Gold price and fear

Via @theEconomist, I understood that there might be connections between the price of Gold (which is said to be extremely high nowadays) and the VIX SP500 index (the option volatility index, i.e. the so-called “fear index“, as discussed – in French- a few months ago). This has been discussed also on several blogs, e.g. or Via Yahoo quotes, it is possible to get also easily the

SP500 VIX index.

> library(tseries)
> X=get.hist.quote("^VIX")
> T=time(VIX)
> Y=as.POSIXlt(T)$year+1900
> X2011=X[Y==2011,]
> VIX=X2011[,4]
> VIX100=as.numeric(VIX)/VIX[1]*100
> T2011=T[Y==2011]
> plot(T2011,VIX100,lwd=2,col="red",type="l",
+ xlab="",ylab="",ylim=c(60,290))

And a huge xls file can give us the price of gold (on a daily basis). But we can extract only one series (with the price in USD, which is the series of interest here)

> goldprice=read.table(
+ "
+ header=TRUE,sep=";",dec=",")
> T=as.Date(goldprice$Name,"%d/%m/%y")
> GP=goldprice$USdollar
> Y=as.POSIXlt(T)$year+1896
> GP2011=GP[Y==2011]
> GP100=GP2011/GP2011[1]*100
> T2011=T[Y==2011]
> lines(T2011-4*365.25,GP100,lwd=2,col="blue")

We can see that scales are quite different on those two series (starting at 100 at the beginning of January 2011),

An alternative might be not to consider the price of gold, but something more psychological, like Internet researches. It is possible to download the csv file for queries on gold price on Google, via google insight.

> google=read.table(
+ "",
+ skip=4,header=TRUE,sep=",",nrows=51)
> W=as.Date(substr(as.character(google$Semaine),1,10))
> G=google$gold.price
> G100=G/G[1]*100
> lines(W,G100,lwd=2,col="blue")

which gives the following graph (again, starting at 100 at the beginning of January 2011),

Here, we can clearly observe that the two series are related, maybe cointegrated. Nice isn’t it ?

Normalité des estimateurs dans une régression

Cette semaine en cours, on a évoqué la normalité des estimateurs dans une régression linéaire. En supposant la normalité des résidus, on a vu en cours que était un estimateur Gaussien. En particulier, chacun des estimateurs est alors Gaussien, au sens où, ce qui peut se visualiser sur le graphique suivant (la constante est en abscisse, et la pente en ordonnée), avec un intervalle de confiance à 95%,

En fait, la variance étant inconnue (mais pouvant être estimée), si on remplace la variance des résidus par leur estimateur, la loi de l’estimateur est une loi de Student. De même pour l’estimateur de la pente de la droite de régression,

Le fait que le couple soit également Gaussien fait que l’on peut construire non plus un intervalle de confiance (on n’est plus en dimension 1) mais une ellipse de confiance. On retient une forme elliptique car c’est la plus petite région dans laquelle on se trouvera avec une probabilité de 95% (comme discuté dans un vieux billet).

Mais pour mieux comprendre cette notion d’ellipse de confiance, le plus simple est d’avoir recours à du rééchantillonnage dans la base de données. En effet, comme on ne dispose que d’un jeu de données, on a un estimateur. Et la discussion sur la loi de notre estimateur est purement théorique. Pour visualiser la loi de notre estimateur, il faudrait des centaines, voire des milliers de bases similaires. Que l’on n’a pas. La solution est alors de tirer au hasard des points de notre échantillon (avec remise), i.e. de faire du bootstrap,

for(s in 1:nrow(COEF)){

Si on regarde ce qui se passe, tirage après tirage, on génère un paquet d’échantillon, et pour chaque échantillon, on ajuste une droite de régression.

Les distributions – sur tous ces échantillons – de la constante et de la pente semblent effectivement Gaussiennes,

hist(COEF[,1],col="light blue",prob=TRUE)
hist(COEF[,2],col="light blue",prob=TRUE)

pour la distribution de l’estimateur de la pente, alors que pour l’estimateur de la constante, on a la distribution suivante,

Si on regarde maintenant la loi jointe,

on retrouve une forme elliptique pour le nuage des points (et une forte corrélation négative entre les deux estimateurs). Mais si on creuse un peu

polygon(e,col="light blue")

l’ellipse ne coïncide pas (tout à fait) avec celle obtenu théoriquement. Ce qui laisse à penser que l’hypothèse de normalité des résidus est peut-être à revoir..,

Fisher-Tippett theorem with an historical perspective

A couple of weeks ago, Rafael asked me if I had something on the history of extreme value theory. Since I will get back to fundamental results about extremes in my course, I promised I will write down a short post on all that issue.

To start from the beginning, in 1928, Ronald Fisher and Leonard Tippett formulated the three types of limiting distributions for the maximum term of a random sample (Fisher & Tippett (1928)). The problem was to characterize function such that

where where‘s are i.i.d. with cumulative distribution function They had supporting arguments, but no (rigorous) proof. Nevertheless, the obtained that the only possible types for G were

i.e. Fréchet type (Pareto-type tails), or

i.e. Weibull type (bounded distribution type), or

i.e. Gumbel type (exponential-type tails). Emil Gumbel has been intensively using the so-called Gumbel distribution on river flows, since (as he explained in 1958), “it seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis“.
Independently of that work (published in 1928), Maurice Fréchet considered in 1927 (in Sur la loi de probabilité de l’écart maximum) possible limits of

and obtained only as possible limit. Richard von Mises gave in 1936 sufficient, but not necessary conditions for their (max) domain of attraction, i.e. characterization of function such that the maxima converges to some specific function (von Mises (1936)). E.g. he noticed that a sufficient condition on be in the (max) domain of attraction of the Gumbel distribution is that

Then in 1943, Boris Gnedenko gave a complete characterization of those three types, with a complete characterization for two of them (heavy tails, i.e. Fréchet type and bounded support, i.e. Weibull) but his necessary and sufficient condition was based on a function that was not explicitly defined (see Gnedenko (1943)). Laurens de Haan in the 70’s derived checkable condition for Gumbel’s type.
Boris Gnedenko proved (in Section 4 of his paper) that F is the (max) domain of attraction of if and only if is regularly varying at infinity, with index (even if the term “regular variation” was not mentioned in the paper). Similar results were derived to characterize functions in the (max) domain of attraction of Weibull. For the (max) domain of attraction of, Boris Gnedenko obtained that a necessary and sufficient condition was that there exists a function such goes to 0 at infinity and

Several papers have discussed what function could be e.g. David Mejzler in 1949 (in Russian, but see also his 1965 paper), and Laurens de Hann in 1970 and 1971 (following the dramatic flood in the Netherlands in 1953, researchers in the Netherlands have focuses on dikes, and extreme value applications).

Mejzler’s idea was to work on quantiles, and not on the cumulative distribution function. I.e. define

Then a necessary and sufficient condition for F to be in the (max) domain of attraction of is that

Laurens de Haan proved in 1971 that function can be – in general – given by

And in 1976, Laurens de Haan obtained a three-type convergence working on quantile function (with a much shorter proof).
There have been many many papers extending Fisher-Tippett’s theorem, e.g. on non-independent sequences, like exchangeable ones (in a paper by Simeon Berman in 1962, or on stationary Gaussian sequences in 1964).