# How to Analyze the Volatility of stock returns with the (G)ARCH model part V

Final part of the Series.

January 6, 2024

15

min

Final part of the Series.

The following analysis shows whether the AR models have ARCH effects. A suitable test for this is the Lagrange multiplier test, LM test for short. This test can also be perceived as a test for autocorrelation. The hypotheses are as follows:

So if the p-value is less than our alpha, we discard H(0) and we have an autocorrelation in the model and the existence of ARCH effects (Schmelzer 2009: 37). Applying the LM test to the two AR models, AR(9) for the Nasdaq100 and AR(17) for gold, yields the following result:

In both tests, the p-value is virtually zero, which is why we have no conditional heteroscedacity in our model (Lamoureux/Lastrapes 1990: 221). This result shows that a simple AR model is not sufficient for volatility analysis. The data structure of the return series is characterized by heteroscedasticity and volatility clusters. Also, these are not normally distributed, but have a leptokurtic distribution.

Volatility generally describes the fluctuation range of the annualized standard deviation of the observed returns on financial markets (Schmelzer 2009: 43). In order to progress in the analysis of volatility, the application of an sGARCH, i.e. a standard GARCH model, is necessary. As in the AR models, it is also advisable for the GARCH models to use the first 9lags for the Nasdaq100 and the first 17 lags for gold. An sGARCH(1,1) model is used as theGARCH model. An sGARCH(1,1) model makes sense precisely because numerous studies have shown that simple models have proven themselves for financial market data(Bera/Higgens 1993: 317). Figure 2.5 shows the GARCH model for the Nasdaq100 andFigure 2.6 shows the GARCH model for gold. In both GARCH models, the ARCH parameter alpha1 is significantly smaller than the GARCH parameter beta1. Alpha1 is responsible for the extent of the immediate reaction to new messages of the error term, while beta1 describes the duration for the effect to wear off (Schmelzer 2009: 47). Since beta1 is significantly larger than alpha1, the conditional variance of the model can only slowly adjust back to the equilibrium level (Schmelzer 2009: 47). The sum of alpha1 and beta1 is less than one in both models. From this result it can be deduced that a high persistence of past shocks lies in the conditional variance of both models (Schmelzer 2009: 47).

The volatility of both returns returns to equilibrium only slowly (Schmelzer 2009: 47).Figures 2.7 and 2.8 show the empirical density of the standardized residuals of the twoGARCH models. The orange line shows the density of a normal distribution. A significantly increased kurtosis of the GARCH residuals can be seen. Around the zero region, the residuals exceed the normal distribution, while a flattening can be seen at the edge. The kurtosis can not be eradicated in the GARCH models either.

To determine whether there are ARCH effects in the GARCH model, an LM test can also be carried out here. In R, an LM test on different lags is displayed directly in the output for the GARCH model:

We see high p-values as a result of both models. This suggests that there is no autocorrelation in either model and that both models are subject to conditional heteroscedacy (Lamoureux/Lastrapes 1990: 221).

Since both models have no ARCH effects, an analysis of the residuals is necessary. Figure 3.1shows the squared residuals from the GARCH model for the Nasdaq100 and gold. The result is similar to the result of the squared residuals of the AR models, but the scaling is slightly higher. The maximum swing of the Nasdaq100 goes above 0.015 here. With the AR models, the scaling only went up to 0.012. Furthermore, it can be seen that gold has different maxima than the Nasdaq100, which is consistent with the previous analysis.

Figure 3.2 shows the comparison of the conditional variance from the GARCH models. Gold is blue and the Nasdaq100 is black. This figure shows again that gold has a very low volatility compared to the Nasdaq100. It can also be clearly seen that the maximum point from Figure3.2 has decreased significantly. The reason for this is that there was an extreme sell-off in gold on this one day. This extreme sell-off has a strong impact on the residual analysis.However, the effect becomes significantly smaller when analyzing the conditional variance in the GARCH model, since the effect of yield shocks is weakened.

After an analysis of the Nasdaq100 and the gold price using AR and GARCH models, the question now arises as to which relevant results can be drawn from the overall analysis. One finding that runs through the entire analysis is that gold has significantly lower volatility and has anti-cyclical behavior to the Nasdaq100. The analysis of the conditional variance of theGARCH models is very meaningful. This shows the lower volatility of gold in relation to theNasdaq100. Also in this analysis, the extreme rash that was seen in gold in the previous residual analysis is smoothed out significantly. The statement that gold fluctuates less and has an anti-cyclical behavior to the Nasdaq100 can therefore be made. Another consideration might be whether one can extract price predictions from the AR model's estimators. At gold, most of the AR model's estimates are not significant, and when they are, they are still hovering at alpha. No price predictions can therefore be made from the AR model, since the estimators are not very significant. The Nasdaq has significantly more significant estimators in the AR model. Nevertheless, it is questionable whether a reasonable trading strategy can be built on a negative estimator like: (ar1 with -0.1100). I would not make a specific statement about the price development with these results. The left-skewing of the distribution and the increased kurtosis also run through all models. This could not be corrected in any model. A critical examination of the models used is also required. AR models and the classic GARCH model were mainly used for the analysis. An extended analysis could be improved by furtherGARCH model variants such as ApARCH, MGARCH or an eGARCH model. These could, for example, underpin or put into perspective the statement that gold fluctuates less than theNasdaq100. The AR models could also be expanded. It would be possible to use the ARMA model here to show a broader analysis. The use of new statistical methods such as neural networks could also improve the results, since the classic models are based on regressions and have less flexibility than neural networks. Neural networks could also offer a more flexible forecast when analyzing price developments. Finally, it can be said that the focus of this work is on the application of the classic AR model and the classic GARCH model. Which is why an extended analysis with further statistical models exceeds the scope. Nevertheless, a statement can be made about the relationship between the volatility of the Nasdaq100 and the price of gold. This statement implies that gold fluctuates less than the Nasdaq100 indices and has a counter-cyclical relationship.

Bera, Anil K, Higgins,Matthew L,(1993): ARCH models: properties, estimation and testing,in: Journal of Economic Surveys 7, 1993, S. 305-366.

Christopher G. Lamoureux, William D. Lastrapes (1990): Heteroskedasticity in Stock ReturnData: Volume versus GARCH Effects, The Journal of Finance, Vol. 45, No.1, S. 221-229

Mandelbrot, Benoit, (1963): The Variation of Certain Speculative Prices, in: Journal ofBusiness, Vol. 36, 1963, S. 394-419.

Marcus Schmelzer, (2009), Die Volatilität von Finanzmarktdaten, Theoretische Grundlagenund empirische Analysen von stündlichen Renditezeitreihen und Risikomaßen, UniversitätKöln

Reinhold Kosfeld, (2007), https://www.uni-kassel.de/fb07/fileadmin/datas/fb07/5-Institute/IVWL/Kosfeld/lehre/zeitreihen/PartielleACF.pdf

Stock, J.H. and Watson, M.W. (2019), *Introduction to Econometrics*, 4th Global edition,Pearson Education Limited.

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