# 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

1 2 3 |

### Arguments

`x` |
a |

`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 |

`ar.method` |
character. The |

`weights` |
The smoothing weights can be computed by |

`approx` |
a character specifying the approximation method if the
bandwidth has to be chosen by |

`tol` |
numeric. Weights that exceed |

`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 1*vector 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 http://www.jstatsoft.org/v16/i09/.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
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])
``` |