SpeTest {SpeTestNP}

Nonparametric specification test

Description

SpeTest tests a parametric specification. It returns the test statistic and its p-value for five different heteroskedasticity-robust nonparametric specification tests

Usage

SpeTest(eq, type="icm", rejection="bootstrap", norma="no", boot="wild", 
nboot=50, para=FALSE, ker="normal",knorm="sd", cch="default", hv="default", 
nbeta="default", direct="default", alphan="default")

Arguments

eq

A fitted model of class lm or nls

type

Test type

If type = "icm" the test of Bierens (1982) is performed (default)

If type = "zheng" the test of Zheng (1996) is performed

If type = "esca" the test of Escanciano (2006) is performed, significantly increases computing time

If type = "pala" the test of Lavergne and Patilea (2008) is performed

If type = "sicm" the test of Lavergne and Patilea (2012) is performed

rejection

Rejection rule

If rejection = "bootstrap" the p-value of the test is based on the bootstrap (default)

If rejection = "asymptotics" and type = "zheng" or type = "esca" or type = "sicm" the p-value of the test is based on asymptotic normality of the normalized test statistic under the null hypothesis

If type = "icm" or type = "esca" the argument rejection is ignored and the p-value is based on the bootstrap

norma

Normalization of the test statistic

If norma = "no" the test statistic is not normalized (default)

If norma = "naive" the test statistic is normalized with a naive estimator of the variance of its components

If norma = "np" the test statistic is normalized with a nonparametric estimator of the variance of its components

boot

Bootstrap method to compute the test p-value

If boot = "wild" the wild bootstrap of Wu (1986) is used (default)

If boot = "smooth" the smooth conditional moments bootstrap of Gozalo (1997) is used

nboot

Number of bootstraps used to compute the test p-value, by default nboot = 50

para

Parallel computing

If para = FALSE parallel computing is not used to generate the bootstrap samples to compute the test p-value (default)

If para = TRUE parallel computing is used to generate the bootstrap samples to compute the test p-value, significantly decreases computing time, makes use of all CPU cores except one

ker

Kernel function used in the central matrix and for the nonparametric covariance estimator

If ker = "normal" the central matrix kernel function is the normal p.d.f (default)

If ker = "triangle" the central matrix kernel function is the triangular p.d.f

If ker = "logistic" the central matrix kernel function is the logistic p.d.f

If ker = "sinc" the central matrix kernel function is the sine cardinal function

knorm

Normalization of the kernel function

If knorm = "sd" then the standard deviation using the kernel function equals 1 (default)

If knorm ="sq" then the integral of the squared kernel function equals 1

cch

Central matrix kernel bandwidth

If type = "icm" or type = "esca" then cch always equals 1

If type = "zheng" the "default" bandwidth is the scaled rule of thumb: cch = 1.06*n^(-1/(4+k)) where k is the number of regressors

If type = "sicm" or type = "pala" the "default" bandwidth is the scaled rule of thumb: cch = 1.06*n^(-1/5)

The user may change the bandwidth when type = "zheng", type = "sicm" or type = "pala".

hv

If norma = "np" or rejection = "bootstrap" and boot = "smooth", hv is the bandwidth of the nonparametric errors covariance estimator, by "default" the bandwidth is the scaled rule of thumb hv = 1.06*n^(-1/(4+k))

nbeta

If type = "pala" or type = "sicm", nbeta is the number of "betas" in the unit hypersphere used to compute the statistic, computing time increases as nbeta gets larger

By "default" it is equal to 20 times the square root of the number of exogenous control variables

direct

If type = "pala", direct is the favored direction for beta, by "default" it is the OLS estimator if class(eq) = "lm"

If type = "sicm", direct is the initial direction for beta. This direction should be a vector of 0 (for no direction), 1 (for positive direction) and -1 (for negative direction)

For ex, c(1,-1,0) indicates that the user thinks that the first regressor has a positive effect on the dependent variable, that the second regressor has a negative effect on the dependent variable, and that he has no idea about the effect of the third regressor

By "default" no direction is given to the hypersphere

alphan

If type = "pala", alphan is the weight given to the favored direction for beta, by "default" it is equal to log(n)*n^(-3/2)

Details

To perform a nonparametric specification test the only argument needed is a model eq of class lm or of class nls. But other options can and should be specified: the test type type, the rejection rule rejection, the normalization of the test statistic norm, the bootstrap type boot and the size of the vector being generated which is equal to the number of bootstrap samples nboot, whether the vector is generated using parallel computing para, the central matrix kernel function ker and its standardization ker, the bandwidths cch and hv. If the user has knowledge of the tests coined by Lavergne and Patilea he may choose a higher number of betas for the hypersphere (which may significantly increase computational time) and an initial "direction" to the hypersphere for the SICM test (none is given by "default") or a starting beta for the PALA test (which is the OLS estimator by "default" if class(eq) = "nls").

The statistic can be normalized with a naive estimator of the conditional covariance of its elements as in Zheng (1996), or with a nonparametric estimator of the conditional covariance of its elements as in in Yin, Geng, Li, Wang (2010). The p-value is based either on the wild bootstrap of Wu (1986) or on the smooth conditional moments bootstrap of Gozalo (1997).

Value

SpeTest returns an object of class STNP.

summary and print can be used on objects of this class.

An object of class STNP is a list which contains the following elements:

stat

The value of the test statistic used in the test

pval

The test p-value

type

The type of test which was used

boot

The type of bootstrap which was used to compute the p-value

nboot

The number of bootstrap samples used to compute the p-value

ker

The central matrix kernel function which was used

knorm

The kernel matrix standardization: "sq" if the second moment equals 1 or "sd" if the standard deviation equals 1

cch

The central matrix kernel function bandwidth

hv

The nonparametric covariance estimator bandwidth

nbeta

The number of directions in the unit hypersphere used to compute the test statistic if type = "pala" or type = "sicm"

direct

The preferred / initial direction in the unit hypersphere if type = "pala" or type = "sicm"

alphan

The weight given to the preferred direction if type = "pala"

Note

The data used to obtain the fitted model eq should not contain factors, factor variables should be transformed into dummy variables a priori

Requires the packages stats (already installed and loaded by default in Rstudio), foreach, parallel and doParallel (if parallel computing is used to generate the test p-value) to be installed

For more information and to be able to use the package to its full potential see the references

Author(s)

Hippolyte Boucher <[e-mail]>

Pascal Lavergne <[e-mail]>

References

H.J. Bierens (1982), "Consistent Model Specification Test", Journal of Econometrics, 20 (1), 105-134

J.C. Escanciano (2006), "A Consistent Diagnostic Test for Regression Models using Projections", Economic Theory, 22 (6), 1030-1051

P.L. Gozalo (1997), "Nonparametric Bootstrap Analysis with Applications to Demographic Effects in Demand Functions", Journal of Econometrics, 81 (2), 357-393

P. Lavergne and V. Patilea (2008), "Breaking the Curse of Dimensionality in Nonparametric Testing", Journal of Econometrics, 143 (1), 103-122

P. Lavergne and V. Patilea (2012), "One for All and All for One: Regression Checks with Many Regressors", Journal of Business and Economic Statistics, 30 (1), 41-52

C.F.J. Wu (1986), "Jackknife, bootstrap and other resampling methods in regression analysis (with discussion)", Annals of Statistics, 14 (4), 1261-1350

J. Yin, Z. Geng, R. Li, H. Wang (2010), "Nonparametric covariance model", Statistica Sinica, 20 (1), 469-479

J.X. Zheng (1996), "A Consistent Test of Functional Form via Nonparametric Estimation Techniques", Journal of Econometrics, 75 (2), 263-289

See Also

print and print.STNP applied to an object of class STNP print the specification test statistic and its p-value

summary and summary.STNP applied to an object of class STNP print a summary of the specification test with all the options used

SpeTest_Stat is the function which only returns the specification test statistic

SpeTest_Dist generates a vector drawn from the distribution of the test statistic under the null hypothesis using the bootstrap

Examples

n <- 100
k <- 2
x <- matrix(rnorm(n*k),ncol=k)
y<-1+x%*%(1:k)+rnorm(n)

eq<-lm(y~x+0)

summary(SpeTest(eq=eq,type="icm",norma="naive",boot="smooth"))

eq<-nls(out~expla1*a+b*expla2+c,start=list(a=0,b=4,c=2),
data=data.frame(out=y,expla1=x[,1],expla2=x[,2]))

print(SpeTest(eq=eq,type="icm",norma="naive",boot="smooth"))