# Machine Learning in Statistical Arbitrage

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2 2 Linear Regression We consider the data of 101 days from Apr.28 to Sep.18, The reason we choose this time period is that given 100 feature variables, we need at least 101 observations to train the 101 parameters in the linear regression model. To avoid overfitting parameters, we only use 101 training examples. Denote P t to be target asset and Q it to be the ith stock constituent at time t. The linear model can be written as 100 P t = θ 0 + θ i Q it. i=1 Implementing the ordinary least squares algorithm in MATLAB, we get parameters θ and training errors, i.e. residuals. we apply Principal Component Analysis to reduce the dimension of the model. Figure 2: Generalization error on 30 testing days 3 Principal Component Analysis (PCA) Now we apply PCA to analyze the 100 stocks. The estimation window for the correlation matrix is 101 days. The eigenvalues at the top of the spectrum which are isolated from the bulk spectrum are obviously significant. The problem becomes evident by looking at eigenvalues of the correlation matrix in Figures 3. Apparently, the Figure 1: Residuals of linear regression over 100 constituents From Figure 1, we see that the empirical error is acceptable. However, when we test this linear model using 30 examples not in the training set, i.e. prices of each asset from Sep. 21 to Oct. 30, 2009, the generalization error is far from satisfying, as shown in Figure 2. This implies that we are facing an overfitting problem, even though we ve used the smallest possible training set. There are so many parameters in the linear model, which gets us into this problem. Hence, Figure 3: eigenvalue of the correlation matrix eigenvalues within the first twenty reveal almost 2

3 all information of the matrix. Now we apply validation rules to find how many principal components to use that can give us the least generalization error. Given that the dimension of the model is reduced, we reset our window size to 60 days to avoid overfitting problem. After running multivariate linear regression within the first 20 components using 60 days training set, we find that the first 12 components give the smallest generalization error on the 30 testing days. The out-of-sample residual is shown in Figure 4. Figure 5: Empirical error on the first 12 components over 60 training days 4 Support Vector Regression (SVR) Figure 4: Generalization error on the first 12 components over 30 out-of-sample days We apply SVR on the 12 feature attributes obtained through PCA. We use a Gaussian kernel and empirically decide on the kernel bandwidth, cost and epsilon (slack variable) parameters. (We used SVM and Kernel Methods MAT- LAB toolbox, available online for implementation). We do not notice any improvement in this approach,as seen in the plots of training error and test error, Figure 6 and Figure 7. A main issue here is to be able to determine appropriate SVR parameters. We take a quick look back at the empirical error of these 12 components over 60 training days. From Figure 5, we see that the residuals are not as satisfying as in Figure 1, in terms of magnitude, but residuals here successfully reveal the trend of residuals using 100 constituents. Thus, by applying PCA to reduce the dimension of the model, we avoid overfitting parameters. We will use the in-sample residuals from regression on principal components to generate trading signals in a following section. 5 Mean Reverting Process We want to model the price of the target asset such that it accounts for a drift which measures systematic deviations from the sector and a price fluctuation that is mean-reverting to the overall industry level. We construct a trading strategy when significant fluctuations from equilibrium are observed. We introduce a parametric mean reverting model for the asset, the Ornstein-Uhlembeck 3

4 Figure 6: Empirical error on the first 12 components over 60 training days using SVR process, dx(t) = κ(m X(t))dt + σdw (t), κ > 0. The term dx(t) is assumed to be the increment of a stationary stochastic process which models price fluctuations corresponding to idiosyncratic fluctuations in the prices which are not reflected in the industry sector, i.e. the residuals from linear regression on principal components in the previous section. Note that the increment dx(t) has unconditional mean zero and conditional mean equal to E{dX(t) X(s), s t} = κ(m X(t))dt. The conditional mean, i.e. the forecast of expected daily returns, is positive or negative according to the sign of m X(t). This process is stationary and can be estimated by an auto-regression with lag 1. We use the residuals on a window of length 60 days, assuming the parameters are constant over the window. In fact, the model is shown as X t+1 = a + bx t + ɛ t+1, t = 1, 2,, 59, where we have a = m(1 e κ t ) Figure 7: Generalization error on the first 12 components over 30 testing days using SVR Hence, we get b = e κ t V ar(ɛ) = σ 2 1 e κ t. 2κ κ = log(b) 101 = a m = 1 b = V ar(ɛ) 2κ σ = 1 b 2 = V ar(ɛ) σ eq = 1 b 2 = Signal Generation We define a scalar varaible called the s-score s = X(t) m σ eq. The s-score measures the distance of the cointegrated residual from equilibrium per unit standard deviation, i.e. how far away a given stock is from the theoretical equilibrium value associated with our model. According to empirical studies in this field, our basic trading signal based on mean-reversion is 4

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