# GARCH model is better for index than stock

We have used a standard GARCH(1,1) model with t distributed innovations for daily data of S&P index and JPM stock.

Question: is there any financial or statistical reason why the GARCH model would be more suitable for an index than for a stock? My initial guess was that returns "behave better" with indicies than stocks (for example JPM vary around +20% and -20% whereas S&P vary around +10% -10%) and since the return data is the input into the GARCH model this would make

# Details

To forecast the volatility we used a standard GARCH(1,1) model with t distributed innovations. Via the formula $VaR_t = \hat \mu_t + t \hat \sigma_t$ the volatility forecast was used in the calculation of the forecasted VaR.

We had two different datasets: returns on S&P index prices and returns on JPM stock prices. We used the same GARCH model for both return series, and labelled them $m.index$ and $m.stock$ respectively.

The VaR forecasts were backtested using standard backtesting procedures, the most important one is Christoffersen's test of independece (also called a cc test). The results from applying this test to our VaR forecasts showed that $m.index$ was better than $m.stock$ with respect to VaR backtesting. It was better because the p-value from Christoffersen's test of independece was $0.603$ for $m.stock$ but only $0.095$ for $m.stock$ suggesting that VaR violations were closer to being independent and correct number of violations when using index returns rather than stock returns.

• Can you provide other tests/empirical analyses on indeces and stocks? It is too poor a reason that indeces returns "behave better" than stocks' and your results may be just a case. – simmy May 14 '16 at 13:30
• Yes, you may look at section 5 and 6 in this pdf [link] I'm asking this question on quant.SE to be able to write section 7 conclusions – jacob May 14 '16 at 16:20
• @simmy Here is the link dropbox.com/s/nlfrlrrlasl5vtc/mmain_su%20160514_18.35.pdf?dl=1 – jacob May 14 '16 at 16:35
• yes there are see my answer to my quesion (tommorow) : quant.stackexchange.com/questions/3638/… in short averaging two identical GARCH processes with alfa and beta produce GARCH with alfa2+beta2>alfa1+beta1 – Qbik Sep 5 '16 at 20:05

The reason is earnings and other idiosyncratic corporate actions like takeovers, major product releases, etc.

There are three terms in garch(1,1), the constant, term proportional to previous day's volatility, and a term proportional to "stock noise". Earnings jump is much larger than previous "regular" volatility, and also much larger than "regular" noise. When a stock jumps, it is not the type of return that is explained by the model, so the fit is worse.

Indexes, by averaging component stocks returns, smoothes out idiosyncratic jumps. Note that you can add terms to a garch model to model earnings, so you can get a better fit, possible comparable to those of an index.

• By "add terms to a garch model" you mean adding external regressors? – jacob May 15 '16 at 14:03
• Yes, e.g. earnings day indicator. – onlyvix.blogspot.com May 16 '16 at 1:52
• I use "adjusted closing" prices not "closing" prices but maybe that does not account for dividends but only macroeconomic factors? – jacob May 16 '16 at 15:05
• Adjusted means adjusted for splits and (maybe) dividends. There is no adjustment for macroeconomic factors. Macroeconomic factors, almost by definition will effect all stocks, including stocks indexes, so will not explain differences in your fit. – onlyvix.blogspot.com May 16 '16 at 18:12

Jacob, you conclude that "The main finding is that VaR is more suited for our index portfolio GSPC than for our stock JPM." This conclusion is not surprising. However I think that is not the VaR in itself "more suited," but the underlying GARCH(1,1) model. You introduce the standardized error in section 4.6. That is the right way. I did not study your pdf in depth, but I'm almost sure that the quality of fit of GARCH(1,1) model for index are, in some sense, better than than those of the same model on a single stock. Here is the problem: the result of Kupiec and or Christoffersen's test are only the transposition of the same thing. The problem for single stock is not the VaR in itself, but finding the "good model," as ARMA(?,?)-GARCH(?,?), for it. Probably if you fit the same GARCH(1,1) model for a single stock much smaller and less liquid that JPM the resulting VaR will be worse than JPM's.

As liquidity I mean the usual sense in stock market as explained here https://en.wikipedia.org/wiki/Market_liquidity#Stock_market if the stock A in less liquid than stock B, probably is easier to find a good model for A than for B. That's all.

• You say "...but the underlying GARCH(1,1) model..." and of course this is correct since $VaR = \hat \mu_t + quantile \cdot \sigma_t$ the the volatility is changing much more than mu or the quantile. – jacob May 15 '16 at 14:05
• @markovitz please elaborate on your liquidity idea. – jacob May 15 '16 at 14:05