better we use the Fama Model, Morkowitz Portfolio model and CAPM Approach this approach

ARIMA and GARCH need very long series, and they don't usually yield good models for stock values.

It depends on whether you believe that history is a good guide for the future. If you do then econometric methods are useful for portfolio construction. If you do not (I don't) then you need to adopt a different approach.

Time series models based on GARCH time series have good forecasting properties, in particular for volatility and for short term horizons. They also have been applied to portfolio optimization, see for example:

Jaroslava Hlouskova, Kurt Schmidheiny, Martin Wagner: Multistep predictions for multivariate GARCH models: Closed form solution and the value for portfolio management, Journal of Empirical Finance 2009.

The performance can be tested on past data, and you can choose if the method is able to match your personal performance criterion.

You can make your data set stationary and model it by ARIMA process. The residuals can be modelled by GARCH, first of all you must test the presence of heteroscedacity

individually, on high frequency data you want find any significant AR component for the stock returns. GARCH models in all its variations are grand. The multidimensional estimation presents several issues. Many years ago a tried a BEKK form, but even with only 3 dimension you will get very slowly convergent likelihood. I suggest to use a constant correlation matrix and mount individual GARCH, just as a starter. The portfolio otimizition that you will get is then a dynamic efficient frontier. DO NOT forget that this problem is extremely sensitive to expected return vector and without an AR component you won't have big fun. I do not have good suggestions here.... You might try with low frequency data or overparameterized ARMA...... O_o

How efficient is COINTEGRATION concept in portfolio construction? Can long term relationship possible between portfolio constructed and the Index?

Why don't you try a dynamic factor model to estimate the covariance matrix and calculate the optimal portfolio in a markovitz framework?

These two models are the models which can test the volatility and their independence.

It further provide the time series analysis for the stock market data. It will help in the knowing the volatility of the stock prices. It further help you to understand the undervalued stock. It will help you to construct the optimum portfolio.

You can use the forecasted GARCH variance as the assets' risk instead of the unconditional variance usually employed in these models.

Thank you all for your suggestions and views

Dear Renukadevi Palaniswamy

Cointegration tests enable modeling long-term relationship between two or more entities. GARCH models and its variants (of different orders and different error term distributions such as Normal, T, GED) enable modeling short-term volatility linkages. You could try modeling both long-term and short-term relationship between the portfolio under consideration and the broader market index using an ECM-DCCGARCH model. Research indicates that an Error correction model is a stronger test for cointegration as opposed to static regression tests such as Engle- Granger test. Further the magnitude of the error correction term would also give an indication of the sensitivity of the bivariate system to disequilibriating shocks. The DCC GARCH model will enable you to model the dynamic conditional correlation between the portfolio returns under consideration and the broader market index.

I hope this is helpful.

In my opinion, the model should be matched with the data frequency. Do you want to identify an optimal portfolio on a daily basis? then, GARCH-type, or latent factor models might be useful. On the contrary, if you are willing to allocate the portfolio, say, monthly, GARCH effects are vanishing, and thus GARCH-type models might not be helpful, while cointegration can provide some intuition for portfolio construction.

Optimal portfolio of stocks! Well, if you have a set of equities and you are interested. In finding the optimal set, then you must use portfolio optimization concept.

The problem is the ingredients needed in the optimization problem: the expected mean, the variance, and/or the skewness and kurtosis of the returns series that does not have " autocorrelation" and "heteroscedasticity"!

ARIMA and GARCH models are used to filter the returns series in order to remove autocorrelation and heteroscedasticity in the series. This means that the residuals of a ARIMA-GARCH model are the best series (filtered returns) to use in order to estimate the inputs to portfolio optimization.

Now your portfolio allocations will depend on both the filtered returns (how good is the filtering model: ARIMA-GARCH) as well as on the optimization model used.

In the presence of high excess kurtosis and skew ness, the mean variance of Markowitz (1952) will not work perfectly as it depends on normally of returns! Therefore a good exercise of a good quantitative analyst is to start by fitting individual return series of each asset you hold in your portfolio to a theoretical distribution, including extreme value distributions. If ALL series are identically distributed, then a multivariate distribution is used, also a multivariate GARCH is used to model simultaneously these return series based either on normal or student t multivariate distributions.

However, in case where individual equity returns are not identically distributed, the copulas MUST be used to built a multivariate distribution of these returns series in the framework of Sklar (1959) and thereafter optimize the utility.

Another way is to consider multi objective portfolio optimization in order to include all four moments of the return series.

Bayesian portfolio selection gives an forward looking approach than traditional optimization problem. Therefore , the performance of your portfolio will depend on the filtering model as well as on the optimization model.

Hope this helps

John MM

They are some serious issues with Garch and ARIMA models. I suggest you to use SVR methods proposed by Shen et al (see http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2008-014.pdf ). Recently new SVR models were proposed by Santamaria Bonfil and myself:

http://link.springer.com/article/10.1007%2Fs10614-013-9411-x#page-1

I hope this helps

Juan Frausto-Solis

I desagree with Juan,

The aim of ARIMA -GARCH model is to filter the return series by removing autocorrelation and heteroscedasticity in the data. Why use a SVR model? That is the problem with many econometricians, they think that any tool at hand is a hammer to nail with. There are variant models of both ARIMA and GARCH with different types of distributions for both symmetric and asymmetric data. Use the correct empirical distribution for your model (asymmetric and/or symmetric GARCH), get the filtered returns and move to portfolio optimization problem. Evaluate the portfolio performance based on good performance measures.

Good Luck

John

An optimal Garch is the best solution.

I do not agree with the comment by John when he suggests to use returns filtered from mean and variance dynamic (what he calls filtered returns) to recover the inputs for portfolio optimization following Markowitz.

If I'm using returns are a not too high frequency, say hot higher than daily, mean dynamic captured by ARMA models would signal some form of predictability. This might be present, but I have doubts it will be beneficial as I guess it will capture a limited part of the series movements. I do not want to enter the topic of having other variables in the model, and stick to the use of the returns own past. ARMA models would thus have, in general, a limited or zero impact. Filtering from ARMA would mean that the residuals will have zero mean, to taking ARMA residuals for portfolio allocation would imply have all assets with zero expected returns. In terms of Markowitz, assts would be on a straight horizontal line in the risk-return space.

Consider now GARCH model, that would capture the heteroskedasticity and would allow forecating the marginal variances (if fitting univariate models) or the entire covariance (with multivariate models). If you filter ARMA residuals by univariate GARCH then the resulting series will be homoskedastic but still correlated. Thus using them for allocation would imply you focus only on the correlations without considering variances. But your inputs in the risk-return plane will be on the same point. Nevertheless, correlations would play a role in portfolio optimization. If you standardize by multivariate GARCH, then you series will be uncorrelated and homoskedastic, say, in terms of Markowitz, they will be perfect substitute one to the other, and the optimal portfolio is the equally weighted one.