CgmmParamsEstim: Estimate parameters of stable laws using a Cgmm method

In StableEstim: Estimate the Four Parameters of Stable Laws using Different Methods

Estimate parameters of stable laws using a Cgmm method

Description

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.

Usage

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,θ)=g(t,X;θ)= e^{itX} - φ_{θ}(t)

If one has a sample x_1,…,x_n of i.i.d realisations of the same random variable X, then:

\hat{g}_n(t,θ) = \frac{1}{n}∑_{i=1}^n g(t,x_i;θ) = φ_n(t) -φ_θ(t),

where φ_n(t) is the eCF associated with the sample x_1,…,x_n, defined by φ_n(t)= \frac{1}{n} ∑_{j=1}^n e^{itX_j}. Objective function

Following \insertCite@@carrasco2007efficientCont, Proposition 3.4;textualStableEstim, the objective function to minimise is given by:

obj(θ)=\overline{\underline{v}^{\prime}}(θ)[α_{Reg} \mathcal{I}_n+C^2]^{-1}\underline{v}(θ)

where:

\underline{v} = [v_1,…,v_n]^{\prime};

v_i(θ) = \int_I \overline{g_i}(t;\hat{θ}^1_n) \hat{g}(t;θ) π(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{θ}^1_n) g_j(t;\hat{θ}^1_n) π(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.

Note

nlminb as used to minimise the Cgmm objective function.

References

\insertRef

carrasco2000generalizationStableEstim

\insertRef

carrasco2002efficientStableEstim

\insertRef

carrasco2003asymptoticStableEstim

\insertRef

carrasco2007efficientContStableEstim

\insertRef

carrasco2010efficientStableEstim

See Also

Estim, GMMParametersEstim, IntegrateRandomVectorsProduct

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)
twoS

StableEstim documentation built on Aug. 7, 2022, 5:17 p.m.