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Data adjustment and forecast performance.

DATA ADJUSTMENT AND FORECAST PERFORMANCE(1) by Andy Blake and Nigel Pain

This article demonstrates how the use of balanced data can affect a forecasting exercise. The forecast published by the National Institute in November 1989 is revised with the constant and current price residual errors in the National Accounts being assigned to component parts in a manner consistent with the overall pattern of imbalances within the accounts at the time. One of the main results is that preliminary estimates for some components of the accounts, available at the time of the forecast, may also need to be adjusted. This is required to ensure that the forecast is consistent with both balanced data over the future and the structure of the model itself 1. introduction In recent years considerable concern has been expressed at the quality of British official statistics, with critics frequently drawing attention to increases in the discrepancies between both the alternative measures of Gross Domestic Product (GDP) and between the national and financial accounts. A number of steps have been taken by the government to remedy this situation. These include an investigation by the Central Statistical Office (CSO) into the sources of error in these accounts, originally published in CSO (1989) and now updated in the companion piece to this article, and the recommendations summarised in the Pickford Report of April 1989.

This article addresses two main issues; we provide an overview of the recent literature on approaches to balancing accounts and examine the extent to which the prior construction of balanced accounts can alter forecasting performance in the short-term. The second part of our exercise involved the imposition of a number of data adjustments aimed at improving the overall coherence of the accounts over 1987Q3-1989Q2 and recasting a forecast originally published in this Review in November 1989. The article also extends the balancing methodology by developing a means of assessing bias in the preliminary estimates of the main economic aggregates which appear during the course of the forecast. Since these estimates only provide a partial overview of the accounts at the time they are published it is important to establish whether they are consistent with a coherent and balanced complete set of accounts.

The plan of this article is as follows; Section 2 illustrates the scale of imbalance in the accounts and reviews a number of methods proposed as a means of optimally adjusting the data. Section 3 reviews the potential difficulties that may arise from the use of preliminary partial estimates for a particular quarter and proposes a means of assessing the extent of any bias. Results from our application to the institute forecast are reported in Section 4. The final section offers some concluding remarks. 2. Reconciling past data We begin this section by illustrating the scale of accounting imbalances over the 1980s. There are two important sources of discrepancies within the accounts, differing estimates of GDP and the inconsistency between the national and the financial accounts. Both may cause considerable difficulty to the interpretation of recent economic developments.

Subsequent to the Pickford Report the CSO has begun to incorporate a number of statistical adjustments into the published figures in an attempt to improve the overall coherence of the national accounts by correcting for known sampling defects. There are two sets of adjustments; the first, to profits and the main expenditure components from 1988 onwards, is designed to reduce the residual errors between the different measures of GDP, whilst the second, a set of quarterly adjustments to stockbuilding and profits from 1983 onwards, is designed to bring the quarterly growth rates of the income and expenditure estimates more closely into line with that of the output estimate. Such adjustments may be viewed as one part of a complete balancing exercise.

The resulting estimates of the national accounts residual errors (defined as in Box A) at both constant and current prices are illustrated in Charts 1 and [sup.2]. For both series we show the errors with and without the prior adjustments. in each instance incorporation of the adjustments has reduced the size of the published residual error and altered its behaviour over time. However removal of the adjustments indicates that there has been little improvement in the extent to which the accounting constraints are satisfied in the unadjusted data.

Both charts indicate the problems that arise in interpreting recent data since it can clearly be seen that the growth of the unadjusted expenditure estimate over the last couple of years is considerably below that of either the output or the income estimates. Despite the decline in the size of the aggregate residual error sizable imbalances remain at the sectoral level. This is illustrated in Chart 3 which shows the net financial surplus and the balancing item for industrial and commerical companies. Although the chart indicates that the company sector has apparently run a sizable financial deficit over the last year or so, interpretation is hampered by the sudden downturn in the balancing item. This could be consistent with either lower expenditure on tangible and financial assets or unrecorded income. Unfortunately the downstream implications of these two explanations are quite different.

Thus even though the National Accounts Statistical Adjustments (NASAS) have helped to reduce errors in the published data, considerable discrepancies still remain in both the unadjusted data and at the sectoral level. It is important to note that the NASAs are only a partial approximation to the outcome from a full balancing exercise as by construction they take no account of possible covariances between the measurement errors of data that is independently collected. Thus balancing, for example, would allow for the possibility of simultaneous underrecording of both exports and corporate income. This suggests that more attention needs to be given to the appropriate means of constructing balanced accounts in order to provide a fuller assessment of the sources of error within the published accounts.

Balancing the data can only be carried out once a number of prior assumptions have been made about the reliability of particular data series, the form of the measurement error and whether the data can be properly treated as observable. Given particular reliabilities for the component series, supplied,. for example, by the statistician who collects the data, and the assumption of a mean zero random measurement error the method proposed by Stone et al. (1942) has considerable appeal.

The problem can be conveniently seen as one of constrained optimization as shown by Byron (1978). Let V be an n*n matrix of the reliabilities of the component data series, x. Thus V is the covariance matrix of the measurement errors. The problem is to obtain a vector of 'true' data x* which satisfies the k accounting constraints given by Ax* =0. Thus the constrained optimization problem can be written: [Mathematical Expression Omitted] where /. is a kx 1 vector of Lagrange multipliers. The solution is given by the first order conditions [Mathematical Expression Omitted] rearranged to give [Mathematical Expression Omitted] as the optimal estimator. The covariance matrix of the balanced data is [Mathematical Expression Omitted] At its simplest, where two time series are to be reconciled, a constraint matrix might be A= [1, 1 1, with V a 2x2 (typically diagonal) matrix of reliabilities.

The method is akin to that used in a standard weighted least squares problem, with observations being weighted using some prior estimate of their reliability, in this case the matrix V of measurement errors. The residual errors in the accounts, described in Box A, are allocated over the entire set of national accounts in proportions determined by the V matrix and the vector of accounting constraints.

To illustrate the underlying methodology, consider a simple example where two estimates are made of the same aggregate, for example the income and expenditure estimates of GDP. If the measurement error in both series is random and independent of the 'true' data (so that cov(x*[sub.t],e[sub.t]=0, and V is diagonal) then the two estimates can be expressed as: [Mathematical Expression Omitted]

In this example if the variance of the published data for the income estimate is greater than that of the expenditure estimate then it must be the case that the variance of the measurement error of income is greater than that for expenditure. The Stone et al. method makes use of these comparative reliabilities in constructing the balanced data. Applying the formulas described above results in an estimate for x', that may be expressed as: [Mathematical Expression Omitted] i.e the two separate observations are weighted according to the variance of the measurement errors.

Although the method outlined above is both simple and intuitive it remained largely unused for nearly forty years due in part to computational difficulties. The approach was revived by Byron (1978) who proposed an algorithm to generate the solution iteratively. This avoided the need to invert large matrices and permitted larger sets of accounts to be balanced. The exercise described in CSO (1989) was a direct application of the Stone et al. (1942) method using their own assessments of component reliabilities with all off-diagonal elements in the V matrix being set to zero.

In practice it may be that the reliabilities of the component series are unknown. Weale (1989) deals with the problem of finding an estimate of V where genuine prior information is unavailable. He demonstrates that for the case described above post multiplying the covariance matrix of the unadjusted data by the accounting constraints purges the effects of data rather than measurement error volatility, giving an estimate of the covariance matrix of the measurement error alone. Thus: [Mathematical Expression Omitted]

is the data covariance matrix) (9) Although it is not possible to estimate V directly, an estimate of VA' is sufficient to balance the accounts. Bias adjustment can be incorporated in this framework. Blake (1990) shows how Kalman Filter methods can be used to produce maximum likelihood estimates of more general autocorrelated error processes.

Use of these methods requires that the measurement error is not correlated with the true data. There are circumstances where this assumption is likely to be unwarranted. An alternative approach to estimating the reliabilities was introduced by Dunn and Egginton (1990) who suggested that measurement error correlated with the actual time series at low frequency could be dealt with by using the deviations of the component series from their trend values to form an estimate of VA'. Perfect correlation with the true data, as discussed by Maravall and Pierce (1 986), cannot be dealt with using any of the methods which attempt to estimate reliabilities from the data itself.

There are several important extensions to the basic approach to balancing the accounts. First, some parts of the accounts may be treated as unobservable. This might be appropriate for certain financial accounts, which balance by construction due to the inclusion of otherwise unidentified items as a separate category. It is straightforward to extend the balancing technique in such cases provided there are sufficient linearly independent accounting constraints. We return to this problem when dealing with the appropriate treatment of incomplete preliminary estimates and therefore incomplete accounts in the initial forecast period.

Two further extensions are to reconcile values, volumes and prices jointly and to analyse the adjustment of sequences of accounting matrices when the error processes are more general. Weale (1988) proposed a method to deal with the former, while van der Ploeg (1982) presented a balanced set of UK accounts allowing for systematic errors. A fuller account of the basic adjustment problem and these extensions may be found in Stone (1 988).

All of these issues impinge on the exercise we carry out. We eliminate both constant and current price residual errors in a consistent manner. Although it is a simple matter to balance the accounts over the past once reliabilities and the form of the measurement error process have been determined, a second difficulty arises in the first quarter of the forecast due to the existence of prior estimates for some of the components of the output-expenditure system. These estimates should also be, in some sense, balanced. 3. Adjusting preliminary data This section deals with the issue of data revisions to preliminary estimates of output and expenditure that become available for the first quarter of the forecast. The problem here is essentially one of bias in the preliminary National Accounts estimates arising from the initial projection of individual components of the accounts.

As both Patterson and Heravi (1989) and Hibberd (1990) demonstrate, revisions to initial estimates can be quite substantial. Inaccurate initial estimates may also send misleading signals about the current state of the economy. For example, at the time of the November 1989 forecast initial data for the production industries suggested that the level of manufacturing output in the third quarter of 1989 was some 0.93 per cent above that of the second quarter. The most recently available data suggests that output actually fell in the third quarter.

Some evidence of potential data inaccuracy was provided by the model itself at the time of the forecast. The mean of the percentage residual on manufacturing output over the eight quarters prior to the start of the forecast was 0-0957, with sample standard deviation of 0.9505. Imposing the preliminary data required a residual of 2.3134 per cent, well outside the notional error bounds.

Preliminary third quarter data for consumers expenditure and the visible and invisible trade balances was also available at the time of the forecast. However since estimates are only ever available for part of the output-expenditure framework there is no a priori way of knowing whether initial estimates are in any sense consistent with balanced accounts. Indeed the more likely expectation is that they are not consistent, at least to the extent that the preliminary figures are based on the extrapolation of past trends extracted from unbalanced accounts.

It is to be expected that any discrepancies arising from bias in the initial estimates will show up in the first quarter forecasts of the remaining components of the output-expenditure system, and, in particular, in the residual component the model uses to ensure that the accounting identities hold in any particular period.

Although the output measure is typically regarded as the most reliable indicator of short-run trends in GDP, see for example Britton and Savage (1984), many macro-models typically forecast the main expenditure components and allow output to reflect expenditure plus some measurement error. This course is adopted in the Institute model. Given total output the behavioural output system, as described in Box B, determines (n-1) of the sectoral output measures with output in the rest of industry' (OREST) being used as the residual category. Thus any bias arising from the preliminary estimates should ultimately show up in OREST. A related problem is discussed in Keating (1985) who notes that the use of what he terms 'conjunctural analysis' may also generate inconsistencies in the initial forecasts of output and expenditure.

The solution to this problem is complicated by the need to forecast the residual error in the National Accounts. One reason why this is often required is to ensure consistency in the growth rates of total output and expenditure over the course of the forecast. As Box B explains detection and elimination of bias arising from preliminary estimates can only be as good as the forecast of the residual error. It may be clear that the behaviour of the residual component (OREST) is in some sense implausible, but there is little way of knowing whether this arises from biased preliminary estimates or a poor projection of the residual error, particularly as econometric evidence in Darby and Wren-Lewis (1990) suggests that output in the chosen residual category is sensitive to the level of the residual error.

For the exercise we report below, detection of first-quarter bias is eased somewhat. This is because the exercise is conducted on the premise that the (constant price) accounts are balanced and therefore a zero residual error is projected. In this instance a substantial ex-ante change in the residual output category provides firm evidence of inaccuracies in the first quarter preliminary estimates. An estimate of potential bias in the residual output category can be obtained by imposing the calculated data for OREST in the omitted n'th share equation (see Box B) and comparing the implicit residual with the equation error bound.

The problem of adjusting preliminary estimates can be formally approached by assuming that the model provides an appropriate way of determining the bias in the initial forecast. With the residual errors set to zero, the model by construction ensures that the accounts balance over the future. However the initial estimates for the first quarter may, in a sense, be unbalanced. Since there are additional stochastic constraints that they should satisfy, i.e. the model equations, it is possible to formulate the problem as one where the initial estimates have to be balanced treating the elements of the accounts for which there are no estimates as unobserved components. Thus adjustments are made to both data that is known to be unreliable and to data whose use generates residuals outside the past error bounds on the model equations. Appendix A illustrates the formal nonlinear optimisation problem, extending the method proposed by Byron (1978). To do this fully is computationally expensive. A rather simpler procedure is to allow the model itself to generate alternative predictions for the first quarter of the forecast by setting the residuals on these equations to their mean values. The resulting predictions can be compared to the initial estimates to check for obvious discrepancies. This is the approach we adopt in Section 4 below. 4. Results The results we report below are the outcome from a variant forecast conducted on balanced data obtained by a means which can be seen as an approximation to the procedures outlined above in Section 2. It is not possible to use the Stone et al. method directly, as estimates of the appropriate reliabilities of quarterly data at constant prices are not available from any published studies, although Dunn and Egginton (1990) do provide some estimates for annual data.

The alternative to using arbitrary reliabilities would be to try to estimate the reliabilities from the data, following Weale (1989) and Blake (1990). We have not done this for two reasons. First, as explained above, these approaches have so far been developed where the measurement error in a particular series is independent of the 'true' data. This requirement will be violated if there is some element of projection in the most recently available quarterly data, either because information is unavailable or because it is of lower frequency. Two examples here are the initial estimate of the services component of consumers expenditure and the projection of income from self-employment. This suggests that estimates of unreliability obtained for a period two or three years prior to the forecast would have to be used in assessing the unreliability of the most recent pre-forecast data. Even these estimates may only be used under the assumption that sampling practice has remained unchanged subsequent to the period over which the estimates were made. Secondly, as the ultimate interest in balanced accounts lies in the use to which such accounts may be put, there seemed a good case for deriving a simple procedure which would provide an initial indication of the change in short-term forecasting performance that might arise from the use of balanced data. Below we report the change in the short-term forecast arising from the elimination of the National Accounts residual errors over the eight quarters prior to the initial forecast quarter. The prior adjustments we have made are consistent with the overall pattern of imbalances within the accounts, and therefore the results should be regarded as illustrative of those that might arise from the use of balanced data. Contrary to the ex-post forecasting analysis of Wallis et al. (1987) our analysis uses the data available to the forecasters at the time of the forecast. A number of important simplifying assumptions were made: - The output measure of GDP was assumed to be

the most reliable short term estimate of GDP. The

constant price residual error was therefore eliminated

by adjusting the expenditure estimate of

GDP. - All the main expenditure deflators were assumed

to be reliably measured. This allowed us to translate

the constant price expenditure adjustments

into current prices as discussed in Box A. The

current price residual error was then eliminated

by adjusting the income estimate of GDP so as to

bring it into line with the new expenditure

estimate. - No adjustments were made to the financial

accounts for the reason explained in Box A. Thus

the sectoral balancing items only change to the

extent that estimates of sectoral income and

expenditure change. - All policy related variables and projected exogenous

variables used in the published forecast (see

NIER November 1989) were retained. Thus interest

rates, exchange rates, government expenditure

and variables such as world trade were kept

at the values used in the base forecast. - New residuals were set as appropriate, following

the guidelines used by the forecasters at the time.

These may be found in Box A of Anderton et al.

(1989). Existing residuals on earnings, consumption

and retail, wholesale and consumer prices

were retained. 4.1 Balancing output and expenditure The first step of the exercise involved the elimination of the constant price residual error. This was done by allocating the residual error over 198703-1989Q2 (the eight quarters prior to the start of the forecast) to business investment. An investment adjustment was chosen for two main reasons; first, because forecasts of fixed investment in recent years have often exhibited a sizable error. This is demonstrated by Patterson (1990) who suggests that statistical discrepancies may be partly to blame. The second motivation for adjusting investment was that such an adjustment was consistent with errors elsewhere in the accounts at the time of the forecast, most notably the balancing item allocated to industrial and commercial companies (ICCS) which averaged nearly 3 pounds billion per quarter over this period. For a given set of financial accounts a rise in corporate expenditure will lower the estimate of the Nafa and lower the balancing item.

An additional somewhat model specific reason for the adjustment to investment was that, as discussed above, preliminary estimates of consumption, imports and exports were already imposed for the first quarter of the forecast. Given that interest lay in the changes arising from the prior adjustments it seemed sensible to adjust a category of expenditure that was allowed to vary in the first quarter.

Having adjusted investment volumes and carried through accounting changes elsewhere (a total of 17 adjustments to values and volumes in all) the next step involved re-running the original forecast of November 1989, taking on board all the original policy and residual settings. Although many of the residual settings are now inappropriate, the outcome provides a useful indication of the extent to which prior data adjustments may alter a short-term forecast.

Results for some important variables including GDP and manufacturing output growth, inflation and the visible trade balance are summarised in Table 1. Following Artis (1988) four summary statistics are presented, the mean growth over 1989Q3-199002, the mean absolute error, the Root Mean Squared Error (RMSE) and a Theil inequality Statistic. This latter figure gives the ratio between the RMSE of the particular run and the RMSE from a 'no-change' forecast with all variables projected flat from their immediate pre-forecast (i.e. 1989Q2) level. The statistics are best interpreted as a means of assessing the changes arising from each individual adjustment, rather than as an ex-post assessment of forecasting performance, particularly as they are all conditioned on the data available to the forecasters at the time.

Figures for the original November 1989 forecast are reported in column (1) of Table 1. It can be seen that output growth was underestimated (particularly in manufacturing) as was inflation. In contrast the forecast of the visible trade deficit was somewhat over-pessimistic. The Theil statistics suggest that performance was worst on manufacturing output and the trade deficit as in both cases the statistic is greater than unity.

The outcome from running the original forecast on balanced data is summarised in the column (2) data in Table 1. In most cases performance is worse than in the original forecast. Following Wallis et al. (1987) an additional summary of comparative forecast performance is given by the 'diamonds' illustrated in Figure 1. The two axes report the RMSEs for the forecasts of whole economy and maufacturing output growth, RPI inflation and the visible trade balance. A 'diamond' that lies entirely within another represents superior forecast performance. The figure indicates that the column (2) forecast is not wholly inferior to the original column (1) forecast as the estimate of the visible trade balance has improved, despite the deterioration in the forecast of both growth and inflation.

The primary reason for the slowdown in growth in the column (2) forecast is the retention of the original forecast residuals on the behavioural investment equations. The importance of residual judgements can be illustrated using a model of the form-. [Mathematical Expression Omitted] where Y, is an N x 1 vector of investment categories, II a N x K matrix of estimated coefficients, Z[sub.1], a K x 1 vector of explanatory variables and u[sub.t]t a N x 1 vector of disturbance terms. Normally we would expect the error terms to satisfy the classical distributional assumptions i.e. E(u[sub.t])=0 etc. However if data revisions have occurred subsequent to estimation or if the equation is used outside the estimation period the assumptions about the error term may be inappropriate. In such cases a neutral' forecast often involves the initial projection of the mean of the recent residuals over the forecast period. Our prior adjustments are a form of data revision and alter the pre-forecast errors as they adjust the investment data but do not adjust the explanatory variables used in the behavioural investment equations. Thus we have: [Mathematical Expression Omitted] where

[Mathematical Expression Omitted] and [Mathematical Expression Omitted] vector of prior adjustments. If the mean of the residual errors in the accounts is non-zero then it follows that [Mathematical Expression Omitted], and therefore new non-zero residuals have to be set once the prior adjustments are made. Failure to adjust residuals may result in sharp changes to the growth profile across the start of the forecast.

In the case of the variant summarised in column (2) retention of the original investment residuals helps to bring about a sharp cut in growth in the first two quarters of the forecast. The mean growth forecast for GDP falls to .14 per cent per quarter from the original .48 per cent. This deterioration is rectified in the run reported in column (3) of Table 1. Here new residuals have been set on the investment related variables in the model, using the neutral' rule described above where appropriate. The short-term forecast performance is now quite close to that of the original forecast, with a slightly improved forecast of both manufacturing output and the visible trade deficit.

So far the elimination of the constant price residual error has induced little change in the short-term forecasts of some of the main economic indicators, although this is only true provided the data changes made over the past are reflected in the residual settings over the forecast period. However we are now in a position from which we can begin to address the possible existence of bias in the preliminary estimates of the output expenditure components. 4.2 Bias adjustments to preliminary data Although the forecast performance with balanced data appears adequate there is clear evidence of an implausible combination of output and expenditure estimates elsewhere in the model since the residual component of the output expenditure system, OREST, falls by 12.2 per cent in the first quarter of the forecast. Given that the residual settings for the remaining components of GDP for which no preliminary data exists do not appear implausible, the fall in the residual category is suggestive of bias in one or all of the preliminary estimates of consumption, trade and manufacturing output. Imposing the calculated value of OREST in its omitted behavioural equation yielded an implicit residual of - 13.9 per cent, nearly eight sample standard deviations away from the mean value of the residual over the eight quarters prior to the inital forecast quarter.

On further investigation it was found that a similar problem arose in the original forecast, where it was dealt with by raising the size of the residual error to boost output growth relative to expenditure. This option was no longer available since the natural projection for the residual error when forecasting from balanced accounts is zero. Thus we used the technique outlined above in Section 3 to obtain an estimate of bias in the initial data.

The outcome of this procedure suggested that the preliminary data was underestimating the volume of expenditure and overestimating the growth of manufacturing output. Making the calculated adjustments to the preliminary data generated a set of results summarised in column (4) of Table 1 and also illustrated in Figure 1. The adjustment of the preliminary data clearly has a considerable impact on forecasting performance in the short-term, with the Theil statistic for the forecast of total output halving in size. In addition the 'diamond' of (4) lies entirely within those of (1) and (2), indicating a superior short-term forecast.

The net effect of removing the initial estimates was to raise expenditure by some P500 million and to revise the estimate of manufacturing output down by some 1.5 per cent. Moreover the behaviour of the residual component in the output expenditure system now appeared far more plausible with the fall in the first quarter reduced from 12.2 per cent to .42 per cent.

The final adjustment we report involves small changes to the residuals on the bias adjusted series from the second quarter of the forecast onwards. These were required as the residuals produced by the use of the preliminary data had originally been included in the set of 'true' residuals from which a judgement was made regarding the setting of residuals subsequent to the initial quarter. The outcome from the additional residual adjustments, shown in column (5), appears relatively neutral in that it improves the forecasts of manufacturing output and inflation but contributes little to the forecast of total GDP growth and the trade deficit. In all cases the statistics in column (5) are a significant improvement to those in column (1) from the original forecast, although it is interesting to note that a 'no-change' forecast still appears the best forecast for manufacturing output. The results suggest that sizable gains may result, at least in the short-term, from using a data-coherent macroeconomic model and balanced accounts as a source of information with which to assess the reliability of flash estimates of the main accounting aggregates. 4.3 Balancing income and expenditure The next, and thus far final, step we report is the impact of removing the current price residual error by balancing the income and expenditure estimates of GDP. Here we have chosen to allocate the residual error over 1987Q3-1989Q2 to income from self-employment. The implied upward adjustment of 5.5 per cent to the published data is consistent with the behaviour of the personal sector balancing item which averaged some 25-1 billion per quarter over the adjustment period. The negative balancing item implies an underestimate of income given the financial accounts. As before the adjustments were carried out consistently across the accounting framework, resulting in a total of seven adjustments to the Institute model.

One interesting feature of these adjustments is that they were only possible due to the way we have approached balancing, dealing firstly with volumes and secondly with values. Balancing volumes by adjusting expenditure led to an upward shift in the value of expenditure of sufficient magnitude to change the sign of the nominal residual error. In contrast to the originally published accounts Economic Trends October 1989) nominal expenditure now exceeds nominal income, requiring an upward adjustment to income. The CSO in the exercise reported in CSO (1989) were unable to make such an adjustment as they dealt solely with a current price data set where income exceeded expenditure.

The newly revised data was then combined with the residual settings used to generate the figures reported in column (5). As in the earlier example a small number of residuals were subsequently readjusted, reflecting, for example, the change in the pre-forecast residuals on the behavioural equation for self-employment income. The outcome of these changes is summarised in both column (6) of Table 1 and 'diamond' (6) in Figure 1. The interesting feature of these results is that elimination of the current price residual error, at least in the way we have chosen, appears to make little difference to short-term forecasting performance. Indeed, performance on three out of the four reported indicators has marginally deteriorated. One reason for this is that the primary impact of a change to self-employment incomes, at least in the Institute model, is to raise personal disposable income and therefore boost consumption. However the adjustments reported above only imply a small change to total personal income and therefore the impact on consumer's expenditure is only marginal. The resulting changes were not felt to be large enough to induce a departure from the original forecast assumption of zero residuals over the course of the forecast. 5. Conclusions The purpose of this paper has been to provide an overview of the variety of statistical methods proposed as a means of balancing the national and financial accounts and to investigate the short-term forecasting implications of the construction of balanced accounts over the period immediately prior to the start of a forecast.

The methodology discussed formally in Sections 2 and 3 and illustrated in Section 4 makes use of the behaviour of the forecasting model over the period in which the accounts are in balance to assess the bias that might arise from the imposition of initial estimates of some of the components of output and expenditure in the first quarter of the forecast. Our results suggest that in some circumstances inclusion of such data may impart a considerable degree of bias into the initial forecast projections.

It is important to emphasise that the combination of adjustments made in constructing balanced accounts can be expected to reflect the particular imbalances that exist over the period in which adjustments are made. For example, over the period in which we applied our adjustments the combination of apparently unrecorded expenditure and a positive company sector balancing item was suggestive of unobserved corporate expenditure. However the situation over the latter half of 1989 and the first half of 1990 (at the time of writing) differs in that while output growth still exceeds that of expenditure the company sector balancing item is now negative. Additional upward adjustments to expenditure would therefore push the balancing item away from equilibrium. A more plausible and coherent set of adjustments over this period might instead include an upward adjustment to the volume of exports. Such an adjustment would bring both the positive overseas sector balancing item and the company sector one closer to zero as it also carries the implication of unrecorded corporate income. NOTES (1) We would like to thank Ray Barrell, Nick Oulton, Martin Weale and Peter Westaway for helpful comments and suggestions. (2) All data for Charts 1-3 is taken from Economic Trends, October 1990.

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Society, Series A Vol. 151 pt. 1),. Weale M. (1989), 'Asymptotic maximum-likelihood estimation of national income and expenditure', DAE Cambridge

Working Paper No. 8913. Appendix A. Adjusting initial estimates This appendix expands on section 3 above to explain how the initial estimates could be revised consistently with the balanced set of accounts. In section 2, the balancing problem was written as the solution to the constrained optimisation problem [Mathematical Expression Omitted] This is appropriate for balancing the data over the past, but in the first forecast period needs to be generalised to take account of the initial estimates being incomplete.

It is simplest to explain the procedure in stages. If we assume that there are no components of x without initial estimates, then these estimates must be adjusted so that they both balance in the first forecast period and are consistent with the model. Writing a linear model without rational expectations as Cx+0 = r, where r is a vector of equation residuals and 0 represents all predetermined parts of the model equations, the appropriate generalisation of [il is [Mathematical Expression Omitted] where we have kept the accounting constraints separate from the rest of the model. The relevant extra constraints that the data are required to satisfy for the output-expenditure example in Box B would include the share equations as well as the expenditure equations. D, the weighting matrix for the residuals (the variance-covariance matrix), would either be given from the estimation error or calculated from the model prediction error over the past if this seemed more appropriate.r* would typically be a zero vector. The solution to this problem is analogous to the usual balancing problem, with the addition of stochastic constraints. Details of a similar problem are given by Byron (1978) and van der Ploeg (1985).

The problem is slightly different when there are some elements of x for which we have no preliminary estimates. Denote as x[sub.2] the variables of the partitioned x vector for which we have initial estimates and as x[sub.1], those that are 'unobserved'. The Lagrangian in this case is given by [Mathematical Expression Omitted] where A and C have been partitioned conformably. x[sub.1] can be recovered straightforwardly after the observed data have been adjusted provided there are sufficient stochastic constraints.

When the model is nonlinear then the constrained optimisation problem will rarely have an analytic solution and must be found numerically. This would be a computationally expensive exercise, particularly in a model with forward expectations. TABULAR DATA OMITTED
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Author:Blake, Andy; Pain, Nigel
Publication:National Institute Economic Review
Date:Feb 1, 1991
Words:6756
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