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vol. 3 /, NO. I

ECONOMIC REVIEW

The Sources and Nature of Long-Term
Memory in Aggregate Output
by Joseph G. Haubrich and Andrew W. Lo

FEDERAL RESERVE BANK
OF CLEVELAND

1
http://clevelandfed.org/research/review/

E C O N O M I C

R E V I E W

2001 Quarter 2
Vol. 37, No. 2

Unitary Thrifts: A Performance Analysis

by James B. Thomson
Title IV of the Gramm-Leach-Bliley Act of 1999 closed the unitary thrift holding company loophole, which allowed a limited
commingling of banking and commerce. This paper examines
whether eliminating this loophole was beneficial by empirically
comparing the performance of thrifts in unitary thrift holding
companies (UTHCs) with other thrifts and UTHCs owned by
nondepository institutions with those owned by banks.
Important differences between these two types of thrifts are
found. UTHC thrifts tend to outperform the other thrifts during
the sample period studied and appear to be less risky—possibly
because the UTHC thrifts appear to have more diversified revenue streams, loan and asset portfolios, and funding sources
than do other thrifts. No evidence is found to suggest that limited
commingling of banking and commerce, in the form of the UTHC
loophole, poses undue risks to the federal financial safety net.

The Sources and Nature of Long-Term 15
Memory in Aggregate Output
by Joseph G. Haubrich and Andrew W. Lo
This paper examines the stochastic properties of aggregate
macroeconomic time series from the standpoint of fractionally
integrated models and focuses on the persistence of economic
shocks. We develop a simple macroeconomic model that
exhibits long-range dependence, a consequence of aggregation
in the presence of real business cycles. To implement these
results empirically, we employ a test for fractionally integrated
time series based on the Hurst-Mandelbrot rescaled range. This
test is robust to short-range dependence and is applied to quarterly and annual real GDP to determine the sources and nature of
long-range dependence in the business cycle.

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ISSN 0013-0281

Unitary Thrifts:
A Performance Analysis
by James B. Thomson

James B. Thomson is a vice president
and economist at the Federal Reserve
Bank of Cleveland.

Introduction

Title IV of the Financial Services Modernization Act
of 1999 closed an important loophole in U.S. banking law: the unitary thrift holding company exemption from laws that prevented the mingling of banking and commerce.1 The loophole had allowed
nondepository institutions, including nonfinancial
corporations, to own unitary thrift holding companies (UTHCs). Only depository institutions could
own multiple bank and thrift holding companies or
unitary bank holding companies. The period over
which the exemption was in force provides us with
the opportunity to examine the implications of commingling banking and commerce.
The focus of this paper is a comparison of the
performance of thrifts in UTHCs with that of other
thrifts. Because little stock market data on UTHCs
is available, this study constructs performance measures using balance sheet and income statement data
from Thrift Financial Reports.2
Overall, we find important differences between
UTHC subsidiaries and other thrifts. UTHC thrifts
tend to outperform other thrifts during the sample
period studied. Moreover, UTHC thrifts appear to
have more diversified revenues streams, loan and
asset portfolios, and funding sources than do nonUTHC thrifts. Furthermore, differences do not

suggest UTHC thrifts pose a greater risk to the federal financial safety net; UTHC thrifts produce similar returns on book equity as non-UTHC thrifts
but with higher levels of capitalization. Therefore,
we find no evidence consistent with a need to close
the UTHC loophole, which could reduce the contestability of banking markets. Overall, these results
suggest that some limited commingling of banking
and commerce might not pose undue risks to the
federal financial safety net. But because thrifts have
limited commercial lending powers, these results
must be interpreted cautiously when evaluating less
restrictive environments.

I. History of the Unitary Thrift Loophole

Holding companies grew in importance in the
1940s and early 1950s as banks used this organizational form to circumvent regulatory restrictions on
branching and other activities. Most thrifts could
not adopt the holding company form. Subsidiary
depository institutions of the holding company had
to be stock chartered, but most savings institutions—and all federally chartered thrifts—were
■

1 Public Law No. 106-102.

■ 2 Most UTHCs are wholly owned subsidiaries of larger organizations, so stock market data do not exist for these firms.

organized as mutuals. Nonetheless, thrifts that were
stock-chartered used the holding company form.
Concerned about the growth of multibank holding companies, Congress passed the Bank Holding
Company Act of 1956 to regulate them and
extended the regulation to unitary bank holding
companies in 1970. The regulations restricted the
ownership of holding companies, their interstate
expansion, and the activities permitted to nonbank
subsidiaries. The Reigle-Neal Act of 1994 and the
Financial Services Modernization Act of 1999
reduced or eliminated these restrictions, except the
restrictions on ownership.
Congressional attempts to reign in the growth of
thrift holding companies began with the Spence Act
of 1959. This act placed a moratorium on the formation of multiple thrift holding companies and the
acquisition of additional thrifts by existing holding
companies. The act did not extend to multithrift
holding companies that already existed, and it did
not prohibit the formation of UTHCs or regulate
their behavior. The 1967 Savings and Loan Holding
Company Amendments removed the moratorium
and subjected multithrift holding companies to regulation. Despite the objection of the Federal Home
Loan Bank Board, the federal regulator of thrifts
and thrift holding companies, Congress exempted
UTHCs from regulation.3
Three factors likely played into the decision to
exempt UTHCs from regulation. First, at the time
thrift holding company regulations were enacted,
unitary bank holding companies were also exempt
from holding company regulation. Second, thrifts
had very restricted lending powers, and allowing
nondepository firms to own unitary thrift holding
companies would not give rise to the concerns about
conflict of interest and concentration of power that
the commingling of commercial banking and commerce would.4 Finally, rising interest rates in the
second half of the 1960s threatened the solvency of
a number of thrift institutions. Instead of acting to
resolve the statutorily induced duration mismatch
between thrift assets and liabilities, policymakers
took a series of actions aimed at providing thrifts
with access to inexpensive deposits—including
extending Regulation Q ceilings to savings deposits
in thrifts. In other words, policymakers limited the
ability of small savers to earn positive real rates of
return on their savings in order to assure the funding of the housing finance industry.5 In this environment, it is not surprising that policymakers
would continue to allow nondepository financial
firms and nonfinancial firms (henceforth, nonbanks) to own UTHCs. The UTHC exemption
would be viewed as a method of providing

additional sources of equity capital for the thrift
industry as it provided a limited form of entry into
banking by nondepository institutions.

II. Empirical Strategy and Sample

This paper investigates whether thrifts in UTHCs
perform differently than other types of thrifts, and if
so, why. It is possible that the UTHC form is a
more efficient arrangement because it gives the firm
greater flexibility in arranging links between commercial and banking activities. On the other hand,
the links could give the thrifts an advantage only at
the expense of taxpayers, by extending the federal
financial safety net subsidy to the nondepository
owners.6
We compare the earnings, leverage, and portfolio
risk of UTHC thrifts and non-UTHC thrifts.7
We also compare the performance of thrifts owned
by depository and nondepository institutions.
Ideally, we would directly test for differences in
risk-adjusted return on equity and the likelihood of
failure. However, using balance sheet and income
data from Thrift Financial Reports precludes us
from constructing direct measures of risk variables,
and the sample period does not contain sufficient
numbers of thrift closings to fit a hazard function to
the data. Hence, our tests measure dimensions of
risk and return using proxy variables that are drawn
(in large part) from the bank performance and bank
failure and early warning literature.8 The list of variables used and their definitions appears in table 1.
The sample consists of all SAIF-insured thrift
institutions that filed at least one complete OTS
Thrift Financial Report between March 31, 1993,
and December 31, 1999, and also were not in conservatorship. The sample period begins after the
Thrift Financial Report was substantially revised as
a result of the Federal Deposit Insurance Corporation Improvement Act of 1991 and ends before
the UTHC exemption was closed by the Financial
Services Modernization Act of 1999. Thrifts are
divided into three groups, those in UTHCs, those in
■

3 See Office of Thrift Supervision (1998a, 1997).

■ 4 See Office of Thrift Supervision (1998a) and Boyd, Chang, and
Smith (1998).
■

5 See Kane (1970, 1977, and 1985, chapter 4).

■ 6 The empirical strategy is similar to that used by Valnek (1999) to
compare the performance of stock retail banks and mutual building societies in the United Kingdom.
■ 7 For a discussion of alternative ways to gauge performance using
accounting data, see Sinkey (1998, chapter 3).
■

8 See Cole (1972), Gillis et al. (1980), and Thomson (1991, 1992).

T A B L E

Variable Definitions
CAPA

Equity capital/total assets

NTCHRGOF

Net charge-offs/earning assets

LAGCAPA

CAPA lagged one quarter

OVRHD

Operating expense/total assets

CLASSASM

Classified assets/earning assets

QTL

Mortgage-related assets/total assets

FEESHR

Fee income/operating income

FIXEDA

Fixed assets/total assets

REGLIQR
(Regulatory
liquidity)

Liquid assets meeting criteria set
forth by the Office of Thrift
Supervision/total liabilities

FHLBADVA

Federal Home Loan Bank advances/
total assets

ROA

Net income after taxes and
extraordinary items/total assets

INSDEPA

Insured deposits/total assets

ROE

INSLNA

Loans to insiders/total assets

Net income after taxes and
extraordinary items/common equity

INTSHR

Interest income/total income

SBLA

Total small business loans/total assets

SIZE

Natural log of total assets

SUBDEBTA

Subordinated debt/total assets

TIER1CAP

Tier 1 capital/total assets

TRUSTA

Trust assets/total assets

LGAP
Total cash, deposits due from depository
(Liquidity gap) institutions, and investment securities
minus borrowings/total assets
LIABHERF
(Liability
Herfindahl
index)

10000•[(deposits/total liabilities)2
+ (borrowings/total liabilities)2
+ (other liabilities/total liabilities)2]

LNHERF
(Loan
Herfindahl
index)

10000 •
[(credit card receivables/total loans)2
+ (other consumer loans/total loans)2
+ (home equity loans/total loans)2
+ (commercial loans/total loans)2
+ (1–4 family mortgage loans/
total loans)2
+ (other mortgage loans/total loans)2
+ (all other loans/total loans)2]

Intercept and Slope Dummy Variables

MPSA

Mortgage pool securities/total assets

MDERA

Mortgage derivative securities/total assets

NIM

Interest income minus interest expense/
earning assets
multithrift holding companies, and independent
thrifts. Thrifts in the UTHC sample were further
partitioned into two groups according to affiliation,
using the Office of Thrift Supervision’s holding company file. The first consists of UTHCs owned or
controlled by depository institutions (henceforth,
banks), and the second consists of thrifts in UTHCs
that are affiliated with nonbanks.9

DUMUT

1 if thrift is in a UTHC,
0 otherwise

DUMUD

1 if thrift is in a UTHC owned
by a depository institution,
0 otherwise

DUMUN

1 if thrift is in a UTHC owned
by a nondepository institution,
0 otherwise

XXXXUT
XXXXUD
XXXXUN

XXXX • DUMUT
XXXX • DUMUD
XXXX • DUMUN

III. Results
Differences in Return on Equity
The simplest method for evaluating performance is
to examine differences in return on equity (ROE)
across the different types of thrifts in the sample.
As seen in table 2, the ROE of thrifts owned by
UTHCs does not differ significantly from that of
other thrifts during the full sample period. But
when assessing whether the UTHC loophole represented a dangerous mixing of banking and commerce, differences across UTHCs by ownership
■

9 The thrift holding company database can be found on the Office of
Thrift Supervision Web site, <www.ots.treas.gov/applications.html>.

T A B L E

Return on Equity
Non-UTHC
Thrifts

UTHC
Thrifts

Nonbank
UTHCs

Bank
UTHCs

Mean

0.0181

0.0200

0.0251

0.0195

Std. error

0.0013

0.0016

0.0020

0.0017

P-value

0.3566

Differences in Return on Assets and
the Ratio of Capital to Assets

0.0375

SOURCES: Office of Thrift Supervision, Thrift Financial Reports; and author’s calculations.

T A B L E

(1)

ROA
Non-UTHC
Thrifts

UTHC
Thrifts

Nonbank
UTHCs

Bank
UTHCs

Mean

0.0018

0.0019

0.0028

0.0018

Std. error

0.0000

0.0001

0.0008

0.0001

0.3680

0.2561

CAPA
Non-UTHC
Thrifts

UTHC
Thrifts

Nonbank
UTHCs

Bank
UTHCs

Mean

0.1037

1.1110

0.1621

0.1061

Std. error

0.0004

0.0010

0.0079

0.0007

P-value

The first step in sorting out the different possible
influences on ROE is to decompose it into its main
components: return on assets (ROA) and the
amount of capital per dollar of assets (CAPA).

Components of ROE

P-value

does not provide sufficient information to understand
the underlying factors driving the performance of
earnings measures such as ROE. For example, the
higher ROE for UTHCs owned by nondepository
firms may reflect a conflict of interest arising from the
commingling of banking and commerce (albeit on a
limited basis), a superior set of real options held by
the nondepository UTHCs, or simply a higher degree
of leverage employed by these firms.

<0.0001

SOURCES: Office of Thrift Supervision, Thrift Financial Reports; and author’s calculations.

type are probably more relevant. After all, ownership
of a UTHC expands the set of activities a nondepository institution can engage in, while UTHC ownership by banks and thrifts represents a more modest
expansion of their existing franchise. Thrifts in nonbank-UTHCs outperformed thrifts in bankUTHCs during the sample period, and the difference is significant.
Unfortunately, simple comparisons of ROE can
be misleading. Given the positive relationship
between risk and required return, a higher ROE may
simply reflect higher risk. Even after controlling for
risk, comparing the ROE of the different subsamples

ROE =

ROA
CAPA

The optimum degree of risk borne by a firm is a
function of the risks emanating from its asset portfolio and the degree of leverage employed. Equation
(1) illustrates how ROE is driven by these two
simultaneous sets of decisions by the thrift’s management, with the investment/operating decision
represented by ROA and the financing decision
embodied on CAPA.
Overall, the first stage of ROE decomposition
reported in table 3 suggests that thrifts in UTHCs,
particularly those in UTHCs owned by nonbanks,
perform better because they have higher ROA, not
higher leverage. While we can’t reject the hypothesis
that higher risk produced the higher ROE, these
univariate results are not consistent with increased
leverage driving the results. The first column of
table 3 shows that the difference in ROA between
UTHC thrifts and non-UTHC thrifts is not statistically significant. UTHC thrifts have higher capitalto-asset ratios than non-UTHC thrifts, and the difference is significant during the sample period.
Insignificant differences in ROE and the significantly higher level of CAPA for the UTHC thrifts
relative to other thrifts suggest that performancerelated differences between UTHC thrifts and nonUTHC thrifts are due to differences in ROA. That
is, differences in performance across these two samples are driven by differences in operating/investment decisions and not by leverage. A comparison
of tables 2 and 3 shows that in all three samples,
higher ROE is accompanied by lower leverage
(higher CAPA).
Similar results hold for the two UTHC subsamples.
Thrifts in nonbank UTHCs have higher ROA,
although the difference is not statistically significant,
and they have significantly higher CAPA. Taken
together, these findings suggest that important differences in performance exist across the UTHC

subsamples, and these differences are related to operating and investment decisions. Decomposing ROE further should tell us more about how ROA affects ROE.

Differences in the Factors
That Affect Return on Assets
The second stage of the ROE decomposition examines the underlying controllable factors that drive
ROA—business mix, income production, asset
quality, expense control, and tax management—by
specifying and estimating the following equation:
(2) ROAit = β 0 + β1NIMit + β 2NTCHRGOFit
+ β 3OVRHDit + β 4SIZEit
+ β 5LAGCAPAit + β6FEESHRit
+ β 7LGAPit + β 8QTLit + β9INSDEPAit
+ β 10DUMUTit + β11NIMUTit
+ β 12NTCHRGOFUTit
+ β 13OVRHDUTit + β14SIZEUTit
+ β 15LAGCAPAUTit + β 16FEESHRUTit
+ β 17LGAPUTit + β18QTLUTit
+ β 19INSDEPAUTit + εit
Table 1 explains how these variables were constructed. To proxy for business mix, we included
variables for the proportion of a thrift’s assets that are
mortgage related (QTL), the proportion of its funding
that comes from insured deposits (INSDEPA), the
proportion of its income that comes from fee-based
services (FEESHR), and the size of its liquidity gap
(LGAP). Asset quality is captured by the proportion
of net charge-offs to interest-earning assets
(NTCHRGOF), although this variable may also
be related to tax management or expense preference
behavior—where the reporting of losses is delayed
until profits are higher than average to reduce taxes
(see Greenwalt and Sinkey [1988]). Income production is proxied for primarily by the spread between
interest income and interest expense scaled by average
assets (NIM), although income from fee-based
services (FEESHR) may also be related. The ratio of
operating expenses to assets (OVRHD) is included to
capture expense control. We included a measure of the
relative size of the thrift’s overall balance sheet (SIZE)
because economies of scale may play a role in providing a number of financial products and services. The
previous quarter’s capital-to-asset ratio (LAGCAPA)
is included because the investment/operating
decision and the financing decision of depository
institutions are not independent. To test the hypothesis that earnings-related measures of performance don’t
differ across the groups of thrifts, we include intercept
and slope dummy variables for the thrift type. For
thrifts in the UTHC sample, the effect of each
independent variable is measured by combining its
coefficient and the coefficient on the corresponding
dummy variable.

Equation (2) is estimated using ordinary least
squares. The results are presented in table 4 and
appear to reject the hypothesis that no performance
differences exist, as the coefficients on all the
dummy variables except the proportion of mortgage-related assets (QTLUT) are significant at the
1 percent level. A closer inspection of the results,
however, is needed to ascertain whether performance-related differences stem from extensions of
the federal safety net or an increased set of options
available to UTHCs. To do this, we examined
differences between UTHC and non-UTHC thrift
ROA as captured by the proxies for business mix,
asset quality, and the financing decision.
The regression results show business mix is
important in determining the earnings performance
of thrifts; all of the variables that proxy for this factor are significant. Three of the four corresponding
dummy variables are significant and of opposite
sign, indicating that differences in business mix
between thrifts in UTHCs and other thrifts drive
differences in earnings performance. In fact, the
dummy variables for fee income and liquidity gap
are larger in absolute value, meaning these variables
affect ROA in opposite ways and to different degrees
depending on the type of thrift.
Non-UTHC thrifts appear to have lower ROA if
they rely more heavily on fee income for revenues
(FEESHR is negative and significant). One should
be careful, however, in interpreting these results in
terms of performance. If fee income and interest
income are sufficiently uncorrelated, thrifts with
higher levels of fee income may have less variable
revenues and higher risk-adjusted returns. However,
the coefficient on FEESHRUT and the combined
effect of FEESHR for UTHC thrifts is significantly
positive. The different relationship between
FEESHR and earnings for UTHC thrifts may trace
to scale economies in the production of fee-based
lines of business—the mean of SIZE for UTHC
thrifts is significantly larger than for non-UTHC
thrifts during the sample period—or to economies
of scope and cross-selling opportunities with other
businesses conducted in the holding company or by
its parent firm. If we accept the argument that fee
income is not highly correlated with interest
income—that is, fee income reduces the variability
of revenues—then the fact that fee-based services
generate better performance for UTHC thrifts than
for other thrifts is not consistent with the hypothesis
that performance differences trace to increased risk
and safety-net subsidies.
The positive and significant sign on LGAP
suggests that non-UTHC thrifts benefit from the
flexibility option associated with liquidity (as measured by the difference between liquid assets and
short-term nondeposit liabilities), and they perform
better the higher it is. For UTHC thrifts, liquidity

T A B L E

OLS Estimation of Equation 2
Dependent Variable: ROA
Coefficient
Intercept
NIM
NTCHRGOF
OVRHD
SIZE
LAGCAPA
FEESHR
LGAP
QTL
INSDEPA
DUMUT
NIMUT
NTCHRGOFUT
OVRHDUT
SIZEUT
LAGCAPAUT
FEESHRUT
LGAPUT
QTLUT
INSDEPAUT

–0.0014
0.6308
0.0895
–1.0328
0.0001
0.0127
–0.0002
0.0023
0.0013
–0.0012
–0.0131
0.4028
–0.2689
1.0474
0.0004
–0.0177
0.0026
–0.0050
0.0007
0.0007

t-Statistic Prob > t
–2.26
83.58
13.00
–126.68
2.90
23.69
–1.95
8.38
3.62
–8.40
–9.59
13.70
–31.15
101.56
5.70
–14.06
4.68
–11.06
1.06
2.70

0.024
<0.0001
<0.0001
<0.0001
0.0037
<0.0001
0.0517
<0.0001
0.0003
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.2892
0.007

R-Square
0.6265
Root MSE
0.0063
Dependent mean 0.0014
Coeff Var
438.5867
F value
2,259.58
Prob > F
<0.0001
Number of
observations
25,614
SOURCES: Office of Thrift Supervision, Thrift Financial Reports; and author’s calculations.

has the opposite effect on ROA; the coefficient on
LGAPUT and the net effect of LGAP on ROA for
UTHC thrifts is significantly negative. There are two
possible explanations for why this is so. On one hand,
the parent holding company may serve as a source of
strength and liquidity for its thrift subsidiary, so
UTHC thrifts do not need as much on-balance-sheet
liquidity to conduct their operations. That is, holding
companies may provide their thrift subsidiaries with
access to other funding sources which, at the margin,
may be less expensive than raising additional retail
deposits.10 In addition, UTHC thrifts are larger and
more likely to face deposit constraints than nonUTHC thrifts. On the other hand, UTHC thrifts
may find it desirable to take on more liquidity risk to
increase the value of their federal deposit guarantees.
Obviously, the first explanation would be consistent
with the greater-options hypothesis, while the latter
would be consistent with the safety-net-subsidy

hypothesis. But differences in leverage and the
composition of nondeposit funding between the two
samples of thrifts do not support the safety-netsubsidy hypothesis. After all, UTHC thrifts are
significantly less leveraged than non-UTHC thrifts,
and a large part of the funding difference arises
because UTHC thrifts rely more on Federal Home
Loan Bank advances.11
It is somewhat curious that for non-UTHC
thrifts, the proportion of assets funded by insured
deposits is significantly negatively related to ROA.
Moreover, while the dummy variable for this factor is
positive, it is smaller in absolute value than
INSDEPA, and the relationship between INSDEPA
and ROA for percentage of funding remains negative
and significant for UTHC thrifts. Care needs to be
taken in interpreting these results since insured
deposits are a stable funding source and are likely
to reduce profit variability. UTHC thrift earnings
appear to be less sensitive to changes in the insured
deposit base than other thrifts (the coefficient on
INSDEPAUT is positive). This may be due to
funding advantages associated with holding
company affiliation, which reduces the marginal cost
of funding additional assets. In other words, differences in earnings performance based on INSDEPA
are not likely to trace to increased safety-net subsidies
but to greater availability of deposit substitutes
(enhanced funding options) associated with using
the holding company organizational form.
ROA is positively related to the concentration of
thrift assets in mortgage-related loans and securities
(QTL). This positive relationship may indicate thrifts
have specialized expertise in mortgage-related assets
and a competitive advantage in housing-finance
markets. On the other hand, the Competitive
Equality Banking Act established a qualified-thriftlender requirement that required a minimum level
of investment in qualified assets (primarily housingfinance-related assets and, after 1996, small business
and agricultural loans). Given the qualified-thriftlender requirement, the positive and significant
coefficient on QTL may be proxying for regulatory
taxes—that is, thrifts with high QTL would be less
subject to regulatory interference. The coefficient on
QTLUT is not significant, however, and this sheds
some doubt on that interpretation. If a thrift in a
UTHC fails to meet the QTL test, the UTHC is
subject to more stringent and invasive bank holding
company regulation. In the case of nonbank-UTHC
■

10 Most of the difference in the liquidity gap between samples
derives from differences in funding, as liquid assets made up similar
portions of both group’s assets—20 percent for non-UTHC thrifts and
18 percent for UTHC thrifts.

■

11 Over the sample period, thrifts in the UTHC sample financed
12 percent of their assets with Federal Home Loan Bank advances on average, while advances were 4 percent of non-UTHC thrifts’ assets.

thrifts, penalties for failing the qualified-thriftlender test could include forced divestiture of the
thrift subsidiary.
The positive and significant effect of the proportion of a thrift’s net charge-offs to interest-earning
assets suggests that non-UTHC thrifts perform better the more they engage in expense preference
behavior—the timing of loss recognition to smooth
income for tax purposes. Note that the opposite is
true for UTHC thrifts (the coefficient on the
dummy variable is negative and significant, and the
overall effect for UTHC thrifts is significantly negative), which is consistent with low net charge-offs as
an indicator of asset quality. The net charge-off variable might operate differently across the subsamples
because holding companies are able to utilize leverage at the parent-company level to arbitrage taxes.
Thus, thrifts in holding companies have less of an
incentive to engage in expense preference behavior.
Unfortunately, because non-UTHC thrifts also
engage in this behavior, it is impossible to interpret
differences in the coefficients on NTCHRGOF
across the two groups of thrifts as an indicator of
differences in asset quality.
The coefficient on LAGCAPA is significantly
positive—which is consistent with the findings in
table 3 that firms with high ROE also had high
capital-to-asset ratios. The negative coefficient on
LAGCAPAUT indicates that UTHC-thrift earnings
are less sensitive to the level of capital than nonUTHC thrifts. Two factors likely drive these results.
First, UTHC thrifts have higher levels of capital
than independent thrifts. Second, to the extent that
the parent holding company serves as a source of
strength to its thrift subsidiary, we would expect the
thrift-level financing decision to have less of an
impact on performance.
Three other differences emerge from the ROA
decomposition. First, UTHC-thrift earnings performance is significantly more responsive to changes in
the net interest margin than non-UTHC thrifts. The
coefficient on the proxy for efficiency, OVRHD, has
a large negative effect on ROA. However, this earnings factor has no effect on earnings performance for
UTHC thrifts (the coefficient on OVRHDUT is of
the same magnitude and of opposite sign as the
coefficient on OVRHD). Finally, larger thrifts have
higher earnings, and this effect is significantly greater
for UTHC thrifts (the coefficients on SIZE and
SIZEUT are both positive and significant).

Differences between Bank- and
Nonbank-UTHC Thrifts
Overall, the results of the ROA decomposition over
the full sample of thrifts are consistent with the
hypothesis that performance-related differences exist

between UTHC thrifts and non-UTHC thrifts.
However, the nature of the differences across these
two samples does not provide sufficient information
to decide which of the two possible causes is responsible. Given that the UTHC exemption may be
more valuable, or at least the bundle of real options
associated with owning a UTHC thrift is likely different, for nonbank owners of UTHCs than for
depository institutions, we perform the ROA
decomposition again, but over the UTHC sample
only. We re-estimate equation (2), including intercept and dummy variables for type of UTHC ownership to explore differences between thrifts owned
by nonbank UTHCs and those owned by bank
UTHCs. The results of this regression are presented
in table 5.

Business Mix
Because we are exploring differences based on
ownership type, we focus on the coefficients of the
nonbank ownership dummy variables. We first
consider whether different choices of business mix
are responsible for differences in performance. The
results of this regression suggest that nonbank
UTHC thrifts use a different business mix than
bank UTHC thrifts. Interest income appears to be
a less important determinant of ROA for thrifts
owned by nonbank firms (NIMUN is negative and
significant, but of smaller magnitude than NIM).
Fee income seems more important (the coefficients
on FEESHRUN and SIZEUN are positive and significant, and the overall relationship between the
share of fee income and ROA for nonbank-UTHC
thrifts is positive). These results are consistent with
nonbank UTHCs holding a different set of real
options than depository institutions. In other words,
as the logit regression analysis that follows will confirm, it is the options other than lending powers
afforded by depository institution charters—such as
access to the payments system—that have the most
value to the nonbank acquirers of UTHCs.
Several other important differences emerge
between nonbank UTHCs and bank UTHCs. First,
asset quality matters more to earnings performance for
nonbank-UTHC thrifts than for bank-UTHC thrifts
(the signs on NTCHRGOF and
NTCHRGOFUN are significantly negative).
Second, the coefficients on OVRHD and
OVRHDUN are significant and of opposite sign. The
overall impact of efficiency for nonbank-UTHC
thrifts is positive and significant. In other words, the
lack of a significant efficiency effect on earnings performance in the first estimation of equation (2) for
UTHC thrifts was due to conflicting effects within
the UTHC sample. Given that OVRHD is constructed as the ratio of operating expense to total
assets, the differences in the effect of this proxy on

T A B L E

OLS Re-estimation of Equation 2
Dependent Variable: ROA
Coefficient
Intercept
NIM
NTCHRGOF
OVRHD
SIZE
LAGCAPA
FEESHR
LGAP
QTL
INSDEPA
DUMUN
NIMUN
NTCHRGOFUN
OVRHDUN
SIZEUN
LAGCAPAUN
FEESHRUN
LGAPUN
QTLUN
INSDEPAUN

0.0010
1.2667
–0.1240
–0.3590
0.0001
–0.0105
–0.0038
–0.0036
–0.0014
–0.0004
–0.0872
–0.2580
–1.1784
0.5198
0.0040
0.0105
0.0347
0.0097
0.0012
–0.0062

R-Square
Root MSE
Dependent mean
Coeff Var
F value
Prob > F
Number of
observations

0.5007
0.0069
0.0020
350.51
375.37
<.0001

t-Statistic Prob > t
0.67
34.60
–19.57
–23.63
1.52
–6.36
–5.74
–8.35
–1.78
–1.51
–19.69
–3.08
–8.08
28.96
15.42
3.62
17.40
5.76
0.72
–2.93

0.5028
<0.0001
<0.0001
<0.0001
0.1298
<0.0001
<0.0001
<0.0001
0.0755
0.1301
<0.0001
0.0021
<0.0001
<0.0001
<0.0001
0.0003
<0.0001
<0.0001
0.4729
0.0034

7,132

SOURCES: Office of Thrift Supervision, Thrift Financial Reports; and author’s calculations.

earnings across the nonbank- and bank-UTHC samples may reflect differences in business mix and strategy. If nonbank-UTHC thrifts place greater emphasis
on fee-based products and services, higher levels of
OVRHD may be picking up increased activity in
these areas, and increases in operating expenses per
dollar of assets would be positively related to
earnings performance. Hence, the significantly positive sign on OVRHDUN is consistent with the
hypothesis that nonbank UTHCs hold a different set
of real options than bank UTHCs and non-UTHC
thrifts.
Bank-UTHC thrifts exhibit a significantly negative relationship between ROA and the previous
quarter’s capital-to-assets ratio (LAGCAPA) and a
negative but not significant relationship between
ROA and the share of insured deposits (INSDEPA).

For nonbank-UTHC thrifts, ROA is significantly
negatively related to the share of insured deposits
(the coefficient on INSDEPAUN is negative and
significant). However, the net effect of the capital-toassets ratio on earnings is not significantly different
from zero for nonbank-UTHC thrifts (the coefficient
on LAGCAPAUN is positive and significant and of
the same magnitude as the coefficient on LAGCAPA). Finally, liquidity has opposite effects on ROA
in the two groups. It is negative for bank-UTHC
thrifts and positive for nonbank-UTHC thrifts (the
coefficients on LGAP and LGAPUN are significantly
negative and positive, respectively). The impact of
differences in the financing decision on performance
as proxied for by ROA for the bank-UTHC thrifts
and the nonbank-UTHC thrifts is not consistent
with the hypothesis that this commingling of banking
and commerce increases the loss exposure of the taxpayer to the federal financial safety net.
Finally, the fact that the concentration of mortgage-backed assets (QTL) is negative and marginally
significant and QTLUN is positive but not significant is not consistent with our earlier interpretation
of QTL as a regulatory variable. Bank-UTHCs
would already be subject to the more stringent bank
holding company regulation, and thus penalties
associated with the violation of the qualified-thriftlender test would have less impact on bank UTHCs
than nonbank ones.

Level of Risk
The second part of examining differences in
performance is to look at the level of risk of the
institutions according to their organizational structure. A lack of market data and a relatively short
time series for the accounting data make a direct
examination of risk problematic. Consequently, we
pursue an alternative strategy of examining differences in a number of risk proxies constructed from
thrift balance sheet data. The approach is to devise
an empirical model that explains organizational type
using proxy variables for different risk characteristics
and business strategies constructed from thrift balance sheet and income statement data. Equations
(3) and (4) specify the model.
To control for thrift-level structural effects that
may be related to holding company affiliation, such
as scale of operation and geographic presence, we
include, regressors for business volume (SIZE) and
proportion of fixed-to-total assets (FIXEDA). (See
table 1 for a description of the variables.) Because
measures of capital adequacy and liquidity have been
shown to be related to the probability that a bank will
be closed, we include regressors for the ratio of capital
to assets (TIER1CAP) and regulatory liquidity
(REGLIQR). These are also included to capture
regulatory restrictions on leverage and liquidity.

To capture the diversification of loan portfolio and
funding sources, the Herfindahl indicies LNHERF
and LIABHERF are included. Higher levels of these
variables suggest higher levels of balance sheet risk.
Diversification of revenue streams is proxied for by
the proportion of interest income to total income
(INTSHR)—we assume that interest income and
noninterest income are not highly correlated. Higher
levels of INTSHR would suggest higher variability
of revenues. Subordinated debt has been held up by
some as a potential source of market discipline for
depository institutions, so we included the proportion of it to total assets (SUBDEBTA). The ratio of
Federal Home Loan Bank advances to total assets
(FHLBADVA) is included to proxy for funding
strategy because this type of funding represents a
subsidized alternative to deposits for funding assets,
albeit to the extent the thrift has sufficient eligible
collateral in the form of mortgage assets. Two regressors are included to proxy for asset quality, the ratio
of insider loans to total loans (INSLNA) and the
ratio of classified assets to total earning assets
(CLASSASM). Both loans to insiders as a percent
of assets and measures of problem assets have been
shown to be positively related to bank closings. To
capture thrifts’ use of alternative business lines, we
include two proxies. The proportion of small business
loans to total assets (SBLA) represents thrifts’ new
small business and agricultural lending powers, and
the proportion of trust assets to total assets
(TRUSTA) represents fee-based activities. Finally,
because mortgage derivative securities represent
potential hedges against risks arising from mortgage
lending, the thrift’s proportion of these securities
(MDERA) is included, and because mortgage pool
securities represent an alternative to direct mortgage
holdings—an asset that is typically more liquid but
riskier than a traditional home mortgage loan—the
proportion of these (MPSA) is included as well.
Equation (3) seeks to explain differences between
UTHC and non-UTHC thrifts, and (4) examines
differences between bank-UTHC and nonbankUTHC thrifts. Using the logistic regression procedure in SAS, we estimate equations (3) and (4) over
the full sample and the UTHC sample, respectively.
The results appear in table 6.
(3) DUMUTit = φ0 + φ1SIZEit + φ2TIER1CAPit
+ φ 3LNHERFit + φ4LIABHERFit
+ φ5FIXEDAit + φ6SBLAit
+ φ 7INTSHRit + φ8SUBDEBTAit
+ φ 9FHLBADVAit + φ10INSLNAit
+ φ11REGLIQRit + φ12TRUSTAit
+ φ13MDERAit + φ14MPSAit
+ φ15CLASSASMit + µit

(4) DUMUDit = φ0 + φ1SIZEit + φ2TIER1CAPit
+ φ 3LNHERFit + φ4LIABHERFit
+ φ 5FIXEDAit + φ6SBLAit
+ φ 7INTSHRit +φ 8SUBDEBTAit
+ φ9FHLBADVAit + φ10INSLNAit
+ φ11REGLIQRit + φ 12TRUSTAit
+ φ13MDERAit + φ14MPSAit
+ φ15CLASSASMit + µ it
The results from equation 3 show important
differences between UTHC thrifts and others in the
proxies for balance sheet structure and risk. Twelve
of the 15 regressors and the intercept term are significant, a result that is not consistent with the
hypothesis that no differences in performance exist
between the two types of thrifts. UTHC thrifts are
significantly larger and hold significantly more tier 1
capital than thrifts in the non-UTHC sample. In
addition, the negative and significant coefficient on
LNHERF is consistent with UTHC thrifts having a
more diversified loan portfolio. None of these
results is consistent with the hypothesis that
UTHCs increase taxpayer risk.
It is also interesting to note that the coefficient
on LIABHERF is negative and significant, which is
consistent with UTHC thrifts having more diversified funding sources. However, to the extent that
the lower-liability Herfindahl results from a higher
dependence on Federal Home Loan Bank advances
–—as indicated by the positive and significant
coefficient on FHLBADVA—UTHC thrifts do not
necessarily rely less on funding sources subsidized by
implicit U.S. government guarantees.
UTHC thrifts appear to have a higher ratio of
fixed assets to total assets, which may indicate that
they maintain larger branching networks (FIXEDA
is positive and significant). Given that thrifts have
more liberal branching powers than banks, both
before and after the Reigle-Neal Act of 1994, this
result is not surprising. Moreover, the positive and
significant coefficient on SBLA is also consistent
with this explanation. That is, it is commonly held
that an office presence in the community is needed
to make the relationship-based small business loan.
Therefore, we would expect thrifts with higher
investments in fixed assets, presumably branches, to
also have higher levels of small business loans.
Thrifts in UTHCs have significantly lower ratios
of qualifying liquid assets12 to liabilities than nonUTHC thrifts. In other words, the negative and
significant coefficient on REGLIQR is consistent
with the higher levels of liquidity risk undertaken by
UTHCs. However, some caution should be used
in interpreting this result. First, to the extent that
UTHC thrifts have larger branch networks than
■

12 Liquid assets for purposes of regulatory liquidity must conform to
the eligibility criteria as expressed in OTS Regulations 566.1(g).

T A B L E

Logit Regression Results
Equation 4: Full Sample

Equation 5: UTHC Sample

Dependent Variable: DUMUT

Dependent Variable: DUMUD

Intercept
SIZE
TIER1CAP
LNHERF
LIABHERF
FIXEDA
SBLA
INTSHR
SUBDEBTA
FHLBADVA
INSLNA
REGLIQR
TRUSTA
MDERA
MPSA
CLASSASM

Parameter
estimate

ChiSquare

Prob >
Chi Square

–5.3877
0.6370
10.0303
–0.0002
–0.0003
7.7109
1.1632
–0.0384
7.7712
1.7655
–6.8246
–0.0035
0.1395
1.1045
–0.7827
–0.1634

218.4102
1457.7063
686.6470
407.9386
183.2916
26.0477
5.3536
0.6350
2.1190
21.3132
5.3927
6.7873
28.4154
19.5965
24.7668
0.1335

<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
0.0207
0.4255
0.1455
<0.0001
0.0202
0.0092
<0.0001
<0.0001
<0.0001
0.7149

“–2 LOG L”
AIC
Likelihood ratio
Prob > Chi Square
Number of observations

21,931.422
21,963.422
8,376.3025, DF: 15
<0.0001
25,655

Parameter
estimate
–0.3226
0.0327
–1.1386
–0.0001
0.0001
7.7869
1.2726
2.5124
–19.6167
1.7099
46.0876
–0.0005
–0.0134
–0.3963
2.3233
–2.9886

ChiSquare

Prob >
Chi Square

0.1428
0.5675
2.1779
36.1343
2.6080
2.8527
0.6252
40.8492
15.8421
6.1526
9.6365
0.0223
0.5456
0.7169
23.6342
11.5626

0.7055
0.4513
0.1400
<0.0001
0.1063
0.0912
0.4291
<0.0001
<0.0001
0.0131
0.0019
0.8812
0.4601
0.3972
<0.0001
0.0007

3,777.77
3,753.194
84.5255, DF: 15
<0.0001
7,122

SOURCES: Office of Thrift Supervision, Thrift Financial Reports; and author’s calculations.

non-UTHC thrifts, UTHC thrifts would need a
secondary reserve less. Second, because holding
companies are a source of liquidity for their subsidiary thrifts, we would also expect thrifts in
holding companies to need liquid assets as a
secondary reserve less.
The positive and significant sign on the proportion of trust assets (TRUSTA) indicates that UTHC
thrifts are more active in services that generate fee
income. Note, proxies for other fee-related business
lines were omitted from this regression because of
the high degree of colinearity between these variables and TRUSTA—that is, thrifts that engaged in
one fee-based activity tend to be engaged in the others. This result is consistent with FEESHR being
more important for UTHC thrifts as a determinant
of ROA in the equation (2) regression. Moreover,
increased reliance on nonlending business is likely to
reduce the variability of revenues and the risk of loss
to depositors and deposit insurance funds. This

result does not support the hypothesis of no performance-related differences.
Three other regression coefficients suggest that
UTHC thrifts may hold less risk than their nonUTHC counterparts. First, UTHC thrifts hold more
mortgage derivative contracts (MDERA is positive
and significant). To the extent that these are used to
hedge risks, such as prepayment risks, from the mortgage portfolio, UTHC thrifts would hold less risk
than their less-hedged counterparts. Second, nonUTHC thrifts have higher holdings of mortgage pool
securities than UTHC thrifts (MPSA is negative and
significant). Finally, UTHC thrifts appear less likely
to have risk exposure to insiders (INSLNA is negative
and significant). Thomson (1991, 1992) interprets
the ratio of inside loans to total loans as an indicator
of fraud and shows that INSLNA is related to the
probability of bank failure. Hence, the INSLNA
result could indicate potential problems in asset
quality for non-UTHC thrifts.

Finally, it is important to note that the data
could not discriminate between UTHC and nonUTHC thrifts in three areas: the share of revenues
represented by interest income (INTSHR), the
reliance on subordinated debt as a funding source
(SUBDEBTA), and the proportion of assets classified by regulators as problems (CLASSASM).
Overall, the results of the logit regression over
the full sample suggest that some important differences exist between UTHC thrifts and other thrifts.
However, none of the significant differences
between the two samples suggests that UTHC
thrifts engage in riskier activities or pose a greater
expected loss to the financial safety net than nonUTHC thrifts. In fact, just the opposite appears to
be true. UTHC thrifts are more diversified in terms
of their lending portfolios and sources of revenue.
In addition, UTHC thrifts are less leveraged and
appear to hedge more of their nondiversifiable risks
than do non-UTHC thrifts.
As noted above, the UTHC loophole would be
of most value to nonbank owners of UTHC thrifts.
Therefore, before we draw definitive conclusions,
we need to examine differences between UTHC
thrifts by ownership type. To do this, we estimate
equation (4) over the UTHC sample. Interestingly,
eight of the fifteen regression coefficients and the
intercept term are not significantly different from
zero—as opposed to three for equation (3). Moreover, all three of the insignificant regressors from
equation (3) are significant explanatory variables in
equation (4).
The regression results suggest that bank-UTHC
thrifts hold slightly more diversified loan portfolios
than nonbank-UTHC thrifts (LNHERF is negative
and significant). But bank-UTHC thrifts also hold
higher levels of mortgage pool securities (MPSA is
positive and significant), and this likely offsets
increased loan portfolio diversification and results in
a higher concentration of their mortgage-related
risks. In addition, nonbank-UTHC thrifts rely less
on interest income than bank-UTHC thrifts
(INTSHR is positive and significant). Overall, these
results suggest that the asset portfolios of nonbankUTHC thrifts are less exposed to mortgage-related
risks and that their revenue streams are more diversified than bank-UTHC thrifts. These results are not
inconsistent with the hypothesis that performance
differences trace to different real options held by
nonbank UTHCs.
Two important differences between our two samples of UTHC thrifts emerge from the funding side
of their balance sheets. First, bank-owned UTHCs
rely more heavily on subsidized Federal Home Loan
Bank advances (FHLBADVA is positive and significant). Second, nonbank-UTHCs appear to be more
influenced by market-based forms of discipline than
bank-owned UTHCs (SUBDEBTA is negative and

significant). To the extent that subordinated debt
serves as a source of market discipline, the positive
and significant coefficient (albeit at the 90 percent
level) on SUBDEBTA is not consistent with the
hypothesis of no performance-related differences.13
However, it is possible that this variable is picking
up a preference by banks to issue subordinated debt
at the holding-company level instead of the level of
the thrift subsidiary.
Finally, two conflicting effects are found for asset
quality. As noted before, previous research finds
INSLNA to be positively related to the probability of
failure. Hence, the positive and significant coefficient
on INSLNA for bank-UTHC thrifts indicates they
have potential asset quality problems. On the other
hand, the negative and significant coefficient on
CLASSASM suggests nonbank-UTHC thrifts have
higher levels of substandard assets on their books.
In general, the results from equations (3) and (4)
do not appear to be consistent with the hypothesis
that the UTHC loophole increases the risk to taxpayers by extending the safety-net subsidy to nondepository firms. We find no evidence from balancesheet proxies that the asset and liability decisions of
UTHC thrifts, particularly nonbank-UTHC thrifts,
pose higher risks. In fact, we tend to see that nonbank-UTHC thrifts rely less on mortgage-related
assets and more on nonsubsidized sources of funds
than their counterparts.

IV. Policy Conclusions

Debates over the merits of commingling banking
and commerce have focused on the potential efficiency gains associated with enhanced bundles of
real options afforded by cross-industry mergers.
These efficiency gains are weighed against potential
efficiency losses due to possible conflicts of interest,
extension of the financial safety net (and the attendant moral hazard incentives) to nonfinancial activities, and increased concentration of economic power.
The UTHC exemption represents the commingling of banking and commerce in the United
States. The degree of commingling is limited
because the commercial lending powers of thrift
institutions are restricted and because nonbanks
must satisfy the qualified-thrift-lender requirement
to own UTHCs. The Financial Services Modernization Act of 1999 effectively eliminated this exemption, preventing the formation of additional nonbank-owned UTHCs.
This paper examines the merits of the UTHC
exemption as a limited form of commingled banking
and commerce. It tests for differences in performance
of non-UTHC thrifts and two categories of UTHC
thrifts. While performance differences are found,
■

13 See Federal Reserve System (1999).

none suggests nonbank ownership of thrift holding
companies poses a risk to the federal financial safety
net, at least no greater a risk than a bank-owned
UTHC or an independent thrift poses. Furthermore,
we find evidence of performance differences that
suggest nonbank-UTHC thrifts hold a different set
of real options than unaffiliated thrifts and hence
have the potential for gains in economic efficiency.
Therefore, our results do not provide an economic
justification for the Financial Services Modernization
Act’s elimination of the UTHC exemption. Given the
limited scope of powers afforded to nonbank owners
of UTHCs, these results do not extend to proposals
for more general forms of universal banking.

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The Sources and Nature of
Long-Term Memory in
Aggregate Output
by Joseph G. Haubrich and Andrew W. Lo

Introduction

Questions about the persistence of economic shocks
currently occupy an important place in economics.
Much of the recent controversy has centered on “unit
roots,” determining whether aggregate time series are
better approximated by fluctuations around a deterministic trend or by a random walk plus a stationary
component. The mixed empirical results reflect the
general difficulties in measuring low-frequency
components. Persistence, however, has richer and
more relevant facets than the asymptotic behavior at
the heart of the unit root debate. In particular, fractionally differenced stochastic processes parsimoniously capture an important type of long-range
dependence midway between the quick decay of an
ARMA process and the infinite persistence of a
random walk. Fractional differencing allows something of a return to the classical NBER business cycle
program exemplified by Wesley Claire Mitchell, who
urged examinations of stylized facts at all frequencies.
Though useful in areas such as international
finance (Diebold, Husted, and Rush [1991] and
Baillie, Bollerslev, and Mikkelsen [1993]), fractionally
differenced processes have had less success in macroeconomics. For GDP at least, it is hard to estimate the
appropriate fractional parameter with any precision.

Joseph G. Haubrich is an economic
advisor at the Federal Reserve Bank
of Cleveland and Andrew W. Lo is the
Harris & Harris Group Professor at
the MIT Sloan School of Management. This research was partially
supported by the MIT Laboratory for
Financial Engineering, the National
Science Foundation (SES–8520054,
SES–8821583, SBR–9709976), the
Rodney L. White Fellowship at the
Wharton School (Lo), the John M.
Olin Fellowship at the NBER (Lo),
and the University of Pennsylvania
Research Foundation (Haubrich).
The authors thank Don Andrews,
Pierre Perron, Fallaw Sowell, and
seminar participants at Columbia
University, the NBER Summer Institute, the Penn Macro Lunch Group,
and the University of Rochester for
helpful comments, and Arzu Ilhan
for research assistance.

One promising technique (Geweke and Porter-Hudak
[1983] and Diebold and Rudebusch [1989]) also has
serious small-sample bias, which limits its usefulness
(Agiakloglou, Newbold, and Wohar [1993]).
Although both Deibold and Rudebusch (1989) and
Sowell (1992) find point estimates that suggest longterm dependence, they cannot reject either extreme of
finite-order ARMA or a random walk. The estimation
problems raise again the question posed by Christiano
and Eichenbaum (1990) about unit roots: do we
know and do we care? In this paper we provide an
affirmative answer to both questions.
Do we know? Applying the modified rescaled
range (R/S) statistic confronts the data with a test
at once both more precise and more robust than
previous estimation techniques. The R/S statistic
has shown its versatility and usefulness in a variety
of different contexts (Lo [1991] and Haubrich
[1993]). Operationally, we can determine what we
know about our tests for long-range dependence by
using Monte Carlo simulations of their size and
power. Not surprisingly, with typical macroeconomic sample sizes we cannot distinguish between
fractional exponents of 1.000 and 0.999, but we
can distinguish between exponents of 0 and 0.333.
Do we care? Persistence matters directly for making predictions and forecasts. It matters in a more

The Sources and Nature of
Long-Term Memory in
Aggregate Output
by Joseph G. Haubrich and Andrew W. Lo

Introduction

subtle way when making econometric inferences.
This is especially true for fractionally differenced
processes, which though stationary, are not “strongmixing” and so in a well-defined probabilistic sense
behave very differently than more standard ARMA
processes. Furthermore, because the optimal filter
depends on the characteristics of the underlying
process (Christiano and Fitzgerald [1999] and Baxter
and King [1999]), long-term persistence (sometimes
called long memory) matters even for estimates of
higher frequency objects such as business cycle
properties. While for some purposes persistence is less
important than how agents decompose shocks into
permanent and temporary components (Quah
[1990]), the results still depend on the persistence of
the time series decomposed and on the persistence of
the temporary component. In addition, the univariate approach, in contrast to Quah, has the advantage
that it does not assume agents observe more than the
econometrician.
Finally, beyond the value of purely statistical
explorations, a compelling reason to search for
fractional differencing comes from economic theory. We show how fractional differencing arises as a
natural consequence of aggregation in real business
cycle models. This holds out the promise of more
specific guidance in looking for long-range dependence, and conversely, adds another criterion to
judge and calibrate macroeconomic models.
This paper examines the stochastic properties of
aggregate output from the standpoint of fractionally integrated models. We introduce this type of
process in section I, reviewing its main properties,
advantages, and weaknesses. Section II develops a
simple macroeconomic model that exhibits longrange dependence. Section III employs the modified rescaled range statistic to search for long-range
dependence in the data. We conclude in section IV.

I. Review of Fractional Techniques
in Statistics
Macroeconomic time series look like neither a random walk nor white noise, suggesting that some
compromise or hybrid between white noise and its
integral may be useful. Such a concept has been given
content through the development of the fractional
calculus, that is, differentiation and integration to
noninteger orders.1 The fractional integral of order
between 0 and 1 may be viewed as a filter that
smooths white noise to a lesser degree than the ordinary integral; it yields a series that is rougher than a
random walk but smoother than white noise.
Granger and Joyeux (1980) and Hosking (1981)
develop the time-series implications of fractional differencing in discrete time. For expositional purposes
we review the more relevant properties in this section.

Perhaps the most intuitive exposition of
fractionally differenced time series is via their
infinite-order autoregressive and moving-average
representations. Let Xt satisfy:
(1) (1 – L)d Xt = ε t ,
where εt is white noise, d is the degree of differencing, and L denotes the lag operator. If d = 0, then
Xt is white noise, whereas Xt is a random walk if d =
1. However, as Granger and Joyeux (1980) and
Hosking (1981) have shown, d need not be an integer. Using the binomial theorem, the AR representation of Xt becomes:
∞

(2) A(L) Xt = ∑ Ak
k=0
∞

= ∑ Ak Xt–k = ε t ,

(3) Lk Xt

k=0

()

d
where Ak ≡ (–1)k k . The AR coefficients are often
re-expressed more directly in terms of the gamma
function:

Γ (k – d )
(4) Ak ≡ (–1)k d =
.
k
Γ (– d )Γ (k + 1)

()

By manipulating equation (1) mechanically, Xt may
also be viewed as an infinite-order MA process
since:
(5) Xt = (1 – L)–d εt = B(L) ε t
Bk =

Γ (k + d ) .
Γ (d )Γ (k + 1)

The particular time-series properties of Xt
depend intimately on the value of the differencing
parameter d. For example, Granger and Joyeux
(1980) and Hosking (1981) show that when d is
less than 1 , Xt is stationary; when d is greater than
1

– 2 , Xt is invertible. Although the specification in
equation (1) is a fractional integral of pure white
noise, the extension to fractional ARIMA models
is clear.
The AR and MA representations of fractionally
differenced time series have many applications and
illustrate the central properties of fractional
processes, particularly long-range dependence.
The MA coefficients, Bk, give the effect of a shock
k periods ahead and indicate the extent to which
current levels of the process depend on past values.
■

1 The idea of fractional differentiation is an old one, dating back to
an oblique reference by Leibniz in 1695, but the subject lay dormant until
the nineteenth century when Abel, Liouville, and Riemann developed it
more fully. Extensive applications have only arisen in this century; see,
for example, Oldham and Spanier (1974), who also present an extensive
historical discussion. Kolmogorov (1940) was apparently the first to
notice its applications in probability and statistics.

Autocorrelation
1.0
Fractionally differenced series X t = (1 – L) –d e t , d = 0.475
0.8

0.6

0.4

0.2
AR(1) with coefficient 0.90
0
30

60
Lag

120

SOURCE: Authors’ calculations.

(7) f ( ω) =

σ2
1
1 – z2d 2π

z ≡ e–iω,

where

σ 2 ≡ E [εt2] .

How fast this dependence decays furnishes valuable
information about the process. Using Stirling’s
approximation, we have:

The identity |1– z|2 = 2[1 – cos(ω )] implies that for
small ω we have:

d–1
(6) Bk ≈ k
Γ (d)

(8) f (ω ) = c ω –2d,

for large k. Comparing this with the decay of an
AR(1) process highlights a central feature of fractional processes: they decay hyperbolically, at rate
kd–1, rather than at the exponential rate of ρ k for
an AR(1). For example, compare in figure 1 the
autocorrelation function of the fractionally differenced series (1 – L)–0.475Xt = εt with the AR(1)
Xt = 0.9Xt–1 + εt. Although they both have firstorder autocorrelations of 0.90, the AR(1)’s autocorrelation function decays much more rapidly.
These representations also show how standard
econometric methods can fail to detect fractional
processes, necessitating the methods of section III.
Although a high order ARMA process can mimic
the hyperbolic decay of a fractionally differenced
series in finite samples, the large number of parameters required would give the estimation a poor
rating from the usual Akaike or Schwartz criteria.
An explicitly fractional process, however, captures
that pattern with a single parameter, d. Granger
and Joyeux (1980) and Geweke and Porter-Hudak
(1983) provide empirical support for this by showing that fractional models often outpredict fitted
ARMA models.
For large k, the value of Bk measures the
response of Xt+k to an innovation at time t, a
natural metric for persistence. From equation 6,

2
c≡ σ .
2π

This encompasses the two extremes of a white
noise (or a finite ARMA) process and a random
walk. For white noise, d = 0, and f ( ω ) = c, while
for a random walk, d = 1, and the spectrum is
inversely proportional to ω 2. A class of processes of
current interest in the physics literature, called 1/f
noise, matches fractionally integrated noise with
d = 21 .

■ 2 There has been some confusion in the literature on this point. Geweke
and Porter-Hudak (1983) argue that limk ∞ Bk > 0, which, in their terminology, is expressed as C (1) > 0. They correctly point out that Granger and Joyeux
(1980) have made an error, but then incorrectly claim that C (1) = 1/ Γ (d ).
If our equation 6 is correct, then it is apparent that C (1) = 0 (which agrees
with Granger [1980] and Hosking [1981]). Therefore, the focus of the conflict
lies in the approximation of the ratio Γ (k + d )/Γ (k + 1) for large k. We have
used Stirling’s approximation. However, a more elegant derivation follows
from the functional analytic definition of the gamma function as the solution
to the following recursive relation (see, for example, Iyanaga and Kawada
[1980, section 179.A]):
←

it is immediate that for 0 < d < 1, limk ∞ Bk = 0,
and asymptotically there is no persistence in a
fractionally differenced series, even though the
autocorrelations die out very slowly.2
This holds true not only for d < 21 (the stationary
case), but also for 21 < d < 1 (the nonstationary case).
From these calculations, it is apparent that the
long-run dependence of fractional processes relates
to the slow decay of the autocorrelations, not to
any permanent effect. This distinction is important; an IMA(1,1) can have small but positive
persistence, but the coefficients will never mimic
the slow decay of a fractional process.
The spectrum, or spectral density (denoted
f (ω )) of a fractionally differenced process reflects
these properties. It exhibits a peak at 0 (unlike the
flat spectrum of an ARMA process), but one not as
sharp as the random walk’s. Given Xt = (1 – L)–d εt,
the series is clearly the output of a linear system
with a white-noise input, so that the spectrum of
Xt is:
←

F I G U R E
Autocorrelation Functions
pj for (1 – L)–.475 εt

Γ (x + 1) = x Γ (x)
and the conditions:

Γ (1) = 1

lim Γ(x + n) = 1.
n →∞ nx
Γ (n )

II. A Simple Macroeconomic Model with
Long-Term Dependence

Economic insight requires more than a consensus
on the Wold representation of GDP; it demands a
falsifiable model based on the tastes and technology
of the actual economy. As Wesley Claire Mitchell
(1927, p. 230) wrote, “We stand to learn more
about economic oscillations at large and about
business cycles in particular, if we approach the
problem of trends as theorists, than if we confine
ourselves to strictly empirical work.”
Thus, before testing for long-run dependence,
we develop a simple model where aggregate output
exhibits long-run dependence. This model presents
one reason that macroeconomic data might show
the particular stochastic structure for which we test.
It also shows that models can restrict the fractional
differencing properties of time series, so that our
test holds promise for distinguishing between
competing theories. Furthermore, the maximizing
model presented below connects long-range dependence to central economic concepts of productivity,
aggregation, and the limits of the representative
agent paradigm.

A Simple Real Model
One plausible mechanism for generating long-run
dependence in output, which we will mention here
and not pursue, is that production shocks themselves
follow a fractionally integrated process. This explanation for persistence follows that used by Kydland and
Prescott (1982). In general, such an approach begs
the question, but in the present case evidence from
geophysical and meteorological records suggests that
many economically important shocks have long-run
correlation properties. Mandelbrot and Wallis
(1969), for instance, find long-run dependence in
rainfall, riverflows, earthquakes, and weather (measured by tree rings and sediment deposits).3
A more satisfactory model explains the time-series
properties of data by producing them despite whitenoise shocks. This section develops such a model
with long-run dependence, using a linear quadratic
version of the real business cycle model of Long and
Plosser (1983) and aggregation results due to
Granger (1980).4 In our multisector model, the
output of each industry (or island) will follow an
AR(1) process. Aggregate output with N sectors will
not follow an AR(1) but rather an ARMA(N, N–1).
This makes dynamics with even a moderate number
of sectors unmanageable. Under fairly general conditions, however, a simple fractional process will
closely approximate the true ARMA specification.
Consider a model economy with many goods
and a representative agent who chooses a production and consumption plan. The infinitely lived

agent inhabits a linear quadratic version of the real
business cycle model. The agent has quadratic
utility, with a lifetime utility function of U =
∑ β tu (Ct ), where Ct is an N × 1 vector denoting
period-t consumption of each of the N goods in
our economy. Each period’s utility function, u(Ct ),
is given by
(9) u(Ct ) = C't ι –

1
2

C't BCt ,

where ι is an N × 1 vector of ones. In anticipation
of the aggregation considered later, we assume B to
be diagonal so that C't BCt = ∑bii Cit2. The agents
face a resource constraint: total output Yt may be
either consumed or saved, thus:
(10) Ct + St ι = Yt ,
where the i,j-th entry Sijt of the N × N matrix St
denotes the quantity of good j invested in process i
at time t, and it is assumed that any good Yjt may
be consumed or invested. Output is determined by
a random linear technology:
(11) Yt = ASt + εt ,
where A is the matrix of input–output coefficients
aij, and ε t is a (vector) random production shock,
whose value is realized at the beginning of period
t + 1. To focus on long-range dependence we
restrict A’s form. Thus, each sector uses only its
own output as input, yielding a diagonal A matrix
and allowing us to simplify notation by defining
ai ≡ aii. This might occur, for example, with a
number of distinct islands producing different
goods. To further simplify the problem, all commodities are perishable and capital depreciates at
a rate of 100 percent.
In this case, the dynamic programming problem
of solving for optimal consumption and investment
policies reduces to the familiar optimal stochastic
linear regulator problem (see Sargent [1987], section
1.8, for an excellent exposition). Given the simple
diagonal form of the A matrix, which corresponds to
an assumption that each sector uses only its own
output as input, the problem simplifies even further.

■ 3 For a related mechanism creating fractional intergration by
aggregating shocks of differing duration, see Parke (1999). Abadir
and Talmain (undated) use aggregation over heterogeneous firms in a
setting of monopolistic competition.
■ 4 Dupor (1999) is skeptical of the ability of multisector models
to match aggregate time series data but does not consider long-range
dependence.

The chosen quantities of consumption and
investment/savings have the following closedform solutions:
bi

β qi ai – 1
bi – 2 βPi ai2

(12) Sit =

Y +
bi – 2 βPi ai2 it

(13) Cit =

β qi ai – 1
2 βPi ai2
Y +
,
2 β Pi ai2 – bi
2 β Pi ai2 – bi it

where:
(14) Pi ≡ bi

[

ai –

(1 + 4 β )ai2 – 4
4β ai

]

and qi are fixed constants given by the matrix
Riccati equation that results from the recursive
definition of the value function.
The simple form of the optimal consumption and
investment decision rules comes from the quadratic
preferences and the linear production function. Two
qualitative features bear emphasizing. First, higher
output today will increase both current consumption
and current investment, thus increasing future output. Even with 100 percent depreciation, no durable
commodities, and independent and identically distributed (i.i.d.) production shocks, the time-to-build
feature of investment induces serial correlation.
Second, the optimal choices do not depend on the
uncertainty present. This certainty-equivalence feature is an artifact of the linear-quadratic combination.
The time series of output can now be calculated
from the production function, equation (11), and
the decision rule, equation (12). Quantity
dynamics then come from the difference equation:
(15) Yit+1 =

ai bi

bi – 2 βPi ai2

Yit + Ki + εit+1

or
(16) Yit+1 = αiYit + Ki + εit+1,
where αi is a function of the utility parameters and
of ai , the input-output coefficient of the industry,
and Ki is some fixed constant. The key qualitative
property of quantity dynamics summarized by
equation (16) is that output, Yit , follows an AR(1)
process. Higher output today implies higher output
in the future. That effect dies off at a rate that
depends on the parameter α i , which in turn depends
on the underlying preferences and technology.
The simple output dynamics for a single industry or island neither mimics business cycles nor
exhibits long-run dependence. However, aggregate
output, the sum across all sectors, will show such
dependence, which we demonstrate here by applying the aggregation results of Granger (1980,1988).

It is well-known that the sum of two series, Xt
and Yt , each AR(1) with independent error, is an
ARMA(2,1) process. Simple induction then implies
that the sum of N independent AR(1) processes with
distinct parameters has an ARMA(N, N–1) representation. With over six million registered businesses in
America, the dynamics can be incredibly rich, and
the number of parameters unmanageably huge. The
common response to this problem is to pretend that
many different firms (or islands) have the same
AR(1) representation for output, which reduces the
dimension of aggregate ARMA process. This “cancelling of roots” requires identical autoregressive
parameters. An alternative approach, due to Granger,
reduces the scope of the problem by showing that
the ARMA process approximates a fractionally integrated process, and thus summarizes the many
ARMA parameters in a parsimonious manner.
Though we consider the case of independent sectors,
dependence is easily handled.
Consider the case of N sectors, with the productivity shock for each serially uncorrelated and independent across sectors. Furthermore, let the sectors
differ according to the productivity coefficient, ai.
This implies differences in α i, the autoregressive
parameter for sector i’s output, Yit. One of our key
results is that under some distributional assumptions
on the α i’s, aggregate output, Yta, follows a fractionally integrated process, where:
N

(17) Yta ≡ ∑ Yit .
i=1

To show this, we approach the problem from
the frequency domain and apply spectral methods,
which often simplify problems of aggregation.5
Let f (ω ) denote the spectrum (spectral density
function) of a random variable, and let z = e–iω.
From the definition of the spectrum as the Fourier
transform of the autocovariance function, the
spectrum of Yit is:
(18) fi (ω ) =

σi2
1
.
|1 – αi z|2 2 π

Similarly, independence implies that the spectrum
of Yta is
N

(19) fi (ω ) = ∑ fi (ω ).
i=1

■

5 See Theil (1954).

The αi’s measure an industry’s average output
for given input. This attribute of the production
function can be thought of as a drawing from
nature, as can the variance of the productivity
shocks, εit, for each sector. Thus, it makes sense to
think of the ai’s as independently drawn from a
distribution G(a) and the αi’s as drawn from F(α ).
Provided that the εit shocks are independent of the
distribution of αi’s, the spectral density of the sum
can be written as:
(20) fi (ω) = N E [ σ 2]
2π

•

∫ |1 – 1α z | dF (α ).
2

If the distribution F(α ) is discrete, so that it
takes on m(< N) values, Yta will be an ARMA
(m,m–1) process. A more general distribution leads
to a process no finite ARMA model can represent.
To further specify the process, take a particular
distribution for F, in this case a variant of the beta
distribution.6 In particular, let α 2 have a beta
distribution β (p,q), which yields the following
density function for α :
(21) dF (α )

{

2 α 2p–1(1 – α2)q–1dα,
= β (p,q)
0,

0 ≤ α ≤ 1;
otherwise.

with p,q > 0.7 Obtaining the Wold representation
of the resulting process requires a little more work.
First note that:
(22) 1/|1 – α z|2
=

1 + α z + 1 + αz
1
_ ,
2 (1 – α 2) 1 – α z 1 – α z

[

]

where z denotes the complex conjugate of z, and the
terms in parentheses can be further expanded by
long division. Substituting this expansion and the
beta distribution, equation (21), into the expression
for the spectrum and simplifying (using the relation
_
z + z = 2cos( ω)) yields:
(23) f (ω ) =

∫

1
0

simplifies to β (p + k /2, q – 1)/ β (p,q). Dividing by
the variance gives the autocorrelation coefficients,
which reduce to
(25) ρ (k) =

Γ (p + q – 1) Γ ( p + 2k )
,
Γ ( p) Γ ( p + 2k + q + 1)

which, again using the result from Stirling’s approximation, Γ (a + k)/ Γ (b + k) ≈ ka–b, is proportional
(for large lags) to k1–q. Thus aggregate output Yta
follows a fractionally integrated process of order
d = 1 – 2q . Furthermore, as an approximation for
long lags, this does not necessarily rule out interesting correlations at higher frequencies, such as those
of the business cycle. Similarly, co-movements can
arise, as the fractionally integrated income process
may induce fractional integration in other observed
time series. Two additional points are worth
emphasizing. First, the beta distribution need not
be over (0,1) to obtain these results, only over (a,1).
Second, it is indeed possible to vary the ai’s so that
α i has a beta distribution.
In principle, all parameters of the model may be
estimated, from the distribution of production
function parameters to the variance of output
shocks. Empirical estimates of production function
parameters (such as those in Jorgenson, Gollop,
and Fraumeni [1987]) reveal a large dispersion,
suggesting the plausibility and significance of the
simple model presented in this section.
Although the original motivation of our real
business cycle model was to illustrate how longrange dependence could arise naturally in an economic system, our results have broader implications for general macroeconomic modeling. They
show that moving to a multiple-sector real business
cycle model introduces not unmanageable complexity, but qualitatively new behavior that can be
quite manageable. Our findings also show that calibrations aimed at matching only a few first and
second moments can similarly hide major differences between models and the data, missing longrange dependence properties. While widening the
theoretical horizons of the paradigm, they therefore
also widen the potential testing of such theories.

∞
[2 + 2 ∑ α kcos(k ω )].
k=1

Then the coefficient of cos(kω) is
(24)

∫

1
0

2α k α 2p–1(a – α 2)q–1dα .
β (p,q)

Since the spectral density is the Fourier transform
of the autocovariance function, equation (24) is the
k-th autocovariance of Yta. Furthermore, because the
integral defines a beta function, equation (24)

■ 6 Granger (1980) conjectures that this particular distribution is
not essential.
■

7 For a discussion of the variety of shapes the beta distribution takes
as p and q vary, see Johnson and Kotz (1970).

III. R/S Analysis of Real Output

The results of section II show that simple aggregation may be one source of long-range dependence
in the business cycle. In this section we employ a
method for detecting long memory and apply it
to real GDP. The technique is based on a simple
generalization of a statistic first proposed by the
English hydrologist Harold Edwin Hurst (1951),
which has subsequently been refined by Mandelbrot (1972, 1975) and others.8
Our generalization of Mandelbrot’s statistic
(called the “rescaled range” or “range over standard
deviation” or R/S) enables us to distinguish
between short- and long-run dependence, in a
sense to be made precise below.
We define our notions of short and long memory
and present the test statistic below. Then we present
the empirical results for real GDP; we find longrange dependence in log-linearly detrended output,
but considerably less dependence in the growth rates.
To interpret these results, we perform several Monte
Carlo experiments under two null and two alternative hypotheses and report these results.

The Rescaled Range Statistic
We test for fractional differencing using Lo’s
modification of the modified rescaled range (R/S)
statistic. In particular, we define short-range dependence as Rosenblatt’s (1956) concept of “strongmixing,” a measure of the decline in statistical
dependence of two events separated by successively
longer spans of time. Heuristically, a time series is
strong-mixing if the maximal dependence between
any two events becomes trivial as more time elapses
between them. By controlling the rate at which the
dependence between future events and those of the
distant past declines, it is possible to extend the
usual laws of large numbers and central limit theorems to dependent sequences of random variables.
Such mixing conditions have been used extensively
by White (1982), White and Domowitz (1984),
and Phillips (1987) for example, to relax the
assumptions that ensure consistency and asymptotic normality of various econometric estimators.
We adopt this notion of short-range dependence as
part of our null hypothesis. As Phillips (1987)
observes, these conditions are satisfied by a great
many stochastic processes, including all Gaussian
finite-order stationary ARMA models. Moreover,
the inclusion of a moment condition also allows
for heterogeneously distributed sequences (such as
those exhibiting heteroscedasticity), an especially
important extension in view of the nonstationarities
of real GDP.

Fractionally differenced models, however, possess autocorrelation functions that decay at much
slower rates than those of weakly dependent
processes and violate the conditions of strong
mixing. More formally, let Xt denote the first
difference of log-GDP; we assume that:
(26) Xt = µ + ε t ,
where µ is an arbitrary but fixed parameter. For the
null hypothesis H, assume that the sequence of
disturbances, { ε t }, satisfies the following conditions:
(A1) E [εt ] = 0 for all t.
(A2) supt E [| εt | β ] < ∞ for some β > 2.
(A3) σ 2 = limn→∞ E

[ n1 ( ∑ n ε ) ] exists and σ 2 > 0.
2

j=1 j

(A4) { εt } is strong-mixing with mixing coefficients
α k that satisfy:9
∞

∑ αk1 – β2 < ∞ .

k=1

Condition (A1) is standard. Conditions
(A2)–(A4) allow dependence and heteroskedasticity, but prevent them from being too dominant.
Thus, short-range dependent processes such as
finite-order ARMA models are included in this null
hypothesis, as are models with conditional heteroskedasticity. Unlike the statistic used by Mandelbrot, the modified R/S statistic is robust to shortrange dependence. A more detailed discussion of
these conditions may be found in Phillips (1987)
and Lo (1991).
To construct the modified R/S
_ statistic, consider
a sample X1, X2, …, Xn and let Xn denote the
sample mean 1 ∑ j Xj . Then the modified R/S
n
statistic, which we shall call Qn, is given by:
(27) Qn ≡

_
k
1
Max1 ≤ k ≤n ∑ (Xj – Xn )
(q)
σn
j=1

[

_
k
– Min1≤ k ≤ n ∑ (Xj – Xn ) ,
j=1

■

]

8 See Mandelbrot and Taqqu (1979) for further references.

9 Let { ε t (ω )} be a stochastic process on the probability space
(Ω, F, P ) and define:

■

α (A,B ) ≡ sup

{A ∈ A, B ∈ B}

|P (A ∩ B ) – P (A) P (B )|

A ⊂ F, B ⊂ F

The quantity α (A,B ) is a measure of the dependence between the two
σ -fields A and B in F. Denote by B st the Borel σ -field generated by
{ε s ( ω ),…, εt (ω)}, i.e., B st ≡ σ (ε s (ω ),…, ε t (ω)) ⊂ F. Define the
coefficients α k as:

α k ≡ sup α (B–∞j , Bj+k∞ )
j

Then {ε t (ω )} is said to be strong-mixing if lim k →∞ α k = 0. For further
details, see Rosenblatt (1956), White (1984), and the papers in Eberlein
and Taqqu (1986).

T A B L E

where

1(a)

Fractiles of the Distribution Fv (v )
P (V < v)
v
P (V < v)
v

.005 .025 .050
0.721 0.809 0.861
.543
π
2

.600 .700
1.294 1.374

.100
0.927

.200 .300
1.018 1.090

.400 .500
1.157 1.223

.800
1.473

.900 .950
1.620 1.474

.975 .995
1.862 2.098

q
_
n
∧
(28) σ n2 (q) ≡ n1 ∑ (Xj – Xn )2 + n2 ∑ ω j(q)
j=1
j=1
_
_
n
∑ (Xi – Xn ) (Xi–j – Xn )

{

ω j (q) ≡ 1 –

T A B L E 1(b)
Symmetric Confidence Intervals
about the Mean
π+
2

γ)

.001
.050
.100
.500

0.748
0.519
0.432
0.185

SOURCE: Authors’ calculations.

F I G U R E 2
Distribution and Density Function of
the Range V of a Brownian Bridge
2.0

Brownian bridge
Normal
1.5

1.0

0.5

0
0

SOURCE: Authors’ calculations.

∧

j=1

j
q+1

q<n

∧

(29) = σ x2 + 2 ∑ ω j (q) γ j

SOURCE: Authors’ calculations.

P ( π – γ <V<

}

i=j+1

and σ x2 and γ j are the usual sample variance and
autocovariance estimators of X. Qn is the range of
partial sums of deviations of Xj from its mean,
normalized by an estimator of the partial sum’s
standard deviation divided by n. The estimator
∧
σ n (q) involves not only sums of squared deviations
of Xj , but also its weighted autocovariances up to
lag q; the weights ωj (q) are those suggested by
Newey and West (1987), and always yield a positive
∧
estimator σ n2 (q).10
Intuitively, the numerator in equation (27)
measures the memory in the process via the partial
sums. White noise does not stay long above the
mean: positive values are soon offset by negative
values. A random walk will stay above or below 0 for
a long time, and the partial sums (positive or negative) will grow quickly, making the range large. Fractional processes fall in between. Mandelbrot (1972)
refers to their behavior as the “Joseph effect”—seven
fat and seven lean years. The denominator normalizes not only by the variance but also by a weighted
average of the autocovariances. This innovation over
Hurst’s (1951) R/S statistic provides the robustness
to short-range dependence.
The choice of the truncation lag, q, is a delicate
matter. Although q must increase with (albeit at a
slower rate than) the sample size, Monte Carlo
evidence suggests that when q becomes large relative
to the number of observations, asymptotic approximations may fail dramatically.11 However, q cannot
be too small or the effects of higher-order autocorrelations may not be captured. The choice of q is
clearly an empirical issue and must therefore be
chosen with some consideration of the data at hand.
The partial sums of white noise constitute a
random walk, so Qn(q) grows without bound as n
increases. A further normalization makes the
statistic easier to work with and interpret:
(30) Vn(q) ≡

Qn(q)
n

∧

10 σ n2 (q) is also an estimator of the special density function of Xt
at frequency zero, using a Bartlett window.

■
■

11 See, for example, Lo and MacKinlay (1989).

T A B L E

R/S Analysis of Real GDP
∼

Series

Vn(1)

Vn(2)

∼
US Log First-Difference
∼
Percentage Bias of Vn

1.092

0.973
(6.0)

0.933
(8.2)

S Log Detrended ∼
Percentage Bias of Vn

2.374

1.741
(16.8)

CAN Log First-Difference
∼
Percentage Bias of Vn

1.254

CAN Log Detrended
∼
Percentage Bias of Vn

Vn(3)

Vn(4)

Vn(5)

Vn(6)

Vn(7)

Vn(8)

0.934
(8.1)

0.971
(6.1)

1.032
(2.9)

1.082
(0.5)

1.115
(–1.0)

1.139
(–2.1)

1.479
(26.7)

1.337
(33.2)

1.252
(37.7)

1.198
(40.8)

1.160
(43.1)

1.134
(44.7)

1.116
(45.8)

1.116
(6.0)

1.042
(9.7)

1.018
(11.0)

1.024
(10.7)

1.045
(9.5)

1.073
(8.1)

1.096
(7.0)

1.132
(5.2)

3.410

2.458
(17.8)

2.048
(29.1)

1.813
(37.2)

1.660
(43.3)

1.552
(48.2)

1.472
(52.2)

1.410
(55.5)

1.360
(58.3)

GER Log First-Difference
∼
Percentage Bias of Vn

1.357

1.185
(7.0)

1.159
(8.2)

1.158
(8.3)

1.176
(7.4)

1.203
(6.2)

1.235
(4.8)

1.278
(3.1)

1.325
(1.2)

GER Log Detrended
∼
Percentage Bias of Vn

4.241

3.052
(17.9)

2.539
(29.2)

2.242
(37.5)

2.044
(44.0)

1.903
(49.3)

1.796
(53.7)

1.712
(57.4)

1.643
(60.6)

UK Log First-Difference
∼
Percentage Bias of Vn

1.051

0.907
(7.7)

0.851
(11.1)

0.853
(11.0)

0.887
(8.9)

0.920
(6.9)

0.961
(4.6)

0.993
(2.9)

1.031
(1.0)

UK Log Detrended
∼
Percentage Bias of Vn

4.637

3.327
(18.1)

2.760
(29.6)

2.431
(38.1)

2.213
(44.8)

2.055
(50.2)

1.933
(54.9)

1.837
(58.9)

1.757
(62.5)

GDP Log First-Difference
∼
Percentage Bias of Vn

1.391

1.201
(7.6)

1.107
(12.1)

1.066
(14.3)

1.057
(14.7)

1.069
(14.1)

1.087
(13.1)

1.109
(12.0)

1.132
(10.9)

GDP Log Detrended
∼
Percentage Bias of Vn

5.612

3.999
(18.5)

3.290
(30.6)

2.873
(39.8)

2.592
(47.1)

2.387
(53.3)

2.229
(58.7)

2.103
(63.4)

2.000
(67.5)

NOTE: US, CAN, GER, and UK refer to the annual Maddison series for those countries 1870–1994. GDP refers to the quarterly U.S. series 1947:QI–1999:QI.
∼
The classical rescaled range Vn and the modified rescaled range Vn(q) are reported. Under a null hypothesis of short-range dependence, the limiting distribution of
Vn(q) is the range of a Brownian bridge, which has a mean of π /2. Fractiles are given in table 1; the 95 percent confidence interval with equal probabilities in
∼
both tails is [0.809, 1.862]. Entries in the %-Bias rows are computed as [Vn /Vn(q)1/2 – 1] • 100 and are estimates of the bias of the classical R/S statistic in the
presence of short-term dependence.
SOURCE: Authors’ calculations.

The limiting distribution of Vn(q) is derived by
Lo (1991), and its most commonly-used values are
reported in tables 1(a) and 1(b). Table 1(a) reports
the fractiles of the limiting distribution while table
1(b) reports the symmetric confidence intervals
about the mean. The moments of the limiting
distribution are also easily computed using fV , the
density of the random variable V to which Vn(q)
converges in distribution; it is straightforward to
2
show that E[V ] = 2π , and E[V 2] = 6π ; thus the
mean and standard deviation of V are approximately 1.25 and 0.27, respectively. The distribution
and density functions are plotted in figure 2.
Observe that the distribution is positively skewed,
and most of its mass falls between 34 and 2.

Empirical Results for Real Output
We apply our test to several time series of real
output: quarterly U.S. postwar real GDP from
1947:QI to 1999:QI, and the annual Maddison
(1995) OECD series for the United States, Canada,
Germany, and the United Kingdom from 1870 to
1994. These results are reported in table 2. Entries
in the first numerical column are estimates of the
∼
classical rescaled range, Vn , which is not robust to
short-range dependence. The next eight columns
are estimates of the modified rescaled range Vn(q)
for values of q from 1 to 8. Recall that q is the truncation lag of the estimator of the spectral density at
frequency zero. Reported in parentheses below the

entries for Vn(q) is an estimate of the percentage
∼
bias of the statistic Vn, which is computed as 100 •
∼
[(Vn/Vn(q)) – 1 ].
The first row of numerical entries in table 2
indicate that the null hypothesis of short-range
dependence for the first-difference of log-GDP
cannot be rejected for any value of q. The classical
rescaled range statistic also supports the null
hypothesis, as do the results for the Maddison
series. On the other hand, when we log-linearly
detrend real GDP, the results differ considerably.
Looking at the results for the annual data in table 2
shows that short-range dependence may be rejected
for log-linearly detrended output using the classical
statistic for the United States and with q values
from 1 to 2 for Canada, 1 to 5 for Germany, and
1 to 6 for the United Kingdom. For quarterly U.S.
data, short-term dependence is rejected for all q up
to 8. That the rejections are weaker for larger q is
not surprising since additional noise arises from
estimating higher-order autocorrelations.
The values reported in table 2 are qualitatively
consistent with other empirical investigations of
fractional processes in GNP, such as Diebold and
Rudebusch (1989) and Sowell (1992). For firstdifferences, the R/S statistic falls below the mean,
suggesting a negative fractional exponent, or in level
terms, an exponent between 0 and 1. Furthermore,
though earlier papers produce point estimates, they
do not lead to a rejection of the hypothesis of shortterm dependence because of imprecise estimates.
For example, the 2 standard deviation error bounds
for two point estimates of Diebold and Rudebusch
(1989), d = 0.9 and 0.52, are [0.42, 1.38] and
[–0.06, 1.10], respectively.
Taken together, these results confirm the unit
root findings of Campbell and Mankiw (1987),
Nelson and Plosser (1982), Perron and Phillips
(1987), and Stock and Watson (1986). That there
are more significant autocorrelations in log-linearly
detrended GDP is precisely the spurious periodicity
suggested by Nelson and Kang (1981). Moreover,
the trend plus stationary noise model of GDP is
not contained in our null hypothesis; hence our
failure to reject the null hypothesis is also consistent
with the unit root model.12 To see this, observe
that if log-GDP yt were trend stationary, that is,
(31) yt = α + β t + ηt
where ηt is stationary white noise, then its firstdifference, Xt , is simply Xt = β + εt , where
εt ≡ ηt – ηt–1. But this innovations process violates
our assumption (A3) and is therefore not contained
in our null hypothesis.

Sowell (1992) has used estimates of d to argue
that the trend-stationary model is correct. Following the lead of Nelson and Plosser (1982), Sowell
checks if the d parameter for the first-differenced
series is close to 0 as the unit root specification
suggests, or close to –1 as the trend-stationary
specification suggests. His estimate of d is in the
general range of –0.6 to 0.2, providing some evidence that the trend-stationary interpretation is
correct. Even in his case though, the standard errors
tend to be rather large, on the order of 0.3.
Although our procedure yields no point estimate
of d, our results do seem to rule out the trendstationary case.
To conclude that the data support the null
hypothesis because our statistic fails to reject it is,
of course, premature since the size and power of
our test in finite samples is yet to be determined.
We perform illustrative Monte Carlo experiments
and report the results in the next section.

The Size and Power of the Test
To evaluate the size and power of our test in finite
samples, we perform several illustrative Monte
Carlo experiments for a sample size of 208 observations, corresponding to the number of quarterly
observations of real GDP growth from 1947:QII to
1999:QI.13 We simulate two null hypotheses: independently and identically distributed increments,
and increments that follow an ARMA(2,2) process.
Under the i.i.d. null hypothesis, we fix the mean
and standard deviation of our random deviates to
match the sample mean and standard deviation of
our quarterly data set: 8.221 x 10–3 and 1.0477 x
10–2, respectively. To choose parameter values for
the ARMA(2,2) simulation, we estimate the model:
(32) (1 – φ1L – φ 2 L2 )yt
2
ε t ∼ WN(0, σε )

= µ + (1 + θ 1L + θ 2 L2) εt

using nonlinear least squares. The parameter
estimates are (with standard errors in parentheses):
∧

φ1 =
∧

φ2 =
∧

µ =

∧

1.3423
(0.1678)

θ1=

–0.7065
(0.1198)

θ2=

0.0082
(0.0008)

σε =

∧

1.0554
(0.1839)
–0.5200
(0.1377)
0.0097

■ 12 Of course, this may be the result of low power against stationary but nearintegrated processes, and it must be addressed by Monte Carlo experiments.
■ 13 Simulations were performed on a DEC Alphaserver 2100 4/275 using a
Gauss random number generator; each experiment comprised 10,000 replications.

T A B L E 3
Finite Sample Distribution of the Modified R/S Statistic under IID and
ARMA (2,2) Null Hypotheses for the First-Difference of Real Log GNP
q
IID
0
1.5
1
2
3
4
5
6
7
8

Min.

Max.

Mean

S.D.

Size 1%–Test

Size 5%–Test

Size 10%–Test

0.527
0.525
0.548
0.548
0.555
0.572
0.592
0.622
0.637
0.657

2.468
2.457
2.342
2.251
2.203
2.156
2.098
2.058
2.031
1.981

1.175
1.171
1.177
1.180
1.183
1.187
1.190
1.193
1.197
1.200

0.266
0.253
0.258
0.251
0.245
0.240
0.234
0.228
0.223
0.218

0.002
0.015
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000

0.030
0.069
0.027
0.024
0.021
0.020
0.018
0.015
0.012
0.010

0.061
0.121
0.052
0.054
0.052
0.050
0.046
0.044
0.041
0.038

0.654
0.610
0.570
0.533
0.517
0.522
0.543
0.562
0.587
0.620

2.864
2.200
2.473
2.269
2.190
2.139
2.101
2.035
2.011
1.995

1.411
1.177
1.227
1.134
1.094
1.086
1.099
1.123
1.149
1.171

0.314
0.229
0.269
0.246
0.233
0.226
0.223
0.221
0.220
0.218

0.025
0.009
0.004
0.001
0.000
0.000
0.000
0.000
0.000
0.000

0.152
0.041
0.039
0.014
0.007
0.006
0.005
0.006
0.007
0.008

0.245
0.084
0.083
0.035
0.021
0.016
0.017
0.020
0.024
0.029

Null

ARMA (2,2)
0
6.8
1
2
3
4
5
6
7
8

Null

NOTE: The Monte Carlo experiments under the two null hypotheses are independent and consist of 10,000 replications each, for a sample size n = 208.
Parameters of the i.i.d. simulations were chosen to match the sample mean and variance of quarterly real GNP growth rates from 1947:QII to 1999:QI; parameters of the ARMA (2,2) were chosen to match point estimates of an ARMA (2,2) model fitted to the same data set. Entries in the column labelled “q” indicate the
number of lags used to compute the R/S statistic; a lag of 0 corresponds to Mandelbrot’s classical rescaled range, and a noninteger lag value corresponds to the
average (across replications) lag value used according to Andrews’s (1991) optimal lag formula. Standard errors for the empirical size may be computed using the
usual normal approximation; they are 9.95 x 10–4, 2.18 x 10–3, and 3.00 x 10–3 for the 1, 5, and 10 percent tests, respectively.
SOURCE: Authors’ calculations.

Table 3 reports the results of both null simulations. It is apparent from the i.i.d. null panel of table
3 that when serial correlation is not a problem, the
classical and modified rescaled range statistics perform similarly. The 5 percent test using the classical
statistic rejects 3 percent of the time: the modified
R/S with q = 4 rejects 2 percent of the time. As the
number of lags increases to 8, the test becomes more
conservative. Under the ARMA(2,2) null hypothesis,
however, it is apparent that modifying the R/S by the
∧
spectral density estimator σ n2(q) is critical; the size of
a 5 percent test based on the classical R/S is
15.2 percent, whereas the corresponding size using
the modified R/S statistic with q = 1 is 3.9 percent.
As before, the test becomes more conservative when
q is increased.
Table 3 also reports the size of tests using the
modified rescaled range when the lag length q is
chosen optimally using Andrews’s (1991) procedure. This data-dependent procedure entails com-

puting the first-order autocorrelation coefficient
∧
the lag length to be the
ρ (1) and then setting
_
integer-value of Mn, where:14
∧ 1/3
_
(33) Mn ≡ 3 α n
2

( )

∧

α ≡

4 ρ∧ 2
.
∧
(1 – ρ 2)2

Under the i.i.d. null hypothesis, Andrews’s
formula yields a 5 percent test with empirical size
6.9 percent; under the ARMA(2,2) alternative,
the corresponding size is 4.1 percent. Although
significantly different from the nominal value, the
empirical size of tests based on Andrews’s formula
may not be economically important. In addition
to its optimality properties, the procedure has the
advantage of eliminating a dimension of arbitrariness in performing the test.
■ 14 In addition, Andrews’s procedure requires weighting the autoco_
j
variances by 1 – _ (j = 1,...,[Mn]) in contrast to Newey and West’s
Mn
_
j
(1987) 1 – q + 1 (j = 1,...,q), where q is an integer and Mn need not be.

T A B L E 4
Power of the Modified R/S Statistics under a Gaussian Fractionally
Differenced Alternative with Differencing Parameters d = 1/3, – 1/3
q

Min.

Max.

Mean

S.D.

Power 1%–Test

Power 5%–Test

Power 10%–Test

d = 1/3
0
6.1
1
2
3
4
5
6
7
8

0.888
0.665
0.825
0.752
0.712
0.687
0.675
0.669
0.666
0.663

5.296
2.569
4.112
3.497
3.126
2.877
2.616
2.469
2.350
2.294

2.551
1.577
2.149
1.936
1.799
1.701
1.630
1.571
1.523
1.481

0.673
0.305
0.528
0.452
0.403
0.367
0.344
0.321
0.302
0.281

0.722
0.039
0.511
0.355
0.244
0.156
0.097
0.051
0.020
0.007

0.842
0.193
0.680
0.543
0.427
0.339
0.268
0.203
0.148
0.095

0.890
0.310
0.758
0.638
0.535
0.446
0.379
0.3171
0.256
0.196

d = –1/3
0
3.9
1
2
4
5
6
7
8

0.339
0.467
0.398
0.443
0.518
0.550
0.576
0.613
0.596

1.009
1.598
1.136
1.282
1.468
1.499
1.573
1.633
1.578

0.583
0.814
0.673
0.741
0.844
0.884
0.922
0.957
0.989

0.095
0.132
0.108
0.117
0.129
0.132
0.136
0.139
0.143

0.917
0.257
0.698
0.474
0.167
0.091
0.046
0.021
0.011

0.979
0.525
0.890
0.736
0.430
0.312
0.2134
0.138
0.092

0.993
0.671
0.944
0.848
0.594
0.467
0.358
0.263
0.190

NOTE: The Monte Carlo experiments under the two alternative hypotheses are independent and consist of 10,000 replications each, for sample size n = 208.
Parameters of the simulations were chosen to match the sample mean and variance of quarterly real GDP growth rates from 1947:QII to 1999:QI. Entries in the
column labeled “q” indicate the number of lags used to compute the R/S statistic; a lag of 0 corresponds to Mandelbrot’s classical range, and a noninteger lag value
corresponds to the average (across replications) lag value used according to Andrews’s (1991) optimal lag formula.
SOURCE: Authors’ calculations.

Table 4 reports power simulations under two
fractionally differenced alternatives: (1 – L)d ε t = ηt ,
where d =1/3, –1/3. Hosking (1981) has shown
that the autocovariance function γε (k) of ε t is
given by:
(34) γε (k) =

Γ (1 – 2d) Γ (d + k)
∧2
ση
Γ (d) Γ (1 – d) Γ (1 – d + k)
d ∈ ( – 1 , 1 ).
2 2

Realizations of fractionally differenced time series
(of length 208) are simulated by premultiplying
vectors of independent standard normal random
variates by the Cholesky factorization of the
(208 x 208) covariance matrix whose entries are
given by equation (34). To calibrate the simulations,
σ η2 is chosen to yield unit variance εt’s, the { ε t }
series is then multiplied by the sample standard
deviation of real GDP growth from 1947:QII to
1999:QI, and to this series is added the sample
mean of real GDP growth over the same sample

period. The resulting time series is used to compute
the power of the rescaled range; table 4 reports
the results.
For small values of q, tests based on the modified
rescaled range have reasonable power against both
fractionally differenced alternatives. For example,
using one lag, the 5 percent test has 68 percent
power against the d = 1/3 alternative, and 89 percent
power against the d = –1/3 alternative. As the lag
length is increased, the test’s power declines.
Note that tests based on the classical rescaled
range are significantly more powerful than those
using the modified R/S statistic. This, however, is
of little value when distinguishing between longrange versus short-range dependence since the test
using the classical statistic also has power against
some stationary finite-order ARMA processes.
Finally, note that tests using Andrews’s truncation
lag formula have reasonable power against the
d = –1/3 alternative but are considerably weaker
against the more relevant d = 1/3 alternative.

The simulation evidence in tables 3 and 4 suggests
that our empirical results do indeed support the
short-range dependence of GDP with a unit root.
Our failure to reject the null hypothesis does not
seem to be explicable by a lack of power against longmemory alternatives. Of course, our simulations were
illustrative and by no means exhaustive; additional
Monte Carlo experiments must be performed before
a full assessment of the test’s size and power is complete. Nevertheless, our modest simulations indicate
that there is little empirical evidence in favor of
long-term memory in GDP growth rates.

IV. Conclusions

This paper has suggested a new approach to the stochastic structure of aggregate output. Traditional dissatisfaction with the conventional methods—from
observations about the typical spectral shape of economic time series, to the discovery of cycles at all
periods—calls for such a reformulation. Indeed,
recent controversy over deterministic versus stochastic trends and the persistence of shocks underscores
the difficulties even modern methods have of identifying the long-run properties of the data.
Fractionally integrated random processes provide
one explicit approach to the problem of long-range
dependence; naming and characterizing this aspect is
the first step in studying the problem scientifically.
Controlling for its presence improves our ability to
isolate business cycles from trends and to assess the
propriety of that decomposition. To the extent that it
explains output, long-range dependence deserves
study in its own right. Furthermore, Singleton
(1988) has pointed out that dynamic macroeconomic models often inextricably link predictions
about business cycles, trends, and seasonal effects.
So, too, is long-range dependence linked: a fractionally integrated process arises quite naturally in a
dynamic linear model via aggregation. This model
not only predicts the existence of fractional noise,
but also suggests the character of its parameters.
This class of models leads to testable restrictions on
the nature of long-range dependence in aggregate
data and holds the promise of policy evaluation.
Advocating a new class of stochastic processes
would be a fruitless task if its members were
intractable. In fact, manipulating such processes
causes few problems. We constructed an optimizing
linear dynamic model that exhibits fractionally
integrated noise and provided an explicit test for
such long-range dependence. Modifying a statistic
of Hurst and Mandelbrot gives us a statistic robust
to short-range dependence, and this modified R/S
statistic possesses a well-defined limiting distribution, which we have tabulated. Illustrative computer

simulations indicate that this test has power against
at least two specific alternative hypotheses of
long memory.
Two main conclusions arise from the empirical
work and Monte Carlo experiments. First, the
evidence does not support long-range dependence
in GDP—the greater power of the modified R/S
test may explain why our results contradict earlier
work that purported to find long-range dependence.
Rejections of the short-range dependence null
hypothesis occur only with detrended data, and this
is consistent with the well-known problem of
spurious periodicities induced by log-linear detrending. Second, since a trend-stationary model is not
contained in our null hypothesis, our failure to
reject may also be viewed as supporting the firstdifference stationary model of GDP, with the
additional implication that the resulting stationary
process is weakly dependent at most. This supports
and extends the conclusion of Adelman (1956) that,
at least within the confines of the available data,
there is little evidence of long-range dependence in
the business cycle. Nevertheless, Haubrich (1993)
finds indirect evidence for long-range dependence
using aggregate consumption series, and hence the
empirical relevance of long memory for economic
phenomena remains an open question that deserves
further investigation.

References
Abadir, Karim, and Bariel Talmain. “Aggregation,
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