Cross-Section Unit-Root Tests of The Returns on Equity and ...

August 22, 2000

Cross-Section Unit-Root Tests of The Returns on Equity and Invested Capital:
State-Dependent Mixtures of I(1) and I(0) Dynamics in the ROE and ROIC *




Allan C. Eberhart

Akhtar Siddique

Richard J. Sweeney


McDonough School of Business
Georgetown University
37th and “O” Sts., NW
Washington, DC 20057



Abstract: Cross-section finance studies virtually never test variables for unit roots, though econometric
validity depends on testing. This paper develops a cross-section unit -root test and uses it on annual data for
the return on equity and the return on invested capital, ROE and ROIC, key variables in the Edwards-Bell-
Ohlson (EBO) accounting and Free Cash Flow (FCF) finance valuation models. Conventional wisdom is
that ROE ∼ I(0) ∼ ROIC; Siddique and Sweeney (2000) present panel evidence that ROE ∼ I(1) ∼ ROIC.
For cross sections, this paper tests the null that both returns are I(1) against the alternative of I(0). Because
a firm is unlikely to survive with rates persistently negative, however, another alternative considered is
rates are I(1) or I(0) as the lagged rates are positive or negative. The cross-section test regresses changes in
the rate on lagged levels: It easily accommodates state-dependent mixtures of dynamics as in the second
alternative, and allows various types of cross- and serial-correlation in the data. The test statistic is the
slope’s t -value against zero. For normal errors and N firms, the t-value follows the t-distribution with k =
N-2 degrees of freedom, and is approximately N(0, 1) for k ≥ 60; for errors subject to the central limit
theorem, the t-value is asymptotically N(0, 1). In contrast, panel tests cannot be used on a single cross
section; further, existing panel tests cannot handle state-dependent mixtures of I(1) and I(0) dynamics. For
annual data, ROE ∼ I(1) ∼ ROIC for positive rates, 75-80% of the sample. But ROE ∼ I(0) ∼ ROIC for
negative rates, about 20% of the sample; each year, many of the negative-rate firms have positive rates the
following year. Corrections for heteroskedasticity and error cross correlations do not qualitatively affect
results.


* For helpful discussions, thanks are owed, in particular, to Keith Ord, and to Jim Bodurtha, Bardia
Kamrad, Lee Pinkowitz, Scott Whisenant, Othmar Winckler and Teri Yohn, and participants in the Finance
and Accounting Seminar at The McDonough School of Business. Georgetown University and The
McDonough School of Business provided summer research support, and the Capital Markets Research
Center provided summer research assistance. Part of this paper was written at Göteborgs Univeristet.
.

Eberhart: (O) 202-687- 4584; fax 202-687- 4031; eberhara@msb.edu
Siddique: (O) 202-687- 3771; fax 202-687- 4031; siddqua@msb.edu
Sweeney: (O) 202-687-3742; fax 202-687-7639, - 4031; sweeneyr@msb.edu

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Cross-Section Unit-Root Tests of The Returns on Equity and Invested Capital:
State-Dependent Mixtures of I(1) and I(0) Dynamics in the ROE and ROIC





Abstract: Cross-section finance studies virtually never test variables for unit roots, though econometric
validity depends on testing. This paper develops a cross-section unit -root test and uses it on annual data for
the return on equity and the return on invested capital, ROE and ROIC, key variables in the Edwards-Bell-
Ohlson (EBO) accounting and Free Cash Flow (FCF) finance valuation models. Conventional wisdom is
that ROE ∼ I(0) ∼ ROIC; Siddique and Sweeney (2000) present panel evidence that ROE ∼ I(1) ∼ ROIC.
For cross sections, this paper tests the null that both returns are I(1) against the alternative of I(0). Because
a firm is unlikely to survive with rates persistently negative, however, another alternative considered is
rates are I(1) or I(0) as the lagged rates are positive or negative. The cross-section test regresses changes in
the rate on lagged levels: It easily accommodates state-dependent mixtures of dynamics as in the second
alternative, and allows various types of cross- and serial-correlation in the data. The test statistic is the
slope’s t -value against zero. For normal errors and N firms, the t-value follows the t-distribution with k =
N-2 degrees of freedom, and is approximately N(0, 1) for k ≥ 60; for errors subject to the central limit
theorem, the t-value is asymptotically N(0, 1). In contrast, panel tests cannot be used on a single cross
section; further, existing panel tests cannot handle state-dependent mixtures of I(1) and I(0) dynamics. For
annual data, ROE ∼ I(1) ∼ ROIC for positive rates, 75-80% of the sample. But ROE ∼ I(0) ∼ ROIC for
negative rates, about 20% of the sample; each year, many of the negative-rate firms have positive rates the
following year. Corrections for heteroskedasticity and error cross correlations do not qualitatively affect
results.

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Cross-Section Unit-Root Tests of The Returns on Equity and Invested Capital:
State-Dependent Mixtures of I(1) and I(0) Dynamics in the ROE and ROIC


1. Introduction
Cross-section studies in finance virtually never test variables for unit roots. The validity of their econometric
results, however, depends crucially on the order of integration; typical cross-section-study techniques are appropriate
only for I(0) variables. One reason for omitting unit-root tests is the small number of time-series observations available.
To remedy this problem, this paper presents a cross-section test for unit roots: The cross section of the variables’
changes is regressed on the cross section of the variable’s lagged levels with OLS. The test statistic is the t -value of the
slope; its small and large sample distributions are standard and need not be found from simulations. This cross-section
test dominates comparable panel unit-root tests: It has superior small-sample properties, and the same asymptotic
distribution.
This paper uses the cross-section test on the return on equity, ROE, the ratio of earnings to lagged book
equity, and the return on invested capital, ROIC, the ratio of net operating profits after cash taxes actually paid to the
lagged stock of invested capital. ROE is a crucial variable in the Edwards-Bell-Ohlson (EBO) accounting valuation
model; ROIC is a crucial variable in Free Cash Flow (FCF) finance valuation models. These models typically assume
ROE and ROIC are mean-reverting, or ROE ∼ I(0) ∼ ROIC (Dechow et al. 1999; Myers 1999; Doron and Penman
1999; Penman and XXX 199X, Lee and Penman 1999). A common argument is that competition eliminates economic
profits over time and thus these variables must revert to required rates of return. The view ROE ∼ I(0) ∼ ROIC is rarely
tested formally, however.
More generally, cross-section finance studies frequently include rations and typically make the implicit
assumption that the ratios are I(0). This paper’s results for ROE and ROIC cast severe doubt on this assumption and
demand that ratios be tested for unit roots.
This is the first paper to use cross-section unit -root tests. For panel tests, Siddique and Sweeney (1999) report
that ROE ∼ I(1) ∼ ROIC for 21 observations on annual data for samples with a minimum of 1300 firms, disaggregated
across size deciles and 1-digit SIC code industries.
Previous unit-root tests do not account for restrictions on the data that economic theory implies, a shortcoming
this paper starts to remedy. One restriction is that ROIC and ROE cannot be persistently negative: A negative-profit

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firm either goes out of business or reverts to positive profitability. Similarly, firms cannot persistently have rates below
the required rate of return, RR: for example, firms cannot persistently have ROIC below RRroic. Thus, the hypothesis
that ROE ∼ I(1) ∼ ROIC no matter what the values of ROE and ROIC seems too strong. Further, if ROE (ROIC) is
very large because book (invested capital) has been run down to extraordinarily low, unsustainable levels, then perhaps
this case should be treated as an outlier and handled differently from other positive ROE (ROIC) observations.
This paper tests the null hypothesis that the rate, ROE or ROIC, is integrated, or ROE ∼ I(1) ∼ ROIC, against
the alternative of I(0). Further, to account for issues regarding firms with negative or unsustainably low rates, it also
considers a second alternative: The rate is I(1) or I(0) as its lagged values are greater or less than some minimum non-
negative rate, RRmin ≥ 0.00. In this second alternative, ROE and ROIC have state-dependent mixtures of I(1) and I(0)
dynamics. This second alternative is an example of an I(1) variable with boundaries: Within the boundary ROE ≥
RRmin ≥ 0, ROE ∼ I(1); but outside the boundary, firms with ROE < RRmin have ROE ∼ I(0), and similarly for ROIC.
Test results, from annual data that Siddique and Sweeney use, support this second alternative. (a) For firms with
positive lagged rates above some RRmin, the rates are I(1); approximately 75 to 80 percent of this data set’s firms fall in
this category. (b) For firms with negative lagged rates, the rate is I(0). Negative-rate firms tend to rebound to
profitability next year, and thereafter have rates that are I(1).
In testing, special care is taken to deal with outliers that arise from economic and accounting issues specific to
the data sets used. Firms with negative invested capital or book are likely to behave differently from firms with positive
values. Further, extraordinarily large values of ROIC or ROE (say values larger than 50 percent) cannot be expected to
persist. The large values may arise because of extraordinary values in the numerator or from miniscule values in the
denominator. These possibilities are systematically explored.
Many a priori arguments that a particular variable is I(0) turn on assertions the variable has an intuitively
plausible boundary (boundaries), outside of which the variable “must” be mean-reverting. The results for ROIC and
ROE show that a variable may be I(1) within the boundaries; within the boundaries, the variable must be treated as I(1)
for econometric purposes. Thus, a priori arguments that a variable is subject to boundaries do not mitigate the need to
test for unit roots; rather, against the null of I(1), at least one alternative should be the b oundary case of I(1) - I(0)
mixtures discussed here.
This paper’s cross-section unit -root test uses an OLS regression of the cross-section of a variable’s changes on
its lagged levels. The test statistic is the slope’s t -value, which is shown to follow known small and large sample

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distributions. For a cross-section of N firms, if the errors are normal, then the t-value follows the exact t-distribution
with degrees of freedom k = N - 2, or t ∼ tk, and is approximately N(0,1) for k ≥ 60. If the errors belong to general
classes satisfy the central limit theorem, the t-value is asymptotically N(0, 1).
If multiple cross sections are available, this paper’s cross-section test can be run separately on each, and the
results averaged. For example, for T cross sections of N firms, the average of the t-values weighted by T - 1/2, tav|T, is
approximately N(0, 1) for k ≥ 60. For smaller k, tav|T is mean-zero with the variance of an exact t-distribution with k
degrees of freedom; because t av|T has smaller kurtosis than the tk, tk-critical values are conservative. More generally,
these properties hold for any weighted-average of cross-section t -values, ∑Tt=1 w’t tb,t, if all w’t > 0 and ∑Tt=1 (w’t)2 = 1.
No existing panel unit-root test allows explicit modeling of state-dependent mixtures of I(1) and I(0)
dynamics. If modeling state-dependencies is required, cross-section tests must be used even with multiple time series.
For projects not explicitly considering state dependencies, panel unit-root tests might be used with multiple
time series; examples are Im, Pesaran and Shin (1996) and Levin and Lin (1992, 1993). In such cases, this paper’s
cross-section approach is shown to dominate those panel tests that fit a single intercept and slope: This paper’s test has
superior small-sample properties, and the same asymptotic distribution. Further, cross-section tests’ small and large
sample critical values are exact, but panel tests often require simulation even for relatively large samples.
Section 2 develops the cross-section unit-root test. Section 3 uses the cross-section test to investigate state-
dependent dynamics in ROIC, Section 4 to investigate state-dependent dynamics in ROE. Section 5 compares cross-
section with panel tests (Appendices A and B provide detailed discussion and extensions). Section 6 offers a summary
and some conclusions.
2. A Cross-Section Unit-Root Test 1
The general model is
∆yj,t = yj,t - yj,t-1 = α + β yi,t-1 + ∑pji=1 ρj,i ∆yj,t-i + uj,t, t = 1,T, j = 1,n uj,t ∼ IN(0, Ω).
Under the null, α = 0 = β. Under the alternative: β < 0 and α may be non-zero. The number (pi) and size (ρi,j) of AR
terms may differ across the firms. The error, ui,t, is homoscedastic, may show contemporaneous cross correlation from
fixed-time effects, but shows no serial correlation; the error-covariance matrix, Ω, is positive definite.
2. A. The Basic Model


To start, specialize the data generating process (DGP) for ∆yj,t to

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∆yj,t = α + β yj,t-1 + uj,t,
uj,t ∼ IIN(0, σ2u) for all j,t,
where β ≤ 1. The estimated cross-section slope from OLS is
(1)
b1 = cov(∆yj,2, yj,1)/var(yj,1),
where b1 is the OLS slope estimate associated with the period-1 level, and cov and var are the sample covariance and
variance operators. Denote by ∆y2 the sample mean of the ∆yj,2, ∆y2 = n - 1 ∑nj=1 ∆yj,2, and similarly for other variables.
From examination of (1), the distribution of b1 conditional on the values of the yj,1s is a linear function of the (∆yj,2 -
∆y2) and hence of the ∆yj,2. Note also that the lagged levels yj,1 are independent of the uj,2s. Then, it follows that b1 is
normally distributed; even for small n, b1 is unbiased (Hamilton, 1994, pp. 207-209). Further, it can be shown that the
t-value tb,1 has an exact t-distribution with k = n - 2 degrees of freedom, or tb,1 ∼ tk.2 For later work, note that these
results require no distributional assumptions about the lagged levels, here the y j,1s; in particular, they may be I(0) as in
β < 1 or I(1) as in β = 1. Thus, these results hold under the second alternative of variables that are I(1) within
boundaries and I(0) outside the boundaries: β = 0 in the first case, β < 1 in the second, and hence E b1 = 0 within
boundaries, E b1 < 0 outside the boundaries.

If the DGP contains autoregressive terms in ∆yj,t, similar results hold. Let the DGP be
∆yj,t = α + β yj,t-1 + ∑pi=1 ρi ∆yj,t-i + uj,t, uj,t ∼ IN(0, σ2u) for all j,t,
or the ∆ yj,t-1 have the same ρi across j. (Mutatis mutandis, the discussion below applies for different ρi,j over the cross
section.) Let the OLS cross-section regression be
∆yj,2 = a + b yj,1 + ∑pi=1 ri ∆yj,2-i + zj,2.
Write the DGP in matrix notation as

∆yt = β xt-1 + ut.
and the OLS cross-section regression for the period-1 levels and the period-2 changes as

∆y2 = b1 x1 + z2.
Examination shows that the distribution of b1 conditional on the values of the yj,1s and the ∆yj,2-is is a linear function of
the (∆yj,2 - ∆y2) and hence of the ∆yj,2. Note also that the lagged levels yj,1 and the lagged ∆yj,2-is are independent of the
uj,2s. Then, it follows that b1 is normally distributed; even for small n, b1 is unbiased (Hamilton, 1994, pp. 207-209).
Further, it can be shown that the t -value for each coefficient in b1 has an exact t-distribution with k = n - 2 - p degrees

1 For more details on the test, and its relationship to cross-section tests of cointegration, see Sweeney (2000).

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for freedom. In particular, b1’s t-value, tb,1, has an exact t-distribution with k = n - 2- p degrees of freedom, or tb,1 ∼ tk.3
In a Wald test that a set of m linear restrictions on the slope parameters jointly holds, the test statistic is exactly
distributed F(m, n - 2 - p). Note for later work that these results do no depend on the regressors’ distributions; in
particular, the lagged levels, here the yj,1s, may be I(0) as in β < 1 or I(1) as in β = 1.
More generally, for any DGP where the errors uj,t+1 are mean zero, independent of each other, independent of
the lagged levels and changes, y j,t and ∆yj,t-i for i ≥ 0, and satisfy the central limit theorem, then asymptotically tb,1 →
N(0, 1). An example is ∆yj,t = uj,t ∼ II(0, σ2u) for all j,t, though asymptotic results hold for weaker conditions (Hamilton
1994, pp 185-199).

A Monte Carlo Example. To illustrate the structure of the test, consider a Monte Carlo that assumes the null β
= 0, assumes all ρj = 0, and assumes the errors are uj,t ∼ IIN(0, 1): Then, ∆yj,2 = uj,2. The levels are generated by
cumulated increments in ∆yj,t, starting with an initial level of zero for all j and allowing the process to run for 250
periods: Then, each yj,1 ∼ IIN(0, 250) = 15.81 IIN(0, 1). Thus, E (∆yj,2 yi,1) = 0 for all i,j, and E (∆yj,2 ∆yi,2) = 0 = E (yj,1
yi,1) for j≠i. The cumulative empirical distribution of the t-values is very close to that for the N(0, 1), as expected:
.995
.990
.950
.990
….
.100
.050
.010
.005
sample -2.567 -2.327
-1.682
-1.298
….
1.274
1.649
2.311
2.558
N(0, 1) -2.576
-2.326
-1.645
-1.282
….
1.282
1.645
2.326
2.576
If the Monte Carlo study is continued for a second period, each ∆yj,3 is generated as IIN(0, 1). The yj,2 are from
previously generated values, y j,2 = ∆yj,2 + yj,1; the yj,2 ∼ IIN(0, 251) and are independent of ∆yj,3. For independence of
∆yj,3 and yj,2, the linear dependence of yj,2 on ∆yj,2 and yj,1 is irrelevant.
Results For Following Periods Consider the OLS cross section regression for the second period, ∆yj,3 on yj,2.
Similar to above, b2 is normally distributed and has a t-value with an exact t-distribution, tb,2 ∼ tdf. As n → ∞, then tb,2
→ N(0, 1). Further, for any DGP where the errors uj,t+1 are mean zero, independent of each other, independent of the
lagged levels and changes, y j,t and ∆yj,t-i for i ≥ 0, and satisfy the central limit theorem, then asymptotically tb,2 → N(0,
1). Mutatis mutandis, these results hold for all tb,t.

2 For development of this argument, see Ord and Stuart (1994), especially pp. 269-271.
3 Making the necessary changes for cross-section rather than time-series analysis, this case can be developed from
Hamilton’s (1994, pp. 207-209) discussion of OLS with stochastic regressors

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Aggregating Cross Section Results for t b Across T Periods For any t’ ≠ t, the distribution of tb,t’ conditional
on tb,t, or tb,t’ | tb,t, is independent of tb,t: tb,t’ | tb,t ∼ tdf. Form a weighted average of the tb,t from T separate cross-section
regressions, av(t b | T) = ∑Tt=1 wav tb,t, where wav = T - 1/2. With equal cross sections of n and degrees of freedom k = n-2-
p, each tb,t ∼ tk and for k ≥ 60, each tb,t is approximately N(0,1), and hence so is av(tb | T). For smaller n, av(tb | T) is
mean-zero and has the same variance as an exact t-distribution with k = n-2-p: E av(tb | T) = ∑Tt=1 T -1/2 E tb.t = 0 and E
[av(tb | T)]2 = E [∑Tt=1 T - 1/2 tb.t]2 = T - 1 ∑Tt=1 σ2tb,t = σ2t,k. Turn to asymptotic results. In a large cross-section, as n →
∞, then av(tb | T) → N(0, 1).4 More generally, for any DGP where the errors uj,t+1 are mean zero, independent of each
other, independent of the lagged levels and changes, y j,t and ∆yj,t-i for i ≥ 0, and satisfy the central limit theorem, then
asymptotically tb,t → N(0, 1); thus, as n → ∞, then av(tb | T) → N(0, 1).
2. B. More Complicated Error Structures
Heteroskedasticity. If σ2uj is not constant over the j, then tb is biased upwards. If only one cross section is
available, it is not possible to calculate time-series variances for each variable, a common approach to adjusting for
heteroskedasticity; if only a few cross sections are available, this approach might not be useful. A variety of ad hoc
approaches, however, can be used to adjust the data. In one approach, the sample may be split by some characteristic —
industry, size etc.—and the sub-samples tested for differences in variances. If say industries have different variances,
observations ∆yj,t, yj,t-1 for industry k can be divided by k’s cross-sectional standard deviation. In another approach, the
observations on ∆yj,t may be ranked by their squared values, (∆yj,t)2, divided into groups based on rank, and tested for
constant variances across groups. If the groups’ variances differ, each group’s data can be adjusted by that group’s
standard deviation.
An alternative is to abjure searching for particular types of heteroskedaticity and to use heteroskedasticity-
consistent standard errors; results for this approach are reported below.
Cross-Correlation in the Errors—Fixed Time Effects. For the many finance data sets with important
contemporaneous cross correlation, the assumption ∆yj,t = uj,t ∼ IN(0, σu) is inappropriate. In particular, if the errors are
positively correlated, then tb,t is biased upwards—intuitively, the effective number of cross sections is smaller than N. If
the correlations essentially arise from fixed time effects (FTEs), then, after appropriate adjustments to the data,
asymptotically tb,t ∼ N(0, 1) in the OLS approach above. Suppose the DGP is

4 Similarly, because the tb,t are independent, and because the tb,t → Nt(0, 1) as n → ∞, then asymptotically the sum of

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∆ yj,t = uj,t + φt, uj,t ∼ N(0, σ2u), φt ∼ N(0, σ2φ),
where the φt are FTEs. The error variances are σ2u +σ2φ and the covariances σ2φ. A standard FTE adjustment defines vj,t
= ∆ yj,t - ∆ yt, where ∆ yt = n - 1 ∑nj=1 ∆ yj,t is the cross-section mean of ∆ yj,t. Then, vj,t = (uj,t - φt) - n - 1 ∑nj=1 (uj,t + φt) =
uj,t - n - 1 ∑nh=1 uh,t, with variances and covariances σ2vj = σ2u - n - 1 σ2u and σvj,vh = - n - 1 σ2u. Thus, even for n = 30, the
covariance matrix ∑v is approximately diagonal. As n → ∞, then σ2vj → σ2u and σvj,vh → 0, so ∑v → σ2u I, where I is an
n x n identity matrix. With the yj,t similarly transformed, Vj,t = yj,t - yt, the Vj,t also have an asymptotic covariance
matrix that is diagonal with constant error variances, ∑v = G σ2u I. Thus tb,t →d N(0, 1). OLS estimators use vj,t = (uj,t -
ut), not uj,t, when an intercept is fit: thus, for large n the OLS estimator bt eliminates error correlations from FTEs.5
Remaining Cross-Correlations. After FTE-adjustment, tests may detect remaining correlations, for example,
tests that split the sample by industry, size etc. Under the null, sub-sample k’s average residual, uk,t, has E uk,t = 0. For
uk,t significantly non-zero, FTE adjustment may be reapplied, or the regression may include a separate mean for k.
Suppose the DGP for industry k, with K firms, and all other firms (∼ k) is
∆ yj,∼k,t = uj,t + φt,
∆ yj,k,t = uj,t + φt + φk,t,
uj,t ∼ N(0, σ2u),
φt ∼ N(0, σ2φ)
φk,t ∼ N(0, σ2φk).
If all firms are adjusted for FTEs, then the two sub-samples are
vj,k,t = uj,t + φk,t - n - 1 ∑nj=1 uj,t - n - 1 ∑nj=1 φk,t = uj,t + φk,t - ut - (K/N) φk,t = uj,t - ut + (1 - K/N) φt,
vj,∼k,t = uj,t - n - 1 ∑nj=1 uj,t - n - 1 ∑nj=1 φk,t = uj,t - ut - (K/N) φk,t,
where vj,∼k,t is for firms not in k. Using FTE adjustment again, over the K firms in k, and over all firms,
v*j,k,t = vj,k,t - K - 1 ∑Kj=1 vj,k,t = uj,t - ut + (1 - K/N) φt - K - 1 ∑Kj=1 [- ut + (1 - K/N) φt] = uj,k,t
v*j,∼k,t = vj,∼k,t - (N-K) - 1 ∑N-Kj=1 vj,t = uj,t - ut - (K/N) φt - (N-K) - 1 ∑N-Kj=1 [- ut - (K/N) φt] = uj,∼k,t.
Alternatively, the same asymptotic results are found from the regression
∆yj,2 = a + ak,2 Dk + b yj,1 + zj,2,
where Dk is a dummy variable, Dk = 1 for all firms in k, and zero otherwise. This ensures that uk = 0, as is the average
residual over the N. The null is now a = 0 = b, ak,t >/< 0: Over cross sections ak,t fluctuates, ak,t >/< 0 as φk,t >/< 0; for
any one cross section, E ak,t = 0 as E φk,t = 0. Results for this approach are reported below.
Differences in Means. The cross-section unit -root test necessarily cannot fit firm-specific fixed effects, that is,
an intercept for each firm. Imposing α rather than fitting separate αj may, but need not, cause important bias in b, as

squared t -values is distributed chi-square with T degrees of freedom: as n → ∞, then ∑Tt=1 (tb,t)2 → χ2 (T).

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Appendix C discusses. Separate means can, however, be fit for sub-samples, for example, size deciles. The researcher
may a priori specify
∆yj,2 = a + ∑9k=1 ak,2 Dk + b yj,1 + zj,2,
where Dk is a one-zero dummy for size decile k. For any ak ≠ 0, two interpretations are possible: First, the deciles have
different drift terms, or second, firms in a given decile may have important cross correlations, as discussed above. If
there are multiple cross sections available, the two interpretations can be distinguished: The first says that the ak,t
fluctuate around a non-zero value, ak; the second says that the ak,t fluctuate around zero, as φk,t >/< 0.
Identifying the Correct p When p > 0 in the DGP ∆yj,t = α + β yj,t-1 + ∑pi=1 ρi ∆yj,t-i + uj,t, call the true value
p*. The above results hold when the researcher sets p = p*. If p < p* is chosen, the estimate b 1 is biased; if p > p is
chosen, the estimate b1 is inefficient. The researcher may try to avoid these problems in two ways: one way is to use the
partial autocorrelation function of the ∆yj,t; the other is to overfit the regression by including more lagged ∆yj,t-i than
seem called for and eliminating those that are insignificant.
First, examine the partial autocorrelation function (pacf) of ∆yj,t for the panel data, that is, run ∆yj,t = c0 + c1
∆yj,t-1 + c2 ∆yj,t-2 + c3 ∆yj,t-3 + … + cn ∆yj,t-n for different maximum lags, cn. Suppose the null is true. If the maximum lag
fit is j, then E cj ≠ 0 for p* ≥ j, and E cj = 0 for j > p*. In particular, if the process is an AR1, as in ∆yj,t = α + ρ1 ∆yj,t-1 +
uj,t, then E c1 = ρ1 and E c2 = E c3 = … = 0. If alternative is true, the AR processes omit yj,t-1 when it should be included,
and thus the cj are likely to be biased.
Second, overfit the data. Under both the null and the alternative, if ∆yj,t = a + b1 yj,t-1 + zj,t is run in the AR1
case, then the t -value tb,1 is biased. If ∆yj,t = a + b1 yj,t-1 + r1 ∆yj,t-1 + zj,t is then run, the bias in tb,1 is zero and E r1 = ρ1;
both the change in t b,1 from eliminating bias and the importance of r1 signal the justification for including ∆yj,t-1. Thus,
the pacf and the way t b varies with p help identify p* under the null; at the least, the way tb varies with p helps identify
p* under the alternative.
3. Estimates of the Dynamics in ROIC
For cross-section tests on ROIC, this paper uses annual COMPUSTAT data on NOT and K from 1991 to 1997
for 1374 firms, giving ROIC values from 1992 to 1997, and changes in ROIC from 1993 to1997. Data are selected to
be comparable to Siddique and Sweeney’s (1999). The data are for firms that survive from 1976 to 1997, and thus, as

5 More generally, this approach applies if the cross-correlations ρj,i are approximately equal, ρj,i ≈ ρ.

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