## Modeling the time series – the spread of IBEX 35 # Modeling the time series – the spread of IBEX 35

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# 1. Introduction

The IBEX 35 is the stock index of Spanish stock exchange market. As a time series, if it is not stationary it would induce spurious regression when applied for further analysis (Phillips, 1986). In this report, the IBEX 35 spread series would be applied by time series to check whether the data series is stationary or not, and whether there is autoregressive conditional heteroscedasticity or not. Then a suitable model would be built to prepare the series for further analysis.

# 2. Stationary tests

The stationary test is firstly applied. If the data is stationary, there would not exist unit root for it. The graphic test and the ADF unit root test are among the major ways to test for stationary.

# 2.1 Stationary test using the graphic test

A series is tested first for stationery as a way of avoiding false regression problems. As noted above, the two common method of checking for stationary are the ADF unit root test and the graphic test.

# 2.1.1 The Graphic test – for stationery

Before applying the Graphic test, an overview of the IBEX 35 daily spread during the past 20 years would be shown in the figure 1 below. As shown in the figure, it is indicated that there might not exist stationary issue because there is no more than one trend; however, the volatility clustering problem shown indicates the possible existence of the autoregressive conditional heteroscedasticity problem.

The graphic tests can be used to test for the stationary include the ACF graph, PACF graph and the correlation graph. From the ACF graph in the figure 2, the autocorrelation of the IBEX 35 spread series fluctuates around 0. While the PACF graph in the figure 3 shows that the partial autocorrelation also fluctuates around 0. Moreover, the table 1, the corrgram table, shows that the values of p are greater than 5% which is the significant level. Thusly, the IBEX 35 spread series is stationary.

# 2.1.2 The ADF unit root test – for the stationary of the time series

Despite graphic tests, ADF unit root test can be used to check the stationary effect of a time series. The null hypothesis of the ADF unit root test is to assume that the time series tested has a “unit root”, or “it is not stationary” (Perron, 1988). The ADF unit root test result can be shown in the table 2 below. From the table, it can be figured out that the null hypothesis set for the ADF unit root test is not accepted since the test produces value of p for the z statistics of 5% which is lower than the significant level (5%). Therefore, the IBEX 35 data series is stationary.

# 2.1.3 Evaluation

In this part, the IBEX 35 spread series is tested to be stationary by proof from the graphic test and the ADF unit root test.

# 2.2  A Test for ARCH effect

The ACF graph and the PACF shown in the figure 2 and figure 3 indicates a possible existence of the autocorrelation conditional heteroscedasticity because of the volatility clustering phenomena. In this part, the ARCH – LM test is imperative in checking for the ARCH effect.

# 2.2.1 The ARCH – LM test for the ARCH effect

To apply the ARCH – LM test, the spread series should be run a regress on a constant. Further the ARCH – LM test may be used remaining errors obtained from the regression. The null hypothesis of the ARCM – LM test infers that the series tested has no ARCH effect. As depicted in table 4 below, the result indicates the value of p as 0.0000, which is much lower than 5 % (which is supposed to be the significance level). Thusly, the null hypothesis should not be accepted. As a result, there exists ARCH effect for the IBEX 35 spread series.

# 2.2.2 Evaluation

In this part, the ARCH – LM test indicates an existence of the ARCH effect for the spread series.

# 3. Further suggested modelling

From the analysis above, the IBEX 35 spread series is stationaryhowever with ARCF effect. To be used for further analysis, a GARCH model should be built on the spread series to eliminate the ARCH effect (Lamoureux, 1990).

Different GARCH model with different parameters of p and q are shown below in tables. And similarly the AIC and BIC are depicted in the tables followed. The information criteria suggests that models with the lowest AIC and BIC value but highest LL value are best models to be chosen. In all, comparing all AIC and BIC for all models listed, it turns out that the model GARCH (1/2, 1) is the best model. It has the lowest AIC of -30217.29, and lowest BIC of -30184.66. Looking at the model itself, the coefficients are not significant indicated by larger than 5% significance level. Therefore, the model GARCH (1, 1) should be chosen as the best model with the second lowest AIC or BIC and significant coefficients. Therefore, the GARCH (1, 1) model is a best model in this case. As a result, the following formula can illustrate the model more straightforward.