Arch Garch Modeling And Forecasting Computer Science Essay

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Time series model is built to predict trend for forecasting (Harris, 2003). It is claimed that models employing higher frequency data gain better forecasting power [Bollerslev (1986); Andersen et al.(2001); Pasley and Austin (2003); Engle and Gallo (2006); Xiao and Aydemir (2007)]. In this report, weekly data is employed. Although monthly data could show more stationarity characteristic, it can hardly capture the volatility feature. Comparably, although daily data reflects the volatility clustering property of time series [Nelson (1992); Tashman (2002); Shively (2003); Lu and Perron (2010); Christensen et al. (2010)], it may easily loss the general stationarity property (Brooks, 2008). Therefore, weekly data is used as a compromise between balancing the long-term stationarity property and short-term volatility property. Gujarati (2003) claims that the forecasting power depends on the time span of the predicted period. If the predicted period is too large, the variance would be small, thus the forecasting power will be weak correspondingly. The predicted t period is 15 in this report. Large sample size is crucial since some empirical failures are caused by small sample size problem (Campbell and Yogo, 2006). Data sample in this report is ranging from 10th March 1995 to 12th March 2010 (784 observations) across 15 years. 15 observations are left out for forecasting. Due to the relatively large size of the sample data and starting month (March), January effect is faded away. Meanwhile, since 10th March 1995 is Friday, therefore the Monday effect also is ruled out.

Dow Jones Industrial Price Index

NZX Top 10 Price Index

SBF 120 Price Index

Both Dow Jones Industrial and SBF 120 index have positive market returns. Moreover, Dow Jones index generates higher return with bearing lower risk than SBF 120 index. Comparably, NZX Top 10 generates negative returns, whereas, it has lowest standard deviation which implies its lowest volatility. The negative skewness indicates that the market risks of three indexes are all underestimated. Meanwhile, the normality test shows that three time series are not normally distributed.

Part Two: Unit Root Testing

Dickey-Fuller test is applied for unit root testing. However, it is difficult to identify explanatory variables when using DF test. Mis-identified variables may lead to type I or type II error. In order to avoid this problem, Peter and Perron (1988) provide a "road map" for DF test. Specifically, Augmented Dickey-Fuller (ADF) test is used to solve the problem "how many lags should be set" in the unit root testing.

Prior to constructing the models, from actual serial graphs, three indexes seem to be non-stationary.

The graphical analysis shows that these series need differencing once to be stationary.

Step 1: unit root testing with the null hypothesis using ADF test with lag of 6. The choose criteria is that finding lag with the smallest AIC, if AICs equal, the smallest lag is chosen (Enders, 2010). (Appendix 1)

The choose criteria finds lag 2 for Dow Jones, lag 5 for NZX, lag 0 for SBF. Noticeably, the correspondingly t-adfs of all indexes in the table are all larger than the critical value (-3.42), thus the null hypothesis does not be rejected, which means unit root exists.

Step 2: test the joint null hypothesis based on OLS estimation (Appendix 2).

F= 9.5789 for Dow Jones, F= 6.3066 for NZX Top 10, both of them are rejected (both are larger than = 6.25 at 5% significant level) [1] , which means the "null hypothesis" needing to be tested by using normal distribution. Comparably, F=4.0969 for SBF 120 is not rejected, which means there is a unit root and no trend for SBF 120.

Step 2.1 is to test the null hypothesis based on OLS estimation for Dow Jones and NZX Top 10, who is rejected in the previous step (Appendix 3). The t-value= -2.57 for Dow Jones, which is rejected (is smaller than normal critical -1.96 at 5% significant level). The rejection of Dow Jones means there is no unit root for Dow Jones Industrial. The t-value= -1.91 for NZX Top 10, which is not rejected (is larger than normal critical -1.96 at 5% significant level). NZX Top 10 has a unit root.

Step 3 is to continually test unit root for SBF 120 using ADF test (Appendix 4)

Since the t-adf=-2.320 for SBF 120 is larger than the critical value -2.87, the null hypothesis "unit root existing" cannot be rejected, which means SBF 120 needs further testing.

Step 4 is to test for the drift's coefficient (i.e. constant) for SBF 120 (Appendix 5). The F=6.3367 of SBF 120 is larger than =4.59 at 5% significant level [2] , therefore, the null hypothesis is rejected, which means the "null hypothesis" needing to be tested by using normal distribution.

Step 4.1 is to test the null hypothesis based on OLS estimation for SBF 120 (Appendix 6). The t-value= -2.32 for SBF 120, which is rejected (is smaller than normal critical -1.96 at 5% significant level). The rejection means there is no unit root for SBF 120. The whole Perron's procedure stops.

The result shows that both Dow Jones Industrial and SBF 120 do not have a unit root respectively. This result is consistent with index random walk theory opponents [Poterba and Summers (1988); Lo and MacKinlay (1988); Chaudhuri and Wu (2003)]. NZX Top 10 has unit root which is consistent with index random walk proponents [Fama and French (1988); Nakamura and Small (2007)]

Part Three: Identifying ARMA Model

ARMA model identification contains 3 steps in this report (Box-Jenkins methodology). Although Box-Jenkins methodology is not an automatically procedure [Anderson (1977); Tang et al. (1991);], it also cannot capture seasonality of the data (Brooks, 2008), this method is still a useful, especially dealing with small sample data (Lu and AbouRizk, 2009).

Dow Jones Industrial Index Return

Stage 1: Identification

Prior to ARMA testing, the series (i.e. Dlog Dow Jones) must be stationary (Brooks, 2008).

Actual series graph indicates that the return of Dow Jones is stationary. Moreover, through ADF test, the t-adf are all smaller than critical value (-3.42) (Appendix 7), which means the null hypothesis is rejected, i.e. Dlog Dow Jones is stationary. The ACF and PACF graph suggests the ARMA models could be AR (7), MA (7) or ARMA (7, 7).

Stage 2: Estimation

Some trial models AR (7, 0), MA (0, 7), ARMA (7, 7) are set to compare through Maximum Likelihood Estimation. AR (7, 0) has one significant coefficient with AIC = -4.51852. MA (0, 7) has one significant coefficient with AIC = -4.51815. ARMA (7, 7) has no significant coefficient (Appendix 8.1). Therefore, AR (7, 0) is comparably better.

Stage 3: Diagnostic Checking

In order to test the adequacy of the model, the overfitting process is needed. Since AR (8, 0) and ARMA (8, 7) are all not better than AR (7, 0) (Appendix 8.1), it is concluded that ARMA (7, 0) is the appropriate model for Dlog Dow Jones. The next step is to do Residual Correlogram and Portmanteau statistic test, in order to test whether the residuals are white noise under ARMA (7, 0).

Residual Correlogram shows the residuals are non-autocorrelated. The ACF lag in Q-test is usually set as one-quarter of the total length of the time series (Gujarati, 2003). In this report, Ljung-Box Q test has lag 192: one-quarter of 768 observations.

The Portmanteau statistic for residuals is insignificant, which means the residual is white noise.

NZX Top 10 Index Return

Stage1: Identification

Actual series graph indicates that the return of NZX Top 10 is stationary.

Moreover, through ADF test, the t-adf are all smaller than critical value (-3.42), which means the null hypothesis is rejected, i.e. Dlog NZX is stationary (Appendix 7). The ACF and PACF graph suggests AR (5), MA (5) or ARMA (5.5).

Stage 2: Estimation

Some trial models AR (5, 0), MA (0, 5), ARMA (5, 5) are set to compare through Maximum Likelihood Estimation. AR (5, 0) has one significant coefficient with AIC = -4.75681. MA (0, 5) has two significant coefficient with AIC = -4.75815. ARMA (5, 5) has two significant coefficients with AIC= -4.751133. Therefore, MA (0, 5) is comparably better (Appendix 8.2).

Stage3: Diagnostic Checking

After testing MA (0, 6), ARMA (5, 6), and ARMA (6, 6), it is concluded that ARMA (5, 6) has most significant coefficient (three) with the lowest AIC= -4.761095(Appendix 8.2). ARMA (5, 6) is the appropriate model for Dlog NZX. The next step is to do Residual Correlogram and Portmanteau statistic test, in order to test whether the residuals are white noise under ARMA (5, 6).

Residual Correlogram shows the residuals are non-autocorrelated.

The Portmanteau statistic for residuals is insignificant, which means the residual is white noise.

SBF 120 Index Return

Stage 1: Identification

Actual series graph indicates that the return of SBF 120 is stationary.

Moreover, through ADF test, the t-adf are all smaller than critical value (-3.42), which means the null hypothesis is rejected, i.e. Dlog SBF is stationary (Appendix 7). The ACF and PACF graph suggests AR (6), MA (6) or ARMA (6, 6).

Stage 2: Estimation

Some trial models AR (6, 0), MA (0, 6), ARMA (6, 6) are set to compare through Maximum Likelihood Estimation. AR (6, 0) has one significant coefficient with AIC = -4.171741. MA (0, 6) has two significant coefficient with AIC = -4.171842. ARMA (6, 6) has seven significant coefficients with AIC= -4.198029. Therefore, ARMA (6, 6) is comparably better (Appendix 8.3).

Stage3: Diagnostic Checking

After testing ARMA (6, 7), ARMA (7, 7), it is concluded that ARMA (6, 7) has most significant coefficient (ten) with the lowest AIC= -4.198862 (Appendix 8.3). ARMA (6, 7) is the appropriate model for Dlog SBF. The next step is to do Residual Correlogram and Portmanteau statistic test, in order to test whether the residuals are white noise under ARMA (6, 7).

Residual Correlogram shows the residuals are non-autocorrelated.

The Portmanteau statistic for residuals is insignificant, which means the residual is white noise.

Part Four: Cointegration test

The Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several time series. This test does not require all variables to be in the same order of integration, and hence this test is much more convenient than the Engle-Granger test for unit roots which is based on the Dickey-Fuller (or the augmented) test. In this report, since three series need to be tested for the cointegration, Johansen procedure is chosen.

From the actual series graph, three indexes seem to show a similar trend.

Step one:

Conclusion from previous parts indicates that both ln Dow Jones and ln SBF 120 can be I (0) and I (1). Ln NZX is I (1). Meanwhile, the lag for ln Dow Jones is 2, for ln NZX is 5, and for ln SBF 120 is 0 (from Part 1). Therefore, the lag=5 in VAR process.

Step two:

From graph analysis in PcGive, the cross fitted values of three series shows good fitness. The ACF and PACF demonstrate that the residual of three series has no serial correlation, which follows white noise.

Dow Jones

NZX Top 10

SBF 120

The F-test indicates that the 5th lag is insignificant in all three series (Appendix 9). Nevertheless, after delete the 5th lag, the F-test statistics is significant F(9,1815)= 1.2869 [0.2388], from Appendix 10.

Testing for Vector error autocorrelation from lags 1 to 5

Chi^2(45)=74.797 [0.0035]** and F-form F(45,2172)=1.6529 [0.0043]**

Vector Normality test for Residuals

Vector Normality test: Chi^2(6) =362.19 [0.0000]**

Testing for Vector heteroscedasticity using squares

Chi^2(180)=396.64 [0.0000]** and F-form F(180,4208)=2.2914 [0.0000]**

It shows that residuals are serial correlated (autocorrelation), and not normally distributed, thus have heteroskedasticity. The Johansen procedure stops, since the residuals are not white noise. Therefore, there is no cointegation among three series.

The underlying reason is three series do not have the same integration (Johansen method does not require series to have same integration). Ln DJ is I (1) or I (0), Ln NZX is I (1), Ln SBF is I (1) or I (0). The exact integration for ln DJ and ln SBF is vague. The Johansen result shows three series do not have the same integration.

Johansen test does not require all series to be in the same order of integration, and hence this test is much more convenient than the Engle-Granger test. Johansen method is better when residuals do not follow normal distribution (Ahking, 2002). Moreover, Johansen approach allows testing the vectors of cointegration, when all variables are all endogenous, OLS is applied (Voronkova, 2004). Nevertheless, Johansen method is argued for it does not capture the trend effect (Perron and Campbell, 1993).

Part Five: ARCH/GARCH Modeling and Forecasting (NZX Top 10 Index used)

DLn NZX is proved as stationary, and its model is found as ARMA (5, 6). First of all, run the OLS regression DLNZX_5 on DLNZX, and save the residuals (Appendix 11). The next step is to run the Lagrange Multiplier Test, i.e. testing the relationship between residuals' squared value and their past squared value.

All the coefficients are significant (i.e. reject the null that individual coefficient is 0), whereas the F-test is insignificant (i.e. the whole regression is reasonable) (Appendix 12), which indicates the ARCH effect.

The first trial GARCH (0, 6) shows that most coefficients are significant. The diagnostic test indicates that there are no autocorrelation and heteroskedasticity problems (Appendix 13).

The second trial model is ARMA (5, 6)-GARCH (1, 1) The estimated coefficients forα₀, α₁ and ß₁ are significant (Appendix 14).

The model is: DLNZXt=0.00078 -0.089DLNZXt-5 + ut, ∂²t=7.77402e-006 + 0.085U²t-1 + 0.903 ∂²t-1

The correspondingly Q test indicating the residuals are white noise.

Forecasting

Apply the ARMA (5, 6)-GARCH (1, 1) model to forecast 15 observations, the conditional mean and variance graphs show:

The third trial model is GARCH (1, 1) The estimated coefficients forα₀, α₁ and ß₁ are significant (Appendix 15).

The model: DLNZXt=0.00071 + ut, ∂²t=7.96858e-006 + 0.087U²t-1 + 0.9∂²t-1

The correspondingly Q test indicating the residuals are white noise.

Forecasting

Apply the GARCH (1, 1) model to forecast 15 observations, the conditional mean and variance graphs show:

Comparison the performance of two GARCH models: this is done by running the regression of actual variance of last 15 observations on the forecast conditional variance. The coefficient of ARMA (5, 6)-GARCH (1, 1) is significant, however, coefficient of GARCH (1, 1) is insignificant (Appendix 16), therefore, ARMA (5, 6)-GARCH (1, 1) performs better on forecasting (i.e. DLNZXt=0.00078 -0.089DLNZXt-5 + ut, ∂²t=7.77402e-006 + 0.085U²t-1 + 0.903 ∂²t-1).

Part Six: Asymmetric GARCH models

Since leverage effect is omitted in GARCH model, the asymmetric GARCH models are applied to detect the asymmetric problem. The result indicates that there is no leverage effect because the estimated coefficient of threshold is insignificant (Appendix 17).

To test the asymmetric problem, is applied. Residuals are saved from previous ARMA (5, 6)-GARCH (1, 1) model. The dummy is set correspondingly. Let 未命名.jpg−as an indicator dummy that takes the value 1 if 2.jpgand zero otherwise. Let 7.jpg otherwise equal to 0. Since all the coefficients are insignificant, it is concluded that there is no size bias and sign bias (Appendix 18).

ARCH/GARCH is used for volatility forecasting. ARCH model allows the variance of residuals to be heteroskedastic, which does work for forecasting in reality since errors in real financial time series are not constant. ARCH also allows the volatility clustering, which is a common feature in real financial markets. GARCH model plays as a complement model for ARCH because it constraints the parameters which increase the adequacy of the forecasting model (Bollerslev, 1987). Nevertheless, since both ARCH and GARCH ignore the leverage effect, asymmetric GARCH is applied for it allowing the leverage effect. Asymmetric GARCH models have been proved to outperform the normal GARCH in volatility forecasting when asymmetric is allowed (Bildirici and Ersin, 2009; Liu and Hung, 2010). The asymmetric effect showed in behavior finance area, for example, small-capitalization stocks illustrate higher volatility in price change than bigger capitalization stocks (Liu at el, 1997). Moreover, Mazouz et al (2008) demonstrate that EMH is mostly reflected through GJR-GARCH rather than OLS.

Summary

This report investigates the series characteristic for Dow Jones Industrial, NZX Top 10, and SBF 120 indexes. Dow Jones Industrial and SBF 120 indexes follow the random walk, while NZX Top 10 Index does not. The ARMA model for DL Dow Jones is ARMA (7, 0), for DL NZX is ARMA (5, 6), for DL SBF is ARMA (6, 7). The Johansen procedure indicates that there is no cointegration among three indexes. ARMA (5,6)-GARCH (1, 1) performs better in volatility forecasting for NZX Top 10 Index, nevertheless, there is no leverage effect, no size bias and no sign bias for NZX Top 10 Index after employing the asymmetric GARCH model, i.e. ARMA (5, 6)-TGARCH(1, 1).

Appendices

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