Standard deviation differences between Portfolio 2 large and small cap indexes and Portfolio 3 medium and small cap indexes based on asymmetric data. Positive values indicate that Portfolio 2 assume higher levels of volatility than Portfolio In all cases, positive values would indicate that the volatilities in the first element of the difference is higher than in the second and, therefore, the preference of the investor for the second element which is always Portfolio 3, the combination of the medium and small cap indexes, in its asymmetric version..

## GARCH Model - MATLAB & Simulink

We can clearly observe that in all cases the differences are mostly positive which means that the standard deviations of Portfolio 3, which combines the Ibex Medium and Small cap indexes, are lower than the first element of each difference and, for that reason, is always the preferred one in terms of volatility. At first glance, and consistent with the objective of determining the portfolio which holds the minimum volatility values, we would choose the portfolio where the medium and small cap indexes are combined.

This is consistent with their lower volatility as reported in the descriptive statistics, and with the fact that their conditional variance is less affected by other volatility and shocks than the IBEX However, in order to obtain some more conclusive results and economic implications, we analyze the optimal portfolio responses to good and bad news positive and negative shocks. For this reason, we consider four different scenarios in order to calculate the average risk and return of each portfolio.

The results 12 for the estimated standard deviations and expected returns are reported in Tables 6 and 7 , respectively. Differences among the volatilities are statistically significant due to the fact that the null of equal means is rejected in all cases. Moreover, from the results of the expected returns Table 7 we can consider that the best combination of minimum risk and maximum return is that provided by Portfolio 3 when the negative—positive scenario is considered.

This is because the minimum risk 13 0.

Risk minimizing portfolio responses to good and bad news. Volatility analysis.. Following Kroner and Ng , Ewing and Malik , and Hassan and Malik , we calculate the composition of three risk minimizing portfolios by the combination of each pair of indexes and employing conditional variance and covariance series from both symmetric and asymmetric multivariate GARCH models. Panel A, B, and C show mean, median, maximum and minimum volatility values of each portfolio in each scenario. We also present results for mean equality tests using the ANOVA statistic p -values are reported in parenthesis..

Return analysis.. Panel A, B, and C show mean, median, maximum and minimum expected return values of each portfolio in each scenario. Initially, we find that all the indexes are directly and indirectly affected by the others, being especially significant the bi-directional relationship between the medium and small firms. However, these relationships disappear when asymmetries and structural breaks are taken into account.. In order to analyze the economic implications of our results, we calculate the optimal invested portfolio holdings. This implies that investors should keep a close eye on that portfolio which also obtains the best return results in some of the analyzed situations.

These results add a new point of view to the analysis of volatility because we show that the reduced volatility of medium and small firm indexes makes them an interesting investment option even in times of recession.. To sum up, we want to point out that the results of multivariate GARCH models can be used by financial market participants to make optimal portfolio allocation decisions.

## A practical introduction to garch modeling

These results have important implications for building accurate asset pricing models, forecasting volatility and, consequently, understanding the Spanish stock market.. We also thank participants at the XX Financial Forum in Oviedo for their helpful comments and suggestions.. The IBEX 35 is composed of the 35 largest companies in terms of capitalization and liquidity which are traded on the Spanish stock market. Nowadays, they are a national and international reference. It is also important to note that these indexes are replicable and investable, that is, their calculation allows for the issuance of financial products such as Exchange Traded Funds ETFs , index funds or derivatives..

The data was extracted from Sociedad de Bolsas, which is the owner of the indexes and is in charge of their management, calculation, dissemination as well as of the review of their composition.. We do not include the error correction term in the mean equations since the Johansen tests showed that three series are not cointegrated.. This specification has been the most popular in the literature. In fact, it improves VECH and diagonal representations because it practically assures that H t will be positive definite.

Furthermore, it does not require so many parameters to be estimated as in the VECH case and is more general than the diagonal representation. See Soriano and Climent b for more detail.. However, as these authors also pointed out, this assumption is, in fact, not true because in most cases, the standardized residuals exhibit leptokurtosis. Thus, the application of the normal distribution leads to a so-called Quasi- or Pseudo-ML estimation.

According to Weiss , this application leads to a consistent estimation of the parameters if the equations for the conditional means and variances are specified correctly. Some authors tried to solve the problem by using different distributions. However, it was proven that when a distribution different from the normal one is used and this distribution is not the true one, then the estimates are, in most cases, not consistent. For that reason, we prefer to apply the normal distribution.. Results for the mean equations are not reported for the sake of brevity, although they are available upon request..

These differences can be explained by the fact that different methodologies and samples are used in these studies.. See Bodie et al. In all cases, the null of zero mean for each difference was tested and rejected. Values are not reported, but are available upon request.. We do not provide the values of the portfolio weights and risk minimizing hedge ratios but they are available upon request.. See results from the asymmetric data in Table ISSN: Indexed in: Scopus See more Follow us:.

Discontinued publication For more information click here. Previous article Next article. Issue 1. Pages January - June Download PDF. Corresponding author. This item has received. Article information. Show more Show less. Table 1. Descriptive statistics.. Table 6. Table 7. After calculating the risk minimizing portfolio weights, we show that the minimum-volatility portfolio is composed of medium and small indexes with a higher weight of medium firms for a set of different scenarios.

Portfolio allocation. JEL classification:. Finally, we shed some light on the behavior of the Spanish stock market by providing some clues to better understand portfolio allocations. Our study focuses on the Spanish stock market because in recent years it has become a reference for the main European stock markets due to improvements in the technical, operational and organizational systems supporting the market which have enabled it to channel large volumes of investment and have made it more transparent, liquid, and effective. These results have important implications for building accurate asset pricing models, forecasting volatility, and understanding the Spanish stock market.

In Section 4 we show the principal results and, finally, in Section 5 we provide the main conclusions. Finally, Hassan and Malik and Li and Majerowska use multivariate GARCH models including more than two variables to analyze the transmission of shocks and volatility among different US sector indexes and links between emerging stock markets in Poland and Hungary and established markets in Germany and the US, respectively. The results show that the Spanish stock market is subject to higher, but less persistent, volatility and that trading volume significantly impacts on volatility.

They show that volatility spillovers exist in both directions between those portfolios after bad news. Additionally, in order to obtain more conclusive economic implications, we analyze the optimal portfolio responses to good and bad news. Descriptive statistics. The p -values of these tests are reported in parenthesis. Table 2. Table 3. Sudden changes in volatility. Table 4. Q 20 stands for the Ljung—Box Q statistic for the standardized residuals up to 20 lags while Q 2 20 for the Ljung—Box Q statistic for the squared standardized residuals up to 20 lags.

- Volatility.
- GARCH Process.
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Table 5. Risk minimizing portfolios. Finally, Panel D presents average weights of each index in the risk minimizing portfolios. Standard deviation differences of Portfolio 3 based on symmetric and asymmetric data. Positive values indicate that Portfolio 3 assume higher levels of volatility when is constructed based on the symmetric specification than employing the asymmetric one.

### Submission history

Standard deviation differences between Portfolio 1 and Portfolio 3. Positive values indicate that Portfolio 1 assume higher levels of volatility than Portfolio 3. Standard deviation differences between Portfolio 2 and Portfolio 3. Positive values indicate that Portfolio 2 assume higher levels of volatility than Portfolio 3. Volatility analysis. We also present results for mean equality tests using the ANOVA statistic p -values are reported in parenthesis.

Return analysis. We also thank participants at the XX Financial Forum in Oviedo for their helpful comments and suggestions. Aggarwal, C. Inclan, R. Journal of Financial and Quantitative Analysis, 34 , pp. Influence of structural changes in transmission of information between stock markets: a European empirical study. Journal of Multinational Financial Management, 17 , pp. Sudden changes in variance and time varying hedge ratios.

European Journal of Operational Research, , pp. Bodie, A.

## GARCH Models in R

Kane, A. McGraw-Hill, ,. Chelley-Steeley, J. Volatility, leverage and firm size: the UK evidence. Structural changes in volatility and stock market development: evidence for Spain. Journal of Banking and Finance, 28 , pp. Ewing, F. Re-examining the asymmetric predictability of conditional variances: the role of sudden changes in variance. Journal of Banking and Finance, 29 , pp. Malik, O. Volatility transmission in the oil and natural gas markets. Energy Economics, 24 , pp. Fleming, C. Kirby, B. Information and volatility linkages in the stock, bond and money markets.

Journal of Financial Economics, 49 , pp. Glosten, R. Jaganathan, D.

On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48 , pp. Grieb, M. The temporal relationship between large and small capitalization stock returns: evidence from the UK. Review of Financial Economics, 11 , pp. Hamao, R. Masulis, V. Correlations in price changes and volatility across international stock markets. Review of Financial Studies, 3 , pp. Hassan, F. The Quarterly Review of Economics and Finance, 47 , pp.

Inclan, G. Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association, 89 , pp. Information transmission between small and large stocks in the national stock exchange in India: an empirical study. The Quarterly Review of Economics and Finance, 50 , pp. King, S. Transmission of volatility between stock markets. Kroner, J. Time varying distributions and dynamic hedging with foreign currency futures.

Journal of Financial and Quantitative Analysis, 28 , pp. Kroner, V. Modeling asymmetric comovements of asset returns. Review of Financial Studies, 11 , pp. Li, E. Research in International Business and Finance, 22 , pp. Journal of Business and Economic Statistics, 15 , pp.

trailblazer.outdoorsy.co/2.php Lin, R. Engle, T. Do bulls and bears move across borders? International transmission of stock returns and volatility. Review of Financial Studies, 7 , pp. Sudden changes in variance and volatility persistence in foreign exchange markets. Journal of Multinational Financial Management, 13 , pp. Malik, S.

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Shock and volatility transmission in the oil US and Gulf equity markets. International Review of Economics and Finance, 16 , pp. Meneu, H. Asymmetric covariance in spot-futures markets. The Journal of Futures Markets, 23 , pp. Pardo, H. Journal of Business Finance and Accounting, 34 , pp. Soriano, F. Region vs industry effects and volatility transmission. Financial Analyst Journal, 62 , pp.

Spanish Review of Financial Economics, 4 , pp. Asymptotic theory for ARCH models: estimation and testing. Econometric Theory, 2 , pp. It is also important to note that these indexes are replicable and investable, that is, their calculation allows for the issuance of financial products such as Exchange Traded Funds ETFs , index funds or derivatives.

This is useful for quantifying the time-varying volatility and the resulting risk for investors holding stocks summarized by the index. Furthermore, this GARCH model may also be used to produce forecast intervals whose widths depend on the volatility of the most recent periods. We have discussed how vector autoregressions are conveniently estimated and used for forecasting in R by means of functions from the vars package.

In an application we have found evidence that 3-months and year interest rates have a common stochastic trend that is, they are cointegrated and thus can be modeled using a vector error correction model. Bollerslev, T. Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics , 31 3 , — Engle, R.

Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica , 50 4 , — Preface 1 Introduction 1. Computation of Heteroskedasticity-Robust Standard Errors 5.

Part I Introduction to Econometrics with R. This book is in Open Review. We want your feedback to make the book better for you and other students. You may annotate some text by selecting it with the cursor and then click the on the pop-up menu. You can also see the annotations of others: click the in the upper right hand corner of the page.

Figure Application to Stock Price Volatility Maximum likelihood estimates of ARCH and GARCH models are efficient and have normal distributions in large samples, such that the usual methods for conducting inference about the unknown parameters can be applied. It is straightforward to estimate this model using garchFit. Summary We have discussed how vector autoregressions are conveniently estimated and used for forecasting in R by means of functions from the vars package.