# Kernel smoothing of a matrix of time series 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. ```r 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 $r\times 1$vector (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 $\sum_{t=1}^n g(\theta,x_t)$ is replaced by: $\sum_{t=1}^n g^k(\theta,x_t)$, where $g^k(\theta,x_t)=\sum_{i=-r}^r k(i) g(\theta,x_{t+i})$, 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 $(.25,.5,1,.5,.25)/(.25+.5+1+.5+.25) = (.1,.2,.4,.2,.1)$ ## Returns smoothx: A $q \times q$ matrix containing an estimator of the asymptotic variance of $\sqrt{n} \bar{x}$, where $\bar{x}$ is $q\times 1$vector with typical element $\bar{x}_i = \frac{1}{n}\sum_{j=1}^nx_{ji}$. 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 tools:::Rd_expr_doi("10.18637/jss.v016.i09") . ## Examples ```r 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]) ```

smoothG function