26th International Symposium on Forecasting
International Forecasting Society Santander; June, 11-14, 2006
Multivariate Modeling of Seasonal Time Series
and Temporal Disaggregation
Enrique M. Quilis
1Instituto Nacional de Estadística Rosario Pino, 14-16. Office 15.34 28020 - Madrid (SPAIN)
Abstract
In this paper I generate linear transformations of high frequency economic indicators based on two types of models specially designed for representing multiple seasonal time series: Vector of Autoregressions and Moving Averages (VARMA) and Bayesian Vector of Autoregressions (BVAR). These transformations are used to reformulate the standard Chow-Lin temporal disaggregation model, in order to provide a multivariate, model-based decomposition of the corresponding high frequency estimates, specially tailored for short-term monitoring. I use a real set-up, based on the Spanish Quarterly National Accounts (QNA), to demonstrate the methods.
Keywords: BVAR and VARMA Models, Calibration, Seasonality, Linear
Transformations, Forecasting, Temporal Disaggregation, Economic Indicators, Quarterly National Accounts.
JEL Codes: C11, C32, C51, C53, C82.
1 I am indebted to Ana Abad and Leandro Navarro for their help. I also thank J. Bógalo, J.R.
Cancelo, A. Cuevas, T. Di Fonzo, A. Espasa, R. Frutos, M. Jerez, A. Maravall, D. Peña, and E. Salazar for stimulating discussions on multivariate modeling or temporal disaggregation. Any views expressed herein are my own and not necessarily those of the Instituto Nacional de Estadística.
INDEX
1. Introduction
2. VARMA modeling of seasonal time series 3. BVAR modeling of seasonal time series
4. Linear transformations based on multivariate models 5. Temporal disaggregation
6. Application: monthly estimates of Gross Value Added References
1. INTRODUCTION
Multivariate models and temporal disaggregation methods are widely used in many economic and statistical applications, ranging from forecasting and economic policy analysis to the compilation of monthly indicators and the Quarterly National Accounts. Nevertheless, they are seldom combined2. In this
paper I examine an approach that combines seasonal multivariate models (VARMA and BVAR) with temporal disaggregation methods (the Chow-Lin procedure). The link is provided by linear transformations of high frequency economic indicators based on the multivariate models. These transformations are used to reformulate the standard Chow-Lin temporal disaggregation model, in order to provide a multivariate, model-based decomposition of the corresponding high frequency estimates specially tailored for economic analysis and short-term monitoring. I use a real set-up, based on the Spanish Quarterly National Accounts (QNA), to demonstrate the methods.
The combination of high-frequency, multivariate models with temporal disaggregation and extrapolation tools have been fostered by the increased relevance of early estimates of Gross Domestic Product (GDP)3. These
estimates rely mostly on seasonal monthly indicators that provide the basic information concerning the sign, size and curvature of short-term variation of QNA aggregates that, properly added4, produce the GDP estimates.
I also revise the issue of modeling seasonal behavior in multivariate models. Seasonality is a structural, permanent feature in many economic time series that should be explicitly taken into account in their multivariate modeling, in the same way as it is actually considered in the standard practice of univariate analysis.
However, due to its intrinsic nature, seasonality complicates notably the specification, estimation and analysis of multivariate models. This fact has determined that most econometric applications perform a preliminary seasonal adjustment5 that greatly simplifies the tasks of multivariate modeling. This practice, sensible at first sight, may have adverse effects on the dynamic specification (Wallis, 1974; Sims, 1974) and may have a negative effect on the estimation and inference based on the corresponding multivariate models, see Maravall (1993), Harvey and Scott (1994) and Ghysels and Perron (1993), among others.
In this paper I explore the performance of two types of models that allow an explicit modeling of the seasonal features of a vector time series. The first one,
2 Remarkable exceptions are Zadrozny (1990) and Jerez et al. (2005), among others.
3 These estimates are released 20-45 days after the end of the quarter, and are also known as
‘flash estimates’.
4 Taking into account the non-additivity issues related to chain-linked indexes.
5 The theory, software and practice of seasonal adjustment has experienced a remarkable
development during the last decade, becoming an almost universal tool in the hands of short-term analysts and econometricians.
a multiplicative seasonal VARMA model is specified by means of a stepwise procedure based on a simple while flexible multivariate filtering. The second one, extends the VAR framework to the seasonal case by way of bayesian constraints aimed at lessening the adverse effects of the overparameterization, worsened by the inclusion of seasonal elements in the model.
The main results of the paper may be summarized as follows:
• The relationship between BVAR and VARMA models is complementary, so that the conclusions obtained using one model can be either corroborated or refuted by the other. Therefore, their joint use improves the modeling and analysis of vector time series.
• VARMA and BVAR models provide a sensible, operative way to include seasonal behavior in multivariate models, getting rid of the need to perform preliminary seasonal adjustment to uncover the dynamic interactions in a vector time series.
• BVAR models allow great flexibility to tailor the model to the specific needs of the user: calibration depending on the forecasting horizon6,
links with theoretical models7 and sophisticated modeling of low-frequency and seasonal-frequencies interactions8.
• Linear transformations based on multivariate models play an important role in the use of multivariate model and bridge the gap with their final users, whose backgrounds are often unrelated to econometrics and quantitative modeling.
• Temporal disaggregation and extrapolation is improved when combined with explicit multivariate models, specially for monitoring and analysis. • Available software allows the routine use of these methods on real data
sets in a timely way.
2. VARMA MODELING OF SEASONAL TIME SERIES
In this section I briefly describe the main features of the class of vector autoregressive moving-average (VARMA) models. My purpose is to design an appropriate strategy for dealing with seasonal time series that encompasses the well-known specification, estimation and diagnostic stages of multivariate time series models, see Tiao et al. (1979), Tiao and Box (1981), Liu (1986), Lütkepohl (1991), Reinsel (1993) and Tiao (2001), for an in-depth analysis of such models. Consider a k-dimensional vector, Zt, which evolves following VARMA(c,d) model,
which can be expressed by the following equation:
[2.1] Zt =µ+P1Zt−1+P2Zt−2+...+PcZt−c+Ut−T1Ut−1−T2Ut−2−...−TdUt−d
6 See Tiao and Xu (1993) for a detailed analysis of this issue.
7 See Ingram and Whiteman (1994) and Del Negro (2003).
being c≥s and/or d≥s and s the seasonal frequency. In this expression, the Pi
and Tj are kxk matrices (with i=1..c, j=1..d) that conform, respectively, the
autoregressive (VAR) and moving average vector operators (VMA):
[2.2] d d 2 2 1 d c c 2 2 1 c B T ... B T B T I ) B ( T B P ... B P B P I ) B ( P − − − − = − − − − =
Furthermore, the vector Ut can be characterized by the following distributional
properties:
[2.3] Ut:kx1~N(0,Σ )
being Σ, in general, a non-diagonal matrix.
Additionally, it is assumed that all the roots of the determinantal polynomials |P(B)| and |T(B)| lie either on or outside the unit circle. The series Zt will be
stationary when the roots of |P(B)| are all outside the unit circle, and will be invertible when the roots of |T(B)| are all outside the unit circle.
Aiming at the most parsimonious representation of this model, it is usually assumed, without loss of generality, that the VAR and VMA operators as expressed in equation [2.2] can be factorized in one of the following ways:
[2.4] t s Q q d t q s Q d p s P c s P p c U ) B ( ) B ( ) B ( T or U ) B ( ) B ( ) B ( T ) B ( ) B ( ) B ( P or ) B ( ) B ( ) B ( P Ξ Θ Θ Ξ Φ Γ Γ Φ = = = =
with c=p+sP and d=q+sQ. In this way, we transform the initial additive seasonal VARMA model into a multiplicative seasonal VARMA model. Due to the non-commutative character of the involved matrices, four different, and alternative, multiplicative expressions can be obtained from the VARMA model in equation [2.1]. This fact, compounded to the difficult application of the standard identification tools in the seasonal case, recommends the use of a stepwise specification procedure that disentangles the seasonal/regular and additive/multiplicative issues comprised in the full specification of seasonal VARMA models.
The suggested procedure is an adaptation to the multivariate case of the univariate one proposed by Liu (1989, 1993), included in the SCA (Scientific Computing Associates) software in its automatic specification module SCA-Expert. The adapted procedure consists of three steps.
Step 1: Seasonal and regular multivariate filtering
The residuals of two VARMA(1,1) models are estimated, one incorporating seasonality and the other purely regular or non-seasonal:
[2.5] t R 1 R t t s s 1 s s t Z ) B P I ( ) B T I ( S Z ) B P I ( ) B T I ( R − − = − − = − −
This step is equivalent to applying two filters in a parallel way: a seasonal one (S) and a regular one (R), so that the first one allows a tentative approach for identifying the model underlying to the regular part and the second, correspondingly, for the seasonal one.
Step 2: Specification of seasonal and regular multivariate operators
Thereafter, the usual multivariate model identification procedures are used, Tiao and Box (1981), Tiao and Tsay (1983) and Tsay and Tiao (1985), and the VARMA models for Rt y St, are tentatively identified:
[2.6] t s Q t s P t q t p W ) B ( S ) B ( V ) B ( R ) B ( Ξ Γ Θ Φ = =
Step 3: Selection of final multiplicative model
The combination of the operators in [2.6] can generate up to four multiplicative models, as seen in [2.4]. The selection of one of them is carried out by comparing the values of the likelihood functions of the multiplicative models with the corresponding likelihood function of the additive model in which they can be nested:
[2.7] Pc(B)Zt =Td(B)Ut c= p+sP d =q+sQ
Alternatively, a minimum distance criterion can be used to select the proper multiplicative model.
The most affine multiplicative model to the additive model can be used as a starting point for formulating, after the usual diagnostic tests, a model that is compatible with the observed sample.
The estimation of VARMA models can be carried out by way of the maximization of the corresponding likelihood function, which, in its turn, admits two different specifications: a conditional and an exact one. The exact estimation procedure is more precise than the conditional one in the VARMA case but it is significantly more prone to overparametrization and numerical problems. For that reason, its use is left to the final stages of the analysis. See Hillmer and Tiao (1979) for additional details.
3. BVAR MODELING OF SEASONAL TIME SERIES
Let Zt=(z1, z2, .., zk)t’ be a vector of observations on k variables at time t,
with t=1..n. Also, let Zt evolve following a vector autoregressive model (VAR) of
order p, being p a multiple of s, the observations per year (p=τs, τ≥1), if it can be expressed in the following way:
[3.1] Zt =µ+Φ1Zt−1+Φ2Zt−2 +...+ΦpZt−p +Ut
where µ is a vector of k constant terms and Φh, h=1,…,p, are kxk matrices. The
term Ut represents a vector of zero-mean stochastic disturbances, which are
serially uncorrelated and with constant variance-covariance matrix. Again, it is assumed:
[3.2] Ut :kx1~ N(0,Σ )
VAR models are very general structures and, depending on the nature of the matrices Φh and Σ, diverse particular cases arise. See Tiao et al. (1979),
Sargent (1979), Sims (1980, 1986, 1993), Tiao and Box (1981), Liu (1986), Lütkepohl (1991), Reinsel (1993), Espasa and Cancelo (1993), Enders (1995) and Tiao (2001), for an in-depth review of these models
VAR models are prone to overparameterization and, as a consequence, to overadjustment, imprecise estimation and poor forecasting records. With the aim of solving these problems, Litterman (1984a, 1984b, 1986), Doan et al. (1984) and Todd (1984, 1988) propose imposing probabilistic or inexact constraints, oriented towards shrinking the size of the parametric space and, as a result, lessening the above mentioned problems. These restrictions are amenable to a bayesian interpretation and may include quite different structures, depending on the non-sample information that the analyst wants to incorporate in the model. Due to its origin, this prior if often named as 'Minnesota prior' or 'Litterman prior'.
Raynauld and Simontato (1993) generalize this prior to include seasonality9. For
that purpose, it seems convenient to modify the notation employed in equation [3.1]. The i-th equation of a VAR(p) is:
[3.3]
(
)
i i i p , k , i p , 1 , i 1 , k , i 1 , 1 , i i ) p ( k ) p ( 1 ) 1 ( k ) 1 ( 1 p n i i Z Z Z Z U x U Z + = + = − β φ φ φ φ µ M M M L L L Lbeing Zj(h) the vector containing (n-p) observations of the j-th series lagged by
h periods. Note that the vector of regressors x is the same for all the equations, so that the simultaneous consideration of all the k equations integrating the VAR gives rise to the following expression:
[3.4] Z =(Ik⊗x)β+U =Xβ+U
being ⊗ the Kronecker product. Of course, vector β is related to the Φ matrices according to:
[3.5] β =vec(Φ' ) with Φ =( µ Φ1 L Φp ) Consequently, the variance-covariance matrix of U is: [3.6] ΣU =Σ ⊗In−p with Σ =
{
σí,j i,j=1..k}
Model [3.4] has a similar aspect to standard linear model of the regression analysis.
The bayesian specification of a VAR model considers that the parameters in β are random variables that are distributed according to a multivariate gaussian distribution:
[3.7] ~ N( *,V )
β
β β
The Raynauld-Simonato prior assumes that the underlying model is a random walk with seasonal unit roots and drift term10:
[3.8] (1−B)(1−Bs )zit, =µi +uit ∀i
Consequently, the mean of the prior on the parameters of the VAR(p) model is given by:
[3.9] + ≠ ≠ + = = − = = = ∀ = = 1 s , s , 1 h j i 0 1 s h j i 1 s , 1 h j i 1 i 0 h , j , i i * φ µ β
The variance-covariance matrix of β is assumed to be diagonal, and is controlled by a vector π containing m hyperparameters, which describes the properties of this matrix in a parsimonious way.
[3.10] ∀ ∀ = ∀ = = h j , i )) / ( sd st F ( ) ( v i ) ( ) ( v ) V ( diagonal 2 jj ii h h j , i 1 h , j , i 2 ii i σ σ π φ σ π µ µ β
The functions sth and sdh quantify the degree to which the variance is reduced
as a function of the lag. These depend on two hyperparameters: π3 and π4. The
first function modulates the decay in the interseasonal lags:
[3.11] − = − = − = − − − − = − − − 1 s 3 .. s 2 h 1 s 2 .. s h 1 s .. 1 h . etc )) 1 s 2 ( h ( )) 1 s ( h ( h st 3 3 3 h π π π
The second function controls for the seasonal decay, presenting a constant pattern for non-seasonal lags:
[3.12] d 1 4 h h sd =π − with: [3.13] + = + = = = s 3 .. 1 s 2 h s 2 .. 1 s h s .. 1 h . etc 3 2 1 dh
If the prior in [3.8] is used, then the additional constraint sts+1=sds+1=1 is
imposed. The interaction of both functions allows considering a variety of situations concerning the regular and seasonal dynamics.
Once the prior β~N(β*,V
β) is specified, the estimation is carried out using the
[3.14] βˆ ((Σ 1 (x'x)) V 1) 1((Σ 1 x' )Z V 1β* )
β
β− − − −
− ⊗ + ⊗ +
=
The corresponding variance-covariance matrix of βˆ is:
[3.15] 1 1 1
ˆ (( (x'x)) V )
Vβ = Σ− ⊗ + β− −
The evaluation of [3.14] and [3.15] requires the prior knowledge about Σ. Usually, this matrix is determined by means of the residual variance-covariance matrix of k univariate AR(p) models or that of an unrestricted VAR(p).
It must be underlined that the estimator in [3.14] is valid under symmetric constraints and as well as under asymmetric ones. In practice, full-system estimation is replaced by equation by equation estimation due to limited departures from the symmetric prior and also to estimation burden.
The determination of the proper values for the hyperparameters is a critical step in BVAR modeling. In this paper, BVAR calibration if performed by means of the quasi-likelihood criterion proposed by Doan et al. (1980), numerically implemented using the axial search procedure outlined in Todd (1988).
4. LINEAR TRANSFORMATIONS BASED ON MULTIVARIATE MODELS
Linear transformations of vector time series play an important role in multivariate models in order to simplify its structure or reveal hidden patterns, see Tiao et al. (1993) and Galeano and Peña (2004), among others.
In this paper I apply a linear transformation linked to the conversion of the reduced-form multivariate VARMA or BVAR model into a structural model. I assume, for the sake of simplicity that a VAR(1) model has been selected:
[4.1] ) , 0 ( N iid ~ U U Z Z t t 1 t t Σ Φ + = −
Under appropriate identification assumptions, a structural model may be associated with [4.1]: [4.2] ) , 0 ( N iid ~ E DE AZ Z A t t 1 t t 0 Ω + = −
Usually, the identifying assumptions state that the diagonal elements of A0 and
Equation [4.2] implies that a linear transformation of Zt evolves according to a
VAR(1) model perturbed by orthogonal shocks. These shocks drive the system and may have meaningful economic interpretation.
There are many possibilities to build the structural transformation and this issue is often very controversial, see Stock and Watson (2001) an the references cited therein. In this paper I use an orthogonalization scheme based on a recursive representation of the structural model, because it suits both the nature of the temporal disaggregation model that will be later considered and the dimension of the system (k=2).
If Ψ is the lower triangular matrix derived from the Cholesky decomposition of Σ, it holds that ΨΨ’=Σ. This matrix generates the following structural transformation: [4.3] I D A A H H A 1 0 1 0 = = = − − Φ Ψ
with H=[diag(Ψ)]1/2 and Ω=diag(Ψ).
The corresponding transformation, in the case k=2, is: [4.4] t 2 1 t 2 1 z z 1 0 1 x x = γ
Note that [4.4] allow us to split z2 in two components: one linearly linked to z1
and a purely idiosyncratic factor, that is, the residual variation not explained by z1:
[4.5] z2t, =γz1t,+x2t, =x1t, +x2t,
5. TEMPORAL DISAGGREGATION
In this section I briefly review the main features of the temporal disaggregation procedure used to estimate high-frequency data, taking into account the linearly transformed indicators as input. The basic approach is described in Chow-Lin (1971) and Fernández (1981). See Quilis (2005) for additional references as well as an analysis of its role in the compilation of the Spanish QNA.
The procedure assumes a linear model that links the (observable) high-frequency indicators with the (unobservable) low-high-frequency variable. The model considers a static relationship between y and x, additively perturbed:
The innovation u follows a stationary AR(1) process (therefore, assuming cointegration between y and x) or a non-stationary I(1), random walk process (therefore, excluding cointegration between y and x):
[5.2]
[
]
1 1 )) I ( )' I ( v ) v , 0 ( N ~ u 1 n n 2 + + − < ≤ =σ ρΞ ρΞ − ρThe auxiliary matrix Ξ is defined as:
− − − = 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 L L L L L L L L L Ξ
Initial conditions depend on the value of ρ: [5.3] = < < − − = 1 if 1 1 if 0 )) 1 /( , 0 ( N u 2 2 0 ρ ρ ρ σ
The model includes a temporal constraint that makes y quantitatively consistent with its low-frequency counterpart Y, called “benchmark”:
[5.4] Y =Cy
with a temporal aggregation-extrapolation matrix defined as: [5.5] C=(IN ⊗c|ON,n−fN)
where N is the number of low-frequency observations, ⊗ stands for the Kronecker product, c is a row vector of size f which defines the type of temporal aggregation and f is the number of high frequency data points for each low frequency data point (frequency conversion ratio). If c=[1,1,...,1] we would be in the case of the temporal aggregation of a flow, if c= [1/f,1/f,...,1/f] in the case of the average of an index and, if c=[0,0,...,1], an interpolation would be obtained. When the size of the indicators' sample is not conformable with the one of the low-frequency series (n>fN), a situation of extrapolation shows up.
The estimation of y according to the model [5.1]-[5.3] and satisfying the constraint [5.4]-[5.5] is performed by means of the following Best11 Linear Unbiased Estimator (BLUE):
[5.6] yˆ=xβˆ +vC'V−1Uˆ =xβˆ +LUˆ
with V=C’vC. The low-frequency disturbance term is defined as: [5.7] Uˆ =Y −Xβˆ
with X=Cx.
Equation [5.6] expresses the estimator as the combination of a term linearly linked to the indicator and a temporally disaggregated residual series. The main feature of the estimator is the dependency of the temporal disaggregation filter L on the form adopted by the high-frequency model, and in particular, on the dynamic structure of its disturbance term u. The Generalized Least Squares (GLS) estimator of β is:
[5.8] βˆ =(X'V−1X)−1(X'V−1Y)
The method generates confidence intervals for the high-frequency estimates according to the variance-covariance matrix:
[5.9] Σyˆ =(In−LC)v+(x−LX )Σβˆ(x−LX )'
Expressions [5.6] to [5.9], which fully define the Chow-Lin method, require for its implementation prior knowledge of the variance-covariance matrix v of the high-frequency disturbance term u which depends on ρ, see [5.3]. The estimation of this parameter is accomplished by means of the evaluation of the log-likelihood function of the implied low-frequency model. The function is:
[5.10] ) X Y ( ) ' C ) ( Cv ( )' X Y ( 2 1 |) ' C ) ( Cv ln(| 2 1 ) 2 ln( 2 N ) | , ( 1 2 2 2 β ρ β σ ρ πσ ρ σ β − − − − − − = − l
This optimization is performed by means of a grid search on the stationary domain of ρ, and pinning down the values of β and σ2 that maximizes [5.10]
conditioned on the selected value for ρ, see Bournay and Laroque (1979). Little difference arises between Chow-Lin and Fernández estimates if ρ ≥ 0.8. Consequently, if ρ exceeds an upper bound (e.g., 0.98) the Fernández procedure may be applied:
[
]
[
x'C' C(D'D) C' Cx]
x'C'[
C(D'D) C']
Y ˆ = −1 −1 −1 −1 β [5.11] yˆ=xβˆ+(D'D)−1C'[
C(D'D)−1C']
−1(Y−Cxβˆ) Ξ + =In D6. APPLICATION12: MONTHLY ESTIMATES OF GROSS VALUE
ADDED
In this section I apply the methodology outlined in the previous sections to derive monthly estimates of industrial Gross Value Added (GVA), temporally consistent with the data provided by the Spanish Quarterly National Accounts (QNA). An application of the VARMA and BVAR methodology outlined in this paper to the US Census housing data is fully spelled out in Quilis (2004c).
The benchmarking process uses as high-frequency input a model-based linear transformation of two monthly indicators.
6.1. Data
The high-frequency indicators are the monthly exports of industrial manufactures (z1) and the monthly index of industrial production of
manufactures (z2), both time series exclude energy-related products. The first
one is valued at constant euros of year 2000 and is compiled by the Spanish State Tax Agency. The second one is compiled by Spanish National Statistical Institute and it is a Laspeyres fixed-base index, average 2000=100. More information on the series can be found on the web pages http://www.aeat.es and http://www.ine.es, respectively. The sample ranges from 1995:01 until 2004:12. Data corresponding to 2005 have been preserved for out-of-sample forecasting analysis. The complete data-set and detailed results are available from the author.
Both indicators are routinely used for short-term monitoring and economic analysis on a real-time basis due to their sampling frequency and timeliness, see Espasa and Cancelo (1993) and the references cited therein.
The low-frequency benchmark, provided by the Spanish QNA, is the industrial manufacturing Gross Value Added (GVA). It is a Laspeyres chain-linked index, reference 2000=100. This series is also a basic element in any short-term economic analysis due to its integration in the general framework of the National Accounts. This framework ensures, among other things, the quantitative coherence with the remaining economic aggregates of the system (e.g., demand and employment).
12 The univariate ARIMA estimates as well as the signal extraction, have been computed using
the programs TRAMO and SEATS (Gómez and Maravall, 1996; Caporello & Maravall, 2004). VARMA models have been estimated using the Scientific Computing Associates (SCA) program (Liu and Hudak, 1995). The application of the BVAR methodology has been carried out by way of a specific Matlab library (Quilis, 2004a). In the same way, temporal disaggregation has been applied by means of another specific Matlab library (Quilis, 2004b), integrated with standard spreadsheets (Abad and Quilis, 2005).
Due to their disparate units of measurement, both series are transformed to index form, first observation=100. The next figure displays their different rates of growth as well as their similar seasonal patterns:
Figure 6.1: Levels 50 100 150 200 250 300 95 96 97 98 99 00 01 02 03 04 05 Exports Industrial Output
Standard mean-rank analysis suggests transforming the data taking logarithms. This is the last transformation applied to the data. Hence, calendar, outlier and seasonal effects, if present, remain in the data.
6.2. VARMA model13
Intermediate S-filtering and R-filtering suggest a regular VAR(4) and a seasonal VAR(1), respectively. The multiplicative model more akin to the nesting, restricted additive VAR(16) is a VAR(4)(1)12. The diagnostic stage
reveals the need to include a constrained seasonal VMA(1) operator, in order to take into account some long-lasting seasonal structure in the industrial production indicator. Final estimation, by conditional maximum likelihood14, yields the following results:
13 Detailed intermediate results are available from the author.
Table 6.1: VARMA Model. Maximum likelihood estimation ) , 0 ( N iid ~ U U ) B I ( Z ) B I )( B B B B I ( t t 12 12 t 12 12 4 4 3 3 2 2 1 Σ Ξ Γ Φ Φ Φ Φ − − − − = − −
Matrices Real Imag. Mod.
0.33 -0.38 0.08 0.13 1.00 0.00 1.00 0 0 -- -- 1.00 0.00 1.00 0 0.50 -- 0.10 -0.50 0.69 0.85 0 0.27 -- 0.05 -0.50 -0.69 0.85 0.34 0.57 0.08 0.14 -0.06 0.78 0.78 0 0.73 -- 0.05 -0.06 -0.78 0.78 0.33 -0.67 0.09 0.15 -0.55 0.00 0.55 0 0 -- -- 0.00 0.00 0.00 0.18 0.86 0.05 0.06 0.96 0.00 0.96 0 0.96 -- 0.01 0.18 0.00 0.18 0 0 -- -- 0.35 0.00 0.35 0 0.35 -- 0.34 0.00 0.00 0.00 0.35 0.16 0.44 0.16 0.13 0.05 1 0.75 1.75 0.75 1 0.25
Notes: 0 and -- means restricted parameters.
Σ matrix has been scaled by 102 and Γ is the corresponding correlation matrix.
Ξ12 Σ1 Γ Φ1 Φ2 Φ3 Φ4 Eigenvalues
Estimate Standard Error
Φ12
The hypothesis that z2 (industrial output) Granger-causes z1 (exports) is not
rejected by the data. The likelihood ratio is 0.99 and the corresponding χ2 test
sustains the hypothesis. There is a significative, positive contemporaneous relationship between both series that supports a Cholesky-based, linear transformation of the observed vector Zt.
The residuals obtained from the VARMA model do not show any major inadequacy, as may be seen from their corresponding SCAN15 table:
Table 6.2: VARMA Model SCAN analysis of the residuals
q => 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 p 6.3. BVAR model16
The results derived from the univariate ARIMA analysis and the multivariate VARMA analysis suggest the specification of BVAR(16), with a Raynauld-Simonato prior centered around the ∆∆12 difference operator, without drift. The
corresponding hyperparameters that control the variance of the prior are calibrated by means of an axial search on the log-determinant of the one-step ahead prediction error variance-covariance matrix. The backtesting exercise runs from 2003:01, generating 24 observations. The following table summarizes the results:
Table 6.3: BVAR model. Final axial calibration
πµ π1 F1,2 F2,1 π3 π4 Σ1,1 Σ2,2
Range 0 0-4 0-4 0-1
Calibrate 0 2 0.25 0.07 0.6 0.9 0.64 0.13
Discretization: ∆π=0.01
Note: Σ matrix has been scaled by 102
0-1
--An intermediate axial search was performed to test the symmetric nature of F. As may be seen in the following figure, the hypothesis that z2 (industrial
output) Granger-causes z1 (exports) is not rejected but with less confidence
than in the VARMA case.
Figure 6.2: Function ϕ(β(π))=log(|E|) Evaluation using πµ=0, π1=2, π3=0.6, π4=0.9
As usual, detailed estimation results of the 16 Φ matrices are not reported and the dynamic mechanism is summarized by means of the corresponding orthogonalized impulse-response function and its confidence intervals17:
Figure 6.3: BVAR model
Response function to the orthogonalized impulses Ordering: (z1: Exports) Î (z2: Industrial Output)
1 12 24 36 -0.02 -0.01 0 0.01 0.02 0.03 0.04 1 12 24 36 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0 12 24 36 -0.02 -0.01 0 0.01 0.02 0.03 0.04 1 12 25 36 -0.02 -0.01 0 0.01 0.02 0.03 0.04
The residuals obtained from the BVAR model do not show any major inadequacy18, as may be seen from their corresponding SCAN table:
17 These intervals, associated to two standard deviations, have been calculated using the
analytical procedure outlined in Lütkepohl (1991).
18 Some small, negative autocorrelation appears in the residuals series u
2t. This fact may be
Table 6.4: BVAR Model SCAN analysis of the residuals
q => 0 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 p 6.4. Forecasting performance
A simple exercise has been carried out to gauge the forecasting performance of the VARMA(4,0)(1,1)12 model and the BVAR(16) model. The
data corresponding to 2005 has been selected for out-of-sample forecasts, both 1-step ahead (h=1) and 12-step ahead (h=12). The experiment yields a better record for the BVAR model:
Table 6.5: VARMA and BVAR Models Forecasting performance
Horizon MAPE RMSE MAPE RMSE
z1 5.13 18.52 4.72 14.40 z2 3.60 5.82 3.50 5.75 z1 5.99 17.15 5.32 14.34 z2 3.14 4.71 3.12 5.34 VARMA BVAR 1 12 6.5. Linear transformation
VARMA and BVAR models provide an explicit statistical model to generate a linear transformation of the observed vector Zt , according to the
Cholesky-based decomposition outlined in section 4. I have resampled the residuals by means of 10000 bootstrap replications in order to estimate the transformation and its standard errors:
scalar VMA. Experiments with a BVAR(24) were numerically conflictive, due to the short available sample relative to the number of (shrinked) parameters.
Table 6.6: VARMA and BVAR Models Linear transformation. Bootstrap estimates
1.00 0.00 1.00 0.00 -0.73 1.00 -0.74 1.00 0 0 0 0 0.08 0 0.08 0 s.e. Φ0 VARMA BVAR
Both models yield equivalent transformations. Due to the better forecasting performance of the BVAR model I select γ=0.74 as the basic parameter to achieve a multivariate model-based decomposition of z2:
[6.1] z2t, =x1t, +x2t, =γz1t, +x2t,
The first component (x1) has a stochastic part that evolves according to an
airline model (0,1,1)(0,1,1)12: [6.2] b 0.04 ) B 1 )( B 1 ( ) B 44 . 0 1 )( B 64 . 0 1 ( n 12 1t, b1 12 t, 1 = − − − − = σ
Its evolution closely resembles the common non-stationary factor affecting both series19. This factor explains most of their trend and seasonality:
Figure 6.4: BVAR linear transformation x1: exports-driven industrial output
3.0 3.2 3.4 3.6 3.8 4.0 4.2 95 96 97 98 99 00 01 02 03 04 05
The second component (x2) has a stochastic part that evolves according to a
seasonal IMA(1,1) model:
[6.3] b 0.02 ) B 1 ( ) B 51 . 0 1 ( n 12 2t, b2 12 t, 2 = − − = σ
Its evolution20 is also non-stationary and is linked to a common factor that encapsulates the differential between industrial output and exports. This transformation may be interpreted as a filter that reduce the non-stationary features of the observed data but do not eliminate them completely, yielding the following stable relationship:
[6.4] 1t, t 12 t, 2 12)z 0.45(1 B )z B 1 ( − = − +ς
Since ζt is I(0), equation [6.4] may be stated as a stable link between the
annual rates of growth of exports and industrial production. Figure 6.5: BVAR linear transformation
x2: idiosyncratic industrial output
0.6 0.8 1.0 1.2 1.4 1.6 1.8 95 96 97 98 99 00 01 02 03 04 05
6.6. Temporal disaggregation
Using x=(x1 x2) as the high-frequency indicators and including 2005 data, the
application of the Chow-Lin method yields the following results: Table 6.7: Chow-Lin high-frequency model
Estimate Standard Error
β0 -159.06 11.90 β1 56.88 2.92 β2 40.64 4.72 ρ 0.96 ρ(y,ye) 0.95 ρ(∆12y,∆12ye) 0.83
The likelihood profile is well-behaved and shows a maximum21 close to the
upper bound (0.99). Although the Fernández method provides similar results, the Chow-Lin method ensures a better fit to the quarterly benchmark.
Figure 6.6: Chow-Lin high-frequency model Implied likelihood function
21 A preliminary coarse search rejects the presence of a maximum in the non-positive domain of
The resulting high-frequency estimates and its ±2σ interval are: Figure 6.7: Chow-Lin high-frequency estimates
Point estimate and ±2σ interval
40 50 60 70 80 90 100 110 120 95 96 97 98 99 00 01 02 03 04 05
This model may be used to explore the decomposition of growth in a given period, 2005 for instance:
Figure 6.8: Chow-Lin high-frequency estimates Decomposition of year-on-year growth: 2005
-6 -4 -2 0 2 4 05:01 05:04 05:07 05:10 Aggregate Exports-driven Industrial idyiosincratic Benchm arking residual
During 2005 exports-driven output exerted a negative impact on industrial GVA, specially during the second semester. This pressure was compensated by the idiosyncratic industrial output and the residual term.
APPENDIX A. VMA OPERATORS AND BVAR HYPERPARAMETERS
The addition of VMA operators is a way to lessen the overparameterization risk of VAR models. Instead of imposing bayesian constraints that shrinks the value of VAR parameters, VARMA models expand the “regression space” by incorporating the lagged innovations in a parsimonious way:
To explore this equivalence let us consider the non-seasonal VARMA(1,1) case: [A.1] Zt =ΦZt−1+Ut −ΘUt−1
Assuming invertibility, the inclusion of the VMA operator is functionally equivalent to imposing a series of non-linear constraints on a VAR(∞) representation. This model may be truncated for a large enough value of p as: [A.2] Zt =Π1Zt−1+Π2Zt−2 +...+ΠpZt−p+Ut
The Πh matrices satisfy the following relationship:
[A.3] h( ) h 1..p
h =Θ Φ −Θ =
Π
In an approximate mode, an equivalence can be established between the elements of the Θ matrix and the vector of hyperparameters π of a BVAR:
• The global degree of uncertainty (π1) is represented by the Θ matrix, in
particular due to the Θ (Φ − Θ) operation,
• Elements outside the main diagonal of Θ constrain the cross dynamics, and play a similar role to the one played by the F matrix. As a result, blocks of variables, asymmetries, etc can be taken into account.
• The diagonal elements Θii constrict, for each series, its own dynamics.
Therefore, Θii can be regarded as the analogue to the π3 hyperparameter
under a geometric decay scheme.
Therefore, as can be seen, a VARMA(1,1) is equivalent to a high order VAR subject to non-linear constraints. As a consequence, the elements contained in Θ act functionally as hyperparameters, but their numerical determination is rather different: in the BVAR case, their computation is carried out separately by means of a calibration process, whereas in the VARMA case the numerical computation is determined at the same time than the Φ parameters, by maximizing the exact likelihood function.
Nonetheless, in case a BVAR model is calibrated according to maximum likelihood criteria, its similarity with a VARMA may be high. This similarity increases in the seasonal multiplicative case, since these models automatically introduce a series of exclusion constraints that remove the intermediate parameter matrices of the implied additive representation.
REFERENCES
ABAD, A. and QUILIS, E.M. (2005) "Software to perform temporal disaggregation of economic time series: a Matlab library and its Excel interface", OECD-Eurostat Workshop on Frontiers in Benchmarking Techniques, Luxembourg, April, 7-8.
BOURNAY, J. and LAROQUE, G. (1979) "Réflexions sur la méthode d'elaboration des comptes trimestriels", Annales de l'INSEE, n. 36, p. 3-30.
CANOVA, F. (1992) "An alternative approach to modeling and forecasting seasonal time series", Journal of Business and Economic Statistics, vol. 10, n. 1, p. 97-108.
CANOVA, F. (1993) "Forecasting time series with common seasonal patterns", Journal of Econometrics, vol. 55, n. 1. p. 73-200.
CAPORELLO, G. and MARAVALL, A. (2004) "Program TSW. Revised manual. Version may 2004", Banco de España, Documento Ocasional n. 0408. CASALS, J., JÉREZ, M. and SOTOCA, S. (2005) "Empirical modeling of time
series sampled at different frequencies", OECD-Eurostat Workshop on Frontiers in Benchmarking Techniques, Luxembourg, April, 7-8.
CHOW, G. and LIN, A.L. (1971) "Best linear unbiased distribution and extrapolation of economic time series by related series", Review of Economic and Statistics, vol. 53, n. 4, p. 372-375.
DOAN, TH., LITTERMAN, R.B. and SIMS, CH.A. (1984) "Forecasting and conditional projection using realistic prior distributions", Econometric Review, n. 3, p. 1-144.
ENDERS, W. (1995) Applied econometric time series, John Wiley and Sons, New York.
ESPASA, A. and CANCELO, J.R. (1993) Métodos cuantitativos para el análisis de la coyuntura económica, Alianza Editorial, Madrid.
FERNÁNDEZ, R.B. (1981) "Methodological note on the estimation of time series", Review of Economic and Statistics, vol. 63, n. 3, p. 471-478. GALEANO, P. and PEÑA, D. (2000) "Multivariate analysis in vector time series",
Resenhas, vol. 4, p. 383-404.
GHYSELS, E. and PERRON, P. (1993) "The effect of seasonal adjustment filters on tests for a unit root", Journal of Econometrics, vol. 55, p. 57-98.
GÓMEZ, V. and MARAVALL, A. (1996) "Programs TRAMO and SEATS", Banco de España, Documento de Trabajo n. 9628.
HARVEY, A.C. and SCOTT, A. (1994) "Seasonality in dynamic regression models", Economic Journal, vol. 104, p. 1324-1345.
HILLMER, S.C. and TIAO, G.C. (1979) "Likelihood function of stationary multiple autoregressive moving average models", Journal of the American Statistical Association, vol, 74, n. 367, p. 652-660.
INGRAM, B.F. and WHITEMAN, C.H. (1994) "Supplanting the 'Minnesota' prior. Forecasting macroeconomic time series using real business cycle model priors", Journal of Monetary Economics, vol. 34, p. 497-510.
LITTERMAN, R.B. (1984a) "Specifying vector autoregressions for macroeconomic forecasting", Federal Reserve Bank of Minneapolis, Staff Report n. 92.
LITTERMAN, R.B. (1984b) "Forecasting and policy analysis with bayesian vector autoregression models", Federal Reserve Bank of Minneapolis, Quarterly Review, vol. 8, n. 4, p. 30-41.
LITTERMAN, R.B. (1986) "Forecasting with bayesian vector autoregressions - five years of experience", Journal of Business and Economic Statistics, vol. 4, n. 1, p. 25-38.
LIU, L.M. (1986) "Multivariate time series analysis using vector ARMA models", SCA Monograph.
LIU, L.M. (1989) "Identification of seasonal ARIMA models using a filtering method", Communication in Statistics, Vol. A18, p. 2279-2288.
LIU, L.M. (1993) "Modeling and forecasting time series using SCA-Expert capabilities", SCA Working Paper n. 127.
LIU, L.M. and HUDAK, G.B. (1995) The SCA Statistical System. Vector ARMA modeling of multiple time series, SCA.
LÜTKEPOHL, H. (1991) Introduction to multiple time series analysis, Springer Verlag, Berlín.
MARAVALL, A. (1993) "Stochastic linear trends. Models and estimators", Journal of Econometrics, n. 56, p. 5-37.
NEGRO, M. DEL and SCHORFEID, F. (2003) "Priors from general equilibrium models for VARs", Federal Reserve Bank of Atlanta, Working Paper n. 2002-14a.
QUILIS, E.M. (2004a) "BayVAR: una librería Matlab para el análisis VAR Bayesiano", Instituto Nacional de Estadística, Working Paper.
QUILIS, E.M. (2004b) "A Matlab library of temporal disaggregation methods", Instituto Nacional de Estadística, Working Paper.
QUILIS, E.M. (2004c) "Una aplicación de los modelos BVAR estacionales", CLM.Economía, n. 4, p. 207-247.
QUILIS, E.M. (2005) "Benchmarking techniques in the Spanish quarterly national accounts", OECD-Eurostat Workshop on Frontiers in Benchmarking Techniques, Luxembourg, April, 7-8.
RAYNAULD, J. and SIMONATO, J.G. (1993) "Seasonal BVAR models", Journal of Econometrics, vol. 55, p. 203-229.
REINSEL, G. (1993) Elements of multivariate time series analysis, Springer Verlag, New York.
SARGENT, Th.J. (1979) "Estimating vector autoregressions using methods not based on explicit economic theories", Federal Reserve Bank of Minneapolis, Quarterly Review, vol. 3, n. 3, p. 8-15.
SIMS, Ch.A. (1974) "Seasonality in regression", Journal of the American Statistical Association, vol. 69, n. 347, p. 618-626.
SIMS, Ch.A. (1980) "Macroeconomics and reality", Econometrica, vol. 48, n. 1, p. 1-48.
SIMS, Ch.A. (1986) "Are forecasting models usable for policy analysis?", Federal Reserve Bank of Minneapolis, Quarterly Review, vol. 10, n. 1, p. 2-15. SIMS, Ch.A. (1993) "A nine-variable probabilistic macroeconomic forecasting
model", in Stock, J.H. and Watson, M.W. (Eds.) Business cycles, indicators and forecasting, NBER Studies in Business Cycles, vol. 28, Univ. of Chicago Press, Chicago.
STOCK, J.H. and WATSON, M.W. (2001) "Vector autoregressions", Journal of Economic Perspectives, vol. 15, n. 4, p. 101-115.
TIAO, G.C., BOX, G.E.P., GRUPE, M.R., HUDAK, G.B., BELL, W.R. and CHANG, I. (1979) "The Wisconsin Multiple Time Series program (WMTS). A preliminary guide", Department of Statistics, University of Wisconsin, Working Paper.
TIAO, G.C. and TSAY, R.S. (1983) "Multiple time series modelling and extended sample cross-correlations", Journal of Business and Economic Statistics, vol. 1, n. 1, p. 43-56.
TIAO, G.C. and BOX, G.E.P. (1981) "Modeling multiple time series with applications", Journal of the American Statistical Association, vol. 76, n. 376, p. 802-816.
TIAO, G.C. and XU, D. (1993) "Robustness of maximum likelihood estimates of multi-step predictions: the exponential smoothing case", Biometrika, vol. 80, n. 3, p. 623-641.
TIAO, G.C., TSAY, R.S. and WANG, T. (1993) "Usefulness of linear transformations in multivariate time series analysis", Empirical Economics, p. 567-595.
TIAO, G. C. (2001), "Vector ARMA models". In A course in Time Series Analysis. Peña, D., Tiao, G. C., and Tsay, R. S. (Eds.) John Wiley and Sons, New York.
TODD, R.M. (1984) "Improving economic forecasting with bayesian vector autoregressions", Federal Reserve Bank of Minneapolis, Quarterly Review, vol. 8, n. 4, p. 18-29.
TODD, R.M. (1988) "Implementing bayesian vector autoregressions", Federal Reserve Bank of Minneapolis, Working Paper n. 384.
TSAY, R.S. and TIAO, G.C. (1985) "Use of canonical analysis in time series model identification", Biometrika, vol. 72, n. 2, p. 299-315.
WALLIS, K.F. (1974) "Seasonal adjustment and relations between variables", Journal of the American Statistical Association, vol. 69, n. 345, p. 18-31. ZADROZNY, P.A. (1990) "Estimating a multivariate ARMA model with
mixed-frequency data: an application to forecasting U.S. GNP at monthly intervals", Federal Reserve Bank of Atlanta, Working Paper.
