ARMA(1,1) processes - introduction and examples

In this video I explain what is meant by an ARMA(1,1) process, and provide a couple of examples of processes which could be modelled as thus.
Check out www.oxbridge-tu... for course materials, and information regarding updates on each of the courses. Check out ben-lambert.co... for course materials, and information regarding updates on each of the courses. Quite excitingly (for me at least), I am about to publish a whole series of new videos on Bayesian statistics on youtube. See here for information: ben-lambert.co... Accompanying this series, there will be a book: www.amazon.co....

[@JesseMaurais]

For an assignment I was asked to write some R code to simulate an ARMA(p,q) process given the AR and MA coefficients. Working solely from the expression of the model in terms of Xt I came up with the following code in about an hour. Hopefully someone finds it useful -- or better! can point out any mistakes before I have to hand it in.
#!/usr/bin/Rscript --vanilla
# The code is simplified when we build {Xt} in reverse then return it re-reverse.
# Otherwise this ARMA simulation function is straight forward.
ARMA = function(phi, theta, n=100, init=0, burn=0)
{
m = n + burn # Optional burn-in
z = rnorm(m) # White noise sample
x = init
for (i in 2:m) # Generate the time series
{
# Autoregression component
p = min(length(x), length(phi)) # Prevent out-of-bounds
AR = sum(phi[1:p] * x[1:p])
# Moving-average component
q = min(i-1, length(theta)) # Prevent out-of-bounds
MA = sum(theta[1:q] * z[(i-1):(i-q)])
# Combined AR+MA component
new = AR + MA + z[i]
x = c(new, x) # x is built in reverse
}
# Truncate the burn-in
# Re-reverse x vector
# Return time series
ts(rev(x[1:n]))
}

[@SpartacanUsuals]

Hi Jesse, many thanks for your generous uploading of this code. I look forward to running this code in R. Many thanks again, Best, Ben

[@double_j3867]

+Jesse Maurais Thanks -- this is exactly why folks shouldn't disable comments for their instructional videos!

[@marchern8526]

the ARMA model is explained differently for each different source i looked at. So confusing

[@qiuxiaomu6746]

the funny thing is, TSA is more of an applied mathematical technique, compared to pure mathematical theories, it is so true that every instructor seems to find their own approach self-explanatory, however leaving us confused as **** when we try to cross-read multiple sources explaining the same concept.

[@rareclassicmovies6429]

I have a question. It is obvious that at time t, we have a shock due to E(t), but in the next time step and for the second example, can we strongly say that the curve goes up and then goes down by exponential decay? I mean in the next time step there are two different forces, rho and teta which are forcing in the opposite directions, so can't we say that the behavior of the curve depends on the relative amount of these two parameters? Thanks

[@hediyehb9082]

Thank you for this amazing video. I finally have an idea why ARMA is a good process to model ocean waves. Now JONSWAP is another model but it can be fitted with CARMA. I am reading this paper on it and I am very confused. Could you make a video about CARMA and how it expands on ARMA? And if I want to model something as a CARMA process, like say stochastic friction, where do I start? Do I start, like you did here by deciding what my rho and theta is? Can I then just write them in state space format and do normal calculations on them!

[@napolnzo6747]

if rho < 1, then why the chart line just decrease to what it was instead of keeping decreasing(rho

[@charliean9237]

Great video series about TS! Question: the y-axes of the two plots in the video are both the first order difference of the sales? or are they just sales? TIA!

[@lastua8562]

Just sales.

[@mdenayethossain6194]

@@lastua8562 Then, why does sales not decline below Xt given Xt = pX(t-1) and p is less than 1?

[@lastua8562]

@@mdenayethossain6194 I am not sure what you mean, but x does indeed decline compared to the previous period, and asymptotically approaches the starting value (or 0)

[@pierre-oliviermathieu5951]

Thanks a lot. Im currently doing my master degree in economics. Your videos saved my life a couple of times during my studies.

[@lastua8562]

upcoming. Any recommendations what to study specifically and what is important to study ahead?

[@guiriamine1536]

thank you for your explanation, but I have a question please .i want to model my time series ( stock returns) with ARMA-GARCH. IN residual analysis, I found some p-values for the Q test (Ljung-Box), are significant and others not, should I consider or reject H0? It's normal ??? I continue my work ?? I must differentiate the series? to pass to the GARCH modeling. THANK YOU

[@alfredalfredovic3076]

Thanks from Germany!

[@sudiptabanerjee6262]

MA contains the error term, how come you assigned it another parameter ?

[@lamineoussama5298]

HI Ben, thank you so much for this great video. I just have a question:
I'm using this model :
Y(t)=ρY(t-1)+β0+∑βi*X(i,t)+ε(t)
Which model is most appropriate to estimate it ?
Emergency ! and THANK YOU :)

[@insomniacdreamer5520]

ure a genius for writing out that equation dude

[@chrislam1341]

Would you kind also doing an video to explain what is arch and garch please?-?

[@SpartacanUsuals]

Hi Chris, thanks for your message. Yes, I am going to cover GARCH and ARCH. It will be part of the time series lectures which I put up in the graduate course. I will try to keep you informed when I do eventually put these up. Best, Ben

[@aleksanderdashi3645]

Ben Lambert please ben...have you done this upload of arch and garch?... can you also say smth about Multivariate garch models please?

[@FB-tr2kf]

Still waiting...

[@aidanmurphy4304]

great videos thanks a lot

[@KE61090]

good examples.

[@patty9399]

Thank you!

[@adeyinkaojedokun4411]

I have an assignment on this which is proving painful, is there anyway i can email you for some assistance please

[@rahulstat]

thanks

[@ArunKumar-yb2jn]

Need practical lessons through Excel or by showing data.

[@Game_Lab_Germany]

Meow

[@exiled9880]

i thought u were discussing arma game lol

[@milescooper3322]

Drop the frequent use of "sort of."

[@aq8827]

Thanks very much. Very helpful videos.... Could u please in next videos demonstrate ARFIMA and MS-ARFIM models?
Thanks

[@adithyapai6158]

What is exactly (roh) ¶ in AR and theta in MA? Correlation coefficient? Slope?If yes how is it caluclated for each lagged values ?

[@lastua8562]

Does this not disregard the effect of E_t in the previous period (contained in pho*x_t-1)?

[@MyMpc1]

Hi Ben, what do you think it would mean if roe and theta were equal? I saw it on a past exam paper and I didn't know the answer, so I came to this video.

[@ethanmay4573]

I believe that a property of this model is that for any time t-j, rho +theta= 1. So if they are equal that means they both =.5. Fact check this Im not completely sure

[@insomniacdreamer5520]

you came to this video, I came on this video