^{a}

^{a}

In this paper, the authors propose a method to obtain explicit solutions for simultaneous benchmarking and reconciliation problems for a system of variables when the cross-restrictions use time-varying coefficients. The method is based on a hierarchical Bayesian model with a normal-gamma specification for the prior distributions. The proposed solution provides explicit (not sequential) feasible estimations, including measurements for its statistical accuracy. One interesting feature of the proposed procedure is that it allows users to include one or several performance indicators and to estimate disaggregated values for incomplete years. The method is applied to obtain Quarterly Regional Accounts for the Spanish economy.

In Applied Statistics, the term

Even though the problem was first posed over seventy years ago (

Mention should be made of two highly innovative contributions. First,

Second, the book by

Present-day interest in studying the main topics concerning benchmarking and reconciliation for time series is evidenced by two recent publications: Firstly, the new version of the Quarterly National Accounts (QNA) Manual (

In this paper, the authors propose a method applicable to problems simultaneously involving both reconciliation and temporal benchmarking. The technique is herein applied to a Laspeyres-type volume index by employing a method derived from a proposal by

The proposed method allows several indicators to be used, and does not require them to be approximations for the value to be estimated. It should also be pointed out that the stochastic nature of the model proposed by the authors enables the dispersion of the solution obtained to be estimated, thereby providing Bayesian confidence intervals, see

In the following section, a detailed description is given for the assumptions and development of the proposed methodology, obtaining the explicit expression of the solution. The third section applies the proposed solution to obtain Quarterly Regional Accounts (QRA) for Spanish regions, with both the Annual National Accounts (ANA) and QNA for the whole of Spain being known. The method may obviously be applied to any country’s QNA and ANA. This example uses the same data as in the

Let

We also assume the annual variables resulting from the disaggregation over the classification,

The high or sub-annual frequency (usually monthly or quarterly)^{1}

Other specifications, such as weekly or daily frequencies, are also possible.

aggregated series is also assumed to be known,where

The aim is to estimate the sub-annual series for each disaggregated area,

including, if any, the high-frequency values relative to the incomplete year,

We assume that the cross-aggregation links at the sub-annual level are the same as are used for the annual series; that is to say

considering, if appropriate, the same relation for the incomplete year.

Finally, we should establish the temporal aggregation scheme for both levels of cross-aggregation. The arithmetic mean has been taken, although other classical schemes (sum, first or last values) are developed in an analogous fashion. Specifically, we assume

and

All of the above classical schemes are consistent with the

Before building the relevant distributions, we should stack the relevant series in matrix form. Specifically, we denote by

In addition, we denote by

Before continuing, one clarification concerning the dimensions of the matrices involved should be made; we denote by

In sum, we have the annual and sub-annual series for the aggregated area, respectively,

As usual, Bayesian strategy initially states the prior distributions for the parameters and variables involved. These distributions include a certain number of non-random hyperparameters, whose values shall be stated by using allocation procedures. In addition, we then establish a behavioural linear model explaining the sub-annual variables as a function of one or more relevant indicators or approximated series. This linear model allows the likelihood function to be established and, consequently, the posterior distribution for the quarterly series and other relevant parameters to be derived.

Following

obeying restrictions (^{2}

Series differentiation is often used to eliminate the tendency of non-stationary temporal series. In this work, on the other hand, it is used to penalize volatile behaviour in the series to be estimated.

(with rankWe assume a gamma^{3}

We use the version for a gamma distribution from

In sum, we obtain a normal-gamma prior joint distribution for

where

The prior distributions include a large number of parameters and, generally, a limited amount of data. These parameters are

Some authors suggest the use of Jeffreys’ rule to assign ‘

Non-informative priors are accurate for problems in which we could take large-sized samples, allowing for a dominance of likelihood over priors. By contrast, for many Macroeconomic and Psychological studies, or for numerous analyses related with Natural Sciences, we have only a small amount of statistical information and, consequently, prior distributions dominate likelihood. The authors have chosen the option of using ‘

In order to establish the likelihood function, we assume a linear model relating the disaggregated sub-annual series with

The model has a similar expression if the last year is incomplete.

We now write a matricial version for the proposed likelihood, which will provide more compact reasoning. We denote

Denoting the perturbations for the models as

The likelihood is completely established by defining the prior for

Hence, the likelihood function is given by

being

We assume a normal prior distribution for

with

Using

under restrictions (

A well-known result (

with

Note, however, that the restricted distribution for

Specifically, our aim is to write

Denoting by

Note S1 in

Now, integrating density (

being

Posterior density (^{4}

A random variable

The posterior first and second order moments for

and

Posterior density for

with

and being

The posterior distribution (

We now present an example, which illustrates the above method. The resulting estimated variables are the quarterly regional chained volume series for Spanish regional Gross Domestic Product (GDP)^{5}

Temporal restrictions impose consistency among the regional estimations (quarterly regional chained volume series) and annual chained volume series provided by the annual regional accounts (ARA) from official Spanish regional statistics.

We also force the transversal consistency of these quarterly regional series with the Spanish national chained series provided by the QNA, a consistency which states that the QNA chained series is a weighted sum of the regional chained volume indices.

Many national statistics institutes have used annual chain-linking series for ANA and corresponding quarterly series for QNA. Specifically, the US Bureau of Economic Analysis (BEA) has used quarterly chain-linking volume series since 1996. In the European Union (EU), a European task force was set up in 2007, co-chaired by Eurostat and the European Central Bank. The growing popularity of chain-linking series for both QNA and ANA has led to the need for efficient tools in the reconciliation and benchmarking of quarterly chain-linking series.

More specifically, the Spanish National Statistics Institute (INE) provides the annual GDP (chain-linked volumes, reference year 2010^{6}

The INE also provides total GDP at market prices in euros.

) for the 17 autonomous regions and for the two autonomous cities (hereinafter, 19 regions) for the nineteen-year period 2000-2018. The INE also estimates total quarterly GDP (from QNA), both at market prices (in euros) and by the volume-chained series (the annual overlap method is used here). In both cases, the raw series as well as the seasonally and working-day adjusted series (SA) are presented. At the time of writing this paper, the quarterly series was composed of 76 quarters^{7}

INE, quarterly Spanish national accounts, series from quarter 1/1995 up to last published (4/2018). Data were extracted on May 2019, from

As mentioned before, the authors’ aim is to estimate the 19-quarterly regional chained series, all being consistent with the annual regional chained series and with the total quarterly one. We only estimate the SA regional series, with the estimation for raw series following a similar development. We apply the procedure obtained in the second section for the period from quarter 2000:1 to 2018:4.

Taking into account that the estimated quarterly regional series will be seasonally adjusted, the SSC series have first been seasonally adjusted using the X12 method, as implemented in Eviews (Version 6) software. The procedure proposed by the authors was implemented in MATLAB (Version R2012b).

Before the method can be applied, an extra adjustment is needed. This is due to the lack of consistency among the whole set of annual regional chained-series and the total Spanish one, resulting from the existence of ‘

To sum up, we have regional series concerning the annual GDP for the 19 regions for the period 2000-2018, both at current prices and in terms of volume. We have also estimated Spanish quarterly GDP (without extra-regio), both raw series and SA series, also at current prices and in chained-linking terms. Finally, we know the raw SSC series for each region, and have previously derived the corresponding SA series. As pointed out earlier, our objective is to estimate the Quarterly Regional chained volume series for Spanish regional GDP.

We then apply the method obtained in the second section, whose notations we now describe.

The INE provides regional annual GDP at market prices, in euros,

The ‘

Furthermore, the cross-restrictions are

at the annual level and

at the quarterly one, being

Readers may easily note that the above

with

Some representative tables and charts for the regional chain-linking SA series are shown in the

Table S1 in the ^{8}

Tables S2 to S4 in the

For its part, Figures S1 and S2 offer a graphic overview of said regional growth path.

Table S5 in the

The first and second column show the so-called ‘conformity ratio’, comparing the paired^{9}

For each region, we have paired its turning points with the closest Spanish ones.

turning points of each regional cyclical signal with the individual turning points. R_{x}thus compares such paired turning points as a percentage of the turning points for regional cycles, and R

_{y}compares them as a percentage of national ones. The conformity ratio varies between 0% and 100%, showing the extent to which the paired turning points reflect the overall cyclical signal of the region.

Readers may note that, with some exceptions, the agreement between regional and national cycles shown by indices Rx and Ry is relevant, with the exceptions corresponding to the Spanish regions of Ceuta and Castilla La Mancha.

The third column shows the global median delay (GMD) between regional and national cycles. Thus, the series are classified as coincident, lagged or leading with respect to the national cycle, respectively, for small (we take between -1 and +1), positive or negative GMD values. It should be noted that the Balearic Islands lead the national economy by two and a half quarters, and that Cantabria lags the national economy by two quarters.

The last column shows an index of cyclical coincidence^{10}

Defining

We also performed the estimation with incomplete years. Now

In this article, the authors propose a method to obtain explicit solutions for simultaneous benchmarking and reconciliation problems for a system of time series when the cross-restrictions use time-varying coefficients. The method provides explicit solutions to the estimation problem and deals with concurrently solving temporal restrictions (benchmarking annual and sub-annual frequency series) and contemporaneous ones (reconciliation among disaggregated and aggregated sub-annual frequency series).

The Bayesian model involved belongs to the frequently used normal-gamma family and minimizes a risk function derived from a quadratic loss function. In addition, the design of the method allows users to include one or several performance indicators through the likelihood model, and to estimate quarterly values for incomplete years.

The stochastic nature of the proposed model allows Bayesian confidence intervals to be obtained for each of the values of the high-frequency series estimated. These intervals are particularly interesting when estimating incomplete years.

Comparisons with alternative methods are not, broadly speaking, feasible since the methods have a different statistical base (mathematical methods compared to statistical methods) and are not nested models (none of them is a generalisation of the other). Nevertheless, certain differences may be pointed out.

Compared to the proposal put forward by

As regards the procedure of

Since it is an approximation,

An example related to Spanish quarterly accounts, combining national and regional volume series, is presented, and evidences the method’s feasibility and appropriateness. In addition, individual regional behaviour during the recent twin economic crisis is analysed using the estimated quarterly volume series.

As previously pointed out (and illustrated in Figure S3 and Table S6 in the

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no support to report.