We investigate the efficiency of the Dickey-Fuller (DF) test as a tool to examine the convergence hypothesis. In doing so, we first describe two possible outcomes, overlooked in previous studies, namely Loose Catching-up and Loose Lagging-behind. Results suggest that this test is useful when the intention is to discriminate between a unit root process and a trend stationary process, though unreliable when used to differentiate between a unit root process and a process with both deterministic and stochastic

trends. This issue may explain the lack of support for the convergence hypothesis in the aforementioned literature

Gómez, Manuel

Ventosa-Santaulària, Daniel

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveIncome convergence: the Dickey-Fullertest under the simultaneous presence ofstochastic and deterministic trendsGómez, Manuel and Ventosa-Santaulària, DanielEscuela de Economía, Universidad de Guanajuato2007Online at https://mpra.ub.uni-muenchen.de/58778/MPRA Paper No. 58778, posted 29 Sep 2014 23:59 UTCIncome convergence: the Dickey-Fuller test underthe simultaneous presence of stochastic anddeterministic trendsManuel Gómez∗Daniel Ventosa-Santaulària†School of EconomicsUniversidad de GuanajuatoMarch 16, 2007AbstractWe investigate the efficiency of the Dickey-Fuller (DF) test as a toolto examine the convergence hypothesis. In doing so, we first describe twopossible outcomes, overlooked in previous studies, namely Loose Catching-up and Loose Lagging-behind. Results suggest that this test is useful whenthe intention is to discriminate between a unit root process and a trendstationary process, though unreliable when used to differentiate betweena unit root process and a process with both deterministic and stochastictrends. This issue may explain the lack of support for the convergencehypothesis in the aforementioned literature.Keywords: Divergence, Loose Catching-up/Lagging-behind, Convergence, De-terministic and Stochastic Trends.JEL classification: C32; O401 IntroductionBernard and Durlauf (1996) showed that the cross-sectional notion of conver-gence is weaker than the time-series notion of convergence. Since then, theempirical literature on time-series convergence (hereinafter τ -convergence) hasfocused on differentiating between three cases out of all the possible results thatthe different testing procedures are able to identify: catching-up, convergence∗Escuela de Economı́a, Universidad de Guanajuato†Corresponding Author: Escuela de Economı́a , Universidad de Guanajuato. UCEA-Campus Marfil Fracc. I, El Establo, Guanajuato, Gto. C.P. 36250, México. Phone and Fax(+52) (473) 735-2925 ext. 2925, e-mail: daniel@ventosa-santaularia.com1and divergence; whereas catching-up or stochastic convergence (the weaker def-inition of convergence) is the case where the logarithmic difference in per capitaincome between two economies is related to a trend stationary process; conver-gence or deterministic convergence (the stronger definition of convergence) isassociated to a constant mean stationary process. Finally, divergence is linkedto a process that contains a unit root.Time-series evidence has not been completely supportive of the convergencehypothesis. Studies that used the DF test have found it difficult to reject thenull hypothesis of unit root (see, for example, Carlino and Mills (1993), Oxleyand Greasley (1995), Loewy and Papell (1996), Li and Papell (1999), and Lee,Lim, and Azali (2005) amongst others). As has been shown (Perron 1989),the effectiveness of a unit root test decreases significantly in the presence ofstructural breaks. Hence, in order to find evidence in favor of this hypothesis,researchers have employed tests that allow structural breaks at the intercept, onthe slope of the trend function, or both. Nevertheless, these tests provide mixedresults. Bernard and Durlauf (1995) investigate this issue using cointegrationtechniques; they look for similar long-run trends in per capita output—eitherstochastic or deterministic, finding no evidence in favor of convergence in 15OECD economies.Regardless of the methodological procedure followed, little attention has beenpaid to the correct sign estimation of the parameter of the deterministic trend.This is extremely important because if such a parameter is positive, ratherthan a steady reduction in per capita income difference, we observe a constantincrement in such disparity.We assert that the lack of support for the convergence hypothesis may be dueto two factors: firstly, the range of outcomes so far considered is incomplete;given the empirical evidence, it is necessary to clearly identify that outcomeoccurring most frequently in this literature, namely the simultaneous presenceof stochastic and deterministic trends in the series under analysis. Secondly, thelimited effectiveness of the standard DF methodology to differentiate betweencertain potential results—specifically, between a unit root process and a processwith deterministic and stochastic trends.The article is organized as follows: Section 2 lists the relevant Data GeneratingProcesses (DGP) included in τ -convergence literature. Section 3 analyzes theasymptotic efficiency of the DF test in estimating both the sign and estimatedvalue of the parameter associated with the determinist trend. Section 4 presentsa Monte Carlo exercise to evaluate the performance of this test in finite samples.Finally, the main conclusions are presented in Section 5.2 Relevant DGPs in τ-convergenceAll pertinent DGPs are summarized in Table 1. DGP 1 is associated with di-vergence, i.e. the case where the logarithmic difference in per capita incomebetween two economies follows a random walk; DGP 2 is interpreted as con-vergence: the series under analysis is mean stationary; DGP 3 is related to2the systematic narrowing (widening) of the per capita income gap if the signof the deterministic trend estimator is negative (positive), i.e. catching-up andlagging-behind respectively.Interpretation Representation1 Divergence yt = yt−1 + uyt2 Convergence yt = µy + uyt3 Catching-up or Lagging-behind* yt = µy + βy · t + uyt4 Loose Catching-up or Loose Lagging-behind** yt = Y0 + µy · t +∑ti=1 uyi* Catching-up: where βy < 0 and Lagging-behind: where βy > 0.** Loose Catching-up: where µy < 0 and Loose Lagging-behind: where µy > 0.The process uyt satisfies the conditions stated by Phillip’s(1986) lemma, p.369 with varianceσ2y , long-run variance λ2, and autocovariances ρyi for i = 1, 2, . . .Table 1: Relevant DGPs in τ -convergence analysis.The cases implied by DGP 4 (when the series displays a stochastic and a de-terministic trend simultaneously) have been mentioned only to a limited extentin this literature. Such process represents a weaker notion of catching-up orlagging-behind. Indeed, loose catching-up (loose lagging-behind) suggests thatthe poorer economy is erratically, though inexorably catching up (lagging be-hind) if the sign of the deterministic trend estimator is negative (positive). Thedominance of a deterministic trend over a stochastic is a well-established fact1;hence, finding evidence of both trends indicates an inevitable reduction—orincrease—in income differences in the long-run.3 The performance of the DF testThe standard methodology to test for convergence using time-series is the DFframework. In this case, the relevant auxiliary regression includes a constantand a deterministic trend, as in Equation 12:∆yt = α + δyt−1 + βT + Ut (1)The various possible outcomes that result from this test are shown in Table 2.Each result is presented alongside its corresponding process:1See for example Hasseler (2000).2Generally, the specification of the DF regression includes lags of ∆yt to alleviate potencialautocorrelation, as well as dummy variables to incorporate structural breaks. Nevertheless,we do not focus on these issues since we consider it important to evaluate the performance ofthe DF under its simplest specification, prior to carrying out such an analysis.3δ β Process= 0 = 0 Divergence< 0 = 0 Convergence< 0 < 0 Catching-up< 0 > 0 Lagging-behind= 0 < 0 Loose Catching-up= 0 > 0 Loose Lagging-behindTable 2: Interpretation of the DF test.It has been proved that the DF test correctly identifies a unit root when theseries are generated by DGPs 1, 3 and 4. However, the test fails to produce anadequate estimate of the linear trend parameter (DGPs 3 and 4).Remark 1 Let yt be generated by DGP 3, and be used to estimate regression(1). Hence, the estimated parameter, β̂, does not converge to its true value, βy:β̂p→ βyσ2y − ρy1σ2yT−12 tβ̂p→[σ2y − ρ1yσ2y + ρ1y] 12Remark 1 asserts that the test asymptotically identifies the parameter sign butdoes not correctly estimate the parameter value. On the one hand, the accu-rate sign identification allows differentiation between Catching-up and Lagging-behind. On the other, the imprecise parameter estimation can be regarded as aminor drawback, given that τ–convergence analysis is statistical in nature andnot explicitly tied to a particular growth theory. Hence, the parameter lacksan economic interpretation. Nevertheless, the performance of the DF when therelevant alternative is DGP 4 deserves further attention.Proposition 1 Let yt be generated by DGP 4, and be used to estimate regres-sion (1). Hence, the estimated parameter, β̂, collapses; its associated t-statisticdoes not diverge, and both asymptotic expressions contain nuisance parameters(σ2y and λ2):T β̂d→ µyλ2[∫ω(ω(1) + 3∫rω)−32(ω(1)∫rω +(∫ω)2)+ 18ω(1)2]+ 18σ2yλ2(∫ω)2−14∫ω2 − 3∫rω(∫ω −∫rω)tβ̂= Op (1)4Proof: see AppendixThe nuisance parameters may modify the shape of the distribution of tβ̂. Thiscauses some combinations of nuisance parameter values to significantly increasethe probability of accepting the null of no significance when in fact it is false(error type II). This issue is illustrated in Figure 1, for the chosen parametervalues approximately half of the cases fall outside the standard (−2, 2) nullacceptance zone.−3 −2 −1 0 1 2 3 4 500.10.20.30.40.50.60.7(a)−5 −4 −3 −2 −1 0 1 2 300.10.20.30.40.50.60.7(b)Asymptotic distribution Sample DistributionUsual Acceptancezoneof the NullUsual Acceptance zoneof the NullFigure 1: Asymptotic and Sample non-parametric estimated distributions of tβ̂ under:(a) Loose Lagging-behind (µy = 0.77) and (b) Loose Catching-up (µy = −0.07);uyt ∼ iidN (0, 1); T = 10, 000If the true DGP of the difference in per capita income contains both deter-ministic and stochastic trends, the standard significance test of parameter βunder-rejects the null hypothesis. In other words, we would tend to erroneouslyconclude that two economies are diverging when in fact they are, though some-what sluggishly, catching up or lagging behind. This is a serious mistake giventhat loose catching-up (loose lagging-behind) is in fact asymptotically equal tocatching-up (lagging-behind), whilst its economic significance is entirely differ-ent from divergence.4 Monte Carlo evidenceWe present a Monte Carlo study in which the finite sample behavior of the DFtest is analyzed where the variable under examination contains both determin-istic and stochastic trends. Table 3 shows the proportion of times (for 10,000replications) in which the test correctly identifies the DGP, as well as the sign ofthe deterministic trend parameter. Sample size, T , ranges from 25 to 250; the5Sample Sizeµy No autocorrelation, ρ = 0.0 Autocorrelation, ρ = 0.725 50 100 150 250 25 50 100 150 250-1.50 0.53 0.56 0.56 0.57 0.57 0.50 0.54 0.56 0.56 0.56-1.25 0.54 0.55 0.56 0.57 0.57 0.47 0.52 0.54 0.55 0.56-1.00 0.52 0.55 0.57 0.57 0.57 0.43 0.48 0.54 0.53 0.56-0.75 0.51 0.55 0.55 0.57 0.57 0.38 0.44 0.48 0.52 0.54-0.50 0.48 0.52 0.54 0.57 0.57 0.32 0.36 0.41 0.45 0.49-0.25 0.34 0.41 0.48 0.51 0.54 0.25 0.27 0.29 0.32 0.360.25 0.34 0.41 0.49 0.53 0.54 0.25 0.26 0.29 0.31 0.350.50 0.46 0.52 0.55 0.56 0.56 0.32 0.36 0.41 0.45 0.490.75 0.51 0.56 0.55 0.56 0.57 0.37 0.44 0.49 0.52 0.541.00 0.52 0.55 0.56 0.57 0.57 0.43 0.50 0.53 0.55 0.551.25 0.53 0.56 0.56 0.57 0.57 0.47 0.53 0.54 0.56 0.561.50 0.54 0.56 0.56 0.56 0.57 0.50 0.53 0.56 0.56 0.57Table 3: The DF test and the correct identification of the Loose Catching–up/Lagging–behind DGP.noise is assumed to be either standard normal or autocorrelated, AR(1) withρ1y = 0.7.Results indicate a poor performance of the test; its effectiveness increases thelarger the absolute value of parameter µy, the larger the sample size and thesmaller the autocorrelation. The Monte Carlo experiment suggests that theoverwhelming evidence of divergence in this literature may be due to the factthat this test fails to differentiate between divergence and loose catching-up(loose lagging-behind).5 ConclusionsThe lack of empirical support for the convergence hypothesis may be due totwo factors: 1) the failure of previous studies to give due importance to thecase where the difference in per capita income contains both a determinist anda stochastic trend—the situation defined in this study as loose catching-up orloose lagging-behind; 2) the poor performance of the DF test when analyzingseries with this characteristic. These circumstances may have led practitionersto erroneously conclude that two economies are diverging when they are, in fact,catching up or lagging behind, though somewhat wearily.We are aware that current empirical studies make extensive use of more so-phisticated tests procedures that allow for the possible existence of structuralbreaks. Their results, indicating rejection of the convergence hypothesis, shouldalso be taken with caution because the shortcomings of the DF may likewise beapplicable to these tests.6A AppendixProof of Proposition 1. The expressions needed to compute the asymptoticvalue of tβ̂are:∑yt−1 = Y0T + µy∑t − µyT +∑ξy,t−1︸ ︷︷ ︸Op“T32”∑yt−1t = Y0∑t + µy∑t2 − µy∑t +∑ξy,t−1t︸ ︷︷ ︸Op“T52”∑∆yt = µyT +∑uy,t︸ ︷︷ ︸Op“T12”∑∆ytt = µy∑t +∑uy,tt︸ ︷︷ ︸Op“T32”∑y2t−1 = Y20 + µ2y∑t2 + µ2yT +∑ξ2y,t−1︸ ︷︷ ︸Op(T 2)+2Y0µy∑t − 2Y0µyT + ...2Y0∑ξy,t−1 − 2µ2y∑t + 2µy∑ξy,t−1t − 2µy∑ξy,t−1∑∆ytyt−1 = Y0µyT + µ2y∑t − µ2yT + µy∑ξy,t + Y0∑uy,t + ...µy∑uy,tt − µy∑uy,t +∑uy,tξy,t−1︸ ︷︷ ︸Op(T )∑∆yt = µ2yT +∑u2y,t︸ ︷︷ ︸Op(T )+2µy∑uy,twhere ξy,t =∑ti=1 uy,i and all the other summations range from 1 to T . Theorders in probability can be found in Phillips (1986), Phillips and Durlauf(1986) and Hamilton (1994). These expressions can be written in Mathemat-ica 4.1 code; the software computes the asymptotics of the classical OLS for-mula (X ′X)−1X ′Y as well as the asymptotic value of the variance estimator:σ̂2u = T−1∑Tt=1 û2t whereX ′X =T∑yt−1∑t∑yt−1∑yt−12∑yt−1t∑t∑yt−1t∑t27and,Y =∑∆yt∑∆ytyt−1∑∆yttAs indicated previously, the proof was achieved with the aid of Mathematica 4.1software3.ReferencesBernard, A., and S. Durlauf (1995): “Convergence in International Out-put,” Journal of Applied Econometrics, 10(2), 97–108.Bernard, A., and S. Durlauf (1996): “Interpreting tests of the ConvergenceHypothesis,” Journal of Econometrics, 71, 161–173.Carlino, G. A., and L. O. Mills (1993): “Are U.S. Regional Incomes Con-verging? A Time Series Analysis,” Journal of Monetary Economics, 32, 335–46.Hamilton, J. (1994): Time Series Analysis. Princeton University Press.Hasseler, U. (2000): “Simple Regressions with Linear Time Trends,” Journalof Time Series Analysis, 21, 27–32.Lee, H. A., K. Lim, and M. Azali (2005): “Income Disparity between Japanand ASEAN-5 Economies: Converge, Catching-up or Diverge?,” EconomicsBulletin, 6, 1–20.Li, Q., and D. Papell (1999): “Convergence of International Output TimeSeries Evidence for 16 OECD Countries,” International Review of Economicsand Finance, 8, 267–280.Loewy, M. B., and D. H. Papell (1996): “Are U.S. Regional Incomes Con-verging? Some Further Evidence,” Journal of Monetary Economics, 38, 587–98.Oxley, L., and D. Greasley (1995): “A Time-Series Perspective on Conver-gence: Australia, UK and USA since 1870,” Economic Record, 71, 259–270.Phillips, P. (1986): “Understanding Spurious Regressions in Econometrics,”Journal of Econometrics, 33(3), 311–40.Phillips, P., and S. Durlauf (1986): “Multiple Time Series Regression withIntegrated Processes,” The Review of Economic Studies, 53(4), 473–495.3The corresponding code is available at:http://www.ventosa-santaularia.com/VSG 07a.zip8

Income convergence: the Dickey-Fuller test under the simultaneous presence of stochastic and deterministic trends

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Income convergence: the Dickey-Fuller
test under the simultaneous presence of
stochastic and deterministic trends
Manuel Go´mez and Daniel Ventosa-Santaula`ria
Escuela de Economı´a, Universidad de Guanajuato
2007
Online at http://mpra.ub.uni-muenchen.de/58778/
MPRA Paper No. 58778, posted 29. September 2014 23:59 UTC
Income convergence: the Dickey-Fuller test under
the simultaneous presence of stochastic and
deterministic trends
Manuel Go´mez∗
Daniel Ventosa-Santaula`ria†
School of Economics
Universidad de Guanajuato
March 16, 2007
Abstract
We investigate the efficiency of the Dickey-Fuller (DF) test as a tool
to examine the convergence hypothesis. In doing so, we first describe two
possible outcomes, overlooked in previous studies, namely Loose Catching-
up and Loose Lagging-behind. Results suggest that this test is useful when
the intention is to discriminate between a unit root process and a trend
stationary process, though unreliable when used to differentiate between
a unit root process and a process with both deterministic and stochastic
trends. This issue may explain the lack of support for the convergence
hypothesis in the aforementioned literature.
Keywords: Divergence, Loose Catching-up/Lagging-behind, Convergence, De-
terministic and Stochastic Trends.
JEL classification: C32; O40
1 Introduction
Bernard and Durlauf (1996) showed that the cross-sectional notion of conver-
gence is weaker than the time-series notion of convergence. Since then, the
empirical literature on time-series convergence (hereinafter τ -convergence) has
focused on differentiating between three cases out of all the possible results that
the different testing procedures are able to identify: catching-up, convergence
∗Escuela de Economı´a, Universidad de Guanajuato
†Corresponding Author: Escuela de Economı´a , Universidad de Guanajuato. UCEA-
Campus Marfil Fracc. I, El Establo, Guanajuato, Gto. C.P. 36250, Me´xico. Phone and Fax
(+52) (473) 735-2925 ext. 2925, e-mail: daniel@ventosa-santaularia.com
1
and divergence; whereas catching-up or stochastic convergence (the weaker def-
inition of convergence) is the case where the logarithmic difference in per capita
income between two economies is related to a trend stationary process; conver-
gence or deterministic convergence (the stronger definition of convergence) is
associated to a constant mean stationary process. Finally, divergence is linked
to a process that contains a unit root.
Time-series evidence has not been completely supportive of the convergence
hypothesis. Studies that used the DF test have found it difficult to reject the
null hypothesis of unit root (see, for example, Carlino and Mills (1993), Oxley
and Greasley (1995), Loewy and Papell (1996), Li and Papell (1999), and Lee,
Lim, and Azali (2005) amongst others). As has been shown (Perron 1989),
the effectiveness of a unit root test decreases significantly in the presence of
structural breaks. Hence, in order to find evidence in favor of this hypothesis,
researchers have employed tests that allow structural breaks at the intercept, on
the slope of the trend function, or both. Nevertheless, these tests provide mixed
results. Bernard and Durlauf (1995) investigate this issue using cointegration
techniques; they look for similar long-run trends in per capita output—either
stochastic or deterministic, finding no evidence in favor of convergence in 15
OECD economies.
Regardless of the methodological procedure followed, little attention has been
paid to the correct sign estimation of the parameter of the deterministic trend.
This is extremely important because if such a parameter is positive, rather
than a steady reduction in per capita income difference, we observe a constant
increment in such disparity.
We assert that the lack of support for the convergence hypothesis may be due
to two factors: firstly, the range of outcomes so far considered is incomplete;
given the empirical evidence, it is necessary to clearly identify that outcome
occurring most frequently in this literature, namely the simultaneous presence
of stochastic and deterministic trends in the series under analysis. Secondly, the
limited effectiveness of the standard DF methodology to differentiate between
certain potential results—specifically, between a unit root process and a process
with deterministic and stochastic trends.
The article is organized as follows: Section 2 lists the relevant Data Generating
Processes (DGP) included in τ -convergence literature. Section 3 analyzes the
asymptotic efficiency of the DF test in estimating both the sign and estimated
value of the parameter associated with the determinist trend. Section 4 presents
a Monte Carlo exercise to evaluate the performance of this test in finite samples.
Finally, the main conclusions are presented in Section 5.
2 Relevant DGPs in τ-convergence
All pertinent DGPs are summarized in Table 1. DGP 1 is associated with di-
vergence, i.e. the case where the logarithmic difference in per capita income
between two economies follows a random walk; DGP 2 is interpreted as con-
vergence: the series under analysis is mean stationary; DGP 3 is related to
2
the systematic narrowing (widening) of the per capita income gap if the sign
of the deterministic trend estimator is negative (positive), i.e. catching-up and
lagging-behind respectively.
Interpretation Representation
1 Divergence yt = yt−1 + uyt
2 Convergence yt = µy + uyt
3 Catching-up or Lagging-behind* yt = µy + βy · t+ uyt
4 Loose Catching-up or Loose Lagging-behind** yt = Y0 + µy · t+
∑t
i=1 uyi
* Catching-up: where βy < 0 and Lagging-behind: where βy > 0.
** Loose Catching-up: where µy < 0 and Loose Lagging-behind: where µy > 0.
The process uyt satisfies the conditions stated by Phillip’s(1986) lemma, p.369 with variance
σ2y , long-run variance λ
2, and autocovariances ρyi for i = 1, 2, . . .
Table 1: Relevant DGPs in τ -convergence analysis.
The cases implied by DGP 4 (when the series displays a stochastic and a de-
terministic trend simultaneously) have been mentioned only to a limited extent
in this literature. Such process represents a weaker notion of catching-up or
lagging-behind. Indeed, loose catching-up (loose lagging-behind) suggests that
the poorer economy is erratically, though inexorably catching up (lagging be-
hind) if the sign of the deterministic trend estimator is negative (positive). The
dominance of a deterministic trend over a stochastic is a well-established fact1;
hence, finding evidence of both trends indicates an inevitable reduction—or
increase—in income differences in the long-run.
3 The performance of the DF test
The standard methodology to test for convergence using time-series is the DF
framework. In this case, the relevant auxiliary regression includes a constant
and a deterministic trend, as in Equation 12:
∆yt = α+ δyt−1 + βT + Ut (1)
The various possible outcomes that result from this test are shown in Table 2.
Each result is presented alongside its corresponding process:
1See for example Hasseler (2000).
2Generally, the specification of the DF regression includes lags of ∆yt to alleviate potencial
autocorrelation, as well as dummy variables to incorporate structural breaks. Nevertheless,
we do not focus on these issues since we consider it important to evaluate the performance of
the DF under its simplest specification, prior to carrying out such an analysis.
3
δ β Process
= 0 = 0 Divergence
< 0 = 0 Convergence
< 0 < 0 Catching-up
< 0 > 0 Lagging-behind
= 0 < 0 Loose Catching-up
= 0 > 0 Loose Lagging-behind
Table 2: Interpretation of the DF test.
It has been proved that the DF test correctly identifies a unit root when the
series are generated by DGPs 1, 3 and 4. However, the test fails to produce an
adequate estimate of the linear trend parameter (DGPs 3 and 4).
Remark 1 Let yt be generated by DGP 3, and be used to estimate regression
(1). Hence, the estimated parameter, βˆ, does not converge to its true value, βy:
βˆ
p
→ βy
σ2y − ρy1
σ2y
T−
1
2 t
βˆ
p
→
[
σ2y − ρ1y
σ2y + ρ1y
] 1
2
Remark 1 asserts that the test asymptotically identifies the parameter sign but
does not correctly estimate the parameter value. On the one hand, the accu-
rate sign identification allows differentiation between Catching-up and Lagging-
behind. On the other, the imprecise parameter estimation can be regarded as a
minor drawback, given that τ–convergence analysis is statistical in nature and
not explicitly tied to a particular growth theory. Hence, the parameter lacks
an economic interpretation. Nevertheless, the performance of the DF when the
relevant alternative is DGP 4 deserves further attention.
Proposition 1 Let yt be generated by DGP 4, and be used to estimate regres-
sion (1). Hence, the estimated parameter, βˆ, collapses; its associated t-statistic
does not diverge, and both asymptotic expressions contain nuisance parameters
(σ2y and λ
2):
T βˆ
d
→ µy
λ2
[∫
ω
(
ω(1) + 3
∫
rω
)
−
3
2
(
ω(1)
∫
rω +
(∫
ω
)2)
+ 18ω(1)
2
]
+ 18σ
2
y
λ2
(∫
ω
)2
−
1
4
∫
ω2 − 3
∫
rω
(∫
ω −
∫
rω
)
t
βˆ
= Op (1)
4
Proof: see Appendix
The nuisance parameters may modify the shape of the distribution of t
βˆ
. This
causes some combinations of nuisance parameter values to significantly increase
the probability of accepting the null of no significance when in fact it is false
(error type II). This issue is illustrated in Figure 1, for the chosen parameter
values approximately half of the cases fall outside the standard (−2, 2) null
acceptance zone.
−3 −2 −1 0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a)
−5 −4 −3 −2 −1 0 1 2 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(b)
Asymptotic distribution Sample Distribution
Usual Acceptance
zone
of the Null
Usual Acceptance
 zone
of the Null
Figure 1: Asymptotic and Sample non-parametric estimated distributions of tβˆ under:
(a) Loose Lagging-behind (µy = 0.77) and (b) Loose Catching-up (µy = −0.07);
uyt ∼ iidN (0, 1); T = 10, 000
If the true DGP of the difference in per capita income contains both deter-
ministic and stochastic trends, the standard significance test of parameter β
under-rejects the null hypothesis. In other words, we would tend to erroneously
conclude that two economies are diverging when in fact they are, though some-
what sluggishly, catching up or lagging behind. This is a serious mistake given
that loose catching-up (loose lagging-behind) is in fact asymptotically equal to
catching-up (lagging-behind), whilst its economic significance is entirely differ-
ent from divergence.
4 Monte Carlo evidence
We present a Monte Carlo study in which the finite sample behavior of the DF
test is analyzed where the variable under examination contains both determin-
istic and stochastic trends. Table 3 shows the proportion of times (for 10,000
replications) in which the test correctly identifies the DGP, as well as the sign of
the deterministic trend parameter. Sample size, T , ranges from 25 to 250; the
5
Sample Size
µy No autocorrelation, ρ = 0.0 Autocorrelation, ρ = 0.7
25 50 100 150 250 25 50 100 150 250
-1.50 0.53 0.56 0.56 0.57 0.57 0.50 0.54 0.56 0.56 0.56
-1.25 0.54 0.55 0.56 0.57 0.57 0.47 0.52 0.54 0.55 0.56
-1.00 0.52 0.55 0.57 0.57 0.57 0.43 0.48 0.54 0.53 0.56
-0.75 0.51 0.55 0.55 0.57 0.57 0.38 0.44 0.48 0.52 0.54
-0.50 0.48 0.52 0.54 0.57 0.57 0.32 0.36 0.41 0.45 0.49
-0.25 0.34 0.41 0.48 0.51 0.54 0.25 0.27 0.29 0.32 0.36
0.25 0.34 0.41 0.49 0.53 0.54 0.25 0.26 0.29 0.31 0.35
0.50 0.46 0.52 0.55 0.56 0.56 0.32 0.36 0.41 0.45 0.49
0.75 0.51 0.56 0.55 0.56 0.57 0.37 0.44 0.49 0.52 0.54
1.00 0.52 0.55 0.56 0.57 0.57 0.43 0.50 0.53 0.55 0.55
1.25 0.53 0.56 0.56 0.57 0.57 0.47 0.53 0.54 0.56 0.56
1.50 0.54 0.56 0.56 0.56 0.57 0.50 0.53 0.56 0.56 0.57
Table 3: The DF test and the correct identification of the Loose Catching–
up/Lagging–behind DGP.
noise is assumed to be either standard normal or autocorrelated, AR(1) with
ρ1y = 0.7.
Results indicate a poor performance of the test; its effectiveness increases the
larger the absolute value of parameter µy, the larger the sample size and the
smaller the autocorrelation. The Monte Carlo experiment suggests that the
overwhelming evidence of divergence in this literature may be due to the fact
that this test fails to differentiate between divergence and loose catching-up
(loose lagging-behind).
5 Conclusions
The lack of empirical support for the convergence hypothesis may be due to
two factors: 1) the failure of previous studies to give due importance to the
case where the difference in per capita income contains both a determinist and
a stochastic trend—the situation defined in this study as loose catching-up or
loose lagging-behind; 2) the poor performance of the DF test when analyzing
series with this characteristic. These circumstances may have led practitioners
to erroneously conclude that two economies are diverging when they are, in fact,
catching up or lagging behind, though somewhat wearily.
We are aware that current empirical studies make extensive use of more so-
phisticated tests procedures that allow for the possible existence of structural
breaks. Their results, indicating rejection of the convergence hypothesis, should
also be taken with caution because the shortcomings of the DF may likewise be
applicable to these tests.
6
A Appendix
Proof of Proposition 1. The expressions needed to compute the asymptotic
value of t
βˆ
are:
∑
yt−1 = Y0T + µy
∑
t− µyT +
∑
ξy,t−1︸ ︷︷ ︸
Op
“
T
3
2
”
∑
yt−1t = Y0
∑
t+ µy
∑
t2 − µy
∑
t+
∑
ξy,t−1t︸ ︷︷ ︸
Op
“
T
5
2
”
∑
∆yt = µyT +
∑
uy,t︸ ︷︷ ︸
Op
“
T
1
2
”
∑
∆ytt = µy
∑
t+
∑
uy,tt︸ ︷︷ ︸
Op
“
T
3
2
”
∑
y2t−1 = Y
2
0 + µ
2
y
∑
t2 + µ2yT +
∑
ξ2y,t−1︸ ︷︷ ︸
Op(T 2)
+2Y0µy
∑
t− 2Y0µyT + ...
2Y0
∑
ξy,t−1 − 2µ
2
y
∑
t+ 2µy
∑
ξy,t−1t− 2µy
∑
ξy,t−1∑
∆ytyt−1 = Y0µyT + µ
2
y
∑
t− µ2yT + µy
∑
ξy,t + Y0
∑
uy,t + ...
µy
∑
uy,tt− µy
∑
uy,t +
∑
uy,tξy,t−1︸ ︷︷ ︸
Op(T )∑
∆yt = µ
2
yT +
∑
u2y,t︸ ︷︷ ︸
Op(T )
+2µy
∑
uy,t
where ξy,t =
∑t
i=1 uy,i and all the other summations range from 1 to T . The
orders in probability can be found in Phillips (1986), Phillips and Durlauf
(1986) and Hamilton (1994). These expressions can be written in Mathemat-
ica 4.1 code; the software computes the asymptotics of the classical OLS for-
mula (X ′X)−1X ′Y as well as the asymptotic value of the variance estimator:
σ̂2u = T
−1
∑T
t=1 û
2
t where
X ′X =
 T ∑ yt−1 ∑ t∑ yt−1 ∑ yt−12 ∑ yt−1t∑
t
∑
yt−1t
∑
t2

7
and,
Y =
 ∑∆yt∑∆ytyt−1∑
∆ytt

As indicated previously, the proof was achieved with the aid of Mathematica 4.1
software3.
References
Bernard, A., and S. Durlauf (1995): “Convergence in International Out-
put,” Journal of Applied Econometrics, 10(2), 97–108.
Bernard, A., and S. Durlauf (1996): “Interpreting tests of the Convergence
Hypothesis,” Journal of Econometrics, 71, 161–173.
Carlino, G. A., and L. O. Mills (1993): “Are U.S. Regional Incomes Con-
verging? A Time Series Analysis,” Journal of Monetary Economics, 32, 335–
46.
Hamilton, J. (1994): Time Series Analysis. Princeton University Press.
Hasseler, U. (2000): “Simple Regressions with Linear Time Trends,” Journal
of Time Series Analysis, 21, 27–32.
Lee, H. A., K. Lim, and M. Azali (2005): “Income Disparity between Japan
and ASEAN-5 Economies: Converge, Catching-up or Diverge?,” Economics
Bulletin, 6, 1–20.
Li, Q., and D. Papell (1999): “Convergence of International Output Time
Series Evidence for 16 OECD Countries,” International Review of Economics
and Finance, 8, 267–280.
Loewy, M. B., and D. H. Papell (1996): “Are U.S. Regional Incomes Con-
verging? Some Further Evidence,” Journal of Monetary Economics, 38, 587–
98.
Oxley, L., and D. Greasley (1995): “A Time-Series Perspective on Conver-
gence: Australia, UK and USA since 1870,” Economic Record, 71, 259–270.
Phillips, P. (1986): “Understanding Spurious Regressions in Econometrics,”
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3The corresponding code is available at:
http://www.ventosa-santaularia.com/VSG 07a.zip
8


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Income convergence: the Dickey-Fuller test under the simultaneous presence of stochastic and deterministic trends

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