Generation of Time Series
This section describes how to generate a time series such as AR(1) with non-linear structure of dependence, defined by using a two-dimensional copula C(u, v). The series is stationary, but the procedure can be easily generalized to non-stationary series. Thus, let u = F(xt), v = F(xt -1). According to (5) the joint distribution of the time series is as follows:
F (xi, X2... Xt) = C(X1,X2)1C(X2,X3)1 . ..C(xr-2 ,XT-l):C (XT-1,xr) (14)
Let known realization xt. Joint distribution xt and xt+1 are settings per copula C(F(xt +1), F(xt)) = C(u, v) where the denoted u = F(xt +1), v = F(xt). Then conditional distribution xt + 1 is set as
It is worth noting we can take advantage of copulas’ remarkable property that allows us to model the joint distribution of C(u1, ... uk) separately from the marginal F1(x1), ... Fk(xk), and move from observations x1 ... xk to pseudo-observations—probabilities, obtained by marginal distribution functions U1 = F1(x1), ... Uk = Fk(xk).
Thus, the process of generation is constructed as follows:
1. First observation x1 generates from marginal distribution F(x).
2. For current observation of the time series, xt derives a pseudo-observation with marginal distribution function: v* = F(xt).
3. Then it obtains a conditional distribution function with a conditional copula:
G(u) = C(u|v)
4. Generate number a from uniform distribution on segment [0,1]
5. Then we find solution u* of equation G(u|v) = a. For this next procedure, sequentially enumerated values 0, 0.1, 0.2 ... 0.9. We find the number 01 such, that G(^1) — a<0 and G(^1+0.1) — a >0. Further, sequentially enumerated values p1, p1 + 0.01, p1 + 0.02, ... p1 + 0.09find such ^2,thatG(^2) — a<0 and G(^2 + 0.01) — a > 0 and so on. Thus, you could easily find a solution of G(u|v) = a up to any number of digits after the decimal point.
6. Knowing the value of u*, we restore the value of time series with the quantile function F(x): xt +1 = F - 1(u*).
It is worth nothing that if we generate the entire time series at once, we could not recalculate every observation and pseudo-observation. If we only work with pseudo-observations, then the first observation is generated from a uniform distribution in [0, 1], and then we use only steps (3)-(5), then we use the quantile
function to find the number of observations. Moreover, this procedure could be easily generalized to arbitrary dimension k, generating a pseudo-observation of the conditional distribution function
For generating the time series we can use six types of copulas: Clayton, Frank, Gumbel, independent, Farlie-Gumbel-Morgenstern (FGM) and Plackett. The first three are Archimedean copulas: they satisfy a certain analogy between the axiom of Archimedes (for any positive numbers a and b could find natural number n, in that n*a > b); for any u and v from [0; 1] we could find natural number n, in that MC(u) > v where operation MnC(t) is defined as follows via Archimedean copula C: for any k MkC+ 1(t) = C(t, MkC(t)). Functionally, the Archimedean copula have the function generator'(t) and are of the form C(u1 ... uk) = '_ 1('(u1) + ... + '(uk)). The fourth copula is an independent copula or product copula; it is the simplest example and specifies independent distribution of random variables, i. e. their joint distribution is the precise product of the partial distribution functions. The last two copula can’t be assigned to any of the conventional classes. Farlie-Gumbel-Morgenstern (FGM) copula, is often used to model the random variables with small absolute values of the rank correlation. Plackett copula is also often used in applied research. Every copula (except product) is characterized by parameter в, which takes values from some range of admissible values. Each value of the parameter 9 corresponds to a specific value of rank correlation, but not necessarily to each value of the rank correlation that corresponds to a parameter 9 of range of admissible values. In the two-dimensional case, Spearman’s pho and Kendall’s tau are expressed through copula C(u, v) = C(F(x), G(y)) = H(x, y) as follows:
ps (x, y) = 12 I (C (u, v) — u * v) du dV (17)
tk (x, y) = 4 if C (u, v) dC (u, v) - 1. (18)
The annex contains the formulas of used copulas for two-dimensional cases, the formulas of conditional copulas (for Archimedean copulas-specified generator functions) and ranges of admissible values for parameters. It also contains some examples of generated time series for the above copulas and scatter plots of dependence between pseudo-observation and lagged pseudo-observation.