Response Surface Regressions for Critical Value Bounds and Approximate p‐values in Equilibrium Correction Models1
| Published date | 01 December 2021 |
| DOI | http://doi.org/10.1111/obes.12377 |
| Author | Daniel C. Schneider,Sebastian Kripfganz |
| Date | 01 December 2021 |
1456
©2020 TheAuthors. OxfordBulletin of Economics and Statistics published by Oxford University and John Wiley & Sons Ltd.
Thisis an open access article under the ter ms of the CreativeCommons Attribution License, which permits use, distribution and reproduction in any medium, provided
the original work is properlycited.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 6 (2020) 0305–9049
doi: 10.1111/obes.12377
Response Surface Regressions for CriticalValue
Bounds and Approximate p-values in Equilibrium
Correction Models*
Sebastian Kripfganz,†Daniel C. Schneider‡
†Department of Economics, University of Exeter Business School, Streatham Court, Rennes
Drive, Exeter, Devon EX4 4PU, UK (e-mail: S.Kripfganz@exeter.ac.uk)
‡Max Planck Institute for Demographic Research, Konrad-Zuse-Straße 1, Rostock,
Mecklenburg-Vorpommern 18057, Germany (e-mail: schneider@demogr.mpg.de)
Abstract
Weconsider the popular ‘bounds test’ for the existence of a level relationship in conditional
equilibrium correction models. By estimating response surface models based on about 95
billion simulated F-statistics and 57 billion t-statistics, we improve upon and substantially
extend the set of available critical values, covering the full range of possible sample sizes
and lag orders, and allowing for any number of long-run forcing variables. By computing
approximate P-values, we find that the bounds test can be easily oversized by more than 5
percentage points in small samples when using asymptotic critical values.
I. Introduction
The empirical analysis of time series data is often confronted with test statistics that have
non-standard distributions in the presence of a unit root. While the asymptotic distributions
can be characterized as functions of stochastic processes such as Brownian motions, the
corresponding quantiles that are needed to compute critical values (CVs) for hypothesis
testing are usually obtained with stochastic simulations. As an additional complication, the
distributions of the test statistics generally depend on the specific assumptions about the
data-generating process (DGP) and the specification of the estimated model, in particular
whether an intercept or time trend are allowed. In a multivariable model, the dimension
of the variable space and the cointegration rank matter. Importantly, the finite-sample
distributions of the test statistics depend on further characteristics of the estimation. While
augmenting the regression model with additional stationary variables does not affect the
asymptotic distributions of unit-root and cointegration tests, their influence on the finite-
sample distributions can be non-negligible. Given the vast number of empirically relevant
regression specifications that lead to possibly different distributions, the tabulation of CVs
JEL Classification numbers: C12; C15; C32; C46; C63
*Weare grateful for comments by Mehdi Hosseinkouchack and the anonymous referees. This study uses simulated
data only.Details to reproduce the data are provided within this paper and the Supplementary Appendix.
Critical value bounds in EC models 1457
quickly approaches space limits and is usuallyonly done for a selected number of situations.
This leaves blank areas that can be interpolated only to a limited extent.
All of these remarks apply to the Pesaran, Shin and Smith (2001) bounds test for
the existence of a level relationship in an unrestricted conditional equilibrium correction
model. This test is highly prominent among empirical researchers, not least because it
evades the necessity of pretesting for the existence of unit roots, assuming that all variables
are integrated at most of order one. The test yields conclusive evidence if the value of the
test statistic falls outside of the critical value bounds established for the situations where
all long-run forcing variables are purely integrated of either order zero, I(0), or order one,
I(1).1Because the bounds procedure does not require that all variables are individually
I(1), the considered concept of a level relationship is broader than that of cointegration.
Pesaran et al. (2001) derive the asymptotic distributions of their test statistics under
the null hypothesis of no level relationship and then use stochastic simulations to compute
near-asymptotic CVs. However, the asymptotic distributions might be poor approximations
of the actual distributions in small samples. Finite-sample CVs are tabulated by Mills and
Pentecost (2001), Narayan and Smyth (2004), Kanioura and Turner (2005), and Narayan
(2005), but they cover onlya limited por tion of the set of possiblemodel specifications and
sample sizes. Moreover, the precision of these CVs suffers from a relatively small number
of replications in the respective simulations.
In this paper, we set out to systematically approximatethe finite-sample and asymptotic
distribution functions for the Pesaran et al. (2001) bounds test statistics. We fill the gaps
regarding the CVs by estimating response surface (RS) models that predict the quantiles
of the distributions as a function of the sample size, lag order and number of long-run
forcing variables. The RS technique was introduced into the field of unit-root testing and
cointegration analysis by MacKinnon (1991) for a range of Dickey and Fuller (1979) and
Engle and Granger (1987) tests, and has since been applied numerous times.
Ericsson and MacKinnon (2002) provide RS estimates for the cointegration t-statistic
in single-equation conditional error correction models that comprise the Dickey–Fuller
statistic as a special case. Both asymptotic and finite-sample CVs can be obtained from
these estimates.2As an important extension, Cheung and Lai (1995a) estimate RS models
for the augmented Dickey–Fuller unit-root test, acknowledging the influence of the lag
order on the finite-sample distributions.3As a complement to the generalized Dickey–
Fuller t-statistic, Pesaran et al. (2001) propose a related F-statistic to test for the existence
of a level relationship in a conditional equilibrium correction model.4So far, the only RS
1McNown et al. (2018) propose a bootstrap procedure for the Pesaran et al. (2001) test that allowsfor conclusive
inference when the test statistic falls within the two bounds.
2Previously tabulated CVs for a small set of sample sizes can be found in Fuller (1976) and Dickey (1976) for the
univariable and Banerjee, Dolado and Mestre (1998) for the multivariablesetting.
3Cook (2001) compares the response surfaces from Cheung and Lai (1995a) with those from MacKinnon (1991)
and concludes that adjusting for the lag order leads to a gain in power.RS estimates for finite-sample CVs of other
unit-root tests are provided by Cheung and Lai (1995b), Harveyand van Dijk (2006), Otero and Smith (2012, 2017),
and Otero and Baum (2017). All of them take the lag order into account. Further related applications of the RS
methodology include Sephton (1995, 2008, 2017), Carrion-i-Silvestre, Sans´o Rossell´o and Art´
is Ortu˜no (1999), and
Presno and L´opez (2003).
4In the univariable model with restricted intercept or time trend, this statistic reduces to the Dickey and Fuller
(1981) unit-root F-statistic.
©2020 The Authors. Oxford Bulletin of Economics and Statistics published by Oxford University and JohnWiley & Sons Ltd.
To continue reading
Request your trial