a simple nonstationary-volatility robust panel unit root test

A Simple Nonstationary-Volatility Robust Panel Unit Root Test

Matei Demetrescu∗ Christoph Hanck†

May 12, 2011

Abstract

We propose an IV panel unit root test robust to nonstationary error volatility. Its finite-sampleperformance is convincing even for many units and strong cross-correlation. An application toGDP prices illustrates the inferential impact of nonstationary volatility.

Keywords: I(1) series, Time-varying volatility, Cross-dependent panel, Nonlinear IVJEL classification: C12 (Hypothesis Testing), C23 (Models with Panel Data)

1 Motivation

Standard unit root tests such as the augmented Dickey and Fuller (1979) [ADF] test do not provide

valid inference if the errors driving the series exhibit time-varying volatility, see e.g. Hamori and

Tokihisa (1997). Such nonstationary volatility arises for instance when the error variance is trending

downward (as during the Great Moderation), or undergoes structural breaks.

Panel unit root tests suffer from analogous shortcomings: Demetrescu and Hanck (2011b) demon-

strate how size distortions accumulate in panels under time-varying volatility (see also Hanck, 2008),

such that popular panel tests (surveyed in Breitung and Pesaran, 2008) are even more size-distorted.

Demetrescu and Hanck (2011b) show that the standard normal instrumental variables [IV] test

of Shin and Kang (2006) is robust to nonstationary volatility. The test relies on an orthogonalized

nonlinear IV procedure. The Achilles’ heel of the test is the orthogonalization: to obtain reasonable

size under estimated orthogonalization, the number of time periods T has to be much larger than

the number of panel units N . This is known at least since Beck and Katz, 1995; so Demetrescu and

Hanck (2011b) advocate shrinkage estimators (Ledoit and Wolf, 2004) when N is comparable to T.

A popular alternative for N T (where even shrinkage reaches its limits) is to use panel-

corrected standard errors for a pooled estimator (as do e.g. Breitung and Das, 2005, under weak

cross-correlation). This note argues that the usual panel-corrected standard errors do not lead to

robustness of the pooled IV test, but a White heteroskedasticity-robust version thereof does.∗Institute of Econometrics and Operations Research, Department of Economics, University of Bonn, Adenauerallee

24-42, D-53113 Bonn, Germany, tel. +49 228 733925, fax +49 228 73 9189, email: [email protected].†Rijksuniversiteit Groningen, Department of Economics, Econometrics and Finance, Nettelbosje 2, 9747AE Gronin-

gen, Netherlands, tel. +31 50 3633836, fax +31 50 363 7337, email: [email protected].

1

2 Model and test statistic

Let the panel be generated as

yi,t = µi + xi,t with xi,t = ρixi,t−1 + ui,t ,

where xi,0 are fixed and ui,t are stable AR(p) processes, ui,t =∑p

j=1 ai,jui,t−1 + εi,t. The cross-

correlated, heteroskedastic shocks are modeled as εt = Ω0.5 (t/T ) εt, where Ω (s) is a N ×N matrix

of piecewise Lipschitz functions, positive definite ∀s, and εt ∈ RN is a martingale difference sequence

with E (εtε′t) = IN and uniformly bounded 4th-order moments. This allows for time-varying cross-

correlation in addition to univariate heteroskedasticity, e.g. in a heteroskedastic factor structure.

The Shin and Kang (2006) test statistic is essentially given by

τ IV =1√N

N∑i=1

∑t sgn (yi,t−1) ε∗i,t√

T

where ε∗i,t are orthogonalized residuals, ε∗t = Σ−0.5ε εt with the vector εt being OLS-estimated un-

der the null, εi,t = ∆yi,t−j −∑p

j=1 ai,j∆yi,t−j , and with Σε the sample covariance matrix, Σε =

T−1∑

t εtε′t, and where yi,t−1 are recursively demeaned, yi,t−1 = yi,t−1 − 1

t−1

∑t−1j=1 yi,j−1 = xi,t−1.

To avoid the orthogonalization procedure, we consider robust panel standard errors. The idea is

to look at the pooled IV estimator (see e.g. Jonsson, 2005)

ρ = 1 +∑

i

∑t sgn (yi,t−1) εi,t∑i

∑t |yi,t−1|

,

and to use standard errors σbρ = (

∑i

∑t |yi,t−1|)−1

√∑t sgn (yt−1)′ Σε sgn (yt−1). Simulation results,

not given here to save space and available upon request, show that the resulting test statistic is

affected by nonstationary volatility too. But if we resort to White standard errors,

σWbρ =

√∑t sgn (yt−1)′ εtε′t sgn (yt−1)∑

i

∑t |yi,t−1|

,

Section 3 indicates that an approximative N (0, 1) distribution results for the panel statistic

ρ− 1σWbρ

=∑

i

∑t sgn (yi,t−1) εi,t√∑

t sgn (yt−1)′ εtε′t sgn (yt−1).

An explanation for this finding is as follows. Begin by noting that 1√T

∑[sT ]t=1 εt ⇒ B (s) where

B (s) =´ 10 Ω

0.5 (s) dW (s) for T → ∞, with W (s) a vector of N independent standard Wiener

processes, and that 1√T

x[sT ] ⇒ diag([1−

∑pj=1 ai,j ]

−1)B (s). Define tIV = (t1,IV , . . . , tN,IV )′ where,

with σ2εi

= T−1∑T

t=p+2 ε2i,t, the individual-unit IV tests are

ti,IV =1

σεi

√T

∑t

sgn (yi,t−1) εi,t.

2

Then, with ω2i =

´ 10 Ωi,i (s) ds + op (1) , B (s) = B (s) − 1

s

´ s0 B (r) dr, B (0) = 0 a.s., and the

elementwise product, we have under the null that εi,t = εi,t + op(1) and thus

tIVd→ diag

(ω−1i

) ˆ 1

0sgn

(B (s)

) dB (s) ;

see Demetrescu and Hanck (2011a). Let Ξ denote the sample covariance matrix of sgn (yt−1)εt, and

note that, due to the martingale difference property of εt, Ξ is also the sample quadratic covariation

of tIV . The elements ti,IV are marginally standard normal (cf. So and Shin, 1999; Demetrescu and

Hanck, 2011a), but not jointly, since the sample covariation of ti,IV and tj,IV , i 6= j, converges weakly,

Ξi,jd→ Ξi,j :=

1ωiωj

ˆ 1

0sgn

(Bi (s)

)sgn

(Bj (s)

)Ωi,j (s) ds,

where the functional on the r.h.s. is random under cross-dependence (i.e. if Ωi,j (s) 6= 0). Some

algebra shows the proposed test statistic to be nothing else than

tb

:=ρ− 1σWbρ

=∑

i ti,IV√∑i,j Ξi,j

=ı′tIV√ı′Ξı

+ op (1) ;

i.e., the sum∑

i ti,IV is standardized by its (random) standard deviation√

Var (∑

i ti,IV ) in the limit.

Considering the approximative standard normality of tb

(see the following section), it appears that

tIV is asymptotically mixed Gaussian with random covariance matrix Ξ.

3 Simulation evidence

Since tb

is invariant to µi we take µi = 0 to get the DGP yi,t = ρiyi,t−1 + εi,t. We consider two

patterns of dependence among the εi,t: (A) Independence. Let εi,t = εi,t for independent normal

εt = (ε1,t, . . . , εN,t)′; (B) Factor Structure: Let εi,t := λi · νt + εi,t, where νti.i.d∼ N (0, 1) and

λi ∼ U(−1, 3). We generate a break in the normal εi,t, where Var(εi,t) = 1 for t = 1, . . . , bζiT c and

Var(εi,t) = 1/δ2 for t = bζiT c+ 1, . . . , T . We consider ζi = ζ ∈ 0.1, 0.5, 0.9 (early, middle and late

breaks), and δ ∈ 1/5, 1, 5. This generates positive (δ = 1/5) and negative (δ = 5) breaks; δ = 1

is the benchmark homoskedastic case. When ρ = 1, we study the size of tb. To analyze power, we

draw ρi uniformly from [0.9, 1] for all i. We simulate without short-run dynamics but still fit p = 1

to capture the effect of not knowing the true lag order in practice.

Table 1 reports rejection rates for tb

at α = 0.05 (analogous results are available for other α).

Size is excellent under independence and cross-sectional dependence, under homoscedasticity (which

we only report for ζ = 0.1) as well as under time-varying variance, and for all configurations N T ,

T N and N ≈ T . As to power, tb

is consistent as T → ∞ for any ζ and δ; power also increases

in N for T sufficiently large. The test is relatively more powerful for early positive and late negative

breaks; interestingly, tb

is more powerful in the empirically relevant cross-sectional dependent case.

3

Table 1: Size and Power of tb, standard normal critical values

Independence Factor StructureSize Power Size Power

ζ δ T N 16 26 56 106 16 26 56 106 16 26 56 106 16 26 56 106

50 .046 .045 .051 .053 .156 .170 .167 .177 .051 .050 .049 .046 .707 .887 .994 1.001/5 100 .060 .056 .056 .054 .337 .348 .361 .368 .048 .047 .049 .052 .975 .999 1.00 1.00

200 .060 .058 .056 .062 .690 .712 .719 .735 .049 .050 .046 .048 1.00 1.00 1.00 1.0050 .045 .046 .048 .048 .150 .152 .162 .166 .050 .047 .043 .046 .402 .466 .533 .562

0.1 1 100 .055 .054 .057 .056 .334 .327 .333 .343 .051 .054 .058 .047 .761 .814 .868 .902200 .061 .062 .056 .053 .653 .687 .696 .694 .054 .059 .059 .058 .980 .989 .996 .99750 .023 .023 .025 .020 .040 .042 .038 .040 .040 .043 .044 .041 .118 .115 .119 .122

5 100 .027 .030 .023 .030 .099 .098 .101 .103 .050 .048 .055 .055 .270 .282 .300 .283200 .035 .037 .029 .034 .254 .269 .280 .276 .055 .053 .053 .052 .639 .668 .683 .688

50 .047 .045 .048 .052 .135 .140 .134 .146 .047 .054 .056 .048 .515 .677 .887 .9511/5 100 .054 .058 .057 .058 .277 .284 .287 .292 .050 .046 .054 .052 .920 .981 .999 1.00

200 .063 .065 .063 .065 .554 .586 .582 .588 .055 .049 .048 .058 .999 1.00 1.00 1.000.5 50 .029 .034 .035 .041 .088 .086 .086 .095 .046 .049 .048 .043 .238 .258 .265 .287

5 100 .048 .051 .044 .041 .186 .189 .189 .199 .050 .053 .056 .055 .515 .556 .601 .606200 .058 .048 .057 .050 .397 .417 .427 .435 .051 .062 .060 .056 .882 .914 .947 .945

50 .044 .049 .051 .050 .108 .096 .104 .113 .048 .056 .062 .064 .271 .336 .431 .4781/5 100 .052 .060 .055 .051 .201 .202 .212 .206 .051 .058 .065 .061 .624 .728 .842 .873

200 .058 .055 .060 .057 .401 .413 .418 .425 .060 .057 .063 .055 .961 .989 .997 .9990.9 50 .044 .045 .042 .044 .130 .135 .138 .143 .045 .048 .051 .049 .345 .389 .459 .487

5 100 .044 .050 .052 .050 .296 .298 .318 .315 .046 .054 .053 .052 .706 .774 .835 .849200 .056 .052 .059 .063 .628 .653 .669 .670 .053 .054 .060 .058 .971 .983 .993 .997

Nominal 5% level. 5000 replications.

4 A robust analysis of GDP prices

We study the stationarity of a panel of GDP price levels. The Price Level of GDP is the Purchasing

Power Parity (PPP) over GDP divided by the exchange rate (from the Penn World Tables, Heston

et al., 2009, item 8), a real exchange rate measure. The PPP of GDP is the national currency value

divided by the real value in international dollars. International dollars are a hypothetical currency

whose purchasing power is the same as the US $’s in the USA in some benchmark year. The PPP

and the exchange rate are expressed as national currency units per US dollar (Heston et al., 2009).

Figure 1 plots the price of GDP for all OECD countries for which complete data from 1950 to 2007

are available. Overall, the series do not appear to mean-revert, such that well-designed unit root tests

ought not reject. Most series are quite tranquil until the early 1970s, with volatility markedly larger

after. This break can be dated at the breakdown of Bretton Woods, after which flexible exchange

rates produced higher GDP price volatility. (Conveniently, no dating is necessary for tb.) The panel

is thus to be modeled as driven by heteroskedastic errors. Moreover, cross-sectional dependence

robust tests are needed as series clearly comove due to global shocks and the common reference

country. Also, the series do not trend.

Figure 2 reports sorted ti,IV and ADF unit root statistics (both accounting for a constant) and

4

Figure 1: OECD GDP prices, 1950 to 2007

p-values, sorted by the ti,IV . Lags are chosen with information criteria. None of the individual tests

ti,IV rejects; ADF rejects only for Canada. That ADF is more rejective under such variance breaks

is consistent with the Monte Carlo evidence of e.g. Demetrescu and Hanck (2011b). The p-values

show a high, but less than one, correlation between single ADF and ti,IV tests, 86.3%.

We compute tb

and compare it with popular second-generation panel tests. Consistent with

eyeballing Figure 1, tb

= −0.02 does not reject. The panel tests trob (Breitung and Das, 2005),

t∗a and t∗b (Moon and Perron, 2004), tξ∗,κ (Demetrescu et al., 2006) and CIPS ∗ (Pesaran, 2007)

have statistics −0.804, −1.203, −1.371, 2.634 and −2.36, respectively. Hence, consistent with the

undersizedness found in Demetrescu and Hanck (2011b) t∗a does not reject; t∗b , trob and tξ∗,κ do not,

either. The oversizedness of CIPS ∗ under a Bretton Woods-type upward variance break yields a

spurious rejection. The robust tb

test supports GDP price nonstationarity.

We also consider all PWT countries for which complete data from 1960 to 2007 are available,

Left Panel: Test Statistics, Right Panel: p-values (sorted according to ti,IV ).Solid: ti,IV . Dashed: ADF. The horizontal lines are 5% critical values.

Figure 2: Single-Country Unit Root Test Statistics for OECD GDP Prices

5

giving N = 111 (of a possible total of 190) and T = 48. This wider sample is not obviously

characterized by a variance break, as the many developing countries now contained in the panel

were less affected by the Bretton Woods breakdown (plots are available upon request). Hence, it is

difficult to a priori advocate adoption of any particular test here. Now, a mixed result arises, with

tb

= −2.61 rejecting. Two of the second generation tests reject, viz. trob = −3.78 and t∗b = −1.93.

The other tests accept with t∗a = −1.43, tξ∗,κ = 0.17 and CIPS ∗ = −2.05.

Both ti,IV and ADF only find eight single time series as stationary. The number is roughly

consistent with the expected rejections if all single nulls are true in such a multiple testing situation,

viz. 0.05 · 111 ≈ 6. Hence, evidence for stationarity of GDP prices appears moderate.

References

Beck N, Katz JN. 1995. What to do (and not to do) with time-series-cross-section data in comparative politics. TheAmerican Political Science Review 89: 634–647.

Breitung J, Das S. 2005. Panel unit root tests under cross sectional dependence. Stat Neerl 59: 414–433.

Breitung J, Pesaran MH. 2008. Unit roots and cointegration in panels. In Matyas L, Sevestre P (eds.) The Economet-rics of Panel Data: Fundamentals and Recent Developments in Theory and Practice, chap. 9, Dordrecht: KluwerAcademic Publishers, 3rd edn., pages 279–322.

Demetrescu M, Hanck C. 2011a. The Cauchy unit root test under nonstationary volatility. mimeo .

Demetrescu M, Hanck C. 2011b. Unit root testing in heteroskedastic panels using the Cauchy estimator. mimeo .

Demetrescu M, Hassler U, Tarcolea AI. 2006. Combining significance of correlated statistics with application to paneldata. Oxford Bulletin Econ Stat 68: 647–663.

Dickey DA, Fuller WA. 1979. Distribution of the estimators for autoregressive time series with a unit root. J AmerStatist Assoc 74: 427–431.

Hamori S, Tokihisa A. 1997. Testing for a unit root in the presence of a variance shift. Econ Lett 57: 245–253.

Hanck C. 2008. The error-in-rejection probability of meta analytic panel tests. Econ Lett 101: 27–30.

Heston A, Summers R, Aten B. 2009. Penn world table version 6.3. Tech. rep., Center for International Comparisonsof Production, Income and Prices, University of Pennsylvania.

Jonsson K. 2005. Cross-sectional dependency and size distortion in a small-sample homogeneous panel-data unit roottest. Oxford Bulletin Econ Stat 67: 369–392.

Ledoit O, Wolf M. 2004. A well-conditioned estimator for large-dimensional covariance matrices. Journal of MultivariateAnalysis 88: 365–411.

Moon HR, Perron B. 2004. Testing for a unit root in panels with dynamic factors. J Econometrics 122: 81–126.

Pesaran MH. 2007. A simple panel unit root test in the presence of cross section dependence. J Appl Econometrics 22:265–312.

Shin DW, Kang S. 2006. An Instrumental Variable Approach for Panel Unit Root Tests under Cross-Sectional Depen-dence. Journal of Econometrics 134: 215–234.

So BS, Shin DW. 1999. Cauchy estimators for autoregressive processes with applications to unit root tests and confidenceintervals. Econometric Theory 15: 165–176.

5 Referee Appendix. Not for Publication

The normal QQ plots in Figures 3 to 5 confirm the mixed Gaussianity of tIV (see the end of Section

2). In particular we randomly draw a (N × 1) from a N (0, I) distribution and calculate

tb,a

=a′tIV√a′Ξa

6

for the different τ , δ, N and T considered in the Monte Carlo study. (Cases not presented for brevity

are available upon request.) We observe that each linear combination closely tracks the standard

normal distribution up to minor small-sample distortions visible for T = 100 and simulation uncer-

tainty in the sparsely populated tails, such that (a is redrawn for each plot) the Cramer-Wold device

supports mixed Gaussianity.

First row: T = 100, second row: T = 200. Left column: δ = 1/5, right column: δ = 5.

Figure 3: Normal QQ-plots of the distributions of tb,a

for τ = 0.5, N = 6, 5000 replications

7







8

a simple nonstationary-volatility robust panel unit root test

Documents