Modeling data revisions: Measurement error and dynamics of “true” values

Journal of Econometrics 161 (2011) 101–109

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Modeling data revisions: Measurement error and dynamics of ‘‘true’’ values Jan P.A.M. Jacobs a,b , Simon van Norden c,∗ a

University of Groningen, The Netherlands

b

CAMA and CIRANO, Canada HEC Montréal, CIRANO and CIREQ, Canada

c

article

info

Article history: Received 8 September 2007 Received in revised form 1 July 2009 Accepted 22 April 2010 Available online 28 December 2010 JEL classification: C22 C53 C82 Keywords: Real-time analysis Data revisions

abstract Policy makers must base their decisions on preliminary and partially revised data of varying reliability. Realistic modeling of data revisions is required to guide decision makers in their assessment of current and future conditions. This paper provides a new framework with which to model data revisions. Recent empirical work suggests that measurement errors typically have much more complex dynamics than existing models of data revisions allow. This paper describes a state-space model that allows for richer dynamics in these measurement errors, including the noise, news and spillover effects documented in this literature. We also show how to relax the common assumption that ‘‘true’’ values are observed after a few revisions. The result is a unified and flexible framework that allows for more realistic data revision properties, and allows the use of standard methods for optimal real-time estimation of trends and cycles. We illustrate the application of this framework with real-time data on US real output growth. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Data revisions have haunted economists for decades.1 Such revisions complicate forecasts and estimates of current economic conditions, since the most recent data are usually the least reliable (e.g. Koenig et al., 2003). Optimal forecasts and indicators require a model of the data revision process (see Croushore (2006) for a survey). Some authors cast the data revision process in state-space

∗ Corresponding address: HEC Montréal, 3000 Chemin de la Côte Sainte Catherine, Montreal QC, H3T 2A7, Canada. Tel.: +1 514 340 6781. E-mail address: [email protected] (S. van Norden). 1 McKenzie (2006) notes eight reasons for revisions of official statistics. 1. Incorporation of source data with more complete or otherwise better reporting (e.g. including late respondents) in subsequent estimates. 2. Correction of errors in source data (e.g. from editing) and computations (e.g. revised imputation). 3. Replacement of first estimates derived from incomplete samples (e.g. subsamples) judgmental or statistical techniques when firmer data become available. 4. Incorporation of source data that more closely match the concepts and/or benchmarking to conceptually more accurate but less frequent statistics. 5. Incorporation of updated seasonal factors. 6. Updating of the base period of constant price estimates. 7. Changes in statistical methodology (such as the introduction of chain-linked volume estimates), concepts, definitions, and classifications. 8. Revisions to national accounts statistics arising from the confrontation of data in supply and use tables. 0304-4076/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2010.04.010

form, which then allows the use of standard filtering techniques for forecasting, estimation, inference, smoothing, estimation of missing data, etc. Part of our motivation for this paper is the work of Aruoba (2008) and Siklos (2008), who note that data revisions typically show more complex behavior than standard models allow. This paper describes a more general state-space model that allows for more flexibility in the dynamics of revisions. Measurement errors in our model may have any combination of several key characteristics:

• they may behave as ‘‘noise’’, so that the measurement errors of consecutive vintages are mutually uncorrelated.

• they may have a ‘‘news’’ component, in which the measurement errors of consecutive vintages behave like a set of rational forecast errors. • they may have a ‘‘spillover’’ component, in which measurement errors within a given data vintage are serially correlated. • they may have important periodic components. For example, seasonal adjustment factors are often revised once a year. This may be four quarters after the initial release for some observations, but five or six quarters for others. • measurement errors may be important even years after initial estimates are published. In addition, our formulation of the state-space model is novel in that it defines the measured vector as a set of estimates for a given point in time rather than a set of estimates from the same

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J.P.A.M. Jacobs, S. van Norden / Journal of Econometrics 161 (2011) 101–109

Fig. 1. The revision triangle.

vintage. We find this leads to a more parsimonious state-space representation and a cleaner distinction between various aspects of measurement error. It also allows us to augment the model of published data with forecasts in a straightforward way. The resulting model provides us with a unified framework to estimate trends or cycles with data subject to revision. We think such a framework is indispensable for the proper formulation and conduct of monetary and fiscal policy. The paper begins with an extensive survey of the literature on state-space models of data revision in Section 2. We relate this literature to a more empirical literature which typically characterizes measurement errors as ‘‘news’’ or ‘‘noise’’. We then describe our state-space model in Section 3, detailing its ability to capture different important types of measurement error dynamics as well as different assumed dynamics for the ‘‘true’’ underlying series. In Section 4 we illustrate our methodology with the US real GNP/GDP series made available by the Federal Reserve Bank of Philadelphia. This paper concludes in Section 5 with an evaluation of the model’s ability to improve the accuracy with which we can model measurement errors. 1.1. Notation Throughout the paper we use the notation that is standard in this literature: superscripts refer to vintages while subscripts to time periods. For example, yt1 is the estimate available at time t of the value of variable y at time 1. Real-time data are typically displayed in the form of the Revision Triangle (see Fig. 1). We move to later vintages as we move across columns from left to right and we move to later points in time when we move down the rows. Note that the frequency of vintages need not necessarily correspond to the unit of observation; for example, statistical agencies often publish monthly vintages of quarterly observations. Although the Figure shows that first available estimates are published without a lag, this assumption is typically relaxed so that the typical entry in the j +1 j +l j last diagonal may be yj or yj rather than yj . We also note there is no important distinction in this framework between ‘‘published data’’ and ‘‘forecasts’’. For example, if we choose to incorporate forecasts up to an h period horizon, our lower diagonal will have typical element ytt +h . 2. Literature review Current interest in the analysis of real-time data dates from the publication of Croushore and Stark’s ‘‘A Real-Time Data Set for Macroeconomists’’ (Croushore and Stark, 1999, 2001). Real-time data sets are increasingly available for the United States (from the Federal Reserve Banks of St. Louis and Philadelphia), the euro zone (EABCN Real Time Database, RTDB), and several other countries (the OECD real-time database). The literature on real-time data analysis is also expanding rapidly.2

2 Dean Croushore’s real-time data bibliography see (http://facultystaff.richmond. edu/∼dcrousho/docs/realtime_lit.pdf), has been very helpful in preparing this review.

However, acknowledgement of data revisions dates at least back to the first issue of the first volume of the Review of Economic Statistics (Persons, 1919) while Kuznets (1948) also explicitly mentioned data revisions in his seminal paper on national income statistics. Zellner (1958, page 58) begins by noting that ‘Many . . . economic policy decisions are formulated or constructed on the basis of preliminary or provisional estimates of . . . our national accounts.’ Zellner (1958) and Morgenstern (1963) analyzed the relationship between preliminary and revised data. Cole (1969) and Denton and Kuiper (1965) investigated the sensitivity of parameter estimates to data revisions. Stekler (1967), Denton and Kuiper (1965), and Cole (1969) examined the loss in forecast accuracy when using preliminary data instead of ‘‘true’’ data. The intervening decades have seen continuing innovations in the study of data revisions. This review focuses on the modeling of measurement error, in particular on state-space models. We also review the empirical literature on the properties of measurement errors, and briefly document recent research on modeling data revisions. 2.1. State-space models State-space models provide a convenient framework for modeling measurement errors in economic data.3 This was recognized early by Howrey (1978, 1984), although he did not mention the method by name. Conrad and Corrado (1979) were the first to apply the Kalman filter to data revisions, soon followed by Harvey et al. (1983). For a typical state-space model, we can gather the l most recent estimates of vintage t for a variable y as y t = [ytt −1 , . . . , ytt −l+1 , ytt −l ]′ , where we have assumed a one-period publication lag. These estimates may be related to the corresponding ‘‘true’’ values y˜ t = [˜yt −1 , . . . , y˜ t −l+1 , y˜ t −l ]′ by defining measurement errors ut ≡ y t − y˜ t = [utt −1 , . . . , utt −l+1 , 0]′ , where we assume that the ‘‘true’’ values are obtained after l releases. For white noise measurement errors, the state vector α is defined as α = y˜ t and the linear statespace model may be written as y t = Z y˜ t + εt y˜ t +1 = T y˜ t + ηt

measurement equation

(1)

transition equation,

(2)

with the usual assumptions on the error processes: εt ∼ i.i.d N (0, H ); ηt ∼ i.i.d N (0, Q ); and E(εt , ητ ) = 0 for all t and τ . The measurement equation models the relationship between the vector of observed variables and the state vector. The transition equation captures the dynamics of the state vector, usually a simple autoregressive process. At a given time τ , y τ contains yτt = y˜ t for t ≤ τ − l and optimal estimates of true values for τ − l + 1 to τ can be obtained with the Kalman filter. This framework also allows us to form optimal forecasts of future true values given the available data, to estimate the standard errors associated with recent observations, and to calculate the relative weights that should be attached to recent data when forecasting or estimating true values. All of these activities have important practical applications. Variants of this state-space framework have been used to deal with special cases. For example, serial correlation in measurement errors (cf. Howrey, 1978) can easily be captured by including the measurement error ut in the state vector and adapting the transition equation, as shown in Harvey (1989, Section 6.4.4). Similar models are applied by Trivellato and Rettore (1986),

3 For general introductions to state-space modeling see the textbooks of Harvey (1989) and Hamilton (1994, Chapter 13).

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Bordignon and Trivellato (1989), Patterson (1994, 1995a,b,c), Mariano and Tanizaki (1995) and Busetti (2006). An alternative approach postulates that vintages are cointegrated and driven by a single common stochastic trend or ‘‘factor’’, while measurement errors and revisions are strictly stationary. Gallo and Marcellino (1999) and a series of papers by Patterson (2000, 2002a,b, 2003) and Patterson and Heravi (1991a,b) adopt this approach, while Siklos (2008) casts doubt on the underlying cointegration assumptions near benchmark revisions. 2.2. Measurement errors: ‘‘news’’ or ‘‘noise’’? While it is acknowledged that measurement errors can be correlated across time, their behavior across vintages has been the subject of much research. Boschen and Grossman (1982), Mankiw et al. (1984), Mankiw and Shapiro (1986), Maravall and Pierce (1986), Mork (1987, 1990), Patterson and Heravi (1992), Croushore and Stark (2001, 2003), Faust et al. (2005), Swanson and van Dijk (2006), and Aruoba (2008) debate whether data revisions are best modeled as ‘news’ or ‘noise’. The two polar views are: (i) Published data contain noise (ζtt +i ): Measurement errors are said to be noise when the errors ζtt +i are orthogonal to the true values y˜ t , but correlated with the available vintage, so that ytt +i = y˜ t + ζtt +i ,

cov(˜yt , ζtt +i ) = 0. (ytt +i+1

(3) ytt +i )

are generally − Noise implies that revisions forecastable. (ii) Published data contain news (νtt +i ): Measurement errors are described as ‘‘news’’ if and only if they match the properties associated with rational forecast errors. t +j+1 This requires that revisions (yt − ytt +j ) are unpredictable given the information set at time t + j. Since this information set contains all previously published vintages, this implies that y˜ t = ytt +i + νtt +i ,

t +j

cov(yt , νtt +i ) = 0 ∀j ≤ i.

(4)

Unlike the previous case, the covariance restriction now relates the measurement error to previously published vintages rather than true values. This condition also implies that the variability of the measurement errors for a given point in time t cannot increase as we move to more recent vintages. de Jong (1987) presents necessary and sufficient conditions for revisions to be rational. Sargent (1989) models a statistical agency that collects economic data as the sum of a vector of ‘‘true’’ variables and a vector of measurement errors, and then uses optimal filtering methods to construct and report least-squares estimates of the ‘‘true’’ variables. Kapetanios and Yates (2010) start from the Sargent (1989) world to model the publication process of the statistical agency. The Mincer and Zarnowitz (1969) test of the ‘‘noise’’ specification regresses the measurement error ytt +i − y˜ t on a constant and the final release utt +i = α1 + β1 y˜ t + ζtt +i .

(5)

The null hypothesis that measurement errors are independent of true values (α1 = 0, β1 = 0) may be tested with a Wald test. The analogous test of the ‘‘news’’ model regresses the measurement error ytt +i − y˜ t on a constant and the ith release ytt +i − y˜ t = α2 + β2 ytt +i + νtt +i .

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2.3. Modeling of data revisions Most of the relevant literature on data revisions can be grouped around three different themes. Data description. These studies focus on detailing the often complex nature of data revisions. Much of the above literature on the news versus noise debate falls into this category, as does Siklos (2008) and Garratt and Vahey (2006). Optimal forecasting and inference. These papers focus on deriving the optimal forecasts or smoothed estimates of ‘‘true’’ values for a postulated data revision process. Examples of such work include the Harvey et al. (1983), Sargent (1989) and Kapetanios and Yates (2010) papers cited above. Cycle-trend decompositions. These papers focus on the accuracy of estimated trends or cycles near the end of sample. While some (such as Laubach (2001), Orphanides and van Norden (2002), Rünstler (2002), or van Norden (2005)) take account of data revision, they ignore its structure and analyze each data vintage independently. The links between these three strands of the literature are weak, but becoming stronger. For example, de Antonio Liedo and Carstensen (2006) improve the links between the first two strands by allowing for both simple news and noise effects in data revision, while Cunningham et al. (forthcoming) also allow for serial correlation in measurement errors. Kishor and Koenig (2005) allow for a general VAR model of data vintages. Similarly, Garratt et al. (2008) better integrate the second two strands by showing that simple revision forecasts can improve HP-filtered trends and cycles. In the next section we present a state-space representation that further integrates all three strands of this literature. In particular, we relax the assumption imposed above that ‘‘true’’ or final values become available after l releases; we accommodate richer and more realistic dynamics for the measurement errors; we use conventional state-space methods to construct optimal forecasts and estimates of the underlying true values; and we show how to optimally estimate trends or cycles from multiple data vintages. 3. Our state-space model 3.1. Structure Following Durbin and Koopman (2001) we write a generic (time-invariant) state-space model as yt

α t +1

= Z αt + εt = T αt + R ηt

(7) (8)

where yt is l × 1, αt is m × 1, εt is l × 1 and ηt is r × 1; εt ∼ N (0, H ) and ηt ∼ N (0, Ir ). Both error terms are i.i.d. and orthogonal to one another.4 For convenience we omit constants from the model in this exposition. In our framework, the data yt is a l × 1 vector of l different vintage estimates ytt +i , i = 1, . . . , l, for a particular observation t,

′

so yt ≡ ytt +1 , ytt +2 , . . . , ytt +l simply stacks the first through the lth estimate of yt . The superscript denotes the period at which the vintage becomes available.5 We will denote the unobserved ‘‘true’’



(6)

The similar null hypothesis (α2 = 0, β2 = 0) now tests whether data revisions are predictable. The two null hypotheses are mutually exclusive but they are not collectively exhaustive, i.e. we may be able to reject both hypotheses, particularly when the constant in both test equations differs from zero (see Aruoba, 2008).

4 For more detailed assumptions, see Durbin and Koopman (2001, Section 3.1 and 4.1). 5 The assumption that the first estimate becomes available with a one-period lag is arbitrary and innocuous; the assumption that first estimate is available at t − j (i.e. a j-period forecast) would make no difference to our analysis.

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value of this variable y˜ t , and its measurement error ut ≡ yt − ιl y˜ t , where ιl is an l × 1 vector of ones. Note that this differs from the conventional state-space modeling framework discussed above, which is specified in terms of y t instead of yt . In addition, unlike other models in this literature, we do not require the assumption that the last available estimate has no measurement error (i.e. that ytt +l = y˜ t ). Instead, we simply assume that y˜ t belongs to a desired class of dynamic models (such as an ARMA(p, q) process, or a particular structural time series model that can be expressed in state-space form). For tractability, we partition the state vector αt into four components

 ′ ′

αt = y˜ t , φt , νt , ζ t , 

′

′

(9)

of length 1, b, l and l respectively, where φt corresponds to the dynamics of the true values, νt to news, and ζt to noise, as will become clear below. It will be convenient to similarly partition Z = [Z1 , Z2 , Z3 , Z4 ]

(10)

where Z1 = ιl (a l × 1 vector of 1s), Z2 = 0l×b (an l × b matrix of zeros), Z3 = Il , and Z4 = Il (both l × l identity matrices). We also impose H = 0, so that there is no error term associated with the measurement equation. The measurement equation (7) then simplifies to

T11 T21 T = 0 0

T12 T22 0 0

0 0 T3 0

R1 R2 R= 0 0

R3 0 −U1 · diag(R3 ) 0

T =

0 0 , 0



R1 R2 0

 R=

0 0 , R4



σζ 2 .. .

··· ··· .. .

0

···

0

0

0 0



..  . . σζ l

The assumption that estimates become more precise over time could be imposed by restricting σζ l < σζ ,l−1 < · · · < σζ 2 < σζ 1 . 3.2.2. Pure news In the pure news case we may drop ζ t from the state vector and simplify Eqs. (7) and (8) to obtain

  y˜ t αt = φ t , νt

T11 T21 0

 T =

T12 T22 0

0 0 0

 and

Z = [Z1 , Z2 , Z3 ] .

Imposing the ‘‘news’’ properties then requires R1 R2 0

R3 0 . −U1 · diag(R3 )



This means that (8) becomes



0 0 , 0 T4

(11)

where T11 is a scalar, and {T12 , T21 , T22 , T3 , T4 } are 1 × b, b × 1, b × b, l × l and l × l; 0 is a conformably defined matrix of zeros. We similarly partition R into an (1 + b + 2l) × r matrix



 σζ 1  0 R4 ≡   .. .

R=

Consistent with the above, we partition matrix T as

T12 T22 0

T11 T21 0



and Z = [Z1 , Z2 , Z4 ]. In the simplest case of ‘‘pure’’ noise, measurement errors are independent of the measurement errors in neighboring vintages, so E(ut u′t ) is a diagonal matrix. We can impose this property by setting



yt = Z αt = y˜ t + νt + ζ t = ‘Truth’ + ‘News’ + ‘Noise’.



  y˜ t αt = φ t , ϵt



0 0 , 0 R4

(12)

where U1 is a l × l matrix with zeros below the main diagonal and ones everywhere else, R3 = [σν 1 , σν 2 , . . . , σν l ], where σν i is the standard error of the incremental news error associated with i-th estimate ytt +i , diag(R3 ) is a l × l matrix with elements of R3 on its main diagonal, and R4 is an l × l matrix. Finally, we partition the error term associated with the transition equation as ηt = [η′et , η′ν t , η′ζ t ]′ , where ηet refers to errors associated with the true values, and ην t and ηζ t are the errors for news and noise, respectively. In the next subsection, we explore the specification of the measurement error and thereafter we consider the specification of the dynamics of y˜ t . 3.2. Measurement error

y˜ t +1 = T11 · y˜ t + T12 · φt + R1 ηet + R3 η2t

σ

ν1

  0 νt +1 = −U1 · diag(R3 ) · η2t ≡ −  .  . . 0

and

σν 2 σν 2 .. . ...

··· .. . .. . 0

σν l  ..  .   · η2t .  σν l σν l

In this way, adding news shocks νt to the true values y˜ t removes some of information (shocks) driving y˜ t . As subsequent vintages peel away some of these news shocks, our estimates of y˜ t improve. 3.2.3. Spillovers Information which arrives at t + j and causes the statistical t +j agency to revise an estimate yt may also cause them to revise t +j t +j nearby estimates, such as yt +1 , yt −1 , etc.6 Such a relationship between measurement errors in different economic time periods (spillovers) is independent of the characterization of measurement error as news or noise. Accordingly, the presence or absence of spillovers has no implications for the form of R3 or R4 , but instead is captured via the specification of T3 or T4 . In the simplest case of noise and spillovers, we can capture the effect of measurement errors for time t affecting estimates at t − 1 by setting T4 = ρ Il , where ρ is the correlation coefficient between ut and ut −1 . Propagation over longer time periods can be accommodated by stacking successive values of ζ t into the state vector and (if desired) specifying richer intertemporal dynamics.

In this subsection, we show how each of three special cases of measurement errors (pure noise, pure news and simple spillovers) may be captured in our state-space framework. We then describe how all three types can be combined in a general model.

3.2.4. General models of measurement errors Since their effects enter via different system matrices, we may freely combine any desired pattern of spillover effects with either

3.2.1. Pure noise In the pure noise case we may drop νt from the state vector and simplify Eqs. (7) and (8) to obtain

6 For example, data from annual tax returns become available only long after the end of the tax year, but may lead to the conclusion that income in each month or quarter of the previous year should be revised upwards or downwards.

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pure noise or pure news models of measurement errors. We may also combine both news and noise effects in the same model. In addition spillovers may differ for news and noise. Still more general dynamics can be obtained by allowing R4 to be an unrestricted upper-triangular matrix so that noise need not be i.i.d. across vintages. 3.3. Dynamics of true values The dynamics of the ‘‘true’’ value y˜ t are jointly determined by the first two blocks ofthe statevector [˜yt , φ′t ]′ , the upper block of T12 T22

T11 T21

the transition matrix

, and the associated blocks of the

error weighting matrix R1 and R2 . This framework is sufficiently general to accommodate a variety of popular dynamic models for y˜ t ; we provide two popular examples below. We first examine the case where y˜ t follows an ARMA(p, q) process. Thereafter, we describe the case where y˜ t is assumed to follow the structural time series model of Harvey and Jaeger (1993). 3.3.1. ARMA dynamics Letting φt = [˜yt −1 , . . . , y˜ t −p+1 , et , . . . , et −q+1]′ we can capture the ARMA(p, q) dynamics of y˜ t by specifying

σe , 01×(q−1) ]′ and  ρ′ [ ] T11 T12  Ip−1 = T T 0 21

22

1 ×p

0(q−1)×p

θ′

R1 R2

= [σe , 01×(p−1) ,



0(p−1)×(q+1)   01×q Iq−1 |0(q−1)×1

where ρ is the p × 1 vector containing the autoregressive coefficients and θ is the q × 1 vector containing the moving average coefficients. 3.3.2. A structural time series model To obtain a trend level and drift rate subject to stochastic shocks, as well as a stochastic cycle with period 2π /λ, we can use the following specification:

• φt = [τt , µt , ct , ct∗ ]′ where τt is the level of the trend component of y˜ t at time t, µt is its growth rate, and ct and ct∗ are the stochastic cycle components of y˜ t at time t, with standard deviations στ , σµ and σc , respectively; • T11 = 0;  ′ • T21 = 0 0 0 0 ; 1 1  0 0 • T22 =

0 0 0

1 0 0

0 cos λ − sin λ

0 sin λ cos λ

, where λ is the frequency of the

cyclical component.7

• T12 = [1, 1, cos λ, sinλ]; στ 0 σc   σ0τ σ0 00  R1 • R2 =  µ . 0 0

0 0

σc σc

The definitions of T12 and R1 simply follow from the trend-cycle decomposition y˜ t ≡ τt + ct . 3.4. Identification and estimation It is well-known that in the absence of restrictions on the parameter matrices of the general state-space model, the

7 We could also prefix the trigonometric constants by a constant ρ subject to |ρ| < 1 to produce dampened cycles.

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parameters are unidentified—more than one set of values for the parameters can give rise to the identical value of the likelihood function, and the data give us no guide for choosing among these (Hamilton, 1994, Section 13.4). Local identification may be verified by translating the state-space representation into a vector ARMA process and checking the conditions for identification given in Hannan (1971, 1976). Another approach is to work directly with the state-space representation, see e.g. Burmeister et al. (1986) or Otter (1986). We adopt the latter approach in a companion paper: Jacobs and van Norden (2007) provide sufficient conditions under which the parameters of our model are identified. We summarize some of them here. We begin by assuming that the parameters of the dynamic model for the true values y˜ are identified. A second condition is that the dynamics of the measurement errors and true values are not too similar. When the dynamics implied by Tφ , Tζ and Tν are sufficiently distinct, this can be critical in distinguishing noise and news from movements in true values. As a result, identification is generally easier in our general model than in a restricted model which imposes the same dynamics on all three components. Even in the case where these components are identical (for example, when the dynamics of news and noise are identical), we can distinguish news and noise if we have sufficient restrictions on the persistence of noise shocks across subsequent vintages. Therefore, although our model allows for rich dynamics, the richness of the data generally allows us to identify quite complex behaviors. While the parameters of our general model may be identified, this need not imply that conventional state space methods are the only (or even the best) way to estimate them. As de Antonio Liedo and Carstensen (2006) note, the presence of multiple vintages implies that many revisions are directly observed and their behavior may be consistently estimated by GLS. This suggests the use of an EM algorithm or a concentrated likelihood function to simplify estimation of the full model. Even in situations where the model’s parameters may be estimated without recourse to its state-space form (such as when we assume that true values are observed after a small number of revisions), the state-space form of the model may still be useful for the construction of optimal forecasts, smoothed estimates, and determining the optimal weights to be placed on data of varying vintages. 4. Illustration This section demonstrates the feasibility of our state-space framework by presenting estimates for five simple models of US real output estimates. We begin by describing our data source and reviewing the characteristics of the data revision process. We then describe, present and compare five alternative state-space models for these data: pure noise, pure news, noise plus spillovers, news and spillovers, and a mixture of news, noise and spillovers. 4.1. Data and data properties To illustrate the uses of our model of data revisions, we use the real output series of the Federal Reserve Bank of Philadelphia, which consists of quarterly vintages from 1965Q4 up to and including 2006Q2 and provides quarterly information from 1947Q1 onwards. The vintages have a publication lag of one quarter, i.e. our last vintage (2006Q2) spans the period 1947Q1 to 2006Q1.8

8 The vintages 1992Q1–1992Q4 do not have observations for 1947Q1–1958Q4, vintages 1996Q2–1997Q1 provide no information for 1947Q1–1959Q2, and vintages 1999Q4 and 2000Q1 begin only in 1959Q1.

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the first four vintage estimates in our y vector. This captures the most important revisions shown in the above figure (including the annual revision each summer) but also implies that important measurement error may be present in the ‘‘final’’ vintage. Our most general model with news, noise and spillovers may then be written as  t +1  yt yt +2    t   t +3  = ι4 yt 

01×4

  y˜ t   y˜  t −1  I4  ,  νt  ζt 

I4

t +4 yt

  ρ1 y˜ t   y˜  t −1   1 =   νt  04×1 04×1 ζt 

σe 0  0  0  0  + 0  0  0  

Fig. 2. News and noise in US GDP growth?

Real output estimates are typically revised once per quarter for the first few quarters after they are released to incorporate the arrival of more and better information. Annual revisions (in the summer) commonly affect up to the last three years of observations. In addition comprehensive or benchmark revisions, carried every few years, typically incorporate conceptual, methodological or benchmark changes and therefore can change every published estimate from the first to the last observation. Benchmark revisions in the real output series occurred in 1976Q1, 1981Q1, 1986Q1, 1992Q1, 1996Q1 and 1997Q2, 1999Q4 and 2000Q2, 2004Q1. To mitigate any level effects of benchmark revisions we follow the common practice of analyzing the first differences of the natural logarithms of the raw data. To better describe the properties of the data, we confine ourselves to periods for which at least 12 different vintages are available. This prevents estimation of the model for the most recent time periods, but still leaves 153 time-series observations for analysis.9 To illustrate the complex character of measurement errors in this commonly studied series, Fig. 2 shows correlations between t +j t +j −1 different vintage revisions (yt − yt ) on the one hand, and t +j 2006Q 2 current and final estimates (yt , yt ) on the other. We observe many non-zero correlation coefficients, suggesting that the news and noise null hypotheses are rejected at times. Interestingly, different revisions appear to display different properties. For the first revision, neither correlation coefficient is close to zero, suggesting that both the pure news and pure noise models would be rejected. For the second revision, however, the correlation with the current vintage is approximately zero, suggesting that news model would be a good fit. The opposite is true for the third revision, suggesting that it may be better characterized as noise. The figure also shows that the dynamics continue to vary for many quarters after the initial release of the data. These results are only meant to be suggestive; we leave a more formal consideration of the characteristics of data revisions to the model results, presented below.

ρ2

01×4

0

01×4

 y˜ t −1    01×4  y˜ t −2    04×4  νt −1  ζ t −1 T4 01×4

04×1

T3

04×1

04×4

σν 1

σν 2

σν 3

σν 4

0

0

0

0

0

0

0

0

0

0

−σν 1

−σν 2 −σν 2

0

0

0

0

0

0

0

0

0

0

0

−σν 4 −σν 4 −σν 4 −σν 4

0

0

−σν 3 −σν 3 −σν 3

0

0

0

0

0

0

0

0

0

σζ 1

0

0

0

0

0

0

0

σζ 2

0

0

0

0

0

0

0

σζ 3

0

0

0

0

0

0

0

0

0

0   ηet  0   ην1 t   0    ην t   2  0   η    ν t 0   3   · ην t  , 0   4   ηζ t   1  0   η    ζ t 0   2   ηζ3 t 0 ηζ4 t σζ 4

where T3 and T4 are diagonal 4 × 4 matrices. The model has 19 free parameters (ρ1 , ρ2 , σe , σν 1 , σν 2 , σν 3 , σν 4 , σζ 1 , σζ 2 , σζ 3 , σζ 4 and the eight diagonal elements of T3 and T4 ) to be estimated on 153 observations of four data vintages.10 We also considered four simpler models, each of which maintained the assumption of an AR(2) process for the true values of real output growth.11 Two of these assumed the absence of news shocks and take the form. ytt +1 yt +2    tt +3  = ι4 y  t ytt +4







y˜ t y˜ t −1



01×4

I4

ρ1

ρ2

1 04×1

0 04×1 0 0

 =

ζt



σ e 0 0 + 0 





ζt

01×4 y˜ t −1 y˜ t −2 01×4 ζ t −1 T4    0 0 0 ηet 0 0 0  ηζ1 t   0 0 0    · η  . σζ 2 0 0   ζ2 t   ηζ3 t 0 σζ 3 0 ηζ4 t 0 0 σζ 4



σζ 1

0 0

y˜ t y˜ t −1 ,

0 0 0



The other two assumed the absence of noise shocks and take the form ytt +1 yt +2    tt +3  = ι4 y  t ytt +4





 01×4

I4



 y˜ t ˜yt −1 , νt

4.2. Estimation results We assume a simple AR(2) process for the dynamics of ‘‘true’’ output growth. The estimates presented below use only

9 This restriction could be relaxed using filtering methods for missing observations (see Harvey, 1989, Section 3.4.7).

10 We investigated the properties of the maximum likelihood (ML) estimators reported below by simulating data from the above general model and comparing the resulting ML parameter estimates to the true values used to create the data. Results were encouraging; ML estimates were close to true values and the robust standard errors used below seemed to be a reasonable guide to their precision, particularly for the elements of the T matrix. 11 Estimates assuming an AR(1) process produced very similar results.

J.P.A.M. Jacobs, S. van Norden / Journal of Econometrics 161 (2011) 101–109

107

Table 1 Estimation results: AR(2) model, four vintages. Parameter

Pure noise

Pure news

Estimate

ρ1 ρ2

AR(1) AR(2) Spill 1 - news Spill 2 - news Spill 3 - news Spill 4 - news Spill 1 - noise Spill 2 - noise Spill 3 - noise Spill 4 - noise AR shock News shock 1 News shock 2 News shock 3 News shock 4 Noise shock 1 Noise shock 2 Noise shock 3 Noise shock 4

T331 T332 T333 T334 T441 T442 T443 T444

σe × 10 σν 1 × 10 σν 2 × 10 σν 3 × 10 σν 4 × 10 σζ 1 × 10 σζ 2 × 10 σζ 3 × 10 σζ 4 × 10

Log likelihood function (llf) AIC BIC

Std. err.

0.410 0.043

0.082 0.085

9.074

0.523

2.387 1.028 0.726 1.569

0.152 0.097 0.112 0.105

Estimate 0.217 0.133

5.666 2.302 1.258 1.612 24.777

Std. err. 0.166 0.088

1.112 0.132 0.072 0.092 14.445

Noise +spillovers

News +spillovers

News +noise +spillovers

Estimate

Std. err.

Estimate

Std. err.

Estimate

Std. err.

0.410 0.043

0.082 0.082

0.198 0.116 −0.008 −0.030 −0.037 −0.025

0.118 0.053 0.044 0.046 0.048 0.047

−0.130 0.035 −0.084 −0.008 9.075

0.087 0.107 0.164 0.042 0.522

4.893 2.199 1.241 1.575 30.087

1.185 0.134 0.073 0.094 15.240

2.364 1.030 0.720 1.570

0.151 0.096 0.112 0.105

0.177 0.147 −0.055 −0.079 −0.088 −0.077 – 0.850 – 0.305 4.761 2.141 1.195 1.430 27.012 0.000 0.183 0.000 0.633

0.146 0.071 0.064 0.065 0.068 0.071 – 0.169 – 0.311 1.086 0.130 0.141 0.330 14.118 – 0.150 – 0.455

−48.533

−16.887

−47.087

−9.365

−7.597

111.067 132.280

47.774 68.987

116.175 149.510

40.730 74.065

53.194 110.772

Notes: Standard errors are based on QML estimates. The Akaike Information Criterion (AIC) is calculated as −2 · llf + 2 · k, where k is the number of parameters. The Bayes Information Criterion (BIC) is calculated as −2 · llf + 2 · ln(T ) · k, where T is the number of observations.

y˜ t y˜ t −1



νt



ρ1

ρ2

 1 =  0

0



4×1

σ e 0 0 + 0  0 0

04×1

σν 1



 01×4  y˜ 01×4  t −1 y˜ T3  t −2 ν t −1

σν 2

σν 3

0

0

0

−σν 1

−σν 2 −σν 2

−σν 3 −σν 3 −σν 3

0 0 0

0 0

0

 σν 4   ηet 0  ην1 t  −σν 4   · ην t  .  −σν 4   η 2  ν3 t −σν 4 ην4 t −σν 4

For both pairs of models, we estimated both a version with the first-order spillover effects described above (where T3 or T4 are diagonal) and a version without spillover effects (where T3 = T4 = 04×4 ). Table 1 lists the parameter estimates for all five of the above models; pure noise, pure news, noise plus spillovers, news plus spillovers, and the general or ‘‘news plus noise plus spillovers’’ model. Prior to estimation, all vintages were approximately standardized using the mean and standard deviation of the final (i.e. 4th) vintage.12 All parameters were estimated by maximum likelihood (ML) and their standard errors are robust estimates based on the cross-product of the score matrix. The third and the fourth columns list the parameter estimates and standard deviations for the pure noise model. The AR parameters ρ1 and ρ2 sum to approximately 0.45. The AR shock has a standard deviation (σe ) slightly less than one. The estimated standard deviations of the noise shocks are large relative to their robust standard errors and they decrease until that of the final measurement error σζ4 . The degree of measurement error remaining after the first year of revisions is slightly more than

12 This provided vintages that each had means very close to zero and standard deviations very close to one, while ensuring that the change from one vintage to the next was entirely due to a revision in the official estimates and not a change in factors used to standardize the data.

half of that of the first revision. The largest of the noise shocks has a standard deviation roughly one-quarter the size of the AR shock. All these characteristics seem reasonable given the dynamic properties of the data. The results of the pure news model listed in the fifth and the sixth columns of Table 1 show an AR(2) process with slightly less persistence than before; the sum of the AR parameters is now only 0.35. However, the AR shock is now about one-third smaller than in the previous model. The standard deviations of the pure news shocks are of the same magnitude as those of pure noise shocks with the exception of the final measurement error, which is now much larger but very imprecisely estimated. Columns seven through ten show the results of adding spillover effects to the above pure news and pure noise models. At first glance, spillover effects appear to be unimportant. Individual estimates of the spillover parameters all appear small relative to their standard errors and estimates of the other model parameters are little changed from their values in the pure noise and news models. However, this neglects the strong correlation between estimated spillover parameters, which jointly appear to be important for each model. For example, a Likelihood Ratio (LR) test statistic for the null hypothesis of no spillovers should have a χ 2 (4) distribution under the null hypothesis, but has a value of 2.89 for the noise model and 15.04 for the news model.13 The final two columns of the table show the parameter estimates of the most general model we estimated: the mixture of news, noise and spillovers. The estimates of the AR parameters and all the parameters of the noise shocks are very similar to those of the news plus spillover model presented in the two previous columns. The noise shocks, however, are very different. Two of

13 Of course, likelihood ratio statistics also have interpretations in terms of posterior odds. If λ is the likelihood ratio statistic, then eλ/2 is the factor by which we increase the prior odds of the spillover model to arrive at the posterior odds. This is roughly 1.2 × 1096 for the noise models and something in excess of 1 × 10500 for the news models.

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J.P.A.M. Jacobs, S. van Norden / Journal of Econometrics 161 (2011) 101–109

the four noise shocks are estimated to have standard deviations of zero, while the remaining two seem much smaller than before.14 However, the estimated spillover effects of the two remaining noise shocks are by far the largest of any model. While the most general model provides an increase in the value of the log likelihood function relative to any of the other models, the use of likelihood ratio tests for model comparisons is not justified in this case as several parameters (the standard deviations of the news or noise shocks) are constrained to lie on the boundary of the parameter space (i.e. =0) under the null hypothesis. The Akaike Information Criterion (AIC) prefers the simpler News & Spillovers model as the best characterization of the data while the Bayes Information Criterion (BIC) prefers the Pure News model. The BIC values parsimony more highly than the AIC and has more desirable asymptotic properties.15 , 16 Most of the models estimated above are not particularly parsimonious; for example, our above discussion of the estimated spillover parameters suggests that a more parsimonious representation may still provide a reasonable description of the data.17 These results illustrate the feasibility of our state-space framework for modeling measurement errors. While they provide relatively little evidence of noise in GDP revisions, they show the importance of the spillover effects of news across time-series observations. These results are consistent with the view that the US Bureau of Economic Analysis has efficiently incorporated available information into its preliminary estimates and revisions of real GDP. 5. Conclusion This paper presented a state-space framework which allows for general dynamics of ‘‘true’’ values and three types of measurement errors (news, noise and spillovers) as well as published estimates which never converge to ‘‘true’’ values. Analysis of the revisions of US real GDP presented here confirms the previous findings that their dynamics are complex. Our novel formulation of the statespace model enables us to flexibly manipulate and estimate these complex dynamics. Our results above confirm that these methods are feasible even using modest, realistic numbers of observations and vintages. They also demonstrate the relative ease with which we can separate the news and noise components in measurement errors. We think this state-space framework for modeling data revisions has great potential. Applications include estimating and constructing confidence intervals for productivity trends and cycles for the formulation and conduct of monetary and fiscal policy. Future research will also address multivariate processes. Acknowledgements This paper was written during visits of the first author to HEC Montréal and CIRANO, of the second author to the research school

14 When the data revision shocks are equal to zero, the corresponding spillover parameters are not identified. As is the case with the standard deviations of other shocks, zero is on the boundary of the parameter space so conventional standard errors are not valid. 15 The BIC is consistent and has the same asymptotic properties independent of the choice of prior. Moreover, it is guaranteed to be maximal at the model with the highest posterior odds. 16 We also examined the Hannan–Quinn Information Criterion (HQIC), which has a variable penalty for lack of parsimony indexed by c. The HQIC is consistent for c > 2; it selects the News + Spillovers models for c < 2.36 and otherwise selects the Pure News model. 17 Additional results (available by request from the authors) confirmed that more parsimonious parameterizations of the spillover effects in the News + Spillovers models were preferred by the BIC to the pure News model.

SOM of the University of Groningen, and of both authors to KOF Zurich. The hospitality and support of these institutions, as well as that of CREF and CIREQ, is gratefully acknowledged. The second author would also like to thank the INE program of the Canadian SSHRC for financial support. We would like to thank participants at the 2006 (EC)2 Conference, Rotterdam, December 2006, the 2007 International Symposium on Forecasting, New York, June 2007, International Conference Measurement Error, Econometrics and Practice, Birmingham, July 2007, the CGBCR Conference, Manchester, July 2007, the 2007 Joint Statistical Meetings, Salt Lake City, Utah, July 2007, the 2008 AEA Meetings, New Orleans, LA, January 2008, ESEM2008, Milan, August 2008, and the 5th Eurostat Colloquium on Modern Tools for Business Cycle Analysis, Luxembourg, September–October 2008, and several seminars and workshops for their helpful comments. 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