How useful are no-arbitrage restrictions for forecasting the term structure of interest rates?

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How useful are no-arbitrage restrictions for forecasting
the term structure of interest rates?
Andrea Carriero, Raffaella Giacomini
To cite this version:
AndreaCarriero,RaffaellaGiacomini. Howusefulareno-arbitragerestrictionsforforecastingtheterm
structureofinterestrates?. Econometrics,2011,164(1),pp.21. ￿10.1016/j.jeconom.2011.02.010￿. ￿hal-
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Accepted Manuscript
Howusefulareno-arbitragerestrictionsforforecastingtheterm
structureofinterestrates?
AndreaCarriero,RaffaellaGiacomini
PII: S0304-4076(11)00037-6
DOI: 10.1016/j.jeconom.2011.02.010
Reference: ECONOM3456
Toappearin: JournalofEconometrics
Pleasecitethisarticleas: Carriero,A.,Giacomini,R.,Howusefulareno-arbitragerestrictions
forforecastingthetermstructureofinterestrates?. JournalofEconometrics(2011),
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How useful are no-arbitrage restrictions for forecasting
the term structure of interest rates?
Andrea Carriero Ra⁄aella Giacomini
(cid:3)
Queen Mary, University of London University College London and CEMMAP
July 2010
Abstract
We develop a general framework for analyzing the usefulness of imposing parameter restric-
tions on a forecasting model. We propose a measure of the usefulness of the restrictions that de-
pends on the forecaster(cid:146)s loss function and that could be time varying. We show how to conduct
inference about this measure. The application of our methodology to analyzing the usefulness of
no-arbitrage restrictions for forecasting the term structure of interest rates reveals that: 1) the
restrictions have become less useful over time; 2) when using a statistical measure of accuracy,
the restrictions are a useful way to reduce parameter estimation uncertainty, but are dominated
byrestrictionsthatdothesamewithoutusinganytheory;3)whenusinganeconomicmeasureof
accuracy, the no-arbitrage restrictions are no longer dominated by atheoretical restrictions, but
for this to be true it is important that the restrictions incorporate a time-varying risk premium.
Keywords: Forecast Combination; Encompassing; Loss functions; Instability; A¢ ne Term
Structure Models
Acknowledgments: We would like to thank Caio Almeida, Gianni Amisano, Albert Chun
and seminar participants at the 2008 Forecasting in Rio conference, 2009 London and Oxbridge
Time Series workshop and Queen Mary University for useful comments and discussions.
J.E.L. Codes: C52, C53, E43, E47
(cid:3)Correspondingauthor,DepartmentofEconomics,UniversityCollegeLondon-GowerStreet-WC1E6BTLondon.
Tel: [phone] Fax: [phone] Email: [email protected].
1 Introduction
In recent years the (cid:133)nance literature has produced major advances in modeling the term structure
of interest rates, building on the assumption of absence of arbitrage opportunities in bond markets.
While the no-arbitrage approach has produced good results in terms of in-sample (cid:133)t, see e.g. De
Jong (2000) and Dai and Singleton (2000), the papers focusing on out-of sample forecasting have
documented a mixed performance of these models. Du⁄ee (2002) shows that beating a random walk
withatraditionalno-arbitragea¢ netermstructuremodelisdi¢ cult. AngandPiazzesi(2003)show
that imposing no-arbitrage restrictions and an essentially a¢ ne speci(cid:133)cation of market prices of risk
improves out-of-sample forecasts from a VAR(12), but the gains with respect to a random walk
forecast are small. Carriero (2007) shows that the no-arbitrage restrictions provide better results
if they are imposed on the data as prior information rather than as a set of restrictions. More
encouraging results have been obtained by Almeida and Vicente (2008), Moench (2008), and Favero
et al. (2009).
A drawback of the above conclusions is that they are based on informal comparisons of mean
squared forecast errors computed over a particular out-of-sample period. In this paper, we develop
a formal framework for investigating the usefulness of parameter restrictions in general - and no-
arbitrage restrictions in particular - when a model is used for forecasting. We achieve several goals:
1) we propose a measure of the usefulness of the restrictions that is tailored to the forecaster(cid:146)s
decision problem; 2) the measure can be time-varying; 3) we show how to perform inference about
the proposed measure. Our framework can be used to answer questions such as "are no-arbitrage
restrictions useful for forecasting the term structure of interest rates?", "are the restrictions useful
for bond portfolio allocation?", and "have the restrictions become more or less useful over time?",
which are not readily answered using conventional model evaluation and hypothesis testing tools.
Our main idea is to cast the problem in an out-of-sample forecast combination framework, in
whichthereisonlyoneforecastmodel,buttheforecasterhastheoptionofimposingsomerestrictions
on its parameters or to forecast with the unrestricted model. We consider a forecast combination
and estimate the optimal combination weight in an out-of-sample framework. We say that the
restriction is "useful for forecasting" when the optimal weight is large, and we can formally test the
hypothesis that the restrictions are useless by an out-of-sample encompassing test. Optimality of
the weight is de(cid:133)ned with respect to a general forecast loss function, but we show how to specialize
the results to either the commonly used quadratic loss or to a loss based on (minus) the utility of
a bond portfolio constructed using the model. The latter example of an economically meaningful
loss has not been considered before for evaluating no-arbitrage models, and we show how its use can
lead to substantially di⁄erent conclusions than those based on conventional statistical measures of
accuracy.
We further generalize the techniques to an environment with possible instability and provide a
2
test to assess whether the usefulness of the restrictions is time-varying. To gain some intuition for
why the usefulness of restrictions could be time-varying, consider the case of a quadratic loss, for
which it can be shown that the measure of usefulness captures the bias/variance tradeo⁄between a
possibly misspeci(cid:133)ed restricted model and the unrestricted model. In this case, time variation could
be due to the variance of the unrestricted model changing or the restrictions becoming more or less
misspeci(cid:133)ed over time. We should point out that our methods do not allow one to disentangle the
two sources of time variation.
We stress that our techniques are not only applicable to the comparison between an unrestricted
and a restricted forecast, but they can be more generally used for measuring the usefulness of two
alternative sets of restrictions imposed on the same forecasting model. For example, the random
walkmodelthatisoftenusedasabenchmarkinforecastingcanalsobeviewedasasetofrestrictions
on a VAR, and one could ask whether the no-arbitrage restrictions are useful relative to the random
walk restrictions. Finally, our framework can be used to compare and combine forecasts from nested
models, which is similar to the problem considered by Clark and McCracken (2009) in a di⁄erent
asymptotic context.
From the perspective of forecast combination, our problem is non-standard because we do not
combine forecasts from di⁄erent models, but forecasts from the same model that are based on
di⁄erent estimators. This in principle poses challenges for the econometric methodology in that the
unrestricted and restricted forecasts may be perfectly correlated in large samples if the restrictions
aretrue. Weovercomethisproblembyconsideringanout-of-sampleenvironmentwithnon-vanishing
estimation uncertainty, as that considered by Giacomini and White (2006) in the di⁄erent context
of equal predictive ability tests, and use it to derive out-of-sample encompassing tests.
Encompassingtestsareappealinginourcontextbecauseincaseofrejectionofthenullhypothesis
they provide as a by-product a combination weight that can be naturally interpreted as a measure
of usefulness of the restrictions, whereas a test of equal predictive ability would force one to choose
either the restricted or the unrestricted forecast. This combination weight can further be used to
produce combined forecasts that exploit the information contained in the economic restrictions in a
way that is optimal for the loss function of interest. Finally, the encompassing approach naturally
lends itself to extensions to unstable environments, since the combination weight can be postulated
tobetime-varying. Ourcontributiontotheliteratureinthisrespectistoprovideavalidasymptotic
theory for testing hypotheses about the time-varying weight.
Note that our problem is also di⁄erent from testing the restrictions in-sample, since we allow for
the possibility that the restrictions are not true, but are still useful for out-of-sample forecasting for
a given loss function.
3
2 A measure of the usefulness of economic restrictions
2.1 Set-up and notation
Let y =(x ;z ) indicate the vector of observables, which include the (scalar) variable of interest x
t t t 0 t
andthevectorofpredictorsz :Weassumethattheuserhasobtainedtwosequencesofh stepahead
t
(cid:0)
out-of-sample forecasts for x , by (cid:133)rst estimating the model without imposing the restrictions (the
t
"unrestricted forecast") and then re-estimating the model subject to the restrictions (the "restricted
forecast"). Iftheinterestisincomparingtwoalternativesetsofrestrictions,theunrestrictedforecast
will be replaced by the alternative restricted forecast, but for simplicity we will continue to refer to
the forecasts as "restricted" and "unrestricted".
The forecasts are obtained by a rolling window estimation scheme, which entails estimating the
model using data indexed t m+1;:::;t for each t = m;:::;T h and using the estimated model
(cid:0) (cid:0)
at time t to produce a forecast for x . This gives two sequences of n T h m+1 forecasts
t+h
T h T h (cid:17) (cid:0) (cid:0)
fU (cid:0) and fR (cid:0) ; denoting respectively the unrestricted and the restricted forecasts.
t;h t;h
t=m t=m
n Tohe asymptnotic foramework considers the in-sample size m (cid:133)xed and lets the out-of-sample size n
growtoin(cid:133)nity,sothatallresultsareimplicitlyconditionalonthechoiceofm,whichisuser-de(cid:133)ned.
The computation of the time-varying measure of usefulness further requires choosing a smoothing
window of size d; which is a constant fraction (cid:25) of the out-of-sample size n.
The user must (cid:133)nally choose a forecast loss function L(x ;f ): We consider in particular two
t+h t;h
types of loss functions, a quadratic loss and a portfolio utility loss. The quadratic loss is de(cid:133)ned as
L(x ;f )=(x f )2; (1)
t+h t;h t+h t;h
(cid:0)
where x in our application will be the yield on a zero coupon bond of maturity (cid:28). The portfolio
t
utility loss considers the asset allocation problem of an investor who is buying a portfolio of q assets
inperiodtandthensellsitinperiodt+1. Inourapplicationsuchassetswillbeq zerocouponbonds
of maturities (cid:28) ;(cid:28) ;:::;(cid:28) . De(cid:133)ning x as the vector of returns on each asset: x = (x ;x ;:::;x ),
1 2 q t t 1 2 q 0
and w as a vector of optimal weights, the return on such a portfolio is given by w x . The portfolio
(cid:3) (cid:3)0 t
utility loss is similar to that considered by West, Edison and Cho (1993), and is given by
(cid:13)
L(x ;f )= w (f )x + w (f )(cid:6)w (f ); (2)
t+h t;h (cid:3) t;h 0 t+h (cid:3) t;h 0 (cid:3) t;h
(cid:0) 2
where w (f ) are the optimal portfolio weights for a quadratic utility, and are linear functions of
(cid:3) t;h
the forecasts (the exact expression is given in (18) below). Note that in this case f is a vector
t;h
containing the forecasts of each element in x . The matrix (cid:6) is the variance-covariance matrix of
t
x ; and (cid:13) is a user-de(cid:133)ned parameter related to the coe¢ cient of relative risk aversion (cid:14) by the
t+h
relationship (cid:13) =(cid:14): Our empirical results are obtained by setting (cid:14) =1; so that (cid:13) =:5:
1 (cid:13)
(cid:0)
4
2.2 Methodology for a general loss function
Consider a combination of the restricted and unrestricted forecast, f = fR +(1 (cid:21))(fU fR );
t(cid:3);h t;h (cid:0) t;h(cid:0) t;h
so that (cid:21) is the weight on the restricted forecast.
The optimal weight (cid:21) minimizes the expected out-of-sample loss of the combined forecast:
(cid:3)
T h
1 (cid:0)
(cid:21) = argminE L x ;fR +(1 (cid:21))(fU fR ) (3)
(cid:3) (cid:21) R "n t+h t;h (cid:0) t;h(cid:0) t;h #
2 t=m
X (cid:0) (cid:1)
= argminE[Q ((cid:21))];
n
(cid:21) R
2
and is estimated by
T h
1 (cid:0)
(cid:21) = argmin L x ;fR +(1 (cid:21))(fU fR ) (4)
(cid:21) R n t+h t;h (cid:0) t;h(cid:0) t;h
2 t=m
X (cid:0) (cid:1)
b = argminQn((cid:21)):
(cid:21) R
2
The estimated optimal weight (cid:21) is our measure of the usefulness of the economic restrictions for
forecasting, for a given loss function L(). A small (cid:21) indicates that the restrictions are not useful
b (cid:1)
for forecasting, whereas a large (cid:21) suggests that the economic restrictions can be usefully imposed to
b
obtain more accurate forecasts. (cid:21) in (4) can be computed for a general loss function using numerical
b
methods, butweshowhowtoderivesimpleanalyticalexpressionsforthespecialcasesofaquadratic
b
and portfolio loss functions in Section 3 below.
The asymptotic distribution of (cid:21) is obtained by recognizing that (cid:21) is an M-estimator, which
minimizes the (typically well-behaved) objective function Q ((cid:21)): A similar remark was made by
n
b b
ElliottandTimmermann(2004),inanenvironmentwheretheforecastsarebasedondi⁄erentmodels
and are taken as given. The fact that in our context the forecasts are based on the same model
and depend on in-sample data and estimated parameters introduces some complications, which we
handleusingageneralizationofthekeyinsightinGiacominiandWhite(2006). Speci(cid:133)cally,weshow
that an asymptotic theory for (cid:21) can still be derived by relying on laws of large numbers, central
limit theorems and functional central limit theorems for the objective function and its derivatives in
b
spite of the fact that such functions depend in a complex nonlinear manner on the in-sample data
through fR and fU : This is because we assume that the in-sample estimation window is (cid:133)nite,
t;h t;h
so that the objective function and its derivatives become functions of the (cid:133)nite history of "short
memory" (mixing) processes, and are thus themselves short memory and plausibly satisfy laws of
large numbers and central limit theorems.
We rely on the asymptotic properties of (cid:21) to obtain formal methods for testing the usefulness of
the restrictions, both in an environment where such usefulness is constant over time (Section 2.2.1)
b
and in an environment with possibly time-varying usefulness (Section 2.2.2).
5
2.2.1 Testing the global usefulness of parameter restrictions
We (cid:133)rst consider an environment in which (cid:21) is constant over time, and can thus be interpreted as
(cid:3)
a "global" measure of the usefulness of the restrictions.
Proposition 1 below shows how to construct formal tests for whether the unrestricted forecast is
useless(HU :(cid:21) =1)orwhethertherestrictedforecastisuseless(HR :(cid:21) =0),whichareessentially
0 (cid:3) 0 (cid:3)
out-of-sample encompassing tests. The tests are derived under the following assumptions.
Assumption A. (1) E[Q ((cid:21))] is uniquely minimized at (cid:21) < ;
n (cid:3)
1
(2) L x ;fR +(1 (cid:21))(fU fR ) is convex and twice continuously di⁄erentiable with re-
t+h t;h (cid:0) t;h(cid:0) t;h
spect to (cid:21)(cid:16); (cid:17)
(3) x is mixing with (cid:30) of size r=(r 1) or (cid:11) of size 2r=(r 2); r >2;
t
f g (cid:0) (cid:0) (cid:0) (cid:0)
(4) E L x ;fR +(1 (cid:21))(fU fR ) r=2 < for all t and all (cid:21);
j t+h t;h (cid:0) t;h(cid:0) t;h j 1
(5)E (cid:16)L x ;fR +(1 (cid:21) )(fU f(cid:17)R ) 2r < forallt;where indicatesthe(cid:133)rstderiv-
jr(cid:21) t+h t;h (cid:0) (cid:3) t;h(cid:0) t;h j 1 r(cid:21)
ative with resp(cid:16)ect to (cid:21); (cid:17)
2
(6) (cid:10)=E (pn Q ((cid:21) )) >0 for all n;
(cid:21) n (cid:3)
r
(7) H =Eh[ Q ((cid:21) )]>0ifor all n;
(cid:21)(cid:21) n (cid:3)
r
(8) sup Q ((cid:21)) E[ Q ((cid:21))] p 0; where (cid:3) indicates a neighborhood of (cid:21) and
(cid:21) (cid:3) (cid:21)(cid:21) n (cid:21)(cid:21) n (cid:3) (cid:21)(cid:21)
2 kr (cid:0) r k! r
the second derivative with respect to (cid:21);
(9) m< ; h< ; n :
1 1 !1
Assumption A(1) is satis(cid:133)ed by the quadratic and the portfolio utility loss functions considered
inSection3, whicharebothquadraticpolynomialsin(cid:21):AssumptionA(2)isstrongerthannecessary
and is only imposed for convenience and because it is satis(cid:133)ed by the loss functions in Section 3.
FollowingNeweyandMcFadden(1994),itisstraightforwardtoextendtheresultstoanenvironment
with non-convex and non-di⁄erentiable objective functions. Assumptions A(3) to A(7) are the
familiar primitive conditions guaranteeing applicability of laws of large numbers and central limit
theorems for the objective function and its derivatives. Note that these conditions, while ruling out
the presence of unit roots, allow the data to be heterogeneous and dependent. Assumption A(7)
could be violated if the forecasts were perfectly correlated in large samples. To see why, consider
2
for simplicity the quadratic loss case, where H = E 2 fR fU : If the restrictions were true,
t;h(cid:0) t;h
(cid:20) (cid:21)
the forecasts would become perfectly correlated as the e(cid:16)stimation s(cid:17)ample grows, making H converge
to zero. This occurrence is however ruled out in our context by A(9), which assumes that the
estimationsampleis(cid:133)xed,thuspreventingestimationuncertaintyfromdisappearingasymptotically.
Assumption A(8) requires a uniform law of large numbers for the second derivatives of the objective
function. Primitive conditions for A(8) could easily be found, but we do not specify them here
because A(8) becomes considerably simpler for the loss functions considered in Section 3, since in
both cases the second derivative of the objective function does not depend on (cid:21): For these loss
functions, A(8) can be replaced with the condition that Q has (cid:133)nite r=2 th moments, which,
(cid:21)(cid:21) n
r (cid:0)
6
together with A(3), guarantees that a law of large numbers can be invoked for Q . Assumption
(cid:21)(cid:21) n
r
(9) shows that the asymptotic distribution is obtained by letting the out-of-sample size n grow to
in(cid:133)nity, whereas the in-sample size m and the forecast horizon h are (cid:133)nite.
Proposition 1 (Tests of global usefulness) Suppose Assumption A holds. Let
pn (cid:21) 1
tU = (cid:0) ; (5)
(cid:16)(cid:27) (cid:17)
b
pn(cid:21)
tR = ;
(cid:27) b
b
where (cid:27) is given by
b
b (cid:27) = H 1(cid:10)H 1; (6)
(cid:0) (cid:0)
H = p Q (cid:21) ;
b r(cid:21)b(cid:21) nb b
(cid:10)b = pn(cid:0)1 (cid:16)b1(cid:17) j n 1 T(cid:0)h s (cid:21) s (cid:21) ;
(cid:0) t t j
(cid:0)jpnj (cid:0)
j=(cid:0)Xpn+1(cid:18) (cid:19) t=Xm+j (cid:16) (cid:17) (cid:16) (cid:17)
s (cid:21)b = @L x ;fR +(1 (cid:21))(fU fbR ) ; b
t (cid:21) t+h t;h t;h t;h
r (cid:0) (cid:0)
(cid:16) (cid:17) (cid:16) (cid:17)
b b
where p is a bandwidth that increases with the sample size (Newey and West, 1987).
n
ThenthehypothesesHU :(cid:21) =1andHR :(cid:21) =0arerejectedatasigni(cid:133)cancelevel(cid:11) respectively
0 (cid:3) 0 (cid:3)
when tU >c and tR >c ; with c indicating the 1 (cid:11)=2 quantile of a N(0;1) distribution.
(cid:11)=2 (cid:11)=2 (cid:11)=2
(cid:0)
(cid:12) (cid:12) (cid:12) (cid:12)
Th(cid:12)e b(cid:12)andwidth p (cid:12)us(cid:12)ed in the construction of the test statistic must be appropriately chosen to
n
account for the possible serial correlation in the (cid:133)rst derivatives of the loss function. In practice, the
accuracy of the estimate of (cid:10) can be an issue, particularly for long-horizon forecasts (see, e.g., Kim
and Nelson, 1993 and Harvey, Leybourne and Newbold, 1998). In our application, we follow Kim
and Nelson (1993)(cid:146)s recommendation and set p =2(h 1):
n
(cid:0)
2.2.2 Testing the usefulness of parameter restrictions in the presence of instability
A question that may be of further interest to forecasters is whether the usefulness of the restrictions
varies over time. To answer this question, we extend the previous analysis to the case of time-
varying forecast combination weights. These time-varying weights can be interpreted as measuring
the "local" usefulness of the restrictions, and solve the problem
(cid:21) =argminE L(x ;fR +(1 (cid:21) ) fU fR ; t=m;:::;T h: (7)
(cid:3)t t+h t;h t t;h t;h
(cid:21)t R (cid:0) (cid:0) (cid:0)
2
(cid:2) (cid:0) (cid:1)(cid:3)
A simple nonparametric estimator of (7) can be obtained by computing rolling average weights
7
over windows of size d:
t
(cid:21) =argmin L(x ;fR +(1 (cid:21) ) fU fR ; t=m+d 1;:::;T h: (8)
t;d j+h j;h t j;h j;h
(cid:21)t R (cid:0) (cid:0) (cid:0) (cid:0)
2 j=t d+1
X(cid:0) (cid:2) (cid:0) (cid:1)(cid:3)
b
Instead of adopting a standard asymptotic approximation to conduct inference about (8), which
would require the bandwidth d=n to go to zero as d and n grow to in(cid:133)nity, we follow a similar
approach as Giacomini and Rossi (2010) and obtain a distribution theory for (cid:21) that has better
t;d
(cid:133)nite-sample properties by using a non-standard asymptotic approximation with (cid:133)xed bandwidth.
b
Note that in this (cid:133)xed-bandwidth approximation, however, (cid:21) is no longer a consistent estimator
t;d
of (cid:21) ; but it consistently estimates a "smoothed" version of (cid:21) :
(cid:3)t (cid:3)t
b
t
(cid:21) =argmin E L(x ;fR +(1 (cid:21) ) fU fR ; t=m+d 1;:::;T h: (9)
(cid:3)t;d j+h j;h t j;h j;h
(cid:21)t R (cid:0) (cid:0) (cid:0) (cid:0)
2 j=t d+1
X(cid:0) (cid:2) (cid:0) (cid:1)(cid:3)
Further note that, as a result of adopting a non-standard (cid:133)xed bandwidth approximation, stan-
dard results for optimal bandwidth selection obtained in the nonparametric literature do not apply
here. Instead, in our framework di⁄erent choices of bandwidth result in a di⁄erent null hypothesis
being tested.
T h
A plot of the sample path of (cid:21) (cid:0) in (8) can uncover possible time-variation in the
t;d
t=m+d 1
usefulnessoftheeconomicrestrictinons. oProposi(cid:0)tion2belowfurthershowshowtotestthehypothesis
b
that the unrestricted forecast was consistently useless (HU :(cid:21) =1 for all t) or that the restricted
0 (cid:3)t;d
forecast was consistently useless (HR : (cid:21) = 0 for all t) over time. We control the overall size of
0 (cid:3)t;d
the procedure by deriving uniform con(cid:133)dence bands that have the desired coverage under the null
hypothesis.
The proposition relies on the following set of assumptions.
Assumption B. Let (cid:28) [0;1]: Under the hypothesis that (cid:21) is constant and equal to (cid:21) ,
2 (cid:3)t;d (cid:3)
(1) n 1=2 m+[(cid:28)n] L x ;fR +(1 (cid:21) )(fU fR ) obeys a Functional Central Limit
(cid:0) j=m r(cid:21) j+h j;h (cid:0) (cid:3) j;h(cid:0) j;h
Theoremn with P(cid:10)=limn!1E((cid:16)n(cid:0)1=2 Tj=(cid:0)mhr(cid:21)L xj+h;fjR;h+(1(cid:17)o(cid:0)(cid:21)(cid:3))(fjU;h(cid:0)fjR;h) )2 >0;
(2) d=n (cid:25) (0; ) as d ;n : m(cid:16)< and h< ; (cid:17)
P
! 2 1 !1 !1 1 1
(3) (cid:27) p (cid:27) and (cid:21) p (cid:21) :
(cid:3)
! !
Primitive conditions for B(1) and B(3) analogous to those listed in Assumption A could be
b b
similarly speci(cid:133)ed here.
Proposition 2 (Tests of time variation in usefulness) Suppose Assumption B holds. For a
signi(cid:133)cance level (cid:11); (cid:133)rst construct the bands:
(cid:27) (cid:27)
((cid:21) k ;(cid:21) +k ); t=m+d 1;:::;T h; (10)
t;d (cid:11);(cid:25) t;d (cid:11);(cid:25)
(cid:0) pd pd (cid:0) (cid:0)
b b
b b
8