Efficient Model in Pair Trading - Statistics Project Example

To determine the efficient model in pair trading between correlation matrix model, co-integration model, and the CAPM model Introduction The pair trading is a common thing in the trading market. Literature on pair trading, market efficiency and return volatility behavior is plenty for a developed stock market. The study involves pair trading using the following three models; correlation matrix and co-integration model or CAPM. In this study, we investigate which of these models is efficient in investigating the pair trading of the stock. The study investigates the stock of USA companies using three models that include correlation matrix, co-integration model and CAPM model. Theoretical frame work Efficient markets are necessary prerequisite if it is desired that funds should be allocated to the highest valued projects (Dempsey 2013). A stock market is termed as efficient if the price fully reflects all information in the markets (Fama and French 2004). Market efficiency and the risk return behavior in a number of emerging stock market economies have been examined by Bekaert and Harvey (91997). Volatility is the tendency of assets prices fluctuating either up or down. Increased volatility is perceived as indicating a rise in the financial risk which can adversely affect investor’s assets and wealth. According to Glosten et al. (1993), they indicated that there is a negative and significant relationship between the volatility and return. Stock volatility has received a great attention from practitioners because it can be used as a measure of risk in financial markets. The Chinese stock market forecast of volatility using the Garch model was done by Hongyu and Zhichao (2006). Design and methods In this study, the forecasting of the stock market of USA stock market was undertaken using the models that include correlation matrix, co- integration, and CAPM. Research question 1. Which among the three models is the most efficient in forecasting the stock? 2. Which is the best model to model the volatility of the stock market using the CAPM, Co- integration and correlation matrix? Hypothesis Null hypothesis: there is no significant evidence to conclude that there is significant difference in the three models to forecast the volatility in the stock market Alternative hypothesis: there is significant evidence to conclude that there is significant difference in the three models to forecast the volatility in the stock market Data analysis Figure 1.The returns of each stock along with the S & P500 Index According to the above figure, it is observed that pairing exists in the returns of the stock with the S& P 500. Table 1. S& P500 regressed on Apple Excess Return Dependent Variable: APPLE__EXCESS_RETURN_ Method: Least Squares Date: 04/13/15 Time: 09:59 Sample (adjusted): 1/09/2012 12/01/2014 Included observations: 152 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.934211 0.187397 4.985201 0.0000 R-squared 0.124142     Mean dependent var 0.530643 Adjusted R-squared 0.124142     S.D. dependent var 3.763642 S.E. of regression 3.522290     Akaike info criterion 5.362657 Sum squared resid 1873.385     Schwarz criterion 5.382551 Log likelihood -406.5619     Hannan-Quinn criter. 5.370739 Durbin-Watson stat 2.035858 As it can be observed in the above table, the variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power and the significance of the model can be suggested to be very poor. The Durbin-Watson stat is 2 and diagnostic tests is undertaken in the next section to know if there is serial correlation in the model Table 2. Heteroskedasticity Test: White F-statistic 0.167956     Prob. F(1,150) Obs*R-squared 0.170005     Prob. Chi-Square(1) Scaled explained SS 0.298308     Prob. Chi-Square(1) It can be observed in the above table that there is no evidence of heteroscedasticity in the residuals. Table 3. Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.062019     Prob. F(1,150) 0.8037 Obs*R-squared 0.000000     Prob. Chi-Square(1) 1.0000 The above table indicates the first order serial correlation and it can be noted that there is no suffering of the residuals from the first order serial correlation. Table 4. S& P500 regressed on Bank of America Excess Return Dependent Variable: BANK_OF_AMERICA_EXCESS_R Method: Least Squares Date: 04/13/15 Time: 10:25 Sample (adjusted): 1/09/2012 12/01/2014 Included observations: 152 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 1.693632 0.171372 9.882752 0.0000 R-squared 0.372192     Mean dependent var 0.745834 Adjusted R-squared 0.372192     S.D. dependent var 4.065279 S.E. of regression 3.221096     Akaike info criterion 5.183878 Sum squared resid 1566.694     Schwarz criterion 5.203772 Log likelihood -392.9747     Hannan-Quinn criter. 5.191959 Durbin-Watson stat 1.821529 Basing on the above table, the variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power and the significance of the model is considered to be poor. The Durbin-Watson stat is 1.8 and diagnostic test is undertaken in the next section to know if there is serial correlation in the model Table 5. Heteroscedasticity Test: White F-statistic 0.105080     Prob. F(1,150) 0.7463 Obs*R-squared 0.106406     Prob. Chi-Square(1) 0.7443 Scaled explained SS 0.396430     Prob. Chi-Square(1) 0.5289 According to Heteroscedasticity Test: White, above, it can be observed that there is no evidence of heteroscedasticity in the residuals Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.961648     Prob. F(1,150) 0.3284 Obs*R-squared 0.378610     Prob. Chi-Square(1) 0.5383 Table 6. Wald Test Equation: Untitled Test Statistic Value df Probability t-statistic  4.047510  151  0.0001 F-statistic  16.38234 (1, 151)  0.0001 Chi-square  16.38234  1  0.0001 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1)  0.693632  0.171372 Restrictions are linear in coefficients. Wald Test was carried out to investigate if the estimated coefficient that is associated with S & P 500 is statistically equal to 1.In this case we consider a null hypothesis that include Ho:β1=1.Basing on the test results, the null hypothesis is to be rejected. Hence, the associated coefficient is statistically different from 1 and this implies that the stock is not as risky as the market Table 7. S& P500 regressed on Ford Excess Return Dependent Variable: FORD_EXCESS_RETURN_ Method: Least Squares Date: 04/13/15 Time: 10:38 Sample (adjusted): 1/09/2012 12/01/2014 Included observations: 152 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.909388 0.393291 2.312251 0.0221 R-squared 0.034185     Mean dependent var 0.026404 Adjusted R-squared 0.034185     S.D. dependent var 7.521937 S.E. of regression 7.392252     Akaike info criterion 6.845299 Sum squared resid 8251.454     Schwarz criterion 6.865193 Log likelihood -519.2427     Hannan-Quinn criter. 6.853381 Durbin-Watson stat 2.190117 According to the regression summary above, the variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power and the significance of the model are very poor. The Durbin-Watson stat is 2.2 and diagnostic test aim in the next section was to investigate if there is serial correlation in the model Table 8. Breusch-Godfrey Serial Correlation LM Test: Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.847017     Prob. F(1,150) 0.1762 Obs*R-squared 1.652080     Prob. Chi-Square(1) 0.1987 The first order serial correlation analysis above shows that there is no suffering of the residuals from the first order serial correlation. Table 9. Wald Test Equation: Untitled Test Statistic Value df Probability t-statistic -0.230395  151  0.8181 F-statistic  0.053082 (1, 151)  0.8181 Chi-square  0.053082  1  0.8178 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1) -0.090612  0.393291 Restrictions are linear in coefficients. By considering a null hypothesis of Ho: β1=1, the null hypothesis can’t be rejected according to the above results. Hence, the associated coefficient is not statistically different from 1 and this implies that the stock is as risky as the market Table 10. S& P500 regressed on Wells Fargo Excess Return Dependent Variable: WELLS_FARGO_EXCESS_RETUR Method: Least Squares Date: 04/13/15 Time: 11:01 Sample (adjusted): 1/09/2012 12/01/2014 Included observations: 152 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 1.157175 0.075554 15.31591 0.0000 R-squared 0.589380     Mean dependent var 0.486517 Adjusted R-squared 0.589380     S.D. dependent var 2.216150 S.E. of regression 1.420100     Akaike info criterion 3.545890 Sum squared resid 304.5195     Schwarz criterion 3.565783 Log likelihood -268.4876     Hannan-Quinn criter. 3.553971 Durbin-Watson stat 2.307756 Basing on the above table, the variable S & P500 excess return is seen not to be statistically significant. The regression(R squared) explanatory power and the significance of the model are good. The Durbin-Watson stat is 2.3 and diagnostic test is undertaken in the next section to know if there is serial correlation in the model Table 11. Heteroskedasticity Test: White F-statistic 0.502098     Prob. F(1,150) 0.4797 Obs*R-squared 0.507095     Prob. Chi-Square(1) 0.4764 Scaled explained SS 0.661503     Prob. Chi-Square(1) 0.4160 The Heteroscedasticity Test: White, above shows that there is no evidence of heteroscedasticity in the residuals Table 12. Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.896587     Prob. F(1,150) 0.0502 Obs*R-squared 2.886027     Prob. Chi-Square(1) 0.0894 The first order serial correlation analysis above shows that there is no suffering of the residuals from the first order serial correlation. Table 13. Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic  2.080311  151  0.0392 F-statistic  4.327694 (1, 151)  0.0392 Chi-square  4.327694  1  0.0375 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1)  0.157175  0.075554 Restrictions are linear in coefficients. By considering a null hypothesis of Ho: β1=1, the null hypothesis is to be rejected and this as well means that there is no enough evidence to accept the null hypothesis. Hence, the associated coefficient is statistically different from 1 and this implies that the stock is not as risky as the market Table 14. S& P500 regressed on Yahoo Excess Return Dependent Variable: YAHOO__INC_EXCESS_RETURN Method: Least Squares Date: 04/13/15 Time: 11:05 Sample (adjusted): 1/09/2012 12/01/2014 Included observations: 152 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.999263 0.183093 5.457689 0.0000 R-squared 0.120873     Mean dependent var 0.838562 Adjusted R-squared 0.120873     S.D. dependent var 3.670355 S.E. of regression 3.441389     Akaike info criterion 5.316184 Sum squared resid 1788.317     Schwarz criterion 5.336078 Log likelihood -403.0300     Hannan-Quinn criter. 5.324266 Durbin-Watson stat 2.100191 The regression summary above shows that the variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power and the significance of the model are very poor. The Durbin-Watson stat is 2.1 and diagnostic test can be done as in the next section to know whether there is serial correlation in the model Table 16. TaHeteroskedasticity Test: White F-statistic 0.577886     Prob. F(1,150) 0.4483 Obs*R-squared 0.583344     Prob. Chi-Square(1) 0.4450 Scaled explained SS 0.732048     Prob. Chi-Square(1) 0.3922 The Heteroscedasticity Test: White, above shows that there is no enough evidence of heteroscedasticity in the residuals Table 17. Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.417796     Prob. F(1,150) 0.5190 Obs*R-squared 0.000000     Prob. Chi-Square(1) 1.0000 The first order serial correlation analysis above shows that there is no suffering of the residuals from the first order serial correlation. Co- integration model S& P500 regressed on Apple Excess Return Dependent Variable: APPLE__EXCESS_RETURN_ Method: Fully Modified Least Squares (FMOLS) Date: 04/13/15 Time: 15:14 Sample (adjusted): 1/16/2012 12/01/2014 Included observations: 151 after adjustments Cointegrating equation deterministics: C Long-run covariance estimate (Bartlett kernel, Newey-West fixed bandwidth         = 5.0000) Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.977854 0.189267 5.166525 0.0000 C 0.224241 0.289182 0.775431 0.4393 R-squared 0.127941     Mean dependent var 0.538239 Adjusted R-squared 0.122088     S.D. dependent var 3.774997 S.E. of regression 3.537057     Sum squared resid 1864.105 Durbin-Watson stat 2.052975     Long-run variance 12.07902 Just like for CAPM model, the variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power and the significance of the model can be suggested to be very poor. The Durbin-Watson stat is 2 and diagnostic tests is undertaken in the next section to know if there is serial correlation in the model Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic -0.117007  149  0.9070 F-statistic  0.013691 (1, 149)  0.9070 Chi-square  0.013691  1  0.9069 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1) -0.022146  0.189267 Restrictions are linear in coefficients. Using the Ho: β1=1 as the null hypothesis, the null hypothesis can’t be rejected. Hence, the associated coefficient is not statistically different from 1 .Hence; the stock is not as risky as the market Date: 04/13/15 Time: 15:20 Sample: 1/02/2012 12/01/2014 Included observations: 151 Autocorrelation Partial Correlation AC   PAC  Q-Stat  Prob*        .|. |        .|. | 1 -0.048 -0.048 0.3519 0.553        .|. |        .|. | 2 0.015 0.013 0.3881 0.824 *Probabilities may not be valid for this equation specification. The autocorrelation results above indicate that the residual does not suffer from autocorrelation Hence the residuals are not auto correlated. S& P500 regressed on Bank of America Excess Return Dependent Variable: BANK_OF_AMERICA_EXCESS_R Method: Fully Modified Least Squares (FMOLS) Date: 04/13/15 Time: 15:33 Sample (adjusted): 1/16/2012 12/01/2014 Included observations: 151 after adjustments Cointegrating equation deterministics: C Long-run covariance estimate (Bartlett kernel, Newey-West fixed bandwidth         = 5.0000) Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 1.672797 0.183790 9.101673 0.0000 C 0.170137 0.280814 0.605871 0.5455 R-squared 0.376215     Mean dependent var 0.704960 Adjusted R-squared 0.372029     S.D. dependent var 4.047349 S.E. of regression 3.207307     Sum squared resid 1532.736 Durbin-Watson stat 1.860249     Long-run variance 11.39002 The variable S & P500 excess return is statistically significant. The regression(R squared) explanatory power is seen to be poor. The Durbin-Watson stat is 1.8 and diagnostic tests is undertaken in the next section to know if there is auto correlation in the model Date: 04/13/15 Time: 15:39 Sample: 1/02/2012 12/01/2014 Included observations: 151 Autocorrelation Partial Correlation AC   PAC  Q-Stat  Prob*        .|. |        .|. | 1 -0.056 -0.056 0.4786 0.489        .|. |        .|. | 2 0.027 0.024 0.5956 0.742 *Probabilities may not be valid for this equation specification. The above auto correlation results shows that the residuals are not auto correlated Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic  3.660682  149  0.0003 F-statistic  13.40060 (1, 149)  0.0003 Chi-square  13.40060  1  0.0003 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1)  0.672797  0.183790 Restrictions are linear in coefficients. The null hypothesis is to be rejected basing on the null hypothesis; the Ho: β1=1.Thus, the associated coefficient is statistically different from 1 .Hence; the stock is not as risky as the market S& P500 regressed on Ford Excess Return Dependent Variable: FORD_EXCESS_RETURN_ Method: Fully Modified Least Squares (FMOLS) Date: 04/13/15 Time: 15:41 Sample (adjusted): 1/16/2012 12/01/2014 Included observations: 151 after adjustments Cointegrating equation deterministics: C Long-run covariance estimate (Bartlett kernel, Newey-West fixed bandwidth         = 5.0000) Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.977559 0.354627 2.756586 0.0066 C -0.392815 0.541836 -0.724971 0.4696 R-squared 0.034488     Mean dependent var -0.076516 Adjusted R-squared 0.028008     S.D. dependent var 7.438812 S.E. of regression 7.333899     Sum squared resid 8014.124 Durbin-Watson stat 2.229460     Long-run variance 42.40561 The regression results above shows that the variable S & P 500 value is statistically significant. Also the explanatory power can be suggested to be very poor. The DW stat is 2.22 and diagnostic tests can be carried out to determine if there is correlation in the model. Date: 04/13/15 Time: 15:56 Sample: 1/02/2012 12/01/2014 Included observations: 151 Autocorrelation Partial Correlation AC   PAC  Q-Stat  Prob*        .|. |        .|. | 1 0.074 0.074 0.8335 0.361        .|. |        .|. | 2 -0.052 -0.058 1.2593 0.533 *Probabilities may not be valid for this equation specification. The above results shows that the residuals are not auto correlated Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic -0.063281  149  0.9496 F-statistic  0.004004 (1, 149)  0.9496 Chi-square  0.004004  1  0.9495 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1) -0.022441  0.354627 Restrictions are linear in coefficients. Basing on the Wald test, it can be concluded that the associated coefficient is not statistically different from 1 .Hence; the stock is as risky as the market S& P500 regressed on Wells Fargo Excess Return Dependent Variable: WELLS_FARGO_EXCESS_RETUR Method: Fully Modified Least Squares (FMOLS) Date: 04/13/15 Time: 15:53 Sample (adjusted): 1/16/2012 12/01/2014 Included observations: 151 after adjustments Cointegrating equation deterministics: C Long-run covariance estimate (Bartlett kernel, Newey-West fixed bandwidth         = 5.0000) Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 1.140641 0.066608 17.12476 0.0000 C 0.109315 0.101770 1.074139 0.2845 R-squared 0.592213     Mean dependent var 0.474487 Adjusted R-squared 0.589477     S.D. dependent var 2.218540 S.E. of regression 1.421465     Sum squared resid 301.0638 Durbin-Watson stat 2.340225     Long-run variance 1.495994 The above results show that the variable S & P 500 value is statistically significant. The explanatory power can be suggested to be good. The DW stat is 2.22 and diagnostic tests can be carried out to determine if there is correlation in the model. Date: 04/13/15 Time: 15:54 Sample: 1/02/2012 12/01/2014 Included observations: 151 Autocorrelation Partial Correlation AC   PAC  Q-Stat  Prob*        .|. |        .|. | 1 0.059 0.059 0.5427 0.461        .|* |        .|* | 2 0.082 0.079 1.5829 0.453 *Probabilities may not be valid for this equation specification. The above results shows that the residuals are not statistically auto correlated Wald Test: Equation: Untitled Test Statistic Value df Probability t-statistic  2.111487  149  0.0364 F-statistic  4.458376 (1, 149)  0.0364 Chi-square  4.458376  1  0.0347 Null Hypothesis: C(1)=1 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. -1 + C(1)  0.140641  0.066608 Restrictions are linear in coefficients. Since there is no enough evidence to accept the null hypothesis. It can be concluded that the associated coefficient is statistically different from 1 .Hence; the stock is not as risky as the market S& P500 regressed on Yahoo Excess Return Dependent Variable: YAHOO__INC_EXCESS_RETURN Method: Fully Modified Least Squares (FMOLS) Date: 04/13/15 Time: 15:59 Sample (adjusted): 1/16/2012 12/01/2014 Included observations: 151 after adjustments Cointegrating equation deterministics: C Long-run covariance estimate (Bartlett kernel, Newey-West fixed bandwidth         = 5.0000) Variable Coefficient Std. Error t-Statistic Prob.   P___S_500_EXCESS_RETURN_ 0.850109 0.179104 4.746463 0.0000 C 0.577821 0.273653 2.111508 0.0364 R-squared 0.141549     Mean dependent var 0.845832 Adjusted R-squared 0.135787     S.D. dependent var 3.681470 S.E. of regression 3.422407     Sum squared resid 1745.217 Durbin-Watson stat 2.141604     Long-run variance 10.81657 The variable S & P 500 value is statistically significant. The explanatory power can be suggested to be good. The DW stat is 2.22 and diagnostic tests can be carried out to determine if there is correlation in the model. Date: 04/13/15 Time: 16:00 Sample: 1/02/2012 12/01/2014 Included observations: 151 Autocorrelation Partial Correlation AC   PAC  Q-Stat  Prob*        .|* |        .|* | 1 0.098 0.098 1.4851 0.223        .|. |        .|. | 2 -0.039 -0.049 1.7225 0.423 *Probabilities may not be valid for this equation specification. The above results shows that there is no enough evidence to show that the residuals are auto correlated Conclusion The study indicates that there is no better model out of the three models for modeling the volatility of the stock market using the CAPM, Co- integration and correlation matrix. The study therefore empirically contributes to the literature available on the stock market by investigating the market form of efficient (Moosa 2013; Partington 2013; Johnstone 2013; Graham and Harvey 2005) Bibliography Bekaert. G and Harvey. C. R (1997). Emerging equity market volatility. Journal of financial Economics 43: 29 -77 Dempsey, M., (2013),“The Capital Asset Pricing Model (CAPM): The History of a Failed Revolutionary Idea in Finance?,”Abacus,49(S1), 7 23. Fama, E.F., and French, K.R., (2004),“The Capital Asset Pricing Model: Theory and Evidence,”The Journal of Economic Perspectives,18, 25-46. Glosten L. R, Jagannathan, R, and Runkle D. E (1993). “ on the relation between the epected value and the volatility of the nomimal ecess return on stock” journal of financial 48: 1779- 1801 Graham, J.R., and Harvey, C.R., (2005),“The Long-Run Equity Risk Premium,”Finance Research Letters2(4), 185-194. Hongyu. P and Zhichao, Z (2006). Forecasting financial volatility: evidence from Chinese stock Johnstone, D., (2013),“The CAPM Debate and the Logic and Philosophy of Finance,”Abacus 49(S1), 1-6. Market. Working paper in econonics and financial no 0602 university of Durham. Moosa, I.A., (2013),“The Capital Asset Pricing Model (CAPM): The History of a Failed Revolutionary Idea in Finance? Comments and Extensions,”Abacus, 49(S1), 62-68 Partington, G., (2013),“Death Where is Thy Sting? A Response to Dempseys Despatching of the CAPM,”Abacus,49(S1), 69-72 Read More