fmQ: Fully-Modified OLS Estimator with Time Polynomial

In COINT: Unit Root Tests with Structural Breaks and Fully-Modified Estimators

Fully-Modified OLS Estimator with Time Polynomial

Description

Computes the Phillips-Hansen (1990) Fully-Modified estimator for cointegrated regressions with Time Polynomial, using OLS for the first stage regression.

Usage

fmQ(y,x,type=c("trend","all"),v=15,
    q=2, ker_fun="parzen",aband=0,filter=0,sb_start=0.15)

Arguments

y	The data of dependent variable in a regression.
x	The data of independent variables in a regression.
type	The deterministic parts in the regression. trend = lnear trend; all = all three options, which requires ts format.
v	Number of autocovariance terms to compute the spectrum at frequency zero, default=15.
q	degree of time polynomial, default=2.
ker_fun	Set kernel function to one of the available kernels, default="parzen". See section details below.
aband	Whether to activate the automatic bandwidth selection. aband=1. To activate. aband=0. Do not activate.The default.
filter	Whether to use an AR(1) filter to compute the spectrum at frequency zero. filter=1. To use. filter=0. Do not use. The default.
sb_start	The percentage specifies the beginning of sub-sample for stability test, and the end sample is (1-sb_start).

Details

1. Available kernels. Technical details are referred to Brillinger (1981,P.55)
"parzen"=Parzen kernel
"bartlett"=Bartlett kernel
"dchlet"= Dirichlet kernel
"mdchlet"= Modified Dirichlet kernel
"tukham"=Tukey-Hamming kernel
"tukhan"=Tukey-Hanning kernel
"cauchy"=Cauchy kernel
"bohman"=Bohman kernel
"reisz"=Riesz,Bochner kernel
"gw"= Gauss-Weierstrass kernel
"qs"= Andrews (1991) Quadratic-Spectral

These kernels are written for fmQ procedures, technically different from those used in pp and kpss tests.

2. Andrews (1991) has developed data based (or automatic) bandwidth procedures for computing the spectrum. COINT implements these procedures for the Parzen, Bartlette, Tukey-Hamming, and the Quadratic-Spectral kernels. When aband is active, COINT ignores the value you specify for the band-width parameter and automatically substitutes the data-based value.
3. The aim of the AR(1) filter is to flatten the spectrum of residual around the zero frequency, thereby making it easier to estimate the true spectrum by simple averaging of the periodogram.

Value

coefTable	Coefficients table.
vcov	Variance-covariance matrix for the parameter estimates.
sigma	Standard error of the residuals.
rss	Residual sum of squares.
fit	The fitted values, or conditional mean, of the regression.
resid	Estimated residuals.

Author(s)

Ho Tsung-wu <[e-mail]>, College of Management, National Taiwan Normal University.

References

Andrews, D. W. K. (1991) Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.Econometrica, 59: 817-858.
Brillinger, David R. (1981) Time Series Data Analysis and Theory. San Francisco, CA: Holden-Day.
Hansen, B. E. (1992) Tests for Parameter Instability in Regressions with I(1) Processes. Journal of Business and Economic Statistics, 10, 321-335.
Phillips, P. C. B. and Hansen B. E.(1990) Statistical Inference in Instrumental Variables Regression with I(1) Processes. Review of Economic Studies, 57, 99-125.

Examples

data(macro)
y=macro[,1]
x=macro[,-1]
out=fmQ(y,x,type=c("trend","all")[1],v=15,q=3,ker_fun="parzen")
out$coefTable
out$vcov
out$stests
tail(out$fit)
tail(out$resid)

COINT documentation built on Sept. 9, 2025, 5:51 p.m.