Choosing GARCH Models for Intraday Returns
Summary
The document describes a researcher’s difficulty fitting GARCH models to five-minute stock index returns. A GARCH(1,1) model with ARMA(1,1) terms produced an estimated sum of volatility coefficients of 0.81, while several optimizers and software packages struggled to converge. Attempts to adjust for overnight returns and intraday volatility patterns did not resolve the issues. Other tested specifications, including AFRIMA, EGARCH, and t-GARCH, also offered little improvement.
The sole proposed direction is to consider ARMA-FIGARCH, a fractionally integrated volatility model associated with higher-frequency data and long-memory behavior. The document does not provide a fitted example, comparative evidence, or implementation details, so this is a suggestion rather than a demonstrated solution. Its account is an individual modeling experience, and the cause of the convergence problems or the meaning of the coefficient sum is not established.
Key ideas
- Intraday return data can be difficult to fit with standard GARCH specifications.
- A GARCH(1,1)-ARMA(1,1) fit yielded a volatility coefficient sum of 0.81 in the example.
- Adjusting for overnight returns and intraday seasonal patterns did not improve the reported fit.
- ARMA-FIGARCH is suggested as a higher-frequency alternative, but no results are supplied.
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Full text
# Problems with dealing with GARCH models and intra-day data # Problems with dealing with GARCH models and intra-day data A Short question would be "Which type of model from GARCH family is most suitable for modeling 5-minute data returns ?" but I've added some story to it. A Long time ago I was preparing my thesis, one part of it was to model 5-minute log-returns of the stock market index. That was the first time when difficulties of modeling real data hit me quite hard. That time I was using STATA. I set overnight returns as missing values, but optimization of GARCH was painful due to constant lack of convergence of BFGS and DFP algorithms (error like: "flat part of likelihood surface approached" = constant value of log-likelihood per many iterations ), I've overridden this by setting max number of iterations = 50. I've find out that best model was GARCH(1,1)-ARMA(1,1) but alfa+beta=0.81 was far from the usual value which is one (perhaps due to ARMA part - but I don't think so) and for my experience alfa=0.14 was strangely high because it's usually about 0.05 for daily log-returns. 0.81 is very low because I've read numerous times that alfa+beta tends to one in probability as the sampling frequency is increased so IGARCH should be tried. I was searching for intraday patterns of seasonal volatility as a cause of my trouble, but I only spotted it in first and last 15 minutes of each session - I've first removed it, but it didn't improve convergence of algorithm and value of alfa+beta, then I've added exogenous variable (which contained pattern of seasonality) to variance equation, it also didn't work. I've tried to deal with overnight returns in many different ways thinking that maybe these cause the problem. Today I think that alfa+beta=0.81 isn't a problem at all, but maybe this gap of 0.19 makes place for stochastic volatility process? The lack of convergence in STATA ( and also in SAS and EViews which I've tried) still looks strange for me, could it be only due to numerical limitations of PC or maybe something else ? (I've also tried to fit model for shorter periods of times, I've tried many different things in many combinations). If you have some interesting general remarks or stories please write them - it's only soft question. (AFRIMA didn't work at all, EGARCH was very weak and t-GARCH gave no improvement at all, value of `t` was high which is really surprising, without some kind of GARCH there was always autocorrelation of residuals, modeled stock market index consisted of 20 stocks with 5000 trades per day in total , I could provide data). ## Answer by pyCthon (score 4, accepted) https://quant.stackexchange.com/a/7309 To quickly answer and address your first question. ``` ARMA - Fractionally integrated GARCH or FIGARCH ``` is one of the more common methods used at higher frequencies, it handles some properties required for higher frequency that standard `ARMA-GARCH` does not There are also a few other so called long memory volatility models, and there are other models which i am not familiar with, that take a different approach.
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