Empirical Evidence of Market Inefficiency: Predicting Single-Stock Returns

Abstract

Although it is widely assumed that the stock market is efficient, some empirical studies have already tried to address the issue of forecasting stock returns. As far as is known, it is hard to find a paper involving not only the forecasting statistics but also the forecasting profitability. This paper aims to provide an empirical evidence of the market inefficiency and to present some simple realistic strategies based on forecasting stocks returns. In order to achieve this study, some linear and non linear algorithms are used to prove the predictability of returns. Many regularization methods are introduced to enhance the linear regression model. In particular, the RIDGE method is used to address the colinearity problem and the LASSO method is used to perform variable selection. The different obtained results show that the stock market is inefficient and that profitable strategies can be computed based on forecasting returns. Empirical tests also show that simple forecasting methods perform almost as well as more complicated methods.

Appendices

Appendices

1.1 Appendix 1: Binary Classification

See Tables 3, 4, 5, 6, 7, 8, 9, 10 and 11.

Table 3 The quality of the binary prediction: 1-min prediction AUC and accuracy per stock

Table 4 The quality of the binary prediction: the daily gain average and standard deviation for the 1-min prediction (without trading costs)

Table 5 The quality of the binary prediction: the daily gain average and standard deviation for the 1-min prediction (with trading costs)

Table 6 The quality of the binary prediction: 5-min prediction AUC and accuracy per stock

Table 7 The quality of the binary prediction: the daily gain average and standard deviation for the 5-min prediction (without trading costs)

Table 8 The quality of the binary prediction: the daily gain average and standard deviation for the 1-min prediction (with 0.5 bp trading costs)

Table 9 The quality of the binary prediction: 30-min prediction AUC and accuracy per stock

Table 10 The quality of the binary prediction: the daily gain average and standard deviation for the 30-min prediction (without trading costs)

Table 11 The quality of the binary prediction: the daily gain average and standard deviation for the 30-min prediction (with 0.5 bp trading costs)

1.2 Appendix 2: Four-Class Classification

See Tables 12, 13, 14, 15, 16, 17, 18 and 19.

Notice that the nans on the tables of the Appendix 2 correspond to the cases where $|\widehat{Y}|$ is always lower than $\theta$ thus no positions are taken.

Table 12 The quality of the 4-class prediction: 1-min prediction AUC and accuracy per stock

Table 13 The quality of the 4-class prediction: the daily gain average and standard deviation for the 1-min prediction (without trading costs)

Table 14 The quality of the 4-class prediction: the daily gain average and standard deviation for the 1-min prediction (with trading costs)

Table 15 The quality of the 4-class prediction: 5-min prediction AUC and accuracy per stock

Table 16 The quality of the 4-class prediction: the daily gain average and standard deviation for the 5-min prediction (without trading costs)

Table 17 The quality of the 4-class prediction: the daily gain average and standard deviation for the 1-min prediction (with 0.5 bp trading costs)

Table 18 The quality of the 4-class prediction: 30-min prediction AUC and accuracy per stock

Table 19 The quality of the 4-class prediction: the daily gain average and standard deviation for the 30-min prediction (without trading costs)

Table 20 The quality of the binary prediction: the daily gain average and standard deviation for the 30-min prediction (with 0.5 bp trading costs)

1.3 Appendix 3: OLS Method

See Tables 20, 21, 22 and 23.

Table 21 The quality of the OLS prediction: the AUC and the accuracy per stock for the different horizons

Table 22 The quality of the OLS prediction: the daily gain average and standard deviation for the different horizons (without trading costs)

Table 23 The quality of the OLS prediction: the daily gain average and standard deviation for the different horizons (with 0.5 bp trading costs)

1.4 Appendix 4: Ridge Method

Notice that:

For any $A \in \mathbb {R}^{p,p}$, $X \in \mathbb {R}^{p}$ such $||X||_2\ne 0$ :

$$\begin{aligned}&\frac{||AX||_2}{||X||_2} \le {\displaystyle \max \limits _{||Y||_2 \ne 0}}\frac{||AY||_2}{||Y||_2}=||A||_2 \nonumber \\&\qquad => ||AX||_2 \le ||A||_2||X||_2 \end{aligned}$$

(1)

For any $A \in \mathbb {R}^{p,p}$, $B \in \mathbb {R}^{p,p}$ :

$$\begin{aligned}&||AB||_2= {\displaystyle \max \limits _{||X||_2 \ne 0}}\frac{||ABX||_2}{||X||_2} = {\displaystyle \max \limits _{||BX||_2 \ne 0}} \frac{||ABX||_2}{||BX||_2} \frac{||BX||_2}{||X||_2} \nonumber \\&\qquad \qquad \quad \le {\displaystyle \max \limits _{||Y||_2 \ne 0}} \frac{||AY||_2}{||Y||_2}{\displaystyle \max \limits _{||X||_2 \ne 0}}\frac{||BX||_2}{||X||_2} =||A||_2||B||_2 \nonumber \\&\qquad \qquad \qquad => ||AB||_2\le ||A||_2||B||_2 \end{aligned}$$

(2)

Proof 1

Let $A,B,X$ such that

$$\begin{aligned} AX=B \end{aligned}$$

(3)

and

$$\begin{aligned} A(X+\delta X)=B + \delta B \end{aligned}$$

(4)

From (3) and (4) $\delta X = A^{-1}\delta B$ and using (1)

$$\begin{aligned} ||\delta X ||_2 = ||A^{-1}\delta B||_2 \le ||A^{-1}||_2||\delta B||_2 \end{aligned}$$

(5)

From (3) $||B||_2=||AX||$ and using (1)

$$\begin{aligned} ||B||_2 \le ||A||_2||X||_2 \end{aligned}$$

(6)

From (5) and (6), $||\delta X ||_2||B||_2\le ||A^{-1}||_2||\delta B||_2||A||_2||X||_2$

Thus $\frac{||\delta X||_2}{||X||_2} \le K(A) \frac{||\delta B||_2}{||B||_2}$

Proof 2

Let $A,B,X$ such that $AX=B$ (3) and

$$\begin{aligned} (A+\delta A)(X+\delta X)=B \end{aligned}$$

(7)

From (3) and (7), $\delta X =-A^{-1}\delta A(X+\delta X)$.

Using (1) and (2) follows $||\delta X ||_2 \le ||A^{-1}||_2||\delta A||_2||X+\delta X||_2$

Thus $\frac{||\delta X||_2}{||X+\delta X||_2} \le K(A) \frac{||\delta A||_2}{||A||_2}$ (See Tables 24, 25, 26, 27, 28 and 29.)

Table 24 The quality of the Ridge HKB prediction: the AUC and the accuracy per stock for the different horizons

Table 25 The quality of the Ridge HKB prediction: the daily gain average and standard deviation for the different horizons (without trading costs)

Table 26 The quality of the Ridge HKB prediction: the daily gain average and standard deviation for the different horizons (with 0.5 bp trading costs)

Table 27 The quality of the Ridge LW prediction: the AUC and the accuracy per stock for the different horizons

Table 28 The quality of the Ridge LW prediction: the daily gain average and standard deviation for the different horizons (without trading costs)

Table 29 The quality of the Ridge LW prediction: the daily gain average and standard deviation for the different horizons (with 0.5 bp trading costs)

1.5 Appendix 5: LASSO Method

See Tables 30, 31 and 32.

Table 30 The quality of the LASSO prediction: the AUC and the accuracy per stock for the different horizons

Table 31 The quality of the LASSO prediction: the daily gain average and standard deviation for the different horizons (without trading costs)

Table 32 The quality of the LASSO prediction: the daily gain average and standard deviation for the different horizons (with 0.5 bp trading costs)