# Lissage exponentiel et séries temporelles

L’idée du lissage exponentiel est de supposer que , où on suppose que le bruit est ici un bruit indépendant (imprévisible). On va alors supposer que où désigne un poids attribué à la nouvelle observation dans la fonction de lissage .

On parle de lissage pour moyenne mobile à poids exponentiel (ewma, exponentially weighted moving average, évoqué ici ou  sur ce blog). En effet i.e., en itérant (même s’il faut faire attention à cette décomposition qui, en pratique, s’arrête avec la première observation).

La méthode de Holt-Winters propose de généraliser ce lissage, en introduisant deux (voire trois) composantes.

• sera interprété comme un niveau
• correspondant à la variation du processus
• qui correspond à un cycle (éventuel)

On suppose désormais, dans cette méthode (dite de Holt-Winters), que et D’un point de vue plus pratique, la programmation se fait comme suit,

```> HW=HoltWinters(X,alpha=.2,beta=0)
> plot(HW)``` Cette trois composantes seront utilisées pour faire une prévision de la série: la prédiction sera de la forme , à la date , pour un horizon (avec les notations du précédant billet).

```> P=predict(HW,24,prediction.interval=TRUE)
> plot(HW,xlim=range(c(time(X),time(P))))
> polygon(c(time(P),rev(time(P))),c(P[,2],rev(P[,3])),
+ col="yellow",border=NA)
> lines(P[,1],col="red",lwd=3)``` # Short selling, volatility and bubbles Yesterday, I wrote a post (in French) about short-selling in financial market since some journalists claimed that it was well-known that short -selling does increase volatility on financial market. Not only in French speaking journals actually, since we can read on http://www.forbes.com that  « in a market with restrictions on short-selling, volatility is reduced ». But things are not that simple. For instancehttp://www.optionsatoz.com/ explains it from a theoretical point of view. But we can also look at the data. For instance, we can compare the stock price of Air China, exchanged in Shanghai in blue (where short-selling is forbidden) and in Hong Kong in rouge (where short-selling is allowed), since @Igor gave me the tickers of those stocks

```library(tseries)
X<-get.hist.quote("0753.HK")
Y<-get.hist.quote("601111.SS")
plot(Y[,4],col="blue",ylim=c(0,30))
lines(X[,4],col="red")``` But as @alea_ pointed out, one asset is expressed here in Yuans renminbi, and the other one in HK dollars. So I downloaded the exchange rate fromhttp://www.oanda.com/ ```Z=read.table("http://freakonometrics.blog.free.fr/public/
D=as.Date(as.character(Z\$date),"%d/%m/%y")
z=as.numeric(Z\$CNY.HKD)
plot(D,z,type="l")
X2=X[,4]
for(t in 1:length(X2)){ X2[t]=X2[t]*z[D==time(X2[t])]}
X2=X[,4]
plot(Y[,4],col="blue",ylim=c(0,30))
lines(X2,col="red")``` Now both stocks are expressed in the same currency. To compare returns volatility, a first idea can be to use GARCH models,

```RX=diff(log(X2))
RY=diff(log(Y[,4]))
Xgarch = garch(as.numeric(RX))
SIGMAX=predict(Xgarch)
Ygarch = garch(as.numeric(RY))
SIGMAY=predict(Ygarch)
plot(time(Y)[-1],SIGMAY[,1],col="blue",type="l")
lines(time(X2)[-1],SIGMAX[,1],col="red")``` But volatility is here too eratic. So an alternative can be to use exponentially-weighted moving averages, where simple recursive relationships are considered or equivalently The code is not great, but it is easy to understand,

```moy.ew=function(x,r){
m=rep(NA,length(x))
for(i in 1:length(x)){
m[i]=weighted.mean(x[1:i],
rev(r^(0:(i-1))))}
return(m)}
sd.ew=function(x,r,m){
sd=rep(NA,length(x))
for(i in 1:length(x)){
sd[i]=weighted.mean((x[1:i]-m[i])^2,
rev(r^(0:(i-1))))}
return(sd)}
q=.97
MX=moy.ew(RX,q)
SX=sd.ew(RX,q,MX)
MY=moy.ew(RY,q)
SY=sd.ew(RY,q,MY)
plot(time(Y)[-1],SY,col="blue",type="l")
lines(time(X2)[-1],SX,col="red")``` And now we have something less erratic, so we can focus now on the interpretation.
It is also possible to look on the difference between those two series of volatility, areas in blue means that in Shanghai (again, where short-selling is forbidden) returns are more volatile than in Hong Kong, and areas in red are periods where returns are more volatile in Hong Kong,

```a=time(X2)[which(time(X2)%in%time(Y))]
b=SY[which(time(Y)%in%time(X2))]-
SX[which(time(X2)%in%time(Y))]
n=length(a)
a=a[-n];b=b[-n]
plot(a,b,col="black",type="l")
polygon(c(a,rev(a)),c(pmax(b,0),rep(0,length(a))),
col="blue",border=NA)
polygon(c(a,rev(a)),c(pmin(b,0),rep(0,length(a))),
col="red",border=NA)``` So clearly, there is nothing clear that can be said… Sometimes, volatility is higher in Hong Kong, and sometimes, it is higher in Shanghai. But if we look at the price, instead of looking at volatility,

```a=time(X2)[which(time(X2)%in%time(Y))]
b=as.numeric(Y[which(time(Y)%in%time(X2)),4])-
as.numeric(X2[which(time(X2)%in%time(Y))])
n=length(a)
a=a[-n];b=b[-n]
plot(a,b,col="black",type="l")
polygon(c(a,rev(a)),c(pmax(b,0),rep(0,length(a))),
col="blue",border=NA)
polygon(c(a,rev(a)),c(pmin(b,0),rep(0,length(a))),
col="red",border=NA)``` Here, it looks like bans on short-selling creates bubbles. Might not not be a goodthing.