Bootstrap forecast of multivariate VAR models without using the backward representation

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In this paper, we show how to simplify the construction of bootstrap prediction densities in multivariate VAR models by avoiding the backward representation. Bootstrap prediction densities are attractive because they incorporate the parameter uncertainty a any particular assumption about the error distribution. What is more, the construction of densities for more than one-step unknown asymptotically. The main advantage of the new simple without loosing the good performance of bootstrap procedures. Furthermore, by avoiding a backward representation, its asymptotic validity can be proved without relying on the assumption of Gaussian errors as proposed in this paper can be implemented to obtain prediction densities in models without a backward representation as, for example, models with MA components or GARCH disturbances. By comparing the finite sample performance of the proposed procedure with those of alternatives, we show that nothing is lost when using it. Finally, we implement the procedure to obtain prediction regions for US quarterly future inflation, unemployment and GDP growth

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Publié le 01 octobre 2011
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Working Paper 11-34 Departamento de Estadística
Statistics and Econometrics Series 026 Universidad Carlos III de Madrid
October 2011
CCaallllee MMaaddrriid,d, 112266

28903 Getafe (Spain)
Fax (34) 91 624-98-49
BOOTSTRAP FORECAST OF MULTIVARIATE VAR MODELS WITHOUT
USING THE BACKWARD REPRESENTATION

♦ ♠ ♣Lorenzo Pascual , Esther Ruiz and DiegoFresoli


Abstract

In this paper, we show how to simplify the construction of bootstrap prediction densities in
multivariate VAR models by avoiding the backward representation. Bootstrap prediction
densities are attractive because they incorporate the parameter uncertainty and do not rely on
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unknown asymptotically. The main advantage of the new procedure is that it is computationally
simple without loosing the good performance of bootstrap procedures. Furthermore, by avoiding
a backward representation, its asymptotic validity can be proved without relying on the
assumption of Gaussian errors as needed by alternative procedures. Finally, the new procedure
proposed in this paper can be implemented to obtain prediction densities in models without a
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of alternatives, we show that nothing is lost when using it. Finally, we implement the procedure
to obtain prediction regions for US quarterly future inflation, unemployment and GDP growth.


Keywords: Non-Gaussian VAR models, Prediction cubes, Prediction density, Prediction
rreeggiionsons, P, Prreeddiiccttiionon e elllliipspsooiidsds, R, Reessaammplpliingng mmeetthodshods..







EDP-Energías de Portugal, S.A., Unidade de Negócio de Gestao da Energía, Director Adjunto.
♠ Corresponding author: Dpt. Estadística and Instituto Flores de Lemus, Universidad Carlos III de Madrid, C/ Madrid
126, 28903 Getafe, Spain. Tel: 34 916249851, Fax: 34 916249849, e-mail: ortega@est-econ.uc3m.es.
♣Dpt. de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail:
dfresoli@est-econ.uc3m.es.
Acknowledgements. The last two authors are grateful for financial support from project ECO2009-08100 of the Spanish
Government. We arree aallssoo ggrraatteeffuull ttoo GGlloorriiaa GGoonnzzáález-RRiivveerraa ffoorr hheerr uusseeffuull ccoommmmeennttss.. TThhee uussuuaall ddiissccllaaiimmss aappppllyy.. Bootstrap Forecast of Multivariate VAR Models without
using the backward representation
y zxLorenzo Pascual Esther Ruiz Diego Fresoli
October 2011
Abstract
In this paper, we show how to simplify the construction of bootstrap prediction densities
in multivariate VAR models by avoiding the backward representation. Bootstrap prediction
densities are attractive because they incorporate the parameter uncertainty and do not rely
on any particular assumption about the error distribution. What is more, the construction of
densities for more than one-step-ahead is possible even in situations when these densities are
unknown asymptotically. The main advantage of the new procedure is that it is computa-
tionally simple without loosing the good performance of bootstrap procedures. Furthermore,
by avoiding a backward representation, its asymptotic validity can be proved without relying
on the assumption of Gaussian errors as needed by alternative procedures. Finally, the new
procedure proposed in this paper can be implemented to obtain prediction densities in models
without a backward representation as, for example, models with MA components or GARCH
disturbances. By comparing the nite sample performance of the proposed procedure with
those of alternatives, we show that nothing is lost when using it. Finally, we implement the
procedure to obtain prediction regions for US quarterly future in ation, unemployment and
GDP growth.
KEYWORDS: Non-Gaussian VAR models, Prediction cubes, Prediction density, Predic-
tion regions, Prediction ellipsoids, Resampling methods.
EDP-Energ as de Portugal, S.A., Unidade de Neg ocio de Gest~ ao da Energ a, Director Adjunto.
yCorresponding author: Dpt. Estad stica and Instituto Flores de Lemus, Universidad Carlos III de Madrid, C/
Madrid 126, 28903 Getafe, Spain. Tel: 34 916249851, Fax: 34 916249849, e-mail: ortega@est-econ.uc3m.es.
zDpt. de Estad stica, Universidad Carlos III de Madrid, C./ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail
dfresoli@est-econ.uc3m.es.
xAcknowledgements. The last two authors are grateful for nancial support from project ECO2009-08100
of the Spanish Government. We are also grateful to Gloria Gonz alez-Rivera for her useful comments. The usual
disclaims apply.
11 Introduction
Bootstrap procedures are known to be useful when forecasting time series because they allow
the construction of prediction densities without imposing particular assumptions on the error
distribution and simultaneously incorporating the parameter uncertainty. Note that when the
errors are non-Gaussian the prediction densities are usually unknown when the prediction hori-
zon is larger than one-step-ahead. However, the bootstrap can be implemented in these cases
to obtain the corresponding prediction densities. They are also attractive because their compu-
tational simplicity and wide applicability. However, these advantages are limited by the use of
the backward representation that many authors advocate after the seminal paper of Thombs and
Schuchany (1990). In particular, Kim (1999) extends the procedure of Thombs and Schuchany
(1990) to stationary VAR(p) models. Later, Kim (2001, 2004) considered bias-corrected predic-
tion regions by employing a bootstrap-after-bootstrap approach. On the other hand, Grigoletto
(2005) proposes two further alternative procedures based on Kim (1999) that take into account
not only the uncertainty due to parameter estimation but also the uncertainty attributable to
model speci cation. In any case, the bootstrap procedures conceived by Kim (1999, 2001, 2004)
and Grigoletto (2005) use the backward representation to generate the bootstrap samples used to
obtain replicates of the estimated parameters. Using the backward representation has three main
drawbacks. First, the resulting procedure is computationally complicate and time consuming.
Second, given that the backward residuals are not independent it is necessary to use the relation-
ship between the backward and forward representation of the model in order to resample from
independent residuals; see Kim (1997, 1998) for this relationship which can be rather complicate
for high order models. Consequently, Kim (1999) resamples directly from the dependent back-
ward residuals. However, the asymptotic validity of the bootstrap resampling can only be proved
by imposing i.i.d. and, as a result, it requires assuming Gaussian errors. Finally, these bootstrap
alternatives can only be applied to models with a backward representation which excludes their
implementation in, for example, multivariate models with Moving Average (MA) components or
with GARCH disturbances.
In an univariate framework, Pascual et al. (2004a) show that the backward representation
can be avoided without loosing the good properties of the bootstrap prediction densities. When
dealing with multivariate systems it is even more important avoiding the backward representation
due to its larger complexity; see Kim (1997, 1998). In this paper, we propose an extension of the
bootstrap procedure proposed by Pascual et al. (2004a) for univariate ARIMA models to obtain
2joint prediction densities for multivariate VAR(p) models avoiding the backward representation
and, consequently, overcoming its limitations. We prove the asymptotic validity of the proposed
procedure without relying on particular assumptions about the prediction error distribution.
We focus on the construction of multivariate prediction densities from which it is possible to
obtain marginal prediction intervals for each of the variables in the system and joint prediction
1regions for two or more variables within the system. Monte Carlo experiments are carried out
to study the nite sample performance of the marginal prediction intervals obtained by the new
bootstrap procedure and compare it those of alternative procedures available in the literature. We
also compare their corresponding elliptical and Bonferroni regions. We show that, although the
bootstrap procedure proposed in this paper is computationally simpler, its nite sample properties
are similar to those of previous more complicate bootstrap approaches and clearly better than
those of the standard and asymptotic prediction densities. We also show that when the errors
are non-Gaussian, the bootstrap elliptical regions are inappropriate with the Bonferroni regions
having better properties. The procedures are illustrated with an empirical application which
consists of predicting future in ation, unemployment and growth rates in the US.
The rest of the paper is organized as follows. Section 2 describes the asymptotic and bootstrap
prediction intervals and regions previously available in the literature. In Section 3, we propose a
new bootstrap procedure, derive its asymptotic distribution and analyze its performance in nite
samples. We compare the new bootstrap densities and corresponding prediction intervals and
regions with the standard, asymptotic and alternative bootstrap procedures. Section 4 illustrates
the results with an empirical application. Finally, section 5 concludes the paper with suggestions
for further research.
2 Asymptotic and bootstrap prediction intervals and re-
gions for VAR models
In this section, we describe the construction of prediction regions in stationary VAR models based
on assuming known parameters and Gaussian errors. We also describe how the parameter uncer-
tainty can be incorporated by using asymptotic and bootstrap approximations of the nite sample
distribution of the parameter estimator. The bootstrap procedures can also be implemented to
deal with non-Gaussian errors.
1Previous paper focus on the Bonferroni regions and not in the bootstrap densities themselves.
32.1 Asymptotic prediction intervals and regions
Consider the following multivariate VAR(p) model
(L)Y = +a ; (1)t t
where Y is the Nx1 vector of observations at time t; is a Nx1 vector of constants, (L) =t
pI L ::: L withL being the lag operator andI theNxN identity matrix. TheNxNN 1 p N
parameter matrices, ;i = 1;:::;p; satisfy the stationarity restriction. Finally,a is a sequence ofi t
Nx1 independent white noise vectors with nonsingular contemporaneous covariance matrix given
by .a
It is well known that if a is an independent vector white noise sequence then the pointt
predictor of Y that minimizes the Mean Square Error (MSE) is its conditional mean whichT+h
depends on the model parameters. In practice, these parameters are unknown and the predictor
of Y is obtained with the parameters substituted by consistent estimates as followsT+h
b b b b bY =b + Y +::: + Y (2)T+hjT 1 T+h1 jT p T+hpjT
b bwhere Y = Y ; j = 0; 1;::: Furthermore, the MSE of Y is usually estimated asT+jT+jjT T+hjT
follows
h1X
0b b b b (h) = ; (3)b j a jY
j=0
0babab bwhere are the estimated matrices of the MA representation of Y and = wherej t a TNp1
ba = (ba ;:::;ba ) with1 T
b bba =Y b Y ::: Y : (4)t t 1 t1 p tp
If a is further assumed to be Gaussian, then the marginal prediction density of the ntht
variable in the system is also Gaussian and the standard practice is to construct the (1-)100%
prediction interval for the nth variable in the system as follows

GI = y jy 2 yb z b ; (5)T+h n;T+h n;T+hjT n;T+hjT =2 n;h
bwhere yb is the nth component of Y , b is the square root of the nth diagonaln;T+hjT T+hjT n;h
belement of (h) andz is the-quantile of the standard Gaussian distribution. The GaussianitybY
of the forecast errors can also be used to obtain the following (1-)100% joint ellipsoid for all the
4variables in the system
h i h i0
1 2b b bGE = Y j Y Y (h) Y Y < (N) ; (6)T+h T+h T+h T+hjT b T+h T+hjT Y
2 2 2where (N) is the -quantile of the distribution with N degrees of freedom. Constructing
the ellipsoids in (6) can be quite demanding when N is larger than two or three. Consequently,
Lutk epolh (1991) proposes using the Bonferroni method to construct the following prediction
cubes with coverage at least (1-)100%

NGC = Y jY 2[ yb z b ; (7)T+h T+h T+h n;hn=1 n;T+hjT
where where = 0:5(=N). However, note that the prediction intervals and regions above have
two main drawbacks. First, they are constructed using the MSE in (3) which does not incor-
porate the parameter uncertainty. As a consequence, if the sample size is small the uncertainty
bassociated with Y is underestimated and the corresponding intervals an regions will haveT+hjT
lower coverages than the nominal. The second problem, is related to the Gaussianity assumption.
When this assumption does not hold, the quadratic form in (6) is not adequate as well as the
width of the intervals in (5) and (7). Even more, when the prediction errors are not Gaussian,
the shape of their densities for h>2 is in general unknown.
As an illustration, consider the following VAR(2) bivariate model
2 3 2 32 3 2 32 3 2 3
y 0:9 0 y 0:5 0:7 y a1;t 1;t1 1;t2 1;t6 7 6 76 7 6 76 7 6 7
= + + (8)4 5 4 54 5 4 54 5 4 5
y 0:4 0 y 0:8 0:1 y a2;t 2;t1 2;t2 2;t
0where a = (a , a ) is an independent white noise vector with contemporaneous covariancet 1;t 2;t
0
matrix given by vec = (1; 0:8; 0:8; 1) where vec denotes the column stacking operator. Thea
2distribution of a is a (4). Panel (a) of Figures 1 and 2 display the joint one-step-ahead andt
eight-steps-ahead densities of y and y respectively, which have clear asymmetries, although1;t 2;t
more pronounced in the former. After generating a time series of size T = 100, the VAR(2)
parameters are estimated by Least Squares (LS). Panel (b) of Figures 1 and 2 plot kernel estimates
of the joint density obtained as usual after assuming that the prediction errors are jointly Gaussian
2The same argument can be applied when the interest lies on only a subset of components of Y .t
For example, if the focus is on the rst J components of Y , the prediction ellipsoid is given byt h i h i0 1
0 0 2b b bfY j Y Y C C (h)C C Y Y < (J)g, where C = [I 0].T+h T+h b T+h JT+hjT T+hjT Y
53with zero mean and convariance matrix given by (3). Comparing panels (a) and (b), it is obvious
that the Gaussian approach fails to capture the asymmetry of the error distribution. It is usual in
practice to construct prediction regions fory andy . From the joint Gaussian density plotted1;t 2;t
in panel (b) of Figure 1, it is possible to obtain the corresponding 95% one-step-ahead ellipsoids
and Bonferroni regions. They are shown in Figure 3 together with a realization of Y . We canT+1
observe that the shape of both regions is not appropriate to construct a satisfactory prediction
region for Y . Finally, as we may also being interested in forecasting only one variable inT+1
the system, Figure 4 displays the true marginal one-step-ahead density of y together with the1;t
Gaussian approximation. Once more, it is clear that the Gaussian approach fails to capture the
skewness of the prediction density.
Consider rst the problem of incorporating the parameter uncertainty. As pointed out above,
the MSE in (3) underestimates the true prediction uncertainty and, consequently, they can be
inappropriate in small samples sizes. Granted that a good estimator is used, the importance
of taking into account the parameter uncertainty could be small in systems consisting in few
variables; see Riise and Tjostheim (1984). But, in empirical applications we often found VAR(p)
models tted to large systems; see, for example, Simkins (1995) for a VAR(6) model for a system
of 5 macroeconomic variables, Waggoner and Zha (1999) who t a VAR(13) model to a system
of 6 macroeconomic variables, Chow and Choy (2006) who t a VAR(5) model to a system of
5 variables related with the global electronic system, G omez and Guerrero (2006) for a VAR(3)
model tted to a system of 6 macroeconomic variables and Chevillon (2009) for a VAR(2) model
for a system of 4 macroeconomic variables just to mention a few empirical applications. Addi-
tionally, as these examples show, when dealing with real systems of time series, their adequate
representation often requires a rather large order p. If the number of variables in the system
and/or the number of lags of the model are relatively large, the estimation precision in nite
samples could be rather low and predictions based on VAR(p) models with estimated parame-
ters may su er severely from the uncertainty in the parameter estimation. In these cases, it is
important to construct prediction intervals and regions that take into account this uncertainty;
see, for instance, Schmidt (1977), Lutk epolh (1991), West (1996), West and McCracken (1998),
Sims and Zha (1998, 1999) for existing evidence on the importance of taking into account param-
eter uncertainty in unconditional forecasts and Waggoner and Zha (1999) for the same result in
conditional forecasts.
3The smoothed density is obtained by applying a Gaussian kernel density estimator with a diagonal bandwidth
matrix with elements given by the Gaussian \rule of thumb".
6To incorporate the parameter uncertainty, Lutk epolh (1991) suggests approximating the sam-
4ple distribution of the estimator by its asymptotic distribution density. In this case, the MSE
bof Y that incorporates the parameter uncertainty can be approximated byT+hjT
1lb b b (h) = (h) + (h); (9)bbY Y T
where
h1 h1 h iXX
0 h1i 1 h1 j 0b b b b b b b b (h) = tr (B ) B ; (10)i a j
i=0 j=0
0ZZb bwith = and B is the following (Np + 1)x(Np + 1) matrix
T
2 3
1 0 0 ... 0 0
6 7
6 7
6 b b b b 7b ... 1 2 p1 p6 7
6 7
b 6 7B = :0 I 0 ... 0 06 N 7
6 7
6 7
6 ... ... ... ... ... ... 7
4 5
0 0 0 ... I 0N
In order to asses the e ect of the parameter uncertainty on the MSE given by (9), consider the
b bcase of one-step-ahead predictions, i.e. when h = 1. In this situation, (1) = ( Np + 1) , anda
lb (1) can be approximated bybY
T +Np + 1lb b (1) = :abY T
This expression shows that the contribution of the parameter uncertainty to the one-step-ahead
MSE matrix depends on the dimension of the system, N, the VAR order, p, and the sample
size, T . As long as N and/or p, or both, are big enough compared to the sample size T , the
e ect of parameter uncertainty can be substantial. Obviously, as the sample size gets larger then
T+Np+1
lim = 1 and the parameter uncertainty contribution to the MSE in (9) vanishes.T!1 T
Once the MSE is computed as in (9), the corresponding prediction intervals, ellipsoids and
cubes are constructed using the Gaussianity assumption as follows

lAI = y jy 2 yb z b ; (11)T+h n;T+h n;T+h n;T+hjT =2 n;h
4Alternatively, some authors propose using Bayesian methods which could be rather complicated from a com-
putational point of view; see, for example, Simkins (1995) and Waggoner and Zha (1999) who need the Gaussianity
assumption to derive the likelihood and posterior distribution.
7n h i h i o
l 1 2b b bAE = Y j Y Y (h) Y Y < (N) ; (12)T+h T+h T+h T+hjT T+h T+hjTb Y

N lAC = Y jY 2[ yb z b ; (13)T+h T+h T+h n;T+hjT n=1 n;h
l lbwhere b is the square root of the nth diagonal element of (h).n;h bY
Panel (c) of Figure 1 which plots the density of y and y for the same example as1;T+1 2;T+1
above constructed assuming that the forecast error are Gaussian with zero mean and covariance
matrix given by (9), shows that this density is not very di erent from that plotted in panel (b)
and, obviously, it is not able to capture the asymmetries of the error distribution. Similarly,
the joint density of y and y in panel (c) of Figure 2 does not look di erent from that1;T+8 2;T+8
in panel (b). The similarity between the standard and the asymptotic densities is even more
clear in Figure 3 that plots the elliptical and Bonferroni regions constructed using (12) and (13),
respectively. As we can observe they are slightly larger than the standard but still located very
close to them. They cannot cope with the lack of symmetry of the prediction error distribution.
This similarity could be expected as we are estimating 8 parameters with T = 100. Similar
comments deserve Figure 4, where we can observe that the asymptotic marginal density for the
rst component of the system y only di ers from the standard density in the variability,1;T+1
which is slightly larger in the former.
Note that the asymptotic approximation of the distribution of the LS estimator can be inad-
equate in small samples depending on the number of parameters to be estimated and the true
distribution of the innovations.
2.2 Bootstrap procedures for prediction intervals and regions
To overcome the limitations of the Gaussian densities described before, Kim (1999, 2001, 2004)
and Grigoletto (2005) propose using bootstrap procedures which incorporate the parameter un-
certainty even when the sample size is small and do not rely on the Gaussianity assumption.
In order to take into account the conditionality of VAR forecasts on past observations, Kim
(1999) proposes to obtain bootstrap replicates of the series based on the following backward
recursion
b bY =!b + Y +::: + Y +b (14)1 pt t+1 t+p t
where Y = Y for i = 0; 1;:::;p 1 are p starting values which coincide with the lastTiTi
b bvalues of the original series, !b; ;:::; ; are LS estimates of the parameters of the backward1 p
representation, andb are obtained by resampling from the empirical distribution function of thet
8centered and rescaled backward residuals. Then, bootstrap LS estimates of the parameters of the
forward representation are obtained by estimating the VAR(p) model in (1) usingfY ;:::;Y g.1 T
^ b bDenote these estimates by B = (b ; ;:::; ). The bootstrap forecast for period T +h is then1 p
given by
b b b b bY =b + Y +::: + Y +ba (15)T+hjT 1 T+h1jT p T+hpjT T+h
whereba are random draws from the empirical distribution function of centered and rescaledT+h
bforward residuals. Having obtained R bootstrap replicates of Y , Kim (2001) de nes the
T+hjT
bootstrap (1-)100% prediction interval for the nth variable in the system as follows
n h io KI = y jy 2 q ;q 1 (16)T+h n;T+h n;T+h K K
2 2
where q () is the empirical -quantile of the bootstrap distribution of the nth component ofK
bY approximated by G (x) = #(yb < x)=R. Similarly, Kim (1999) proposes toT+hjT n;K n;T+hjT
construct bootstrap prediction ellipsoids with probability content (1 )100% are given by
h i h i0
K 1 b bKE = Y j Y Y S (h) Y Y <Q (17)T+h T+h T+h T+h^ KT+hjT T+hjTY
(r) Kb bwhere Y is the sample mean of the R bootstrap replicates Y and S (h) is the cor-^T+hjT T+hjT Y
5 responding sample covariance. The quantity Q in (17) is the (1 )100% percentile of theK
bootstrap distribution of the following quadratic form
h i h i0
K 1 b b b bY Y S (h) Y Y : (18)^T+hjT T+hjT T+hjT T+hjTY
Furthermore, Kim (1999) proposes using the Bonferroni approximation to obtain prediction cubes
with nominal coverage of at least (1-)100% which are given by
n h io N KC = y jy 2[ q ;q 1 (19)T+h n;T+h n;T+h n=1 K K
2 2
6where ==N.
5 KKim (1999) does not explicitly show how S (h) should be de ned. Alternatively, one can obtain S (h) by^^ YY
substituting the parameters in (9) by their bootstrap estimates and computing the average through all bootstrap
replicates. By calculating it with the sample covariance or by substituting the bootstrap parameters in the
corresponding expressions we get similar results.
6Actually, what Kim (1999) de nes as KC is slightly di erent from (19) as he uses the percentile and percentile-t
methods of Hall (1992). Here we prefer to use the Bonferroni prediction regions in (19) because they are better
suited to deal with potential asymmetries of the error distribution; see Hall (1992).
9