Estimate parameters of stable laws using a Cgmm method

Estimate the four parameters of stable laws using generalised method of moments based on a continuum of complex moment conditions (Cgmm) due to Carrasco and Florens. Those moments are computed by matching the characteristic function with its sample counterpart. The resulting (ill-posed) estimation problem is solved by a regularisation technique.

CgmmParametersEstim(x, type = c("2S", "IT", "Cue"), alphaReg = 0.01,
                    subdivisions = 50,
                    IntegrationMethod = c("Uniform", "Simpson"),
                    randomIntegrationLaw = c("unif", "norm"),
                    s_min = 0, s_max = 1,
                    theta0 = NULL,
                    IterationControl = list(),
                    pm = 0, PrintTime = FALSE,...)

Arguments

x: Data used to perform the estimation: a vector of length n.
type: Cgmm algorithm: "2S" is the two steps GMM proposed by Hansen(1982). "Cue" and "IT" are respectively the continuous updated and the iterative GMM proposed by Hansen, Eaton et Yaron (1996) and adapted to the continuum case.
alphaReg: Value of the regularisation parameter; numeric, default = 0.01.
subdivisions: Number of subdivisions used to compute the different integrals involved in the computation of the objective function (to minimise); numeric.
IntegrationMethod: Numerical integration method to be used to approximate the (vectorial) integrals. Users can choose between "Uniform" discretization or the "Simpson"'s rule (the 3-point Newton-Cotes quadrature rule).
randomIntegrationLaw: Probability measure associated to the Hilbert space spanned by the moment conditions. See Carrasco and Florens (2003) for more details.
s_min,s_max: Lower and Upper bounds of the interval where the moment conditions are considered; numeric.
theta0: Initial guess for the 4 parameters values: vector of length 4.
IterationControl: Only used with type = "IT" or type = "Cue" to control the iterations, see Details.
pm: Parametrisation, an integer (0 or 1); default: pm = 0 (Nolan's ‘S0’ parametrisation).
PrintTime: Logical flag; if set to TRUE, the estimation duration is printed out to the screen in a readable format (h/min/sec).
...: Other arguments to be passed to the optimisation function and/or to the integration function.

Details

The moment conditions The moment conditions are given by: $$g_t(X,\theta)=g(t,X;\theta)= e^{itX} - \phi_{\theta}(t)$$ If one has a sample $x_1,\dots,x_n$ of i.i.d realisations of the same random variable $X$, then: $$\hat{g}_n(t,\theta) = \frac{1}{n}\sum_{i=1}^n g(t,x_i;\theta) = \phi_n(t) -\phi_\theta(t),$$ where $\phi_n(t)$ is the eCF associated with the sample $x_1,\dots,x_n$, defined by $\phi_n(t)= \frac{1}{n} \sum_{j=1}^n e^{itX_j}$. Objective function

Following Carrasco et al. (2007, Proposition 3.4) , the objective function to minimise is given by: $$obj(\theta)=\overline{\underline{v}^{\prime}}(\theta)[\alpha_{Reg} \mathcal{I}_n+C^2]^{-1}\underline{v}(\theta)$$ where:

$\underline{v} = [v_1,\ldots,v_n]^{\prime}$;: $v_i(\theta) = \int_I \overline{g_i}(t;\hat{\theta}^1_n) \hat{g}(t;\theta) \pi(t) dt$.
$I_n$: is the identity matrix of size $n$.
$C$: is a $n \times n$ matrix with $(i,j)$th element given by $c_{ij} = \frac{1}{n-4}\int_I \overline{g_i}(t;\hat{\theta}^1_n) g_j(t;\hat{\theta}^1_n) \pi(t) dt$.

To compute $C$ and $v_i()$ we will use the function IntegrateRandomVectorsProduct. The IterationControl If type = "IT" or type = "Cue", the user can control each iteration using argument IterationControl, which should be a list which contains the following elements:

NbIter:: maximum number of iterations. The loop stops when NBIter is reached; default = 10.
PrintIterlogical:: if set to TRUE the values of the current parameter estimates are printed to the screen at each iteration; default = TRUE.
RelativeErrMax:: the loop stops if the relative error between two consecutive estimation steps is smaller then RelativeErrMax; default = 1e-3.

Value

a list with the following elements:

Estim: output of the optimisation function,
duration: estimation duration in numerical format,
method: character describing the method used.

References

Carrasco M, Florens J (2000). “Generalization of GMM to a continuum of moment conditions.” Econometric Theory, 16(06), 797--834.

Carrasco M, Florens J (2002). “Efficient GMM estimation using the empirical characteristic function.” IDEI Working Paper, 140.

Carrasco M, Florens J (2003). “On the asymptotic efficiency of GMM.” IDEI Working Paper, 173.

Carrasco M, Chernov M, Florens J, Ghysels E (2007). “Efficient estimation of general dynamic models with a continuum of moment conditions.” Journal of Econometrics, 140(2), 529--573.

Carrasco M, Kotchoni R (2010). “Efficient estimation using the characteristic function.” Mimeo. University of Montreal.

Note

nlminb as used to minimise the Cgmm objective function.

Examples

## general inputs
theta <- c(1.45, 0.55, 1, 0)
pm <- 0
set.seed(2345)
x <- rstable(50, theta[1], theta[2], theta[3], theta[4], pm)

## GMM specific params
alphaReg <- 0.01
subdivisions <- 20
randomIntegrationLaw <- "unif"
IntegrationMethod <- "Uniform"

## Estimation
twoS <- CgmmParametersEstim(x = x, type = "2S", alphaReg = alphaReg, 
                          subdivisions = subdivisions, 
                          IntegrationMethod = IntegrationMethod, 
                          randomIntegrationLaw = randomIntegrationLaw, 
                          s_min = 0, s_max = 1, theta0 = NULL, 
                          pm = pm, PrintTime = TRUE)
#> [1] "CgmmParametersEstim_2S :duration= 0  h, 0  min, 2  sec. "
twoS
#> $Estim
#> $Estim$par
#> [1]  1.3193378  0.6995646  0.7875786 -0.1791584
#> 
#> $Estim$all
#> $Estim$all$par
#>      alpha       beta      gamma      delta 
#>  1.3193378  0.6995646  0.7875786 -0.1791584 
#> 
#> $Estim$all$objective
#> [1] 0.02720932
#> 
#> $Estim$all$convergence
#> [1] 0
#> 
#> $Estim$all$iterations
#> [1] 9
#> 
#> $Estim$all$evaluations
#> function gradient 
#>       13       44 
#> 
#> $Estim$all$message
#> [1] "both X-convergence and relative convergence (5)"
#> 
#> 
#> 
#> $duration
#> [1] 1.554
#> 
#> $method
#> [1] "Cgmm_type=2S_alphaReg=0.01_OptimAlgo=nlminb_subdivisions=20_IntegrationMethod=Uniform_randomIntegrationLaw=unif_s_min=0_s_max=1"
#>