GARCH Model : Time Series Talk

[Music] hi everyone how's it going so in this video we're finally gonna start talking about the GARCH model so I made the video on the arch model a long time ago and the most frequent comment on that video was please make a video on the GARCH model it's taken me a little while but we're finally here and as long as it's taken between making the arch model video and this one the truth is the GARCH model is not that much more complicated than the arch model but that said I would understand the arch model first so if you haven't seen the arch model video please go ahead and watch that video and understand the arch model if you really don't want to watch it I will explain the basic basic idea of the arch model here but it would really help if you had a more in-depth understanding so how we're gonna understand the GARCH model to make it most easy on us is going to be starting from the relationship between the AR 1 model and the ARMA 1 1 model because that's going to be a very similar logical jump to going from the arch 1 model to the GARCH 1 1 model and that'll make our lives a lot easier so let's start talking again at the AR 1 role the AR 1 model the logical idea behind it even without talking about the math is saying that if I want to predict the time series on some given day I can use the time series lagged 1 day prior so basically I can predict where I'm gonna be tomorrow based on where I am today that is the basic idea of the AR 1 model and an equation we basically say that the value of the time series on any given day is some coefficient times the value of the time series prior one day and we add of course this epsilon sub T which is called a random error or sometimes a random innovation in a causal diagram which is in this red box here we say that if I knew two quantities if I knew those two quantities for sure I could tell you for sure what the value of the time series will be on any given day those two quantities are first the value of the time series on the previous day and second the random error or innovation on the current day of course the issue in time series modelling is we don't usually know the random error or innovation on a given day that's the difficulty but if we knew that error innovation and if we knew the value of the time series on the previous day I could tell you with full certainty what the value of the time series would be today keep that in mind now let's jump to the ARMA 1 1 model now remember ARMA 1 is a combination of the AR 1 model and the ma 1 model so if we look at the equation it says that the value of a time-series is equal to some coefficient times the value of the time series one day prior same thing as the AR one the new part is the MA one which is we also add some coefficient times the random error or innovation from the previous day and then we have to add the same thing which is the random error from today so looking at the causal diagram for the ARMA 1 1 model it's just one extra thing in there which is the random error or innovation from the previous day so what I say is if I knew three things if I knew the value of the time series yesterday the value of the innovation from yesterday and the value of the innovation from today I could tell you with full certainty what the time series would be today so that's the logical jump from AR to ARMA now let's go ahead and look at the arch 1 model the arch 1 model remember is basically a model that takes into account the volatility or how far away that time series is jumping and it takes that information into account to help predict what the next value will be of the time series the basic idea of the arch 1 model or the arch model in general is saying if I'm jumping a lot today like if my time series is very volatile today it's probably gonna be also pretty volatile tomorrow if it's not very volatile today it's a very steady then it'll probably be very steady tomorrow as well that's the basic idea of the arch 1 model so the arch 1 model is saying that the value of a time series is given by epsilon sub T which remember is white-noise times the square root of some constant plus another constant times the value of the time series yesterday squared and as we saw in the arch video this guy as complicated as it looks can be shown to be the standard deviation of the time series or the volatility of the time series on the given day which is why we're allowed to write a T the value of the time series as a function of the white noise and white noise multiplied by the volatility of the time series today so if I draw that in a causal diagram I basically say that the value of the time series today is affected by the random error today and the volatility today and since the volatility today is a function of the time so yesterday I have another little causal piece right here which says volatility today is affected by the value of the time series yesterday so go ahead and pause or rewind and convince yourself of this equation and this causal diagram keeping in mind the logical idea behind both of them now that means if I trace back in this diagram I can show you I can say that the time series today is a function of the innovation today and the time series yesterday because I'm just tracing back to all the leaves of the causal diagram so basically it's affected by these two things which are the same two things of the AR 1 model it's just that we're putting these things together in a slightly different way now here's a problem with the arch 1 model because why would we even need a GARCH model whatever that means if there wasn't some deficiency in the arch 1 model in the same way that why would we need an ARMA model if there wasn't something wrong with the AR model so the thing the issue with the arch model is that it can be something called bursty so this is a term that time series people like to use bursty just means that if I'm modeling a r21 process it'll be kind of constant and then it'll jump and then it'll kind of just go back to the regular state as that it'll jump again it'll go back to the regular State it was that so it has these kind of like bursts of volatility rather than persistent volatility which means if I'm trying to model something that has more persistent volatility let's say that like an example we used before was like ice cream sales let's say you're an ice cream salesman and you find that usually your returns are something like this but some once in a while they'll jump to a high value and kind of stay there for a couple days and then they'll go back to normal sometimes they'll jump to a really low value and kind of stay there for a couple days or maybe a couple weeks and then they'll go back to normal arch by itself isn't super great at dealing with that because it's good at modeling these bursts so we need something more we need something better that's where GARCH comes in so the GARCH 1:1 model looks like this and I have to fill in one piece but the part I've written so far looks exactly like the arch one model here's the new part plus some other constant times the volatility of the time series yesterday squared so this is fundamentally different from what we were doing before because before we were taking into account the volatility today and we're still doing that here because this piece is still Sigma sub T okay so the equation still holds but now if we look at the causal diagram we see that the value of the time series today is affected by this white noise of course it's affected by Sigma sub T which is here but Sigma sub T is made up of two different components now it's affected by two different things because if we look at this guy this square root which is Sigma sub teen we see that it's affected by the value of the time series yesterday but also any volatility yesterday so I'm gonna fill in here so we see that the basically the story in this part of the causal diagram is that your volatility today is affected by the value of your time series yesterday but also your volatility yesterday which logically makes sense right because it's saying that if your time series was really high yesterday and it was also really volatile yesterday it's probably gonna be really volatile today so that's why this makes sense logically now if I do the same thing here if I look at all the leaves of the causal diagram then I can say that my time series today is a function of the random error innovation today the time series yesterday which is what I had before but also the volatility yesterday and the fact that I now include the volatility from yesterday is going to make my resulting time series a lot less bursty and so it'll look more like this which means that when it kind of jumps it sort of stays there for a while before returning to normal when it jumps in the negative Direction is sort of stays there for a while before returning to normal and to kind of get an intuition for why that is based on the equations we are not only taking into account the value of the time series yesterday we're also taking into account the volatility yesterday so if it was really if it went far away from its average yesterday then we take that into account in to the volatility today so that it's still gonna remain volatile today and maybe the next day in the next week until eventually it goes back to its average near its mean and then maybe it goes in the negative direction and it kind of stays there for a while so because we take into account the volatility or standard deviation from previous time periods we are able to kind of propagate this volatility over time rather than just getting rid of it after one or two time periods as in the h1 model the last two things I want to talk about are where do these coefficients come from so this is called the arch 1/1 model because we take into account one time period from yesterday for the time series itself and the volatility so if I had a arch to to model it would be taking into account a t minus 1 and 18 minus 2 and then also Sigma t minus 1 and Sigma t minus 2 so basically just taking more previous time periods into account I think you're comfortable with that idea and the last thing I want to talk about is of course what does GARCH actually stand for I didn't want to lead with that because I think most time series models are named not very helpfully but remember arch stands for autoregressive conditional heteroscedasticity so auto regressive because we're taking to account previous volatility and conditional heteroscedasticity because we are dealing with volatility GARCH the last four letters are the same G just stands for generalized so basically it's just saying that we're not only taking to account the value of the time series at previous time periods we're also taking into account values of the volatility at previous time periods so that's what the generalized means although by itself is not very helpful in explaining anything so that is your GARCH model if you have any questions please leave them in the comments below thanks for watching

Channel: ritvikmath

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Id: inoBpq1UEn4

Length: 10min 25sec (625 seconds)

Published: Mon Jan 13 2020