MLE and Quantile Evaluation for a Clayton AR(1) Model with Student Marginals

Original demo from Roeger Koenker; by Marius Hofert, Ivan Kojadinovic, Martin Mächler, Jun Yan

2016-09-24

require(copula)
set.seed(271)

First, let’s fix some parameters.

mu <- 0
sigma <- 1
df <- 3
alpha <- 10

For the marginals, we will use location scale transformed Student distributions.

rtls <- function(n, df, mu, sigma) sigma * rt(n,df) + mu
ptls <- function(x, df, mu, sigma) pt((x - mu)/sigma,df)
qtls <- function(u, df, mu, sigma) sigma * qt(u,df) + mu
dtls <- function(u, df, mu, sigma) dt((x - mu)/sigma,df)/sigma

Generating the data

Let’s generate some data.

rclayton <- function(n, alpha) {
    u <- runif(n+1) # innovations
    v <- u
    for(i in 2:(n+1))
            v[i] <- ((u[i]^(-alpha/(1+alpha)) -1)*v[i-1]^(-alpha) +1)^(-1/alpha)
    v[2:(n+1)]
}
n <- 200
u <- rclayton(n, alpha = alpha)
u <- qtls(u, df=df, mu=mu, sigma=sigma)
y <- u[-n]
x <- u[-1]

We now estimate the parameters under known marginals

fitCopula(claytonCopula(dim=2),
          cbind(ptls(x,df,mu,sigma), ptls(y,df,mu,sigma)))
## fitCopula() estimation based on 'maximum pseudo-likelihood'
## and a sample of size 199.
##       Estimate Std. Error
## param    11.08      0.858
## The maximized loglikelihood is 226.5 
## Optimization converged
## Number of loglikelihood evaluations:
## function gradient 
##       17        6

Estimation under unknown marginal parameters

## Identical margins
M2tlsI <- mvdc(claytonCopula(dim=2), c("tls","tls"),
               rep(list(list(df=NA, mu=NA, sigma=NA)), 2), marginsIdentical= TRUE)
(fit.id.mar <- fitMvdc(cbind(x,y), M2tlsI, start=c(3,1,1, 10)))
## The Maximum Likelihood estimation is based on 199 observations.
## Identical margins:
##         Estimate Std. Error z value Pr(>|z|)    
## m.df     3.90978    0.88657   4.410 1.03e-05 ***
## m.mu     0.40952    0.05586   7.331 2.29e-13 ***
## m.sigma  0.73199    0.05238  13.974  < 2e-16 ***
## Copula:
##       Estimate Std. Error z value Pr(>|z|)    
## param   5.9379     0.5904   10.06   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## The maximized loglikelihood is -338.2248 
## Optimization converged
## Number of loglikelihood evaluations:
## function gradient 
##       46       15
## Not necessarily identical margins
M2tls <- mvdc(claytonCopula(dim=2), c("tls","tls"),
              rep(list(list(df=NA, mu=NA, sigma=NA)), 2))
fitMvdc(cbind(x,y), M2tls, start=c(3,1,1, 3,1,1, 10))
## The Maximum Likelihood estimation is based on 199 observations.
## Margin 1 :
##          Estimate Std. Error
## m1.df      3.7851      1.033
## m1.mu      0.4150      0.057
## m1.sigma   0.7288      0.059
## Margin 2 :
##          Estimate Std. Error
## m2.df      4.0760      1.198
## m2.mu      0.4046      0.058
## m2.sigma   0.7358      0.060
## Copula:
##       Estimate Std. Error
## param    5.944      0.591
## The maximized loglikelihood is -338.0503 
## Optimization converged
## Number of loglikelihood evaluations:
## function gradient 
##       63       18

Plot some true and estimated conditional quantile functions

u.cond <- function(z, tau, df, mu, sigma, alpha)
    ((tau^(-alpha/(1+alpha)) -1) * ptls(z,df,mu,sigma)^(-alpha) + 1) ^ (-1/alpha)
y.cond <- function(z, tau, df, mu, sigma, alpha) {
    u <- u.cond(z, tau, df, mu, sigma, alpha)
    qtls(u, df=df, mu=mu, sigma=sigma)
}
plot(x, y)
title("True and estimated conditional quantile functions")
mtext(quote("for" ~~  tau == (1:5)/6))
z <- seq(min(y),max(y),len = 60)
for(i in 1:5) {
    tau <- i/6
    lines(z, y.cond(z, tau, df,mu,sigma, alpha))
    ## and compare with estimate:
    b <- fit.id.mar@estimate
    lines(z, y.cond(z, tau, df=b[1], mu=b[2], sigma=b[3], alpha=b[4]),
          col="red")
}