????= -0.0074 + 0.0152D is obtained with a coefficient of determinant R2 = 0.4230 as shown in Table 2. Figure 3 does show that Buy/Sell Days do have prediction capacity on daily rate of returns despite the market is efficient or not.

In this study, we observe that residuals from the regression error terms were neither independent with constant variance, nor normally distributed, which are key assumptions for applying the Student T-test for testing H0: μs = μb v.s. Ha: μs ≠ μb where μs and μb are the means of daily returns for Buy and Sell Days, respectively. It indirectly proves the inappropriateness of using the Student T-test for independent two samples to claim that Ha: μs ≠ μb, then conclude that the market is inefficient. Even though the purpose of this study is not to prove the market is efficient or not, an alternative way of reclassifying the Buy and Sell Days by using MA(S, L, α) on returns do improve the predictability of returns by about five times than the old way of reclassifying the Buy and Sell Days by MA(S, L, α) on prices according to the coefficient of determinant R2.

1. Brock W, Lakonishok J, Baron LB. Simple technical trading rules and the stochastic roperties of stock returns. J Finance. 1992; 47: 1731-1764.
2. Fama E. The behavior of stock market prices. J Bus. 1965; 38: 34-105.
3. Metghalchi M, Gomez GX, Chen C, Monsef S. Market Efficiency for S&P 500: 1954-2004. Int Bus Econ Res J. 2005; 4: 23-30.
4. Pistole T, Metghalchi M. Comparison of three technical trading methods vs. buy- and-hold for the S&P 500 market. 2010 SWDSI Conference Proceedings. 2010; 153: 1-19.
5. Pring MJ. Technical Analysis Explained: The Successful Investor's Guide to Spotting Investment Trends and Turning Points, 5^th edition, New York: McGraw-Hill Education. 2014.

Figure 1: Normality plot.

Figure 2: Residuals plot.

Figure 3: Regression Line Fit Plot.

Table 1: Table 1: Regression line fit summary output.

Table 2: Regression line fit summary output.