In this FMBF essay, three equity market price indices, including FTSE350, TOPIX and OMX HELSINKI (OMXH), will be studied based on time series. For the accuracy of conclusion, this essay will use last decade data because latest available information and large sample can contribute to the precision of the study. Because the tradeoff between significant ARCH effects with very high order from daily data and low order but less significant ARCH effects due to the monthly data, the frequency of weekly data is more appropriate. In addition, closing data on Friday can avoid some evitable bias and result in higher reliability of the information. Therefore, the weekly data from Jan 2th 1998 to April 10th 2009 downloaded from DataStream will be used in this report. Instead of themselves, the natural logs of FTSE350, TOPIX and OMX HELSINKI (OMXH) equity market price indices will be used in this essay. This report will first test the unit root of three stock markets use Augmented Dickey-Fuller (ADF) test based on perron’s testing procedure, and then it will find out the appropriate univariate model for estimating equity market returns of the three. After that, EG approach will used for testing cointegration between three indices. Where after, it will test whether there are the ARCH effects exist in the OMXH index, and re-estimate the appropriate univariate model mentioned above for OMXH index with a GARCH model. And, this report will use the last 15 observations to forecast the conditional mean and the conditional variance. Finally, it will critically compare the performance of the forecasts of different GARCH processes (eg, EGARCH, h_t GARCH) under same model.

Keeping the stationary of variables is an important condition of regression. When the data is not stationary, there is one or some unit roots exist in time series data. The presence of unit root will cause the problem of spurious regression. Usually, people can use one of two methods to test unit roots, either Augmented Dickey-Fuller (ADF) test approach (Said and Dickey, 1984), which developed from Dickey-Fuller (DF) approach (Dickey and Fuller, 1979), or Philip-Perron test created by Perron in 1998. The Perron’s testing procedure increase restriction step by step, so that eliminate unnecessary parameters like trend and constant (these reductions of parameters will change the critical values of testing and reduce bias) and finally get precise conclusions (Harris and Sollis, 2003).

C:DOCUME~1ADMINI~1LOCALS~1Temp7QWX`ZNVV)$8DZW6Y7_RD{P.jpgSource: Harris and Sollis, 2003 In this essay, it will use ADF test by using Perron’s testing procedure to reduce bias. The only different between DF test and ADF test is the regressions of latter one have a new parameterA“AA¢ iA”žyt-1.

C:DOCUME~1ADMINI~1LOCALS~1Temp@YW8X7BH]J9K_61XW(E6)S3.jpg Source: Harris and Sollis, 2003 At first step, unit root will be tested based on . This essay will choose the model with appropriate number of lags based on lowest Akaike’s Information Criterion (AIC) values for three indices. Then, compare t values of selected model with critical value at 95% significant level. From the results given by Pc Give, the null hypothesis of unit root cannot be rejected, the test process should continue. The appropriate number of lags is 7 for FTSE 350 and OMXH, 2 for TOPIX. Table 2. Step 1 in Perron’s testing procedure

(1) t-value -1.336 -0.9436 -2.031 (table) 5% critical value -3.42 Accept Accept Accept At second step, this report will do OLS by using regression used in first step of procedure and regression without trend. Based on F statistic value and of two regressions, we can found that the later one is better. All three indices cannot reject null hypothesis and move on to next step. Table 3. Step 2 in Perron’s testing procedure

Contents

The purpose of this part is to identify an appropriate univariate model to estimate equity market returns in time series. This report will use Box-Jenkins methodology (Box and Jenkins, 1970) for time series model building. The procedure includes three stages: identification, model estimation and diagnostic checking. The first step is identification: the first order difference of the nature log of PI is market return. Because only stationary data can used in ARMA time series models, before ARMA model estimation, this essay will use ADF test to check the unit roots of market returns. The ACF and PACF diagrams may help to find appropriate ARMA model. The second step is model estimation: we should pick up the most appropriate model from a number of set possible ARMA models estimated by using Pc Give. In this stage, we should compare p-values and choose the model that most coefficients of parameters are significant from zero. The third step is diagnostic checking, it will test the efficiency of ARMA model selected in the second stage. It should test whether both additional lags are significant or not. According to Harris and Sollis (2003), only all additional parameters are insignificant, the original ARMA(p,q) model is efficient. Moreover, the residuals should be white noise for an adequate model. In this report, AIC will be used as one criteria to select appropriate model.

Identification According to the results of ADF test from Pc Give, market returns of FTSE350 are stationary. The actual series, ACF and PACF diagrams prove the results from ADF test that the FTSE 350 market return has no unit root at all. Table 7. ADF test on market return of FTSE350 index FTSE350 t-value -9.526 5% critical value -3.42

Figure 2. Actual series and ACF, PACF of DLFTSE350 GiveWin Graphics 1.emf Estimation Compared with other models, ARMA (1, 1) seems to be appropriate. Based on p-values, most coefficients are significant. And residuals are white noise. DLFTSE350t=-0.000257715–0.913851DLFTSE350t-1+0.854708 Diagnostic Overfitting Information from table indicated that ARMA (1, 1) is the correct one. The additional parameters AR2 and MA2 in ARMA (2, 1) and ARMA (1, 2) are insignificant. Both ARMA (2, 1) and ARMA (1, 2) should be rejected. Table 8. Overfitting test on ARMA (1,1) model Model Statistic t-value p-value result ARMA(2,1) AR_2 0.503 0.615 insignificant ARMA(1,2) MA_2 0.503 0.615 Insignificant Residual analysis Under Ljung-Box test statistic (Qk), Chi^2(21) = 28.353 [0.1304] in model ARMA (1, 1) provides evidence that residuals are white noise and independent. Figure 3. Actual series, ACF and PACF of DLFTSE350’s residuals GiveWin Graphicsftse350.emf AIC Value When compared the AIC values of ARMA (1, 1), ARMA (2, 1) and ARMA (1, 2), we can found that ARMA (1, 1) is the model with lowest AIC value and should be selected. It is consistent with the results above. Table 9. AIC of ARMA (p, q) ARMA(1,1) ARMA(2,1) ARMA(1,2) AIC -4.52046242 -4.5174036 -4.51740249

Identification The results from Pc Give shows that market return series of TOPIX is stationary. Figure 4 provides the actual series, ACF and PACF diagrams from Pc Give. All information is consistent with the idea of no unit-root in TOPIX market return. Table 10. ADF test on market return of TOPIX index

t-value -16.14 5% critical value -3.42

Figure 4. Actual series and ACF and PACF of DLTOPIX GiveWin Graphics2.emf Estimation We can found that ARMA (1, 1) is the best one in ARMA models. Most of coefficients of parameters in ARMA (1, 1) are significant from zero based on P values. DLTOPIXt=-0.000570574-0.530279 DLTOPIXt-1+0.448276 Diagnostic Overfitting The p-values of both additional parameters AR_2 and MA_2 are greater than 0.05, it means that the coefficients of AR_2 and MA_2 are not significant at 95% significant level. So we should reject ARMA (2, 1) and ARMA (1, 2) in favor of ARMA (1, 1). Table 11. Overfitting test on ARMA (1,1) model Model Statistic t-value p-value result ARMA(2,1) AR_2 1.05 0.296 insignificant ARMA(1,2) MA_2 -0.00 1.000 insignificant Residual analysis From Portmanteau(23): Chi^2(21)= 21.585 [0.4238] in ARMA (1, 1) model, we can find that there is no serial correlation between residuals and they are white noise. None of bar chart in ACF or PACF exceeds standard error bounds. Figure 5. Actual series, ACF, PACF of DLTOPIX’s residuals GiveWin Graphicstopix.emf AIC value The result of AIC is same as the outcome mentined above. Based o AIC, we can found that ARMA (1, 1) is the best model with lowest AIC value. Table 12. AIC of ARMA (p, q) ARMA(1,1) ARMA(2,1) ARMA(1,2) AIC -4.24323212 -4.24138739 –4.23974138

Identification Market return of OMXH is stationary from ADF test. And the figure 6 shows the actual series, ACF and PACF of DLOMXH by using Pc Give. It provide positive evidence that there is no unit root in DLOMXH (market return) Table 13. ADF test on market return of OMXH index

t-value -7.920 5% critical value -3.42

Figure 6. Actual series, ACF, PACF of DLOMXH’s residuals GiveWin Graphics3.emf Estimation Based on results from Pc Give, we can found that the most appropriate model is ARMA (2, 2) for OMX HELSINKI (OMXH). Most coefficients of parameters are significant from zero at 95% significant level with evidence that most P values of them are less than 0.05. DLOMXHt=0.000804126+1.52249DLOMXHt-1-0.662672DLOMXHt-2-1.53479 +0.708503 Diagnostic Overfitting Looking at p-values, we can found that ARMA (3, 2) and ARMA (2, 3) should be rejected because the coefficients of additional parameters AR_3 and MA_3 are both insignificant from zero. Table 14. Overfitting test on ARMA (2,2) model Model Statistic t-value p-value result ARMA(3,2) AR_3 0.932 0.352 insignificant ARMA(2,3) MA_3 0.604 0.546 insignificant Residual analysis We can see that Portmanteau (23): Chi^2(19) = 28.866 [0.0681]. When translating numerical result to written words, it means residuals are not serial correlated and close to white noise. ACF and PACF are within two standard error bounds. Figure 7.Actual series, ACF, PACF of DLMOXH’s residuals GiveWin Graphicsresidualomxh.emf AIC value Under the method that chooses the model with lowest AIC value, still ARMA (2, 2) should be the best model. Table 16. AIC of ARMA (p, q) ARMA(2,2) ARMA(3,2) ARMA(2,3) AIC -3.34659081 -3.33848893 -3.34351931

There are two common methodologies on cointegration test: Engle-Granger approach and Johansen approach. EG approach can but can only adopted in single equation with two variables, and Johansen approach is used in multivariate system. In this report, it will pursue EG approach. Firstly, it will outline the procedure of EG approach. And then, it will test the cointegration between each two indices. In theory, C:DOCUME~1ADMINI~1LOCALS~1TempKXB{][email protected]/* */)W.jpgis the only static model that necessary to be estimated to exam the long-run relationship between Yt and Xt. Before do the cointegration test, the order of integration of each other index should be test. If all data are belong to I(0), there will no cointegtaion between them; if variables are belong to I(1), it estimate the model mentioned above by using OLS regression. In the second step, it will test unit roots in the residuals with ADF test. When there is no cointegration between two variables, residual is belongs to I(1). If the residuals is belongs to I(0), it means that cointegration relationship exist in that single equation. What is important here is the critical values used by ADF are calculated by MacKinnon (1991). Critical value can be calculated by following the model C:DOCUME~1ADMINI~1LOCALS~1Temp641N%ER][email protected]/* */[DB.jpg. It can get the appropriate critical value for test including residuals from OLS equation if the number of regressors (constant and trend are not included) from 1 to 6. If the cointegration existed in long-run relationship between two variables, we should estimate the simple dynamic model of short-run adjustments. In this report, it will use equilibrium correlation model (ECM) of dynamic model. In this step, the speed of adjustment for regression will be obtained. (Harris and Sollis, 2003)

Three indices are all integrated of order 1, or we can call I (1), because the natural logs of them are not stationary and the first order difference of the natural logs of them are stationary.

Firstly, it regress LFTSE350 on LTOPIX and a constant by using OLS regression. Then, save the cointegration regression residual in database as residual1. After that, test residual1 with ADF. We can see that the number of regressors is 1 (excluding constant). We choose model based on the idea of the lowest AIC value. From the table of ADF test, residual 1 is stationary and thus variables are cointegrated. Also, figure 8 from Pc Give shows persistent but stationary residuals. In short, these two indices are cointegrated. Constant and no trend, n=1 C(p)=-2.8621-2.738/563-8.36/563^2= = -2.86699 Table 16. ADF test on residual 1

t-value -3.181 5% critical value -2.86699

Figure 8. Residuals from Cointegration regression between FTSE350 and TOPIX GiveWin GraphicsEG1.emf ECM The coefficient of residual provides information on the speed of adjustment. The number 0.0448954, which is significant from zero, means the adjustment for the regression. (See Appendix)

Regress LTOPIX on LOMXH and a constant by using OLS. Then, save the cointegration regression residual in database as residual2. After that, use ADF to test residual. The number of regressors is 1 (excluding constant). We choose model based on the idea of the lowest AIC value and the t-value is-2.622. We found out that -2.86699 is less -2.622, the residual2 is non-stationary and two indices are not cointegrated. In addition, from graph, the residual2 provides clear information about stationary or non-stationary of data. In conclusion, there is no cointegration between TOPIX and OMXH. Constant and no trend, n=1 C(p)=-2.8621-2.738/563-8.36/563^2= = -2.86699 Table 17. ADF test on residual 2

t-value -2.622 5% critical value -2.86699

Figure 9. Residuals from Cointegration regression between TOPIX and OMXH GiveWin GraphicsEG2.emf

Regress LFTSE on LOMXH and a constant by using OLS. Then, save the cointegration regression residual in database as residual3. After that, use ADF to test residual3. The number of regressors is 1 (excluding constant). From table of ADF test on residual 3, two indices are not cointegraed because the residual3 is non-stationary. From figure 10, it does not provide strong evidence for stationary. So, these two indices are not cointegrated. Constant and no trend, n=1 C(p)=-2.8621-2.738/563-8.36/563^2= = -2.86699 Table 18. ADF test on residual 3

t-value -2.278 5% critical value -2.86699

Figure 10. Residuals from Cointegration regression between FTSE350 and OMXH GiveWin GraphicsEG3.emf

This report will only discuss the OMXH stock market index in this section. Due to the limitation of Pc Give, it cannot deal with ARMA model of DLOMXH directly, but change it to AR model at first. Compared with other AR models, the AR (7) model is the best. DLOMXHt=0.000839271-0.00444882DLOMXHt-1+0.0369747DLOMXHt-2+ 0.0307856DLOMXHt-3+0.0258150DLOMXHt-4+0.141961DLOMXHt-5 +0.0693818 DLOMXHt-6 -0.0841123 DLOMXHt-7 Under Ljung-Box test statistic (Qk), Portmanteau (23): Chi^2(16)=15.328 [0.5007] in AR (7) indicates that residuals are independent and white noise. The ACF and PACF support the above result. None of bar chart is passing beyond the significant lines. Figure 11. ACF, PACF of AR (7) model on DLOMXH GiveWin Graphicsomxh ar.emf Before the formal test on ARCH effects in residuals of DLOMXH, the ACF and PACF graphs of squared residuals of DLOMXH provide evidences that ARCH effects exist in DLOMXH residuals. Figure 12. Squared residuals’ actual series, ACF and PACF GiveWin Graphicsomxh42.emf After the informal test of ARCH effects in residuals of DLOMXH based on ACF and PACF, it will test ARCH effects more academic. This report use Pc Give to test ARCH effect in lower order and higher order, and the results point out that there are ARCH effects in lower and higher order. ARCH 1-1 test: F(1,566) = 12.071 [0.0006]** ARCH 1-6 test: F(6,556) = 4.8959 [0.0001]** After ARCH effect test, the report tries to estimate an appropriate ARCH model. In this stage, it selects Volatility Models and chooses 7 as the lags of length. From ARCH1 to ARCH8, we can found that the Chi^2 becomes to be better and better, while the p-value of parameters are becomes to be worse and worse. None of the ARCH model can fit the data well.

According to Brooks (2008), GARCH models can overcome some problems faced in the process of using ARCH models. GARCH (p, q) model with p and q values can fit the data better than ARCH (m) model with a high m value. It still uses Volatility Model with 7 lags as ARCH model used. At first, it gets the numerical information of GARCH (1, 1) model, GARCH (2, 1) model and GARCH (1, 2) from Volatility package of Pc Give. When checking the F distribution values and Chi^2 values of three models, it can found that all are acceptable (values are significant from zero). Unfortunately, there are some data missing related to parameters in all three models. So, it cannot compare p values of parameters to choose best model. Instead, it will use information includes results of ARCH tests, Portmanteau statistic of squared scaled residuals, log-likelihood and AIC values to find a better one. Table19.Compareable information of GARCH(p, q) GARCH(1,1) GARCH(2,1) GARCH(1,2) AIC -3.59989604 -3.60415615 -3.59636247 log-likelihood 1029.77058 1031.97619 1029.77058 Portmanteau(23): Chi^2(21)=16.696 [0.7293] Chi^2(20)=14.329 [0.8135] Chi^2(20)= 16.696 [0.6726] ARCH 1-2 test: F(2,551)=0.10648 [0.8990] F(3,548)=0.075153 [0.9734] F(3,548) = 0.15031 [0.9295] Compared with other GARCH models, we can find that GARCH (2, 1) model is better; even neither model can fit data very well. GARCH (2, 1) is the model with lowest AIC value. The highest log-likelihood value in this model indicates that it can fit data best in the three. All of three Chi^2 are significant from zero, which means there are no relationships between residuals and they are white noise. The numbers in brackets are p values of Chi^2. The higher the value the greater evidence of white noise of squared scaled residuals. The ARCH effect test F distribution values are related to whether there are ARCH effects in squared scaled residuals. Similar with the Portmanteau statistic, a higher number in bracket means greater support of none ARCH effects in squared scaled residuals.

In this section, the last 15 observations will used to forecast the conditional mean and variances. SD (Standard Derivation Error) RMSE (Root Mean Square Error) and MAPE (Mean Absolute Percentage Error) will adopted to evaluate the accuracy of forecasting based on different GARCH models. A model with better data fitting is always has lower SD (Error). A low MSE indicates a better fit of GARCH model on time series data. MAPE is another indicator of accuracy on data fitting of GARCH model. When the value decreases, the model can fit the data better. In addition, the model with lower MAPE can predict the trend of data. The power to measure the data fitting of models is increasing from SD (Error), RMSE to MAPE. EGARCH (2, 1), GJR GARCH (2, 1) and H_TGARCH (2, 1) models were established to evaluate the efficiency of GARCH (2, 1). All can pass the processes of ARCH effect tests and white noise tests. Because GARCH model used to measure volatility of data, the ability of forecasting on conditional mean is limited, and the key capability is volatility measurement. All models are doing badly in conditional mean forecasting: the predict graph can provide very limited information about conditional means that all are close to zero in the forecasting period and. When compared with forecasting capability, we can see that GJR GARCH (2, 1) is the model with lowest MAPE and second small RMSE. It indicates that GJR GARCH (2, 1) has strongest power for forecast. Table 20. RMSE and MAPE SD (Error) RMSE MAPE GARCH(2,1) 0.055276 0.055772 91.742 EGARCH(2,1) 0.055429 0.055814 91.422 GJR GARCH(2,1) 0.055220 0.055645 90.988 H_TGARACH(2,1) 0.055236 0.055396 94.793 Figure 13. GARCH(2, 1) forecast on conditional mean and variance THGARCH(2,1).emf Figure 14. EGARCH(2, 1) forecast on conditional mean and variance EGARCH(2,1).emf Figure 15. GJR GARCH(2, 1) forecast on conditional mean and variance th2,1.emf Figure 16. h_tGARCH (2, 1) forecast on conditional mean and variance GARCH(2,1)FORECAST.emf

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