1. A time series approach to analyzing stock market volatility and returns. By: Arteid Memaj & Talal Butt

2.  Introduction  Methodology  Results  Limitations  References

4.  Develop a Model which predicts future values of S&P 500’s and DJI’s Returns and Volatility in a response time series as a linear combination of its past values and past errors.  Importance of study: This information concerns economists and investors since the model developed could allow them to predict future risk and hedge against it.

5.  Hypothesis 1: There is a time series model (ARMA(p,q)) that fits our data for Returns.  Hypothesis 2: There is a time series model(ARMA(p,q)) that fits our data for Volatility.

7.  Identification Stage  Estimation and Diagnostic Stage  Forecasting Stage

8.  Identification stage: 1.Autocorrelation check for white noise: -Ho: none of the autocorrelations up to a given lag are significantly differently from 0. 2.Stationarity Test : -Stationarity in the variance must be present. 3.Autocorrelation: -Corr (Z t,Z t+k ) 4. Partial Autocorrelations: - Corr (Z t,Z t+k |Z t+1,Z t+2,….Z t+k-1 )

9. To LagChi-Square DFPr > Chi SqAutocorrelations 610.8760.09240.042-0.060.049-0.0060.105-0.051 To LagChi-Square DFPr > Chi SqAutocorrelations 63.2360.78-0.032-0.1520.003-0.0040.0690.034 Autocorrelation Check for White Noise (S&P500) Autocorrelation Check for White Noise (DJI)

10. To LagChi-Square DFPr > Chi SqAutocorrelations 6511.26<0.0001.632.4980.4260.3080.3070.303 To LagChi-Square DFPr > Chi SqAutocorrelations 6511.26<0.0001.632.498 0.4260.3080.3070.303 Autocorrelation Check for White Noise (S&P500) Autocorrelation Check for White Noise (DJI)

11. LagCovarianceCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1Std Error 02.91E-051 | |******************** |0 11.84E-050.63154|. |************* |0.046829 21.45E-050.49828|. |********** |0.062788 31.24E-050.42648|. |********* |0.070931 48.95E-060.30794|. |***** |0.076348 58.92E-060.30688|. |***** |0.079025 68.81E-060.30309|. |***** |0.081596

12. LagCovarianceCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1Std Error 07.53E-051 | |******************** |0 15.74E-050.76195|. |*************** |0.144338 23.76E-050.49996|. |********** |0.212188 32.71E-050.36022|. |*******. |0.235455 42.22E-050.29418|. |******. |0.246669 51.42E-050.18918|. |****. |0.253873 66.40E-060.08495|. |**. |0.256793

13. LagCorrelation - 1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 10.76195|. |************* 2-0.19219|. ****|. 30.12995|. |***. 40.0354|. |*. 5-0.1303|. **|. 6-0.01816|. DJI S&P500 LagCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 10.63154|. |************* | 20.1654|. |*** | 30.10158|. |** | 4-0.05112|. *|. | 50.11552|. |** | 60.07894|. |** |

14. ProcessACFPACF AR(p)Tails off as exponential decay or damped sine wave. Cuts off after lag p MA(q)Cuts off after lag qTails off as exponential decay or damped sine wave ARMA(p,q)Tails off after lag (q – p)Tails off after lag (p – q ) LagCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 10.63154|. |************* | 20.1654|. |*** | 30.10158|. |** | 4-0.05112|. *|. |

15. ParameterEstimateStandard Errort Value Pr > |t| Lag MU0.009310.0005317.56<.00010 AR1,10.632820.036417.39<.00011 To LagChi-SquareDFPr > ChiSqAutocorrelations 628.215<.0001-0.1050.0480.145-0.0540.0720.136 1237.7911<.0001-0.0370.1240.030.0130.0270.043 1845.74170.00020.050.0040.0460.058-0.0680.064 2456.28230.00010.0770.0010.0090.025-0.1030.068 3064.47290.0002-0.072-0.0710.0620.013-0.0320.04 3668.65350.0006-0.027-0.0470.0150.004-0.0450.058 4271.44410.0023-0.047-0.0250.01-0.051-0.0080.001 4874.54470.0064-0.0370.018-0.0560.014-0.013-0.029

16. ProcessACFPACF AR(p)Tails off as exponential decay or damped sine wave. Cuts off after lag p MA(q)Cuts off after lag qTails off as exponential decay or damped sine wave ARMA(p,q)Tails off after lag (q – p)Tails off after lag (p – q ) LagCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 01 | |******************** 10.63154|. |************* 20.49828|. |********** 30.42648|. |********* 40.30794|. |***** 50.30688|. |***** 60.30309|. |***** LagCorrelation-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 10.63154|. |************* 20.1654|. |*** 30.10158|. |** 4-0.05112|. *|. 50.11552|. |**

17. To LagChi-SquareDFPr > ChiSqAutocorrelations 62.8410.0918-0.004-0.0070.004-0.0240.0130.073 127.9970.3335-0.070.068-0.012-0.027-0.020.016 1812.96130.4510.014-0.010.0270.028-0.0790.05 2420.98190.33770.0720.017-0.018-0.002-0.0970.038 3029.65250.2377-0.091-0.0730.0450.029-0.020.032 3632.52310.3921-0.025-0.0360.0090.014-0.0260.054 4233.85370.6175-0.032-0.0170.003-0.035-0.0080.01 4835.22430.7948-0.0210.019-0.0390.0180.001-0.007

18. To LagChi-SquareDFPr > ChiSqAutocorrelations 60.5410.46150-0.002-0.005-0.0080.0320.05 123.5270.8326-0.0740.06-0.027-0.054-0.0040.079 1810.62130.6423-0.0870.06-0.010.065-0.1620.04 2416.25190.64070.1190.042-0.066-0.027-0.054-0.094

19. ObsForecastStd Error 95% Confidence Limits Jan 20110.00630.004100.0143 Feb 20110.0070.004600.016 March 20110.00740.004900.0169 April 20110.00790.005100.0179 May 20110.0080.005100.0181 June 20110.00810.005200.0183 July 20110.00820.005200.0185 Aug 20110.00830.005300.0186 Sept 20110.00840.005300.0187 Oct 20110.00840.005300.0189 Nov 20110.00850.005300.019 Dec 20110.00860.005400.019 ObsForecastStd Error95% Confidence Limits Jan 20110.01120.004200.0194 Feb 20110.01070.005400.0212 March 20110.01060.005800.0219 April 20110.01050.00600.0223 May 20110.01050.006200.0225 June 20110.01040.006300.0227 July 20110.01040.006400.0228 Aug 20110.01040.006400.0229 Sept 20110.01030.006500.023 Oct 20110.01030.006500.023 Nov 20110.01030.006500.023 Dec 20110.01030.006500.0231 S&P500DJI

22. We can conclude that a ARMA(1,4) process forecasts volatility for both S&P500 and DJI with high precision. Returns could not be forecasted with an AR(p) MA(q) or an ARMA(p,q) process.

24. [1] Yahoo Finance (finance.yahoo.com) [2] W.S. Wei, William. Time Series Analysis. New York: Greg Tobin, 2006. Print. [3] O’Rourke, Norm. Hatcher, Larry. Stepanksi, Edward. A Step-By-Step Approach to Using SAS for Univariate and Multivariate Statistics. 2005. Print. [4] SAS Online Doc: Version 8, Chapter 7.

25. Special Thanks to… Dr. Chapman & Dr. Wolff for their valuable contributions and guidance during the process. Xiana Clarke & Yanira Pichardo for contributions to data collections.