# MAT8181

## Stationarity of ARCH processes

In the context of AR(1) processes, we spent some time to explain what happens when is close to 1. if the process is stationary, if the process is a random walk if the process will explode Again, random walks are extremely interesting processes, with puzzling properties. For instance, as , and the process will cross...

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## Inference for ARCH processes

Consider some ARCH() process, say ARCH(), where with a Gaussian (strong) white noise . > n=500 > a1=0.8 > a2=0.0 > w= 0.2 > set.seed(1) > eta=rnorm(n) > epsilon=rnorm(n) > sigma2=rep(w,n) > for(t in 3:n){ + sigma2=w+a1*epsilon^2+a2*epsilon^2 + epsilon=eta*sqrt(sigma2) + } > par(mfrow=c(1,1)) > plot(epsilon,type="l",ylim=c(min(epsilon)-.5,max(epsilon))) > lines(min(epsilon)-1+sqrt(sigma2),col="red") (the red line is the conditional variance...

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## Independence and correlation

A short post to get back on a property I gave briefly in the MAT8595 class in January, and again in the MAT8181 class this week (to illustrate the distinction between weak and strong white noises). Recall that (real-valued) random variables and are independent if for all , Another characterization, for integrable variable is...

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## Seasonal Unit Roots

As discussed in the MAT8181 course, there are – at least – two kinds of non-stationary time series: those with a trend, and those with a unit-root (they will be called integrated). Unit root tests cannot be used to assess whether a time series is stationary, or not. They can only detect integrated time...

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## Linear ‘Prediction’ for AR Time Series

In the exercises for the MAT8181 course, there are two Exercises (16 and 17) about prediction and extrapolation based on MA(1) and AR(1) time series. But before discussing those exercises (I had some request for hints), I wanted to recall the definition of the linear prediction, where As discussed previously on this blog, we...

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## Seasonal, or periodic, time series

Monday, in our MAT8181 class, we’ve discussed seasonal unit roots from a practical perspective (the theory will be briefly mentioned in a few weeks, once we’ve seen multivariate models). Consider some time series , for instance traffic on French roads, > autoroute=read.table( + "http://freakonometrics.blog.free.fr/public/data/autoroute.csv", + header=TRUE,sep=";") > X=autoroute$a100 > T=1:length(X) > plot(T,X,type="l",xlim=c(0,120)) > reg=lm(X~T) >...

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## Filtering a Stationary Time Series

In the first part of the MAT8181 course, on linear (univariate) time series, I forgot to mention an important theorem. Let be a stationary time series, and a sequence of real numbers such that then the time series defined as is a stationary time series. Further, one can get easily that This result can be used, if necessary...

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## Identification of ARMA processes

Last week (in the MAT8181 course) in order to identify the orders of an ARMA process, we’ve seen the eacf method, and I mentioned the scan method, introduced in Tsay and Tiao (1985). The code below – to produce the output of the scan procedure – has been adapted from an old code by Steve...

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## Temperatures Series as Random Walks

Last year, I did mention in a post that unit-root tests are dangerous, because they might lead us to strange models. For instance, in a post, I did obtain that the temperature observed in January 2013, in Montréal, might be considered as a random walk process (or at leat an integrated process). The code...

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## Unit Root Tests

This week, in the MAT8181 Time Series course, we’ve discussed unit root tests. According to Wold’s theorem, if is (weakly) stationnary then where is the innovation process, and where is some deterministic series (just to get a result as general as possible). Observe that as discussed in a previous post. To go one step...

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## Inference for ARMA(p,q) Time Series

As we mentioned in our previous post, as soon as we have a moving average part, inference becomes more complicated. Again, to illustrate, we do not need a two general model. Consider, here, some process, where is some white noise, and assume further that . > theta=.7 > phi=.5 > n=1000 > Z=rep(0,n) >...

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## Inference for MA(q) Time Series

Yesterday, we’ve seen how inference for time series was possible. I started with that one because it is actually the simple case. For instance, we can use ordinary least squares. There might be some possible bias (see e.g. White (1961)), but asymptotically, estimators are fine (consistent, with asymptotic normality). But when the noise is...

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## Inference for AR(p) Time Series

Consider a (stationary) autoregressive process, say of order 2, for some white noise with variance . Here is a code to generate such a process, > phi1=.25 > phi2=.7 > n=1000 > set.seed(1) > e=rnorm(n) > Z=rep(0,n) > for(t in 3:n) Z=phi1*Z+phi2*Z+e > Z=Z > n=length(Z) > plot(Z,type="l") Here, we have to estimate two...

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## Defining Properly MA(∞) Time Series

In order to properly define series, we need to get back on some properties of infinite sequences, as briefly mentioned yesterday in the MAT8181 course. Consider some sequence . The sequence is said to be summable if is convergent, i.e. if the limit of exists when . From Cauchy criterion, converges if and only...

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## Triangle for Parameters of AR(2) Stationary Processes

We’ve seen yesterday conditions on so that the canonical process, , satisfying The condition is rather simple, since should be a triangular region. But the proof is a bit more tricky… Recall that we want to parametrize the region Since we have a true process, then . Our polynomial is here where ‘s are...

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