GARCH(1,1) Model : Long_Run_Volatility

[gargi.adhikari]

Can anyone explain what a "Long Run" Volatility is...? I am studying the GARCH Model but not sure what the "LONG RUN" Volatility means....is it Volatility "Over a Longer Span of Time" ..? Am not able to find much articles/resources to explain Long Run Volatility online either... :-(

 

[David Harper CFA FRM]

Hi @gargi.adhikari It’s a great question, as usual from you! I would like to acknowledge there exist some technical nuances (which might be explored in C. Alexendar at https://forum.bionicturtle.com/reso...ractical-financial-econometrics-volume-ii.91/, or for an even deeper resource, one of my favorites on this topic is Stephen Taylor’s Asset Price Dynamics http://amzn.to/2qoaBGl). Given that caveat …

The long-run average variance (aka, σ^2) is also called the unconditional variance in GARCH. As you know, I like examples. Say yesterday’s volatility was 1.0%, σ(n-1) = 0.010, and conveniently the most recent price return is +1.0%, µ(n-1) = +1.0%. EWMA updates the variance with the latest squared return, but in this example, they are the same such that, for any λ, EWMA will return an updated variance of 0.01^2. What does this EMWA model forecast for the DAILY volatility in ten days? Or thirty days? (note: this is not a ten-day volatility, but rather a forecast of the the one-day volatility forward in time ten days!). Well, I think you can either say “EMWA does not forecast” or, maybe you can say “the best EWMA can do is forecast the same current 1.0% volatility forward, as a flat line, so to speak.”

This is the key difference of the GARCH model, which generalizes the EWMA by adding the unconditional (aka, long term average) variance. Let’s say we have the same σ(n-1) = µ(n-1) = 1.0% but additionally our long-run average volatility is 2.0%. In my view, we can almost work backwards from the 2.0%; ie, we can actually START here. When we fit GARCH(1,1) to our series, we are asserting that regardless of the current situation, the long-term forecast for the daily volatility is 2.0% (and gamma, γ, is the speed to which today’s variance reverts toward this 2.0%). This is not about scaling the 1-day to 10- or 30-day or annualized volatility; this is an assumption about the "true" unconditional one-day volatility. Hence this commonly discussed distinction between the conditional and unconditional variance. GARCH is a model that updates the conditional variance; i.e., GARCH’s updated daily variance is time-varying and depends on the current day because each day has a different historical window. At the same time, this conditional (updated) variance is assumed to be trending toward the unconditional variance (2.0%^2 in my example) which is unconditional because it applies at all time (at all days; so it’s analogous to a population parameter). So, to finish my example under these assumptions, while EMWA will update our volatility estimate to 1.0%, GARCH will update today's volatility to something greater than 1.0%; eg, 1.15%; to assign some weight towards the unconditional 2.0% volatility. Further, technically the switch to GARCH arms us with an actually forecasting model, such that as we forecast further in the future, the daily volatility forecast trends toward the unconditional 2.0%. I hope that's a helpful frame!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much as usual for putting together this explanation...got the GARCH the EMWA - couldn't be any clearer !!

 

[David Harper CFA FRM]

@gargi.adhikari Thank you, I always enjoy trying to improve my explanations of concepts. I noticed that I tend to commingle volatility and variance, in my head I think of them similarly. I trust this audience knows that volatility is a synonym for the standard deviation of returns and squaring the standard deviation, σ, gets you the variance, σ^2. GARCH of course is model that updates the variance, σ^2(n) = α*u^2(n-1) + β*σ^2(n-1) + γ*σ^2(long-run; aka, unconditional). Thanks!

 

[lRRAngle]

Hi There. My first post

I think my question is similar and you may have already answered it above, but just to confirm:
So in the GARCH (1,1) equation, the σ^2 (n-1) will always equal u^2(n-1)? (i.e. the formula referencing to a ONE DAY variance for the prior day, and that will always be the same as the one day return for prior day?) That's what I understood from your answer, but it seems strange to be looking at variance over a one day period in the context of a one day return...I must be missing something.

The Long Run variance term in the GARCH, that is just the variance of daily returns over an interval of xyz number of days (based on the data set), right?

Thanks

 

[David Harper CFA FRM]

Hi @lRRAngle Thank you for your first post! I don't think it's quite what you are saying. Maybe this will help. Here is the GARCH(1,1) which gives us an updated variance estimate: σ^2(n) = γ*σ^2(L.R.) + α*µ^2(n-1) + β*σ^2(n-1). You are basically correct about the long-run variance, it is the "unconditional variance" and represents a long-term variance that is not conditional to the current volatility situation. Let's say it is equal to σ^2(L.R.) = 1.0%^2 = 0.00010. Then maybe yesterday our GARCH(1,1) model estimated the (conditional) daily variance was 2.0%^2 = 0.00040. As of yesterday, that was the then-current variance estimate. But today, we have an additional observation in the form of a daily return that is used to update the variance estimate. Maybe the stock price plunged dramatically from $10.00 to $9.20 for an -8.00% drop. A variance is basically an average squared return, so the daily return of -8.0% enters the variance formula naturally by squaring itself. In this way, yesterday's variance estimate of 2.0%^2 is "re-averaged" or updated by including the new -8.0%^2 but also included is the long-run (unconditional) variance of 1.0%^2. This GARCH(1,1) can be simplistically viewed as a weighted average of these three variances: yesterday's variance estimate (most of the weight), the most recent squared return (which can be viewed as the variance of a series of one observation!) and the long-run variance. We are basically using yesterday's variance estimate and instead of using it (weighting it) 100%, we are "averaging in" a little bit two other variances: the variance that happened yesterday (ie., squared return) and our assumption for the long-run variance two which we are reverting (getting pulled toward). Maybe our GARCH(1,1) gives 80% weight to yesterday's variance estimate, 10% to the latest squared return and 10% to the long run. In which case we have σ^2(n) = γ*σ^2(L.R.) + α*µ^2(n-1) + β*σ^2(n-1) = 0.10*1.0%^2 + 0.10*-8.0%^2 + 0.80*2.0%^2 = 3.1145%^23.153%^2; i.e., yesterday's 2.0% estimate was "pulled up" by averaging in the new high -8.0%^2 which was slightly offset by the gravitational pull toward the long-run 1.0%^2. I hope that's a helpful perspective, thank you!

 

[lRRAngle]

Thanks David. It makes sense now. So yesterdays Variance (n-1) is really the conditional GARCH (1,1) variance and the LR variance is a simple variance if you will (or as you described 'unconditional'. In your example, I suppose you could have updated the LR average to be slightly over 1.0% for illustration purposes, to account for the -8% return, so say something like 1.04%, yes?

Regards

 

[David Harper CFA FRM]

Hi @lRRAngle Yes, except for updating the L.R. variance (which per its definition is not typical practice) that is basically correct although each of today's updated variance estimate and yesterday's variance (upon with today's depends per the auto-regressive feature) are conditional variances. I don't want to pull you too far into the theory, but let me just replace the Greek weighs with α = A, β = B, and γ = G, in case plainer identification of the weights makes this easier. The GARCH (1,1) then is given by σ^2(n) = G*σ^2(L.R.) + A*µ^2(n-1) + B*σ^2(n-1); and, as mentioned, we can view this is a weighted average (where A, B and G are the weights that must sum to one) of three variances: yesterday's conditional variance, σ^2(n-1), the unconditional (aka, long-run variance), and the "newest piece of information" (aka, the innovation) which is the one-day variance given by µ^2(n-1).

This is my current favorite way to explain GARCH(1,1): given weights A + B+ G = 1.0, GARCH(1,1) is an updated estimate of today's conditional variance as a weighted average = A*[today's 1-day variance; i.e., squared return] + B*[yesterday's conditional variance] + G*[the long-run/unconditional variance]

GARCH(1,1) is a model for the conditional variance, by which we mean the variance is conditional on the latest "information set." (To use Carol Alexander's phrase). This also means that the variance is not i.i.d.: it is neither identical from one day to the next, nor is it independent (the A in GARCH stands for autogressive). So yesterday's variance was conditional, but today's updated variance is also conditional. The L.R. variance is not conditional on a day to day basis, it is constant over the sample: in a typical analysis, we would collect the historical window and estimate the average variance over the sample (just as we would take a sample and compute the sample variance). Typically, it would not be updated day-to-day as, in theory, it represents the "steady state" value. Sure, you could recalculate the weights and parameters periodically, but the meaning of "unconditional" in the model is that, unlike the conditional variance, the long-run variance would not generally change each day (nor would the weights). Believe it or not, this still scratches the surface of the model, it has been extensively researched! I hope that's helpful! (I think my previous post contained a calculation error that i think i fixed, fyi).

 

[lRRAngle]

I see what you are saying. Thank you very much!