Hello,Can anyone recomend to me a way of removing autocorrelation (as measure by the Q test) from a GARCH regression?I have tried adding in an extra lagged term to the mean equation, I dont want to mess around with the variance equation too mcu h because its quite sensitive...(i know this probably doesnt show a robust model, but its only for a piece of work).ThanksJim

Did you prepare your data properly before the estimation?

As far as I am aware..yes.I didnt do much though - just took the returns using:rt = [logpt-logpt-1]*100Is there something more I could have done?I have since discovered that adding an MA(1) or (2) term to my mean GARCH equations (purely thorugh expirmnetation) helps to remove the aoutocorrelation in some of the regressions. However #I am worried about just 'plonking' extra terms in when I am not exactly sure of what I am doing..Any help would be great...

I would usually make a table with all possible descriptive statistics for both the prices and the returns (or any other variables I am modelling). The same goes for ACF and PACF functions - generally all kinds of things to get the feel of the data, which helps in choosing the appropriate model. Then I'd probably choose the number of lags to use in the later stages and checked for seasonality (what data are you using BTW - weekly / daily / hourly ?). Then a stationarity check and appropriate transformations when needed.Autocorrelation might mean that your model is misspecified. Even if not, it often appears in time series models simply because of the nature of the data. You might want to use autocorrelation consistent estimators, otherwise autocorrelation could make your t-statistics more or less useless, depending on its strength. Refer to Davidson and MacKinnon (1998 if I remember correctly) for more. For this kind of modelling I would heartily recommend gretl, which is an open-source econometric apllication, very handy.

What do you mean by removing the autocorrelation? Do you mean get uncorelated data from correlated one or remove the seasonality from the time series?

I understand fmwanabe meant residual autocorrelation. Is that right?

Autocorrelation in your GARCH residuals is almost always caused by autocorrelation in your data. The GARCH fit should not introduce it and will eliminate it if you include AR(n) terms at the lags of the autocorrelation.We do not expect to see autocorrelation in return series, because normally that would permit market-beating trading. There are exceptions. For example, short-term bond returns are autocorrelated because interest rates move up and down. But measured properly against a cost of funds, the autocorrelation should disappear. Also, returns could have autocorrelations over long periods of time due to changing levels of risk preference or inflation. Over very short intervals, transaction costs could prevent traders from eliminating all autocorrelation.If you're searching for autocorrelations in order to beat the market, then the last thing you want to do is remove them. In most other cases, autocorrelation results from a measurement error. One common example is the return on a global equity portfolio will show autocorrelation because the market moves between Asia's and New York's close will be reflected in today's New York quotes, tomorrow's Asia quotes and mixed for Europe. A similar effect causes measured autocorrelation for illiquid securities. Or, hedge fund returns show autocorrelation, possibly in part because managers have incentives to smooth returns.In all these cases, the key is to remove the autocorrelation from the initial returns in a financially meaningful way. Removing it in a statistical way is pointless, you won't end up with an actionable result. If you cannot remove the autocorrelation in a financially meaningful way, then just go ahead and leave it in.

On a related note...say I have two time series (both AR(1)) and I regress one against another. The residuals from the regression are also found to be AR(1)...What is the relationship between the two series? From textbook theory I guess they are not cointegrated but can I say something more than that???

Thanks for all the replies - all extremely helpful!I have a 3 more inter-related follow up questions if I may.First of all yes I am talking about the AC in the residuals. However I feel it is important to clarify that my GARCH models are two stage. Firstly I am calculating mean returns and variances for 3 markets sperately, then including the residuals from market B and C into market A's equation to analyse spillovers.My first stage GARCH models all have no auto correlation...it is the second stage when it arises.I am using daily data - however I am using FTSE indices for each country. This is because I am using sectoral indices so I had to use only FTSE as there is a difference between industry classifications accross world indices - US uses GICS - FTSE uses ICB.1. I want to include a dummy for all the mondays and days following market closures in all markets - to capture day of the week effect. I have found that when I calculate the returns they come out as zero across all sectors (i.e. the index was closed on this day) on days of all the market closure days in the UK....this is fine.However for the 'foreign' markets, i.e. US and Japan the returns are zero only on christmas day and new years day - but no other days. Is there a quick way I can figue out holiday days for these two countries. Or do you think the FTSE indices may have accounted for this?2. Secondly Im not sure as to when I should include the dummy variable and exactly how to interpret it - if it is not statistically significant should I then take it out or leave it in so that it captures any effect on other indices.3. Finally because I will have 3 seperate dummy variables (because countries holiday on different days) - should I include all three in the regressions where I am looking at all three countries? Or by including ithem in the first stage of my model, will they capture the effect thus making them surplus to requirement in the second stage?Thanks for all the help again its invaluable!Jim

Fmwanabe, a few more things. How do you test for the autocorrelation, e.g. which test do you use? Could you also give the equations for all your first-stage and second-stage models? Did you test for colinearity?

QuoteOriginally posted by: playerOn a related note...say I have two time series (both AR(1)) and I regress one against another. The residuals from the regression are also found to be AR(1)...What is the relationship between the two series? From textbook theory I guess they are not cointegrated but can I say something more than that???Let X_i and Y_i be the original AR(1) series. Say we can write them as:X_i = a*X_i-1 + x_iY_i = b*Y_i-1 + y_iwhere the series x_i and y_i have no autocorrelation.Now you regress Y on X. The residual is:e_i = Y_i - c*X_i - d = b*Y_i-1 - c*a*X_i-1 + y_i - c*x_i - d Y_i and X_i are autocorrelated, so in general the linear combination b*Y_i-1 - c*a*X_i-1 will be autocorrelated as well, unless c is chosen precisely to eliminate the autocorrelation (there is a c that will do this, but least squares regression will not choose it in general). y_i and x_i individually have no autocorrelation. They can only introduce autocorrelation into the residual if they are correlated with each other at a lag.In general, you expect the residuals to be autocorrelated. If they are not, then either the regression coefficient c happened to be near the value that eliminated the autocorrelation in X and Y, or the lagged cross-correlation between x_i and y_i just offset any residual autocorrelation in X_i and Y_i.

QuoteOriginally posted by: fmwanabeThanks for all the replies - all extremely helpful!I have a 3 more inter-related follow up questions if I may.1. Look to your data provider. I assume you're getting returns on market close days because domestic stocks trading on foreign or electronic exchanges are being counted. The provider should be able to help you with the market closed days. If not, you could guess from the volume statistics, or find an on-line source.2. For most purposes, you only keep the dummy if it is statistically significant. You interpret it as an excess return on days following a closed market.3. Don't even think about including them in the second stage, unless this is specifically a study of cross-market effects of market closures. I suspect your autocorrelation is the result of non-synchronous data measurement.

Thanks again for advice.I will be handing in the work in the next few days, so I dont have time to resitimate all the models now unfortunately. I left the dummy variable for modays in the mean and variance equation unfortunately, as I didnt read the post saying not to in time. I have however seen them left in studies which I have used, so I will cite in such a way of 'they did it so Ive done it'...if your interested ill let you know which ones.My models are quite simple so dont laugh!! its only an undergrad dissertation so we arent required to get too clever, Im hoping what Ive done isnt complete nonsense though!! Ill put a heavy caveat in the conclusion that higher level (i.e. asymmetric) GARCH models are used more often nowdays..andthat this is a starting point.Ive lifted it straight out of my script so its not formatted perfectly (the twos are sibscript not 2 x rt)...and I didnt put the dummy variables in yet - ive just put one in each equation (including the second stage (doh!!)STAGE 1: rt = φ1 + φ2rt-1 + ε,t (1) ht = a + αht-1 + βε²t-1 (2)Where rt are the daily returns on an index at time t; εt is a white noise error term; and ht is the conditional variance of rt. STAGE 2: rt,D = φ1 + φ2rt-1 + γ1εt-1,F1 + γ2εt-1,F2 + γ2εt,F1 + γ2εt,F2 + ε,tD (3) ht,D = a + αht-1 + βε²t-1 + λ1ε²t-1,F1 + λ1ε²t-1,F2 + λ1ε²t,F1 + λ1ε²t,F1 (4)Where rt,D is the daily return of the domestic stock index at time t,. εt,F1 and εt,F2 are the standardized residuals series from the two foreign markets at time t. In order to analyse the volatility spillovers the exogenous variables ε²t,F1 and ε²t,F1 (which are the squared standardized residual series for each of the foreign markets) are added to the variance equation for each index....I tested for AC with the Ljung Box-Q test as this seemed to be the most widely used diagnostic test. Anything that exhibited AC I've not really included in my final conclusions.I also added a GARCH in mean to the mean equation afterwards on recomendation of a tutor. This is just incase any significant spillovers as measured by the residuals included in the formula were picking up a 'general volatility effect' - the GARCH-M didnt change much. So ive said the results are robust.If you have time, let me know what you think..hopefully its not too terrible! Im a bit worried now! If you dont mind ill mention (and fully acredit) a few criticisms in the conclusion. No worries if not.Thanks again for the help. Ill let you know my grade (if its not awful!).