Blog Archives

Unit Root Tests

February 12, 2014
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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 further, there is also the...

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Personal Analytics with RSS Feeds

February 7, 2014
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Personal Analytics with RSS Feeds

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 hours, then it should mean that, indeed, I consider my blog as...

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Inference for ARMA(p,q) Time Series

January 30, 2014
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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) > set.seed(1) > e=rnorm(n) > for(t...

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Inference for MA(q) Time Series

January 29, 2014
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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 (auto)correlated, then it is more...

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Inference for AR(p) Time Series

January 28, 2014
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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 sets of parameters: the autoregressive...

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Bias of Hill Estimators

January 28, 2014
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Bias of Hill Estimators

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, i.e. (under additional assumptions on the...

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Causal Autoregressive Time Series

January 21, 2014
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Causal Autoregressive Time Series

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 (using simple Analysis arguments on series). From that...

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Visualizing Autoregressive Time Series

January 21, 2014
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Visualizing Autoregressive Time Series

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 <- layout(matrix(c(1,1,1,1,2,3,4,5), 2, 4, byrow=TRUE), respect=TRUE) + e=rnorm(n) + X=rep(0,n) + for(t in 2:n) X=phi*X+e + plot(X,type="l",ylab="") + abline(h=mean(X),lwd=2,col="red") + abline(h=mean(X)+2*sd(X),lty=2,col="red") + abline(h=mean(X)-2*sd(X),lty=2,col="red") + u=seq(-1,1,by=.001) + plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,ylim=c(-2,2),xlim=c(-2.5,2.5)) + polygon(c(u,rev(u)),c(sqrt(1-u^2),rev(-sqrt(1-u^2))),col="light yellow")...

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Statistical Interests in Large Cities

January 10, 2014
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Statistical Interests in Large Cities

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 did extract information about articles puslished in about 35 journals...

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Sequences defined using a Linear Recurrence

January 6, 2014
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Sequences defined using a Linear Recurrence

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 most simple case is the first order recurence, where...

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