Modeling and Forecasting Time Series Sampled at Different Frequencies José Casals (jcasalsc@cajamadrid.) Miguel Jerez (mjerez@ccee.ucm.es) † Sonia Sotoca (sotoca@ccee.ucm.es) Departamento de Fundamentos del Analisis Economico II Universidad Complutense de Madrid Campus de Somosaguas S/N 28223 Madrid (SPAIN) † Corresponding author. The telephone numbers for the three authors are: Voice: (+34) 91 394 23 61; Fax: (+34) 91 394 25 91 1 2 Modeling and Forecasting Time Series Sampled at Different Frequencies 3 Modeling and Forecasting Time Series Sampled at Different Frequencies Abstract: This paper discusses how to specify an observable high-frequency model for a vector of time series sampled at high and low frequencies. To this end we first study how aggregation over time affects both, the dynamic components of a time series and their observability, in a multivariate linear framework. We find that the basic dynamic components remain unchanged but some of them, mainly those related to the seasonal structure, become unobservable. Building on these results, we propose a structured specification method built on the idea that the models relating the variables in high and low sampling frequencies should be mutually consistent. After specifying a consistent and observable high-frequency model, standard state-space techniques provide an adequate framework for estimation, diagnostic checking, data interpolation and forecasting. Our method has three main uses. First, it is useful to disaggregate a vector of low- frequency time series into high-frequency estimates coherent with both, the sample information and its statistical properties. Second, it may improve forecasting of the low-frequency variables, as the forecasts conditional to high-frequency indicators have in general smaller error variances than those derived from the corresponding low-frequency values. Third, the resulting forecasts can be updated as new high-frequency values become available, thus providing an effective tool to assess the effect of new information over medium term expectations. An example using national accounting data illustrates the practical application of this method. Keywords: State-space models, Kalman filter, temporal disaggregation, observability, seasonality 4 1. Introduction This paper discusses how to specify an observable high-frequency model for a vector of time series sampled at high and low frequencies. To avoid cumbersome wordings, we will refer to the low-frequency variables as “annual” and to the high-frequency variables as “quarterly”. The results however are valid for any combination of sampling frequencies. The need for building such a model arises in two main situations: 1) An organization samples quarterly and annual data for several variables. To make a clear presentation to statistically unsophisticated audiences, it wants to estimate the unobserved quarterly values using all available (aggregated and disaggregated) information. 2) An important time series is measured once per year, but indicators about its performance are sampled quarterly. To assess the evolution of the target variable, one wants to: (a) compute a quarterly indicator of its fluctuations and (b) forecast its annual value, exploiting in both cases the information provided by the quarterly indicators. Such needs arise in different frameworks. For example, a statistical agency may want to disaggregate, monitor and forecast annual GDP using quarterly indicators. An analogous problem called “rainfall disaggregation” arises in hydraulic resources management, where high-frequency rainfall data should be inferred from low-frequency records, see e.g., Onof et al. (2000). These problems can be addressed by different methods. Some of them adopt a benchmarking approach to bridge the gap between the information contained in the quarterly indicators and the annual variables. This is typically done by means of a system of equations which describe the aggregation and accounting constraints relating the different variables. An important feature of such a “bridge model” approach is that there is no explicit assumption about the high-frequency dynamic relationships between the variables. Some relevant examples of the work in this area are those of Baffigi, Golinelli & Parigi (2004) or Di Fonzo & Marini (2005). Other methods assume (explicitly or implicitly) a given high-frequency model and use it to specify a uniquely determined system of equations relating both, the sample data with the unknown quarterly figures. Typically, this system is then solved by an extended least-squares procedure. Important examples of this model-based approach are those of Denton (1971), Chow-Lin (1971), Fernández (1981), Litterman (1983) or Santos-Silva & Cardoso (2001). Most of the models assumed by these authors are encompassed by: 1 2 1; (1 )(1 )t t ty B B ε ε ϕ ϕ = + = − − T tx β ta a iid (1.1) where denotes the target variable in quarter t, is a vector of quarterly indicators, txty 5 2(0, )t aσ∼ , 0, 1, 2,k t t kB w w k−= = ± ± … , and B denotes the backshift operator, such that for any sequence : tw Table 1 summarizes the restrictions on (1.1) assumed by different methods. Note that all of them: (a) impose a static linear relationship between the indicator (cause) and the target variable (effect), (b) assume different orders of integration for the variables, often implying cointegration between the indicator and the target variable, and (c) include no seasonal factors so, either seasonality has been removed beforehand, or it is a common feature between and , such that the linear combination txty ty − T tx β has no seasonal structure. [Insert Table 1] When looking at these widely different structures it is natural to ask: how would a specification error affect the disaggregates and forecasts resulting from these models? As time series interpolation and forecasting are particular cases of the same basic inference problem, a natural answer to this question arises by analogy: disaggregates computed using model (1.1) instead of the true data generating process (or a good approximation) will have in general the same flaws as the forecasts computed with (1.1) in comparison with optimal forecasts. On the other hand, predictive accuracy is critical for time series disaggregation, as large forecast errors generate important revisions of the disaggregates. In this paper we implement a state-space (SS) approach to model, interpolate and forecast a vector of time series observed at different frequencies. Our starting point consists of breaking up the global problem in four basic questions: How could one specify a quarterly model on the basis of a mixture of quarterly and annual data? How could such model be estimated? How to compute within-the-sample estimates of the unobserved quarterly values? How to forecast the annual variables exploiting the quarterly information available? All these issues, except the first, have been effectively addressed by the SS literature, see e.g., Ansley & Kohn (1983), Harvey & Pierse (1984) and Terceiro (1990, chapters 2 and 5). Due to this wealth of powerful and ready to use tools, other authors, such as Durbin and Quenneville (1997), Nunes (2005) or Proietti (2006), adopted the SS approach to implement different disaggregation proposals. Therefore we will concentrate in the specification problem. Our approach builds on two ideas. 6 First, quarterly and annual models should be mutually consistent, given the aggregation constraint. Second, a statistically adequate annual model, built by standard techniques, could provide clear clues about the specification of a quarterly model. To develop this idea, Section 2 analyzes the effect of aggregation on the dynamics of a linear system. We find that the system dynamics are not altered by aggregation, but some components may become unobservable. Section 3 characterizes which components loose observability after aggregation and, combining this result with those in Section 2, defines an algorithm to derive the annual model corresponding to a general quarterly representation. Building on previous results about quarterly model aggregation, Section 4 discusses how to specify an observable model for the quarterly values, building on a previously fitted annual model. We find that enforcing consistency between the annual and quarterly models and imposing observability on the latter, is enough to determine a useful initial specification, to be estimated and tested using standard SS techniques. This discussion results in a structured model-building procedure, which practical application is illustrated in Section 5 using macroeconomic data. This example also shows that the resulting high-frequency model may provide better forecasts than those resulting from mainstream methods. Section 6 provides some concluding remarks and indicates how to obtain, via Internet, a MATLAB toolbox for time series modeling, which implements all the computational procedures required. The proofs of formal results can be found at [URL to be completed in a final version]. 2. The effect of aggregation on the dynamics and forecasting accuracy of a time series model tzLet be an mx1 random vector of quarterly values. Without loss of generality (Casals, Sotoca & Jerez, 1999, Theorem 1) we will assume that these values are the observable output of a steady-state innovations SS model (hereafter, innovations model): Φ Γ E= + +t+1 t t tx x u a (2.1) = + +t t t tz Hx Du a (2.2) where: tx is a nx1 vector of state variables or dynamic components, tu is a rx1 vector of exogenous indicators, ta is a mx1 vector of errors, such that . ( ) iid , 0∼ta Q H 7 and the terms , , , Φ Γ E and are real-valued matrices of dimensions nxn, nxr, nxm, mxn and mxr, respectively. D We will also assume that model (2.1)-(2.2) is minimal. This is a non-restrictive hypothesis meaning that n is the smallest number of states required to describe the system dynamics. 2.1. The quarterly model in stacked form It is difficult to discuss aggregation using model (2.1)-(2.2). To this end, it is more convenient the following “stacked” representation. Let S be the seasonal frequency, defined as the number of high-frequency sampling periods (quarters) yielding a single low-frequency (annual) observation. Consider the stacked signal, indicator and error vectors: 1 1 1 ⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ t+ t :t+ S - t+ S - ⎡ ⎤ ⎢ ⎥tz z z z ; ; 1 1 1 ⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ t+ t :t+ S - t+ S - uu u ⎡ ⎤ ⎢ ⎥tu ⎡ ⎤ = 1 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ t+ t :t+ S - t+ S - aa a ta (2.3) Without loss of generality we will assume that the aggregation period coincides with the seasonal frequency, so the stacked vectors in (2.3) include all the values subject to aggregation. Under these conditions, the following Proposition holds: Proposition 1. Model (2.1)-(2.2) can be written equivalently as: Φ Γ EΤ+1 Τ t :t+ S -1 t :t+ S -1x = x + u + a (2.4) = + +t :t+ S -1 Τ t :t+ S -1 t :t+ S -1z H x D u C a (2.5) where the index T refers to the aggregated (annual) time unit, such as: , , =Τ+1 t+ Sx x =Τ tx x ( , )iid 0∼t :t+ S -1a Q and the matrices in (2.4)-(2.5) are related to those in (2.1)-(2.2) by: 8 Φ ΦS = ; , , … ,Γ Γ Γ ΓΦ Φ⎡ ⎤⎣ ⎦ S -1 S -2 = ; = , , … , Φ Φ⎡ ⎤E ⎣ ⎦ S -1 S -2 E E E (2.6) = -1 Φ Φ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ S H H H ⎡ ⎤ ⎢ ⎥H …0 0⎡ ⎤ ⎢ ⎥ ⎦ D Γ … = …-2 -3 0 Φ Φ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ S S H D D H Γ H Γ D ; ; (2.7) …0 0⎡ ⎤ ⎢ ⎥ S S I HE I C … = …-2 -3 0 Φ Φ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ; H E H E I = ⊗Q I Q + S - + S - Proof. Equation (2.5) is obtained by successively substituting (2.1) in (2.2) and writing the resulting system in matrix form. On the other hand (2.4) is immediately obtained by propagating (2.1) from t to t+S. ■ 2.2. Aggregation relationships Assume now the aggregation relationships: 1 A A T t :tz = J z (2.8) 1 P P T t:tz = J z (2.9) A Tz P Tz and where denote, respectively, the vectors of annual and partially aggregated data observed in year T, including the quarterly values at t, t+1, …, t+S-1. By “partially aggregated data” we mean a sample combining all the observed annual and quarterly values. Note also that there always exists a relationship between the annual and partially aggregated series, see Lütkepohl (1987, Chapter 6): P T=A * Tz J z (2.10) where (2.8)-(2.10) imply: . =* P AJ J J The structure of the aggregation matrices in (2.8), (2.9) and (2.10) obviously depends on the aggregation constraints and the nature of the variables. Following Di Fonzo (1990), there are four basic types of quarterly variables: flows, indices, beginning-of-period stocks or end-of-period stocks. Accordingly, the annual samples will be sums of quarterly values, averages or discrete beginning-of-period/end-of-period values. The following examples illustrate some common aggregation structures: Example 2.1: If all the variables considered are flows then , meaning that each annual figure is the sum of the corresponding quarterly values. On the other hand, implies that the variables are end-of-period stocks, such that is observed while the corresponding values at t, t+1, …, t+S-2 are not. = [ , , … , ]AJ I I I ]I 2 1 2) ( )S m S m + S m [ = , , … ,AJ 0 0 t+ S -1z 9 Example 2.2: Assume that the first m1 variables in , see (2.3), are annual flows, while the remaining m t :t+S -1z variables (m + m2 1 2=m) are quarterly values. In this case the partial aggregation matrix would be: ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [ 1(m + ] ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⋅ × ⋅ ⋅ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 m m mP m m I I I J I (2.11) I mIwhere is the identity and the vectors of quarterly and partially observed values have the following structures: m m× ; [ ]1 2( ) 1S m + S m 1 1 1 1 1 ⎢ ⎥ ⎢ ⎥= ⎢ ⎥⋅ ⋅ × ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ 2 1 2 t+ t :t+ S - m t+ m t+ S - m t+ S - z z z z ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 2 1 m t m t m z z z [ ]1 2( ) 1m + S m 1 1 1 1 ⎡ ⎤+ + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥⋅ × ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ …1 1 1 2 2 2 m m m t t+ t+ S - m P t T m t+ m t+ S - z z z z z z z (2.12) Example 2.3: Assuming again that all the variables are flows, the aggregation matrix in (2.10), transforming partially aggregated values into annual values, would be: *J 10 ⎥ ⎥ 2 [ ]1 2 1 2( ) ( )m + m m + S m ⎡ ⎤ ⎢= ⎢× ⋅ ⎢ ⎥⎣ ⎦ 0 0 0 0 1 2 2 * m m m m IJ I I I (2.13) 2.3. Relationship between the quarterly, annual and partially aggregated models Under the previously stated conditions, the following result formalizes the relationship between the quarterly data model in the form (2.4)-(2.5) and the corresponding models for the annual and partially observed data: A Tz P TzProposition 2. The models for and have the state equation (2.4) with the observation equations: = + +A A A A T T t:t+ S -1 t :tz H x D u C a + S -1 1 (2.14) (2.15) 1= + +P P P P T T t:t+ S - t :t+ S -z H x D u C a with: = = = ; ;A A A A A AH J H D J D C J C (2.16) ; ;= = =P P P P P PH J H D J D C J C (2.17) AJProof. Trivial, as pre-multiplying (2.5) by and immediately yields (2.14) and (2.15) respectively. ■ PJ A Tz As a Corollary to Proposition 2, note that the observer of can be alternatively obtained by aggregation of . Pre-multiplying (2.15) by J*P Tz yields: (2.18) 1= = + +A * P * P * P * P T T T t:t+ S - t :t+ S -z J z J H x J D u J C a 1 2.4. The effect of aggregation on predictive accuracy The following Proposition states that a model efficiently combining annual and quarterly data may provide better forecasts for the annual values than a model using only annual data. Proposition 3. The forecast error variance of the annual model, given by (2.4) and (2.14), is greater or equal than the error variance of the annual forecasts computed by: tz derived from the quarterly model (2.4)-(2.5) and Case (1) aggregation of the forecasts for P TzCase (2) aggregation of the forecasts for derived from the partially aggregated model (2.4) and (2.15). Proof: The proof of this result can be found at [URL to be completed in a final version]. Proposition 3 compares the ability of three models to predict the observable annual values. These are: (a) the model for the annual sample, (b) the “true” quarterly data generating process, and (c) the model for the partially aggregated sample. Assuming that these models are mutually consistent, given the aggregation constraint, model (b) would provide optimal forecasts, but cannot be empirically specified. Model (a) can be specified by standard methods, but cannot predict better than (b) or (c). Finally, model (c) can be inferred from data using, e.g. the method defined in Section 3, and may provide better forecasts than model (a). It is also more flexible, as it has the ability to update the annual forecast as new quarterly information becomes available. Proposition 3 generalizes previous results. Wei (1978) gave a proof of Proposition 3, Case (1) for univariate processes, also showing that the loss in forecasting efficiency due to aggregation can be substantial if the nonseasonal component of the model is nonstationary. Conversely, there is no loss in efficiency if the quarterly model is a pure seasonal process. Lütkepohl (1987) also discussed Case (1) in a multivariate stationary framework. 11 3. The effect of aggregation on the observability of a time series model 3.1. Characterization of the modes that become unobservable after aggregation As shown in Section 2, the models for quarterly, annual and partially aggregated data share the state equation (2.4). Therefore, aggregation does not affect the states governing a dynamic system. However, it may reduce its observability. Comparing the observation equations (2.5) and (2.14)-(2.15) it is immediate to see that observability loss occurs in two ways: (a) the quarterly observer (2.5) has more observable signals than the aggregated observers (2.14)-(2.15), and (b) the matrices in (2.14)-(2.15) are non-linear functions of the matrices in (2.5), potentially decreasing the observability of some states. To further discuss this issue, Definition 1 particularizes the general concept of observability (Anderson & Moore, 1979) to the notation defined in Section 2. Definition 1 (observability in the quarterly model). All the components in the quarterly model (2.1)-(2.2) are said to be observable i.i.f. there is no real vector , with 0≠w λ Φ w = w such that , where is the eigenvalue associated to the eigenvector w. = 0Hw λ Definition 2 (unobservable modes in the annual model). Assuming that the variables are flows, so , the annual model (2.4) and (2.14) has unobservable modes i.i.f. there exists a vector , with such that = [ , , … , ]AJ I I I =0 S-1 i 0Φ∑ iH w = 0AH w =0≠w λΦS w = w or, equivalently, , see (2.6)-(2.7) and (2.16). Under these conditions, Theorem 1 characterizes what components of the quarterly model become unobservable after annual aggregation. Theorem 1 (loss of observability in the annual model). Assuming that the quarterly model (2.1)-(2.2) is observable, the annual model (2.4) and (2.14) includes unobservable modes in any of the following cases: Case (1) when the annual variables are flows, such that , if there exists an eigenvalue of , , such that: = [ , , … , ]AJ I I I 2 -11 + λ + λ + … + λ = 0SλΦ ( )kλ ΦCase (2) if there is a subset of k eigenvalues of Φ , denoted by , such that for any pair , it holds that: S S i j=λ λ , ( ) , i j k i λ λ λ λ λ∈ Φ j≠ 12 Case (3) when has a kxk Jordan block with null eigenvalues, such that the geometric multiplicity of this block grows with its S-th order power. Φ Proof: The proof of this result can be found at [URL to be completed in a final version]. 13 -1 Theorem 1 characterizes the situation where some modes of the quarterly model, associated to different transition eigenvalues, collapse to a smaller number of modes in the annual model. Consider, e.g., a quarterly univariate model where all the dynamics of a flow variable are given by a seasonal difference . Case (1) implies that, after aggregation, this structure is indistinguishable from a nonseasonal difference 2= ( 1 )( 1 )S S B B B … B∇ − + + + + ( 1 )B∇= − . Then observability of S-1 dynamic components of the quarterly model is lost and the dynamics of the corresponding annual model simplify accordingly. Under Case (2), aggregation reduces the number of system modes, while maintaining the dimension of the state vector. The number of unobservable modes is (k rank )− ⋅H W , where k is the number of elements in ( )kλ Φ and W is a matrix whose columns are the eigenvectors that generate the k-order subspace S, 1,2, ,i = …iw k k. In the univariate case aggregation keeps only one mode, which corresponds with the larger Jordan block. In the multivariate case the reduction in the number of modes depends on how they affect the different observable time series so, in this case there is no way to compute a priori how many modes will become unobservable, but the maximum number of unobservable modes is 1k − . Case (3) occurs when the seasonal structure is a pure moving average. In this situation a systematic rule for mode elimination only exists in the univariate case, where there are distinguishable modes, being [] the integer part operator and k the dimension of the corresponding Jordan block. 1 [( 1) / ]k+ − S Two important works connected with Theorem 1 are those of Stram & Wei (1986) and Wei & Stram (1990). Specifically, the loss of observability has a clear relationship with the concept of “hidden periodicity” defined by Stram & Wei (1986, Definition 4.1) for univariate models. In a univariate framework, Case (2) of Theorem 1 is equivalent to hidden periodicity because aggregation creates undistinguishable modes that affect a single time series. However, in the multivariate case hidden periodicity not always implies loss of observability; consider e.g., the situation where two modes with hidden periodicity affect two different time series. Also, the mode elimination rules discussed for cases (2) and (3) are coherent, in the univariate case, with Stram & Wei (1986, Theorems 4.1 and 3.1). 14 After discussing the effect of aggregation over the quarterly model dynamics and its observability, it is important to characterize the uniqueness of the correspondence between the models for disaggregated and aggregated data. This is done in the following Theorem. Theorem 2 (correspondence between quarterly and annual models). If the quarterly model (2.4)-(2.5) is minimal and all its components are observable from annual data, then the annual model (2.4) and (2.14) is unique, allowing for similar transformations, minimal and observable. Proof: The proof of this result can be found at [URL to be completed in a final version]. The reciprocal proposition is not true in general. There are minimal and observable annual models that necessary correspond to quarterly models with unobservable components. Assume e.g., that the model for annual data is an AR(1) with a negative parameter and that the seasonal frequency S is even. In this case, there is no high-frequency ARMA(1,1) process that adds up to the annual AR(1) model because the S-th power of the transition matrix will always be positive, see (2.6). In the case of ARIMA models this result collapses to Lemma 2 in Wei & Stram (1990). 3.2. Observability and fixed-interval smoothing Observability of a state obviously affects our ability to estimate it. In a SS framework, the method of choice for efficient state estimation is the fixed-interval smoother (Anderson & Moore, 1979), which is a two-sided symmetric filter providing estimates of the first and second-order moments of the states conditional on all the information in the sample. The uncertainty of smoothed estimates critically depends on a property called “detectability”. Definition 3 (detectability). A system is said to be detectable if its unobservable modes are stationary. Under these conditions, the following result characterizes the effect of undetectable modes on smoothed estimates. Proposition 4. The variance of fixed-interval smoothing estimates of the states in models (2.4) and (2.14) or (2.15) is finite if and only if all the states are detectable. Proof: The proof of this result can be found at [URL to be completed in a final version]. 15 In time series disaggregation smoothing is typically employed to estimate the unobserved quarterly values. Therefore detectability is a necessary and sufficient condition to estimate these values with bounded uncertainty while observability is a sufficient (not necessary) condition. In time series disaggregation infinite smoothed variances would arise, for example, when the target variables are flows and their quarterly model includes seasonal roots in the unit circle. In this case the aggregated model has undetectable components and the variances of the estimates of seasonal components would be infinite. In practice this means that the annual series does not contain information about seasonal components and, therefore, if the quarterly indicators have seasonal component, it is advisable to remove them before disaggregation. 3.3. An algorithm to obtain the annual representation corresponding to a quarterly model Combining Propositions 1 and 2 with Theorem 1 and other results from the SS literature, one can devise an algorithm to obtain the reduced-form model for a vector of annual data corresponding to any linear model for the quarterly values, allowing for a general aggregation constraint. This algorithm proceeds as follows: Step (1) Consider any linear and fixed-coefficients model for the quarterly data. Write the model in the innovations form (2.1)-(2.2). If the model can be written in VARMAX form, this can be done using the expressions given by Terceiro (1990, Section 2.1). In any other case, write the model in a general (non-innovations) SS form and obtain the equivalent innovations representation (Casals, Sotoca & Jerez, 1999, Theorem 1). Step (2) Obtain the equivalent quarterly representation (2.4)-(2.5) and the annual representation (2.4) and (2.14). Step (3) If the annual representation is not observable, reduce (2.4) and (2.14) to an equivalent minimal SS realization applying the staircase algorithm (Rosenbrock, 1970). Step (4) Transform the model obtained in Step (3) to the corresponding innovations form (Casals, Sotoca & Jerez, 1999, Theorem 1). Step (5) If required, transform the innovations model to the Luenberger observable canonical form (Petkov et al. 1991). Translation from this form to other common representations, such as VARMAX, is then straightforward. 16 Tables 2.a and 2.b show the aggregation of several univariate and bivariate models, illustrating the application of this algorithm and some previous results. Specifically: 1) Models # 1-3 are examples of the observability loss described in Theorem 1 cases (1), (2) and (3), respectively. In particular, aggregation of Model 1 shows how a seasonal difference collapses to a nonseasonal unit root. This result is coherent with Granger & Siklos (1995) and Stram & Wei (1986). 2) Models # 1, 4, 5 and 7-11 show that aggregation does not affect the number of unit roots in a time series, so I(0), I(1) or I(2) quarterly flows yield I(0), I(1) or I(2) annual aggregates. A straightforward implication of this is that, if the quarterly variables are cointegrated, the corresponding annual aggregates will be also cointegrated. This is consistent with the findings of Pierse & Snell (1995), Granger (1990) and Marcellino (1999). 3) Models # 4-5 assume that the quarterly variables have both, regular and seasonal unit roots. In this case, often found when analyzing seasonal data, the corresponding annual model has two unit roots. This suggests that many annual variables should be I(2) while, in practice, annual models are often specified with a single unit root. This apparent contradiction is easy to explain because aggregation may induce a MA root close to unity, see e.g. Model # 5, therefore compensating an AR unit root. 4) Aggregation induces additional MA structure and maintains the order of the stationary AR structure (allowing for observability loss) and typically reducing its persistency. Models # 2 and 8 are clear examples of this. This result is consistent in the univariate case with Amemiya & Wu (1972), Wei (1978) and Stram & Wei (1986). In the multivariate stationary case, it is consistent with the findings of Lütkepohl (1987, Chap. 4 and 6) and Marcellino (1999). 5) Models # 8-11 show that, if there is feedback in the quarterly frequency, there is feedback in the annual frequency. 6) Model # 6 shows the annual model corresponding to a quarterly Chow-Lin AR(1) regression. Therefore, this method is empirically justified only if the annual model relating the target variable and the indicator is a static regression with ARMA(1,1) errors. 7) Models # 7 and 11 show that the algorithm is not restricted to VARMAX or transfer functions, as it can be applied to structural time series models (Harvey, 1989) and VARMAX echelon models (Hannan & Deistler, 1988). In general, it supports any model with an 17 equivalent SS representation. 8) Finally, model # 10 shows that the algorithm can be applied to a general combination of sampling and aggregation frequencies, as it shows how a monthly model aggregates to a quarterly VARMA. [Insert Tables 2.a and 2.b] 18 4. An empirical method to specify a high-frequency model Assume that a linear model has been fitted to all the available variables in the annual frequency. The problem now reduces to devise a systematic method enforcing consistency between the annual model and the unknown quarterly model, given the aggregation constraint and the partially aggregated sample. Without loss of generality, we will refer to a VARMA specification process, consisting of the successive determination of unit roots, AR and MA dynamics. The basic ideas can be mapped to other model-building methods such as, e.g., structural time series modeling (Harvey, 1989). 4.1. Feasibility of an exact correspondence between the annual and quarterly models The most rigorous way to specify the quarterly model would consist of obtaining a numerical solution to the equations relating the known annual model and the unknown quarterly model, using the algorithm defined in Section 3.3. We tried this approach and found it unpractical because it is difficult, unrealistic and may be impossible in some cases. First, it is difficult because the equations relating the SS matrices of the annual and quarterly models are highly nonlinear. Perhaps they can be solved, but we have not been able to devise a procedure to do it consistently. Second, it is unrealistic because achieving an exact match between the true quarterly data generating process and an empirical annual model would require the ability to model very weak parameters in the annual frequency. For example, consider the models # 2 and 8-10 in Table 2.a. Obviously some parameters in the MA factors are small to be detected by a realistic analysis of the annual time series, so an exact fit between the annual and quarterly models cannot be expected in practice. Third it may be impossible in some cases because, as stated in the discussion of Theorem 2, a statistically adequate model for the annual data may not have a mathematically consistent quarterly representation. 4.2. A method to enforce approximate consistency If an exact correspondence between the quarterly and annual models cannot be expected to be found in practice, the only way forward consists of devising a systematic process to achieve an approximate fit and a diagnostic method to assess whether the quarterly model obtained is statistically adequate or not. The following procedure can be used to these purposes. Step (1) Annual modeling. Specify and estimate a model relating the target annual variable(s) with the annualized values of the quarterly indicator(s). Any model having an equivalent linear SS representation, such as e.g., a transfer function or VARMAX, is adequate for this purpose. Step (2) Decomposition of the quarterly indicator. Adjust the quarterly indicator(s) to suppress undesired features, such as seasonality and calendar effects. Step (3) Model specification. Step (3.1) Set the VAR factor order of the quarterly model to be equal to that of the annual model and, particularly, constrain the number of unit roots to be the same. The foundation of this step results from comparison of the quarterly model (2.4)-(2.5) and the annual model given by (2.4) and (2.14). As both models share the same state equation, they will have the same (stationary and nonstationary) autoregressive components. q n≤Step (3.2) Add a VMA(q) structure, with , being n the size of the state vector in the annual model. This bound to MA dynamics results from the fact that, in a minimal SS representation, the size of the state vector is the maximum of p and q, being p the order of the VAR factor and q the order of the VMA factor. Step (4) Estimation. Estimate the model specified in Steps (2) and (3) by maximum likelihood and prune insignificant parameters to obtain a parsimonious parametrization. Step (5) Diagnostics. Check the final quarterly model by obtaining the corresponding annual representation, using the algorithm described in Section 3.3, and then: Step (5.1) compare this model with the one specified in Step (1) and Step (5.2) check whether it filters the annual data to white noise residuals. Step (6) Forecast accuracy check. If the sample is long enough, compare out-of-the-sample forecasts for the annual values produced by both, the tentative quarterly model and the annual model specified in Step (1). 19 20 4.3. Practical suggestions We have applied the method described above to several real and simulated time series. These exercises provided some useful insights about the practical application of our method: First, a good characterization of unit roots in Step (1) is critical, as misspecification of these components impacts severely over the quality of final results (Tiao, 1972). When in doubt, over- differencing is safer than under-differencing in our opinion. Second, the number of MA parameters specified in Step (3.2) may be excessive, depending on the sample size and number of time series. In this case, it is a good idea to constrain the MA matrices to be diagonal and, later, add off-diagonal parameters in a sequence of overfitting experiments. Third, as Nunes (2005) points out, there are a number of situations where missing observations may occur in a time series disaggregation frameworks. Some of these are: different release dates of some variables, changes in the sampling frequency or non-conformable sample, i.e., when the series considered have different starting dates. In any of these situations, the ability of SS methods to take care of missing values is an important asset, as it allows using all the information available in the dataset. Last, the forecasting accuracy check proposed in Step (6) can be implemented by setting some within-the-sample values to missing and estimating them afterwards. We have found this alternative useful when the sample is too short to reserve some values for out-of-the-sample forecasting. The example in Section 5 illustrates how the last two ideas can be applied in practice. 5. Disaggregation of Value Added by Industry in Spain (1980-2001). This Section illustrates the application of the method proposed and its practical advantages in comparison with alternative procedures. To do this, we will disaggregate and forecast the annual series of Value Added by the Industry in Spain (VAI), from 1980 to 2001, using as indicator the quarterly values of a re-balanced Industrial Production Index (IND), from 1980 1st Quarter to 2001 4th Quarter. The latter series is the indicator actually employed by the Agency in charge of the Spanish national accounts. To clarify when a series is expressed in annual or quarterly frequency we will use an uppercase/lowercase notation, so and A TVAI A TIND are the values of VAI and the annual average of the indicator in year T (T=1980, 1981, … , 2001), while and denote the values of both variables in quarter t (t=1980;1, 1980;2, … , 2001;4). tvai tind 5.1. Step (1) Annual model According to our method, the first step in the analysis consists of building a model relating the target variable and the indicator in the annual frequency. After a standard analysis (Jenkins & Alavi, 1981) we found the following model to be statistically adequate: 21 2 2 1 1.85 740(1 ) (.06) .02 ( 1980 1981 2001) 1.48 0 (1 ) A VAI T T A IND T T B . BB VAI a T = , , , BIND aB ( ) ˆ ; ˆ = ⎡ (.05) 2.64 - - = 5.53 13.08 a ˆ ⎤−⎡ ⎤− ⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ 1 … ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦ Σ 4.84 4.31 ; ( 5 ) = 6.23 4.17 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦ Q (5.1) ˆ aΣwhere the figures in parentheses are the standard errors of the estimates, represents the estimate of the error covariance matrix and Q(5) is the matrix of Ljung-Box statistics, computed using the first five residual auto and cross-correlations. Note that this model implies the existence of feedback between VAI and the indicator. Therefore, if model (5.1) is statistically adequate any procedure assuming unidirectional causality, such as those in Table 1, would be unsuitable for these series. Model (5.1) provides precise clues about the dynamics of both variables in the quarterly frequency. According to the method outlined in Section 4.2, a consistent quarterly model must have a non-stationary AR structure with two unit roots for each series and an MA term with a maximum order of 4, because the minimal SS representation of model (5.1) requires four state variables. 5.2. Step (2) Decomposition of the quarterly indicator The second step requires modeling the indicator in the quarterly frequency to estimate the components useful for disaggregation. To this end, we will use the following model: ˆ1.02 1.85 5.94 ; ( 1980;1 1980;2 2001;4)92.4 t t t t tL E S N t = , , , B B N B a Qσ = − − + − − = − = … (5.3) 2 2 (1 ) 1 1.48 .36 .044 .06 - - (.06) (.06) (.003) ; .40 3.64.84 1 .78(1 ) (.06) (.02) ( 1980;1 198 2 vai t t ind t t B B B B vai a B Bind aB t = , ˆ ˆ ˆ ⎡ ⎤− − +⎢ ⎥ ⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤ ind 4 4 2 (.03) (.35) (1.72) ˆ ˆ ˆ(1 )(1 ) (1 .83 ) ; 3.21 ; (15) = 12.80 (.06) t t a (5.2) where is the number of non-holidays in quarter t, is a dummy variable to account for Easter effects and is an intervention variable capturing a persistent level change from 1992 4 tL tE th92.4 tS Quarter onwards. Model (5.2) implies that the indicator can be split into: (a) trend, (b) seasonal component, (c) irregular component, (d) calendar effects associated to the number of non-holidays and Easter and (e) the step 1992 4th Quarter effect, see Casals, Jerez & Sotoca (2002). Figure 1 shows these components. In the light of previous results, components (b) and (d) are useless for disaggregation. By adding the remaining components we obtain a seasonally and calendar-adjusted quarterly indicator series. [Insert Figure 1] 5.3. Steps (3) and (4) Specification and estimation of the quarterly model Building on the results of previous steps, we now estimate a doubly integrated VMA(4) process for the value added and the adjusted indicator in the quarterly frequency. After pruning insignificant parameters we obtained the following model: ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− −− ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ 0 0 aΣ 0;2 2001;4), , … 22 tindwhere denotes the unobserved value of VAI in quarter t and tvai is the corresponding adjusted indicator. Finally, by applying a fixed-interval smoothing to the sample (Casals Jerez & Sotoca 2000) we obtain the disaggregates and forecasts shown in Figure 2. [Insert Figure 2] 5.4. Step (5) Diagnostics We now obtain the annual representation corresponding to model (5.3) using, the algorithm described in Section 3.3. It is: 2 2 (1 ) 1.52 .23 70 .10 (1 ) 1.18 .19 .14 .07 A VAI T T A IND VAIB B B . B + B a B B B B B ˆ = ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − − + − ⎦ 2 2 2 2 0 1 0 1 ( 1980 1981 2001)T = , , , ⎥ … 2.54 - - 7.38 7.81 = ; ( 5 ) = 5.53 13.79 9.49 8.48 T TIND â ˆ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎣ ⎦⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ a Q (5.4) Σ A TINDwhere denotes the annual average of the adjusted indicator in year T. Models (5.4) and (5.1) differ mainly in the additional second-order MA parameters in (5.4). We tried to fit an annual model including these parameters and the corresponding estimates resulted insignificant. Perhaps this additional MA structure is due to the disaggregated indicator information included in (5.3) and excluded in (5.1). On the other hand, the residuals obtained by filtering the annual series using (5.4) are stationary, normal and do not show important autocorrelations. Therefore, we accept that model (5.3) is statistically adequate and roughly conformable with (5.1). Previous results in this example show that our method can be applied to real disaggregation problems. The remaining Subsections highlight its unique advantages when dealing with non- conformable samples and in terms of forecasting power. 23 5.5. Step (6) Forecast accuracy and non-conformable samples Statistical Bureaux typically have long records for the target variable, say GDP, and shorter ones for the indicators. In this situation, standard techniques constrain the analysis to the common time window, thus assuming a substantial information loss. However the SS methods employed here allow for missing values, so all the available information can be used. To illustrate this idea, we delete the first four observations of the quarterly indicator and the last annual value of VAI. Re- estimating model (5.3) with this non-conformable sample yields the following results: 24 2(1 ) 1 1.48 .34 .045 .06 - - (.04) (.03) (.003) 2 vai B B B B vai â ⎡ ⎤− − +⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− −−⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ 2 ; = .41 3.74.89 1 .78(1 ) (.04) (.02) ( 1980;1 19 t t ind t tB Bind aB t = , ˆ ˆ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ 0 0 aΣ 80;2 2001;4), , … (5.5) Estimates in (5.5) are remarkably close to those in (5.3). Table 3 shows the original sample information and the numerical results obtained with models (5.3) and (5.5). Note that: (a) the forecast provided by (5.5) for the 2001 annual value is very accurate, (b) quarterly interpolations and out-of-sample forecasts obtained with both models are very similar, and (c) the indicator retropolations computed with (5.5) and the non-conformable sample have acceptable errors. [Insert Table 3] 5.6 Comparison with mainstream methods Table 4 shows the root mean squared errors (RMSEs) obtained by forecasting the last five end-of-year values of VAI, using Model (5.3) and the main alternative methods. In all cases, the forecasts are conditional to the true indicator values for the same years. For example, to compute the 1997 values each model was estimated using annual VAI and quarterly indicator values up to 1996, and was then used to predict the end-of-year value of VAI using: (a) the past of VAI up to 1996 and (b) past values of the indicator up to 1997. The same procedure was applied for 1998, 1999, 2000 and 2001 extending the sample in each case with one, two, three and four additional years of data. [Insert Table 4] Note that Model (5.3) produced the best forecasts, with a 8-9% advantage over the second- best and much larger gains in comparison with the remaining methods. Obviously, this comparison is not fair, as model (5.3) was carefully fitted to the data while the alternative forecasts are mechanical. Therefore previous results should probably be seen as a re-statement of the idea that a model fitted by a trained human is usually able to beat automatic forecasting systems. 5.7 Partially aggregated model The previous analysis concentrates on time series disaggregation. On the other hand many users, such as macroeconomic analysts, are not interested in disaggregation but in: (a) forecasting the annual values taking advantage of the information contained in quarterly indicators, (b) updating these forecasts as new indicator values are published and (c) computing confidence intervals for this sequence of forecasts. Note that the quarterly model (5.3) could be used to solve the issues (a) and (b), by accumulating the resulting forecasts to annual values. However, it does not provide directly the standard errors for the annual forecast needed for (c). The specific needs of these users could be better addressed by computing a partially aggregated version of (5.3), which would be a model relating the annual value of VAI with the values of the indicators in the four quarters of the same year. Using the algorithm described in Section 3.3, we obtained this model in VARMA echelon form (Hannan & Deistler, 1988). )2 ;4 T T VAI ind ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T T a a ˆ ˆ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ 1 0 0 0 0 0 1 0 0 0 2 # 1 (5.6) ( ) (2B B B B# # # # # # 0 1 2 0 1 2− − = Θ −Θ −ΘΦ Φ Φ T Ty a where the variables and the errors are given by: ; (5.7) ;3 ;2 ;1 T T T ind ind ind ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ Ty ;4 ;3 ;2 ;1 T T T a a a ˆˆ ˆ ˆ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ Ta 25 Note that the subscripts T;4, T;3, T;2 T;1 denote the 4th, 3rd, 2nd and 1st quarters of year T. On the other hand, the coefficient matrices in (5.6) are given by the following expressions: ⎡ ⎤ ⎡ ⎤− ; # # 0 0 ⎢ ⎥ ⎢ ⎥=Θ = ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥−⎢ ⎥⎣ ⎦ 0 0 1 0 0 0 1 2 1 0 0 2 3 0 1 Φ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ 0 0 0 0 0 3 4 0 0 0 4 5 0 0 0 0 0 0 0 0 0 0 0 0 Φ ; # 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ 1 0 0 0 0 0 0 0 0 0 (5.8) 26 . . . . . . . . . . ⎡ ⎤−⎢ ⎥ ⎢ ⎥− − − 1 35 044 030 007 500 310 3 03 3 98 0098 1 158 .25 .075 .038 .004 .035 ⎡ ⎤− − − − ⎢ ⎥ # 1 . . . . . ⎢ ⎥ ⎢ ⎥Θ = − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ 610 4 05 4 96 0196 1 316 0 0 0 0 0 0 0 0 0 0 ⎢ ; # 2 ⎥ ⎢ ⎥ ⎢ ⎥Θ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (5.9) 27 6. Concluding remarks This paper makes three main contributions to the literature about aggregation of time series: First, it encompasses and extends many previous results about: (a) the effect of aggregation on the dynamics of ARIMA (Amemiya & Wu, 1972; Brewer, 1973; Stram & Wei, 1986) and VARMA processes (Marcellino, 1999), (b) the observability loss due to aggregation (Wei & Stram, 1990) and (c) the negative effect of aggregation on forecasting accuracy in the univariate (Wei, 1978) and multivariate cases (Lütkepohl, 1987). Second, it proposes an algorithm to aggregate any quarterly model to the corresponding annual reduced-form. This method is very general, as the only formal requirement for the quarterly model is that it has an equivalent state-space representation. Third, it proposes a method to build an observable high-frequency model from a partially aggregated sample, allowing for general sampling frequencies and aggregation constraints. This method emphasizes the idea that the quarterly model should be both, consistent with a statistically adequate model fitted to annual data, and compliant with standard diagnostic tests. At this point, some readers may wonder: why should I use this complex method instead a simpler mechanical procedure? We think that simple methods are adequate in complex situations, e.g., when one has to disaggregate thousands of time series with limited resources, to produce official statistics according to a rigid and exigent calendar. This situation, which is very common in statistical agencies, requires the virtues of robust, fast and mechanic methods. On the other hand, many forecasters concentrate in a smaller set of time series and, for different reasons, may want to obtain the model with the higher forecasting power. The example in Section 5 shows that, for these needs, our modeling approach has the potential to beat mechanical methods. Of course, there is no guarantee that this will happen in all the cases. The procedures described in this article are implemented in a MATLAB toolbox for time series modeling called E4, which can be downloaded at www.ucm.es/info/icae/e4. The source code for all the functions in the toolbox is freely provided under the terms of the GNU General Public License. This site also includes a complete user manual and other materials. 28 Acknowledgements We are deeply indebted to Tommaso Di Fonzo, who devoted much effort to testing our methods and read carefully previous drafts of this paper. Enrique Quilis made helpful remarks and kindly provided the dataset for the empirical example. We are also indebted for the feedback received from Toni Espasa and the participants in the research seminars of Universidad Carlos III, Universidad Autónoma de Madrid, the Italian Office of Statistics (ISTAT) and the 2005 Eurostat Workshop on Benchmarking Techniques. 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Christov, N.D. & Konstantinov, M.M. (1991). Computational Methods for Linear Control Systems, Englewood Cliffs (NJ): Prentice-Hall. Pierse, R.G. & Snell, A.J. (1995). Temporal Aggregation and the Power of Tests for a Unit Root, Journal of Econometrics, 65, 333-345. Proietti, T. (2006). Temporal Disaggregation by State Space Methods: Dynamic Regression Methods Revisited, Econometrics Journal, 9, 357-372. Rosenbrock, M.M. (1970). State-Space and Multivariable Theory, New York: John Wiley. Santos-Silva, J.M.C. & Cardoso, F. (2001). The Chow-Lin Method using Dynamic Models, Economic Modelling, 18, 269-280. Stram, D.O. & Wei, W.W.S. (1986). Temporal Aggregation in the ARIMA Process, Journal of Time Series Analysis, 7, 4, 279-292. Tiao, G.C. (1972). Asymptotic Behavior of Time Series Aggregates, Biometrika, 59, 521-531. Terceiro, J. (1990). Estimation of Dynamic Econometric Models with Errors in Variables, Berlin: Springer-Verlag. Wei, W.W.S. (1978). Some Consequences of Temporal Aggregation in Seasonal Time Series Models” In: Zellner, A. (ed.), Seasonal Analysis of Economic Time Series, Washington DC: Bureau of the Census, 433-448. Wei, W.W.S. & Stram, D.O. (1990). Disaggregation of Time Series Models, Journal of the Royal Statistical Society, B Series, 52, 3, 453-467. 32 Biographies José CASALS is senior analyst in a Spanish Savings Bank and part-time Associate Professor of Econometrics at the Universidad Complutense de Madrid. He also served as consultant to several Spanish and international organizations. He obtained his Ph.D. at Universidad Complutense in 1997. Miguel JEREZ is Associate Professor of Econometrics at the Universidad Complutense de Madrid and freelance consultant. During six years he was Executive Vice-President in a Spanish bank. He obtained his Ph.D. at Universidad Complutense in 1989. Sonia SOTOCA is Associate Professor of Econometrics at the Universidad Complutense de Madrid. She obtained her Ph.D. at Universidad Complutense in 1992. These authors are engaged in a long-term project to apply state-space techniques to standard econometric problems. They have published several articles on these topics. Further details about this project can be found at http://www.ucm.es/info/icae/e4/ http://www.ucm.es/info/icae/e4/ Figure 1. Decomposition of the quarterly indicator. The adjusted indicator includes the trend, irregular and level change components and excludes seasonality and calendar affects. Disaggregates based in this indicator can be interpreted as calendar and seasonally adjusted quarterly estimates of VAI. 33 81.4 84.4 87.4 90.4 93.4 96.4 99.4 0 20 40 60 80 100 120 140 Original data v s. trend Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 0 20 40 60 80 100 120 140 Original data v s. trend Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 55 60 65 70 Eff ect of exogenous v ariables Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 55 60 65 70 Eff ect of exogenous v ariables Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 -15 -10 -5 0 5 10 15 Seasonal component Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 -15 -10 -5 0 5 10 15 Seasonal component Date Irregular component 81.4 84.4 87.4 90.4 93.4 96.4 99.4 -15 -10 -5 0 5 10 15 81.4 84.4 87.4 90.4 93.4 96.4 99.4 -15 -10 -5 0 5 10 15 Irregular component Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 0 20 40 60 80 100 120 140 Seasonality and calendar adjusted indicator v s. original data Date 81.4 84.4 87.4 90.4 93.4 96.4 99.4 0 20 40 60 80 100 120 140 Seasonality and calendar adjusted indicator v s. original data Date Figure 2. Standardized plot of quarterly estimates of VAI (thick line) and adjusted indicator (thin line). The last values are forecasts computed from 2002.1 to 2002.4 quarters using model (5.3). 81.4 84.4 87.4 90.4 93.4 96.4 99.4 02.4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 S ta nd ar d er ro rs Quarter 81.4 84.4 87.4 90.4 93.4 96.4 99.4 02.4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 S ta nd ar d er ro rs Quarter 34 35 Table 1: Restrictions on model (1.1) assumed by different methods. The symbol (*) means that the corresponding parameter is to be estimated. Method β ϕ1 ϕ2 Denton (1971) 1 1 0 Chow-Lin (1971) (*) 0 0 Chow-Lin (1971) / AR(1) (*) (*) 0 Fernández (1981) (*) 1 0 Litterman (1983) (*) 1 (*) Table 2.a: Aggregation of several univariate models. The columns labeled “states” show the number of dynamic components in the minimal SS representation of the corresponding model. Therefore, the difference between the number of states in the quarterly and annual models is the number of dynamic components that become unobservable after aggregation. The quarterly variables tz 1tz and are assumed to be flows, so their annual aggregate is the sum of the corresponding quarterly values. If the quarterly indicator in model # 6 ( 2tz ) were to be aggregated as an annual average, the coefficients in the annual transfer function should be multiplied by 4. # Quarterly Model States Annual Model States 36 1 4 2(1 ) ; 1t t aB z a σ− = = 4 2(1 ) ; 4.00A A t t AB z a σ− = = 1 2 2(1 .8 )(1 .8 ) ; 1t t aB B z a σ− + = = 2 2(1 .410 ) (1 .160 ) ; 7.99A A t t A B z B a σ− = + = 1 3 4 2(1 .6 ) ; 1t t az B a σ−= = 4 2(1 .600 ) ; 4.00A A t t Az B a σ−= = 1 4 4 2(1 )(1 ) ; 1t t aB B z a σ− − = = 5 2 2(1 ) (1 .240 ) ; 41.60A A t t AB z B a σ− = + = 2 4 4 2(1 )(1 ) (1 .8 )(1 .6 ) ; 1t t aB B z B B a σ− − − −= =5 5 2 2 2(1 ) (1 .997 238 ) ; 7.05A A t t AB z B B a σ− −= +. = 2 6 2 1 2 1.5 ; 1 1 .8t t t az z a B σ+ − = = 1 2 1 2 1 1 228.500 ; 23.103 1 .410 A A A t t t a Bz z a B σ − +.= + = 1 2 1 2 3 2 1 2 (1 ) ; 01 (1 ) ; 01 ; 1 t t u t t v B T u B B B S v z T S σ σ ε σ + + − + + = =. + + + = =. = =t t t t ε 2(1 ) (1 .669 ) ; 5.844A A t t AB z B a σ− −= =7 4 1 37 Table 2.b: Aggregation of several bivariate models. All the quarterly variables 1tz and 2tz are assumed to be flows, so their annual aggregate is the sum of the corresponding quarterly values. If a variable were to be aggregated as an average of the quarterly values, the model would require an appropriate re-scaling. # High-frequency Model States Low-frequency Model States 8 4 111 .7 .8 1 .5(1 ) ;tt aB B B z− − ⎡ ⎤− ⎡ ⎤⎡ ⎤ ⎡ ⎤ 4 220 1 .5 1(1 ) a tt aB z⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢−⎣ ⎦ ⎣⎣ ⎦⎣ ⎦ = =Σ ⎥ ⎦ 10 1 1 2 2 1 .240 0 1 .039 1.461(1 ) 0 1 0 1(1 ) 35.36 7.62 7.62 4.00 A A t t A A t t A B B BB z a B z a ⎡ ⎤ ⎡ ⎤− −−⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ Σ = = 3 9 4 11 1 .8 1 .5(1 ) tt aBB z⎡ ⎤ −− ⎡ ⎤⎡ ⎤ ⎡ ⎤ 4 22 ; 1.2 1 .5 1(1 ) a tt aBB z⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ = =Σ 2 2 ; .289 1 .015 1.41 13.00(1 ) AA A t tB BB z a 8 1 11 .080 .053 4.09 1.41(1 ) A A t tB BB z a⎡ ⎤ ⎡ ⎤− −− ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ = = + Σ 2 10 12 11 1 .8 1 .5(1 ) ;tt a aBB z⎡ ⎤ −− ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =Σ12 22 1.2 1 .5 1(1 ) tt aBB z− ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ 4 2 2.365 1 .024 1.03 9.27(1 ) A A t tB BB z a− ⎣ ⎦ ⎣⎣ ⎦ ⎣ ⎦ = = + Σ 24 4 1 11 .100 .075 3.22 1.03(1 ) ; A A t t A B BB z a⎡ ⎤ ⎡ ⎤− −− ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎦ 8 11 1 11 0 1 .4 .8 1 .5 ;t tz aB B B− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =Σ 2 2.5 1 .5 1 .5 1a t tz a− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ 2 2.5 1 .5 1 29.55 19.58AA A t tz a 1 1 11 0 1 .745 1.808 51.91 29.55 ; A A t tB B Bz a⎡ ⎤ ⎡ ⎤− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ = =Σ 1 ⎣ 38 Table 3. Original data and interpolations obtained with full and non-conformable samples. Underlined values correspond to interpolations, retropolations or forecasts. The column “Indicator” refers to the calendar and seasonally adjusted values computed using model (5.2). Original sample Full sample Non-conformable sample Obs. Annual VAI Indicator Interpolation Annual sum Indicator Interpolation Annual sum Indicator 1980.1 81.421 14.993 81.421 15.009 82.869 1980.2 80.451 14.810 80.451 14.896 82.237 1980.3 80.524 14.744 80.524 14.778 81.584 1980.4 59.341 81.877 14.794 59.341 81.877 14.658 59.341 80.968 1981.1 80.196 14.474 80.196 14.502 80.196 1981.2 80.710 14.531 80.710 14.519 80.710 1981.3 80.823 14.483 80.823 14.474 80.823 1981.4 57.895 80.299 14.407 57.895 80.299 14.400 57.895 80.299 ... 2000.1 123.520 23.326 123.520 23.342 123.520 2000.2 124.005 23.417 124.005 23.431 124.005 2000.3 122.033 23.374 122.033 23.371 122.033 2000.4 93.620 121.147 23.502 93.620 121.147 23.476 93.620 121.147 2001.1 121.823 23.753 121.823 23.690 121.823 2001.2 121.550 23.801 121.550 23.718 121.550 2001.3 121.006 23.830 121.006 23.734 121.006 2001.4 94.711 115.986 23.327 94.711 115.986 23.217 94.359 115.986 2002.1 23.414 115.664 23.311 115.723 2002.2 23.295 115.343 23.199 115.461 2002.3 23.177 115.021 23.088 115.198 2002.4 23.059 92.944 114.699 22.976 92.574 114.936 39 Table 4: Ranking of RMSEs for end-of year forecasts of annual VAI from 1997 to 2001. RMSEs are computed for forecasts of both, VAI levels and growth rates. The columns RMSE% show the corresponding RMSEs normalized so that the RMSE of Model (5.5) is 100% Levels Annual growth rates Rkg Model RMSE RMSE % RMSE RMSE % 1 Model (5.5) 0.65 100.0% 0.71 100.0% 2 Litterman (1983) with constant 0.72 109.6% 0.77 108.4% 3 Fernández (1981) with constant 0.96 147.5% 1.03 145.2% 4 Chow Lin (1971) AR(1) with constant 1.21 185.2% 1.30 182.5% 5 Fernández (1981) without constant 1.44 219.8% 1.54 216.0% 6 Litterman (1983) without constant 1.47 224.9% 1.57 220.9% 7 ADL(1,0) without constant 1.53 234.0% 1.73 242.4% 8 ADL(1,0) with constant 1.61 247.0% 1.75 245.3% 9 Chow Lin (1971) AR(1) without constant 1.82 278.9% 1.98 277.2% 10 Boot, Feibes and Lisman (1967) 3.04 466.2% 3.39 475.2% 11 Denton (1971) 18.01 2758.4% 20.26 2842.3% Appendices of: Modeling and Forecasting Time Series Sampled at Different Frequencies Proofs of Proposition 3, Theorem 1, Theorem 2 and Proposition 4 José Casals (jcasalsc@cajamadrid.) Miguel Jerez (mjerez@ccee.ucm.es) † Sonia Sotoca (sotoca@ccee.ucm.es) Departamento de Fundamentos del Analisis Economico II Universidad Complutense de Madrid Campus de Somosaguas S/N 28223 Madrid (SPAIN) † Corresponding author. The telephone numbers for the three authors are: Voice: (+34) 91 394 23 61. Fax: (+34) 91 394 25 91. 40 APPENDIX A: PROOF OF PROPOSITION 3 A.1. Previous results Result 1. Consider the algebraic Riccati equation of the Kalman filter: = −T T t t tt+1 t t t -1P P + EQE K BΦ Φ TK (A.1) where: (= -1T t t t -1K P + EQΦ Η ) tB (A.2) = T t t t -1B P +Η Η Q (A.3) 41 Under these conditions, if ≥* t t -1 t t -1P P and then ≥*Q Q ≥* t+1 t t+1 tP P . See the proof in Bitmead et al. (1985) Result 2. Let be M a ( ) matrix where with , and A, a symmetric positive-definite ( ) matrix, with . Under these conditions: ( *rank m) =M*m m× *m m< (rank m) =Am m× (A.4) − ( ) ≥-1A Μ ΜA Μ ΜΤ Τ 0 )T ) (A.5) *[rank m m− ( ) ] = −-1A Μ ΜA Μ ΜΤ Τ Proof. It is immediate to see that: (A.6) ( ) ( − ( ) = − = − −-1 TA Μ ΜA Μ Μ A CAC I C A I CΤ Τ where . Therefore, positive-definiteness of A assures that = ( )-1C Μ ΜA Μ ΜA−1 −1Τ Τ ( ) (− − TI C A I C is a positive-definite matrix also. A.2. Proof of Proposition 3 Case 1) Consider the high-frequency model in the stacked form (2.4)-(2.5). The corresponding k-step ahead forecast error variance is given by: ( )var Ω = 1 T K T T + K - Te P + CΗ Η TQC (A.7) where T is the forecast origin and Ω denotes the quarterly information available at time T. According T to (A.7), the variance of the annual forecast obtained by aggregation would be: ( )( ) ( )var varΩ = Ω TA A K T K Te J e J A (A.8) 1T + K - TP On the other hand, the Kalman filter covariance matrices in (A.7) result from the recursion: = 1 T T + K T T + K - TP P + EQΦ Φ TE (A.9) and successive substitution in (A.9) yields immediately: 2 0 K i − = = ( ) ( )∑1 T i T T + K T T + TP P + EQE−1 −1Φ Φ Φ ΦΚ Κ i T (A.10) see, e.g., Anderson & Moore (1979). On the other hand, in the low-frequency model (2.4)-(2.14) the k-step ahead forecast error variance is: ( ) ( )( )var Ω = 1 TA A A T A A T A K T T + K - Te J P J + J CQC JΗ Η T (A.11) where denotes the annual information available at time T. ΩA T Expressions (A.7) and (A.11) have the same mathematical structure. Then, any difference between both variances will be due to the covariances 1T + K - TP . Assuming that the initial condition for the propagation of P in both cases is the same, 1P , we obtain for the quarterly model (2.4)-(2.5): = −12 1 T TP P + EQE K B KΦ Φ 1 1 1 T (A.12) and for the annual model (2.4) and (2.14): ( ) ( ) -1 ⎡ ⎤= − ⎢ ⎥⎣ ⎦1 1 1 12 1 T TA T T A A A AP P + EQE K B J J B J J B KΦ Φ 1 1 T (A.13) where is the Kalman filter gain in t=1. Therefore, from (A.12) and (A.13): 1K ( ) ( ) -1 -1⎧ ⎡ ⎤− = −⎨ ⎢ ⎥⎣ ⎦⎩ ⎭ A 1 1 1 1 1 12 1 21 T T ⎫ ⎬ A A A A TP P K B B J J B J J B K (A.14) 0− ≥A 2 1 2 1P Pand, using Result 1, it is immediate to see that because 42 ( ) ( ) -1 -1 0⎡ ⎤− ≥⎢ ⎥⎣ ⎦1 1 T TA A A AB J J B J J . Therefore, for t=2: ( ) ( ) -1 ⎡ ⎤≥ − ⎢ ⎥⎣ ⎦ A 2 2 2 2 23 2 2 1 T TT T A A A AP P + EQE K B J J B J J B KΦ Φ T (A.15) ( ) ( ) -1 -1 0⎧ ⎫⎡ ⎤− = − ≥⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ A 2 2 2 2 2 23 2 3 2 T TA A A A TP P K B B J J B J J B Kand Result 2 assures that: Therefore, by induction we obtain: ( ) ( ) -1 -1 0⎧ ⎡ ⎤− = − ≥⎨ ⎢ ⎥⎣ ⎦⎩ ⎭ A -1 -1 -1 -1 -1 -1-1 -1 T TA A A A T t t t t t tt t t tP P K B B J J B J J B K⎫ ⎬ ⎫ ⎬ T TA A A A A T E E E E E E E EP P K B B J J B J J B K (A.16) and, as the Riccati equation of the Kalman filter converges to its steady-state solution and it holds that: (A.17) ( ) ( ) -1 -1 0⎧ ⎡ ⎤− ≥ − ≥⎨ ⎢ ⎥⎣ ⎦⎩ ⎭ ■ Case 2) Consider the partially aggregated model (2.4)-(2.15). Its k-step ahead forecast error variance is given by: ( ) ( ) (var Ω = 1 )P P P P T P P K T T + K - Te Η P Η + C Q C T (A.18) ΩP Twhere denotes the partially aggregated information available at time T. As (a) the partially aggregated model and the annual model share the same state equation and (b) the underlying quarterly model is the same, Expression (A.11) is also valid in this case. Substituting in (A.11) yields: =A *J J J P ( ) ( ) ( ( ) ( ) ( ) ( ) var Ω = = 1 1 K T T + K - T * P P T * T * P P T * T T + K - T e J J P J J + J J CQC J J J P J + J C Q C J Η Η Η Η )A P * P T * P T * P T * P T (A.19) and, consequently: { }-1-1 ( ) ( ) 0⎡ ⎤− ≥ − ≥⎣ ⎦ A P * T * * T * T E E E E E E E EP P K B B J J B J J B K (A.20) ■ 43 APPENDIX B: PROOF OF THEOREM 1 According to definitions 1) and 2), the annual model (2.4) and (2.14) has unobservable modes if there exists at least a vector λ=S w wΦ≠ 0w such that and: 1 =0 = S- i 0∑ iH w Φ (B.1) B.1 Proof of Case 1). Expression (B.1) implies: + + + = 0… SHw H w H w H w2 −1Φ Φ Φ (B.2) If w is an eigenvector of , then it is also an eigenvector of the powers of , so and, therefore: Φ Φ λ ( = 1, 2, )i i i …w = wΦ 2 1 2 -1+ + + = 1+ + + =S - Sλ λ λ λ λ λ 0⎡ ⎤+⎢ ⎥⎣ ⎦…Hw H w H w H w Hw … (B.3) but, as the quarterly model is assumed to be observable, then and (B.3) holds i.i.f. ■ 0≠Hw 2 -11+ + + = 0Sλ λ λ+… B.2 Proof of Case 2). Under the conditions of Case 2), the k eigenvectors associated to generate a subspace, denoted by ( )λk Φ S λw = wΦλ,kS λ,k∈w S, such that for any , it holds that , where . λ λ λ ( )λS ki i,= ∀ ∈ Φ IS and the null space of matrix H, then λ,kSLet be the intersection between I λ,kdim dim null k rank( ) ( ) (⎡ ⎤= ∩ = −⎣ ⎦ )S S H HW , where W is a matrix which columns are the eigenvectors spanning λ,kS . In the univariate case, the number of unobservable modes is exactly , because H is a row matrix and the quarterly model is observable. In the multivariate case there is no exact rule but, in general, k Idim k rank k( ) ( )= − = −1S HW rank(− )HW modes become unobservable and the maximum number of unobservable modes is k . ■ −1 44 B.3 Proof of Case 3). 45 ] Under the conditions of Case 3), it is easy to see that the S-th power of any k-dimension Jordan block with null eigenvalues breaks into several blocks. The dimension of the larger sub- block will then be [ , where “[]” denotes the integer part of a real argument. 1 1)+ k - /S( Fragmentation of the Jordan block implies an increase in the geometric multiplicity associated to the null eigenvalues, so there are several linearly independent eigenvectors associated to each null eigenvalue. This situation is therefore similar to the one considered in Case 2). ■ APPENDIX C: PROOF OF THEOREM 2 ( , , , , , )=1R E H DΦ Γ Q and Consider the matrices in (2.1)-(2.2) and (2.4)-(2.5), denoted by ( , , , , , , )=2R E H D C QΦ Γ . If (2.1)-(2.2) is minimal, then there exists a biunivocal correspondence between both representations, such that ( )F=2R R1 , being F() a biyective application. 46 Assume that this is not true. Then there are at least two realizations, and 1R 1 *R , with the same canonical representation, such that ( ) ( )F F= =2 1 * 1R R R . Then and 1R 1 *R are output- equivalent. However, if is minimal, then 1R 1 *R can only be a similar transformation of . As and 1R 1R 1 *R have the same canonical representation, then the only similar transformation is identity and =1 1 *R R . Also, there exists a biunivocal correspondence between and the matrices characterizing the annual model (2.4)-(2.14), denoted by 1R ( , , , , , , )=3 A A AR E H D C QΦ Γ , such that , being G() a biyective application. ( )G=3 1R R Assume again that this is not true. Then there would be at least two quarterly realizations, and 1R 1 *R ( ) ( )G G= =3 1 * 1R R R, with the same canonical representation, such that . However we know that ( ) ( )F F≠1 * 1R R and, as and share the same state equation, then and ( )F 1R ( )G 1R ( )F 1R ( )F 1 *R should have different quarterly observation equations that yield the same annual observation equation. Then there are two quarterly models with the same annual realization, so: and ( , , , , , )=1R E H DΦ Γ Q *( , , , , , )=1 * *R E H D QΦ Γ 1( ) ( , , , , , , )F =R E H D C QΦ Γ, such that: , 1( ) ( , , , , , ,F =* * * )*R E H D C QΦ Γ ( ) ( ) ( , , , , , , )G G= = =* A A 3 1 1 AR R R E H D CΦ Γ Q and , where: A A A *H = J H = J H (C.1) A A AC = J C = J C* (C.2) A A AD = J D = J D* I (C.3) and, assuming that the variables are flows, . = [ , , … , ]AJ I I 1 1 0 0 S- S -i i i= i= =∑ ∑A *H = H HΦ ΦCondition (C.1) implies that , so . This implies that, if 1 0 ( ) S- i i= − =∑ 0*H H Φ ≠ *H H , then is a rank-deficient matrix and there exists an eigenvalue of , λ , such that . Therefore a contradiction arises as the annual model would not be observable (Theorem 1, Case 1). 1 0 S - i i=∑ Φ 1 0 S- i i= λ =∑ 0ΦΦ Finally, conditions (C.2) and (C.3) are now easy to prove. As and 1R 1 *R share the matrices =*C = C I, , ,E QΦ Γ and H, see (C.1), then and , see (2.7). ■ *D = D APPENDIX D: PROOF OF PROPOSITION 4 D.1. Previous results Result 1 (observability staircase form). This Result, due to Kalman (1963), states that for any SS model, characterized by the pair ( , )H Φ , there exists a similar transformation T, such that 47 * -1= HT * -1= T TΦ )H , Φ , that results in a model ( , ∗*H Φ with the following structure: (D.1) ⎡ ⎤ ⎢ ⎥ ⎥ ⎥ 11 12Φ Φ 0 0Ν Ν ⎤⎦ Ε 2 Φ Φ 0 Φ Ν Ν Ν ⎤ ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ⎥⎣ ⎦ * = 22 11 12 22 0 Φ 0 0 Φ 0 0 Φ Φ 0 0 0 Φ Ν Ε Ε Ε (D.2) ⎡⎣ * = ΝΗ Η Η20 0 where: 1) 22 Φ , ⎡ ⎤ ⎢ ⎥⎣ ⎦ N = 11 12 ⎡ ⎢⎣ ⎦ E = 11 12 22 Φ ΦΦ 0 Φ Ε Ε Ε characterize, respectively, the nonstationary and stationary subsystems, so: , ) 1λ( ≥NΦ ) 1λ( Π 0 S ⎡ ⎤ ⎢ ⎥ ⎥ ⎥ 2 0 0 0 0 ⎥ ⎦ S S S 2 2 ΝΝ ΝΕ ΕΝ ΕΕ (D.9) D.2. Proof of Proposition 4 Using Result 1, the aggregated models (2.4) and (2.14), or (2.4) and (2.15), can be written in observability staircase form. If this form has a non-stationary subsystem, the smoother must be initialized according to (D.9) and the matrix S, see (D.5), has the following structure: (D.10) ⎢ ⎢ ⎢ ⎥ ⎣ ⎦ S S S = S S 2 2 2 0 0 0 0 0 0 0 0 ΝΝ ΝΕ ΕΝ ΕΕ where ⎣S2 2 is positive-definite. Denoting ⎡ ⎤ ⎢ −1=R Π −1=V N, and partitioning , R, N and V as: Π ; ; ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ 11 12 21 22 Π Π Π Π Π ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ R R R R R 11 12 21 22 ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ N N N N N 11 12 21 22 ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ V V V V V 11 12 21 22 ; (D.11) 1 NPthe matrix , see (D.4), can be written in the following block-form: 49 11 1 k k 1 1 + k k − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎥ ⎥ 11 12R R 0 0 0 ⎢ +⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ 21 221 11 12 21 22 N R R S SP = V V S V V S 2 2 2 2 0 0 0 0 ΝΝ ΝΕ ΕΝ ΕΕ (D.12) with . Applying the partitioned-matrix inversion lemma to (D.12) yields: k ⎡ ⎤ ⎢ ⎢ ⎥⎣ ⎦ 1 1 1 1 1 N N N N N P P P = P PΕΝ ΕΕ ⎥ ΝΝ ΝΕ (D.13) where: -1 -1-11 1 + + ⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞⎢ ⎥− −⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎠ -1 -122 22 11 12 21 11 12 RR R W R R R WΠ 2 -1 k k k k + k ⎢ ⎥⎝ ⎠⎢ ⎥ ⎝⎣ ⎦⎢ ⎥ ⎢ ⎥⎛ ⎞ =⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠⎣ ⎦ 1 -1 22 NN NP = WΠ (D.14) ( ) ( ) ( )-1 + + + − −⎢⎣ ⎦ ⎢ =⎢ ⎥⎣ ⎦ 11 12 22 21 11 12 22 1 -1 22 EE N V V V Y V V V Ν Y P = Ν Y -1 -1-1⎡ ⎤⎡ ⎤ ⎥ ⎥ -1 -1 (D.15) ( ) ( ) ( ) ( ) -1 -1 -1 -1 + + + + ⎡ ⎤⎛ ⎞ ⎛ ⎞ −⎢ ⎥⎜ ⎟ ⎜ ⎟ -1 -1 -1 -1 -1 -1 -122 22 11 12 22 2 2 21 11 11 12 22 2 2 EE EN EE ENV V N S S W R R V V Ν S S WΠ Π -1 -1 -1 -1 k k + + + + k k ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ 1 -1 -1 -1 -1 -122 22 22 2 2 21 11 22 2 2 EN N EE EN EE EN P = N S S W R R Ν S S WΠ Π (D.16) 1 k ⎛ ⎞− +⎜ ⎟ ⎝ ⎠ 22 2 2 2 2 − EE EN NN NRY = S S S S Ebeing and 1( )−− +2 2 22 2 2 NN NE SS ENW = S S V S S In these conditions the limit values of the blocks in (D.13), as k tends to infinity, are: 1 1 1 lim 1k →∞ ⎡ ⎤− ⎢ ⎥ ⎣ ⎦ 11 11 12 1 N R R R W P = = W − − − ΝΝ − (D.17) k 50 1 1−− − →∞ − ⎡ ⎤⎡ ⎤− + − +⎣ ⎦⎢ ⎥ ⎢ ⎥+⎣ ⎦ 11 12 22 21 11 12 22 1 22 N V V V Y V V V N YP = = N Y − − ΕΕ − 1 1 1 1 1 ( ) ( ) lim ( ) (D.18) k 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) lim ( ) ( )k − − − − − − − − − − − −→∞ ⎡ ⎤− + ⎢ ⎥+ +⎣ ⎦ 11 12 22 2 2 21 11 11 12 22 2 2 1 22 2 2 21 11 22 2 2 EE EN EE EN N EE EN EE EN V V N S S W R R V V N S S W P = N S S W R R N S S W − ΕΝ − − 1− (D.19) Therefore, the blocks (1,2), (2,1) and (2,2) in (D.13) converge to finite values. Only the (1,1) block diverge to infinity but this is irrelevant to Proposition 4, as it corresponds to the non- detectable modes, see Result 1. ■ Note that the initial state has a persistent effect over the smoothed estimates of the states because the (1,1) block of t NP , see (D.3), is: (1,1) * (1,1) (1,1)t t−1 −1= + ( ) ( ) +11 111 …N N t N t N NP P PΦ Φ (D.20) taking into account (D.7)-(D.8) and the fact that when the system is in observability staircase form the matrix is: tR (D.21) ⎡ ⎤ ⎢ ⎥ ⎥ ⎥ t 2 0 0 0 0 ⎢ ⎢ ⎢ ⎥ ⎢ ⎥⎣ ⎦ t t t t R R R = R R 2 2 2 0 0 0 0 0 0 0 0 ΝΝ ΝΕ ΕΝ ΕΕ As , it is immediate to see that initial conditions will affect all the sequence of smoothed estimates, no matter the sample size. Therefore, the infinite variance of a diffuse prior would be propagated to the smoothed estimates along the whole sample. ) 1λ( ≥N 11Φ where denotes the annual average of the adjusted indicator in year T. Models (5.4) and (5.1) differ mainly in the additional second-order MA parameters in (5.4). We tried to fit an annual model including these parameters and the corresponding estimates resulted insignificant. Perhaps this additional MA structure is due to the disaggregated indicator information included in (5.3) and excluded in (5.1). On the other hand, the residuals obtained by filtering the annual series using (5.4) are stationary, normal and do not show important autocorrelations. Therefore, we accept that model (5.3) is statistically adequate and roughly conformable with (5.1). Previous results in this example show that our method can be applied to real disaggregation problems. The remaining Subsections highlight its unique advantages when dealing with non-conformable samples and in terms of forecasting power. Figure 1. Decomposition of the quarterly indicator. The adjusted indicator includes the trend, irregular and level change components and excludes seasonality and calendar affects. Disaggregates based in this indicator can be interpreted as calendar and seasonally adjusted quarterly estimates of VAI. Table 1: Restrictions on model (1.1) assumed by different methods. The symbol (*) means that the corresponding parameter is to be estimated. Table 2.a: Aggregation of several univariate models. The columns labeled “states” show the number of dynamic components in the minimal SS representation of the corresponding model. Therefore, the difference between the number of states in the quarterly and annual models is the number of dynamic components that become unobservable after aggregation. The quarterly variables and are assumed to be flows, so their annual aggregate is the sum of the corresponding quarterly values. If the quarterly indicator in model # 6 ( ) were to be aggregated as an annual average, the coefficients in the annual transfer function should be multiplied by 4. Quarterly Model Table 2.b: Aggregation of several bivariate models. All the quarterly variables and are assumed to be flows, so their annual aggregate is the sum of the corresponding quarterly values. If a variable were to be aggregated as an average of the quarterly values, the model would require an appropriate re-scaling. High-frequency Model Full sample RMSE % A.1. Previous results Result 1. Consider the algebraic Riccati equation of the Kalman filter: According to definitions 1) and 2), the annual model (2.4) and (2.14) has unobservable modes if there exists at least a vector such that and: (B.1) B.1 Proof of Case 1). B.2 Proof of Case 2). B.3 Proof of Case 3). (C.1) (C.2) (C.3) Condition (C.1) implies that , so . This implies that, if , then is a rank-deficient matrix and there exists an eigenvalue of , , such that . Therefore a contradiction arises as the annual model would not be observable (Theorem 1, Case 1). Finally, conditions (C.2) and (C.3) are now easy to prove. As and share the matrices and H, see (C.1), then and , see (2.7). ■ D.1. Previous results Result 1 (observability staircase form). This Result, due to Kalman (1963), states that for any SS model, characterized by the pair , there exists a similar transformation T, such that , , that results in a model with the following structure: