Funciones package descomponer 1.3
Francisco Parra Rodríguez
Documento tecnico:
http://econometria.wordpress.com/2013/07/29/estimacion-con-parametros-dependientes-del-tiempo/
Packages necesarios
library(taRifx)
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## tarlibrary(sapa)
library(bspec)##
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## prewhitenlibrary(descomponer)Función gdf(a)
Transforma los datos del dominio del tiempo al dominio de la frecuencia pre-multiplicandolos por la matriz ortogonal,\(W\), sugerida por Harvey (1978)
Nerlove (1964) y Granger (1969) fueron los primeros investigadores en aplicar el analisis espectral a las series de tiempo en economía. El uso del analisis espectral requiere un cambio en el modo de ver las series económicas, al pasaser de la perspectiva del tiempo al dominio de la frecuencia. El analisis espectral parte de la supsición de que cuanquier serie {Xt}, puede ser transformada en ciclos formados con senos u cosenos:
\(X_t=\eta+\sum_{j=1}^N[a_j\cos(2\pi\frac{ft}n)+b_j\sin(2\pi\frac{ft}n)]\) (1)
donde \(\eta\) es la media de la serie, \(a_j\) y \(b_j\) son su amplitud,\(f\) son las frecuencias que del conjunto de las \(n\) observaciones,\(t\) es un indice de tiempo que va de 1 a N, siendo N el numero de periodos para los cuales tenemos observaciones en el conjunto de datos, el cociente \(\frac{ft}n)\) convierte cada valor de \(t\) en escala de tiempo en proporciones de \(2n\) y rango \(j\) desde \(1\) hasta \(n\) siendo \(n=\frac{N}2\) (es decir, 0,5 ciclos por intervalo de tiempo). Las dinámica de las altas frecuencias (los valores más altos de f) corresponden a los ciclos cortos en tanto que la dinámica de la bajas frecuencias (pequeños valores de f) van a corresponder con los ciclos largos. Si nosotros hacemos que \(\frac{ft}n=w\) la ecuación (1) quedaría, asi :
\(X_t=\eta+\sum_{j=1}^N[a_j\cos(\omega_j)+b_j\sin(\omega_j)]\)(2)
El analisis espectral puede utilizarse para identificar y cuantificar en procesos aparentemente aperiodicos, sucesiones de cicos de periodo de corto y largo plazo. Una serie dada \({X_t}\) puede contener diversos ciclos de diferentes frecuencias y amplitudes, y esa combinación de frecuencias y amplitudes de carcter cíclico la hacer aparecer como un serie no periodica e irregular. De hecho la ecuación (2), muestra que cada observación \(t\) de una serie de tiempo, es el resultado sumar los valores en \(t\) que resultan de \(N\) ciclos de diferente longitud y amplitud, a los que habría que añadir si cabe un termino de error.
Una manera practica de pasar desde el dominio del tiempo al dominio de la frecuencia es pre-multiplicando los datos originales por una matriz ortogonal, \(W\), sugerida por Harvey (1978), con el elemento (j,t)th :
\[\begin{equation} w_{jt} = \left\lbrace\begin{array}{ll}\left(\frac{1}T\right) ^\frac{1}2 & \forall j=1\\ \left(\frac{2}T\right) ^\frac{1}2 \cos\left[\frac{\pi j(t-1)}T\right] & \forall j=2,4,6,..\frac{(T-2)}{(T-1)}\\ \left(\frac{2}T\right) ^\frac{1}2 \sin\left[\frac{\pi (j-1)(t-1)}T\right] & \forall j=3,5,7,..\frac{(T-2)}T\\ \left(\frac{1}T\right) ^\frac{1}2 (-1)^{t+1} & \forall j=T\end{array}\right.\end{equation}\] (3)
La matriz \(W\) tiene la ventaja de ser ortogonal por lo que \(WW^T=I\).
Matriz \(W\)
MW <- function(n) {
# Author: Francisco Parra Rodr?guez
# Some ideas from: Harvey, A.C. (1978), Linear Regression in the Frequency Domain, International Economic Review, 19, 507-512.
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
uno <- as.numeric (1:n)
A <- matrix(rep(sqrt(1/n),n), nrow=1)
if(n%%2==0){
for(i in 3:n-1){
if(i%%2==0) {
A1 <- matrix(sqrt(2/n)*cos(pi*(i)*(uno-1)/n), nrow=1)
A <- rbind(A,A1)}
else {
A2 <- matrix(sqrt(2/n)*sin(pi*(i-1)*(uno-1)/n), nrow=1)
A <- rbind(A,A2)
}}
AN <- matrix(sqrt(1/n)*(-1)^(uno+1), nrow=1)
A <- rbind(A,AN)
A
} else {
for(i in 3:n-1){
if(i%%2==0) {
A1 <- matrix(
sqrt(2/n)*cos(pi*(i)*(uno-1)/n), nrow=1)
A <- rbind(A,A1)}
else {
A2 <- matrix(sqrt(2/n)*sin(pi*(i-1)*(uno-1)/n), nrow=1)
A <- rbind(A,A2)
}}
AN <- matrix(
sqrt(2/n)*sin(pi*(n-1)*(uno-1)/n), nrow=1)
A <- rbind(A,AN)
}
}gdf <- function(y) {
a <- matrix(y,nrow=1)
n <- length(y)
A <- MW(n)
A%*%t(a)
}Función gdt(a)
Transforma los datos del dominio de frecuencias al dominio del tiempo pre-multiplicandolos por la matriz ortogonal, A, sugerida por Harvey (1978)
gdt <- function(y) {
# Author: Francisco Parra Rodr?guez
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
a <- matrix(y,nrow=1)
n <- length(y)
A <- MW(n)
t(A)%*%t(a)
}Función cdf(a)
Obtiene la matriz auxiliar para operaciones con vectores en dominio de tiempo y dominio de la frecuencia, pre-multiplica un vector por la matriz ortogonal, \(W\) y por su transpuesta, Parra F. (2013) La multiplicación de dos series armónicas de diferente frecuencia:
\([a_j\cos (\omega_j)+b_j\sin (\omega_j)]x [a_i\cos (\omega_i)+b_i\sin (\omega_i)]\)
da como resultado la siguiente suma: \[\begin{equation} \begin{array}{c} a_ja_i\cos(\omega_j)\cos(\omega_i)+a_jb_i\cos (\omega_j)\sin (\omega_i)\\ +a_ib_j\sin (\omega_j)\cos (\omega_i)b_i\sin (\omega_i)+b_jb_i\sin(\omega_j)\sin(\omega_i) \end{array} \end{equation}\]
considerando las identidades del producto de senos y cosenos, quedaría:
\[\begin{equation} \begin{array}{c} \frac{a_ja_i+b_jb_i}{2} \cos(\omega_j- \omega_i)+\frac{b_ja_i-b_ja_j}{2}\sin(\omega_j- \omega_i)\\ +\frac{a_ja_i-b_jb_i}{2}\cos(\omega_j+ \omega_i)+\frac{b_ja_i+b_ja_i}{2}\sin(\omega_j+ \omega_i) \end{array} \end{equation}\]
La circularidad de \(\omega\) determina que la serie producto de dos series en \(t\), resulte una nueva serie cuyos coeficientes de Fourier sean una combinación lineal de los coeficientes de Fourier de las series multiplos.
Partiendo de las dos series siguientes:
\[\begin{equation} \begin{array} {cc} y_t=\eta^y+a_0^y\cos(\omega_0)+b_0^y\sin(\omega_0)+a_1^y\cos(\omega_1)+b_1^y\sin(\omega_1)+ a_2^y\cos(\omega_2)+b_2^y\sin(\omega_2)+a_3^y\cos(\omega_3)\\ x_t=\eta^x+a_0^x\cos(\omega_0)+b_0^x\sin(\omega_0)+a_1^x\cos(\omega_1)+b_1^x\sin(\omega_1)+ a_2^x\cos(\omega_2)+b_2^x\sin(\omega_2)+a_3^x\cos(\omega_3) \end{array} \end{equation}\]
Dada una matriz \(\Theta^{\dot x\dot x}\) de tamaño 8x8 :
\[ \Theta^{\dot x\dot x} = \eta^x I_8+\frac{1}2\left( \begin{array}{cccccccc} 0& a_0^x& b_0^x & a_1^x & b_1^x & a_2^x & b_2^x& 2a_3^x \\ 2a_0^x& a_1^x& b_1^x & a_0^x+a_2^x & b_0^x+b_2^x & a_1^x+2a_3^x & b_1^x& 2a_2^x \\ 2b_0^x& b_1^x&- a_1^x & -b_0^x+b_2^x & a_0^x-a_2^x &- b_1^x &a_1^x- a_3^x &- 2b_2^x \\ 2a_1^x& a_0^x+a_2^x&- b_0^x+b_2^x & 2a_3^x &0 & a_0^x+a_2^x & b_0^x-b_2^x& 2a_1^x \\ 2b_1^x& a_0^x+b_2^x&- b_0^x-a_2^x &0& -2a_3^x & -b_0^x+b_2^x & a_0^x-a_2^x& -2b_1^x \\ 2a_2^x& a_1^x+2a_3^x&- b_1^x & a_0^x+a_2^x &-b_0^x-b_2^x & a_1^x &- b_1^x& 2a_0^x \\ 2b_2^x& b_1^x& a_1^x-2a_3^x & b_0^x-b_2^x &a_0^x-a_2^x & -b_1^x &- a_1^x& -2b_0^x \\ 2a_3^x& a_2^x& -b_2^x & a_1^x &- b_1^x & a_0^x & -b_0^x& 0 \end{array} \right) \] Se demuestra que:
\(\dot z=\Theta^{\dot x\dot x}\dot y\)
donde \(\dot y = Wy\),\(\dot x = Wx\), y \(\dot z = Wz\).
En el dominio del tiempo:
\(z_t= x_t y_t=W^T\dot x W^T\dot y=W^T Wx_t W^T\dot y=x_tI_nW^T\dot y\)
\(W^T\dot z=x_tI_nW^T\dot y\)
\(\dot z=Wx_tI_nW^T\dot y\)
Entonces:
\(\Theta^{\dot x\dot x}=W^Tx_tI_nW\)
La matriz cuadrada \(\Theta^{\dot x\dot x}\) puede ser utilizada para obtener los resultados en el dominio de la frecuencia de diversas funciones de series de tiempo . Por ejemplo, si se desea obtener el desarrollo de los coeficientes en fourier de \(z_t=x_t^2\), entonces:
\(\dot z= Wx_tI_nW^T\dot x\)
En consecuencia, si \(z_t=x_t^n\)
\(\dot z= Wx_t^{n-1}I_nW^T\dot x\)
Si ahora queremos obtener el desarrollo en coeficientes de fourier de \(z_t=\frac{x_t}{y_t}\), entonces:
\(\dot z= W[\frac{1}y_t]I_nW^T\dot x\)
cdf <- function(y) {
# Author: Francisco Parra Rodr?guez
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
a <- matrix(y, nrow=1)
n <- length(y)
uno <- as.numeric (1:n)
A <- MW(n)
I<- diag(c(a))
B <- A%*%I
B%*%t(A)
}Función periodograma (a)
Calcula y presenta el espectro de la serie “a”
Sea \(a\) un vector n x 1 el modelo transformado en el dominio de la frecuencia esta dado por:
\(\hat a= Wa\)
Denominando \(p_j\) el ordinal del periodograma de \(\hat a\) en la frecuencia \(\lambda_j=2\pi j/n\), y \(\hat a_j\) el j-th elemento de \(\hat a\), entonces
\[ \left\lbrace \begin{array}{ll} p_j=\hat a_{2j}^{2}+\hat a_{2j+1}^{2} & \forall j = 1,...\frac{n-1}{2}\\ p_j=\hat a_{2j}^{2}& \forall j = \frac{n}{2}-1 \end{array} \right . \]
\[p_0=\hat a_{1}^{2}\]
Entonces el cuadrado del \(\hat a\) puede ser utilizado como un estimador consistente del periodograma de \(a\).
periodograma <- function(y) {
# Author: Francisco Parra Rodr?guez
# Some ideas from Gretl
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
cf <- gdf(y)
n <- length(y)
if (n%%2==0) {
m1 <- c(0)
m2 <- c()
for(i in 1:n){
if(i%%2==0) m1 <-c(m1,cf[i]) else m2 <-c(m2,cf[i])}
m2 <-c(m2,0)
frecuencia <- seq(0:(n/2))
frecuencia <- frecuencia-1
omega <- pi*frecuencia/(n/2)
periodos <- n/frecuencia
densidad <- (m1^2+m2^2)/(4*pi)
tabla <- data.frame(omega,frecuencia, periodos,densidad)
tabla$densidad[(n/2+1)] <- 4*tabla$densidad[(n/2+1)]
data.frame(tabla[2:(n/2+1),])}
else {m1 <- c(0)
m2 <- c()
for(i in 1:(n-1)){
if(i%%2==0) m1 <-c(m1,cf[i]) else m2 <-c(m2,cf[i])}
m2 <-c(m2,cf[n])
frecuencia <- seq(0:((n-1)/2))
frecuencia <- frecuencia-1
omega <- pi*frecuencia/(n/2)
periodos <- n/frecuencia
densidad <- (m1^2+m2^2)/(4*pi)
tabla <- data.frame(omega,frecuencia, periodos,densidad)
data.frame(tabla[2:((n+1)/2),])}
}Función gperiodogrma (a)
Presenta gráficamente el espectro de la variable a
gperiodograma <- function(y) {
# Author: Francisco Parra Rodr?guez
# Some ideas from Gretl
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
tabla <- periodograma(y)
plot(tabla$frecuencia,tabla$densidad,
main = "Espectro",
ylab = "densidad",
xlab="frecuencia",type = "l",
col="#ff0000")}Función descomponer(y,frequency,type)
En base al modelo de regresión en el dominio de la frecuencia descompone una serie \(y_t\) en los factores de tendencia \(TD\), estacionales \(ST\), e irregulares \(IR\).
La función se desarrolla en los siguientes pasos:
Se calcula el periodograma de la serie, y se ordena según el vector de frecuencias para crear diferentes indices de orden.
Se obtiene un modelo de tendencia, a partir de las frecuencias mayores que \(\frac{n}{2*frequency}\) si la serie es par y las mayores que \(\frac{n-1}{2*frequency}\) si la serie es impar. Para ello, se realiza la regresión en domininio de la frecuencia entre la serie \(y_t\) y los regresores que se obtienen con la matriz auxiliar \(Wx_tI_nW^T\), donde \(x_t\) es el resultado de ajustar un modelo lineal del tipo \(y_t=a+bt+e_t\) a la serie de datos (tipo=1) ? un modelo cuadr?tico del tipo \(y_t=a+bt+ct^2+e_t\), en donde solo se consideran los regresores correspondientes a las diferentes frecuencias seleccionadas.Una vez obtenidos los par?metros del modelo, se calcula la serie en el dominio de la frecuencia que una vez convierten al dominio del tiempo da como resultado la serie de tendencia \(TD\).
Se obtiene la serie residual \(IRST=y_t-TD\), se y sobre esa serie se realiza una nueva selecci?n de frecuencias, las correspondientes a los factores estacionales es decir:\(\frac{n}{2*frequency}\), \(\frac{2n}{2*frequency}\),\(\frac{3n}{2*frequency}\), etc….. Se realiza la regresión en el dominio de la frecuencia entre \(IRST\) y los regresores correspondientes a las frecuencias seleccionadas obtenidas a partir de a matriz auxiliar \(Wx_tI_nW^T\), donde \(x_t\) es el resultado de ajustar un modelo lineal del tipo \(IRST=a+bt+e_t\) a la serie de datos (tipo=1) ? un modelo cuadr?tico del tipo \(IRST=a+bt+ct^2+e_t\). Una vez obtenidos los parámetros del modelo, se calcula la serie en el dominio de la frecuencia que una vez convierten al dominio del tiempo da como resultado la serie de tendencia \(ST\).
Se obtiene la serie irregular a partir de \(IR=IRST-ST\).
La versión 1.3 de la función descomponer:
descomponer <- function (y,frequency,type) {
# Author: Francisco Parra Rodriguez
# http://rpubs.com/PacoParra/24432
# date:"y", frequency:"frequency".
# Use 7 for frequency when the data are sampled daily, and the natural time period is a week,
# or 4 and 12 when the data are sampled quarterly and monthly and the natural time period is a year.
n <- length(y)
y <- matrix(y,ncol=1)
M <- MW(n) #crea la matriz de harvey para los n datos
f1 <- NULL
if(n%%2==0) {f2 <- n/(2*frequency)} else {
f2 <- (n-1)/(2*frequency)}
#Modelo para obtener serie con tendencia
c <- seq(from=2, to=(2+(n/frequency) ))
#Use the "sort.data.frame" function, Kevin Wright. Package taRifx
i <- seq(1:n)
i2 <- i*i
i <- seq(1:n)
i2 <- i*i
if (type==1)
{eq <- lm(y~i)
z <- eq$fitted} else {
if (type==2) eq <- lm(y~i+i2)
z <- eq$fitted}
cx <- M%*%diag(z)
cx <- cx%*%t(M)
id <- seq(1,n)
S1 <- data.frame(cx)
S2 <- S1[1:(2+(n/frequency)),]
X <- as.matrix(S2)
cy <- M%*%y
B <- solve(X%*%t(X))%*%(X%*%cy)
Y <- t(X)%*%B
BTD <- B
XTD <- t(M)%*%t(X)
TD <- t(M)%*%Y
# Genero la serie residual
IRST <- y-TD
# Realizo la regresion dependiente de la frecuenca utilizando como explicativa IRST.
# modelo para obtener serie con estacionalidad con trunc.
frecuencia <- seq(0:(n/2))
frecuencia <- frecuencia-1
S <- data.frame(f1=frecuencia)
sel <- subset(S,f1==trunc(2*f2))
c <- seq(from=2,to=(n/f2))
for (i in c) {sel1 <- subset(S,f1==i*trunc(2*f2))
sel <- rbind(sel,sel1)}
m1 <- c(sel$f1 * 2)
m2 <- c(m1+1)
c <- c(m1,m2)
n3 <- length(c)
d <- rep(1,n3)
s <- data.frame(c,d)
S <- sort.data.frame (s,formula=~c)
#Use the "sort.data.frame" function, Kevin Wright. Package taRifx
l <- frequency*trunc(n/frequency)
ML <- MW(l)
i <- seq(1:l)
i2 <- i*i
if (type==1)
{eq <- lm(y[1:l]~i)
z <- eq$fitted} else {
if (type==2) eq <- lm(y[1:l]~i+i2)
z <- eq$fitted}
cx <- ML%*%diag(z) #problema
cx <- cx%*%t(ML)
id <- seq(1,l)
S1 <- data.frame(cx,c=id)
S2 <- merge(S,S1,by.x="c",by.y="c")
S3 <- rbind(c(1,1,cx[1,]),S2)
m <- l+2
X1 <- S3[,3:m]
# matriz de regresores a l
X1 <- as.matrix(X1)
# la paso al dominio del tiempo
X2 <- data.frame(t(ML)%*%t(X1))
if (n==l) X3 <- X2 else
X3 <- rbind(X2,X2[1:(n-l),])
# la paso al dominio de la frecuencia
X4 <-M%*%as.matrix(X3)
cy <- M%*%IRST
B1 <- solve(t(X4)%*%X4)%*%(t(X4)%*%cy)
Y <- X4%*%B1
BST <- B1
XST <- M%*%X4
ST <- t(M)%*%Y
TDST <- TD+ST
IR <- IRST-ST
data <- data.frame(y,TDST,TD,ST,IR)
regresoresTD <- data.frame(XTD)
regresoresST <- data.frame(XST)
list(datos=data,regresoresTD=regresoresTD,regresoresST=regresoresST,coeficientesTD=BTD,coeficientesST=BST)
}library(descomponer)
library(quantmod)## Loading required package: xts##
## Attaching package: 'xts'## The following objects are masked from 'package:taRifx':
##
## first, last## Loading required package: TTR## Version 0.4-0 included new data defaults. See ?getSymbols.getSymbols('^IBEX', from = '1990-01-01')## As of 0.4-0, 'getSymbols' uses env=parent.frame() and
## auto.assign=TRUE by default.
##
## This behavior will be phased out in 0.5-0 when the call will
## default to use auto.assign=FALSE. getOption("getSymbols.env") and
## getOptions("getSymbols.auto.assign") are now checked for alternate defaults
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for more details.## [1] "IBEX"t1 <- Sys.time()
ibex.des.a=descomponer(Ad(IBEX)['2012::2016'],256,1)
plot(ts(Ad(IBEX)['2012::2017'],frequency = 256, start = c(2012, 1)))
lines(ts(ibex.des.a$datos$TD,frequency = 256, start = c(2012, 1)),col='red')
t2 <- Sys.time()
print(t2-t1)## Time difference of 1.97058 minsgdescomponer (y,freq,type,year,q))
Gráfico de la función descomponer.
gdescomponer <- function(y,freq,type,year,q) {
# Author: Francisco Parra Rodriguez
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
serie <- descomponer (y,freq,type)
TdsT <- c(serie$datos$TDST)
Td <- c(serie$datos$TD)
sT <- c(serie$datos$ST)
TDST <- ts(TdsT,frequency=freq,start = c(year,q))
TD <- ts(Td,frequency=freq,start = c(year,q))
ST <- ts(sT,frequency=freq,start = c(year,q))
par(mfrow=c(3,1))
plot (TDST)
plot (TD)
plot (ST)
}par(mar=c(1,1,1,1))
gdescomponer(Ad(IBEX)['2012::2016'],60,1,2012,256)
Función td (y,significance)
Realiza una prueba estadística para estudiar la dependencia serial sobre el periodograma acumulado de a, con una significación de 0,1(significance=1); 0,05(significance=2); 0,025(significance=3); 0,01(significance=4) y 0,005 (significance=5) (Durbin; 1969)
El test de Durbin esta basado en el siguiente estadistico: \(s_j=\frac{\sum_{r=1} ^j p_r}{\sum_{r=1}^m p_r}\)
donde \(m=\frac{1}{2}n\) para \(n\) par y \(\frac{1}{2}(n-1)\) para \(n\) impar.
El estadístico \(s_j\) ha en encontrarse entre unos límites inferior y superior de valores críticos que han sido tabulados por Durbin (1969). Si bien hay que tener presente que el valor \(p_o\) no se considera en el cálculo del estadístico esto es, \(p_o=\hat v_1=0\)
En la función se utilizan las bandas de frecuencia de la función cpgram de W. N. Venables and B. D. Ripley.
Leemos los datos del PIB y empleo de Canada.
data("Canada")
summary(Canada)## e prod rw U
## Min. :928.6 Min. :401.3 Min. :386.1 Min. : 6.700
## 1st Qu.:935.4 1st Qu.:404.8 1st Qu.:423.9 1st Qu.: 7.782
## Median :946.0 Median :406.5 Median :444.4 Median : 9.450
## Mean :944.3 Mean :407.8 Mean :440.8 Mean : 9.321
## 3rd Qu.:950.0 3rd Qu.:410.7 3rd Qu.:461.1 3rd Qu.:10.607
## Max. :961.8 Max. :418.0 Max. :470.0 Max. :12.770PIBC <- as.numeric(Canada[, "prod"])
E <- as.numeric(Canada[, "e"])cpgram(PIBC)
xm <- frequency(PIBC)/2
mp <- length(PIBC)-1
crit <- 1.358/(sqrt(mp)+0.12+0.11/sqrt(mp))
plot(c(0, xm*(1-crit)), c(crit, 1),type="l",col= 1, lty = 2)
lines(c(xm*crit, xm), c(0, 1-crit),type="l", col = 2, lty = 2)
a=seq(crit,1,by=(1-crit)/mp)
b=(a-crit/xm)
plot(a,type="l",col= 1, lty = 2)
lines(b,col= 2, lty = 2)
td <- function(y) {
# Author: Francisco Parra Rodríguez
# Some ideas from:
#Harvey, A.C. (1978), Linear Regression in the Frequency Domain, International Economic Review, 19, 507-512.
# DURBIN, J., "Tests for Serial Correlation in Regression Analysis based on the Periodogram ofLeast-Squares Residuals," Biometrika, 56, (No. 1, 1969), 1-15.
# Venables and Ripley, "Modern Applied Statistics with S" (4th edition, 2002).
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
per <- periodograma(y)
p <- as.numeric(per$densidad)
n <- length(p)
s <- p[1]
t <- 1:n
for(i in 2:n) {s1 <-p[i]+s[(i-1)]
s <- c(s,s1)
s2 <- s/s[n]
}
xm <- frequency(y)/2
c <- 1.358/(sqrt(n-1)+0.12+0.11/sqrt(n-1))
min <- -c+(t/length(p))
max <- c+(t/length(p))
data.frame(s2,min,max)
}Función gtd (a)
Presenta gráficamente los resultados de la prueba de Durbin (Durbin; 1969) :
gtd <- function (y) {
S <- td(y)
plot(ts(S), plot.type="single", lty=1:3,main = "Test Durbin",
ylab = "densidad acumulada",
xlab="frecuencia")
}data(ipi)
cpgram(ipi)
gtd(ipi)
Función rbs (a,b)
Realiza la regresión band spectrum del vector de datos “a” con el vector de datos “b”.
Hannan (1963) fue quien propuso la regresión en dominio de la frecuencia (regresión band spectrum). Engle (1974), demostró que dicha regresión no alteraba los supuestos básicos de la regresión clásica, cuyos estimadores eran Estimadores Lineales Insesgados y Optimos (ELIO).
\[\begin{equation} y=X\beta+u \end{equation}\] (4)
donde \(X\) es una matriz \(n x k\) de observaciones de \(k\) variables independientes, \(\beta\) es un vecto \(k x I\) de parámetros, \(y\) es un vecto \(n x 1\) de observaciones de la variable dependiente, y \(u\) en un vector \(n x I\) de pertubacciones de media cero y varianza constante, \(\sigma^2\).
El modelo puese expresarse en el dominio de la frecuencia aplicando una transformación lineal a las variables dependiente e independientes,por ejemplo, premultiplicando todas las variables por la matriz ortogonal \(W\). La técnica de la “regresión band spectrum”,consiste en realizar el analisis de regresión en el dominio de la frecuencia pero omitinedo determinadas oscilaciones periodicas. Con este procedimiento pueden tratarse problemas derivados de la estacionalidad de las series o de autocorrelación en los residuos. Engle (1974) muesta que si los residuos están correlacionados serialmente y son generados por un procieso estacionario estocastico, la regresión en el dominio de la frecuencia es el estimador asintóticamente más eficiente de \(\beta\).
La transformación de la ecuación (4) del dominio del tiempo al dominio de la frecuencia en Engle (1974), se basa en la matriz \(W\), cuyo elemento \((t, s)\) esta dado por:
\(w_{ts}=\frac{1}{\sqrt n} e^{i\lambda_t s}\),\(s= 0,1,...,n-1\)
donde \(\lambda_t = 2\pi \frac{t}n\), \(t=0,1,...,n-1\), y \(i=\sqrt{-1}\).
Premultiplicando las observaciones de (4) por \(W\), obtenemos: \[\begin{equation} \dot y=\dot X\beta+\dot u \end{equation}\] (5)
donde \(\dot y = Wy\),\(\dot X = WX\), y \(\dot u = Wu\).
Si el vector de las perturbaciones en (4) cumple las hipoteis clásicas del modelo de regresión: \(E[u] = 0\) y \(E[uu']=\sigma^2 I_n\). entonces el vector de perturbaciones transformado al dominio de la frecuencia, \(\dot u\), tendrá las mims propiedades. Por otro lado, dado que la matriz W es ortogonal., \(WW^{T}= I\), entonces \(W^T\) sería la transpuesta de la completa conjugada de W. De forma que las observaciones del modelo (5) acaban conteniendo el mismo tipo de información que las observaciones del modelo inicialmente planteado.
Si aplicamos MCO a (5) , dadas las propiedades de \(\dot u\), obtendríamos el mejor estimador lineal insesgando (ELIO) de \(\dot \beta\). El estimador obtenido sería de hecho identico al estimador MCO de (4).
Estimar (5) manteniendo unicamente determinadas frecuencias, puede llevarse a cabo omitiendo las observaciones correspondientes a las restantes frecuencias, si bien, dado que las variables en (5) son complejas, Engle (1974) sugiere la transformada inversa de Fourier para recomponer el modelo estimado en términos de tiempo.
Definiendo una matriz de tamaño \(A\) de tamaño n x n de ceros excepto en las posiciones de la diagonal principal correspondientes a las frecuencias que queremos incluir en la regresión y premultiplicando \(\dot y\) y \(\dot X\) por \(A\) eleminamos determindas observaciones y las reemplazamos por ceros para realizar la regresión band spectrum. Devolver al dominio del tiempo estas observaciones requiere:
\[\begin{equation} y^* = W^{T}A\dot y = W^{T}AWy \\ x^* = W^{T}A\dot x = W^{T}AWx \end{equation}\] (6)
Al regresar \(y^*\) sobre \(x^*\) obtenemos un \(\beta\) identico al estimador que obtendríamos al estimar por MCO \(\dot y\) frente a \(\dot x\).
Cuando se realiza la regresión band spectrum de esta mnera, ocurre un problema asociado a los grados de libertad de la regresión de \(y^*\) sobre \(x^*\) que asumen los programas estadisticos convencionales, \(n - k\), en vez de los grados de libertad reales que serían unicamente \(n'- k\), donde \(n'\) es el numero de frecuencias incluidas en la regresión band spectrum.
Si la regresión espectral va a ser usada para obtener un estimador asintóticamente eficiente de \(\beta\) en presencia de autocorrelación en el termino de error, la matriz \(A\) ha de ser reemplazada por otra matriz diagonal, \(V\). En dicha diagonal principal ha de incluirse el estimador de \(\int_u^{1/2}(\lambda)\), donde \(\int_u (\lambda)\) es el la transformación del termino de error obtenido en (4) al dominio de la frecuencia \(\lambda\). Puede utilizarse un programa convencional para obtener \(\beta\) haciendo una transformación análoga a (6); ver Engle and Gardner [1976]. Sin embargo, si el procedimiento va a ser iterativo, lo que podría llevar a una mejora en las propiededes de las muestra pequeñas, la transformada inversa de fourier debería emplearse en cada iteración.
Consideramos ahora el modelo de regresión siguiente:
\[\begin{equation} y_t=\beta_tx_t+u_t \end{equation}\] (7)
donde \(x_t\) es un vector n x 1 de observaciones de las variable independiente, \(\beta_t\) es un vector de n x 1 parametros, \(y_t\) es un vector de n x 1 observaciones de la variable depenendiente, y \(u_t\) es un vector de errores distribuidos con media cero y varianza constante.
Asumiendo que las series, \(y_t\),\(x_t\),\(\beta_t\) and \(ut\), pueden ser transformadas en el dominio de la frecuencia:
\[y_t=\eta^y+\sum_{j=1}^N[a^y_j\cos(\omega_j)+b^y_j\sin(\omega_j)\]
\[x_t=\eta^x+\sum_{j=1}^N[a^y_j\cos(\omega_j)+b^y_j\sin(\omega_j)]\]
\[ \beta_t=\eta^\beta+\sum_{j=1}^N[a^\beta_j\cos(\omega_j)+b^\beta_j\sin(\omega_j)]\]
Otenemos dichas series pre-multiplicando (7) por \(W\)
\(\dot y=\dot x\dot\beta+\dot u\) (8)
donde \(\dot y = Wy\),\(\dot x = Wx\), \(\dot \beta = W\beta\) y \(\dot u = Wu\)
El sistema (8) puede reescribirse como:
\[\begin{equation} \dot y=Wx_tI_nW^T\dot \beta + WI_nW^T\dot u \end{equation}\]
Si denominamos \(\dot e=WI_nW^T\dot u\), podrían buscarse los \(\dot \beta\) que minimizaran la suma cuadrática de los errores \(e_t=W^T\dot e\).
Una vez encontrada la solución a dicha optimización, se transformarían las series al dominio del tiempo para obtener el sistema (7).
El algoritmo de calculo se realiza en las siguentes fases:
- Calcula el co-espectro de la serie “x” e “y”
Sea \(x\) un vector n x 1 el modelo transformado en el dominio de la frecuencia esta dado por:
\(\hat x= Wx\)
Sea \(y\) un vector n x 1 el modelo transformado en el dominio de la frecuencia esta dado por:
\(\hat y= Wy\)
Denominando \(p_j\) el ordinal del cross-periodograma de \(\hat x\) y \(\hat y\) en la frecuencia \(\lambda_j=2\pi j/n\), y \(\hat x_j\) el j-th elemento de \(\hat x\) y \(\hat y_j\) el j-th elemento de \(\hat y\), entonces
\[ \left\lbrace \begin{array}{ll} p_j=\hat x_{2j}\hat y_{2j}+\hat x_{2j+1}\hat y_{2j+1} & \forall j = 1,...\frac{n-1}{2}\\ p_j=\hat x_{2j}\hat y_{2j}& \forall j = \frac{n}{2}-1 \end{array} \right . \]
\[p_0=\hat x_{1}\hat y_{1}\]
Ordena el co-espectro en base al valor absoluto de \(|p_j|\) y genera un índice en base a ese orden para cada coeficiente de fourier.
Calcula la matriz \(Wx_tI_nW^T\) y la ordena en base a el indice anterior.
Obtiene \(\dot e=WI_nW^T\dot u\), incluyendo el vector correspondiente al parámetro constante, \((1,0,...0)^n\), y calucula el modelo utilizando los dos primeros regresores de la matriz \(Wx_tI_nW^T\) reordenada y ampliadas, calcula el modelo para los 4 primeros, para los 6 primeros, hasta completar los \(n\) regresores de la matriz.
Realiza el test de durbin a los modelos estimados, y selecciona aquellos en donde los \(e_t=W^T\dot e\) están dentro de las bandas elegidas a los niveles de significación que determina la función: \(\alpha=0.1\).
De todos ellos elige aquel que tiene menos regresores. Si no encuentra modelo devuelve la solución MCO.
rdf <- function (y,x) {
# Author: Francisco Parra Rodriguez
# http://rpubs.com/PacoParra/24432
# Leemos datos en forma matriz
a <- matrix(y, nrow=1)
b <- matrix(x, nrow=1)
n <- length(a)
# calculamos el cros espectro mediante la funcion cperiodograma
cperiodograma <- function(y,x) {
# Author: Francisco Parra Rodr?guez
# http://econometria.wordpress.com/2013/08/21/estimation-of-time-varying-regression-coefficients/
cfx <- gdf(y)
n <- length(y)
cfy <- gdf(x)
if (n%%2==0) {
m1x <- c(0)
m2x <- c()
for(i in 1:n){
if(i%%2==0) m1x <-c(m1x,cfx[i]) else m2x <-c(m2x,cfx[i])}
m2x <- c(m2x,0)
m1y <- c(0)
m2y <- c()
for(i in 1:n){
if(i%%2==0) m1y <-c(m1y,cfy[i]) else m2y <-c(m2y,cfy[i])}
m2y <-c(m2y,0)
frecuencia <- seq(0:(n/2))
frecuencia <- frecuencia-1
omega <- pi*frecuencia/(n/2)
periodos <- n/frecuencia
densidad <- (m1x*m1y+m2x*m2y)/(4*pi)
tabla <- data.frame(omega,frecuencia, periodos,densidad)
tabla$densidad[(n/2+1)] <- 4*tabla$densidad[(n/2+1)]
data.frame(tabla[2:(n/2+1),])}
else {m1x <- c(0)
m2x <- c()
for(i in 1:(n-1)){
if(i%%2==0) m1x <-c(m1x,cfx[i]) else m2x <-c(m2x,cfx[i])}
m2x <-c(m2x,cfx[n])
m1y <- c(0)
m2y <- c()
for(i in 1:(n-1)){
if(i%%2==0) m1y <-c(m1y,cfy[i]) else m2y <-c(m2y,cfy[i])}
m2y <-c(m2y,cfy[n])
frecuencia <- seq(0:((n-1)/2))
frecuencia <- frecuencia-1
omega <- pi*frecuencia/(n/2)
periodos <- n/frecuencia
densidad <- (m1x*m1y+m2x*m2y)/(4*pi)
tabla <- data.frame(omega,frecuencia, periodos,densidad)
data.frame(tabla[2:((n+1)/2),])}
}
cper <- cperiodograma(a,b)
# Ordenamos de mayor a menor las densidades absolutas del periodograma, utilizando la función "sort.data.frame" function, Kevin Wright. Package taRifx
S1 <- data.frame(f1=cper$frecuencia,p=abs(cper$densidad))
S <- sort.data.frame (S1,formula=~-p)
id <- seq(2,n)
m1 <- cbind(S$f1*2,evens(id))
if (n%%2==0) {m2 <- cbind(S$f1[1:(n/2-1)]*2+1,odds(id))} else
{m2 <- cbind(S$f1*2+1,odds(id))}
m <- rbind(m1,m2)
colnames(m) <- c("f1","id")
M <- sort.data.frame (m,formula=~id)
M <- rbind(c(1,1),M)
# Obtenemos la función auxiliar (cdf) del predictor y se ordena según el indice de las mayores densidades absolutas del co-espectro.
cx <- cdf(b)
id <- seq(1,n)
S1 <- data.frame(cx,c=id)
S2 <- merge(M,S1,by.x="id",by.y="c")
S3 <- sort.data.frame (S2,formula=~f1)
m <- n+2
X1 <- S3[,3:m]
X1 <- rbind(C=c(1,rep(0,(n-1))),S3[,3:m])
# Se realizan las regresiones en el dominio de la frecuencia utilizando un modelo con constante, pendiente y los armónicos correspondientes a las frecuencias más altas de la densidad del coespectro. Se realiza un test de durbin para el residuo y se seleccionan aquellas que son significativas.
par <- evens(id)
i <- 1
D <- 1
resultado <- cbind(i,D)
for (i in par) {
X <- as.matrix(X1[1:i,])
cy <- gdf(a)
B1 <- solve(X%*%t(X))%*%(X%*%cy)
Y <- t(X)%*%B1
F <- gdt(Y)
res <- (t(a) - F)
T <- td(res)
L <- as.numeric(c(T$min<T$s2,T$s2<T$max))
LT <- sum(L)
D <- LT-(n-1)
resultado1 <- cbind(i,D)
resultado <- rbind(resultado,resultado1)
resultado}
resultado2 <-data.frame(resultado)
criterio <- resultado2[which(resultado2$D==0),]
sol <- as.numeric(is.na(criterio$i[1]))
if (sol==1) {
X <- as.matrix(X1[1:2,])
cy <- gdf(a)
B1 <- solve(X%*%t(X))%*%(X%*%cy)
Y <- t(X)%*%B1
F <- gdt(Y)
res <- (t(a) - F)
datos <- data.frame(cbind(t(a),t(b),F,res))
colnames(datos) <- c("Y","X","F","res")
list(datos=datos,Fregresores=t(X),Tregresores= t(MW(n))%*%t(X),Nregresores=criterio$i[1])
} else {
X <- as.matrix(X1[1:criterio$i[1],])
cy <- gdf(a)
B1 <- solve(X%*%t(X))%*%(X%*%cy)
Y <- t(X)%*%B1
F <- gdt(Y)
res <- (t(a) - F)
datos <- data.frame(cbind(t(a),t(b),F,res))
colnames(datos) <- c("Y","X","F","res")
list(datos=datos,Fregresores=t(X),Tregresores= t(MW(n))%*%t(X),Nregresores=criterio$i[1])}
}Ejemplo: Regresión entre el empleo y PIB de canada
Periodograma del PIB de Canada y representación gráfica por diferentes metodos metodos
gperiodograma(PIBC)
densidad <- Mod(fft(PIBC))^2/length(PIBC)
plot(densidad[2:43],type="l")
# periodogramas acumulados
gtd(PIBC)
cpgram(PIBC)
Analisis del empleo de Canada
gperiodograma (E)
gtd (E)
Regresión dependiente de la frecuencia entre el PIBC y E con datos del mercado de trabajo de Canada
reg1 <- rdf (PIBC,E)
reg1## $datos
## Y X F res
## 1 405.3665 929.6105 401.9374 3.42905707
## 2 404.6398 929.8040 402.0151 2.62470869
## 3 403.8149 930.3184 402.2218 1.59312539
## 4 404.2158 931.4277 402.6674 1.54841692
## 5 405.0467 932.6620 403.1632 1.88353965
## 6 404.4167 933.5509 403.5203 0.89648563
## 7 402.8191 933.5315 403.5125 -0.69332739
## 8 401.9773 933.0769 403.3298 -1.35249142
## 9 402.0897 932.1238 402.9470 -0.85723647
## 10 401.3067 930.6359 402.3493 -1.04262771
## 11 401.6302 929.0971 401.7312 -0.10098695
## 12 401.5638 928.5633 401.5168 0.04699005
## 13 402.8157 929.0694 401.7200 1.09565952
## 14 403.1421 930.2655 402.2005 0.94158810
## 15 403.0786 931.6770 402.7675 0.31110296
## 16 403.7188 932.1390 402.9531 0.76571281
## 17 404.8668 932.2767 403.0084 1.85840531
## 18 405.6362 932.8328 403.2318 2.40441257
## 19 405.1363 933.7334 403.5936 1.54273153
## 20 406.0246 934.1772 403.7718 2.25281967
## 21 406.4123 934.5928 403.9388 2.47349212
## 22 406.3009 935.6067 404.3460 1.95488998
## 23 406.3354 936.5111 404.7093 1.62602699
## 24 406.7737 937.4201 405.0745 1.69922948
## 25 405.1525 938.4159 405.4745 -0.32193731
## 26 404.9298 938.9992 405.7088 -0.77894127
## 27 404.5765 939.2354 405.8036 -1.22709963
## 28 404.1995 939.6795 405.9821 -1.78256532
## 29 405.9499 940.2497 406.2111 -0.26123534
## 30 405.8221 941.4358 406.6876 -0.86546614
## 31 406.4463 942.2981 407.0339 -0.58764686
## 32 407.0512 943.5322 407.5297 -0.47842286
## 33 407.9460 944.3490 407.8578 0.08827177
## 34 408.1796 944.8215 408.0476 0.13202462
## 35 408.5998 945.0671 408.1462 0.45357742
## 36 409.0906 945.8067 408.4433 0.64723710
## 37 408.7042 946.8697 408.8703 -0.16608270
## 38 408.9803 946.8766 408.8731 0.10718535
## 39 408.3287 947.2497 409.0230 -0.69426371
## 40 407.8857 947.6513 409.1843 -1.29857083
## 41 407.2605 948.1840 409.3982 -2.13771517
## 42 406.7752 948.3492 409.4646 -2.68948394
## 43 406.1794 948.0322 409.3373 -3.15785669
## 44 405.4398 947.1065 408.9654 -3.52563421
## 45 403.2800 946.0796 408.5529 -5.27294604
## 46 403.3649 946.1838 408.5948 -5.22994038
## 47 403.3807 946.2258 408.6117 -5.23098045
## 48 404.0032 945.9978 408.5201 -4.51688742
## 49 404.4774 945.5183 408.3275 -3.85005780
## 50 404.7868 945.3514 408.2604 -3.47363823
## 51 405.2710 945.2918 408.2365 -2.96547129
## 52 405.3830 945.4008 408.2803 -2.89726683
## 53 405.1564 945.9058 408.4831 -3.32670883
## 54 406.4700 945.9035 408.4822 -2.01214200
## 55 406.2293 946.3190 408.6491 -2.41980324
## 56 406.7265 946.5796 408.7538 -2.02730670
## 57 408.5785 946.7800 408.8343 -0.25578945
## 58 409.6767 947.6283 409.1750 0.50167840
## 59 410.3858 948.6221 409.5742 0.81153924
## 60 410.5395 949.3992 409.8864 0.65313324
## 61 410.4453 949.9481 410.1069 0.33835629
## 62 410.6256 949.7945 410.0452 0.58042099
## 63 410.8672 949.9534 410.1090 0.75823194
## 64 411.2359 950.2502 410.2283 1.00766359
## 65 410.6637 950.5380 410.3439 0.31979708
## 66 410.8085 950.7871 410.4439 0.36458930
## 67 412.1160 950.8695 410.4770 1.63894280
## 68 412.9994 950.9281 410.5006 2.49884763
## 69 412.9551 951.8457 410.8691 2.08590726
## 70 412.8241 952.6005 411.1723 1.65180389
## 71 413.0489 953.5976 411.5728 1.47602646
## 72 413.6110 954.1434 411.7921 1.81891079
## 73 413.6048 954.5426 411.9525 1.65231676
## 74 412.9684 955.2631 412.2419 0.72648568
## 75 412.2659 956.0561 412.5604 -0.29452558
## 76 412.9106 956.7966 412.8579 0.05271444
## 77 413.8294 957.3865 413.0948 0.73458012
## 78 414.2242 958.0634 413.3668 0.85739527
## 79 415.1678 958.7166 413.6291 1.53863717
## 80 415.7016 959.4881 413.9391 1.76251984
## 81 416.8674 960.3625 414.2903 2.57712569
## 82 417.6104 960.7834 414.4593 3.15105046
## 83 418.0030 961.0290 414.5580 3.44495549
## 84 417.2667 961.7657 414.8539 2.41273563
##
## $Fregresores
## C 1
## X1 1 944.257255028
## X2 0 -0.821047451
## X3 0 -6.332682221
## X4 0 1.189610377
## X5 0 -4.710895918
## X6 0 0.092783361
## X7 0 -1.817896593
## X8 0 0.143048535
## X9 0 -1.845972568
## X10 0 -0.502320083
## X11 0 -1.085803966
## X12 0 -0.451367877
## X13 0 -0.961354541
## X14 0 -0.450139687
## X15 0 -0.936788446
## X16 0 -0.457678076
## X17 0 -1.021053349
## X18 0 -0.526099158
## X19 0 -0.918440492
## X20 0 -0.248582850
## X21 0 -0.920358327
## X22 0 -0.172167811
## X23 0 -0.672872670
## X24 0 -0.237618583
## X25 0 -0.604807230
## X26 0 -0.214969449
## X27 0 -0.480703142
## X28 0 -0.221105460
## X29 0 -0.498840636
## X30 0 -0.398564812
## X31 0 -0.389467803
## X32 0 -0.260406585
## X33 0 -0.425986472
## X34 0 -0.251718657
## X35 0 -0.364087458
## X36 0 -0.198124941
## X37 0 -0.401640033
## X38 0 -0.254505068
## X39 0 -0.287770143
## X40 0 -0.251386539
## X41 0 -0.292282833
## X42 0 -0.271415430
## X43 0 -0.265884902
## X44 0 -0.274013845
## X45 0 -0.253340306
## X46 0 -0.248724863
## X47 0 -0.254191881
## X48 0 -0.246637978
## X49 0 -0.190508230
## X50 0 -0.294006546
## X51 0 -0.205384915
## X52 0 -0.264193397
## X53 0 -0.187906749
## X54 0 -0.285960798
## X55 0 -0.201715381
## X56 0 -0.255648290
## X57 0 -0.165321415
## X58 0 -0.276092593
## X59 0 -0.155624735
## X60 0 -0.287708850
## X61 0 -0.133277651
## X62 0 -0.261103583
## X63 0 -0.111506735
## X64 0 -0.263263618
## X65 0 -0.121437713
## X66 0 -0.289524182
## X67 0 -0.098951038
## X68 0 -0.286230070
## X69 0 -0.084156321
## X70 0 -0.264650452
## X71 0 -0.082419830
## X72 0 -0.280875222
## X73 0 -0.042686146
## X74 0 -0.260687679
## X75 0 -0.065741600
## X76 0 -0.274635485
## X77 0 -0.027613053
## X78 0 -0.285388933
## X79 0 -0.039470690
## X80 0 -0.281213839
## X81 0 -0.023390955
## X82 0 -0.276735336
## X83 0 0.001934752
## X84 0 -0.191962767
##
## $Tregresores
## C 1
## [1,] 0.1091089 101.4288
## [2,] 0.1091089 101.4499
## [3,] 0.1091089 101.5061
## [4,] 0.1091089 101.6271
## [5,] 0.1091089 101.7618
## [6,] 0.1091089 101.8588
## [7,] 0.1091089 101.8566
## [8,] 0.1091089 101.8070
## [9,] 0.1091089 101.7030
## [10,] 0.1091089 101.5407
## [11,] 0.1091089 101.3728
## [12,] 0.1091089 101.3146
## [13,] 0.1091089 101.3698
## [14,] 0.1091089 101.5003
## [15,] 0.1091089 101.6543
## [16,] 0.1091089 101.7047
## [17,] 0.1091089 101.7197
## [18,] 0.1091089 101.7804
## [19,] 0.1091089 101.8787
## [20,] 0.1091089 101.9271
## [21,] 0.1091089 101.9724
## [22,] 0.1091089 102.0831
## [23,] 0.1091089 102.1817
## [24,] 0.1091089 102.2809
## [25,] 0.1091089 102.3896
## [26,] 0.1091089 102.4532
## [27,] 0.1091089 102.4790
## [28,] 0.1091089 102.5274
## [29,] 0.1091089 102.5897
## [30,] 0.1091089 102.7191
## [31,] 0.1091089 102.8132
## [32,] 0.1091089 102.9478
## [33,] 0.1091089 103.0369
## [34,] 0.1091089 103.0885
## [35,] 0.1091089 103.1153
## [36,] 0.1091089 103.1960
## [37,] 0.1091089 103.3119
## [38,] 0.1091089 103.3127
## [39,] 0.1091089 103.3534
## [40,] 0.1091089 103.3972
## [41,] 0.1091089 103.4554
## [42,] 0.1091089 103.4734
## [43,] 0.1091089 103.4388
## [44,] 0.1091089 103.3378
## [45,] 0.1091089 103.2257
## [46,] 0.1091089 103.2371
## [47,] 0.1091089 103.2417
## [48,] 0.1091089 103.2168
## [49,] 0.1091089 103.1645
## [50,] 0.1091089 103.1463
## [51,] 0.1091089 103.1398
## [52,] 0.1091089 103.1517
## [53,] 0.1091089 103.2068
## [54,] 0.1091089 103.2065
## [55,] 0.1091089 103.2519
## [56,] 0.1091089 103.2803
## [57,] 0.1091089 103.3022
## [58,] 0.1091089 103.3947
## [59,] 0.1091089 103.5032
## [60,] 0.1091089 103.5879
## [61,] 0.1091089 103.6478
## [62,] 0.1091089 103.6311
## [63,] 0.1091089 103.6484
## [64,] 0.1091089 103.6808
## [65,] 0.1091089 103.7122
## [66,] 0.1091089 103.7394
## [67,] 0.1091089 103.7484
## [68,] 0.1091089 103.7548
## [69,] 0.1091089 103.8549
## [70,] 0.1091089 103.9372
## [71,] 0.1091089 104.0460
## [72,] 0.1091089 104.1056
## [73,] 0.1091089 104.1491
## [74,] 0.1091089 104.2278
## [75,] 0.1091089 104.3143
## [76,] 0.1091089 104.3951
## [77,] 0.1091089 104.4594
## [78,] 0.1091089 104.5333
## [79,] 0.1091089 104.6046
## [80,] 0.1091089 104.6887
## [81,] 0.1091089 104.7841
## [82,] 0.1091089 104.8301
## [83,] 0.1091089 104.8569
## [84,] 0.1091089 104.9372
##
## $Nregresores
## [1] NAmco<-lm(PIBC~E)
plot(ts(lm(PIBC~E)$fitted),type="l")
plot(ts(reg1$datos$F),type="l")
Regresión dependiente de la frecuencia entre el PIBC y empleo con datos del mercado de trabajo de Canada
reg2 <- rdf (E,PIBC)
reg2## $datos
## Y X F res
## 1 929.6105 405.3665 931.7636 -2.153094777
## 2 929.8040 404.6398 930.7738 -0.969818214
## 3 930.3184 403.8149 929.5617 0.756677040
## 4 931.4277 404.2158 931.6228 -0.195107190
## 5 932.6620 405.0467 934.7974 -2.135347526
## 6 933.5509 404.4167 934.4661 -0.915132065
## 7 933.5315 402.8191 932.3376 1.193909516
## 8 933.0769 401.9773 930.8736 2.203262084
## 9 932.1238 402.0897 931.7271 0.396695094
## 10 930.6359 401.3067 929.7421 0.893823968
## 11 929.0971 401.6302 930.1002 -1.003101101
## 12 928.5633 401.5638 929.3738 -0.810503313
## 13 929.0694 402.8157 931.0588 -1.989395049
## 14 930.2655 403.1421 931.1853 -0.919783073
## 15 931.6770 403.0786 929.8379 1.839098684
## 16 932.1390 403.7188 930.7300 1.408991152
## 17 932.2767 404.8668 932.3535 -0.076767512
## 18 932.8328 405.6362 933.6135 -0.780683828
## 19 933.7334 405.1363 932.3819 1.351491371
## 20 934.1772 406.0246 934.1391 0.038143470
## 21 934.5928 406.4123 935.6584 -1.065583304
## 22 935.6067 406.3009 935.7812 -0.174448203
## 23 936.5111 406.3354 937.1611 -0.649990641
## 24 937.4201 406.7737 939.1433 -1.723172017
## 25 938.4159 405.1525 937.0363 1.379608960
## 26 938.9992 404.9298 937.9482 1.050997386
## 27 939.2354 404.5765 938.0848 1.150566215
## 28 939.6795 404.1995 938.3615 1.318013669
## 29 940.2497 405.9499 942.2151 -1.965440531
## 30 941.4358 405.8221 942.1550 -0.719167365
## 31 942.2981 406.4463 942.8315 -0.533400980
## 32 943.5322 407.0512 943.4625 0.069730378
## 33 944.3490 407.9460 944.6470 -0.298066834
## 34 944.8215 408.1796 944.2002 0.621316773
## 35 945.0671 408.5998 944.9913 0.075868255
## 36 945.8067 409.0906 945.7322 0.074561033
## 37 946.8697 408.7042 945.7234 1.146264263
## 38 946.8766 408.9803 947.1466 -0.269977510
## 39 947.2497 408.3287 947.2235 0.026239799
## 40 947.6513 407.8857 948.0178 -0.366495221
## 41 948.1840 407.2605 948.1775 0.006482568
## 42 948.3492 406.7752 949.0606 -0.711367655
## 43 948.0322 406.1794 948.8036 -0.771411385
## 44 947.1065 405.4398 948.5275 -1.420989144
## 45 946.0796 403.2800 944.2414 1.838113878
## 46 946.1838 403.3649 944.8062 1.377563454
## 47 946.2258 403.3807 944.9164 1.309400950
## 48 945.9978 404.0032 945.8010 0.196782599
## 49 945.5183 404.4774 946.6685 -1.150262922
## 50 945.3514 404.7868 946.3548 -1.003355554
## 51 945.2918 405.2710 946.9114 -1.619566883
## 52 945.4008 405.3830 946.0107 -0.609907574
## 53 945.9058 405.1564 944.5228 1.382979806
## 54 945.9035 406.4700 946.3429 -0.439406342
## 55 946.3190 406.2293 944.4210 1.898023449
## 56 946.5796 406.7265 944.6716 1.907983043
## 57 946.7800 408.5785 947.3354 -0.555357544
## 58 947.6283 409.6767 949.1629 -1.534572363
## 59 948.6221 410.3858 949.8444 -1.222310725
## 60 949.3992 410.5395 949.8093 -0.410161010
## 61 949.9481 410.4453 949.4349 0.513222968
## 62 949.7945 410.6256 949.4690 0.325493795
## 63 949.9534 410.8672 950.1949 -0.241502827
## 64 950.2502 411.2359 950.4887 -0.238492409
## 65 950.5380 410.6637 949.2746 1.263437253
## 66 950.7871 410.8085 948.9946 1.792503069
## 67 950.8695 412.1160 951.4451 -0.575575508
## 68 950.9281 412.9994 953.0067 -2.078537355
## 69 951.8457 412.9551 952.3764 -0.530647333
## 70 952.6005 412.8241 952.3269 0.273560215
## 71 953.5976 413.0489 952.6946 0.902998712
## 72 954.1434 413.6110 954.7529 -0.609511069
## 73 954.5426 413.6048 955.3023 -0.759687139
## 74 955.2631 412.9684 954.8611 0.402077365
## 75 956.0561 412.2659 954.2151 1.840936628
## 76 956.7966 412.9106 955.9376 0.858986106
## 77 957.3865 413.8294 958.3763 -0.989836727
## 78 958.0634 414.2242 958.5083 -0.444907632
## 79 958.7166 415.1678 959.9833 -1.266723686
## 80 959.4881 415.7016 959.6417 -0.153546694
## 81 960.3625 416.8674 960.7214 -0.358930865
## 82 960.7834 417.6104 960.8974 -0.113976788
## 83 961.0290 418.0030 960.2399 0.789142112
## 84 961.7657 417.2667 958.1156 3.650076178
##
## $Fregresores
## C 1 2 3 4
## X1 1 407.820910977 1.101647001 -2.372322e+00 -1.332132e-01
## X2 0 1.101647001 407.726715019 -1.770789e+00 9.942845e-01
## X3 0 -2.372322091 -1.770789239 4.079151e+02 1.352597e+00
## X4 0 -0.133213202 0.994284539 1.352597e+00 4.081443e+02
## X5 0 -2.504274157 -2.002372720 5.636796e-01 -9.588785e-01
## X6 0 0.304483679 0.229214643 8.119107e-01 1.022470e+00
## X7 0 -0.459460566 -2.729667773 -4.176066e-01 -1.776054e+00
## X8 0 0.457371658 0.458790398 2.263185e-01 -1.754348e-01
## X9 0 -1.356059028 -0.423456819 -2.818545e-02 -2.080014e+00
## X10 0 0.344343923 0.242171715 6.496536e-01 1.948338e-01
## X11 0 -0.139397810 -1.268103487 4.046495e-01 -7.182042e-01
## X12 0 -0.114889135 0.223019254 -2.947474e-01 1.696598e-01
## X13 0 -0.437310122 -0.491885700 2.639566e-01 -1.316347e+00
## X14 0 -0.028947069 -0.234989661 -4.824345e-02 3.396932e-02
## X15 0 -0.556233618 -0.666693355 7.251189e-02 -1.980937e-01
## X16 0 -0.217436431 -0.229987276 2.937920e-01 -1.733080e-01
## X17 0 -0.505536663 -0.492841125 1.890499e-01 -5.833212e-01
## X18 0 -0.296304057 -0.245819865 8.337215e-02 -1.645524e-01
## X19 0 -0.140748986 -0.631564650 -6.168169e-02 -6.065592e-01
## X20 0 -0.130205356 -0.353602353 -1.137181e-01 -2.067007e-01
## X21 0 -0.387630631 -0.312767196 -6.543486e-02 -5.116175e-01
## X22 0 -0.203765186 -0.145019002 1.199471e-01 -3.327174e-01
## X23 0 -0.301570624 -0.428245362 -3.911918e-02 -2.843265e-01
## X24 0 -0.074882484 -0.267282497 2.844071e-02 -1.265830e-01
## X25 0 -0.217999768 -0.398044556 -2.088499e-02 -4.682630e-01
## X26 0 -0.174229346 -0.087463802 -4.001763e-02 -1.912143e-01
## X27 0 -0.261349385 -0.348315856 -1.843602e-02 -3.455260e-01
## X28 0 -0.048810011 -0.170329271 5.251851e-02 -9.687923e-02
## X29 0 -0.274593239 -0.317085334 -7.606823e-02 -2.970466e-01
## X30 0 -0.066652619 -0.078443207 5.126926e-02 -2.775376e-01
## X31 0 -0.187076995 -0.337064222 9.415428e-03 -3.414972e-01
## X32 0 -0.062125436 -0.201469323 -2.441183e-02 -1.063564e-01
## X33 0 -0.202087555 -0.288978656 1.072083e-01 -2.464175e-01
## X34 0 -0.218268029 -0.115771789 9.064668e-02 -8.133376e-02
## X35 0 -0.221600539 -0.195148282 2.791315e-02 -1.891347e-01
## X36 0 -0.101600598 -0.188542043 9.984393e-02 -3.748884e-02
## X37 0 -0.073893792 -0.213546557 -1.201356e-01 -2.347506e-01
## X38 0 -0.048370686 -0.065401990 -3.960230e-02 -1.806714e-01
## X39 0 -0.080399897 -0.144103902 -7.828295e-02 -2.141852e-01
## X40 0 0.009108216 -0.060535859 -6.386325e-04 -1.149145e-01
## X41 0 -0.129899900 -0.114341258 -7.870620e-03 -6.895546e-02
## X42 0 -0.037239947 -0.036631583 7.514844e-02 -1.423532e-01
## X43 0 -0.081303060 -0.108557763 4.951255e-02 -9.825609e-02
## X44 0 -0.060913098 -0.134482614 1.608517e-02 -5.547114e-02
## X45 0 -0.023623961 -0.098894719 8.181738e-02 -1.117372e-01
## X46 0 -0.152947190 -0.104983683 -3.179458e-03 -9.969282e-02
## X47 0 -0.058555192 -0.036588783 1.883955e-02 -1.368533e-01
## X48 0 -0.087556250 -0.181510191 -3.795858e-02 -1.164487e-01
## X49 0 -0.028120393 -0.120768128 -3.478980e-02 -4.475588e-02
## X50 0 -0.103746984 -0.135288273 -8.167094e-03 -1.670569e-01
## X51 0 -0.112236732 -0.047935335 1.146504e-02 -2.230761e-02
## X52 0 -0.103770261 -0.132267056 9.846051e-02 -1.603479e-01
## X53 0 -0.039670408 -0.060266194 -1.445333e-02 -5.269803e-02
## X54 0 -0.083306881 -0.171812964 -4.762692e-03 -1.434749e-01
## X55 0 0.027007463 -0.060865122 2.505965e-02 -1.432642e-01
## X56 0 -0.139209964 -0.129021600 -8.299799e-02 -1.062467e-01
## X57 0 -0.046405872 -0.044803665 1.120788e-02 -3.783239e-02
## X58 0 -0.099157215 -0.131306328 2.303273e-02 -1.567347e-01
## X59 0 -0.090369414 -0.042595086 -6.556629e-02 -2.817290e-02
## X60 0 -0.046485226 -0.167942580 1.663076e-02 -1.728131e-01
## X61 0 -0.013832676 -0.111170890 2.771310e-02 -5.037161e-02
## X62 0 -0.138349459 -0.107246797 -7.776522e-03 -1.024701e-01
## X63 0 -0.066849966 -0.027338881 4.150676e-02 -1.226558e-01
## X64 0 -0.105184648 -0.130183177 -1.148491e-02 -1.155256e-01
## X65 0 -0.024830340 -0.106025041 -6.547250e-02 -3.451017e-03
## X66 0 -0.045757356 -0.157032385 2.388786e-02 -1.770188e-01
## X67 0 -0.083092086 -0.011227539 8.278829e-03 -5.837383e-02
## X68 0 -0.116892680 -0.111546313 4.765121e-02 -1.275784e-01
## X69 0 0.008952201 -0.069858744 4.683564e-02 -2.158890e-02
## X70 0 -0.111992952 -0.135857239 -1.036136e-02 -1.167382e-01
## X71 0 -0.015703097 0.002298968 -2.945397e-02 -3.431219e-02
## X72 0 -0.075238471 -0.163573834 3.554655e-02 -1.860626e-01
## X73 0 -0.005700970 0.013339021 5.191882e-03 -9.404575e-03
## X74 0 -0.119335382 -0.156608614 -1.170354e-02 -1.212562e-01
## X75 0 0.034567321 -0.019765932 5.020535e-02 -3.947051e-02
## X76 0 -0.146239555 -0.126448068 -5.280954e-02 -1.196106e-01
## X77 0 -0.022252279 -0.003923961 -4.231765e-02 -9.219608e-04
## X78 0 -0.059489191 -0.169815898 1.884397e-02 -1.197631e-01
## X79 0 -0.040116640 -0.012625504 -3.699806e-02 1.320891e-02
## X80 0 -0.093916392 -0.077445495 1.713288e-02 -1.826664e-01
## X81 0 0.004397121 -0.039600621 -6.684925e-03 -1.573474e-02
## X82 0 -0.050035279 -0.145668288 -3.109234e-03 -7.744550e-02
## X83 0 -0.015887095 0.003109234 1.285045e-02 -1.713288e-02
## X84 0 -0.079259371 -0.050035279 1.588709e-02 -9.391639e-02
## 5 8 9 6 7
## X1 -2.504274e+00 4.573717e-01 -1.356059e+00 0.304483679 -4.594606e-01
## X2 -2.002373e+00 4.587904e-01 -4.234568e-01 0.229214643 -2.729668e+00
## X3 5.636796e-01 2.263185e-01 -2.818545e-02 0.811910704 -4.176066e-01
## X4 -9.588785e-01 -1.754348e-01 -2.080014e+00 1.022469988 -1.776054e+00
## X5 4.074975e+02 1.461564e+00 -1.295707e-02 1.578915901 5.354941e-01
## X6 1.578916e+00 7.585134e-01 -2.070802e+00 407.739672091 -3.092250e-01
## X7 5.354941e-01 1.284168e+00 7.994507e-01 -0.309224953 4.079021e+02
## X8 1.461564e+00 4.076672e+02 -3.574684e-01 0.758513396 1.284168e+00
## X9 -1.295707e-02 -3.574684e-01 4.079747e+02 -2.070801601 7.994507e-01
## X10 -6.842888e-02 5.694635e-01 1.577960e+00 -0.247946733 1.413321e+00
## X11 2.357711e-01 -1.777010e+00 9.885007e-01 -2.128257641 5.955482e-02
## X12 6.014101e-01 -1.862650e-01 1.496693e+00 0.005783867 2.253631e-01
## X13 4.771614e-01 -2.044885e+00 -2.126868e-03 -0.424412244 4.248211e-01
## X14 -9.554254e-04 7.121873e-02 1.116450e-01 0.231341511 6.847823e-01
## X15 4.530065e-01 -5.381303e-01 3.593862e-01 -1.232974782 4.154797e-01
## X16 3.512871e-02 2.704607e-01 8.047294e-01 0.099404178 -1.146735e-01
## X17 1.083020e-02 -1.113028e+00 3.763605e-01 -0.311811770 3.875717e-01
## X18 1.800739e-01 1.202892e-01 -8.623279e-02 -0.134188798 1.550758e-01
## X19 1.236151e-01 -2.833711e-01 3.666867e-01 -0.463374067 -2.828897e-02
## X20 2.033193e-01 -1.157528e-01 1.150582e-01 -0.143667421 2.085146e-01
## X21 -1.008009e-01 -5.033917e-01 -4.672500e-02 -0.578118485 1.027301e-01
## X22 -8.527736e-02 -6.759919e-02 2.610332e-01 -0.188264665 1.633017e-01
## X23 -8.631986e-02 -5.256000e-01 2.666185e-02 -0.551635144 -1.192369e-01
## X24 7.992951e-02 -1.976801e-01 2.145709e-01 -0.256649127 -3.275885e-02
## X25 -5.755520e-02 -5.003659e-01 -1.098215e-01 -0.231807974 -1.623881e-01
## X26 8.095922e-02 -3.638574e-01 -5.717068e-02 -0.135998407 1.311988e-01
## X27 -9.695323e-02 -2.562198e-01 -5.517980e-02 -0.416993728 -4.813977e-02
## X28 1.125163e-02 -1.639116e-01 2.218454e-01 -0.298422549 5.654739e-02
## X29 -9.020595e-03 -3.263470e-01 -2.022662e-02 -0.369937878 1.025506e-02
## X30 2.810668e-02 -1.782870e-01 1.563913e-01 -0.124792384 1.018983e-01
## X31 3.114005e-02 -2.700939e-01 -1.098805e-01 -0.206399916 1.889256e-02
## X32 1.419159e-01 -4.650943e-02 6.229601e-02 -0.157401992 1.279506e-01
## X33 3.732858e-02 -2.460022e-01 -5.939039e-02 -0.241653235 -8.899551e-02
## X34 7.543210e-02 -1.495314e-01 1.273120e-01 -0.028073408 1.023136e-01
## X35 -1.292728e-02 -2.422919e-01 -9.686613e-02 -0.286019842 -4.095437e-02
## X36 5.104438e-02 -7.758595e-02 1.774621e-01 -0.073463138 7.479347e-02
## X37 -5.036980e-02 -2.108714e-01 8.558175e-03 -0.189773357 -2.079790e-02
## X38 9.920530e-02 -1.552805e-01 9.087864e-02 -0.087001382 1.261928e-01
## X39 -1.280062e-01 -1.736882e-01 6.101948e-02 -0.159602143 -8.572525e-04
## X40 3.554614e-02 -1.058409e-01 1.230134e-01 -0.262488799 1.152905e-01
## X41 -2.877041e-02 -1.627816e-01 1.798230e-02 -0.198100017 -4.618881e-02
## X42 1.544654e-02 -2.276990e-01 7.733189e-02 -0.133754090 3.236668e-02
## X43 7.394676e-02 -2.360586e-01 -8.097861e-02 -0.072134922 -9.930854e-03
## X44 7.196898e-02 -1.452191e-01 2.419959e-02 -0.107563435 -2.251204e-02
## X45 6.835210e-02 -8.030202e-02 1.534183e-03 -0.136214667 3.915696e-02
## X46 -2.187341e-02 -9.311010e-02 7.594847e-02 -0.066936173 6.380189e-02
## X47 4.702758e-02 -3.775415e-02 2.470362e-02 -0.119904315 7.981714e-02
## X48 -1.134655e-02 -9.199583e-02 5.903919e-02 -0.085239480 7.658711e-02
## X49 3.030459e-02 -1.246670e-01 1.048768e-01 -0.038392785 3.257424e-02
## X50 6.050193e-02 -9.644736e-02 -6.410880e-03 -0.141508374 -1.610924e-02
## X51 -4.924313e-02 -1.213908e-01 4.378212e-02 -0.049518570 5.536424e-02
## X52 -1.292979e-02 -7.594208e-02 6.923484e-03 -0.178264734 -2.249605e-02
## X53 3.652469e-02 -2.648584e-02 -1.020205e-02 -0.105305599 -3.803526e-02
## X54 1.546253e-02 -2.059778e-01 -5.865291e-03 -0.094781637 1.010294e-02
## X55 -3.245456e-03 -8.867484e-02 -1.032215e-02 -0.029665300 -2.904160e-02
## X56 1.827004e-02 -1.362884e-01 2.326419e-03 -0.171188036 3.209329e-02
## X57 -4.050664e-02 -3.744182e-02 1.246516e-02 -0.126633418 2.446764e-02
## X58 -6.636722e-02 -1.057155e-01 2.060838e-02 -0.147753433 1.049351e-02
## X59 3.892098e-02 -1.381183e-01 -4.100486e-02 -0.045608916 1.000123e-03
## X60 1.525621e-02 -1.560323e-01 3.438138e-02 -0.091262197 -7.785214e-02
## X61 -2.405953e-02 -2.172105e-02 9.278952e-03 -0.039657817 -2.655152e-02
## X62 5.145848e-03 -1.380978e-01 -3.020093e-02 -0.181091916 3.914407e-02
## X63 -3.775940e-02 7.993394e-03 2.028412e-02 -0.026483745 -1.578070e-02
## X64 1.611134e-02 -1.516379e-01 2.878271e-02 -0.149305715 5.279706e-02
## X65 4.978559e-02 -3.684510e-02 -4.523468e-02 -0.075004592 9.076237e-03
## X66 3.616630e-02 -1.544976e-01 8.834361e-02 -0.086071652 5.749985e-03
## X67 -1.863686e-02 -3.945804e-02 1.426812e-02 -0.013812374 2.033161e-02
## X68 1.352651e-02 -1.362770e-01 -5.953558e-03 -0.182210698 7.171285e-02
## X69 -2.117515e-02 -2.551592e-02 7.053696e-02 -0.022827277 -1.344498e-02
## X70 8.319776e-02 -1.398931e-01 1.890332e-02 -0.177783759 1.822964e-03
## X71 5.202752e-02 -7.563681e-02 -5.576263e-02 -0.033292439 2.903020e-02
## X72 -2.206490e-02 -1.407857e-01 2.066694e-02 -0.074420547 3.038823e-02
## X73 2.075137e-02 -1.444847e-02 -7.967861e-03 -0.087121725 9.709874e-03
## X74 -1.726298e-02 -6.773562e-02 4.752110e-02 -0.149064524 -3.220929e-03
## X75 -3.712577e-02 -6.998885e-02 3.024949e-03 0.009439396 -1.624669e-02
## X76 7.140428e-03 -1.619150e-01 -6.330162e-03 -0.114571261 -1.301062e-04
## X77 1.320728e-02 6.330162e-03 -3.396236e-03 -0.022337639 -4.381069e-02
## X78 -3.567666e-02 -1.145713e-01 2.233764e-02 -0.132461004 4.031195e-03
## X79 -4.900257e-02 1.301062e-04 -4.381069e-02 -0.004031195 2.605774e-02
## X80 1.573474e-02 -1.196106e-01 9.219608e-04 -0.119763143 -1.320891e-02
## X81 -2.414761e-02 -7.140428e-03 1.320728e-02 0.035676660 -4.900257e-02
## X82 3.960062e-02 -1.264481e-01 3.923961e-03 -0.169815898 1.262550e-02
## X83 -6.684925e-03 5.280954e-02 -4.231765e-02 -0.018843971 -3.699806e-02
## X84 -4.397121e-03 -1.462396e-01 2.225228e-02 -0.059489191 4.011664e-02
## 72 73 14 15 12
## X1 -0.075238471 -0.005700970 -2.894707e-02 -0.556233618 -0.114889135
## X2 -0.163573834 0.013339021 -2.349897e-01 -0.666693355 0.223019254
## X3 0.035546554 0.005191882 -4.824345e-02 0.072511889 -0.294747426
## X4 -0.186062588 -0.009404575 3.396932e-02 -0.198093699 0.169659826
## X5 -0.022064900 0.020751374 -9.554254e-04 0.453006531 0.601410132
## X6 -0.074420547 -0.087121725 2.313415e-01 -1.232974782 0.005783867
## X7 0.030388229 0.009709874 6.847823e-01 0.415479691 0.225363120
## X8 -0.140785695 -0.014448468 7.121873e-02 -0.538130316 -0.186265048
## X9 0.020666935 -0.007967861 1.116450e-01 0.359386220 1.496692991
## X10 -0.133208126 -0.058503937 -1.471459e-01 -1.924938353 0.634898319
## X11 0.036036192 -0.062447555 1.616640e+00 -0.041246046 1.464242405
## X12 -0.112129390 -0.009781179 6.557833e-01 -1.862286960 407.767961065
## X13 0.009781179 0.046389352 1.492683e+00 0.902180817 -0.154149115
## X14 -0.105495024 -0.052666953 4.077864e+02 -0.194166742 0.655783312
## X15 0.075134698 -0.034734454 -1.941667e-01 407.855424867 -1.862286960
## X16 -0.164845227 -0.035923141 7.318515e-01 1.545201627 -0.128709848
## X17 0.029704673 -0.032027392 -1.809768e+00 0.826112583 -1.964955980
## X18 -0.100972071 0.047463909 -1.381253e-01 1.627891758 0.168171956
## X19 0.009269588 -0.016841651 -1.913687e+00 -0.050266641 -0.457171094
## X20 -0.176783636 -0.012316477 6.096367e-02 0.168192438 0.279481284
## X21 0.043785952 0.030030326 -4.815829e-01 0.369641278 -1.101776015
## X22 -0.157743053 -0.103806142 2.515681e-01 0.906627733 0.089149119
## X23 0.054920567 0.011022662 -1.011129e+00 0.395253073 -0.255264381
## X24 -0.115113251 -0.015852926 2.092847e-01 0.041717824 -0.153081358
## X25 0.023915315 -0.008709985 -1.554204e-01 0.277691163 -0.361475754
## X26 -0.187340971 -0.030301007 -7.479840e-02 0.217371852 -0.054671909
## X27 0.052508540 -0.028959019 -4.010781e-01 -0.087679368 -0.450167876
## X28 -0.125727671 -0.023034825 -4.680129e-02 0.335826618 -0.147310294
## X29 0.010374500 0.039583542 -4.508065e-01 0.005863951 -0.449321503
## X30 -0.058687956 0.001265032 -1.968228e-01 0.340763740 -0.235851227
## X31 0.116244922 0.006022717 -3.741731e-01 -0.110678711 -0.157014507
## X32 -0.067936297 -0.074295399 -3.176686e-01 0.058119789 -0.135141155
## X33 0.109410802 0.080817259 -1.409293e-01 -0.101368613 -0.290800910
## X34 -0.132031080 -0.009581249 -1.539807e-01 0.254212127 -0.252233740
## X35 0.104121377 0.063624601 -2.939804e-01 -0.030157472 -0.254647407
## X36 -0.104712491 -0.042469622 -2.174439e-01 0.133879279 -0.114861530
## X37 0.062031981 -0.038972453 -2.926060e-01 -0.070723549 -0.174033235
## X38 -0.224453543 -0.092794419 -1.263266e-01 0.126097900 -0.196558948
## X39 0.220596069 -0.084224064 -1.822003e-01 0.020426743 -0.264165276
## X40 -0.142365626 -0.110083574 -1.821056e-01 0.203899082 -0.107890544
## X41 0.175711387 0.054506992 -1.657048e-01 -0.064291892 -0.222217956
## X42 -0.106037380 -0.151380572 -1.329502e-01 0.161352835 -0.106037380
## X43 0.113186251 0.011776342 -2.269806e-01 0.063922420 -0.113186251
## X44 -0.107890544 -0.166115527 -1.172453e-01 0.068382586 -0.142365626
## X45 0.222217956 0.038862765 -1.961842e-01 0.022984220 -0.175711387
## X46 -0.196558948 -0.105438568 -7.679934e-02 0.133116301 -0.224453543
## X47 0.264165276 -0.049838557 -1.526787e-01 -0.011059298 -0.220596069
## X48 -0.114861530 -0.134264994 -2.521666e-01 0.109425180 -0.104712491
## X49 0.174033235 0.008961706 -2.039653e-01 -0.056510963 -0.062031981
## X50 -0.252233740 -0.171837860 -1.462192e-01 0.034693100 -0.132031080
## X51 0.254647407 -0.035933750 -6.980850e-02 0.002534306 -0.104121377
## X52 -0.135141155 -0.257391585 -6.655858e-02 -0.001903665 -0.067936297
## X53 0.290800910 -0.048997025 -1.156063e-01 -0.001847903 -0.109410802
## X54 -0.235851227 -0.042034617 -7.621513e-02 0.098183263 -0.058687956
## X55 0.157014507 -0.183185988 -8.552294e-02 0.089096088 -0.116244922
## X56 -0.147310294 -0.265615302 -1.055236e-01 0.046386179 -0.125727671
## X57 0.449321503 -0.160191257 -6.859371e-02 0.052858357 -0.010374500
## X58 -0.054671909 -0.336465250 -9.627370e-02 0.012673468 -0.187340971
## X59 0.450167876 0.013734571 -2.073586e-02 0.010129568 -0.052508540
## X60 -0.153081358 -0.256974151 -1.925329e-01 0.065847561 -0.115113251
## X61 0.361475754 -0.009396415 -1.696199e-02 -0.023767137 -0.023915315
## X62 0.089149119 0.058126107 -1.653186e-01 0.004149383 -0.157743053
## X63 0.255264381 0.397826727 -3.561886e-02 0.041495363 -0.054920567
## X64 0.279481284 -0.815981054 -1.154254e-01 0.050996606 -0.176783636
## X65 1.101776015 0.367339918 -1.077301e-01 -0.031294985 -0.043785952
## X66 0.168171956 -0.192604271 -1.397856e-01 0.031160448 -0.100972071
## X67 0.457171094 0.262432993 -2.494198e-02 -0.006967738 -0.009269588
## X68 -0.128709848 -1.576622497 -9.428715e-02 -0.030331033 -0.164845227
## X69 1.964955980 -0.059682069 7.863288e-03 -0.023526576 -0.029704673
## X70 0.655783312 -1.492683116 -1.776957e-01 0.032813907 -0.105495024
## X71 1.862286960 0.902180817 -3.281391e-02 -0.019176938 -0.075134698
## X72 407.767961065 0.154149115 -1.054950e-01 0.075134698 -0.112129390
## X73 0.154149115 407.873860890 -5.266695e-02 -0.034734454 -0.009781179
## X74 0.634898319 1.890727672 -9.927894e-02 0.006671945 -0.133208126
## X75 -1.464242405 0.923065810 -1.289041e-02 0.033538899 -0.036036192
## X76 -0.186265048 2.044885486 -1.398931e-01 0.075636812 -0.140785695
## X77 -1.496692991 -0.002126868 -1.890332e-02 -0.055762630 -0.020666935
## X78 0.005783867 0.424412244 -1.777838e-01 0.033292439 -0.074420547
## X79 -0.225363120 0.424821082 -1.822964e-03 0.029030203 -0.030388229
## X80 0.169659826 1.316346937 -1.167382e-01 0.034312190 -0.186062588
## X81 -0.601410132 0.477161376 -8.319776e-02 0.052027521 0.022064900
## X82 0.223019254 0.491885700 -1.358572e-01 -0.002298968 -0.163573834
## X83 0.294747426 0.263956592 1.036136e-02 -0.029453974 -0.035546554
## X84 -0.114889135 0.437310122 -1.119930e-01 0.015703097 -0.075238471
## 13 10 11
## X1 -0.437310122 0.344343923 -1.393978e-01
## X2 -0.491885700 0.242171715 -1.268103e+00
## X3 0.263956592 0.649653582 4.046495e-01
## X4 -1.316346937 0.194833806 -7.182042e-01
## X5 0.477161376 -0.068428881 2.357711e-01
## X6 -0.424412244 -0.247946733 -2.128258e+00
## X7 0.424821082 1.413320836 5.955482e-02
## X8 -2.044885486 0.569463457 -1.777010e+00
## X9 -0.002126868 1.577960476 9.885007e-01
## X10 -1.890727672 407.728841887 -2.740962e-01
## X11 0.923065810 -0.274096248 4.079130e+02
## X12 -0.154149115 0.634898319 1.464242e+00
## X13 407.873860890 -1.890727672 9.230658e-01
## X14 1.492683116 -0.147145870 1.616640e+00
## X15 0.902180817 -1.924938353 -4.124605e-02
## X16 1.576622497 0.092103722 1.400858e-01
## X17 -0.059682069 -0.509689604 3.385012e-01
## X18 0.192604271 0.288896712 7.647118e-01
## X19 0.262432993 -1.153045276 3.579245e-01
## X20 0.815981054 0.196357404 -3.371427e-02
## X21 0.367339918 -0.230852548 2.906184e-01
## X22 -0.058126107 -0.125168203 1.663275e-01
## X23 0.397826727 -0.452122433 -3.730957e-02
## X24 0.256974151 -0.174807472 2.366213e-01
## X25 -0.009396415 -0.550011807 1.338701e-01
## X26 0.336465250 -0.225593247 3.052176e-01
## X27 0.013734571 -0.409719204 -8.190830e-02
## X28 0.265615302 -0.243721848 4.267325e-02
## X29 -0.160191257 -0.156375875 -1.753154e-01
## X30 0.042034617 -0.085628608 1.822431e-01
## X31 -0.183185988 -0.365949348 -9.850957e-02
## X32 0.257391585 -0.170416365 1.557527e-01
## X33 -0.048997025 -0.270732579 -1.177511e-01
## X34 0.171837860 -0.096021977 1.374445e-01
## X35 -0.035933750 -0.170853777 -9.877848e-03
## X36 0.134264994 -0.231348747 1.433971e-01
## X37 0.008961706 -0.226206696 -1.504876e-02
## X38 0.105438568 -0.096425507 1.742826e-01
## X39 -0.049838557 -0.214050862 2.739773e-02
## X40 0.166115527 -0.120490714 5.292006e-02
## X41 0.038862765 -0.211646766 2.622968e-02
## X42 0.151380572 -0.117305972 1.148463e-01
## X43 0.011776342 -0.170948695 2.944734e-02
## X44 0.110083574 -0.213245664 1.757924e-01
## X45 0.054506992 -0.137598084 -9.543194e-02
## X46 0.092794419 -0.170278781 1.943689e-02
## X47 -0.084224064 -0.085064708 2.659384e-02
## X48 0.042469622 -0.104317979 -7.049513e-03
## X49 -0.038972453 -0.120752138 3.591150e-02
## X50 0.009581249 -0.026429537 8.207192e-02
## X51 0.063624601 -0.101634280 3.931050e-02
## X52 0.074295399 -0.124160460 1.021988e-02
## X53 0.080817259 -0.104760009 7.149522e-02
## X54 -0.001265032 -0.117448843 -8.530388e-04
## X55 0.006022717 -0.034262364 3.130471e-02
## X56 0.023034825 -0.140505332 -1.735020e-02
## X57 0.039583542 -0.100159751 -7.579466e-02
## X58 0.030301007 -0.144567225 2.621428e-02
## X59 -0.028959019 -0.013553958 2.074399e-02
## X60 0.015852926 -0.152551172 6.825959e-02
## X61 -0.008709985 -0.090467121 5.830780e-03
## X62 0.103806142 -0.126578288 2.402002e-02
## X63 0.011022662 -0.032082409 -2.017502e-02
## X64 0.012316477 -0.143289719 5.345627e-03
## X65 0.030030326 0.043539948 2.547600e-02
## X66 -0.047463909 -0.201843290 1.707917e-02
## X67 -0.016841651 -0.048548644 4.970672e-03
## X68 0.035923141 -0.112179949 3.553408e-02
## X69 -0.032027392 -0.092267574 -2.804953e-02
## X70 0.052666953 -0.099278936 1.289041e-02
## X71 -0.034734454 -0.006671945 3.353890e-02
## X72 0.009781179 -0.133208126 3.603619e-02
## X73 0.046389352 -0.058503937 -6.244756e-02
## X74 0.058503937 -0.153636149 1.755770e-02
## X75 -0.062447555 -0.017557702 4.882593e-03
## X76 0.014448468 -0.067735622 6.998885e-02
## X77 -0.007967861 -0.047521105 3.024949e-03
## X78 0.087121725 -0.149064524 -9.439396e-03
## X79 0.009709874 0.003220929 -1.624669e-02
## X80 0.009404575 -0.121256186 3.947051e-02
## X81 0.020751374 0.017262982 -3.712577e-02
## X82 -0.013339021 -0.156608614 1.976593e-02
## X83 0.005191882 0.011703543 5.020535e-02
## X84 0.005700970 -0.119335382 -3.456732e-02
##
## $Tregresores
## C 1 2 3 4
## [1,] 0.1091089 44.22911 6.254940e+01 1.796827e-13 62.549404
## [2,] 0.1091089 44.14983 6.226269e+01 4.665944e+00 61.739909
## [3,] 0.1091089 44.05982 6.161404e+01 9.286822e+00 59.541731
## [4,] 0.1091089 44.10356 6.080806e+01 1.387904e+01 56.195093
## [5,] 0.1091089 44.19422 5.972336e+01 1.842222e+01 51.639977
## [6,] 0.1091089 44.12548 5.808918e+01 2.279832e+01 45.744532
## [7,] 0.1091089 43.95117 5.600093e+01 2.696863e+01 38.753845
## [8,] 0.1091089 43.85932 5.371648e+01 3.101322e+01 31.013225
## [9,] 0.1091089 43.87159 5.126299e+01 3.495051e+01 22.667142
## [10,] 0.1091089 43.78615 4.841332e+01 3.860834e+01 13.779156
## [11,] 0.1091089 43.82144 4.542934e+01 4.215226e+01 4.631239
## [12,] 0.1091089 43.81420 4.214529e+01 4.542182e+01 -4.630473
## [13,] 0.1091089 43.95080 3.875351e+01 4.859537e+01 -13.830969
## [14,] 0.1091089 43.98641 3.504199e+01 5.139716e+01 -22.726469
## [15,] 0.1091089 43.97948 3.109819e+01 5.386365e+01 -31.098191
## [16,] 0.1091089 44.04933 2.702886e+01 5.612600e+01 -38.840398
## [17,] 0.1091089 44.17459 2.282370e+01 5.815383e+01 -45.795439
## [18,] 0.1091089 44.25854 1.844903e+01 5.981028e+01 -51.715130
## [19,] 0.1091089 44.20399 1.391065e+01 6.094653e+01 -56.323065
## [20,] 0.1091089 44.30092 9.337641e+00 6.195120e+01 -59.867556
## [21,] 0.1091089 44.34321 4.686382e+00 6.253542e+01 -62.010347
## [22,] 0.1091089 44.33107 -3.496882e-13 6.269359e+01 -62.693595
## [23,] 0.1091089 44.33482 -4.685495e+00 6.252359e+01 -61.998611
## [24,] 0.1091089 44.38265 -9.354868e+00 6.206549e+01 -59.978002
## [25,] 0.1091089 44.20577 -1.391121e+01 6.094898e+01 -56.325326
## [26,] 0.1091089 44.18147 -1.841690e+01 5.970613e+01 -51.625075
## [27,] 0.1091089 44.14292 -2.280733e+01 5.811214e+01 -45.762608
## [28,] 0.1091089 44.10178 -2.706104e+01 5.619283e+01 -38.886645
## [29,] 0.1091089 44.29276 -3.131971e+01 5.424733e+01 -31.319711
## [30,] 0.1091089 44.27882 -3.527494e+01 5.173883e+01 -22.877548
## [31,] 0.1091089 44.34693 -3.910280e+01 4.903336e+01 -13.955628
## [32,] 0.1091089 44.41293 -4.272122e+01 4.604253e+01 -4.693750
## [33,] 0.1091089 44.51056 -4.614374e+01 4.281513e+01 4.704068
## [34,] 0.1091089 44.53604 -4.924247e+01 3.926956e+01 14.015142
## [35,] 0.1091089 44.58189 -5.209297e+01 3.551638e+01 23.034138
## [36,] 0.1091089 44.63544 -5.466703e+01 3.156202e+01 31.562022
## [37,] 0.1091089 44.59329 -5.681909e+01 2.736263e+01 39.320029
## [38,] 0.1091089 44.62341 -5.874467e+01 2.305559e+01 46.260724
## [39,] 0.1091089 44.55231 -6.020728e+01 1.857149e+01 52.058401
## [40,] 0.1091089 44.50398 -6.136014e+01 1.400505e+01 56.705295
## [41,] 0.1091089 44.43577 -6.213977e+01 9.366064e+00 60.049786
## [42,] 0.1091089 44.38281 -6.259126e+01 4.690566e+00 62.065716
## [43,] 0.1091089 44.31781 -6.267484e+01 4.473936e-13 62.674844
## [44,] 0.1091089 44.23711 -6.238579e+01 -4.675168e+00 61.861967
## [45,] 0.1091089 44.00145 -6.153242e+01 -9.274520e+00 59.462859
## [46,] 0.1091089 44.01071 -6.068005e+01 -1.384982e+01 56.076797
## [47,] 0.1091089 44.01244 -5.947771e+01 -1.834644e+01 51.427572
## [48,] 0.1091089 44.08036 -5.802978e+01 -2.277501e+01 45.697753
## [49,] 0.1091089 44.13210 -5.623146e+01 -2.707965e+01 38.913381
## [50,] 0.1091089 44.16586 -5.409191e+01 -3.122998e+01 31.229978
## [51,] 0.1091089 44.21869 -5.166857e+01 -3.522704e+01 22.846482
## [52,] 0.1091089 44.23091 -4.890509e+01 -3.900051e+01 13.919119
## [53,] 0.1091089 44.20619 -4.582820e+01 -4.252235e+01 4.671901
## [54,] 0.1091089 44.34952 -4.266022e+01 -4.597679e+01 -4.687048
## [55,] 0.1091089 44.32325 -3.908193e+01 -4.900719e+01 -13.948178
## [56,] 0.1091089 44.37750 -3.535355e+01 -5.185413e+01 -22.928532
## [57,] 0.1091089 44.57957 -3.152252e+01 -5.459860e+01 -31.522516
## [58,] 0.1091089 44.69939 -2.742774e+01 -5.695429e+01 -39.413589
## [59,] 0.1091089 44.77676 -2.313482e+01 -5.894655e+01 -46.419702
## [60,] 0.1091089 44.79353 -1.867204e+01 -6.053327e+01 -52.340263
## [61,] 0.1091089 44.78325 -1.409294e+01 -6.174519e+01 -57.061132
## [62,] 0.1091089 44.80293 -9.443453e+00 -6.265322e+01 -60.545959
## [63,] 0.1091089 44.82929 -4.737753e+00 -6.322092e+01 -62.690086
## [64,] 0.1091089 44.86952 1.530234e-13 -6.345508e+01 -63.455080
## [65,] 0.1091089 44.80708 4.735405e+00 -6.318959e+01 -62.659023
## [66,] 0.1091089 44.82288 9.447659e+00 -6.268112e+01 -60.572928
## [67,] 0.1091089 44.96554 1.415030e+01 -6.199652e+01 -57.293397
## [68,] 0.1091089 45.06193 1.878392e+01 -6.089597e+01 -52.653877
## [69,] 0.1091089 45.05709 2.327966e+01 -5.931560e+01 -46.710321
## [70,] 0.1091089 45.04281 2.763846e+01 -5.739185e+01 -39.716392
## [71,] 0.1091089 45.06733 3.186741e+01 -5.519598e+01 -31.867413
## [72,] 0.1091089 45.12866 3.595197e+01 -5.273185e+01 -23.316636
## [73,] 0.1091089 45.12798 3.979150e+01 -4.989696e+01 -14.201420
## [74,] 0.1091089 45.05855 4.334224e+01 -4.671183e+01 -4.761981
## [75,] 0.1091089 44.98190 4.663237e+01 -4.326851e+01 4.753881
## [76,] 0.1091089 45.05224 4.981321e+01 -3.972471e+01 14.177585
## [77,] 0.1091089 45.15249 5.275970e+01 -3.597095e+01 23.328948
## [78,] 0.1091089 45.19556 5.535303e+01 -3.195809e+01 31.958087
## [79,] 0.1091089 45.29852 5.771767e+01 -2.779536e+01 39.941865
## [80,] 0.1091089 45.35676 5.971010e+01 -2.343449e+01 47.020987
## [81,] 0.1091089 45.48396 6.146630e+01 -1.895984e+01 53.147014
## [82,] 0.1091089 45.56503 6.282307e+01 -1.433896e+01 58.057248
## [83,] 0.1091089 45.60786 6.377886e+01 -9.613116e+00 61.633739
## [84,] 0.1091089 45.52753 6.420561e+01 -4.811545e+00 63.666512
## 5 8 9 6 7
## [1,] -3.312882e-13 62.549404 5.365170e-13 6.254940e+01 2.005123e-13
## [2,] 9.305794e+00 59.663369 1.840371e+01 6.087185e+01 1.389360e+01
## [3,] 1.836619e+01 51.482929 3.510047e+01 5.613936e+01 2.703529e+01
## [4,] 2.706213e+01 38.888211 4.876427e+01 4.876427e+01 3.888821e+01
## [5,] 3.520754e+01 22.833838 5.817967e+01 3.896815e+01 4.886452e+01
## [6,] 4.244472e+01 4.663371 6.222837e+01 2.707559e+01 5.622303e+01
## [7,] 4.859578e+01 -13.831087 6.059795e+01 1.383109e+01 6.059795e+01
## [8,] 5.371648e+01 -31.013225 5.371648e+01 -1.635705e-13 6.202645e+01
## [9,] 5.775494e+01 -45.481318 4.220050e+01 -1.380604e+01 6.048822e+01
## [10,] 6.037043e+01 -55.790665 2.686737e+01 -2.686737e+01 5.579066e+01
## [11,] 6.179959e+01 -61.280695 9.236579e+00 -3.863946e+01 4.845235e+01
## [12,] 6.178937e+01 -61.270561 -9.235051e+00 -4.844434e+01 3.863307e+01
## [13,] 6.059744e+01 -56.000451 -2.696840e+01 -5.600045e+01 2.696840e+01
## [14,] 5.790610e+01 -45.600355 -4.231095e+01 -6.064654e+01 1.384218e+01
## [15,] 5.386365e+01 -31.098191 -5.386365e+01 -6.219638e+01 -1.130693e-14
## [16,] 4.870432e+01 -13.861977 -6.073329e+01 -6.073329e+01 -1.386198e+01
## [17,] 4.249196e+01 4.668561 -6.229762e+01 -5.628560e+01 -2.710572e+01
## [18,] 3.525878e+01 22.867068 -5.826434e+01 -4.893563e+01 -3.902486e+01
## [19,] 2.712376e+01 38.976770 -4.887532e+01 -3.897677e+01 -4.887532e+01
## [20,] 1.846670e+01 51.764654 -3.529254e+01 -2.718323e+01 -5.644657e+01
## [21,] 9.346556e+00 59.924711 -1.848433e+01 -1.395446e+01 -6.113848e+01
## [22,] 5.372072e-13 62.693595 2.192474e-14 3.022834e-13 -6.269359e+01
## [23,] -9.344787e+00 59.913370 1.848083e+01 1.395182e+01 -6.112691e+01
## [24,] -1.850076e+01 51.860152 3.535765e+01 2.723338e+01 -5.655070e+01
## [25,] -2.712485e+01 38.978335 4.887729e+01 3.897833e+01 -4.887729e+01
## [26,] -3.519738e+01 22.827249 5.816288e+01 4.885042e+01 -3.895691e+01
## [27,] -4.246149e+01 4.665214 6.225296e+01 5.624525e+01 -2.708628e+01
## [28,] -4.876231e+01 -13.878483 6.080561e+01 6.080561e+01 -1.387848e+01
## [29,] -5.424733e+01 -31.319711 5.424733e+01 6.263942e+01 4.116698e-13
## [30,] -5.829104e+01 -45.903494 4.259222e+01 6.104970e+01 1.393420e+01
## [31,] -6.114360e+01 -56.505184 2.721146e+01 5.650518e+01 2.721146e+01
## [32,] -6.263374e+01 -62.107842 9.361251e+00 4.910634e+01 3.916100e+01
## [33,] -6.277142e+01 -62.244367 -9.381829e+00 3.924709e+01 4.921429e+01
## [34,] -6.140435e+01 -56.746152 -2.732751e+01 2.732751e+01 5.674615e+01
## [35,] -5.869003e+01 -46.217689 -4.288375e+01 1.402957e+01 6.146757e+01
## [36,] -5.466703e+01 -31.562022 -5.466703e+01 -4.411116e-13 6.312404e+01
## [37,] -4.930576e+01 -14.033156 -6.148327e+01 -1.403316e+01 6.148327e+01
## [38,] -4.292368e+01 4.715994 -6.293057e+01 -2.738111e+01 5.685747e+01
## [39,] -3.549282e+01 23.018854 -5.865108e+01 -3.928390e+01 4.926045e+01
## [40,] -2.730783e+01 39.241282 -4.920701e+01 -4.920701e+01 3.924128e+01
## [41,] -1.852291e+01 51.922220 -3.539997e+01 -5.661838e+01 2.726598e+01
## [42,] -9.354901e+00 59.978217 -1.850083e+01 -6.119307e+01 1.396692e+01
## [43,] -6.299540e-15 62.674844 2.136403e-13 -6.267484e+01 1.574357e-13
## [44,] 9.324191e+00 59.781321 1.844010e+01 -6.099219e+01 -1.392107e+01
## [45,] 1.834186e+01 51.414732 3.505397e+01 -5.606500e+01 -2.699948e+01
## [46,] 2.700516e+01 38.806347 4.866162e+01 -4.866162e+01 -3.880635e+01
## [47,] 3.506272e+01 22.739918 5.794037e+01 -3.880787e+01 -4.866353e+01
## [48,] 4.240132e+01 4.658603 6.216473e+01 -2.704790e+01 -5.616554e+01
## [49,] 4.879584e+01 -13.888025 6.084741e+01 -1.388802e+01 -6.084741e+01
## [50,] 5.409191e+01 -31.229978 5.409191e+01 -4.767029e-13 -6.245996e+01
## [51,] 5.821189e+01 -45.841159 4.253438e+01 1.391527e+01 -6.096680e+01
## [52,] 6.098365e+01 -56.357363 2.714028e+01 2.714028e+01 -5.635736e+01
## [53,] 6.234218e+01 -61.818729 9.317674e+00 3.897871e+01 -4.887775e+01
## [54,] 6.254431e+01 -62.019162 -9.347885e+00 4.903623e+01 -3.910509e+01
## [55,] 6.111096e+01 -56.475020 -2.719694e+01 5.647502e+01 -2.719694e+01
## [56,] 5.842095e+01 -46.005792 -4.268714e+01 6.118575e+01 -1.396525e+01
## [57,] 5.459860e+01 -31.522516 -5.459860e+01 6.304503e+01 -3.738797e-13
## [58,] 4.942308e+01 -14.066547 -6.162957e+01 6.162957e+01 1.406655e+01
## [59,] 4.307119e+01 4.732201 -6.314683e+01 5.705286e+01 2.747521e+01
## [60,] 3.568499e+01 23.143486 -5.896864e+01 4.952717e+01 3.949660e+01
## [61,] 2.747919e+01 39.487528 -4.951579e+01 3.948753e+01 4.951579e+01
## [62,] 1.867596e+01 52.351238 -3.569247e+01 2.749127e+01 5.708620e+01
## [63,] 9.449010e+00 60.581588 -1.868694e+01 1.410742e+01 6.180867e+01
## [64,] -7.960212e-15 63.455080 -4.957276e-14 -7.378126e-14 6.345508e+01
## [65,] -9.444328e+00 60.551570 1.867769e+01 -1.410043e+01 6.177804e+01
## [66,] -1.868427e+01 52.374556 3.570837e+01 -2.750351e+01 5.711163e+01
## [67,] -2.759105e+01 39.648261 4.971735e+01 -3.964826e+01 4.971735e+01
## [68,] -3.589881e+01 23.282158 5.932197e+01 -4.982393e+01 3.973325e+01
## [69,] -4.334084e+01 4.761828 6.354217e+01 -5.741005e+01 2.764722e+01
## [70,] -4.980278e+01 -14.174616 6.210305e+01 -6.210305e+01 1.417462e+01
## [71,] -5.519598e+01 -31.867413 5.519598e+01 -6.373483e+01 -2.123119e-13
## [72,] -5.940982e+01 -46.784518 4.340969e+01 -6.222143e+01 -1.420163e+01
## [73,] -6.222049e+01 -57.500377 2.769072e+01 -5.750038e+01 -2.769072e+01
## [74,] -6.354422e+01 -63.010679 9.497332e+00 -4.982018e+01 -3.973027e+01
## [75,] -6.343613e+01 -62.903491 -9.481176e+00 -3.966268e+01 -4.973543e+01
## [76,] -6.211606e+01 -57.403869 -2.764425e+01 -2.764425e+01 -5.740387e+01
## [77,] -5.944119e+01 -46.809222 -4.343261e+01 -1.420913e+01 -6.225428e+01
## [78,] -5.535303e+01 -31.958087 -5.535303e+01 1.558805e-13 -6.391617e+01
## [79,] -5.008551e+01 -14.255087 -6.245562e+01 1.425509e+01 -6.245562e+01
## [80,] -4.362910e+01 4.793498 -6.396479e+01 2.783110e+01 -5.779188e+01
## [81,] -3.623502e+01 23.500210 -5.987756e+01 4.010538e+01 -5.029056e+01
## [82,] -2.795890e+01 40.176862 -5.038019e+01 5.038019e+01 -4.017686e+01
## [83,] -1.901149e+01 53.291790 -3.633373e+01 5.811183e+01 -2.798518e+01
## [84,] -9.596183e+00 61.525173 -1.897800e+01 6.277136e+01 -1.432715e+01
## 72 73 14 15 12
## [1,] 62.54940 2.673348e-13 6.254940e+01 -2.499743e-13 62.54940
## [2,] -56.25405 2.709052e+01 5.407227e+01 3.121864e+01 56.25405
## [3,] 38.84964 -4.871591e+01 3.115499e+01 5.396203e+01 38.84964
## [4,] -13.87904 6.080806e+01 1.950592e-13 6.237185e+01 13.87904
## [5,] -13.90757 -6.093306e+01 -3.125003e+01 5.412664e+01 -13.90757
## [6,] 38.90755 4.878852e+01 -5.404246e+01 3.120143e+01 -38.90755
## [7,] -56.00093 -2.696863e+01 -6.215634e+01 -2.420395e-14 -56.00093
## [8,] 62.02645 2.792558e-13 -5.371648e+01 -3.101322e+01 -62.02645
## [9,] -55.89952 2.691979e+01 -3.102190e+01 -5.373150e+01 -55.89952
## [10,] 38.60834 -4.841332e+01 -5.641113e-13 -6.192297e+01 -38.60834
## [11,] -13.79026 6.041909e+01 3.098644e+01 -5.367009e+01 -13.79026
## [12,] -13.78798 -6.040910e+01 5.366121e+01 -3.098132e+01 13.78798
## [13,] 38.75351 4.859537e+01 6.215581e+01 2.941933e-13 38.75351
## [14,] -56.04583 -2.699025e+01 5.387213e+01 3.110309e+01 56.04583
## [15,] 62.19638 1.182665e-13 3.109819e+01 5.386365e+01 62.19638
## [16,] -56.12600 2.702886e+01 -1.670886e-14 6.229516e+01 56.12600
## [17,] 38.95084 -4.884281e+01 -3.123615e+01 5.410260e+01 38.95084
## [18,] -13.92781 6.102173e+01 -5.420542e+01 3.129551e+01 13.92781
## [19,] -13.91065 -6.094653e+01 -6.251389e+01 1.141832e-13 -13.91065
## [20,] 39.06224 4.898249e+01 -5.425732e+01 -3.132548e+01 -39.06224
## [21,] -56.50046 -2.720919e+01 -3.135539e+01 -5.430912e+01 -56.50046
## [22,] 62.69359 -9.033052e-14 -3.916624e-13 -6.269359e+01 -62.69359
## [23,] -56.48976 2.720404e+01 3.134945e+01 -5.429885e+01 -56.48976
## [24,] 39.13430 -4.907286e+01 5.435742e+01 -3.138327e+01 -39.13430
## [25,] -13.91121 6.094898e+01 6.251640e+01 -2.027699e-13 -13.91121
## [26,] -13.90356 -6.091547e+01 5.411102e+01 3.124101e+01 13.90356
## [27,] 38.92292 4.880780e+01 3.121376e+01 5.406382e+01 38.92292
## [28,] -56.19283 -2.706104e+01 3.890985e-14 6.236934e+01 56.19283
## [29,] 62.63942 3.102865e-13 -3.131971e+01 5.424733e+01 62.63942
## [30,] -56.41841 2.716967e+01 -5.423026e+01 3.130985e+01 56.41841
## [31,] 39.10280 -4.903336e+01 -6.271602e+01 4.998868e-13 39.10280
## [32,] -13.97640 6.123461e+01 -5.439451e+01 -3.140469e+01 13.97640
## [33,] -14.00712 -6.136921e+01 -3.147372e+01 -5.451408e+01 -14.00712
## [34,] 39.26956 4.924247e+01 2.449221e-13 -6.298348e+01 -39.26956
## [35,] -56.80457 -2.735564e+01 3.152416e+01 -5.460145e+01 -56.80457
## [36,] 63.12404 9.839143e-13 5.466703e+01 -3.156202e+01 -63.12404
## [37,] -56.81909 2.736263e+01 6.306443e+01 -1.973502e-13 -56.81909
## [38,] 39.34659 -4.933906e+01 5.465229e+01 3.155351e+01 -39.34659
## [39,] -14.02026 6.142678e+01 3.150324e+01 5.456522e+01 -14.02026
## [40,] -14.00505 -6.136014e+01 -4.085031e-13 6.293813e+01 14.00505
## [41,] 39.18114 4.913159e+01 -3.142083e+01 5.442248e+01 39.18114
## [42,] -56.55090 -2.723348e+01 -5.435762e+01 3.138338e+01 56.55090
## [43,] 62.67484 -1.137770e-13 -6.267484e+01 -1.962237e-13 62.67484
## [44,] -56.36526 2.714408e+01 -5.417917e+01 -3.128036e+01 56.36526
## [45,] 38.79818 -4.865138e+01 -3.111373e+01 -5.389055e+01 38.79818
## [46,] -13.84982 6.068005e+01 7.243806e-13 -6.224055e+01 13.84982
## [47,] -13.85037 -6.068243e+01 3.112150e+01 -5.390401e+01 -13.85037
## [48,] 38.86776 4.873863e+01 5.398720e+01 -3.116952e+01 -38.86776
## [49,] -56.23146 -2.707965e+01 6.241222e+01 -8.350460e-14 -56.23146
## [50,] 62.45996 6.984066e-13 5.409191e+01 3.122998e+01 -62.45996
## [51,] -56.34179 2.713278e+01 3.126734e+01 5.415662e+01 -56.34179
## [52,] 39.00051 -4.890509e+01 7.022265e-13 6.255195e+01 -39.00051
## [53,] -13.91134 6.094956e+01 -3.125850e+01 5.414130e+01 -13.91134
## [54,] -13.95644 -6.114718e+01 -5.431684e+01 3.135984e+01 13.95644
## [55,] 39.08193 4.900719e+01 -6.268254e+01 -5.574317e-15 39.08193
## [56,] -56.54414 -2.723022e+01 -5.435111e+01 -3.137963e+01 56.54414
## [57,] 63.04503 -5.396725e-13 -3.152252e+01 -5.459860e+01 63.04503
## [58,] -56.95429 2.742774e+01 -3.526346e-14 -6.321449e+01 56.95429
## [59,] 39.48180 -4.950862e+01 3.166195e+01 -5.484010e+01 39.48180
## [60,] -14.09617 6.175937e+01 5.486065e+01 -3.167381e+01 14.09617
## [61,] -14.09294 -6.174519e+01 6.333308e+01 2.515689e-13 -14.09294
## [62,] 39.50488 4.953755e+01 5.487215e+01 3.168045e+01 -39.50488
## [63,] -57.11980 -2.750744e+01 3.169910e+01 5.490444e+01 -57.11980
## [64,] 63.45508 -1.080122e-12 -3.754635e-14 6.345508e+01 -63.45508
## [65,] -57.09149 2.749381e+01 -3.168339e+01 5.487724e+01 -57.09149
## [66,] 39.52248 -4.955962e+01 -5.489660e+01 3.169456e+01 -39.52248
## [67,] -14.15030 6.199652e+01 -6.359087e+01 -4.325891e-13 -14.15030
## [68,] -14.18063 -6.212942e+01 -5.518937e+01 -3.186360e+01 14.18063
## [69,] 39.72899 4.981857e+01 -3.186017e+01 -5.518344e+01 39.72899
## [70,] -57.39185 -2.763846e+01 -3.917144e-13 -6.370015e+01 57.39185
## [71,] 63.73483 1.123095e-12 3.186741e+01 -5.519598e+01 63.73483
## [72,] -57.50124 2.769114e+01 5.527110e+01 -3.191078e+01 57.50124
## [73,] 39.79150 -4.989696e+01 6.382060e+01 -3.615563e-13 39.79150
## [74,] -14.17957 6.212475e+01 5.518522e+01 3.186120e+01 14.17957
## [75,] -14.15545 -6.201907e+01 3.180700e+01 5.509135e+01 -14.15545
## [76,] 39.72471 4.981321e+01 4.530057e-14 6.371349e+01 -39.72471
## [77,] -57.53161 -2.770576e+01 -3.192763e+01 5.530028e+01 -57.53161
## [78,] 63.91617 -3.243843e-12 -5.535303e+01 3.195809e+01 -63.91617
## [79,] -57.71767 2.779536e+01 -6.406178e+01 -1.351850e-13 -57.71767
## [80,] 39.99322 -5.014991e+01 -5.555046e+01 -3.207207e+01 -39.99322
## [81,] -14.31344 6.271130e+01 -3.216202e+01 -5.570625e+01 -14.31344
## [82,] -14.33896 -6.282307e+01 -5.224484e-13 -6.443868e+01 14.33896
## [83,] 40.21463 5.042755e+01 3.224963e+01 -5.585800e+01 40.21463
## [84,] -58.00946 -2.793589e+01 5.575961e+01 -3.219282e+01 58.00946
## 13 10 11
## [1,] -1.215226e-13 6.254940e+01 3.050737e-13
## [2,] 2.709052e+01 5.812123e+01 2.281090e+01
## [3,] 4.871591e+01 4.567645e+01 4.238156e+01
## [4,] 6.080806e+01 2.706213e+01 5.619509e+01
## [5,] 6.093306e+01 4.670636e+00 6.232530e+01
## [6,] 4.878852e+01 -1.839357e+01 5.963047e+01
## [7,] 2.696863e+01 -3.875384e+01 4.859578e+01
## [8,] 2.970332e-13 -5.371648e+01 3.101322e+01
## [9,] -2.691979e+01 -6.135081e+01 9.247147e+00
## [10,] -4.841332e+01 -6.037043e+01 -1.377916e+01
## [11,] -6.041909e+01 -5.120440e+01 -3.491057e+01
## [12,] -6.040910e+01 -3.490479e+01 -5.119593e+01
## [13,] -4.859537e+01 -1.383097e+01 -6.059744e+01
## [14,] -2.699025e+01 9.271350e+00 -6.151139e+01
## [15,] 1.885669e-13 3.109819e+01 -5.386365e+01
## [16,] 2.702886e+01 4.870432e+01 -3.884040e+01
## [17,] 4.884281e+01 5.969683e+01 -1.841403e+01
## [18,] 6.102173e+01 6.241601e+01 4.677433e+00
## [19,] 6.094653e+01 5.632307e+01 2.712376e+01
## [20,] 4.898249e+01 4.261348e+01 4.592641e+01
## [21,] 2.720919e+01 2.291082e+01 5.837581e+01
## [22,] -6.232157e-14 -2.048188e-14 6.269359e+01
## [23,] -2.720404e+01 -2.290648e+01 5.836477e+01
## [24,] -4.907286e+01 -4.269209e+01 4.601113e+01
## [25,] -6.094898e+01 -5.632533e+01 2.712485e+01
## [26,] -6.091547e+01 -6.230732e+01 4.669288e+00
## [27,] -4.880780e+01 -5.965404e+01 -1.840083e+01
## [28,] -2.706104e+01 -4.876231e+01 -3.888664e+01
## [29,] 1.929890e-13 -3.131971e+01 -5.424733e+01
## [30,] 2.716967e+01 -9.332983e+00 -6.192030e+01
## [31,] 4.903336e+01 1.395563e+01 -6.114360e+01
## [32,] 6.123461e+01 3.538178e+01 -5.189554e+01
## [33,] 6.136921e+01 5.200961e+01 -3.545955e+01
## [34,] 4.924247e+01 6.140435e+01 -1.401514e+01
## [35,] 2.735564e+01 6.234412e+01 9.396864e+00
## [36,] 5.951783e-13 5.466703e+01 3.156202e+01
## [37,] -2.736263e+01 3.932003e+01 4.930576e+01
## [38,] -4.933906e+01 1.860112e+01 6.030336e+01
## [39,] -6.142678e+01 -4.708481e+00 6.283031e+01
## [40,] -6.136014e+01 -2.730783e+01 5.670530e+01
## [41,] -4.913159e+01 -4.606620e+01 4.274319e+01
## [42,] -2.723348e+01 -5.842794e+01 2.293128e+01
## [43,] 9.656794e-14 -6.267484e+01 1.727923e-13
## [44,] 2.714408e+01 -5.823613e+01 -2.285600e+01
## [45,] 4.865138e+01 -4.561595e+01 -4.232542e+01
## [46,] 6.068005e+01 -2.700516e+01 -5.607680e+01
## [47,] 6.068243e+01 -4.651424e+00 -6.206895e+01
## [48,] 4.873863e+01 1.837476e+01 -5.956950e+01
## [49,] 2.707965e+01 3.891338e+01 -4.879584e+01
## [50,] -1.278127e-13 5.409191e+01 -3.122998e+01
## [51,] -2.713278e+01 6.183621e+01 -9.320309e+00
## [52,] -4.890509e+01 6.098365e+01 1.391912e+01
## [53,] -6.094956e+01 5.165396e+01 3.521708e+01
## [54,] -6.114718e+01 3.533126e+01 5.182144e+01
## [55,] -4.900719e+01 1.394818e+01 6.111096e+01
## [56,] -2.723022e+01 -9.353782e+00 6.205829e+01
## [57,] -4.973634e-13 -3.152252e+01 5.459860e+01
## [58,] 2.742774e+01 -4.942308e+01 3.941359e+01
## [59,] 4.950862e+01 -6.051059e+01 1.866505e+01
## [60,] 6.175937e+01 -6.317049e+01 -4.733974e+00
## [61,] 6.174519e+01 -5.706113e+01 -2.747919e+01
## [62,] 4.953755e+01 -4.309636e+01 -4.644683e+01
## [63,] 2.750744e+01 -2.316196e+01 -5.901571e+01
## [64,] 2.253224e-13 8.291978e-15 -6.345508e+01
## [65,] -2.749381e+01 2.315048e+01 -5.898647e+01
## [66,] -4.955962e+01 4.311556e+01 -4.646752e+01
## [67,] -6.199652e+01 5.729340e+01 -2.759105e+01
## [68,] -6.212942e+01 6.354900e+01 -4.762339e+00
## [69,] -4.981857e+01 6.088943e+01 1.878190e+01
## [70,] -2.763846e+01 4.980278e+01 3.971639e+01
## [71,] 3.418609e-14 3.186741e+01 5.519598e+01
## [72,] 2.769114e+01 9.512111e+00 6.310873e+01
## [73,] 4.989696e+01 -1.420142e+01 6.222049e+01
## [74,] 6.212475e+01 -3.589611e+01 5.264992e+01
## [75,] 6.201907e+01 -5.256036e+01 3.583505e+01
## [76,] 4.981321e+01 -6.211606e+01 1.417758e+01
## [77,] 2.770576e+01 -6.314205e+01 -9.517133e+00
## [78,] -1.563680e-13 -5.535303e+01 -3.195809e+01
## [79,] -2.779536e+01 -3.994187e+01 -5.008551e+01
## [80,] -5.014991e+01 -1.890682e+01 -6.129440e+01
## [81,] -6.271130e+01 4.806941e+00 -6.414417e+01
## [82,] -6.282307e+01 2.795890e+01 -5.805725e+01
## [83,] -5.042755e+01 4.728130e+01 -4.387064e+01
## [84,] -2.793589e+01 5.993491e+01 -2.352272e+01
##
## $Nregresores
## [1] 18plot(reg2$datos$X,reg2$datos$Y,pch=19,col="blue")
lines(reg2$datos$X,reg2$datos$F,col="red")
gtd (reg2$datos$res)
plot(ts(E, frequency=4))
lines(ts(reg2$datos$F,frequency=4),col="red")
reg3 <- lm(E~PIBC)
reg3##
## Call:
## lm(formula = E ~ PIBC)
##
## Coefficients:
## (Intercept) PIBC
## 171.507 1.895plot(PIBC,E,pch=19,col="blue")
lines(PIBC,reg3$fitted,col="red")
gtd (reg3$resid)
plot(ts(E, frequency=4))
lines(ts(reg3$fitted,frequency=4),col="red")
Se comprueba que el resultado es el mismo realizando la estimación MCO con los regresores seleccionados de la matriz auxiliar \(WX_tIW^T\), una vez convertidos estos en series de tiempo:
regresores1 <- data.frame(reg2$Tregresores)
reg4 <- lm(E~.,data=regresores1)
plot(PIBC,E,pch=19,col="blue")
lines(PIBC,reg4$fitted,col="red")
lines(PIBC,reg2$datos$F,col="green")
summary(reg4)##
## Call:
## lm(formula = E ~ ., data = regresores1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1531 -0.7737 -0.1848 0.8961 3.6501
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 55.9043149 49.2067418 1.136 0.260019
## C NA NA NA NA
## X1 19.9646449 1.1059485 18.052 < 2e-16 ***
## X2 -0.0725800 0.0043367 -16.736 < 2e-16 ***
## X3 -0.0265453 0.0071701 -3.702 0.000438 ***
## X4 0.0336288 0.0032263 10.423 1.39e-15 ***
## X5 0.0166400 0.0074983 2.219 0.029919 *
## X8 -0.0190319 0.0034477 -5.520 6.13e-07 ***
## X9 0.0250884 0.0048733 5.148 2.57e-06 ***
## X6 -0.0127827 0.0033163 -3.855 0.000265 ***
## X7 -0.0186298 0.0034312 -5.430 8.72e-07 ***
## X72 -0.0026821 0.0032123 -0.835 0.406767
## X73 -0.0006815 0.0032051 -0.213 0.832269
## X14 -0.0085513 0.0032064 -2.667 0.009618 **
## X15 0.0058523 0.0035362 1.655 0.102678
## X12 -0.0044438 0.0032188 -1.381 0.172069
## X13 -0.0003543 0.0034091 -0.104 0.917552
## X10 -0.0280127 0.0033461 -8.372 5.73e-12 ***
## X11 -0.0179119 0.0032234 -5.557 5.32e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.307 on 66 degrees of freedom
## Multiple R-squared: 0.9838, Adjusted R-squared: 0.9796
## F-statistic: 235.7 on 17 and 66 DF, p-value: < 2.2e-16gtd (reg4$resid)
plot(ts(E, frequency=4))
lines(ts(reg4$fitted,frequency=4),col="red")
Bibliografia
DURBIN, J., “Tests for Serial Correlation in Regression Analysis based on the Periodogram ofLeast-Squares Residuals,” Biometrika, 56, (No. 1, 1969), 1-15.
Engle, Robert F. (1974), Band Spectrum Regression,International Economic Review 15,1-11.
Harvey, A.C. (1978), Linear Regression in the Frequency Domain, International Economic Review, 19, 507-512.
Parra F (2014): Seasonal Adjustment by Frequency Analysis. Package R Version 1.1. URL:http://cran.r-project.org/web/packages/descomponer/index.html
Venables and Ripley, “Modern Applied Statistics with S” (4th edition, 2002).