I have a sample, log-returns of a finance time series. The sample size is n=252. I'd like to estimate a distribution in order to modeling dependence with a copula.

The binned histogram of frequencies, cumulative frequency counts, and qq-plot are below: enter image description here

I have applied the normality hypothesis test (Anderson-Darling test) and the null hypothesis was rejected at level 0.05 (p-value is 0.003905). I have applied the Kolmogorov-Smirnov test and the null hypothesis can't rejected at level 0.05 (p-value is 0.21769). But the null hypothesis should be rejected based on the Shapiro-Wilk test (p-value is 0.211e-4), the Chen-Shapiro test (5% critical value is 0.00356). Original data is here CFC.csv

mydata = read.csv("CFC.csv")  # read csv file 

log_returns <- diff(log(mydata$Close), lag=1)
n <- length(log_returns)
mydata$log_ret = with(mydata, c(NA, diff(log(mydata$Close))))
hist(log_returns, freq = TRUE, label=TRUE, breaks = 14, xlim=c(-0.08,0.08))
curve(dnorm(x, mean=mean(log_returns), 
               sd=sd(log_returns)), add=TRUE, col="darkblue", lwd=2)

Fn = ecdf(log_returns) 
   main="CDF of log_returns", 
   ylab="Cumulative Frequency") 

# Shapiro-Wilk normality test

# data:  log_returns
# W = 0.97509, p-value = 0.000211

# required 
# Anderson-Darling normality test

# data:  log_returns
# A = 1.1991, p-value = 0.003905

Problem. Data is not normal. Add I don't know how to specify a distribution to modeling dependence with a copula.

My attempt is: I have applied the Johnson's transformation to normalize the data.

# require
#Applying Johnson transformation 

#[1] 0.2304269

#[1] 0.1002716

#[1] 0.03428872

#[1] 0.00456466

#[1] 2.109682

hist(log_returns_JT$transformed, freq = TRUE, label=TRUE)

curve(dnorm(x, mean=mean(log_returns), 
               sd=sd(log_returns)), add=TRUE, col="darkblue", lwd=2)

curve(dnorm(x, mean=mean(log_returns_JT$transformed), 
               sd=sd(log_returns_JT$transformed)), add=TRUE, col="red", lwd=2)

Fn = ecdf(log_returns_JT$transformed) 
   main="CDF of log_returns_JT", 
   ylab="Cumulative Frequency") 

#Shapiro-Wilk normality test

#data:  log_returns_JT$transformed
#W = 0.99465, p-value = 0.5217

#Anderson-Darling normality test

#data:  log_returns_JT$transformed
#A = 0.48099, p-value = 0.2304

# [1] 0.2304269

Here are the binned histogram of frequencies, cumulative frequency counts, and qq-plot after the Johnson's transformation: enter image description here

I have applied the normality hypothesis test (Anderson-Darling test) and the null hypothesis can't reject at level 0.05 (p-value is 0.2304269).

Edit. I'm reading the papers recommended by @user25459 but, unfortunately, I don't find the answer for my question yet.

Questions. Is it possible to use a normal copula to modeling dependence and set the marginal distribution as the normal distribution?

After the modeling I'm going to apply the inverse transformation.

  • 4
    $\begingroup$ From what you have told us we have no reason to believe in a particular form for the density. One thing about goodness of fit tests to keep in mind is that in small samples it is difficult to reject normality if the true distribution is nearly normal . In very large samples it is easy to reject normality because even small departures from normal can be detected. It may be that you only need to be close to normal to meet the requirement for statistical methods that assume normality will work well. In your case 250 is a moderate sample size. The histogram looks approximately normal. $\endgroup$ Jan 10, 2017 at 3:36
  • 1
    $\begingroup$ Also you only barely rejected at the 0.05 level and the Anderson-Darling statistic provide one of the most powerful tests when the null distributionis normal. $\endgroup$ Jan 10, 2017 at 3:38
  • $\begingroup$ what is the data on axis A. Further, may be you should give graph ndicating cumulative frequency. $\endgroup$
    – user10619
    Jan 10, 2017 at 3:43
  • 1
    $\begingroup$ It's easier to see the departure from normality with a QQ plot rather than an arbitrarily binned histogram. Also, please indicate what the analytic goal is because then one can evaluate if the observed departure from normality is even relevant. $\endgroup$
    – gammer
    Jan 10, 2017 at 4:11
  • 2
    $\begingroup$ To add to Glen_b point rejecting normality doesn't tell you that the distribution is not normal either. The fact is models are only approximations and if we say the "true" distribution has to be normal to apply normal theory we would be unable to apply much of parametric statistics. The same can be said for goodness of fit tests for the exponential distribution or any other parametric family. Results of goodness of fit tests deserves careful thought and is not the sole determinant of what inferential statistical tests should follow. $\endgroup$ Jan 10, 2017 at 13:35

2 Answers 2


There is now a proof that derives the complete set of asset and liability classes, though all are mixture distributions. You can find it at https://ssrn.com/abstract=2828744

The reason for the mixture is bankruptcy risk, merger risk, liquidity costs, and a stochastic budget constraint. Some debt instruments can be modeled using the normal or lognormal, but no equity-type asset can have a first moment. Some assets, such as antiques, have the ratio of two Gumbel distributions as its distribution so there is no analytic representation.

Generally, there is no distribution, although for the broad population of US equities, its remarkably close to a truncated Cauchy distribution. Multiply it by a very wide survival function and you'll get an extraordinarily good fit. You can think of a trade as a survivor of the budget constraint because, of course, not all orders fill.

The fundamental argument of the paper is that returns are not data, instead prices are data and returns are a transformation of prices. In particular, it is a future value divided by a present value with an appraisal error upon entry and exit, minus one. As such, all returns are ratio distrubutions. Zero coupons differ in that the payoff is essentially fixed, so part of the ratio is a constant.

For completeness, a population study was done on US trades from 1925-2013 to test the most important claims. It can be found at https://ssrn.com/abstract=2653151

edit I saw after my post it was log returns. It will be a potentially ugly transformation of the hyperbolic secant distribution. Really we should quit using the log approximation, the only reason the process started was the need to use slide rules.

edit 2 I crudely mapped your bins in Excel and it was close to the hyperbolic secant, though probably skewed. You can do the Johnson transformation because you can map any univariate standard distribution to any other univariate standard distribution, but you need to be careful that you do not forget that properties that govern are the properties of the untransformed data.

Consider the simplest possible case, the unrealistic case of Markowitz. The distribution of returns in a universe with only two equal sized equity assets would be $$\frac{1}{2\pi}\frac{\gamma}{(\gamma^2+(r_1-\mu_1)^2+(r_2-\mu_2)^2)^{1.5}}.$$ You could map this distribution to any bivariate distribution, but look closely at its properties.

$\gamma$ is not a vector, it is a scalar. If you add dimensions $\gamma$ will become $\gamma'$, but $\gamma'$ is still a scalar. The likelihood has spherical uncertainty and the density has spherical errors. You can map this to a normal distribution with a covariance matrix, but it will not be unique because you are mapping to a distribution with more parameters than exist in the raw data.

Let's imagine instead you map a Cauchy to a normal, with zero covariance on the off diagonals, that is $\sigma_{i,j}\equiv{0},\forall{i,j}$ and equal variances to assure circularity, that is $\sigma_{i,i}\equiv\sigma_{j,j},\forall{i,j}$. In the normal distribution, asset 1 and asset 2 are independent, but in the raw data they are not independent. You have preserved the shape, but not the dynamics. The Cauchy distribution is an example of a distribution where none of the dimensions are correlated, but none are independent.

There is no theoretical reason to not map the raw data to a copula function, but the "how" to do that is something I cannot answer. I have done very light reading on the topic as I have thought about it myself since the Cauchy distribution doesn't really appear except in a mixture, but I read enough to realize I would have to spend a lot of time on the topic.

My preference to solve the problem, if I ever get to it, would be to note that things such as bankruptcy and liquidity costs do covary with underlying factors and so I would use Bayesian model selection and tools such as marginalization to separate out the components.

  • 3
    $\begingroup$ I take issue with the general call to stop using logarithms. Logarithms have more uses than just ease of use with slide rules. Whenever you think in terms of multiplicative effects rather than additive effects the logarithm automatically pops up. The slide rule was invented because the logarithm is such a useful tool, not the other way around. $\endgroup$ Jan 10, 2017 at 8:55
  • 3
    $\begingroup$ I agree with @Maarten Buis. The usefulness of a logarithmic transformation is completely independent of the technology you might use to implement it. $\endgroup$
    – Nick Cox
    Jan 10, 2017 at 9:01
  • $\begingroup$ @MaartenBuis and Nick I have no issue with the use of the logarithm where it serves a theoretical role, as it does in economic growth models or in studying elasticity, but that isn't why it is generally used in economics. If you asked most "why did you use the log?" the answer would likely be that it was done because the last person did it. If economists were thinking in terms of multiplicative effects, that would be a different issue. $\endgroup$ Jan 10, 2017 at 15:08
  • $\begingroup$ The logarithmic approximation is not harmless in economics, because $\exp(\beta_{log})\ne\beta$ you would hope the shift in values would be very small, but it is not. For the population of returns in the CRSP universe, essentially every end of day trade from 1925-, the different between $\exp(\mu_{log})$ and $\mu$ is two percent per year and it understates risk by $\pm{4}\%$. If you are going to do any transformation, there should be a substantive reason. $\endgroup$ Jan 10, 2017 at 15:11

Questions. Is it possible to use a normal copula to modeling dependence and set the marginal distribution as the normal distribution?

After the modeling I'm going to apply the inverse transformation.

As a basic principle, it's certainly possible to transform margins to approximate normality and then fit a multivariate Gaussian, which is in effect fitting a Gaussian copula to the dependence relationship.

I don't know how to specify a distribution to modeling dependence with a copula.

Perhaps I misunderstand your intent here, I'm not sure; the point of a copula is to fit the dependence structure independent of the marginals. Usually in copula work this is achieved by transforming the marginals to approximate uniformity (via one of several different approaches) then fitting some copula, but in the case of the Gaussian copula that's not necessarily the easiest approach (nor even the most common).

I see no particular issue with using Johnson transformations if they're adequate to achieve very near normality. It shouldn't be worse than some of the other approaches people use.

However, for asset returns usually the Gaussian copula is regarded as unsuitable because the tail dependence in the Gaussian copula goes to 0, while in real asset returns can show substantial tail dependence (in particular, if the market goes south, like say in the GFC, correlated assets will tend to see very extreme results at the same time -- the Gaussian copula can't capture that.

Sometimes the t-copula is used to model this, since it does have tail dependence, but it doesn't capture the asymmetry (the dependence is not equally strong in the upper and lower tail). It may be adequate for some purposes, however.

  • $\begingroup$ Thanks for answer, for instance, in the study: Davari-Ardakani H. et al (2016) Multistage portfolio optimization with stocks and options. International Transactions in Operational Research. 23 (3), 593–622. onlinelibrary.wiley.com/doi/10.1111/itor.12174/pdf the Johnson transformation was used and its is one of the main parts of a simulation. $\endgroup$
    – Nick
    Jan 20, 2017 at 11:50
  • $\begingroup$ @Nick as I suggest in my answer, it's not the Johnson that concerns me ("I see no particular issue"), it's the Gaussian copula when modelling dependence in returns that worries me. $\endgroup$
    – Glen_b
    Jan 20, 2017 at 11:53

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