Statistical Analysis of Stock Indices - Research Paper Example

STATISTICAL ANALYSIS OF STOCK INDICES Question I A very useful tool for the evaluation of performance of the Stock Markets is the regression analysis as has adopted by many studies (such as Fama and French, 1988). The above strategy 'reduces to regressing overlapping returns for various holding periods on a set of predetermined instruments or forecasting variables (Guido et al., 2001, 163). Furthermore, the regression analysis especially the autoregressive model that is of interest in this case has successfully been used during the development of a series of robust tests of the 'intrinsic value measure, along with other traditional measures of value, for the Australian Stock Market'. On the other hand, according to Pelaez (1999, 232) 'there are many ways to forecast economic series, including extrapolation, econometric models, time-series models, and leading indicator models'. For the issue under analysis in this report, the test for unit root is considered as the most appropriate tool for evaluating the given data series from the Stock indices. The methodology applied has been considered as most appropriate after a thorough consideration of the specific subject involved. A technical overview on the nuances of the unit root test is presented followed by the analysis of the Stock indices given in SPSS v14.0. This method will enable the presentation of both the theories and the practical application using reliable software to ease the process and eliminate errors. Guido (2001, 164) says that 'the composite intrinsic value measure does not appear to be an adequate measure of a stock's or portfolio's value' in his experiment to compare the US and the Australian markets. Several possible reasons are offered for this difference, including the differing market structures, the use of a different index or the use of alternate statistical tests'. In the light of the above arguments, it is clear that for the data set under analysis it is essential to use a strong statistical tool to identify the relationship between the given stock indices. Dickey-Fuller statistic tests for the unit root in the time series data. Pt is regressed against Pt-1 to test for unit root in a time series random walk model, which is given as: Pt = r Pt-1 + ut (1) If r is significantly equal to 1, then the stochastic variable Pt is said to be having unit root. A series with unit root is said to be un-stationary and does not follow random walk. There are three most popular Dickey-Fuller tests used for testing unit root in a series. The above equation can be rewritten as: D Pt = d Pt-1 + ut (2) Here d = (r - 1) and here it is tested if d is equal to zero. Pt is a random walk if d is equal to zero. It is possible that the time series could behave as a random walk with a drift. This means that the value of Pt may not center to zero and thus a constant should be added to the random walk equation. A linear trend value could also be added along with the constant to the equation, which results in a null hypothesis reflecting stationary deviations from a trend. To test the validity of market efficiency, random walk hypothesis has been tested. Unit root test has been conducted on Pt, natural log values of indices price data by running the regression equations of the following type: D Pt = d Pt-1 + ut (3) D Pt = a + d Pt-1 + ut (4) D Pt = a + dPt-1 + b t + ut (5) where, a is constant term and b is the coefficient of trend term. The null hypothesis for each is: H0: d = 0 (viii) The null hypothesis that Pt is a random walk can be rejected if calculated t is greater than the tabulated t. From the aforementioned it is clear that the test for unit root is a reliable analytical tool to test the consistency of the data series. In case of the stock market indices we are analysing, the test for unit root is a reliable tool to test the extent to which the index is speculating. The output from the autoregressive analysis for unit root test reveals that the behaviour of the stock indices it is clear that "OMXCOPENHAGEN" and "MADRIDSEGENERAL" have consistent behaviour. The analysis using the Autoregressive model (ARIMA) on the model statistics reveals that FTSE has a greater Root-Mean-Square error as well as stable in both rise and fall of the index as shown in fig 1 below. Fig 1. Autoregressive analysis using ARIMA model for Unit Root The analysis also justifies the arguments made initially that the unit root test is a significant method of analyzing the stability and mainly the fluctuation in the stock market. Furthermore, the predictive analysis on the ARIMA model to investigate the root mean square error in the data series is a critical element on deciding the levels for a given stock index. Since the test provides a snapshot of the actual situation as well as the insight into the errors and inconsistency in data, any discrepancy in data due to manual error in calculations can be identified quickly. Question II Fama and MacBeth (1973) and Fama and French (1992, 1993) [11] have studied the relationship between equity returns, taken as the relative change in equity prices, and a number of financial variables, including CAPM beta's, squared beta's, leverage, size (market equity) and book-to-market. Fama and French (1992) conclude that size ME (the stock's price times shares outstanding) and book-to-market BE/ME (the ratio of a firm's book value of common equity BE to its market value ME), subsume the effects of the other variables in explaining the variation in expected returns. On the other hand, Chiarella et al. (2001) use the Blanchard macromodel (1989) which is a rational expectations model comparing shocks to macroeconomic variables that cause the stock price to fluctuate whilst keeping the output fixed (rather than allowing it to adjust gradually). Thus as the stock price amplifies, there is no feedback effect on the output. Once the stock price is on the stable branch output also then gradually adjusts; the stock price overshoots its steady state value during its jump and then decreases thereafter. Blanchard's macromodel thus predicts that unless unanticipated shocks occur, the stock price moves monotonicly toward a point of rest or if it is there it will stay there; thus, in fact, only exogenous shocks will move stock prices (Chiarella et al., 2001, 7) The study of Blanchard's can be used in order to evaluate a stock market but with the prerequisite that the output prices are fixed. In this context, q is considered as the value of the stock market, y is income, g the index of fiscal expenditure so that aggregate expenditure d is given by Moreover, the output adjusts to changes in aggregate expenditure can be calculated with a delay according to where Furthermore, from the standard assumption of LM equilibrium in the asset market we can write Where 'i' denotes the short term rate of interest, m and p the logarithms of nominal money and prices respectively. Real profit is given by so that is the instantaneous expected real rate of return from holding shares where we use x to denote the instantaneous expected change in the value of the stock market. Hence the instantaneous differential between returns on shares and returns on short term bonds (i.e. the instantaneously maturing bond) which may allow for a (long run constant) risk premium on equity, is given by A key assumption of Blanchard's approach is that this differential is always zero. We further assume that the stock market adjusts to the excess demand with a spread of adjustment which can be expressed as follows: where Kq (> 0) is the speed of adjustment of the stock market to excess demand. for stocks and is itself assumed to be a function of the excess demand. In the light of the aforementioned arguments and the given Stock indices for analysis, a Univariate analysis on the variance of the stock indices is conducted using SPSS. The results are presented in tables 1, 2 and 3 below. Table 1 - Tests of Between-Subjects Effects: FTSE 100 - PRICE INDEX Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 559529159.841(a) 574 974,789.477 . . Intercept 13,879,327,289.573 1 . . Name 559,529,159.840 574 974,789.477 . . Error 0.000 0 . Total 14,438,856,449.413 575 Corrected Total 559,529,159.840 574 a. R Squared = 1.000 (Adjusted R Squared = .) Table 2 : Tests of Between-Subjects Effects OMX COPENHAGEN (OMXC20) - PRICE INDEX Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 3218810.161(a) 574 5,607.683 . . Intercept 29,422,931.112 1 . . Name 3,218,810.161 574 5,607.683 . . Error 0.000 0 . Total 32,641,741.273 575 Corrected Total 3,218,810.161 574 a. R Squared = 1.000 (Adjusted R Squared = .) Table 3 -Tests of Between-Subjects Effects: MADRID SE GENERAL - PRICE INDEX Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 33925632.626(a) 574 59,103.890 . . Intercept 313,816,855.688 1 . . Name 33,925,632.626 574 59,103.890 . . Error 0.000 0 . Total 347,742,488.313 575 Corrected Total 33,925,632.626 574 a. R Squared = 1.000 (Adjusted R Squared = .) From the results of the Univariate analysis it is clear that the equity market returns of three different stock indices having disjoint results for the test of unit root can be compared and analysed on one scale. The results from the Univariate analyses also justified the arguments on Blanchard macromodel of estimating the equity market returns from the stock indices provided. Question III The Engle and Granger technique is based on the following analysis [10]: If Zt is a k 1 vector of variables, then the components of Zt are cointegrated of order (d,b) if i) All components of zt are I(d) ii) There is at least one vector of coefficients such that zt I(d-b) Many time series are non-stationary but "move together" over time. If variables are cointegrated, it means that a linear combination of them will be stationary. There may be up to r linearly independent cointegrating relationships (where r k-1), also known as cointegrating vectors. r is also known as the cointegrating rank of zt. A cointegrating relationship may also be seen as a long-term relationship. Although the method is a successful and reliable one for identifying the cointegration between multivariate variables, this method cannot be applied to the given situation. This is because of the fact that the Engle Granger Method of cointegration can be used only in cases where there is a relationship existing between two series and the variables involved. As the data series in the given case are entirely different to each other, it is difficult to identify the cointegration effectively using this method. Although bivariate analysis using Engle Granger method cannot be accomplished effectively, a bivariate analysis for correlation is viable. As the manual calculations on the aforementioned method is not only time consuming but also prone for human errors, SPSS is deployed for deriving the results of the bivariate analysis. The bivariate analysis is conducted to derive the parametric and non-parametric co-efficient 1. Pearson's co-efficient 2. Kendall's tau_b correlation co-efficient 3. Spearman's rho correlation co-efficient Since the variation of the stock indices cannot be treated as parametric entirely the non-parametric coefficients are also checked to verify any consistency between them. The results for the aforementioned coefficients are tabulated below Table 4: Pearson's Correlation Coefficient FTSE 100 - PRICE INDEX OMX COPENHAGEN (OMXC20) - PRICE INDEX MADRID SE GENERAL - PRICE INDEX FTSE 100 - PRICE INDEX Pearson Correlation 1 .659(**) .802(**) Sig. (2-tailed) 0.000 0.000 N 575 575 575 OMX COPENHAGEN (OMXC20) - PRICE INDEX Pearson Correlation .659(**) 1 .931(**) Sig. (2-tailed) 0.000 0.000 N 575 575 575 MADRID SE GENERAL - PRICE INDEX Pearson Correlation .802(**) .931(**) 1 Sig. (2-tailed) 0.000 0.000 N 575 575 575 **. Correlation is significant at the 0.01 level (2-tailed). Table 5: Nonparametric Correlations FTSE 100 - PRICE INDEX OMX COPENHAGEN (OMXC20) - PRICE INDEX MADRID SE GENERAL - PRICE INDEX Kendall's tau_b FTSE 100 - PRICE INDEX Correlation Coefficient 1.000 .498(**) .647(**) Sig. (2-tailed) . 0.000 0.000 N 575 575 575 OMX COPENHAGEN (OMXC20) - PRICE INDEX Correlation Coefficient .498(**) 1.000 .745(**) Sig. (2-tailed) 0.000 . 0.000 N 575 575 575 MADRID SE GENERAL - PRICE INDEX Correlation Coefficient .647(**) .745(**) 1.000 Sig. (2-tailed) 0.000 0.000 . N 575 575 575 Spearman's rho FTSE 100 - PRICE INDEX Correlation Coefficient 1.000 .653(**) .817(**) Sig. (2-tailed) . 0.000 0.000 N 575 575 575 OMX COPENHAGEN (OMXC20) - PRICE INDEX Correlation Coefficient .653(**) 1.000 .888(**) Sig. (2-tailed) 0.000 . 0.000 N 575 575 575 MADRID SE GENERAL - PRICE INDEX Correlation Coefficient .817(**) .888(**) 1.000 Sig. (2-tailed) 0.000 0.000 . N 575 575 575 **. Correlation is significant at the 0.01 level (2-tailed). The results from the parametric and non-parametric bivariate analyses have revealed that there is significant correlation between the pairs of stock indices. The high level of significance in the correlation analyses proves that the fluctuation in one stock market affects other stock market also making it clear that the stock market behavior is interdependent in the global business community. Furthermore, the consistency in the significance between the non-parametric and parametric methods of analyses further reveals that the stock indices can not only be evaluated on the same platform but also accomplish the process of identifying lucrative investment sources across the globe through monitoring the stock index levels. Question IV The ARMAX(R,M,Nx) model [7] can be used as a conditional mean model for all stock indices: with autoregressive coefficients , moving average coefficients , innovations , and returns . is an explanatory regression matrix in which each column is a time series and denotes the th row and th column. The eigenvalues associated with the characteristic AR polynomial must lie inside the unit circle to ensure stationarity. Similarly, the eigenvalues associated with the characteristic MA polynomial must lie inside the unit circle to ensure invertibility. The aforementioned model although effective in deriving reliable results on stock index analysis, due to computational complexity over the data range under analysis the conditional linear regression method to analyze the behavior of FTSE Stock index is conducted. The results of the conditional linear regression is presented in Table 6 whilst a plot of the stock index under analysis for cubic regression is presented in fig 2 below. Table 6: Linear Regression Analysis Model R R Square Adjusted R Square Std. Error of the Estimate 1 .213(a) 0.045 0.044 965.570 a. Predictors: (Constant), Name ANOVA(b) Model Sum of Squares df Mean Square F Sig. 1 Regression 25,306,821.473 1 25,306,821.473 27.144 .000(a) Residual 534,222,338.368 573 932,325.198 Total 559,529,159.840 574 a. Predictors: (Constant), Name b. Dependent Variable: FTSE 100 - PRICE INDEX Coefficients(a) Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) -22,633.562 5,287.441 -4.281 0.000 Name 0.000 0.000 0.213 5.210 0.000 a. Dependent Variable: FTSE 100 - PRICE INDEX Fig 2: Scatter Plot for FTSE Stock Index with cubic regression fit line From the Analysis of Variance (ANOVA) in table 6, it is clear that the dependent variable - FTSE Stock index has a strong correlation with time (i.e.) date. The use of the Stock index as the dependent variable further justifies the choice of the regression model to analyze the significance of the chosen stock index with respect to time. Question V The conditional variance of the innovations, , is by definition The key insight of GARCH lies in the distinction between conditional and unconditional variances of the innovations process . The term conditional implies explicit dependence on a past sequence of observations. The term unconditional is more concerned with long-term behavior of a time series and assumes no explicit knowledge of the past. The various GARCH models characterize the conditional distribution of by imposing alternative parameterizations to capture serial dependence on the conditional variance of the innovations. The particular GARCH model, which is going to be used for the last 20 observations, is going to be based on the above equation. More specifically, it will be: The general GARCH(P,Q) model for the conditional variance of innovations is with constraints As the GARCH models are predominantly used for analysing the changes in the variance of a function of time further justifies the choice of the model for analysing the stock indices where the stock index varies with time, which is the independent variable in the series. Furthermore, the use of GARCH models instead of the ARCH models is also due to the fact that the data series under analysis do not have common characteristics under normal conditions and a generalised model alone can accommodate the variance of three unique series without compromising their individual characteristics. The fact that the autoregressive conditional Heteroskedasticity is an element that should be analysed on a common platform is the main reason for the introduction of the GARCH model on top of the traditional ARCH Model which is the best-fit model for series with common characteristics. It is also interesting to note that the autoregressive modelling for Heteroskedasticity gives reliable forecasts on volatility and accurate measures that are crucial for pricing in a stock market justifies the choice of the model for forecasting and analysis in this report. An autoregressive time series model for the given stock indices is presented below. The model on the time series is again constructed using SPSS in order to ease the process of analysis and investigation. The three Stock Indices are included in the model as dependent variables whilst time (i.e.) the date is the independent variable for the construction of the model. Model Summary Charts Model Summary Model Fit Fit Statistic Mean SE Minimum Maximum Percentile 5 10 25 50 75 90 95 Stationary R-squared .000 .000 1.55E-015 .000 1.55E-015 1.55E-015 1.55E-015 1.78E-015 .000 .000 .000 R-squared .992 .004 .987 .994 .987 .987 .987 .994 .994 .994 .994 RMSE 45.368 57.080 5.822 110.805 5.822 5.822 5.822 19.478 110.805 110.805 110.805 MAPE 1.766 .143 1.612 1.895 1.612 1.612 1.612 1.790 1.895 1.895 1.895 MaxAPE 12.066 2.606 9.310 14.490 9.310 9.310 9.310 12.398 14.490 14.490 14.490 MAE 33.020 41.708 4.180 80.843 4.180 4.180 4.180 14.038 80.843 80.843 80.843 MaxAE 193.612 227.823 34.139 454.538 34.139 34.139 34.139 92.158 454.538 454.538 454.538 Normalized BIC 6.304 2.962 3.534 9.427 3.534 3.534 3.534 5.950 9.427 9.427 9.427 Model Statistics Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared MAPE MaxAPE Statistics DF Sig. FTSE 100 - PRICE INDEX-Model_1 0 .000 1.612 9.310 15.416 17 .566 0 OMX COPENHAGEN (OMXC20) - PRICE INDEX-Model_2 0 1.55E-015 1.790 14.490 24.058 18 .153 0 MADRID SE GENERAL - PRICE INDEX-Model_3 0 1.78E-015 1.895 12.398 22.126 18 .226 0 Exponential Smoothing Model Parameters Model Estimate SE t Sig. FTSE 100 - PRICE INDEX-Model_1 No Transformation Alpha (Level) .955 .042 22.902 .000 ARIMA Model Parameters Estimate SE t Sig. OMX COPENHAGEN (OMXC20) - PRICE INDEX-Model_2 OMX COPENHAGEN (OMXC20) - PRICE INDEX Natural Log Constant .002 .001 2.499 .013 Difference 1 MADRID SE GENERAL - PRICE INDEX-Model_3 MADRID SE GENERAL - PRICE INDEX No Transformation Constant 1.518 .813 1.867 .062 Difference 1 Residual ACF Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 FTSE 100 - PRICE INDEX-Model_1 ACF -.003 .021 -.038 -.055 .005 -.009 -.038 .003 -.003 .045 -.043 .035 .076 -.084 -.018 .014 .018 .032 -.018 -.055 .018 .088 -.026 -.012 SE .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 OMX COPENHAGEN (OMXC20) - PRICE INDEX-Model_2 ACF .030 .071 .055 -.078 -.040 .043 -.079 .024 .002 .027 -.034 -.011 .106 -.023 .009 .028 -.014 .031 .051 -.044 .037 .136 .002 .064 SE .042 .042 .042 .042 .042 .042 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .044 .044 .044 .044 .044 MADRID SE GENERAL - PRICE INDEX-Model_3 ACF .083 .052 .003 .030 -.010 .043 -.075 -.062 -.024 .053 -.040 .004 .007 -.035 .027 .074 -.042 .042 -.069 -.061 -.016 .028 -.087 -.079 SE .042 .042 .042 .042 .042 .042 .042 .042 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .043 .044 .044 .044 .044 Residual PACF Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 FTSE 100 - PRICE INDEX-Model_1 PACF -.003 .021 -.038 -.055 .007 -.008 -.043 .001 -.001 .041 -.047 .034 .081 -.087 -.024 .031 .024 .014 -.015 -.049 .020 .088 -.037 -.006 SE .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 OMX COPENHAGEN (OMXC20) - PRICE INDEX-Model_2 PACF .030 .070 .051 -.087 -.044 .056 -.067 .019 .000 .039 -.047 -.019 .123 -.027 -.007 .012 .014 .027 .034 -.028 .025 .143 -.005 .045 SE .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 MADRID SE GENERAL - PRICE INDEX-Model_3 PACF .083 .045 -.005 .028 -.015 .043 -.082 -.055 -.007 .059 -.043 .005 .017 -.042 .029 .062 -.047 .046 -.081 -.057 .004 .024 -.075 -.065 SE .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 .042 The autocorrelation function ACF in the aforementioned tables reveal that the FTSE stock index is not consistent with the other two stock indices. Furthermore, it is also clear that the FTSE index is not only demonstrating a unique residual characteristic but also has a higher level of significance which is clear from the Analysis of Variance tables (ANOVA) above. This further justifies the initial argument in the test for unit rot that the FTSE stock index is not consistent with the other two stock indices. Furthermore, the model results have also revealed that the residual autocorrelation function analysis on the stock indices is an effective method of analysing the variance of mean in the series. References Chiarella, C., Semmlery, W., Mittnikz, S., Zhux, P. (2001). Stock Market, Interest Rate and Output: A Model and Estimation for US Time Series Data [http://www.newschool.edu/gf/econ/syllabi/econ6203/Time-series-data.pdf] Chou, R. Y. (2005). Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model. Journal of Money, Credit & Banking, 37(3): 561-580 Ghysels, E., Granger, C. W. J., Swanson, N. R., Watson, M. W. (2001). Essays in Econometrics: Spectral Analysis, Seasonality, Nonlinearity, Methodology, and Forecasting. Volume: 1. Cambridge University Press. Cambridge Granger, C. W. J. (1991). Modelling Economic Series: Readings in Econometric Methodology. Clarendon Press, Oxford Guido, R., Walsh, K. (2001). Equity Market Valuation: Assessing the Adequacy of Value Measures to Predict Index Returns. Australian Journal of Management, 26(2): 163-189 Nelson, C. R., Piger, J., Zivot, E. (2001). Markov Regime Switching and Unit-Root Tests. Journal of Business & Economic Statistics, 19(4): 404-424 Pelaez, R. F. (1999). Net Discount Rate and Below-Market Discount Rate: Two Methods in Forensic Economics. Journal of Forensic Economics, 12(3): 225-235 Tse, Y. K., Tsui, A. K. C. (2002). A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-Varying Correlations. Journal of Business & Economic Statistics, 20(3): 351-373 Granger, C. W. J. (1991). Modelling Economic Series: Readings in Econometric Methodology. 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