Forecasting systematic risk by Least Angel Regression, AdaBoost and Kernel Ridge Regression

In according to importance of risk in financial decision and investment is one of issue that helps to investors is existing tools and appropriate models in order to predict systematic risk. Aim of this research was forecasting systematic risk of companies admitted at Tehran stock Exchange by Least Angel Regression (LARS), AdaBoost and Kernel Ridge Regression (KRR) and comparing ability of the algorithms in order to find the best methods of the test. In this study the financial data of (1159 observations) during 2005-2014. We used MATLAB software vision (R2013b). Results indicated that Kernel Ridge Regression (KRR) with 9.65% error (90.35% confidence) in comparison with Least Angel Regression (LARS) with 12.15% error (87.85% confidence) and AdaBoost with 28.91% error (71.09 confidence) has more ability for forecasting systematic risk. Moreover, ability of forecasting systematic risk in Least Angel Regression (LARS) is more than AdaBoost.


Introduction
Investment is fundamental and necessary in processing grow and economic development of country.Among effective factors in selecting investment is paying attention of investors to risk and investment's return.
Investors try to investment their financial resources in where the highest return and the least risk.Thus, companies should instead of focusing on earnings more pay attention to risk as limited factors as maximize return.
In contrast of return, risk is mental concept and non-quantitative.Therefore, most of financial and economic experts more focus on measuring and identifying risk.
Based on modern portfolio theory, risk is divided into two sections: First section which is related to systematic risk; second section, non-systematic risk which is related to specific condition.
In this theory, risk measures by risk of assets with Beta (criteria of systematic risk).Therefore, beta is one of the most applicable and accepted tools of financial and economic experts in order to evaluate and risk management.
Additionally, beta in field of various financial and accounting sciences like fair value of equity, related research about measuring market reaction to specific decision of a company and related research of price has specific responsibility (Hong and Arker, 2007).Several believe that an accounting as a notification system and some believe that it provides information in order to decision makers can do prior decision.In according to importance risk in financial decision and investment is one of issue can helps investors and it provides tools and prior models for predicting systematic risk.Aim of this research is comparing various method of systematic risk in order to select optimal method.Aim of this research is forecasting systematic risk in companies admitted in Tehran stock exchange by using Least Angel Regression (LARS), AdaBoost and Kernel Ridge Regression (KRR) and comparing ability of the algorithms in order to find the best methods of the test.
Efficiency changes of systematic risk of a particular investment to changes in the market and the investment returns of the index β are measured.The index returns reflect the sensitivity of a stock relative to the market portfolio return.Dividing the covariance between the return on asset i in the market portfolio return and the variance of the portfolio return, the beta of asset i is obtained (βi) and calculate as following (Mahdavi and Godarzi, 2001).
A central issue in portfolio theory in finance is the maximization of returns for people who invest in assets, that is, in firms' stocks (Lintner, 1965;Mossin, 1966;Sharpe, 1964).
The key idea of portfolio theory is that investors can construct a portfolio of stocks with imperfectly correlated returns and thus eliminate nonsystematic (i.e., individualistic) risk associated with those stocks.The remaining variability, the firm's systematic risk, reflects the extent to which its stock's return responds to movement of the average return on all stocks in the market.Beaver, Kettler and Scholes (1970) related systematic risk to variables that describe the financial position of a firm.Specifically, they suggested that greater systematic risk is related to the following: •Higher growth because, in a competitive economy, the excessive earnings opportunities may erode when new firms enter the industry;

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Greater leverage because the earnings stream of common shareholders becomes more volatile as debt increases; Considering two periods (1947-1956 and 1957-1965), BKS (1970) (1) regressed the aforementioned firm characteristics on systematic risk in the first period and (2) examined whether a model of systematic risk from the first period predicted systematic risk in second period better than did systematic risk in first period.
Two diverse streams of empirical research have emerged from BKS's (1970) study.The first stream of research, which is not related to the current study's objectives, focuses on the prediction of the firm's systematic risk in a future period (Elgers, 1980;Eskew, 1979;Ismail and Kim 1989).The second stream of research, which is more pertinent to the current study, augments the predictor variables in BKS's study with additional firm characteristics that may explain systematic risk.Although there are several studies in this stream, our review indicates a lack of cumulative knowledge building in this area.Rather, each study adds some new variables to a subset of the variables in BKS's study.Variables considered in prior research include dividend policy (Bildersee, 1975), operating leverage (Mandelker and Rhee, 1984), earnings funds flow and cash flow (Ismail and Kim 1989), international diversification (Goldberg and Heflin, 1995), and strategic profiles (Veliyath and Ferris, 1997).
Similarly, two studies (Bharadwaj and Menon, 1993;Kroll, Wright and Heiens, 1999) explore the relationship between aspects of a firm's marketing strategy and its risk.
Using service strategic business units from the Profit Impact of Marketing Strategy database, Bharadwaj and Menon (1993) find that some aspects of marketing (e.g., promotional expenditure, sales force expenditure, relative price) are associated with lower variability in return on investment, whereas other aspects of marketing (e.g., advertising, customization) are associated with higher variability in return on investment.Although variability in return on investment, a surrogate for total risk, confounds systematic risk with nonsystematic risk, these results suggest a relationship between firms' marketing activities and their systematic risk.Kroll, Wright, and Heiens (1999) consider a surrogate for systematic risk, the covariance of firms' cash flows relative to a market portfolio of equities.Bougheas et al., (2014) showed that resent progress in financial economy about measuring systematic risk using applicable programs like data mining for analyzing and pricing systematic risk has substantial advantages in concept and practice.Laeven et al., (2014) showed that in according to bigger banks have more systematic risk due to these organizations are involving in activities of banking or they have more organizational complexity in comparison with smaller banks with less capital, traditional banking rules and they are not sufficient for bigger banks.Based on systematic risk is necessary in order to use some rules for avoiding from bankruptcy and financial failures.This action can lead to overinvestment in bigger bank and consequently reduce contribution in the market.Gregor et al., (2014) concluded that merging banks and capital rising in market reduce systematic bank of risk.Acharya and Stephen (2013) did a research about analyzing systematic risk from E-banking concluded that sources of debt on the banks of one of the main causes of increasing systemic risk in European banks.

Kernel Ridge Regression (KRR)
Blades linear regression problem is as follows.

Equitation (1)
In the nonlinear case, it comes in the form below: Where H denotes the Hilbert space and f coefficients in Hilbert space and function φ mapping from input space to feature space is a Hilbert.If the coefficients can be written in the form of dual and we used Kernel trick the shape of the output function is calculated as follows: (Hastie et al., 2009).

AdaBoost
The main idea of boosting is to combine rough and moderately inaccurate weak hypotheses to form a very accurate strong one.AdaBoost (YoavFreund and Robert, 1997) named by its capability of adjusting adaptively to the errors of weak classifiers, has been widely used in a variety of object detection applications (Viola et al., 2003;Miyake et al., 2006;Wei and Weiming, 2005) because of its excellent performance in both speed and accuracy since Viola and Jones (VJ) proposed their successful detection framework (Paul et al., 2004;Paul et al., 2001).
There are 3 main contributions in VJ's work.First, the Integral Image representation, obtained in one pass over the original image, helps evaluate features in constant time at any scale or location on the original image; and then among the hundreds of thousands of features, a learning algorithm based on AdaBoost can effectively select a small set of the most discriminative ones with lowest errors, and boost them into a strong classifier; finally, the Cascaded Classifiers Structure combines those strong classifiers and accelerates the detection procedure by quickly rejecting many negative sub-windows with little processing, such that computations are mainly focused on promising regions.
However, it is noticed that a strong classifier learnt by AdaBoost is suboptimal for applications in terms of error rate (Buhman and Yu, 2000).Some improved variants of the algorithm have been proposed.To reduce false positives, VJ proposed a simple extension of AdaBoost called AsymBoost (Viola and Jones, 2002).It balances the asymmetric costs between false negatives and false positives somewhat by reweighting positive and negative samples at each training round.FloatBoost (Li and Zhang, 2004)

Least Angle Regression
Least Angle Regression (LARS) relates to the classic model-selection method known as Forward Selection, or "forward stepwise regression," described as given a collection of possible predictors, we select the one having largest absolute correlation with the response y, say xj1, and perform simple linear regression of y on xj1.This leaves a residual vector orthogonal to xj1, now considered to be the response.We project the other predictors orthogonally to xj1 and repeat the selection process.After k steps this results in a set of predictors xj1, xj2, . . .,xjk that are then used in the usual way to construct a k-parameter linear model.Forward Selection is an aggressive fitting technique that can be very greedy, perhaps eliminating at the second step useful predictors that happen to be correlated with xj1.Forward Stagewise, as described below, is a much more cautious version of Forward Selection, which may take thousands of tiny steps as it moves toward a final model.It turns out, and this was the original motivation for the LARS algorithm, that a simple formula allows Forward Stagewise to be implemented using fairly large steps, though not as large as a classic Forward Selection, greatly reducing the computational burden.The geometry of the algorithm, described in Section 2, suggests the name "Least Angle Regression." It then happens that this same geometry applies to another, seemingly quite different, selection method called the Lasso (Tibshirani, (1996).The LARS-Lasso-Stagewise connection is conceptually as well as computationally useful.The Lasso is described next, in terms of the main example used in this paper.
Least angle regression (Efron et al., 2004) can be viewed as a version of stagewise that uses mathematical formulas to accelerate the computations.Rather than taking many tiny steps with the first variable, the appropriate number of steps is determined algebraically, until the second variable begins to enter the model.Then, rather than taking alternating steps between those two variables until a third variable enters the model, we jump right to the appropriate spot.Figure 3 shows this process in the case of 2 predictor variables, for linear regression.
The first variable chosen is the one which has the smallest angle between the variable and the response variable; in Figure 3 the angle COX1 is smaller than COX2.We proceed in that direction as long as the angle between that predictor and the vector of residuals Y X1 is smaller than the angle between other predictors and the residuals.
Eventually the angle for another variable will equal this angle, at which point we begin moving toward the direction of the least-squares fit based on both variables.In higher dimensions we will reach the point at which a third variable has an equal angle, and joins the model, etc.
Expressed another way, the (absolute value of the) correlation between the residuals and the first predictor The LAR algorithm in the case of 2 predictors.O is the prediction based solely on an intercept.C = ^ Y = ^ _1X1 + ^ _2X2 is the ordinary least-squares fit, the projection of Y onto the subspace spanned by X1 and X2.A is the forward stepwise fit after one step; the second step proceeds to C. Stagewise takes a number of tiny steps from O to B, then takes steps alternating between the X1 and X2 directions, eventually reaching E; if allowed to continue it would reach C. LAR jumps from O to B in one step, where B is the point for which BC bisects the angle ABD.At the second step it jumps to C. The Lasso follows a path from O to B, then from B to C.
Here LAR agrees with Lasso and stagewise (as the step size for stagewise).In higher dimensions, additional conditions are needed for exact agreement to hold.

Methodology
We used one perspective of two levels by using the algorithms in order to forecast systematic risk.The two levels are as following: 1. Application of stepwise regression to select the input variables to the model 2. LARS algorithms, AdaBoost and Kernel Ridge Regression (KRR) to predict systemic risk

Hypotheses
H1: Least Angel Regression (LARS) has more ability than AdaBoost in order to forecast systematic risk.
H2: Kernel Ridge Regression (KRR) has more ability than AdaBoost in order to forecast systematic risk.
In first level, after collecting data by using stepwise regression among independent variables; we use significant variables in order to enter in the model.In following step, by using LARS, AdaBoost and Kernel Ridge Regression (KRR) provides a model in order to forecast systematic risk.We used MATLAB and this software is one of the most powerful mathematical software and has various applications in other fields.

Selected Data
The first level is selecting data.Financial data includes (1159 years-companies) with 30 independent variables along with dependent variable with systematic risk between 2005 and 2014.In following

Cleaning and preparing data:
Second level is cleaning and preparing the data.In this level some of independent variables will be due to dysfunction of information.We used stepwise in order to clean and prepare data.Methods hypothesis underlying this regression is inevitable.Investigate the hypothesis underlying regression testing is done as follows: A) Homogeneity of variance: The residual plot against predicted values to evaluate the homogeneity of variance in the model is studied.Results of the table above mentioned is Kernel Ridge Regression with 9.65 error percentage (90.35percent confidence) has higher ability in order to forecast systematic risk in comparison LARS with 12.15 percent error (87.85 percent confidence) and AdaBoost with 28.91 percent error (71.09 percent confidence).

Conclusion
In current research, we investigated by using 30 financial ratios in order to forecast systematic risk of companies admitted listed

Figure 2 .
Figure 2. Normality chance of residual , proposed by Stan Z.Li et al.,incorporates the backtrack mechanism from Floating Search, and repeatedly performs a backtrack to remove unfavorable weak classifiers after a new weak classifier is added by AdaBoost, such that a lower error rate and reduced feature set are guaranteed at the cost of about five times longer training time.And another much more complex variant Ent-Boost (Le and Satoh, 2007), partially motivated by Kullback-Leibler Boosting, employs symmetric KL divergence for optimal weak classifier selection.Generally, all these works modified the mechanism of weak classifier selection through error rate evaluation.

Table 5 .
Independent variableIn according to related algorithm of LARS, AdaBoost and Kernel Ridge Regression (KRR) and results of the tests are as following:

Table 6 .
LARS, AdaBoost and Kernel Ridge Regression (KRR) and results Tehran stock exchange of LARS algorithm, AdaBoost and Kernel Ridge Regression.Financial information of 1159 observations during 2005-2014.Results of research indicated that Kernel Ridge Regression (KRR) with 9.65% error (90.35% confidence) in comparison with Least Angel Regression (LARS) with 12, 15% error (87.85% confidence) and AdaBoost with 28.91% error (71.09 confidence) has more ability for forecasting systematic risk.Moreover, ability of forecasting systematic risk in Least Angel Regression (LARS) is more than AdaBoost.