¹ The Measurement of Consumers' Expenditure and Behaviour in the United Kingdom, 1920–1938. I. By Stone, Richard. Cambridge: At the University Press [Toronto: The Macmillan Company of Canada Limited]. 1954. Pp. xxxv, 448. $18.00.Google Scholar

Demand Analysis: A Study in Econometrics. By Wold, Herman. In association with Juréen, Lars. Stockholm: Almqyist & Wiksell; New York: John Wiley & Sons, Inc. [Toronto: University of Toronto Press]. 1953. Pp. xvi, 358. $8.40.Google Scholar

² Allen, R. G. D. and Bowley, A. L., Family Expenditure: A Study of its Variation (London, 1935)Google Scholar; Schultz, Henry, The Theory and Measurement of Demand (Chicago, 1938).Google Scholar

³ As the volumes to be considered are for the most part concerned with the demand for individual commodities or groups of commodities rather than with consumer expenditures as a whole, the expansive literature on the consumption function does not come under review here.

⁴ Especially, Hicks, J. R. and Allen, R. G. D., “A Reconsideration of the Theory of Value,” Economica, XIV, 1934, 52–76, 196–219 CrossRefGoogle Scholar, and Schultz, H., “Interrelations of Demand Price and Income,” Journal of Political Economy, XLIII, 1935, 433–81.CrossRefGoogle Scholar

⁵ Namely, that which permits his marginal rates of substitution to be expressed as the ratio of linear functions of the amounts of each commodity consumed.

⁶ Though the assumption of integrability which he recognized as necessary, even if not of much interest to the economist, might be treated with more finesse by some present-day writers. In addition, as will be mentioned below, from the exponents of the revealed preference approach, we have learned new and perhaps better ways of saying old things in the last two decades.

⁷ Price elasticities with income effects neglected are referred to by many names: “compensated price elasticities,” “Slutsky effects,” “substitution effects.”

⁸ Schultz was also very much interested in classifying commodities as complements or substitutes, and he favoured the compensated price elasticity as the criterion of classification. It could not serve him ideally, however, as the symmetry condition does not always apply to market curves, and its use gives rise to several technical statistical problems. He urged the need for a better criterion. Interest in this subject seems to have declined, however, since the confusion contained in Hick's Value and Capital has been cleared up, and the only positive contribution of the last two decades which the writer knows of is Mr. Ichimura's reinterpretation of the Schultz criterion in terms of changes in tastes. This does not supplant the old criterion, however; it merely gives it a new and useful interpretation. If a writer were, today, to try to classify commodities, he would face the same difficulties, but probably he would regard the matter as less important than Schultz did. See Ichimura, S., “A Critical Note on the Definition of Related Goods,” Review of Economic Studies, XVIII (3), no. 47, 1950–1951, 179–83CrossRefGoogle Scholar; J. R. Hicks, “A Comment on Mr. Ichimura's Definition,” ibid., 184–7.

⁹ Samuelson, Paul A., Foundations of Economic Analysis (Cambridge, 1947), 105, 111.Google Scholar

¹⁰ A function is homogeneous of zero degree if its value does not change when each of its arguments is increased by a given percentage.

¹¹ On p. 261, n. 25 he quotes the Slutsky result that the theory implies equality between the income elasticity of demand for a good and the sum of the elasticities of its demand with respect to its own price and each other price. This result is implied by the homogeneity property mentioned. Moreover, Schultz always used single-valued functions in his statistical work. He did not invariably use functions that were homogeneous of zero degree, but many have as arguments, prices and income, each deflated by the cost of living, and thus have the requisite property of homogeneity.

¹² “A Note on the Pure Theory of Consumers' Behavior,” Economica, XVIII, 1938, 61–71 Google Scholar; 353–4; “Empirical Implications of Utility Analysis,” Econometrica, VI, 1938, 344–56.Google Scholar His later publications on the subject are: Foundations of Economic Analysis, especially chap, v; “Consumption Theory in Terms of Revealed Preference,” Economica, XXVIII, 1948, 243–53Google Scholar; “The Problem of Integrability in Utility Theory,” ibid., XVII (n.s.), 1950, 355–85; “Consumption Theorems in Terms of Overcompensation rather than Indifference Comparisons,” ibid., XX (n.s.), 1953, 1–9.

¹³ Hicks, J. R., Value and Capital (Oxford, 1939).Google Scholar

¹⁴ Houthakker, H. S., “Revealed Preference and the Utility Function,” Economica, XVII (n.s.), 1950, 159–74.CrossRefGoogle Scholar Some corrections to Houthakker's proof have been provided by Corlett, W. J. and Newman, P. K. in “A Note on Revealed Preference and the Transitivity Condition,” Review of Economic Studies, XX (2), no. 52, 1952–1953, 156–8.CrossRefGoogle Scholar See also Ville, Jean, “The Existence Conditions of a Total Utility Function,” Review of Economic Studies, XIX (2), 1951–1952, 123–8.CrossRefGoogle Scholar

¹⁵ The present writer will not take a firm stand on this; the “special circumstances” referred to may be useful as assumptions in some analyses.

¹⁶ Stone, Richard, The Role of Measurement in Economics (Cambridge, 1951), 15–18, and his volume under review, 257–60.Google Scholar

¹⁷ The statement in the text is strictly true only for the version given by Stone in 1951; the later version does not include a convexity assumption used earlier.

¹⁸ Wold has also given some theorems in demand theory for the case of a discrete commodity space. See his volume under review, sections 4.7 and 4.8.

¹⁹ Another interpretation that may be given to the Stone postulates may have some appeal. One might define the unit of a commodity to be that minimum amount which, if added to his present consumption of the good, would lead the consumer to prefer the greater amount to the smaller, and postulate that this unit is a constant physical amount of the commodity at all levels of consumption. Thus the unit is the minimum amount it is worth while (or possible) to distinguish.

²⁰ Klein, Lawrence R. and Rubin, Herman, “A Constant Utility Index of the Cost of Living,” Review of Economic Studies, XV (2) no. 38, 1947–1948, 84–7.CrossRefGoogle Scholar

²¹ “Some Implications of ‘Linearity,’” ibid., 88–90.

²² Stone, Richard, “Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British Demand,” Economic Journal, LXIV, 1954, 511–27.CrossRefGoogle Scholar This appeared after the publication of the book under review.

²³ Frisch, Ragnar, “Linear Expenditure Functions,” Econometrica, XXII, 1954, 505–10.CrossRefGoogle Scholar

²⁴ Stone's predictions, based on this hypothesis, are reported in his article cited in footnote 22.

²⁵ For a survey of this development and references to the literature, the reader may consult Tobin, J., “A Survey of the Theory of Rationing,” Econometrica, XX, 1952, 521–53.CrossRefGoogle Scholar

²⁶ Farrell, M. J., “Some Aggregation Problems in Demand Analysis,” Review of Economic Studies, XXI (3), no. 56, 1954, 193–203.Google Scholar

²⁷ Duesenberry, James S., Income, Saving and the Theory of Consumer Behavior (Cambridge, Mass., 1949), chap, vGoogle Scholar; Modigliani, Franco, “Fluctuations in the Savings-Income Ratio: A Problem in Economic Forecasting,” Studies in Income and Wealth, XI (New York: National Bureau of Economic Research, 1949), 371–443 Google Scholar; Farrell, M. J., “Irreversible Demand Functions,” Econometrica, XX, 1952, 171–86.CrossRefGoogle Scholar

²⁸ Wold, H., “A Synthesis of Pure Demand Analysis,” Skandinavisk Aktuarietidskrift, XXVI, 1943, 85–118, 220–63; XXVII, 1944, 69–120.Google Scholar

²⁹ But see Samuelson, , “The Problem of Integrability in Utility Theory,” 384–5Google Scholar; Wold's, reply, “Demand Functions and the Integrability Theorem,” Skandinavisk Aktuarietidskrift, XXXIV, 1951, 149–51Google Scholar; and Houthakker's, defence of Samuelson in “Demand Analysis,” Journal of the American Statistical Association, XLIX, 1954, 88–96, especially 89–90.CrossRefGoogle Scholar

³⁰ The more recent study of the question cannot be summarized here. Reference is made to the expository article by Koopmans, T. C., “Identification Problems in Economic Model Construction” published as chap, ii in Studies in Econometric Method, ed. Hood, Wm. C. and Koopmans, Tjalling C. (New York, 1953)Google Scholar, and to the technical literature cited therein.

³¹ Econometrica, XI, 1943, 1–12.Google Scholar

³² Discussions of these new procedures and detailed references to the literature may be found in Klein, Lawrence R., A Textbook of Econometrics (Evanston, Ill., 1953)Google Scholar; Tintner, Gerhard, Econometrics (New York, 1952)Google Scholar; Hood and Koopmans, eds., Studies in Econometric Method. See also Bennion, E. G., “The Cowles Commission's ‘Simultaneous Equation Approach’: A Simplified Explanation,” Review of Economics and Statistics, XXXIV, 1952, 49–56.CrossRefGoogle Scholar

³³ Cf. Koopmans, T. C. and Hood, Wm. C., “The Estimation of Simultaneous Linear Economic Relationships,” chapter vi in Hood, and Koopmans, , eds., Studies in Econometric Method, especially pp. 131–43.Google Scholar

³⁴ Stone explicitly chooses not to work with systems of equations, claiming (see p. 295, also pp. 244–9), among other things, that the improvement in estimates to be expected would not warrant the extra expense and time involved, but in effect this decision means he has ignored the question of identifiablility in his own statistical work. Stone adopts the practice of using estimates of income elasticity based on budget data in deriving estimates of other parameters from time series data. He argues that this practice reduces the problem of multicollinearity (arising from correlation among the independent variables in a least squares regression) but he does not discuss its effects on the related problem of the identifia'bility of parameters. In any case the practice is open to some criticism as is indicated below.

³⁵ Wold has consistently held this position in a series of papers beginning with Bentzel, R. and Wold, H., “On Statistical Demand Analysis from the Viewpoint of Simultaneous Equations,” Skandinavisk Aktuarietidskrift, XXIX, 1946, 95–114.Google Scholar It is most exhaustively defended in scattered sections of the book under review. One paper, published since the book, continues the defence: Wold, H., “Causality and Econometrics,” Econometrica, XXII, 1954, 162–77.CrossRefGoogle Scholar

³⁶ Wold nowhere gives a formal definition of exogenous variables in recursive models. A definition for general linear structural models has been given in Hood, and Koopmans, , eds., Studies in Econometric Method, 115–20.Google Scholar The definition requires to be modified in the light of the special assumptions made by Wold in the recursive model.

³⁷ The postulated statistical properties of the random variables are important specifications of the model. In the recursive model the random variable in any equation for time t is specified to be statistically independent of the values of all variables in that equation except the current value of the endogenous variable explained by that equation. The random variables pertaining to time t may or may not be statistically independent; if they are not, the statement in the text requires modification.

³⁸ Namely, the one whose current value appears in the first equation.

³⁹ Namely, the one whose current value is determined by the last equation.

⁴⁰ This technical material is given in chapters xn and xm. These chapters provide a terse, sophisticated statement of the theory of linear regression both as a mathematical problem of linear approximation and as a statistical problem. These chapters follow three others constituting Part III of the book on “Some Topics in the Theory of Stationary Random Processes.” Part III, summarizing part of the author's doctoral thesis on A Study in the Analysis of Stationary Time Series (Uppsala, 1938)Google Scholar, and some more recent work, fits rather uncomfortably into this volume, although it cannot be denied that results from this Part are used in deriving theorems concerning regression in the next.

⁴¹ In exercises 27 and 28 on p. 251, Wold points out the fact that if the random variables in different equations of the recursive model are intercorrelated, least squares estimates of some coefficients of the equations may be biased even in large samples.

The reader is reminded that in a regression model, the explaining variables connot be regarded as fixed variates in so far as they include lagged values of endogenous variables.

⁴² See his summing up of the case for the recursive form on p. 70.

⁴³ Wold's particular kind of causal relationship among variables is not the only kind that may be considered. Simon has shown (for the non-stochastic case only) that some more general types of linear system may also permit one to interpret relations among variables in terms of cause and effect, and, moreover, has suggested that there is an intimate connection between identifiability and “causal ordering” of the variables. H. A. Simon, “Causal Ordering and Identifiability,” chap, iii in Hood and Koopmans, eds., Studies in Econometric Method. See also his “On the Definition of the Causal Relation,” Journal of Philosophy, XLIX, 1952, 517–28Google Scholar; and his “Spurious Correlation: A Causal Interpretation,” Journal of the American Statistical Association, XLIX, 1954, 467–79.Google Scholar

⁴⁴ The present writer finds more satisfactory the type of argument in favour of the application of customary least squares methods to demand analysis invoked by Karl A. Fox of the U.S. Department of Agriculture ( Fox, Karl A., “Structural Analysis and the Measurement of Demand for Farm Products,” Review of Economics and Statistics, XXXVI, 1954, 57–66 CrossRefGoogle Scholar). Mr. Fox does not try to make a case in favour of least squares procedures in general, but argues that with respect to certain specific farm products the relevant facts are such that estimates produced by simultaneous equations methods (the equations that he considers relevant are not in recursive form) do not differ significantly from direct least squares estimates. It is helpful to have this class of situation described and illustrated. Incidentally, Fox's argument also serves to vindicate Schultz's judgment with respect to demand analysis of certain farm products referred to above. An illustration similar to those given by Fox is presented by Tobin, James in “A Statistical Demand Function for Food in the USA,” Journal of the Royal Statistical Society, CXIII, Series A, Part II, 1950, 113–49, especially section 3.4.CrossRefGoogle Scholar

⁴⁵ Wold provides some discussion of this point on p. 204, but it is not conclusive.

⁴⁶ One must speak of “relative” neglect—see Stone's volume, especially pp. 296–302 and references given there.

⁴⁷ One aspect of the concentration on large-sample properties of estimates is that we cannot determine how serious for small samples is the lack of asymptotic unbiasedness in estimates. If more were known of their small sample properties it might be possible to justify on firm theoretical grounds the use of least squares procedures in certain cases in which their use now lacks such justification. See Christ, Carl, “A Test of an Economic Model for the United States, 1921–1947” in Conference on Business Cycles (New York: National Bureau of Economic Research, 1951), especially p. 48.Google Scholar See also the Stone volume, especially p. 249.

⁴⁸ Anderson, R. L., “The Problem of Autocorrelation in Regression Analysis,” Journal of the American Statistical Association, XLIX, 1954, 113–29.CrossRefGoogle Scholar

⁴⁹ Durbin, J. and Watson, G. S., “Testing for Serial Correlation in Least Squares Regression, Parts I and II,” Biometrica, XXXVII, 1950, 409–28, and XXXVIII, 1951, 159–78Google Scholar; Watson, G. S. and Durbin, J., “Exact Tests for Serial Correlation Using Non-Circular Statistics,” Annals of Mathematical Statistics, XXII, 1951, 446–51.CrossRefGoogle Scholar

⁵⁰ See especially, Cochrane, D. and Orcutt, G. H., “Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms,” Journal of the American Statistical Association, XLIV, 1949, 32–61 Google Scholar; and G. H. Orcutt and D. Cochrane, “A Sampling Study of the Merits of Autoregressive and Reduced Form Transformations in Regression Analysis,” ibid., XLIV, 1949, 356–72.

⁵¹ The range of income or total expenditure in the data is especially important in this respect.

⁵² In each case, expenditure per equivalent adult; see below.

⁵³ No results of the application of this formula are given in the chapter on empirical findings with respect to income elasticities obtained from data on family budgets (chap., 16).

⁵⁴ Törnqvist has suggested several hyperbolic functions for several classes of commodities referred to as necessities, relative luxuries, luxuries, and inferior goods. The function utilized mostly in the empirical work reported in Wold is the one for necessities, viz., d = α μ/(μ + β), where d denotes the demand for α particular good either in terms of quantity or expenditure, μ denotes total income, and α and β are constants.

The discussion on the form of the Engel curve continues in the very recent issues of the journals. Prais, for example, mindful of the wide variations of income in the data available from some budget studies, has presented a forceful case for using non-linear Engel curves of variable elasticity, especially for food items, and utilizes in his analysis results of recent studies of the problem of variations in quality which will be referred to below. Aitcheson and Brown also contribute an argument for the non-linear form of the curve, and, in common with Prais, use the notion of a “satiation level” for the consumption of each commodity. One gathers that this work, which is being pursued at the Department of Applied Economics at Cambridge, is related to the general study of pre-war British budget material slated for publication in the Department's Monograph Series as S. J. Prais and H. S. Houthakker, The Analysis of Family Budgets. The works referred to above are: Prais, S. J., “Non-linear Estimates of the Engel Curves,” Review of Economic Studies, XX (2), no. 52, 1953, 87–104 Google Scholar; Aitcheson, J. and Brown, J. A. C., “A Synthesis of Engel Curve Theory,” Review of Economic Studies, XXII (1), no. 57, 1954–1955, 35–46.CrossRefGoogle Scholar

⁵⁵ This, subject to qualifications concerning family size to be mentioned below.

⁵⁶ Houthakker, H. S. and Prais, S. J., “Les Variations de qualité dans les budgets de famille,” Economie appUquée, no 1, 01–mars., 1952, 1–14 Google Scholar; Houthakker, H. S., “Compensated Changes in Quantities and Qualities Consumed,” Review of Economic Studies, XIX (3), no. 50, 1952–1953, 155–64CrossRefGoogle Scholar; Theil, H., “Qualities, Prices and Budget Enquiries,” Review cf Economic Studies, XIX (3), no. 50, 1952–1953, 129–47CrossRefGoogle Scholar; Houthakker, H. S., “The Econometrics of Family Budgets,” Journal of the Royal Statistical Society, Series A (Gen.), CXV, Part 1, 1952, 1–28, especially section 4.Google Scholar

⁵⁷ The adequacy of this technique has been questioned on several occasions. See for example the critical study by Allen, R. G. D., “Expenditure Patterns of Families of Different Types” in Lange, O., McIntyre, F., and Yntema, T. O., eds., Studies in Mathematical Economics and Econometrics (Chicago, 1942), 190–207 Google Scholar, and the comments upon it by H. S. Houthakker in “The Econometrics of Family Budgets,” especially section 5. For another recent Cambridge study of this problem see Prais, S. J., “The Estimation of Equivalent Adult Scales from Family Budgets,” Economic Journal, LXIII, 1953, 791–810.CrossRefGoogle Scholar

⁵⁸ Houthakker, H. S., “The Econometrics of Family Budgets,” 1.Google Scholar

⁵⁹ A time variable is introduced in some equations and not always linearly.

⁶⁰ The deflator for electricity consumption is treated as a special case.

⁶¹ This matter is taken up below.

⁶² Stone, pp. 277–8.

⁶³ It may be noted that Fox, in “Structural Analysis and the Demand for Farm Products,” uses price as the dependent variable.

⁶⁴ Introduced by Frisch, R. in Confluence Analysis by Means of Complete Regression Systems (Oslo, 1934).Google Scholar

⁶⁵ Wold uses no bunch maps; Schultz referred to the Frisch publication in a footnote on p. 750, but did not include it in his bibliography.

⁶⁶ It may be noted that in making calculations for his bunch maps, Stone has computed all possible regressions among the variables in any equation and in this respect has probably been even more systematic and meticulous than Schultz. Schultz, however, gave his results in terms of elasticities, or their reciprocals, whereas Stone presents his in the bunch maps only. The bunch maps give, graphically, regression coefficients that pertain to variables after each has been normalized so as to have zero mean and unit sum of squares. Without further information, it is not possible to translate these regression coefficients into the elasticities of interest. In any event, bunch maps are not presented for all of the time series regressions. As noted above, the question of how to treat errors in the variables is still very much an open one.

⁶⁷ The statistical theory underlying the subject may be found in parts of Wold, especially exercise 11, p. 246; in Stone, pp. 303–5; and in Durbin, J., “A Note on Regression When There Is Extraneous Information about One of the Coefficients,” Journal of the American Statistical Association, XLVIII, 1953, 799–808.CrossRefGoogle Scholar

⁶⁸ Stone's discussion (on pp. 409–10) of his use of the conditional regression technique would not permit one to accuse him of being too sanguine.

⁶⁹ Net national income per equivalent adult at 1938 prices.

⁷⁰ This factor is a “guess” (see Stone, p. 321). It is the factor that, with one exception, is used in all food equations.

⁷¹ Stone, Richard, “The Demand for Food in the United Kingdom Before the War,” Metroeconomica, III, 1951, 8–27 CrossRefGoogle Scholar, especially p. 9 from which the quotation is taken.

Stone also observes, referring to the work of Prais on non-linear Engel functions (cited above), that the form of the Engel curve he (Stone) used in the volume under review may have yielded overstatements of income elasticities of demand.

⁷² Wold thinks this fact unimportant (p. 227).

⁷³ The author first encountered this argument in conversation with Professor Milton Friedman. See also Vickrey, William, “Resource Distribution Patterns and the Classification of Families” in Studies in Income and Wealth, X (New York: National Bureau of Economic Research, 1947 ) 266–97, especially 287–95.Google Scholar A version, elaborated in considerable detail, of the above argument as it applies to budget data, appears in Franco Modigliani and Brumberg, Richard, “Utility Analysis and the Consumption Function: An Interpretation of Cross-Section Data,” chap, xv in Kurihara, Kenneth K., ed., Post-Keynesian Economics (New Brunswick, N.J., 1954).Google Scholar It is of some interest in the present context to note that Schultz was not unmindful of the importance of adjustment lags in consumer behaviour. See his p. 119.

⁷⁴ Modigliani and Brumberg, in the paper cited in the previous footnote, remark: “We must take a dim view of the attempts that have been made at deriving the time-series average and marginal propensity to save from the cross-section relation between saving and income” (p. 431). See also in this connection, Haavelmo, Trygve, “Family Expenditures and the Marginal Propensity to Consume,” Econometrica, XV, 1947, 335–41.CrossRefGoogle Scholar

⁷⁵ Most of the commodities are classes of food; presumably other commodities will be studied in the second volume which has not yet appeared. Stone does not present his basic data on budgets but, as noted above, a detailed analysis of the family budget material utilized by him has been promised in a forthcoming volume by Houthakker and Prais, The Analysis of Family Budgets.

⁷⁶ Schultz's data were not compiled by him or specifically for him; he assembled his data from various, but mostly official sources.

⁷⁷ Wold's material on “empirical findings” summarizes material presented more fully elsewhere.

⁷⁸ A large proportion of these concern the demand for various foods.

⁷⁹ Wold does not give standard errors for his estimates of income elasticity from budget data, on the ground that “in a regression analysis of non-experimental data, the standard errors carry little weight as significance indicators” and being small, they might give a spurious impression of accuracy (pp. 260–1). He does give them, however, for his elasticity estimates based on time series data, which, one may observe with some puzzlement, are not “experimental data.”

⁸⁰ For a similar summary statement on price elasticities see pp. 339–40 in Stone.

⁸¹ This remark, incidentally, follows the earlier reminder that “Our consumption [of sugar] series represents, therefore, the quantity demanded for stocks as well as for direct consumption” (p. 178), and the assertion that the demand curve is to be regarded as a ”wholesaler-dealer-fanner demand curve” (p. 162).

⁸² Since his price elasticities are taken to measure substitution effects only, and not income effects, the requirement becomes that the sum of these price elasticities be zero.

⁸³ Wold, too, has used this homogeneity condition to test his statistical results.

⁸⁴ The ratio of the substitution effect of the price of i on the demand for j to the substitution effect of the price of j on the demand for i. The quotation is from Wold, p. 301. Italics added. See also p. 282 for an application of the test under the specific assumption that the income effects are small.

⁸⁵ PP. 602–3 and 628–31.

⁸⁶ P. 646. Italics in original.

⁸⁷ Stone refrains from commenting on the observed relation between pairs of substitution terms in those few of his cases which allow comparisons.

⁸⁸ Words like “prediction” and “forecast” do not appear in the index to Schultz's volume. But he did make predictions (e.g. pp. 563–4) based, in large measure, on the time trends in the demand equations (cf. Stone, p. 231). They are very general and not particularly suitable for testing the results of the theoretical and statistical analysis.

⁸⁹ Reference may be made to Carl Christ, “A Test of an Econometric Model for the United States, 1921–1947,” and the accompanying comments of Professor Milton Friedman. See also Klein, Lawrence R., A Textbook of Econometrics, 249–75.Google Scholar

⁹⁰ The presentation is hard to follow in that relevant elasticities are given in two widely separated parts of the book with the same elasticity having two different values in some instances.

⁹¹ He has however addressed himself to the problem of prediction on several occasions. See for example “Prediction from Autoregressive Schemes and Linear Stochastic Difference Systems,” Econometrica, XVII, 1949, Supplement, 29–37 Google Scholar; Stone, Richard and Prais, S. J., “Forecasting from Econometric Equations: A Further Note on Derationing,” Economic Journal, LXIII, 1953, 189–95CrossRefGoogle Scholar (this is essentially a commentary on Houthakker, H. S. and Tobin, J., “Estimates of the Free Demand for Rationed Foodstuffs,” Economic Journal, LXII, 1952, 103–18CrossRefGoogle Scholar); and The Role of Measurement in Economics, especially pp. 23 and 27–32. It was noted earlier that in “Linear Expenditure Systems and Demand Analysis,” Stone undertook predictions based on estimates of these linear expenditure systems.