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 ... [Read more...]

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/... [Read more...]

I am still working with @3wen on visualizations of the North Pole. So far, it was not that difficult to generate maps, but we started to have problems with the ice region in the Arctic. More precisely, it was complicated to compute the area of this region (even if we ... [Read more...]

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 ... [Read more...]

On the Monkey Cage blog, Baptiste Coulmont (a.k.a. @coulmont) recently uploaded a post entitled “You can vote twice ! The many political appeals of proxy votes in France“, coauthored with Joël Gombin (a.k.a. @joelgombin), and myself. The study was initially written in French as mentioned in ... [Read more...]

This Monday, in the ACT8595 course, we came back on elliptical distributions and conditional independence (here is an old post on de Finetti’s theorem, and the extension to Hewitt-Savage’s). I have shown simulations, to illustrate those two concepts of dependent variables, but I wanted to spend some time ... [Read more...]

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 ... [Read more...]

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 ... [Read more...]

I am currently working on a paper on Academic Blogging, from my own experience. And I wanted to do something similar to Stephen Wolfram’s personal analytics of my life. More specifically, I wanted to understand when I do post my blog entries. If I post more entries during office ... [Read more...]

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) __ ... [Read more...]

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). ... [Read more...]

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[t]=phi1*Z[t-1]+phi2*Z[t-2]+e[t] __ Z=Z[800:1000] __ ... [Read more...]

In the MAT8595 course, we’ve seen yesterday Hill estimator of the tail index. To be more specific, we did see see that if , with , then Hill estimators for are given by for . Then we did say that satisfies some consistency in the sense that if , but not too fast, ... [Read more...]

In the MAT8181 graduate course on Time Series, we will discuss (almost) only causal models. For instance, with , with some white noise , those models are obtained when . In that case, we’ve seen that was actually the innovation process, and we can write which is actually a mean-square convergent series (... [Read more...]

In the MAT8181 graduate course on Time Series, we started discussing autoregressive models. Just to illustrate, here is some code to plot  – causal – process, __ graphar1=function(phi){ + nf graphar1(.8) or __ graphar1(-.7) (with, on the […] [Read more...]

I always thought that there were some kind of schools in statistics, areas (not to say universities or laboratories) where people had common interest in term of statistical methodology. Like people with strong interest in extreme values, or in Lévy Processes. I wanted to check this point so I ... [Read more...]

In the introduction to the time series course (MAT8181) this morning, we did spend some time on the expression of (deterministic) sequences defined using a linear recurence (we will need that later on, so I wanted to make sure that those results were familiar to everyone). First order recurence The ... [Read more...]

In my previous post, I tried to answer the following question Consider  points uniformly distributed on a sphere. What is the probability that the  points lie on a same hemisphere, for some hemisphere (there is no south or north here) ? If I have been able to use Monte Carlo simulations ... [Read more...]

This week, I spend some time at the Workshop on Nonparametric Curve Smoothing conference at Concordia. Yesterday afternoon, Noël Veraverbeke show an interesting graph, to illustrate conditional copulas (and the derivation of conditional dependence measures, such as Kendall’s tau, or Spearman’s rho). A long time ago, in ... [Read more...]

There is something that I love about mathematics: sometimes, you discover – by chance – a law. It has always been there, it might have been well known by some people (specialized in some given field), but you did not know it. And then, you discover it, and you start wondering how ... [Read more...]