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I have to plot autocorellation function of the rate of return which is given by the set of equations: $$ \left\{\begin{matrix} \varepsilon_t = \sigma_t e_t\\ \sigma_t=\sum_{k=1}^\infty \frac{B(p+k-1,q+1)}{B(p,q)}\frac{|\varepsilon_{t-k}|}{\lambda}\\ r_t = C + D\varepsilon_{t-1}+\varepsilon_t \end{matrix}\right. $$ where $e_t$ is random variable from Laplace distribution with $\mu=0, \lambda=\sqrt{1/2}$, $C, D, p, q$ are known constants.

The problem I face is that to plot autocorrelation defined by: $$ ACF(\tau)=\frac{E\big((|r_t|-\mu')(|r_{t+\tau}|-\mu')\big)}{\sigma^2} $$ (where $\sigma^2$ is a varance of $|r_t|$ and $\mu'$ is mean of $|r_t|$ after time $t$) I have the compute nominator explicitly or store some values in vectors (as I choosed to simulate the process in R).

So my code looks like:

library(smoothmest)

C <- 0.000608 
D <- 0.103
p <- 5.41
q <- 0.597
k <- 1
N <- 200000


lambd = sqrt(1/2)
sigma <- 1

r <- vector(mode = "numeric", length = N)
rnew <- vector(mode = "numeric", length = N)

varepsilon <-vector(mode = "numeric", length = N)
varepsilon[1] = sigma * rdoublex(1, mu = 0, lambda = sqrt(1/2))



for(t in 2:N) {
  sigma <- sigma + beta(p + k - 1, q + 1)/beta(p,q)/lambd * abs(varepsilon[t - k]);
  varepsilon[t] <- sigma * rdoublex(1, mu = 0, lambda = sqrt(1/2));
  r[t] <- C + D * varepsilon[t - 1] + varepsilon[t];
  k = k + 1;
}


rprzes <- vector(mode = "numeric", length = N)
mu <- vector(mode = "numeric", length = N)
Auto <- vector(mode = "numeric", length = N)

tau = 500

for(i in 1:(N-tau)) {
  rprzes[i] <- abs(r[i + tau]) 
}

for(j in 1:N){
  mu[j] <- sum(abs(r[1:j]))/j
}

p1 <- vector(mode = "numeric", length = N)
p2 <- vector(mode = "numeric", length = N)
for(k in 1:N) {
  p1[k] <- (abs(r[k]) - mu[k])
  p2[k] <- abs(rprzes[k]) - mu[k] 
}

for(l in 1:N){
  Auto[l] <- mean(p1[l] * p2[l])/sd(abs(r))^2
}

plot(Auto,type = "h", main = "", xlab="", ylab="")

But unfortunately the resulting plot is totally different form what I expect, and as far as I know the autocorelation function should not be $<0$ anywhere, but it is.

Before implementing by hand I tried to use R functions acf and pacf, but the results were nowhere close to desired one and I had not insight on what is going on, in particular I could not change $\tau$ value.

Anyone can help? Thanks in advance.

  • 0
    `acf(r)` should do exactly what you are looking for. You can change $\tau$ with the `lag.max` argument. E.g. `acf(r,lag.max = 100)` plots the autocorrelation function for $\tau = 1,\ldots, 100$. If you want the absolute value of $r$, simply do `acf(abs(r))`. – 2017-02-08
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    Another comment: consider using `forecast::Acf` instead of `stats::acf` to exclude lag 0 in the plot. – 2017-02-08
  • 0
    The blue dashed lines in the `acf` and `Acf` functions are the confidence bounds under the assumption that the observations are iid. – 2017-02-08

0 Answers 0