Confidence intervals estimation in linear regression models

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Description

This routine obtains a confidence interval for the value a^T * β, by asymptotic distribution and bootstrap, from a sample (Y_i, X_{i1},...,X_{ip}): i=1,...,n, where:

a = (a_1,...,a_p)^T

is an unknown vector,

β = (β_1,...,β_p)^T

is an unknown vector parameter and

Y_i = X_{i1}*β_1+ ... + X_{ip}*β_p + ε_i.

The random errors, ε_i, are allowed to be time series.

Usage

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par.ci(data=data, seed=123, CI="AD", B=1000, N=50, a=NULL, 
p.arima=NULL, q.arima=NULL, p.max=3, q.max=3, alpha=0.05, 
alpha2=0.05, num.lb=10, ic="BIC", Var.Cov.eps=NULL)

Arguments

data

data[, 1] contains the values of the response variable, Y;

data[, 2:(p+1)] contains the values of the explanatory variables, X_1, ..., X_p.

seed

the considered seed.

CI

method to obtain the confidence interval. It allows us to choose between: “AD” (asymptotic distribution), “B” (bootstrap) or “all” (both). The default is “AD”.

B

number of bootstrap replications. The default is 1000.

N

Truncation parameter used in the finite approximation of the MA(infinite) expression of ε.

a

Vector which, multiplied by beta, is used for obtaining the confidence interval of this result.

p.arima

the considered p to fit the model ARMA(p,q).

q.arima

the considered q to fit the model ARMA(p,q).

p.max

if Var.Cov.eps=NULL, the ARMA models are selected between the models ARMA(p,q) with 0<=p<=p.max and 0<=q<=q.max. The default is 3.

q.max

if Var.Cov.eps=NULL, the ARMA models are selected between the models ARMA(p,q) with 0<=p<=p.max and 0<=q<=q.max. The default is 3.

alpha

1 - alpha is the confidence level of the confidence interval. The default is 0.05.

alpha2

significance level used to check (if needed) the ARMA model fitted to the residuals. The default is 0.05.

num.lb

if Var.Cov.eps=NULL, it checks the suitability of the selected ARMA model according to the Ljung-Box test and the t-test. It uses up to num.lb delays in the Ljung-Box test. The default is 10.

ic

if Var.Cov.eps=NULL, ic contains the information criterion used to suggest the ARMA model. It allows us to choose between: "AIC", "AICC" or "BIC" (the default).

Var.Cov.eps

n x n matrix of variances-covariances associated to the random errors of the regression model. If NULL (the default), the function tries to estimate it: it fits an ARMA model (selected according to an information criterium) to the residuals from the fitted regression model and, then, it obtains the var-cov matrix of such ARMA model.

Value

A list containing:

Bootstrap

a dataframe containing ci_inf and ci_sup, the confidence intervals using bootstrap and p_opt and q_opt (the orders for the ARMA model fitted to the residuals).

AD

a dataframe containing ci_inf and ci_sup, the confidence intervals using the asymptotic distribution and p_opt and q_opt (the orders for the ARMA model fitted to the residuals).

pv.Box.test

p-values of the Ljung-Box test for the model fitted to the residuals.

pv.t.test

p-values of the t.test for the model fitted to the residuals.

Author(s)

German Aneiros Perez ganeiros@udc.es

Ana Lopez Cheda ana.lopez.cheda@udc.es

References

Liang, H., Hardle, W., Sommerfeld, V. (2000) Bootstrap approximation in a partially linear regression model. Journal of Statistical Planning and Inference 91, 413-426.

You, J., Zhou, X. (2005) Bootstrap of a semiparametric partially linear model with autoregressive errors. Statistica Sinica 15, 117-133.

See Also

A related function is plrm.ci.

Examples

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# EXAMPLE 1: REAL DATA
data(barnacles1)
data <- as.matrix(barnacles1)
data <- diff(data, 12)
data <- cbind(data[,1],1,data[,-1])

## Not run: par.ci(data, a=c(1,0,0), CI="all")
## Not run: par.ci(data, a=c(0,1,0), CI="all")
## Not run: par.ci(data, a=c(0,0,1), CI="all")



# EXAMPLE 2: SIMULATED DATA
## Example 2a: dependent data

set.seed(123)
# We generate the data
n <- 100
beta <- c(0.5, 2)

x <- matrix(rnorm(200,0,3), nrow=n)
sum <- x%*%beta
sum <- as.matrix(sum)
eps <- arima.sim(list(order = c(1,0,0), ar=0.7), sd = 0.1, n = n)
eps <- as.matrix(eps)
y <-  sum + eps
data_parci <- cbind(y,x)

# We estimate the confidence interval of a^T * beta in the PLR model
## Not run: par.ci(data, a=c(1,0), CI="all")
## Not run: par.ci(data, a=c(0,1), CI="all")

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