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Can $GDP$ measurement be further improved? Data revision and reconciliation
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\\[0.1in] JEL classification: E01, E32 \\ Keywords: national accounts, output, income, expenditure, news, noise
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Unlike many other nations, U.S. national accounts feature distinct estimates of real output based on the expenditure approach ($GDE$) and the income approach ($GDI$), see Figure (ref). As pointed out by Stone, Champernowne and Meade (1942)\nocite{StoneChampernowneMeade1942}, while in theory these two approaches should give identical estimates, measurement errors cause discrepancies to arise.\footnote{ The same applies to the production-based estimate of output. See e.g. the study of Rees, Lancaster and Finlay (2015)\nocite{ReesLancasterFinlay2015} on Australian GDP.} These discrepancies are sometimes important. Chang and Li (2015)\nocite{ChangLi2015} examine the impact of using $GDI$ rather than $GDE$ in nearly two dozen recent empirical papers published in major economic journals; they find substantive differences in roughly 15% of them. Nalewaik (2012)\nocite{Nalewaik2012} finds that $GDI$ leads to quicker detection of U.S. recessions than $GDE$.
While several studies have tried to determine which measure should be preferred in various contexts, Weale (1992)\nocite{Weale1992} and Diebold (2010)\nocite{Diebold2010} argue that reconciling them is a more useful response as it should incorporate more information. Fixler and Nalewaik (2009)\nocite{FixlerNalewaik2009} point out, however, that reconciliation traditionally relies on the assumption that measurement errors are “noise”, which in turn forces the reconciled estimate of the latent variable (“true” $GDP$ in this case) to be less variable than any of the individual series being reconciled. They instead propose that measurement errors may also include a “news” component. While this causes a loss of identification, they glean information from the revision of $GDE$ and $GDI$ to place bounds on relative contributions of news and noise in a least-squares framework. Aruoba et al. (2012)\nocite{Aruobaetal2012} consider the problem from a forecast combination perspective, assuming “news” errors and imposing priors in lieu of identification without revisions, while Aruoba et al. (2016)\nocite{Aruobaetal2016} consider alternative identifying assumptions and propose the addition of an instrumental variable. Almuzara et al. (2018)\nocite{AlmuzaraFiorentiniSentana2018} investigate a dynamic factor model with cointegration restrictions.
Aruoba et al. (2016)\nocite{Aruobaetal2016} is the basis for the $GDP^+$ measure published by the Federal Reserve Bank of Philadelphia.\footnote{See http://www.philadelphiafed.org/research-and-data/real-time-center/gdpplus/} However, while their approach ignores the possibility of data revision, Figure (ref) shows that the published series is subject to important revisions, which complicates its interpretation and use in policy decisions. Separately, Jacobs and van Norden (2011)\nocite{JacobsvanNorden2011} and Kishor and Koenig (2012)\nocite{KishorKoenig2012} propose state-space frameworks that allow estimation of both news- and noise-type measurement errors in data revision, but do not consider problems of data reconciliation. In this paper we extend Jacobs and van Norden (2011\nocite{JacobsvanNorden2011}, henceforth JvN) to consider the problem of reconciliation and identification in which there are multiple estimates of the common underlying variable, all of which are subject to revision. Allowing for both news and noise measurement errors, the result is a modeling framework substantially more general than those previously proposed. We show that identification of these two types of measurement errors is made possible by modeling data revisions as well as the dynamics of the series. We provide a historical decomposition of $GDE$ and $GDI$ into news and noise shocks, and we compare those series to our improved $GDP$ estimate, $GDP^{++}$. We find that $GDP^{++}$ is more persistent than either $GDE$ or $GDI$. While both series appear to contain both news and noise shocks, news shocks have a larger share in $GDE$ than in $GDI$.
The paper is structured as follows. In Section (ref) we present our econometric framework. We show that our system is identified using real-time data and news-noise assumptions. In Section (ref) we describe our data and estimation method. Results are shown in Section (ref) and Section (ref) concludes. Formal proofs of some results related to identification are presented in an Appendix.
In this section, after establishing some notation, we describe our econometric framework. We begin by briefly reviewing the univariate news and noise model of JvN before generalizing it to the problem of data reconciliation. We then compare the results to the $GDP^{+}$ model of Aruoba et al. (2016) and discuss their differences for the identification of news and noise measurement errors.
We follow the standard notation in this literature by letting $y_{t}^{t+j}$ be an estimate published at time $t+j$ of some real-valued scalar variable $y$ at time $t$. We define $\bm y_{t}$ as a $l\times 1$ vector of $l$ different vintage estimates of $y_{t}^{t+i}$, $i=1,\ldots ,l$ so $\bm y _{t}\equiv \left[ y_{t}^{t+1},y_{t}^{t+2},\ldots ,y_{t}^{t+l}\right] ^{\prime }$. \ For state-space models, we follow the notation of Durbin and Koopman (2001)\nocite{DurbinKoopman2001}
where $\bm y_{t}$ is $l\times 1$, $\bm \alpha_{t}$ is $m\times 1$, $\bm \varepsilon _{t}$ is $l\times 1$ and $\bm \eta_{t}$ is $r\times 1$; $\bm \varepsilon_{t}\sim N(0,\bm H)$ and $\bm \eta_{t}\sim $ $N(0,\bm I_{r})$. Both error terms are i.i.d.\ and orthogonal to one another.\footnote{ For more detailed assumptions, see Durbin and Koopman (2001\nocite {DurbinKoopman2001}, Section 3.1 and 4.1. For convenience we omit constants from the model in this exposition. }
JvN denote the unobserved \textquotedblleft true\textquotedblright\ value of a variable as $\tilde{y}_{t}$, so that its measurement error $\bm u_{t}\equiv \bm y_{t}-\bm \iota_{l}\cdot \tilde{y}_{t}$, where $\bm \iota_{l}$ is an $l\times 1$ vector of ones. They model these measurement errors as the sum of “news” and “noise” measurement errors. Measurement errors are said to be noise $ \left( \zeta _{t}^{t+i}\right) $ when they are orthogonal to the true values $\tilde{y}_{t}$, so that
Noise implies that revisions ($y_{t}^{t+i+1}-y_{t}^{t+i}$) are generally forecastable. Measurement errors are described as news $(\nu _{t}^{t+i})$ if and only if
If data revisions are pure news errors, current and past vintages of the series will be of no use in forecasting future data revision.
In their state-space model JvN impose $\bm \varepsilon_{t}\equiv \bm{0}_{l\times 1}$ and partition the state vector $\bm \alpha_{t}$ into four components
of length $1,$ $b$, $l$ and $l$ respectively, where $\bm \phi_{t}$ is used to capture the dynamics of the true values while $\bm \nu_{t}$ and $\bm \zeta_{t}$ are the news and noise measurement errors, respectively. They similarly partition
where $\bm Z_{1}=\bm \iota_{l}$ (a $l\times 1$ vector of 1's), $\bm Z_{2}=\bm{0} _{l\times b}$ (an $l\times b$ matrix of zeros), $\bm Z_{3}=\bm I_{l}$, and $\bm Z _{4}=\bm I_{l}$ (both $l\times l$ identity matrices). Their measurement equation ((ref)) then simplifies to
They conformably partition the matrix $\bm T$ as
where $T_{11}$ is a scalar, and $\left\{ \bm T_{12},\bm T_{21},\bm T_{22},\bm T_{3}, \bm T_{4}\right\} $ are $1\times b$, $b\times 1$, $b\times b$, $l\times l$ and $l\times l$; $\bm{0}$ is a conformably defined matrix of zeros. The $\left( b+1\right) \times \left( b+1\right) $ block in the upper left simply captures the dynamics of $\widetilde{y}_{t}$ while $\bm T_{3}$ and $\bm T_{4}$ capture the dynamics of the news and noise shocks. If measurement errors are independent across time periods (but not vintages), then $\bm T_{3}\equiv \bm T _{4}\equiv \bm{0}_{l\times l}$. \ As we will see below, in the special case where $ \tilde{y}_{t}$ is assumed to follow an $AR\left( p\right) $ process, this will impose $p=b+1$, the row vector $
$ will contain the autoregressive coefficients and the remainder of the upper left $\left( b+1\right) \times \left( b+1\right) $ part will be composed of zeros and ones.\footnote{ For details, see Jacobs and van Norden (2011)\nocite{JacobsvanNorden2011}.}
The essential difference between news and noise errors is captured in the $\left( 1+b+2l\right) \times (1+2l)$ matrix $\bm R,$ which is partitioned as follows
where $\bm U_{l}$ is a $l\times l$ matrix with zeros below the main diagonal and ones everywhere else, $\bm R_{3}=\left[ \sigma _{\nu 1},\sigma _{\nu 2},\ldots ,\sigma _{\nu l}\right] $, where $\sigma _{\nu i}$ is the standard error of the measurement error associated with $i$-th estimate $y_{t}^{t+i}$ , $\operatorname{diag}(\bm R_{3})$ is a $l\times l$ matrix with elements of $\bm R_{3}$ on its main diagonal, and $\bm R_{4}$ is an $l\times l$ matrix. Finally, the error term is partitioned as $\bm \eta_{t}=\left[ \bm \eta_{et}^{\prime },\ \bm \eta_{\nu t}^{\prime },\ \bm \eta_{\zeta t}^{\prime }\right] ^{\prime }$, where $\bm \eta _{et}$ refers to errors associated with the true values, and $\bm \eta_{\nu t}$ and $\bm \eta_{\zeta t}$ are the errors for news and noise, respectively.
JvN note that (if the model is identified, a question we deal with below) this framework permits conventional techniques to be used to estimate the model parameters, allow for missing observations, estimate and forecast the unobserved true values $\widetilde{y}_{t}$ together with their confidence intervals, and test hypotheses.
We now show how the above framework may be adapted to the case where we have two alternative estimates of the same underlying true value $\widetilde{ y}_{t}$, both of which are subject to revision. We define $\bm Y_{t}$ as a $ 2l\times 1$ vector of $l$ different vintage estimates for the $2$ variables $ y1_{t}^{t+i}$ and $y2_{t}^{t+i}$, $i=1,\ldots ,l$, for a particular observation $t$, so $\bm Y_{t}\equiv \left[ y1_{t}^{t+1},y1_{t}^{t+2}\ldots ,y1_{t}^{t+l},y2_{t}^{t+1},y2_{t}^{t+2},\ldots ,y2_{t}^{t+l}\right] ^{\prime },$ a vector of length $2l.$\ Our state-space model now becomes
We again partition the state vector $\bm \alpha_{t}$ into four components
which are now of length $1,$ $b$, $2l$ and $2l$ respectively, and we similarly partition
where $\bm Z_{1}=\bm \iota_{2l}$(a $2l$ vector of ones), $\bm Z_{2}=\bm{0} _{2l\times b}$ (a $2l\times b$ matrix of zeros), and $\bm Z_{3}=$ $\bm Z_{4}=\bm I _{2l}$ (both are $2l\times 2l$ identity matrices). The measurement equation ( (ref)) therefore again simplifies to \[ \bm Y_{t}=\bm Z\cdot \bm \alpha_{t}=\tilde{y}_{t}+\bm \nu_{t}+\bm \zeta_{t}=\mbox{`Truth'} +\mbox{`News'}+\mbox{`Noise'}. \] The matrix $\bm T$ is partitioned much as before
The upper left block (consisting of $T_{11},\bm T_{12},\bm T_{21}$ and $\bm T_{22}$ ) is precisely the same as in ((ref)) above; this is because it solely determines the dynamics of $\widetilde{y}_{t}$, which are unchanged. However, the addition of a new series increases the dimension of $ \bm T_{3}$ and $\bm T_{4}$ from $l\times l$ to $2l\times 2l$.
$\ \bm R$ is now a $ \left( 1+b+4l\right) \times (1+4l)$ matrix where we separate the news and noise measurement errors for the two variables
where the row vector $\bm R_{3}=\left[ \sigma _{\nu _{1}^{1}},\sigma _{\nu _{2}^{1}},\ldots ,\sigma _{\nu _{1}^{1}}\right] $ corresponds to the news in $y1$ while $\bm R_{4}=\left[ \sigma _{\nu _{1}^{2}},\sigma _{\nu _{2}^{2}},\ldots ,\sigma _{\nu _{l}^{2}}\right] $ corresponds to the news in $y2$. $\operatorname{diag}(\bm R_{3})$ and $\operatorname{diag}(\bm R_{4})$ are $l\times l$ diagonal matrices with the elements of $\bm R_{3}$ and $\bm R_{4}$ on their main diagonals, while $\bm R_{5}$ and $\bm R_{6}$ are $l\times l$ diagonal matrices.
Finally, we partition $\bm \eta_{t}=\left[ \bm \eta_{et}^{\prime },\ \bm \eta_{\nu _{1}t}^{\prime },\ \bm \eta_{\nu _{2}t}^{\prime },\ \bm \eta_{\zeta _{1}t}^{\prime },\ \bm \eta_{\zeta _{2}t}^{\prime }\right] ^{\prime }$, where $\bm \eta_{et}$ refers to errors associated with the true values, and $\bm \eta_{\nu it}$ and $ \bm \eta_{\zeta it}$ are the errors for news and noise measurement errors in variable $i.$
To illustrate, consider the following very simple case. Let $y1\equiv GDE$ (the growth rate of real gross domestic expenditure), $y2\equiv GDI$ (the growth rate of real gross domestic income), $l=2$ (we only consider two vintages, the 1st and 2nd releases) and we'll assume that the growth rate of “true" real output $\widetilde{y}$ follows an $AR\left( 1\right) $. Then ( (ref)) becomes
and ((ref)) becomes \[
=
\cdot
+\bm R\cdot \bm \eta_{t}, \] where
Aruoba et al. (2016)\nocite{Aruobaetal2016} consider the problem of identification in a special case of the GDE/GDI example considered above where only a single vintage is available $\left( l=1\right) $. Their unrestricted model may be written as\footnote{ See Aruoba et al. (2016)\nocite{Aruobaetal2016}, equations (A.1) and (A.2). Their model further differs from the model above in that (a) they model only the sum of news and noise shocks, and (b) they assume that $\bm T_{3}=\bm T_{4}=0$, a condition that we will also impose, below.}
and they show that it is not identified. They propose adding a third (instrumental) variable which is correlated with $\widetilde{y}_{t}$ but not with $\eta _{t}^{E}$ or $\eta _{t}^{I}$, suggesting that household survey data may be suitable for this purpose. We argue that the model may be identified instead by increasing the number of vintages analysed and assuming that measurement errors are the sum of news and noise measurement errors as characterized above. We explore this point in the remainder of this section by comparing the available number of sample moments to the number of free parameters in the model. In the Appendix we provide a more rigorous proof of identification in a slightly simpler model using the methods of Komunjer and Ng (2011).
The essential insight comes from the form of the $\bm R$ matrix in ((ref)). News and noise measurement errors have tightly constrained behaviour across successive data vintages; Noise errors are assumed to be uncorrelated across vintages and with innovations in true values, while news errors must be correlated with one another, with innovations in true values, and their variances must be decreasing as series are revised.
If we have two series to reconcile (here $GDE$ and $GDI$) and $l$ vintages of each, we have $2\cdot l\cdot (2\cdot l+1)/2$ observable cross moments as well as $2\cdot l$ first-order autocorrelation coefficients, for a total of $ l\cdot (2\cdot l+3)$ moments. The only free parameters in the above model, however, are the autocorrelation coefficient $\rho $ and the $(1+4\cdot l)$ non-zero elements of $R$, for a total of $2\cdot (1+2\cdot l).$ This implies that the number of available moments increases with $l^{2}$ while the number of free parameters increases only with $l$.\footnote{ Note that we have ignored any free parameters in $\bm T_{3}$ and $\bm T_{4}$ in these calculations. We return to this, below. One must also keep in mind that identification by data revision requires that the data are in fact revised. If not, we effectively return to the underidentified case of $l=1$. }
In the special case where we use only a single data release, $l=1$, we have $2\cdot (1+2\cdot 1)=6$ free parameters to estimate, but only $1\cdot (2\cdot 1+3)=5$ available moments with which to do so. This is consistent with the lack of identification noted by Aruoba et al. (2016)\nocite {Aruobaetal2016}. However, if we use $l=2$ data vintages, we have $2\cdot (1+2\cdot 2)=10$ free parameters and $2\cdot (2\cdot 2+3)=14$ moments with which to identify them. For $l=3$ we have 27 moments with which to estimate 14 parameters and for $l=4$ (the case we consider below) we have 44 moments with which to estimate 18 parameters.
This suggests that as we add more data releases, we potentially have the ability to generalize the model further still. The univariate data revision model of JvN envisages two such types of generalization.
In the Appendix, we briefly explore the possibilities for identification with some of these generalizations. We now turn to consider the revisions in the available data.
We use monthly vintages of quarterly expenditure-based and income-based estimates of GDP from the Bureau of Economic Analysis (BEA) covering the period 2003Q1--2014Q3. For $GDE$ we employ the Advance, the Third, the 12th and the 24th releases and Second/Third, 12th and the 24th releases for $GDI$. Due to a lag in source data availability the BEA does not prepare Advance estimates for $GDI$. The initial estimates for $GDI$ are presented with the Second $GDI$ estimate. Estimates for fourth quarter $GDI$ are presented in the Third estimate only.\footnote{See Fixler et al. (2014)\nocite{FixlerGreenawayGrimm2014} for a more detailed discussion of the $GDE$-$GDI$ vintage history.}
We employ Gibbs Sampling methods to obtain posterior simulations for our model's parameters (see, e.g., Kim and Nelson 1999\nocite{KimNelson1999}). We use conjugate and diffuse priors for the coefficients and the variance covariance matrix, resulting in a multivariate normal posterior for the coefficients and an inverted Wishart posterior for the variance covariance matrix. For the prior for the coefficients restricted to zero we assume the mean to be zero and variance to be close to zero.
Our Gibbs sampler has the following structure. We first initialize the sampler with values for the coefficients and the variance covariance matrix. Conditional on data, the most recent draw for the coefficients and for the variance covariance matrix, we draw the latent state variables $\bm \alpha_t$ for $t=1,...,T$ using the procedure described in Carter and Kohn (1994)\nocite{CarterKohn1994}. In the next step, we condition on data, the most recent draw for the latent variable $\bm \alpha_t$ and for the variance covariance matrix, drawing the coefficients from a multivariate normal distribution. Finally, conditional on data, the most recent draw for the latent variables and the coefficients, we draw the variance covariance matrix from an inverted Wishart distribution. We cycle through 100K Gibbs iterations, discarding the first 90K as burn-in. Of those 10K draws we save only every 10th draw, which gives us in total 1000 draws on which we base our inference. Convergence of the sampler was checked by studying recursive mean plots and by varying the starting values of the sampler and comparing results.
Here we compare our measure of $GDP$ to releases of $GDE$ and $GDI$ in four different ways: (i) in graphs, (ii) looking at historical decompositions, (iii) by investigating dynamics, and (iv) by calculating relative contributions. To distinguish between the true unknown values of $GDP$ and our model's estimates of these values, we refer to our model's estimates as $GDP^{++}$.
In Figure (ref) we compare $GDP^{++}$ and its shaded posterior ranges (90% of probability mass) to the four releases of $GDE$ we employed in the estimation, the Advance, third, the 12th and the 24th release. There is some evidence that the releases are more volatile than the true values of $GDP$. We observe that the releases are outside the posterior bounds for some periods. This observation holds especially for the Advance release and the 24th release; in some periods, like e.g. 2010Q1, the Advance release and the 24th release are on different sides of the posterior range.
Figure (ref) shows $GDP^{++}$ together with shaded posterior ranges (90% of probability mass) and the three releases of $GDI$ we employed in the estimation, the Second/Third, the 12th and the 24th release. The releases fluctuate around the posterior bounds of the true values. The $GDI$ releases are more volatile than our estimates $GDP^{++}$. The releases of $GDI$ are also much more volatile than the releases of $GDE$. Note that the sample paths of $GDP_M$ and $GDE$ and $GDI$ in Aruoba et al. (2016, Figure 3) show a different picture than our Figures (ref) and (ref). $GDE$ differs more from their $GDP$ measure than $GDI$.
Our econometric framework ((ref)-(ref)) allows the historical decomposition of $GDE$ and $GDI$ in terms of news and noise measurement errors. We illustrate the decomposition for $GDE$.
Suppose, we have $l$ releases of $GDE_t$
Then the total revision of $GDE$ can be written as
where every element on the right-hand side of the equation is part of the state vector whose estimates may be recovered using standard techniques.
The outcomes of the historical decompositions are shown in Figure (ref). The top panel shows total revisions in $GDE$ with news and noise shares, the bottom panel total $GDE$ revisions with news and noise shares. We observe that total revisions in $GDI$, the bottom panel, are larger than total revisions in $GDE$, a stylized fact which can also be distilled from the previous two figures. The two panels suggest that the news share in total $GDE$ revisions is larger than the noise share while the opposite seems to hold for total revisions in $GDI$. This observation is consistent with Fixler and Nailewaik (2009), who also reject the pure noise assumption in $GDI$. It also appears that GDI was particularly noisy around the start of 2008 and after 2012.
In Figure (ref) we depict the ($\rho,\sigma^2$) pairs summarizing the dynamics of our true $GDP$ estimate across all draws. We contrast the ($\rho,\sigma^2$) pairs corresponding to our $GDP^{++}$ estimate to the ($\rho,\sigma^2$) pairs obtained when using a news measurement error only or a noise measurement error only version of our model, the benchmark model estimated in Aruoba et al. (2016) and when fitting an AR(1) model to $GDE$ and $GDI$.
Figure (ref) reveals that our real-time data based estimate of $GDP$ is somewhat less persistent than the $GDP^+$ measure of Aruoba et al. (2016), but exhibits a higher persistence than the estimates for $GDE$ and $GDI$.\footnote{We thank Dongho Song for making his Matlab code available online.} We also find that the posterior mean of the innovation variance of our $GDP^{++}$ is much smaller than the innovation variances of $GDE$, $GDI$ and the benchmark model of Aruoba et al. (2016). The innovation variance of $GDP^{++}$ is also smaller than the innovation variance of the models estimated with news and noise measurement errors only, which in turn are higher than the innovation variance of $GDP^+$. The combination of a $\rho$ that is close to those implied by the various models estimated in Aruoba et al. (2016) and a $\sigma^2$ that is much smaller than the ones implied by Aruoba et al. (2016) leads to a higher forecastability of the $GDP^{++}$ measure.
To assess the relative importance of $GDI$ and $GDE$ at different releases, we use the Kalman gains. They represent the weight that the estimated value places on estimates of various releases. The outcomes are listed in Table (ref).
The results show that the weights assigned to different releases vary greatly as we change the assumed structure of the measurement error. When they are assumed to be pure News, the second panel of the table shows that 98% of the weight is put on the last release of $GDE$. Once we allow for the possibility of noise errors, however, more weight is assigned to $GDI$ and weights are spread over more releases. The earliest releases of $GDE$ receive less weight than the later releases, while the opposite is true for $GDI$. In all cases, we also find that $GDE$ releases are more important for explaining $GDP$ than $GDI$ releases, in contrast to Aruoba et al. (2016).
We have described a new approach to data reconciliation that exploits multiple data releases on each series. This helps both with the identification of measurement errors and with optimally extracting information from multiple noisy series. We used this to propose a new measure of U.S. $GDP$ growth using real-time data on $GDE$ and $GDI$. Our measure $GDP^{++}$ is shown to be more persistent than $GDE$ and $GDI$ and has smaller residual variance. In addition it has a similar autoregressive coefficient but smaller residual variance than the $GDP$ measure $GDP^+$ of Aruoba et al. (2016). Historical decompositions of $GDE$ and $GDI$ measurement errors reveal a larger news share in $GDE$ than in $GDI$.