^{page 54 note (1)} Mrs M. Gregory, now at Glasgow University, made a substantial early contribution to the development of the econometric system described in this article. S. W. Davies is now at the University of Warwick.

^{page 54 note (2)} All the data used for this paper were either at 1963 prices or were indices based on 1963 = 100. The recent rebasing of official economic statistics onto a 1970 base is likely to mean changes in some, if not all, of the relationships reported here, and, when re-estimation work has been completed, it is hoped to report briefly on the updated estimates.

^{page 55 note (1)} The two-stage method is not used to forecast the output of the mining and quarrying, aerospace, other vehicle and shipbuilding industries. These sectors, not dealt with in this article, are primarily influenced by special factors best dealt with by ad hoc methods. Shipbuilding output is affected by international freight rates and government policies to support the declining British industry. Mining and quarrying output, predominantly coal, is affected by the government's energy policy. The production of aerospace equipment is similarly influenced by government policy as well as growth in air traffic and the state of British technology. Output of other vehicles, mainly railway equipment, depends to a large extent upon investment expenditure by British Rail.

^{page 55 note (2)} lnput-output tables for the United Kingdom 1963, CSO, 1970.

^{page 56 note (1)} Although all output is eventually accounted for by final demand, there are good reasons for not simply including only categories of final expenditure as explanatory variables in the forecasting equations. If industry A, for example, sells part of its output directly to final consumers and part of it indirectly through industry B, it will immediately feel only part of any upsurge in final demand; the remainder will not be felt until the upsurge has worked through industry B. Depending on the timing of the increase in demand, then, an increase in output of industry A may occur partly or wholly in the current quarter. Similarly, an increase in demand for industry B's output may never be fully felt by A if B's stocks are high or import substitution for B's output takes place. The two group procedure enables factors of this kind to be incorporated in the forecast of industry B's output before it is used as an explanatory variable in the output equation for industry A. Generally there exists a more stable and immediate relation ship between the output of industry B and its demand for A's output than between total final demand and demand for A's output.

^{page 59 note (1)} The symbols are explained in the Appendix. Figures in brackets (thus) denote standard errors of estimates; figures in brackets { thus} denote elasticities calculated at the levels of the variables in 1963.

^{page 59 note (2)} This takes the value of one each quarter until 1966, then increases by one each quarter thereafter to coincide with the rapid rise in imports after 1966.

^{page 61 note (1)} An unsuccessful attempt was made to incorporate inter- industry input-output coefficients more explicitly. This was carried out in the following way. Using both the 1963 and 1968 input-output tables it was possible, by extrapolation and interpolation, to derive a matrix of coefficients for each year between 1958 and 1970. By fitting the actual values of each GDP final expenditure component to these indices it was possible to calculate, for each industry, a series of intermediate demands for its products from other industries. When intermediate output was subtracted from total output, as indicated by each industry's index of production, there remained a series supposedly indicating the volume of output passing directly to the various sectors of final expenditure. In theory an attempt could then be made to regress these on the final expenditure variables to forecast direct output flows. Unfortunately many of these residual series proved to be totally useless, indeed, some of them turned negative before 1963 or after 1968. These results clearly indicate the variability of input-output coefficients.

^{page 62 note (1)} The indices of production for individual intermediate industries are weighted according to their importance as markets for iron and steel.

^{page 64 note (1)} See G. F. Ray and S. W. Davies, ‘Medium-term forecasts reassessed: paper and board’, National Institute Economic Review, no. 62, November 1972, pages 54-55.

^{page 65 note (1)} Problems associated with evaluating forecasts of this kind have been discussed by G. D. N. Worswick in M.J.C. Surrey, The analysis and forecasting of the British Economy, NIESR, 1971, pages 4-9. For an evaluation of the Institute's macro-economic forecasts, see M. C. Kennedy, ‘How well does the National Institute forecast?’, National Institute Economic Review, no. 50, November 1969.

^{page 65 note (2)} The forecasting equations predict quarterly output levels.

^{page 65 note (3)} In this case for all but one of the 22 sectors there are three year-on-year growth rates, the exception being motor vehicles for which only two growth rates are computable as yet (see later) and so 65 observations on which to conduct the overall valuation. Of these 65 observations only 18 were negative.

^{page 65 note (4)} There are, in fact, two variants of this coefficient, an alternative specification being: U=√1/nΣ(P1-Ai)²/[√1/nΣPi² + √/1/nΣAi²] which again = 0, for perfect predictions but a value of 1 this time demonstrates ‘perfect inequality’. This alternative specification is perhaps not as meaningful due to the added inclusion of Pi in the denominator. See Theil: ‘Economic Forecasts and Policy’, 1961; Theil: ‘Applied Econometric Forecasting’, 1966. For an application see also Stekler: ‘Econometrica’ Volume 36, July-October 1968.

^{page 65 note (5)} In some cases—where N1 yields the best forecasts of all four naive models—(a) and (b) amount to the same thing. Of the 22 sectors, N1 performs better than N2, N3 and N4 on nine occasions.

^{page 66 note (1)} Applying the simple statistical test that this result could have come from a binomial distribution with mean 1/2, i.e. that the econometric method is only equally as accurate as N1, it is found with 95 per cent certainty that we may reject the hypothesis (the maximum number of successes compatible with econometric method being no better than N1 is 16). The success rate of 17 out of 22 is therefore statistically meaningful.

^{page 67 note (1)} The index of Industrial Production and other Output Measures, C.S.O., 1970, page 22.