Uncertainty and automatic balancing of national accounts with a Swedish application

1.Introduction

The uncertainty associated with the national accounts (NA), such as the gross domestic product (GDP), is of great interest for decision-makers, researchers, and the public. This information is nevertheless often absent in statistical releases. This is partly due to the complexity of the compilation of NA that uses a large number of data sources. It is difficult to estimate all uncertainties of the initial estimates. In a report [1], the data sources and possible error sources of NA compilation are well documented. The difficulties related to identifying and quantifying the errors, and in particular, the non-sampling errors are discussed. In paragraph 52 of report [1], it is stated, “given the current state of the art, it is not possible to calculate objective error margins for national accounts aggregates.”

Furthermore, NA must comply with the restrictions of accounting systems. There are three approaches for calculating GDP in the NA: the expenditure, the production, and the income approach. Usually, the estimates of these different approaches differ. It is therefore necessary to have a post-adjustment (“balance”) of those estimates. The balancing is usually done within the supply-use framework, i.e., the supply and use of different products CPA (Classification of Products by Activity). The balancing process is important for the compilation of NA. However, this process is not only highly demanding of expertise and time, but it is also to a large extent manual, which further complicates eventual attempts to investigate the uncertainties of the balanced NA aggregates.

In 1942, Richard Stone and others ([2], hereafter SCM) proposed a weighted least square (WLS) approach to balance economic accounts on a limited scale. Other authors, in particular [3, 4], formalized the approach and associated it to a Lagrange Multiplier approach with a quadratic loss function. A few applications [5, 6, 7, 8, 9, 10] have been reported. Among others [7, 8] used the SCM application to reconcile the US Industry Accounts and distribute the aggregate statistical discrepancy to industries in the Bureau of Economic Analysis (BEA). These articles show that the method is feasible and empirically efficient, although it is still difficult to obtain objective estimates to the uncertainties.

In our work, an SCM balancing approach is investigated within a supply-use (SU) framework in the Swedish NA; the income approach has not implemented completely in Sweden. The aim is to construct a framework workable in the real production environment of the NA compilation and balance. Efforts have been made to estimate the uncertainties from not only the sampling errors but also non-sampling errors. Those estimates are used then as weights in the balancing procedure. In Section 2, the framework of generalized least squares and a flexible equivalent optimization setting are described. The estimation of uncertainties of initial NA estimates is presented in Section 3. The Swedish application of this approach, along with the results, is given in Section 4. Discussion of the results, other related and future work are given in Section 5.

2.The framework

Following the description in [3, 8], write the estimates in NA as a vector 𝒙. For the sake of simplicity of the theoretical framework, write the accounting restrictions as 𝑨⁢𝒙=0, where the matrix 𝑨 consists of known constants. Inequality and other types of constraints can be imposed (see e.g. the discussion below Eq. (2) and the reference [9]). Assume that 𝒙* is an initial, unbiased estimate of 𝒙 with a variance-covariance matrix 𝑾, which does not satisfy the accounting restriction. For a balanced estimate 𝒙**, we use the quadratic loss function (𝒙**-𝒙*)′⁢𝑾-1⁢(𝒙**-𝒙*), and the solution by WLS approach is given by

Furthermore, under the condition that 𝑾 is the correct covariance of 𝒙* and that 𝑾 has full row rank, the variance-covariance of 𝒙** is given by

Note that the vector 𝒙 can be large, possibly consisting of thousands of elements. The required computing capacity required to handle this matrix size might be one reason for the limited use of the SCM approach in the past.

In practice, following [3, 7], a more workable formula under the SU framework is used in our study. Consider the following notation. For i=1,⋯,Ni; k=1,⋯⁢Nk;d=1,⋯,4, with Ni as the (known) number of industries and Nk the (known) number of product groups, denote

xi,k: the output (gross production) of product k from industry i.

zi,k: the intermediate consumption of product k for industry i.

xi,∙: the total output (over products) from industry i.

zi,∙: the total intermediate consumption by industry i.

yk,d: the d-th type of domestic final use of product k (the domestic final uses include Household final consumption expenditure, Government spending, Gross fixed capital formation, Changes in inventories).

ek: the exports of product k.

pk: the imports of product k.

y∙d,e∙,p∙: the totals (over products) of the domestic final uses, exports, and imports, respectively.

The corresponding initial estimates are indicated with a superscript 0 and w□0 are the corresponding uncertainties (which will be defined later) of the variable □0 (e.g. xi,k0 or ek0). The problem is therefore equivalent to minimizing

over xi,k, zi,k, xi,∙, zi,∙, yk,d, ek,pk, y∙d, e∙,p∙, under some constraints. In Eq. (2) w□0 is assumed to be positive for those variables to be adjusted. For variables with w□0=0, the initial estimates □0 will not be changed by convention. Note also that both individual variables by products (e.g. xi,k) and the totals over products (e.g. xi,∙) are included as the arguments to the objective Eq. (2). They will be adjusted separately since very often they are estimated using different data sources.

The constraints of Eq. (2) include that for each industry i, the total output from the industry shall be equal to the sums of outputs for all products from this industry,

Similar restrictions apply to the intermediate consumption, the domestic final uses, export and import, respectively:

The NA accounting requires further that the total supply is equal to the total use for each product k (where tk0 denotes the tax rates and mk0 the trade margins),

Note that compared with the theoretical framework Eqs (1) and (2), it is assumed implicitly that there is no covariance in 𝒙 in the system with Eqs (2)–(2). This assumption is of course arguable, but might be plausible. In the application of [7, 8], the same assumption applies. Whilst it is not easy, if not impossible, to satisfy all the conditions required for the framework with Eqs (1) and (2), the system with Eqs (2)–(2) is very flexible. The initial estimates can be expressed in both current and constant prices. Besides restrictions Eqs (4)–(2), it is possible to impose more and other types of restrictions. It is easy to keep some variables unchanged. For instance, the coefficient tk0 in Eq. (2) is included to account for taxes (such as customs, value-added taxes, minus subsidies). These variables are assumed to be proportional to the total output for each product. Due to the complex definition and compilations the trade margin (including the third-party trading), mk0 is included in Eq. (2) for the sake of the accounting, but will not be changed.

It is worth noting that Eqs (2)–(2) is a standard convex optimization problem and that a solution exists and is unique (see e.g. [11]).

Previous studies on the balancing of NA, in particular SU or Input-Output tables, such as [12] have not made use of the uncertainties of the initial estimates. In the BEA application of the US [7, 8], the SCM approach is carried out for the reconciliation and redistribution of the statistical discrepancy after a general revision of the NA estimates. In this application, the expenditure-based estimates are considered final and are not changed. Our settings aim nevertheless to create a framework workable in the real production environment of the NA compilation and balancing.

Efforts are also made in our study to estimate the uncertainties of the initial estimates 𝒙*, with respect to the sampling errors and the non-sampling errors as well; because nothing guarantees that sampling errors make up the main part of total errors (see [1]). Recall that a solution still exists for the system Eqs (2)–(2), even when one uses trivial weights in Eq. (2). Although such weights can by no means satisfy the conditions needed for Eqs (1) and (2), they are of interest as control groups in the application described in Section 4. In our study, the following four alternatives have been tested:

See Section 3 for a description of the last two alternatives.

3.The estimation of the uncertainties

4.A Swedish application

5.Discussion and final remarks

Estimation of the uncertainties and balancing the double entities of NA estimates are known difficult problems in national statistical institutes (NSI). The SCM approach that we generalized from [2] is investigated in our study. It is shown that an automatic balancing approach is possible for the compilation of NA in NSI. With the SCM approach, the balancing is not only more objective, but also fully replicable. Furthermore, the manual balancing procedure existing today requires much more resources and expertise. However, in order to implement the SCM approach in the official statistics production, more experiments have to be carried out and evaluated since there is no obvious evaluation criterion to compare with the manual procedure.

It is a big challenge to estimate the uncertainties in the initial NA estimates. There is little work available in the literature concerning the quantification of the non-sampling errors and how to combine them with the sampling errors. In our study, the sampling errors are used as the basis and a direct expert judgment is done for the non-sampling errors. This approach may be arguable. It is nevertheless a first step to tackle this difficult problem. Recently [16] proposed, instead to handle the variance-covariance matrix for individual input variables, to consider accounting equations as single entities and developed scalar uncertainties measures for those entities. They showed appealing theoretical properties of this approach. It would be of great interest to follow applications of this approach in the NA.

The balancing investigated in our study is only for one period. It is possible to use this approach for multiple-year balancing; see [17] for such an application, where the balanced result satisfies not only the accounting constraints, but also show movements that are as close as possible to the preliminary information. Moreover, in a broader context of balancing, not only accounting constraints, but also temporal constraints (i.e. the sum of quarterly accounts in a year equal to the annual ones) have to be satisfied in systems of time series. Readers who are interested in this area are referred to [18] for approaches for a reconciliation of both accounting and temporal constraints, and to the monograph [19] by Dagum and Cholette for general reconciliation methods.

Our application is done on the current prices. It is not trivial to extend the approach to the constant prices. The possibility to compute the uncertainty of the balanced aggregates, such as GDP, and the theoretical property Eq. (2) are of great interest. However, the conditions are very difficult to verify. All those are possible topics for our future work.