Parametric analysis of covariance (based on linear models)

Description

This routine tests the equality of L vector coefficients, (β_1, ..., β_L), from samples {(Y_{ki}, X_{ki1},...,X_{kip})}: i=1,...,n, k=1,...,L, where:

β_k = (β_{k1},...,β_{kp})

is an unknown vector parameter and

Y_{ki} = X_{ki1}*β_{k1}+ ... + X_{kip}*β_{kp} + ε_{ki}.

The random errors, ε_{ki}, are allowed to be time series. The test statistic used for testing the null hypothesis, H0: β_1 = ...= β_L, derives from the asymptotic normality of the ordinary least squares estimator of β_k (k=1,...,L), this result giving a χ^2-test.

Usage

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par.ancova(data = data, time.series = FALSE, Var.Cov.eps = NULL, 
p.max = 3, q.max = 3, ic = "BIC", num.lb = 10, alpha = 0.05)

Arguments

data

data[, 1, k] contains the values of the response variable, Y_k, for each model k (k=1, ..., L);

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

X_{k1}, ..., X_{kp}, for each model k (k=1, ..., L).

time.series

it denotes whether the data is independent (FALSE) or if data is a time series (TRUE). The default is FALSE.

Var.Cov.eps

Var.Cov.eps[, , k] contains the n x n matrix of variances-covariances associated to the random errors of the regression model k (k=1, ..., L). 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 linear regression model and, then, it obtains the var-cov matrix of such ARMA model.

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.

ic

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

num.lb

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

alpha

if Var.Cov.eps=NULL, alpha contains the significance level (default is 0.05) which the ARMA models are checked.

Details

If Var.Cov.eps=NULL and the routine is not able to suggest an approximation for Var.Cov.eps, it warns the user with a message saying that the model could be not appropriate and then it shows the results. In order to construct Var.Cov.eps, the procedure suggested in Domowitz (1982) can be followed.

The implemented procedure particularizes the parametric test in the routine plrm.ancova to the case where is known that the nonparametric components in the corresponding PLR models are null.

Value

A list with a dataframe containing:

Q.beta

value of the test statistic.

p.value

p-value of the corresponding statistic test.

Moreover, if data is a time series and Var.Cov.eps is not especified:

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.

ar.ma

ARMA orders for the model fitted to the residuals.

Author(s)

German Aneiros Perez ganeiros@udc.es

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

References

Domowitz, J. (1982) The linear model with stochastic regressors and heteroscedastic dependent errors. Discussion paper No 543, Center for Mathematical studies in Economic and Management Science, Northwestern University, Evanston, Illinois.

Judge, G.G., Griffiths, W.E., Carter Hill, R., Lutkepohl, H. and Lee, T-C. (1980) The Theory and Practice of Econometrics. Wiley.

Seber, G.A.F. (1977) Linear Regression Analysis. Wiley.

See Also

Other related functions are np.ancova and plrm.ancova.

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])

data(barnacles2)
data2 <- as.matrix(barnacles2)
data2 <- diff(data2, 12)
data2 <- cbind(data2[,1],1,data2[,-1])

data3 <- array(0, c(nrow(data),ncol(data),2))
data3[,,1] <- data
data3[,,2] <- data2

par.ancova(data=data3)



# EXAMPLE 2: SIMULATED DATA
## Example 2a: dependent data - true null hypothesis

set.seed(1234)
# We generate the data
n <- 100
t <- ((1:n)-0.5)/n
beta <- c(0.05, 0.01)

x1 <- matrix(rnorm(200,0,1), nrow=n)
sum1 <- x1%*%beta
epsilon1 <- arima.sim(list(order = c(1,0,0), ar=0.7), sd = 0.01, n = n)
y1 <-  sum1 + epsilon1
data1 <- cbind(y1,x1)

x2 <- matrix(rnorm(200,1,2), nrow=n)
sum2 <- x2%*%beta
epsilon2 <- arima.sim(list(order = c(0,0,1), ma=0.5), sd = 0.02, n = n)
y2 <- sum2 + epsilon2
data2 <- cbind(y2,x2)

data_eq <- array(cbind(data1,data2),c(100,3,2))

# We apply the test
par.ancova(data_eq, time.series=TRUE)


## Example 2a: dependent data - false null hypothesis
# We generate the data
n <- 100
beta3 <- c(0.05, 0.01)
beta4 <- c(0.05, 0.02)

x3 <- matrix(rnorm(200,0,1), nrow=n)
sum3 <- x3%*%beta3
epsilon3 <- arima.sim(list(order = c(1,0,0), ar=0.7), sd = 0.01, n = n)
y3 <-  sum3 + epsilon3
data3 <- cbind(y3,x3)

x4 <- matrix(rnorm(200,1,2), nrow=n)
sum4 <- x4%*%beta4
epsilon4 <- arima.sim(list(order = c(0,0,1), ma=0.5), sd = 0.02, n = n)
y4 <-  sum4 + epsilon4
data4 <- cbind(y4,x4)

data_neq <- array(cbind(data3,data4),c(100,3,2))

# We apply the test
par.ancova(data_neq, time.series=TRUE) 

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