# Work on balanced accounts in the CSO: history and prospects.

A technique for producing balanced national accounts was described
by Stone, Champemowne and Meade in 1942, but there has so far been
little use of the method in official publications. This article
describes the experimental work undertaken recentlY by the Central
Statistical Office and offers a personal view of the potential for
further development. A variation of the original method of Stone et al,
called partitioned balancing, seems likely to be useful in complex
cases. Introduction

The problems caused by discrepancies between measures of GDP, sector balancing items and other failures of the national accounts to satisfy balancing conventions have been a recurrent feature of work in the CSO for many years. Efforts to deal with these problems have mainly taken the form of investigating possible biases in the data sources, so that the discrepancies could be reduced to an apparently random measurement error of tolerable magnitude. Although we have been aware for many years of procedures advocated by Stone and his followers which can produce a completely balanced set of accounts, the practical problems of implementing these methods have always been sufficient to deter us from trying that approach.

The publication of the 1988 Blue Book showed discrepancies between measures of GDP of a greater magnitude than usual; in the accompanying press notice the CSO undertook to carry out a study aimed at producing more nearly balanced accounts, with a deadline of six months to produce results. At the time of that commitment it was not clear what form of balancing would be used, or what purpose the balanced accounts would be expected to serve. The CSO's customers, including the Treasury, had expressed a wish to have the traditional three measures of GDP; the Pickford Review also recommended that the production of separate measures of GDP should continue. Presumably the intention was that the magnitude of the discrepancies should give some clue to the reliability of the figures. Pickford recommended that balanced accounts should be produced as a diagnostic tool', though without specifying what they might be used to diagnose.

Although there may have been an idea in some minds that the balanced accounts method could be extended to replace all the existing national accounts, which would imply producing a completely balanced version of the entire Blue Book, it was clear that that could not be attempted within the six month timescale laid down in September 1988. Whether that does seem a realistic prospect at any time in the immediate future will be discussed in the final section of this article, after reviewing our experience with the method so far and our proposals for further development of the method. The first balancing exercise The commitment undertaken in 1988 was fulfilled with the publication of an article in Economic Trends (CSO, 1989). Full details of the exercise are given in that article; only a summary version will be given here.

In the time available, it was clearly not possible to develop new theory from scratch. The only readily available method which we were aware of was the constrained least squares method first described by Stone and others (1 942). This method can be interpreted as being based on distributional assumptions about the measurement errors, or alternatively as a purely computational procedure with weights prescribed in a more or less arbitrary fashion. Taking the former approach, we assume that an account is represented by a number of quantities which are measured subject to error. The measurement errors are assumed to have zero means and to follow a multi-variate normal distribution with known covariance matrix. The true values in the account satisfy a number of linear relationships with known coefficients. With this formulation, the Stone method represents a maximum likelihood solution.

To apply this method to an account containing n quantities, we have to supply an n by n matrix of variances and covariances of measurement errors. We also have to ensure that the estimates in the account are free of any known biases. It was therefore clear that, without a massive data collection exercise, it would not be possible to tackle any very large set of accounts. Ideally, we might have wished to balance a large part of the Blue Book figures for several years simultaneously, to take account of our belief that we can measure year-to-year changes more precisely than absolute levels. A brief reflection on whether we could supply and handle the necessary parameters convinced us that something less ambitious would have to be tried. The objective finally chosen was to balance the fairly aggregated set of current price data shown in table A of the Blue Book for each of the years 1985, 1986, and 1987 considered separately. The number of entries in table A is approximately 250, so that we were required to construct a 250 by 250 matrix for each year.

The linear constraints imposed were firstly that the rows of table A should sum to zero where they already do so, and secondly that the columns (sectors) should also sum to zero. There was no explicit constraint that the income and expenditure measures of GDP should balance, since that is implied by the balancing of all the sectors; we learned by experience that it is undesirable to put in constraints which are redundant in this way, since this leads to an attempt to invert a singular matrix. The parameters for the covariance matrix were obtained by approaching those responsible for each component of table A and asking them to supply a 90 per cent confidence interval for the figures. They were also asked to indicate any other figures with which the measurement errors might be correlated. An early problem was that many contributors were unwilling to give a symmetrical confidence interval. They were not required to do so, the assumption being that the difference between the published value and the mid-point of the confidence interval could be regarded as a sort of bias adjustment. Taking the bias adjusted figures as a starting point, and assuming that half the width of the confidence interval could be treated as an estimate of the standard deviation of the measurement error, we obtained a set of figures consistent with the Stone model. Generally, contributors did not suggest any correlations between the measurement errors of different components of the account. In the few cases where they did so, the suggested correlations were small and seemed unlikely to have a major effect on the results. For this reason, we decided to carry out the entire exercise with all correlations set to zero. The computational methods allowed for non-zero correlations, and it would be relatively straightforward to include them when reliable estimates are available.

In addition to the constraints mentioned above, all but one of the variant models considered included a constraint which required the balanced figure for GDP to equal the average estimate of current price GDP as published in the Blue Book. This constraint was included partly to give some representation to the output measure, which otherwise would not affect the outcome, and also to avoid the possible confusion caused by the production of yet another estimate of GDP, even in the context of a research exercise such as this. One of the variant cases considered for the 1987 account removed the GDP constraint, with a fairly small effect on the balanced GDP. As already mentioned, the three years were balanced separately. There was therefore no constraint on the adjustments that might be made by the balancing process to year-to-year growth rates. No attempt had been made to specify confidence limits for any such growth rates but, as already mentioned, there was a view that they were known more reliably than the levels. As a check, the growth rates of a number of major aggregates were calculated from the balanced accounts and compared with those from the published accounts. The subjective judgement of those responsible for the article was that the amendments to growth rates produced by the balancing process were within plausible limits. This must mean, of course, that the adjustments to the different years had tended to be in the same direction. Perceived benefits and deficiencies of the first exercise The first clear benefit of the exercise was to demonstrate the feasibility of the basic approach. For each year, the balanced version of the initial table A figures was produced; despite the large initial discrepancies, few figures needed to be adjusted by more than the range of their 90 per cent confidence limits. It was even found possible to give a realistic representation of those cells in the table for which effectively nothing is known except that they have the right value to enable the table to balance-that is to say the residual items. For the many categories of financial transaction for which personal sector transactions are derived as a residual, balanced figures were obtained with plausible error ranges.

As well as producing a balanced set of accounts, the Stone method also gives a covariance matrix for the balanced figures. Thus, if it were thought desirable, it would be possible to publish a balanced account with a confidence interval attached to every figure. As the theory requires, the confidence intervals for the balanced figures are always narrower or equal in width to the confidence intervals for the original figures. Generally speaking, the greatest reduction in interval width was obtained for those figures which were initially the least well determined. Although it might be considered a rather negative benefit, it was found that the directions of the larger changes were usually in accordance with prior expectations.

The most significant limitation of the exercise was its restricted scope. The balancing model made no reference to constant price data or in any direct way to output measure data. There was also no representation of certain known relationships such as those between capital acquisitions and changes in interest flows. Neither was there any attempt to represent inter-year constraints.

In addition, there was some concern about the reliability of the confidence limits. It was not certain that all the contributors, working independently, had used the same approach in assessing the limits. The factors taken account of, and the method of using them, was necessarily subjective. The exact status of the 'bias adjustments' and the reasons which had led to their inclusion were also not necessarily consistent.

As regards the working of the balancing process, it was noticed that the largest adjustments seem to fall disproportionately on the items in the account with the largest error ranges. it can be shown that in the simplest case, with a single discrepancy to be allocated over uncorrelated measurement errors, the Stone model leads to adjustments which are proportional to the squares of the standard errors. In the model of table A, the effect in each year was that the largest changes were made to company profits and acquisitions of certain capital assets. Some people also thought it implausible that, in balancing a sector with excess income, the adjustment to every income item should be negative.

Some commentators, particularly external ones, found the constraint to published GDP undesirable. Although the motives were understood, it was felt that the inability to see the direction in which the balancing process would try to move GDP and the absence of a meaningful error margin on balanced GDP were unwarrantable losses. Enhancements and modifications of the model Further consideration of the deficiencies identified in the first exercise,. together with further reading in the literature,. led to a number of proposals to modify the model: a. The omission of constant price data and output

information from the model for the first exercise

was not simply due to concern about the size of

the model-, we were also unsure how to handle

constraints which are non-linear. A paper by

Weale (1988) showed an effective way of coping

with this problem. We did not decide to use the

whole of his method, in particular his way of

deriving error margins from incomplete data, but

we found his basic idea of representing multiplicative

constraints to linearising logarithmic versions

of the variables very convincing and

effective. b. It is clear that adjustments to the acquisition of

certain forms of capital should lead also to

changes in the flow of interest produced by that

capital in the year of acquisition and in

succeeding years. If it is assumed that the rate of return

on capital is known, it is possible to represent this

relationship through a series of additional constraints

which operate over more than one year.

An approach which could handle such constraints

was worked out. c. It was clear that some of the items in table A

should have correlations between their

measurement errors, not because of correlations

in the original measurement processes but

because the aggregated items in table A could

include common components. In several areas it

was possible to break down the aggregates into

finer detail representing original and presumably

independent data sources which then entered

into various linear relationships with the aggregates

in table A. Correlations between the table

A items could be deduced from these

relationships.

A consequence of these modifications was necessarily a large increase in the size of the model which needed to be balanced. A simple minded approach to handling this model would have led to unacceptably large computational requirements. Fortunately, it was realised that an unpublished result of Durbin and Fisher gave an indication of how to break down a large model into manageable blocks. Building on this result, we have developed a technique which we describe as partitioned balancing' which enables us to handle very large models without excessive computational loads, although at the expense of some complexity in the structure of the model. (See Appendix A for a more detailed account.)

Although not directly related to the structure of the model, another area which we thought worth trying to improve was the method of setting confidence limits. In particular, we sought ways of making the processs more consistent and objective. Where data are derived from sample surveys, it may be possible to estimate the sampling error by conventional means. However, there will usually be non-sampling errors which may be of a larger magnitude than the sampling ones. Some limited thought was given to ways of obtaining objective, or at least consistent, quantifications of non-sampling errors. Until such a method has been found, the only approach which occurs to us is to give responsibility for assessing the confidence limits to a single person or small group, who obtain their estimates after consultation with the data providers. The second balancing exercise It was decided that an updated version of the first exercise would be produced in the autumn of 1989, following publication of the Blue Book data. This exercise would move on the balancing period to cover the years 1986 to 1988, and would incorporate as many as possible of the enhancements considered above.

In the event, limitations of time and resources prevented us from going as far as we had wished. We had contemplated using the Weale method to introduce constant price data and output information, but there was not enough time to assemble the necessary parameters. A similar limitation prevented us introducing a finer disaggregation of the financial part of table A, although the necessary extensions of the model were produced. The only change made to the model in this exercise was to introduce a relationship between capital acquisitions and interest flows for the overseas sector, along the lines indicated above. This change took account of the cumulative effect of capital acquisitions, and so involved a link between the three years, using the technique of partitioned balancing.

The only other change from the first exercise was to provide updated confidence limits for all components of the account. The method of updating and the information used were left to the discretion of the contributors. In the event, it seemed that there was a tendency to assign confidence limits which were somewhat narrower than those used in the first exercise. in addition, some of the bias adjustments showed strange changes from year-to-year, possibly from an attempt to guess the actual errors that had occurred in particular years.

The results of this exercise showed a similar pattern of changes to those in the first exercise. Because of the generally narrower confidence limits.. it was found that rather more of the adjustments moved the balanced figures outside the initially prescribed limits. It was also found that the initially assigned error margins for the capital and interest items which were linked by the additional constraints were not fully consistent; the balanced figures showed implausibly tight limits on the interest items. In view of the limited changes produced in this exercise, the results were not published. A sample of the results appears in table 1. Retrospect and prospect Two balanced accounts exercises have now been completed, it must be admitted with little effect on the mainstream work of the CSO. No systematic diagnostic use has been made of the results, and it is not clear how they could be used for such a purpose. One facet of the first exercise which was explored was to compare the adjustments made to some major aggregates with the national accounts statistical adjustments (NASAS) for the same aggregates. The results were broadly similar in magnitude, but there has been no enthusiasm to substitute balanced accounts adjustments for NASAS. When we consider why the balanced accounts work has had so little effect, it must be admitted that a major constraint has been the scale of resources needed to develop the balancing models to a realistic scale. Such resources are not easily available, especially in view of the many other developments of the national accounts which are under way. But, more seriously, there is undoubtedly a credibility gap about the use of balanced accounts within the CSO. Enthusiasts argue that balancing methodology could replace NASAS, quarterly path adjustments and all the other devices which are now being used to produce more coherent national accounts. The practitioners in this field are not convinced; they generally prefer the use of statistical judgement to the 'mechanical 'processes of balancing. It is perhaps appropriate here to explore a little further the concept of statistical judgement in the national accounts field. It is clear that, in the present stage of development, it is not possible to have a purely mechanical and objective process to generate the national accounts" some professional judgement will always be necessary. However, we can validly ask what sorts of question are amenable to the exercise of statistical judgement. We can expect the national accounts statistician to have an understanding of the properties of the measurement process which generates his estimates-the mean error (that is, bias), the standard deviation or range of errors, together with any systematic relationship with errors in other measurements. But if a statistician offers a judgement on the exact amount of random error present in a particular observation, one is entitled to ask whether this is more guesswork than judgement.

To try to get a realistic view of what balancing methodology could achieve to meet the needs of users, we first have to define more closely what those needs are. Do the users really want three discrepant measures of GDP? Surely the requirement is for a single measure of GDP with breakdowns into income, expenditure and output components-and preferably with some consistent and intelligible statement of error margins for each component. A suitably developed balanced accounts method is capable of providing this; it could be argued that no other method can do so as effectively, if at all.

Looking at the practicalities, one can question whether the human resources will ever be available to develop a full balancing model, and also whether the computing resources will be available if we have to re-balance the whole of the Blue Book, in quarterly detail, every time a figure changes.

On the first point, we can progressively refine the model and the error estimates. It will be necessary to start at a fairly aggregated level and with fairly crude estimates of error margins, as in the exercises already carried out, and gradually introduce more detail and re-examine each area to produce better estimates of error margins. There will of course be changes in data sources as we improve our surveys, so there will be a regular need to revise the error estimates. Indeed, we could argue that it will be helpful to have the balanced accounts method available to give a direct indication of the effect of improving a data source on the error margins in GDP and the other major aggregates. Such a tool would be very useful in the arguments about costs and benefits and burdens on business which are a feature of any change to the statistical system. On the problems of computation, the technique of partitioned balancing could provide a solution. It is possible to have a hierarchy of balancing models of differing levels of detail, the errors in the aggregated models derived from the more detailed ones. In such a system, it is possible to re-balance only at the required level of detail, while still being sure that the results will be consistent with what could be achieved if the more detailed analysis were carried out. To clarify the significance of the partitioned balancing approach, it may be helpful to give an example. As already mentioned, the first balancing exercise used Blue Book Table A, with about 250 entries. It would be possible to reduce the entries above the line in each column to just three, labelled income, net transfers and expenditure, and the entries below the line to just one, labelled financing. The constraints would be that the net transfers row and the financing row each sum to zero, and ill each column income plus net transfers minus expenditure equals financing. The totals of the income and expenditure rows are of course the two measures of GDP.

Thus, the model would be reduced to just 32 items rather than 250. However, if the covariance matrix of these 32 items were obtained from that for the 250 items by the partitioned balancing methods, the results of balancing the 32 items would be exactly the same as would be obtained by balancing the 250 items and then aggregating the balanced results. Furthermore, the part of the partitioned balancing which we have called 'ripple-back' would enable us to allocate the balancing changes in the income, expenditure and financing rows to each of their component rows, again with exactly the same results as if we had balanced in full detail.

The implications of this for the quarterly and annual national accounts rounds are clear in general terms, though the details remain to be worked out. A balancing model would be developed for all the quarterly and annual components, including current and constant price data and all the inter-year constraints. By partitioned balancing this would be reduced to the level of detail used in the quarterly rounds, using perhaps 12 consecutive quarters and about 20 aggregated items in each quarter. The constraints used would represent in essence the relationships which now underlie the quarterly path adjustments, but without the undesirable feature of forcing all the adjustment on to one or two items. The balanced version of this aggregated model would give all the headline figures for the quarterly accounts, and ripple-back would enable us to give as much detail of the changes as we thought necessary.

At present the CSO has no firm commitments for future work on balanced accounts. The debate on the place of this method in the national accounts continues. The views expressed here are evidently those of a committed enthusiast; others in the Government Statistical Service have different views. Other interested parties are free to join the debate.

REFERENCES Byron, R.P., (1978), 'The estimation of large social account matrices', Journal of the Royal Statistical Society, Series A, vol.

141, pp. 359-67. CSO, (1989), 'An investigation with balancing the UK national and financial accounts, 1985-7', Economic Trends, No. 424,

February. pp. 74-103. Durbin, J. and Fisher, H.R., (undated), 'The adjustment of observations with application to national income statistics', mimeo. Stone, J.R.N., Champernowne, D.G., and Meade, J.E., (1942). 'The precision of national income accounting estimates',

Review of Economic Studies, vol. 9, pp. 111-25. Weale, M.R. (1988), 'The reconciliation of values, volumes and prices in the national accounts', Journal of the Royal

Statistical Society, Series A, vol. 1 51, pp. 211-21. Appendix A. Partitioned balancing The method called partitioned balancing depends on two results, both of which involve a form of partitioning. In the first we partition the constraints, in the second we partition the variables. 1. It is possible to partition the constraints into two groups, in an arbitrary way, and balance for each group separately,

provided the covariance matrix is modified appropriately. Specifically, the requirement is that the initial covariance matrix

for the variables in the second stage balancing shall be the final (balanced) covariance matrix produced by the first stage

balancing. With this condition the two stage balancing produces the same results as balancing for all constraints

simultaneously. Evidently the result can be extended to cover balancing in any number of stages. This is the result due to

Durbin. 2. It may sometimes happen that the set of variables explicitly mentioned in the constraints is smaller than the total set of

variables. In this case. the balanced values of the explicitly mentioned variables can be obtained by means of a balancing

run which involves only those variables. Balancing adjustments to the remaining variables will arise by virtue of their

correlations with the explicitly mentioned variables, and it required can be calculated in a second stage by combining the

first stage adjustments and the appropriate portion of the covariance matrix. The second stage is what we have called

'ripple back', from the metaphor of the changes required to balance the explicitly mentioned variables then rippling

across the remaining variables. It is clear that any of the remaining variables which are uncorrelated with all the explicitly

mentioned variables will be unchanged in the ripple back.

These two results are stated separately, but the greatest benefits are obtained by employing them together, usually by partitioning the constraints in such a way that, in the first or second stage balancing, only a subset of variables will be involved and hence the second result is applicable. Some examples of the application of these results will make the idea clearer. It is useful to note a third, fairly trivial, result. 3. There will sometimes be constraints which are already satisfied by the input data. Such constraints may express the

requirement for transactions to balance across sectors when one sector is calculated as a residual. Alternatively, they

may be used to define additional variables which are the sums of groups of original variables. These constraints have to

be included to ensure that they are not violated while balancing for other unsatisfied constraints. Balancing for such

constraints alone will have no effect on the values of the variables, though in general the covariance matrix will be

changed.

The first simple example of using partitioned balancing is in the case. which is quite common below the line in Table A of the Blue Book, where some class of financial transactions of one sector (usually the personal sector) are obtained as a residual. Following Byron (1978), the appropriate treatment is to give the residual sector an error range which is so large as to be effectively infinite; this expresses in an intuitively satisfying way the fact that we know nothing about the residual sector except that it satisfies the row constraint. If we balance for this row constraint separately. our explicit balancing need involve only the variables in the row. The variables themselves will be unchanged, as will the covariance matrix elements for all but the residual item, for which there will be the appropriate negative correlation with each other variable in the row. Provided there is no correlation between the non-residual items in this row and any items elsewhere in the account we will not need to carry out any ripple back calculation. Hence we may proceed to balance all such row constraints one row at a time. replacing each constraint by an appropriately modified block of the covariance matrix

A second, much more complex, example is the simultaneous balancing of three years with inter-year constraints due to the relationship between capital and income. This involves dividing the constraints into four groups and carrying out four separate balancing exercises on subsets of variables. The first three groups of constraints are those which apply within year 1, year 2 and year 3 respectively; the fourth group is the inter-year constraints. Each of the first three groups is balanced, involving only the variables from the year concerned: because our assumption is that there is no correlation in measurement errors in different years there is no ripple back to other years. The final balance involves explicitly only the capital and income variables included in the inter-year constraints. Because these variables are correlated with other variables from the same year-remembering that the covariance matrix is that resulting from the first three balancing operations-it is necessary to calculate a ripple back effect. However, even this is not as complex as it might be because the ripple back operates only within the separate years. (Note that the mathematically equivalent operation of balancing the inter-year constraints first would be computationally much harder. because of the resulting inter-year correlations. It is necessary to choose the order of operations sensibly to benefit from partitioned balancing.)

The final example discussed here is the reduction of the full Table A to four rows representing income, transfers. expenditure and financing. The first step here, perversely, is to enlarge the problem by adding extra variables to represent these totals and extra constraints to relate each total to its components. These extra constraints will be automatically satisfied by the input data. Balancing is then divided into two stages. the first of which is to balance for all the initially satisfied constraints, both the extra constraints for the new total variables and the row constraints of Table A. Since this first balancing does not affect the variables it can be carried out before their values are known, provided we know the covariance matrix. The second stage is to balance for all the initially unsatisfied constraints. which in this case will be the sector balancing constraints; these constraints will have been re-expressed in terms of the new total variables, so that only those variables need be explicitly balanced. If we wish to apply inter-period constraints such as the alignment adjustments we can proceed analogously to the second example we first balance within each period using the total variables, then apply the inter-period constraints and finally ripple back to all the original variables. In this case however ripple back needs to be applied in two stages and can become rather complex. it may be preferable to do a full simultaneous balance of all the total variables for all periods to apply the inter-period constraints. and then ripple back within each period to recover the detailed variables. TABULAR DATA OMITTED

The problems caused by discrepancies between measures of GDP, sector balancing items and other failures of the national accounts to satisfy balancing conventions have been a recurrent feature of work in the CSO for many years. Efforts to deal with these problems have mainly taken the form of investigating possible biases in the data sources, so that the discrepancies could be reduced to an apparently random measurement error of tolerable magnitude. Although we have been aware for many years of procedures advocated by Stone and his followers which can produce a completely balanced set of accounts, the practical problems of implementing these methods have always been sufficient to deter us from trying that approach.

The publication of the 1988 Blue Book showed discrepancies between measures of GDP of a greater magnitude than usual; in the accompanying press notice the CSO undertook to carry out a study aimed at producing more nearly balanced accounts, with a deadline of six months to produce results. At the time of that commitment it was not clear what form of balancing would be used, or what purpose the balanced accounts would be expected to serve. The CSO's customers, including the Treasury, had expressed a wish to have the traditional three measures of GDP; the Pickford Review also recommended that the production of separate measures of GDP should continue. Presumably the intention was that the magnitude of the discrepancies should give some clue to the reliability of the figures. Pickford recommended that balanced accounts should be produced as a diagnostic tool', though without specifying what they might be used to diagnose.

Although there may have been an idea in some minds that the balanced accounts method could be extended to replace all the existing national accounts, which would imply producing a completely balanced version of the entire Blue Book, it was clear that that could not be attempted within the six month timescale laid down in September 1988. Whether that does seem a realistic prospect at any time in the immediate future will be discussed in the final section of this article, after reviewing our experience with the method so far and our proposals for further development of the method. The first balancing exercise The commitment undertaken in 1988 was fulfilled with the publication of an article in Economic Trends (CSO, 1989). Full details of the exercise are given in that article; only a summary version will be given here.

In the time available, it was clearly not possible to develop new theory from scratch. The only readily available method which we were aware of was the constrained least squares method first described by Stone and others (1 942). This method can be interpreted as being based on distributional assumptions about the measurement errors, or alternatively as a purely computational procedure with weights prescribed in a more or less arbitrary fashion. Taking the former approach, we assume that an account is represented by a number of quantities which are measured subject to error. The measurement errors are assumed to have zero means and to follow a multi-variate normal distribution with known covariance matrix. The true values in the account satisfy a number of linear relationships with known coefficients. With this formulation, the Stone method represents a maximum likelihood solution.

To apply this method to an account containing n quantities, we have to supply an n by n matrix of variances and covariances of measurement errors. We also have to ensure that the estimates in the account are free of any known biases. It was therefore clear that, without a massive data collection exercise, it would not be possible to tackle any very large set of accounts. Ideally, we might have wished to balance a large part of the Blue Book figures for several years simultaneously, to take account of our belief that we can measure year-to-year changes more precisely than absolute levels. A brief reflection on whether we could supply and handle the necessary parameters convinced us that something less ambitious would have to be tried. The objective finally chosen was to balance the fairly aggregated set of current price data shown in table A of the Blue Book for each of the years 1985, 1986, and 1987 considered separately. The number of entries in table A is approximately 250, so that we were required to construct a 250 by 250 matrix for each year.

The linear constraints imposed were firstly that the rows of table A should sum to zero where they already do so, and secondly that the columns (sectors) should also sum to zero. There was no explicit constraint that the income and expenditure measures of GDP should balance, since that is implied by the balancing of all the sectors; we learned by experience that it is undesirable to put in constraints which are redundant in this way, since this leads to an attempt to invert a singular matrix. The parameters for the covariance matrix were obtained by approaching those responsible for each component of table A and asking them to supply a 90 per cent confidence interval for the figures. They were also asked to indicate any other figures with which the measurement errors might be correlated. An early problem was that many contributors were unwilling to give a symmetrical confidence interval. They were not required to do so, the assumption being that the difference between the published value and the mid-point of the confidence interval could be regarded as a sort of bias adjustment. Taking the bias adjusted figures as a starting point, and assuming that half the width of the confidence interval could be treated as an estimate of the standard deviation of the measurement error, we obtained a set of figures consistent with the Stone model. Generally, contributors did not suggest any correlations between the measurement errors of different components of the account. In the few cases where they did so, the suggested correlations were small and seemed unlikely to have a major effect on the results. For this reason, we decided to carry out the entire exercise with all correlations set to zero. The computational methods allowed for non-zero correlations, and it would be relatively straightforward to include them when reliable estimates are available.

In addition to the constraints mentioned above, all but one of the variant models considered included a constraint which required the balanced figure for GDP to equal the average estimate of current price GDP as published in the Blue Book. This constraint was included partly to give some representation to the output measure, which otherwise would not affect the outcome, and also to avoid the possible confusion caused by the production of yet another estimate of GDP, even in the context of a research exercise such as this. One of the variant cases considered for the 1987 account removed the GDP constraint, with a fairly small effect on the balanced GDP. As already mentioned, the three years were balanced separately. There was therefore no constraint on the adjustments that might be made by the balancing process to year-to-year growth rates. No attempt had been made to specify confidence limits for any such growth rates but, as already mentioned, there was a view that they were known more reliably than the levels. As a check, the growth rates of a number of major aggregates were calculated from the balanced accounts and compared with those from the published accounts. The subjective judgement of those responsible for the article was that the amendments to growth rates produced by the balancing process were within plausible limits. This must mean, of course, that the adjustments to the different years had tended to be in the same direction. Perceived benefits and deficiencies of the first exercise The first clear benefit of the exercise was to demonstrate the feasibility of the basic approach. For each year, the balanced version of the initial table A figures was produced; despite the large initial discrepancies, few figures needed to be adjusted by more than the range of their 90 per cent confidence limits. It was even found possible to give a realistic representation of those cells in the table for which effectively nothing is known except that they have the right value to enable the table to balance-that is to say the residual items. For the many categories of financial transaction for which personal sector transactions are derived as a residual, balanced figures were obtained with plausible error ranges.

As well as producing a balanced set of accounts, the Stone method also gives a covariance matrix for the balanced figures. Thus, if it were thought desirable, it would be possible to publish a balanced account with a confidence interval attached to every figure. As the theory requires, the confidence intervals for the balanced figures are always narrower or equal in width to the confidence intervals for the original figures. Generally speaking, the greatest reduction in interval width was obtained for those figures which were initially the least well determined. Although it might be considered a rather negative benefit, it was found that the directions of the larger changes were usually in accordance with prior expectations.

The most significant limitation of the exercise was its restricted scope. The balancing model made no reference to constant price data or in any direct way to output measure data. There was also no representation of certain known relationships such as those between capital acquisitions and changes in interest flows. Neither was there any attempt to represent inter-year constraints.

In addition, there was some concern about the reliability of the confidence limits. It was not certain that all the contributors, working independently, had used the same approach in assessing the limits. The factors taken account of, and the method of using them, was necessarily subjective. The exact status of the 'bias adjustments' and the reasons which had led to their inclusion were also not necessarily consistent.

As regards the working of the balancing process, it was noticed that the largest adjustments seem to fall disproportionately on the items in the account with the largest error ranges. it can be shown that in the simplest case, with a single discrepancy to be allocated over uncorrelated measurement errors, the Stone model leads to adjustments which are proportional to the squares of the standard errors. In the model of table A, the effect in each year was that the largest changes were made to company profits and acquisitions of certain capital assets. Some people also thought it implausible that, in balancing a sector with excess income, the adjustment to every income item should be negative.

Some commentators, particularly external ones, found the constraint to published GDP undesirable. Although the motives were understood, it was felt that the inability to see the direction in which the balancing process would try to move GDP and the absence of a meaningful error margin on balanced GDP were unwarrantable losses. Enhancements and modifications of the model Further consideration of the deficiencies identified in the first exercise,. together with further reading in the literature,. led to a number of proposals to modify the model: a. The omission of constant price data and output

information from the model for the first exercise

was not simply due to concern about the size of

the model-, we were also unsure how to handle

constraints which are non-linear. A paper by

Weale (1988) showed an effective way of coping

with this problem. We did not decide to use the

whole of his method, in particular his way of

deriving error margins from incomplete data, but

we found his basic idea of representing multiplicative

constraints to linearising logarithmic versions

of the variables very convincing and

effective. b. It is clear that adjustments to the acquisition of

certain forms of capital should lead also to

changes in the flow of interest produced by that

capital in the year of acquisition and in

succeeding years. If it is assumed that the rate of return

on capital is known, it is possible to represent this

relationship through a series of additional constraints

which operate over more than one year.

An approach which could handle such constraints

was worked out. c. It was clear that some of the items in table A

should have correlations between their

measurement errors, not because of correlations

in the original measurement processes but

because the aggregated items in table A could

include common components. In several areas it

was possible to break down the aggregates into

finer detail representing original and presumably

independent data sources which then entered

into various linear relationships with the aggregates

in table A. Correlations between the table

A items could be deduced from these

relationships.

A consequence of these modifications was necessarily a large increase in the size of the model which needed to be balanced. A simple minded approach to handling this model would have led to unacceptably large computational requirements. Fortunately, it was realised that an unpublished result of Durbin and Fisher gave an indication of how to break down a large model into manageable blocks. Building on this result, we have developed a technique which we describe as partitioned balancing' which enables us to handle very large models without excessive computational loads, although at the expense of some complexity in the structure of the model. (See Appendix A for a more detailed account.)

Although not directly related to the structure of the model, another area which we thought worth trying to improve was the method of setting confidence limits. In particular, we sought ways of making the processs more consistent and objective. Where data are derived from sample surveys, it may be possible to estimate the sampling error by conventional means. However, there will usually be non-sampling errors which may be of a larger magnitude than the sampling ones. Some limited thought was given to ways of obtaining objective, or at least consistent, quantifications of non-sampling errors. Until such a method has been found, the only approach which occurs to us is to give responsibility for assessing the confidence limits to a single person or small group, who obtain their estimates after consultation with the data providers. The second balancing exercise It was decided that an updated version of the first exercise would be produced in the autumn of 1989, following publication of the Blue Book data. This exercise would move on the balancing period to cover the years 1986 to 1988, and would incorporate as many as possible of the enhancements considered above.

In the event, limitations of time and resources prevented us from going as far as we had wished. We had contemplated using the Weale method to introduce constant price data and output information, but there was not enough time to assemble the necessary parameters. A similar limitation prevented us introducing a finer disaggregation of the financial part of table A, although the necessary extensions of the model were produced. The only change made to the model in this exercise was to introduce a relationship between capital acquisitions and interest flows for the overseas sector, along the lines indicated above. This change took account of the cumulative effect of capital acquisitions, and so involved a link between the three years, using the technique of partitioned balancing.

The only other change from the first exercise was to provide updated confidence limits for all components of the account. The method of updating and the information used were left to the discretion of the contributors. In the event, it seemed that there was a tendency to assign confidence limits which were somewhat narrower than those used in the first exercise. in addition, some of the bias adjustments showed strange changes from year-to-year, possibly from an attempt to guess the actual errors that had occurred in particular years.

The results of this exercise showed a similar pattern of changes to those in the first exercise. Because of the generally narrower confidence limits.. it was found that rather more of the adjustments moved the balanced figures outside the initially prescribed limits. It was also found that the initially assigned error margins for the capital and interest items which were linked by the additional constraints were not fully consistent; the balanced figures showed implausibly tight limits on the interest items. In view of the limited changes produced in this exercise, the results were not published. A sample of the results appears in table 1. Retrospect and prospect Two balanced accounts exercises have now been completed, it must be admitted with little effect on the mainstream work of the CSO. No systematic diagnostic use has been made of the results, and it is not clear how they could be used for such a purpose. One facet of the first exercise which was explored was to compare the adjustments made to some major aggregates with the national accounts statistical adjustments (NASAS) for the same aggregates. The results were broadly similar in magnitude, but there has been no enthusiasm to substitute balanced accounts adjustments for NASAS. When we consider why the balanced accounts work has had so little effect, it must be admitted that a major constraint has been the scale of resources needed to develop the balancing models to a realistic scale. Such resources are not easily available, especially in view of the many other developments of the national accounts which are under way. But, more seriously, there is undoubtedly a credibility gap about the use of balanced accounts within the CSO. Enthusiasts argue that balancing methodology could replace NASAS, quarterly path adjustments and all the other devices which are now being used to produce more coherent national accounts. The practitioners in this field are not convinced; they generally prefer the use of statistical judgement to the 'mechanical 'processes of balancing. It is perhaps appropriate here to explore a little further the concept of statistical judgement in the national accounts field. It is clear that, in the present stage of development, it is not possible to have a purely mechanical and objective process to generate the national accounts" some professional judgement will always be necessary. However, we can validly ask what sorts of question are amenable to the exercise of statistical judgement. We can expect the national accounts statistician to have an understanding of the properties of the measurement process which generates his estimates-the mean error (that is, bias), the standard deviation or range of errors, together with any systematic relationship with errors in other measurements. But if a statistician offers a judgement on the exact amount of random error present in a particular observation, one is entitled to ask whether this is more guesswork than judgement.

To try to get a realistic view of what balancing methodology could achieve to meet the needs of users, we first have to define more closely what those needs are. Do the users really want three discrepant measures of GDP? Surely the requirement is for a single measure of GDP with breakdowns into income, expenditure and output components-and preferably with some consistent and intelligible statement of error margins for each component. A suitably developed balanced accounts method is capable of providing this; it could be argued that no other method can do so as effectively, if at all.

Looking at the practicalities, one can question whether the human resources will ever be available to develop a full balancing model, and also whether the computing resources will be available if we have to re-balance the whole of the Blue Book, in quarterly detail, every time a figure changes.

On the first point, we can progressively refine the model and the error estimates. It will be necessary to start at a fairly aggregated level and with fairly crude estimates of error margins, as in the exercises already carried out, and gradually introduce more detail and re-examine each area to produce better estimates of error margins. There will of course be changes in data sources as we improve our surveys, so there will be a regular need to revise the error estimates. Indeed, we could argue that it will be helpful to have the balanced accounts method available to give a direct indication of the effect of improving a data source on the error margins in GDP and the other major aggregates. Such a tool would be very useful in the arguments about costs and benefits and burdens on business which are a feature of any change to the statistical system. On the problems of computation, the technique of partitioned balancing could provide a solution. It is possible to have a hierarchy of balancing models of differing levels of detail, the errors in the aggregated models derived from the more detailed ones. In such a system, it is possible to re-balance only at the required level of detail, while still being sure that the results will be consistent with what could be achieved if the more detailed analysis were carried out. To clarify the significance of the partitioned balancing approach, it may be helpful to give an example. As already mentioned, the first balancing exercise used Blue Book Table A, with about 250 entries. It would be possible to reduce the entries above the line in each column to just three, labelled income, net transfers and expenditure, and the entries below the line to just one, labelled financing. The constraints would be that the net transfers row and the financing row each sum to zero, and ill each column income plus net transfers minus expenditure equals financing. The totals of the income and expenditure rows are of course the two measures of GDP.

Thus, the model would be reduced to just 32 items rather than 250. However, if the covariance matrix of these 32 items were obtained from that for the 250 items by the partitioned balancing methods, the results of balancing the 32 items would be exactly the same as would be obtained by balancing the 250 items and then aggregating the balanced results. Furthermore, the part of the partitioned balancing which we have called 'ripple-back' would enable us to allocate the balancing changes in the income, expenditure and financing rows to each of their component rows, again with exactly the same results as if we had balanced in full detail.

The implications of this for the quarterly and annual national accounts rounds are clear in general terms, though the details remain to be worked out. A balancing model would be developed for all the quarterly and annual components, including current and constant price data and all the inter-year constraints. By partitioned balancing this would be reduced to the level of detail used in the quarterly rounds, using perhaps 12 consecutive quarters and about 20 aggregated items in each quarter. The constraints used would represent in essence the relationships which now underlie the quarterly path adjustments, but without the undesirable feature of forcing all the adjustment on to one or two items. The balanced version of this aggregated model would give all the headline figures for the quarterly accounts, and ripple-back would enable us to give as much detail of the changes as we thought necessary.

At present the CSO has no firm commitments for future work on balanced accounts. The debate on the place of this method in the national accounts continues. The views expressed here are evidently those of a committed enthusiast; others in the Government Statistical Service have different views. Other interested parties are free to join the debate.

REFERENCES Byron, R.P., (1978), 'The estimation of large social account matrices', Journal of the Royal Statistical Society, Series A, vol.

141, pp. 359-67. CSO, (1989), 'An investigation with balancing the UK national and financial accounts, 1985-7', Economic Trends, No. 424,

February. pp. 74-103. Durbin, J. and Fisher, H.R., (undated), 'The adjustment of observations with application to national income statistics', mimeo. Stone, J.R.N., Champernowne, D.G., and Meade, J.E., (1942). 'The precision of national income accounting estimates',

Review of Economic Studies, vol. 9, pp. 111-25. Weale, M.R. (1988), 'The reconciliation of values, volumes and prices in the national accounts', Journal of the Royal

Statistical Society, Series A, vol. 1 51, pp. 211-21. Appendix A. Partitioned balancing The method called partitioned balancing depends on two results, both of which involve a form of partitioning. In the first we partition the constraints, in the second we partition the variables. 1. It is possible to partition the constraints into two groups, in an arbitrary way, and balance for each group separately,

provided the covariance matrix is modified appropriately. Specifically, the requirement is that the initial covariance matrix

for the variables in the second stage balancing shall be the final (balanced) covariance matrix produced by the first stage

balancing. With this condition the two stage balancing produces the same results as balancing for all constraints

simultaneously. Evidently the result can be extended to cover balancing in any number of stages. This is the result due to

Durbin. 2. It may sometimes happen that the set of variables explicitly mentioned in the constraints is smaller than the total set of

variables. In this case. the balanced values of the explicitly mentioned variables can be obtained by means of a balancing

run which involves only those variables. Balancing adjustments to the remaining variables will arise by virtue of their

correlations with the explicitly mentioned variables, and it required can be calculated in a second stage by combining the

first stage adjustments and the appropriate portion of the covariance matrix. The second stage is what we have called

'ripple back', from the metaphor of the changes required to balance the explicitly mentioned variables then rippling

across the remaining variables. It is clear that any of the remaining variables which are uncorrelated with all the explicitly

mentioned variables will be unchanged in the ripple back.

These two results are stated separately, but the greatest benefits are obtained by employing them together, usually by partitioning the constraints in such a way that, in the first or second stage balancing, only a subset of variables will be involved and hence the second result is applicable. Some examples of the application of these results will make the idea clearer. It is useful to note a third, fairly trivial, result. 3. There will sometimes be constraints which are already satisfied by the input data. Such constraints may express the

requirement for transactions to balance across sectors when one sector is calculated as a residual. Alternatively, they

may be used to define additional variables which are the sums of groups of original variables. These constraints have to

be included to ensure that they are not violated while balancing for other unsatisfied constraints. Balancing for such

constraints alone will have no effect on the values of the variables, though in general the covariance matrix will be

changed.

The first simple example of using partitioned balancing is in the case. which is quite common below the line in Table A of the Blue Book, where some class of financial transactions of one sector (usually the personal sector) are obtained as a residual. Following Byron (1978), the appropriate treatment is to give the residual sector an error range which is so large as to be effectively infinite; this expresses in an intuitively satisfying way the fact that we know nothing about the residual sector except that it satisfies the row constraint. If we balance for this row constraint separately. our explicit balancing need involve only the variables in the row. The variables themselves will be unchanged, as will the covariance matrix elements for all but the residual item, for which there will be the appropriate negative correlation with each other variable in the row. Provided there is no correlation between the non-residual items in this row and any items elsewhere in the account we will not need to carry out any ripple back calculation. Hence we may proceed to balance all such row constraints one row at a time. replacing each constraint by an appropriately modified block of the covariance matrix

A second, much more complex, example is the simultaneous balancing of three years with inter-year constraints due to the relationship between capital and income. This involves dividing the constraints into four groups and carrying out four separate balancing exercises on subsets of variables. The first three groups of constraints are those which apply within year 1, year 2 and year 3 respectively; the fourth group is the inter-year constraints. Each of the first three groups is balanced, involving only the variables from the year concerned: because our assumption is that there is no correlation in measurement errors in different years there is no ripple back to other years. The final balance involves explicitly only the capital and income variables included in the inter-year constraints. Because these variables are correlated with other variables from the same year-remembering that the covariance matrix is that resulting from the first three balancing operations-it is necessary to calculate a ripple back effect. However, even this is not as complex as it might be because the ripple back operates only within the separate years. (Note that the mathematically equivalent operation of balancing the inter-year constraints first would be computationally much harder. because of the resulting inter-year correlations. It is necessary to choose the order of operations sensibly to benefit from partitioned balancing.)

The final example discussed here is the reduction of the full Table A to four rows representing income, transfers. expenditure and financing. The first step here, perversely, is to enlarge the problem by adding extra variables to represent these totals and extra constraints to relate each total to its components. These extra constraints will be automatically satisfied by the input data. Balancing is then divided into two stages. the first of which is to balance for all the initially satisfied constraints, both the extra constraints for the new total variables and the row constraints of Table A. Since this first balancing does not affect the variables it can be carried out before their values are known, provided we know the covariance matrix. The second stage is to balance for all the initially unsatisfied constraints. which in this case will be the sector balancing constraints; these constraints will have been re-expressed in terms of the new total variables, so that only those variables need be explicitly balanced. If we wish to apply inter-period constraints such as the alignment adjustments we can proceed analogously to the second example we first balance within each period using the total variables, then apply the inter-period constraints and finally ripple back to all the original variables. In this case however ripple back needs to be applied in two stages and can become rather complex. it may be preferable to do a full simultaneous balance of all the total variables for all periods to apply the inter-period constraints. and then ripple back within each period to recover the detailed variables. TABULAR DATA OMITTED

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Title Annotation: | British Central Statistical Office |
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Author: | Kenny, P.B. |

Publication: | National Institute Economic Review |

Date: | Feb 1, 1991 |

Words: | 5218 |

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