Original link:tecdat.cn/?p=19688
Original source:Tuo End number according to the tribe public number
When they were introduced, Copulas were generally considered interesting because they allowed modeling of edge distributions and dependent structures separately.
Copula models edges and dependencies
Given some edge distribution functions and a copula, we can generate a multivariate distribution function whose edges are specified above.
Consider a binary lognormal distribution
> library(mnormt)
> set.seed(1)
> Z=exp(rmnorm(25,MU,SIGMA))
Copy the codeWe can start with the edge distribution.
Meanlog sdlog 1.168 0.930 (0.186) (0.131) meanlog sdlog 2.218 1.168 (0.233) (0.165)Copy the codeBased on these edge distributions, and considering the maximum likelihood estimates of copula parameters obtained from this pseudo-random sample, numerically, we obtain
> library(copula) > Copula() estimation based on 'maximum likelihood' and a sample of size 25. Estimate Std. Error z The value (Pr > | z |) rho. 1 0.86530 0.03799 22.77Copy the codeHowever, since the dependence relation is a function of edge distribution, we do not treat the dependence relation separately. If the global optimization problem is considered, the results are different. You can figure out the density
> optim(par=c(0,0,1,1,0),fn=LogLik)$par
[1] 1.165 2.215 0.923 1.161 0.864
Copy the codeNot much difference, but not the same estimator. From a statistical point of view, it is almost impossible to deal with edge and dependent structures separately. The other thing we should keep in mind is that marginal distributions can be misspecified. For example, if we assume an exponential distribution,
Fitdistr (Z[,1],"exponential") Rate 0.222 (0.044) Fitdistr (Z[,2],"exponential")Copy the codeParameter estimation of gaussian Copula
Copula() estimation based on 'maximum likelihood' and a sample of size 25. Estimate Std. Error z value Pr(>|z|) rho.1 0.87421 0.03617 24.17 <2e-16 *** -- signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 '1 The Maximized loglikelihood is 15.4 Optimization convergedCopy the code
Copy the codeBecause we incorrectly specified the edge distribution, we could not get a uniform edge. If we use the code above to generate a sample size of 500,
barplot(counts, axes=FALSE,col="light blue"
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If the edge distribution is well defined, we can clearly see that the dependent structure depends on the edge distribution,
Copula simulates the correlated random walk in the stock market
Next, we use the Copula function to simulate the correlation random walk in the stock market
# * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # load historical data #****************************************************************** load.packages('quantmod') data$YHOO = getSymbol.intraday.google('YHOO', 'NASDAQ', 60, '15d') data$FB = getSymbol.intraday.google('FB', 'NASDAQ', 60, '15d') bt.prep(data, Align = 'remove. Na') # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # generated simulation # * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * rets = diff (log (prices)) # price matplot(exp(apply(rets,2,cumsum)), type='l')Copy the code
Plot (rets[,1], REts [,2], xlab= LABS [1], ylab= LABS [2], col='blue', las=1) points(fit.sim[,1], fit.sim[,2], Print (round(100*temp,2) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) print(round(100*temp,2)) Print (LABS [I]) print(ks.test(rets[, I], fit.sim[I])) exp(apply(fit.sim,2,cumsum)) Main ='Simulated Price path') # simulate Copula load.packages(' Copula ')Copy the code
Apply (rets,2,function(x) list(mean=mean(x), Sd =sd(x))) # Simulate rMvdc from fit distribution (4800, FIT)Copy the code
Actual Simulated Correlation 57.13 57.38 Mean Fb-0.31-0.47 Mean yhoo-0.40-0.17 StDev FB 1.24 1.25 StDev YHOO 1.23 1.23Copy the codeFB
Two-sample Kolmogorov-Smirnov test
data: rets[, i] and fit.sim[i]
D = 0.9404, p-value = 0.3395
alternative hypothesis: two-sided
Copy the codeHO
Two-sample Kolmogorov-Smirnov test data: REts [, I] and fit.sim[I] D = 0.8792, p-value = 0.4222 alternative hypothesis: two-sidedCopy the code
visualize.rets(fit.sim)
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# qnorm(runif(10^8)) and rnorm(10^8) are equivalent uniforms.sim = rCopula(4800, gumbelCopula(gumbel@estimate, dim=n))Copy the code
Actual Simulated Correlation 57.13 57.14 Mean Fb-0.31-0.22 Mean yhoo-0.40-0.56 StDev FB 1.24 1.24 StDev YHOO 1.23 1.21Copy the codeFB
Two-sample Kolmogorov-Smirnov test
data: rets[, i] and fit.sim[i]
D = 0.7791, p-value = 0.5787
alternative hypothesis: two-sided
Copy the codeHO
Two-sample Kolmogorov-Smirnov test data: REts [, I] and fit.sim[I] D = 0.795, p-value = 0.5525 alternative hypothesis: two-sidedCopy the code
vis(rets)
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The standard deviation is very large relative to the mean, close to zero; Therefore, in some cases, we are likely to get unstable results.
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