GARCH Equation Interpretation

[anique]

Dear David,

I just want to start off with a thank you for being active on this site. i read your articles and have found them to be quite interesting. anyway to my point. im currently regestered as a CFA level 1 candidate and am also about to complete my MBA. my question today is in refference to my MBA thesis. my topic of research is "Pace of mean reversion in Pakistani stock market". so the problem is that im using the GARCH(1, 1) model but i just cant seem to find any explanation on the internet that may simply explain (In plain simple English) as to what the GARCH (1, 1) model is all about. i do know that it is used in order to determine if there is any significance related to the ARCH and GARCH effects, for any given dataset, so that we may be able to determine whether mean reversion exists in a markets or not. my concern is that on the day of my final presentation I may not be able to explain in simple language what the GARCH (1, 1) model really is which will negatively impact the panel.
the things i would like for you to kindly explain again in simple language is that:
1)what is GARCH?
2) how do we interpret the GARCH equation (Conditional variance)
3)what is a lag?
4) what is a 1st or 2nd lag squared return?

kindly respond as soon as possible as my submission is in a few days and im currently writing up my conclusions and recommendations.

 

[David Harper CFA FRM]

Hi anique,

Thanks for your support. GARCH is a deep topic; in my opinion, to discuss it "in simple terms" requires much foundation. In the FRM, we skip much of the stochastic time series theory and, following John Hull's chapter on estimating volatility, we treat it rather mechanically: as an ARCH(m) model along with moving average (exponentially weighted moving average, EWMA), it is a model for estimating the current/future conditional variance which, like EWMA, weights recent returns greater than distant returns (to update the estimate) but, additionally, assumes mean reversion. So, at the risk of simplification, a shorthand view can be that it gives a current/future estimate of conditional variance in a manner consistent with EWMA (both weighting returns exponentially) and then just adds a term for mean reversion. This omits much of the theory/assumptions that underlying it as a model (and truly distinguish it from EWMA); i.e., GARCH(1,1) is "just a model," a set of assumptions about a stochastic process, as volatility itself is statistic inferred from a price series.

In the FRM, fwiw, we first consider vanilla volatility (simple standard deviation). But it's weakness is all returns are equally weighted. So EWMA overcomes this with exponentially weighting (the volatility estimate is more influenced by recent returns). Then GARCH generalizes on EWMA by adding the mean reversion term (a weighed unconditional, or long-run, variance) for supposed improved realism.

It would take me too long to address your points in detail, sorry. Here are my favorite references, in order of accessibility (most accessible to least)

• FRM Assigned John Hull Chapter 22 (Estimating Volatility)
• Carol Alexander's MRA Vol II is good mix of practical + theory:
  http://www.amazon.com/Market-Analysis-Practical-Financial-Econometrics/dp/0470998016
• Stephen Taylor's Asset Price Dynamics, is I think the theoretical authority http://www.amazon.com/Asset-Price-Dynamics-Volatility-Prediction/dp/0691134790

Here is an old video I recorded (4+ years ago, yikes)

Good luck!

 

[anique]

Hello David,

Thank u so much for d response and thank u for those references. I m still confused about some things regarding GARCH (1, 1). There are two distributed lags used to explain variance under GARCH models, one on lag squared returns to capture high frequency effects and second on lagged values of variance itself to capture long term effects.
what i dont understand is dat what is the meaning of lag. Secondly what is meant by on lag squared returns to capture high frequency effects and second lag (1, 1).
Third thing which i m really really confused about is the equation of GARCH which is ( variance = omega+ submission alpha error square + submission beta variance)
m sorry for writing in words form, i coudnt write it in actual form. here`s the reference for this equation:
http://www.ijbssnet.com/journals/Vol_2_No_23_Special_Issue_December_2011/13.pdf
on page no. 118 there is last equation which says (The general specification of GARCH is, GARCH (p, q) is as


this equation i have to put in my slides as well. i would be really thankful to you if u can plz throw some light upon this equation. its a humble request. i have tried my level best to understand this equation but all in vain.

so if u can answer these questions then it will be very very helpful for my presentation.

kindly respond me again ASAP.

defense is 2 weeks away.

thank u again

PS: the video was not there. it was blank

 

[anique]

i m attaching a file for ur convenience. all i want u to do is dat kindly explain me what it means. i have to put this material in my thesis as well as presentation

i wud b waiting for ur response again. plz help me out. in need of desperate help regarding the file i have attached

 

[anique]

sorry for bothering u again Mr David, but i m really tensed about the material which is in the file which is attached. just explain me this material and i would be really really thankful to u for this help. submission and presentation is just 2 weeks away

 

[David Harper CFA FRM]

Hi anique,

It's just too time consuming for me to try to explain all of GARCH. Here is a quick way to look at it:

• a "vanilla" historical variance is given by variance(t) = [r(t-1)^2 + r(t-2)^2 + r(t-n)^2]/n = 1/n*r(t-1)^2 + 1/n*r(t-2)^2 + ... + 1/n*r(t-n)^2;
  where r(t-1) is the most recent (i.e., lagged once) return and (n) is the number of returns in the series
  i.e., historical variance is the average squared return, such that each historical return is equally weighted by 1/n
• The glaring problem is that distant returns have the same weight, 1/n, as the most recent. EMWA overcomes this by assigning a smoothing paramater (lambda). For example, if lambda = 94%, then rather than 1/n weights for all returns, the EWMA variance(t) = (1-94%)*r(t-1)^2 + (1-94%)*94%*r(t-2)^2 + (1-94%)*94%^2*r(t-3)^2 + ...
  Under this quick view method, this step here is the key insight (IMO): the variance estimate is now based on returns that are declining in weight, in constant ratio, by lambda (eg, 94%)
  Importantly, this (infinite) series reduces to the recursive, but equivalent, EWMA variance = lambda*variance(n-1) + (1-lambda)*r(t-1)^2.
  We can look at this, just for illustration sake, as EWMA that recursively "lags" on one variance and one return. Nobody does this, but we can superficially call this EWMA(1,1). But again, keep in mind, it is just the mathematical reduction of the infinite series.
• In EWMA, there are two weights. Weights sum to 1.0. In EWMA, lambda + (1-lambda) = 1.0. Now let's do the following to EMWA
  • add a third term for mean reversion, and call the weight gamma
  • call lambda instead beta
  • call (1-lambda) instead alpha
• Now we have: GARCH(1,1) = gamma*long_run_variance + beta*variance(t-1)^2 + alpha*r(t-1)^2
  • The updated variance estimate is a function of an unconditional (long run) variance weighted by gamma, PLUS an (alpha+beta) weight applied to the historical returns series, where the weights are declining in constant ratio by beta (~ lambda in EWMA). Alpha + beta + gamma = 1.0.
  • The alpha*r(t-1)^2 term is merely updating the series to include the most recent return

I hope that helps, the above simplifies by too much suggesting that GARCH is just like EWMA, but it's the quickest way i know to grasp it (and it's illustrated in the YT video, the paper you included is actually (IMO) a good introduction.

Actually, my description above is much like my youtube video, which i can view just fine, but just search youtube for "GARCH(1,1) to estimate volatility"

Thanks,

 

[anique]

hello David,

thank u once again for d response n giving a nice detailed explaination.
i jus wanted to ask that the information u have provided above, is it related to the file which i attached for ur convenience? about the equation?
i will go through this information n c if i m able to grasp it
and if i dont understand anything then i will surely get back to u
thanks

 

[David Harper CFA FRM]

Hi anique,

Yes, the only difference from your file is that your file gives GARCH(1,1) = omega + beta*variance(t-1)^2 + alpha*r(t-1)^2, such that omega = gamma*long-run variance. I think it's easier to break-down the omega term; e.g., we can visually check that total weights (alpha + beta + gamma) should equal 1.0; and then it is "parallel" with the other terms (weight * factor) so we don't confuse omega with the long-run variance. Thanks,

 

[anique]

Hi David,

thank u again.
i will go through this again n again in order to have a clear understanding. lets c how much i can understand. i will get back to u again in a day or two.
thank u so much David