gmm: smoothG – R documentation

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smoothG

Kernel smoothing of a matrix of time series

Description

It applies the required kernel smoothing to the moment function in order for the GEL estimator to be valid. It is used by the gel function.

Usage

smoothG(x, bw = bwAndrews, prewhite = 1, ar.method = "ols", weights = weightsAndrews,
	kernel = c("Bartlett", "Parzen", "Truncated", "Tukey-Hanning"), 
	approx = c("AR(1)", "ARMA(1,1)"), tol = 1e-7)

Arguments

`x`	a n\times q matrix of time series, where n is the sample size.
`bw`	The method to compute the bandwidth parameter. By default, it uses the bandwidth proposed by Andrews(1991). As an alternative, we can choose bw=bwNeweyWest (without "") which is proposed by Newey-West(1996).
`prewhite`	logical or integer. Should the estimating functions be prewhitened? If `TRUE` or greater than 0 a VAR model of order `as.integer(prewhite)` is fitted via `ar` with method `"ols"` and `demean = FALSE`.
`ar.method`	character. The `method` argument passed to `ar` for prewhitening.
`weights`	The smoothing weights can be computed by `weightsAndrews` of it can be provided manually. If provided, it has to be a r\times 1vector (see details).
`approx`	a character specifying the approximation method if the bandwidth has to be chosen by `bwAndrews`.
`tol`	numeric. Weights that exceed `tol` are used for computing the covariance matrix, all other weights are treated as 0.
`kernel`	The choice of kernel

Details

The sample moment conditions ∑_{t=1}^n g(θ,x_t) is replaced by: ∑_{t=1}^n g^k(θ,x_t), where g^k(θ,x_t)=∑_{i=-r}^r k(i) g(θ,x_{t+i}), where r is a truncated parameter that depends on the bandwidth and k(i) are normalized weights so that they sum to 1.

If the vector of weights is provided, it gives only one side weights. For exemple, if you provide the vector (1,.5,.25), k(i) will become (.25,.5,1,.5,.25)/(.25+.5+1+.5+.25) = (.1,.2,.4,.2,.1)

Value

smoothx: A q \times q matrix containing an estimator of the asymptotic variance of √{n} \bar{x}, where \bar{x} is q\times 1vector with typical element \bar{x}_i = \frac{1}{n}∑_{j=1}^nx_{ji}. This function is called by gel but can also be used by itself.

kern_weights: Vector of weights used for the smoothing.

References

Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias. Econometrica, 73, 983-1002.

Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59, 817–858.

Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes. The Annals of Statistics, 25, 2084-2102.

Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. Journal of Statistical Software, 16(9), 1–16. URL https://www.jstatsoft.org/v16/i09/.

Examples

g <- function(tet, x)
	{
	n <- nrow(x)
	u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)])
	f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)])
	return(f)
	}
n = 500
phi<-c(.2, .7)
thet <- 0.2
sd <- .2
x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1)
gt <- g(c(0, phi), x) 
sgt <- smoothG(gt)$smoothx
plot(gt[,1])
lines(sgt[,1])

gmm

Generalized Method of Moments and Generalized Empirical Likelihood

v1.6-6

GPL (>= 2)

Authors

Pierre Chausse <pchausse@uwaterloo.ca>

Initial release

2021-02-07