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A Simple Forecasting Model for the U.S. Economy

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1955
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AN ATTEMPT IS MADE in this paper to construct for the U.S. economy a statistical system for forecasting gross national product, consumption, and private investment (in plant and equipment).1 The system is based on the pattern of relationships observed for the period 1929–52. It has been found that, in spite of the great changes during these 23 years, the magnitudes of the major economic variables can be foretold with great accuracy throughout the period by one set of equations constructed on the basis of the movements of a few key variables. These key variables are so-called “predetermined variables” (both exogenous variables and the values of endogenous variables during preceding periods), whose magnitudes for a subsequent period either may be known at the time the forecasts are made or can be assumed on the basis of alternative assumptions regarding governmental policies.

By constructing a statistical “exploratory” model of the main economic relationships relating to consumption, investment, disposable income, money supply, and gross national product, it has been found that five predetermined variables, which are set out in detail below, satisfactorily explain the movements of the main economic variables. The values of gross national product (GNP), consumption, and gross private investment in plant and equipment computed from these forecasting equations during the sample period 1929–52, together with the actual values, are shown in Chart 1. The estimated values are, on the whole, quite close to the actual values.

Chart 1.U.S. Gross National Product, Consumption, and Investment: Actual and Computed Values, 1929–52

(In billions of U.S. dollars)

The model is limited to a few variables; however, as is argued in more detail in Appendix I, the available data probably do not permit the construction by any method of a model involving more variables. At least for gross product and aggregate consumption and investment, the few predetermined variables included in the model are adequate as a basis for reasonably accurate forecasts.

The 1953 values for GNP, consumption, and investment are predicted under two different assumptions about inventory changes and net foreign investment. While the two assumptions differ substantially, the forecasts based upon them are not far apart (see Table 2).

Forecasts for 1954 required knowledge of the magnitudes of the predetermined variables, which depend mostly on the fiscal plans of the Government. Since fiscal policies for 1954 were far from definite when this paper was written, it appeared most useful to estimate the effects of alternative fiscal policies on the GNP. The results indicate that any given reduction in government purchases of goods and services within a range of up to $20 billion would, if unaccompanied by a reduction in tax rates, bring about a decline in GNP of roughly one and one-half times the amount of the reduction. Therefore, in this case the multiplier of government purchases is about 1.5. If tax rates are cut in such a way that tax revenue falls by an amount equal to the reduction in government purchases, GNP would decline only by an amount equal to about nine tenths of the reduction. The multiplier of an equal reduction in both government purchases and taxes is therefore estimated at about 0.9.2

The “Exploratory” Structural Relationships

1. Method to Be Used

The method used in this paper differs from that recommended in the current literature.3 The latter method centers attention on the measurement of the coefficients in the “structural relationships,” explaining the movements of such endogenous variables of the economy as consumption, investment, etc. Statistical estimates found for these structural coefficients are then used to predict the endogenous variables on the basis of particular values assumed for the predetermined variables (such as government expenditure, taxation, etc.).4

With rare exceptions, however, no reliable estimates can be made of the coefficients in the structural relationships. Basically, this is due to the fact that, in economic reality, there are so many variables which have an important influence on the dependent variable in any structural equation that all structural relationships are likely to be “underidentified.”5 Secondly, even when this lack of identification is somehow overcome, the tendency of many economic variables to follow a similar movement, i.e., to show a high degree of collinearity, makes it quite impossible to say that any estimate of a structural coefficient indicates the influence of one particular economic variable rather than of another.6 This is so, even when the estimates are made by the recently developed technique of simultaneous estimation and are statistically significant. The weakness of the coefficients found for the structural relationships also affects their reliability and usefulness as elements in forecasting.

The forecasting equations in this paper are obtained by correlating directly by the least squares method the economic variables in whose future values we are interested with the predetermined variables of the system.7 By this procedure we rely to the utmost on the collinearity of most economic variables in making our forecasts.

Under this method, however, it may still be essential to do some structural estimation, even though only in an “exploratory” sense, in order to determine the predetermined variables to be used in the forecasting scheme. These exploratory structural relationships, admittedly representing oversimplification of economic reality with the endogenous variables limited as far as possible to those for which forecasts are to be made,8 are useful in revealing the identities of the important predetermined variables. Certain predetermined variables might be brought to our attention only as a result of going through the process of writing out the structural relationships. For instance, it might not be obvious that government transfer payments had to be inserted as a predetermined variable separate from other government expenditures unless the equations for consumption, disposable income, and the definition of the gross national product were written out.9 More generally, it is only through its bearing on one or more of the structural relationships involved in the problem that we can consider a given predetermined variable as a factor relevant to the problem at hand. The reason is that economic theory deals directly with structural relationships but rarely so with reduced forms. In the forecasting equations which are elaborated below, no predetermined variable will be used unless it has been found to have significant coefficients in the exploratory structural relationships.10

For reasons already explained, these exploratory structural relationships are useful only in selecting the predetermined variables, and the estimated structural coefficients found in these equations are not used for forecasting purposes. A way is suggested in Appendix III, however, by which these exploratory equations may be improved so as to throw more light on the structural relationships.

2. A Description of the “Exploratory” Model

The variables used are measured in billions of current U.S. dollars, with the exception of t, which is the ratio of two of the variables expressed as a percentage.

Endogenous variables

GNP:Gross national product
C:Personal consumption expenditures
I:Gross private domestic investment in new construction and producers’ durable equipment
Yd:Disposable personal income
M:Privately held currency outside banks, and bank deposits (including both time and demand deposits but net of interbank deposits)

Predetermined variables

GNP-1:Value of gross national product during the preceding year
K:The sum of change in business inventories, net foreign investment, and government purchases of goods and services11
D:National security expenditures (This forms part of K but, for reasons to be explained, is also entered as a separate variable.)
P:Government transfer payments (including interest on government securities)
t:The level of tax rates measured in terms of the ratio of total tax revenue (T) to GNP, expressed as a percentage
s:The sum of Federal Reserve holdings of government securities and monetary gold stock

To determine the five endogenous variables, we have four functional relations (the consumption function, the investment function, the disposable income function, and the money supply function) and one identity defining GNP. The functional relationships are estimated on the basis of the data for 1929–51, inclusive, by the Cowles Commission method of simultaneous estimation of the structural coefficients.12 Two sets of results are given on page 440. The only difference between systems A and B is that the predetermined variable S is included in system B but not in A.13 These exploratory functions are discussed separately below; certain comments which are relevant to all of them are presented in Appendix I.

Consumption function

It is generally recognized that disposable personal income and cash balances14 are the two most important explanatory variables of consumption. Consumption is found to be an increasing function of both these variables in equations (1a) and (1b).

“Exploratory” Models of Structural Relationships1
System ASystem B
R = 0.998R = 0.998
R = 0.990R = 0.990
R = 0.999R = 0.999
R = 0.982R = 0.984

For the method of estimation, see text footnote 12. The figures given in parentheses under the regression coefficients are the respective asymptotical standard errors. The R’s represent “correlation coefficients” in the sense that they are the square roots of the portions of the variances of the dependent variables explained by the respective equations. They are, however, different from the usual correlation coefficients which are computed through the least squares method.

A term (usually denoted by u), representing random disturbances, is omitted from all the statistical equations in this paper for the sake of simplicity in presentation.

For the method of estimation, see text footnote 12. The figures given in parentheses under the regression coefficients are the respective asymptotical standard errors. The R’s represent “correlation coefficients” in the sense that they are the square roots of the portions of the variances of the dependent variables explained by the respective equations. They are, however, different from the usual correlation coefficients which are computed through the least squares method.

A term (usually denoted by u), representing random disturbances, is omitted from all the statistical equations in this paper for the sake of simplicity in presentation.

During the war years, the increase in consumption did not fully reflect the high levels of disposable income and privately held cash balances. This was due to the efforts to restrain consumption and to free resources for war. Some measurement of the effects of these efforts must be incorporated in the equation if the war years are to be included in the same consumption function. It is reasonable to assume that the intensity of the efforts varied in the same direction with D (the magnitude of defense or security expenditure), a predetermined variable. The relationship, however, is unlikely to have been linear. The marginal strain on the economy must have been progressively greater as defense expenditures increased. It is therefore assumed that consumption varied inversely with D2, with the latter representing a wide variety of considerations, such as wartime rationing, unavailability of goods and services, refraining from buying for patriotic reasons (e.g., increased purchases of savings bonds), and postponing of purchases on account of the inferior quality of wartime goods.15 It is seen from equations (1a) and (1b) that the regression coefficients of the predetermined variable D2 are quite significant.

Investment function

A function explaining private gross investment in plant and equipment is included in systems A and B as equations (2a) and (2b). The explanatory variables included are the gross product, the gross product lagged by one year, the square of national defense expenditures (D2), and an “average rate of taxation” (t=TGNP).

Investment in plant and equipment is assumed to be determined first by the magnitude of the gross product, the latter reflecting the demand for the final goods to be produced from the capital assets. Investment plans, however, have to be made long before the physical plant and equipment take shape. While the lags involved may vary in length from a few months to more than a year, it is certain that a part of current investment is actually decided upon the basis of the rates of gross national output at different intervals during the preceding year. Hence GNP-1 is introduced as a second explanatory variable. With a role similar to that played by it in the consumption function, the D2 term is then added to summarize the competitive nature of defense demands, in relation to ordinary uses, for the available resources and the effects of the restrictive measures placed upon private investment on account of defense activities. The predetermined variable D2 again appeared to be quite significant in equations (2a) and (2b).

On the assumption that a higher “average rate of taxation” would, other things being equal, have an adverse effect on private investment in plant and equipment, the variable t is introduced as a fourth explanatory variable in equations (2a) and (2b). The concept of an “average rate of taxation” is admittedly unsatisfactory. A larger t may not necessarily indicate increased rates of taxation. It may also, for instance, be the result of upward shifts of income toward the higher brackets, which are subject to progressively higher rates of taxation, even though GNP and the rates remain constant. It has been found, however, that the variable t did indicate quite well changes in the rates of important categories of taxes;16t therefore may be used as an approximation to the “average tax rate.”

Disposable income function

With the exception of a few items of minor importance, the following is an identity:

Yd = GNP + PT — (capital consumption allowances + undistributed corporate profits)

On the assumption that the terms in the parentheses in this identity are primarily a function of GNP and of the predetermined variable t, the disposable income functions (3a) and (3b) are merely a linear approximation of this relationship.

Money supply function

The analysis has started with the assumption that the amount of cash balance (currency and bank deposits) held by the private sectors of the economy is determined primarily by their transaction requirements, which may be indicated by the current and the recently past levels of the gross product. To take account of the supply of money, the variable S (the sum of Federal Reserve holdings of government securities and monetary gold stock) is included as a third variable in equation (4b).17 While the S variable is quite significant in the money supply equation (4b), it will be shown later that the nature of the statistical data is such that it cannot be included in our forecasting equation for GNP. Hence S is dropped as a predetermined variable in system A, and for that system a money supply function is obtained in the form of equation (4a).

Definitional relation of GNP

As given in equation (5), the gross product is by definition equal to the sum of C, I, and K. This relationship is, of course, necessary to make both systems A and B determinate.

The Forecasts

1. The Forecasting Equations

Since the predetermined variables have now been selected, direct least squares regressions on the predetermined variables can also be calculated. These calculations will serve as the forecasting equations for the endogenous variables. The forecasting equations obtained for GNP are as follows, the sample period being 1929–51, the same as for the exploratory structural model:

System A

System B

The negative sign of the coefficient of the variable S in equation (7) is unreasonable; it would mean that, other things being equal, an increase in the amount of Federal Reserve holdings of government securities and monetary gold stock would reduce the money value of GNP.18Equation (7) must therefore be rejected as a basis for forecasts.

For comparison, the following forecasting equations that would have been obtained by the method currently in use, i.e., by solving the system of structural equations for GNP, are given:

System A

System B

The coefficients in (8) are quite different from those in (6), and those in (9) from those in (7). Apart from theoretical considerations, the figures in Table 1 show that equation (6) provides much closer estimates of the actual observations than either (8) or (9).

Table 1.Reliability of Estimate for U.S. Gross National Product (GNP)(In billions of U.S. dollars)
Equation
(6)(8)(9)
Standard error of estimate, 1929–516.610.818.6
Residuals for 1952−3.1−12.2−12.7

After system A had been selected for the reasons indicated, forecasting equations for GNP, consumption, and investment were recomputed with the 1952 observations included. Equation (10), of course, differs only insignificantly from equation (6).

All the regression coefficients appear to be quite significant, and the residuals quite random.19 The values computed from the equations are compared with the actual values in Charts 2, 3, and 4; also, the relative importance of the different predetermined variables is indicated. Our method may now be applied to a number of particular forecasts.

Chart 2.Forecasting Equation for U.S. Gross National Product: Actual and Computed Values, 1929–52

(In billions of U.S. dollars)

Chart 3.Forecasting Equation for U.S. Consumption: Actual and Computed Values, 1929–52

(In billions of U.S. dollars)

Chart 4.Forecasting Equation for U.S. Investment: Actual and Computed Values, 1929–52

(In billions of U.S. dollars)

2. Forecasts for the 1953 Values of GNP, Consumption, and Investment

A forecast of the value of GNP for 1953 as a whole can be made by substituting in equation (10) the proper magnitudes of the predetermined variables (Table 2). GNP-1 is, of course, given at the 1952 level of $348.0 billion. Estimated values of the other predetermined variables are given in Table 3. Data for the first half year are computed from the official figures. For the second half year, the figures for government purchases (including security purchases) are given at the levels indicated as the most probable at the end of 1953 under current legislation and commitments. The figures given for the sum of the changes in business inventories and net foreign investment for the second half year are based on two alternative assumptions: (a) that it was equal to the sum for the first half of 1953, (b) that it was zero. P and t are assumed to be the same for the second half year as for the first half year.

Table 2.Actual Values for 1952 and Forecasts for 1953 for U.S. Gross National Product, Consumption, and Investment1(In billions of U.S. dollars)
1953
1952 ActualAssumption (a)Assumption (b)
Gross national product348.0368.6 ± 3.6364.8 ± 3.7
Personal consumption expenditure218.1232.1 ± 2.3230.7 ± 2.3
Private domestic gross investment in construction and producers’ durable equipment48.849.0 ± 1.648.3 ± 1.4

The numbers following the plus and minus signs are the standard errors of the estimates.

The values of GNP, consumption (C), and investment (I) computed from equations (10)-(12), page 444, on the basis of the actual values of the predetermined variables which became available early in 1954, are shown below. These figures are quite close to the actual data; also, they do not diverge greatly from the preliminary calculations analyzed in the text.

The U.S. Department of Commerce subsequently made a thorough revision of the GNP and related data for the whole period 1929–53, the results of which were published in the Survey of Current Business, July 1954. Therefore, equations (10)-(12) can no longer be used without recomputation for making forecasts. The effects of the revision upon the application of these equations can be seen by computing the values of GNP, C, and I from equations (10)-(12) on the basis of the revised data for the predetermined variables. The tabulation below also compares these computed figures with the actual (revised) data. The margins of error of these estimates are greater than those obtained on the basis of the data before revision.

The numbers following the plus and minus signs are the standard errors of the estimates.

The values of GNP, consumption (C), and investment (I) computed from equations (10)-(12), page 444, on the basis of the actual values of the predetermined variables which became available early in 1954, are shown below. These figures are quite close to the actual data; also, they do not diverge greatly from the preliminary calculations analyzed in the text.

The U.S. Department of Commerce subsequently made a thorough revision of the GNP and related data for the whole period 1929–53, the results of which were published in the Survey of Current Business, July 1954. Therefore, equations (10)-(12) can no longer be used without recomputation for making forecasts. The effects of the revision upon the application of these equations can be seen by computing the values of GNP, C, and I from equations (10)-(12) on the basis of the revised data for the predetermined variables. The tabulation below also compares these computed figures with the actual (revised) data. The margins of error of these estimates are greater than those obtained on the basis of the data before revision.

Table 3.Values of Predetermined Variables for 1953

(Annual rates, billions of U.S. dollars, except t which is expressed as a percentage)

First Half YearSecond Half Year

(Estimated)
Average

(Estimated)
D51.852.852.3
Government purchases of goods and services83.084.8
National security purchases excluding government sales51.852.8
Other federal purchases6.36.0
State and local24.826.0
Sum of change in business inventories and net foreign investment3.63.61
02
K86.688.4187.51
84.8285.72
P18.118.118.1
t24.724.724.7

Assumption (a).

Assumption (b).

Assumption (a).

Assumption (b).

3. Effects of Fiscal Changes on GNP

The preceding forecasts for 1953 cannot now be regarded as having any practical value. A much more interesting prediction would be of the value of GNP in 1954. For this purpose, it is again necessary to know the proper magnitudes of the predetermined variables to be used in equation (10). The forecasts given above for the 1953 value of GNP may perhaps be used for the GNP-1 term in the equation. As long as the fiscal and budgetary policies of the Government for the calendar year 1954 had not been definitely determined, magnitudes for the other predetermined variables could not be assumed with any degree of assurance. At the time of writing this paper, it seemed clear, however, that there would be some reductions in government expenditures and in tax rates. Some conjectures on the probable effects of such reductions on the level of GNP were of interest. Such forecasts could also be made on the basis of equation (10). Several points, however, which are related to the use of the different terms in equation (10) for these forecasts, should be discussed first.

(A) In analyzing the effect of changes in government purchases and in tax rates, attention will be focussed on the D2, K, and t terms in equation (10). It should be noted, however, that the GNP-1 term in the equation has a positive coefficient of 0.39. Since GNP was in all likelihood considerably higher in 1953 than in 1952, GNP in 1954 will be greater than in 1953 if there is no change in the other terms. This increase would, according to equation (10), amount to $8.0 billion for assumption (a) or $6.6 billion for assumption (b).20 This “growth” factor will not be included in the following discussion of fiscal effects.

(B) It is observed in equation (10) that government purchases of goods and services for national security purposes enter into the equation through both the K and the D2 terms, whereas government purchases for other purposes affect equation (10) only through the K term. The effect of a change in government purchases on GNP would, according to the equation, be different depending upon whether the cut is in security purchases or in other purchases. There are probably considerable differences between the content of national security purchases and of other government purchases; hence the multiplier effects of these two types of purchases need not be the same. Equation (10), however, is not in fact intended to provide estimates for these possibly different components of the multiplier of government purchases. It is recalled that the D2 term was introduced to indicate the degree of stress and strain placed on the economy by the heavy increases in security purchases during World War II and, to a smaller extent, during 1950–51.21 The consequent restraints on the economy reduced the magnitude of the multiplier of government purchases of goods and services during these years. The K and the D2 terms in equation (10) jointly give effect to this tendency, and provide a basis for estimating the influence on the gross product of a change in government purchases of goods and services as a whole.22

Comparison of Actual and Estimated Figures of GNP, C, and I for 1953(In billions of U.S. dollars)
Based on data available before July 1954 revisionBased on revised GNP and related figures in SCB, July 1954
ActualEstimated from

equations (9)-(12)
ActualEstimated from

equations (9)-(12)
GNP367.2367.4364.9358.7
C229.8232.6230.1226.3
I51.849.449.947.6

(C) In using the K and D2 terms to estimate the effect of a reduction in government purchases on the gross product, we need to know the shares of security and nonsecurity purchases in the total contemplated reductions of government purchases.23 These data would be available for use, if the alternative plans of the Government were given. A study of past data, however, indicates that government purchases have varied throughout the years in accordance with a discernible pattern. This pattern is shown in Chart 5. Before 1941, security purchases were practically zero. Since that year, an increase in total government purchases has always included an increase in security purchases. These two variables have moved up and down together. The pattern, however, is not a rigid one. For instance, on the upward swings of government purchases, the percentage of security purchases in the total has been consistently smaller during the postwar years than it was during the war years 1941–44. Chart 6 shows the scatter relationship between total government purchases and the portion of nonsecurity purchases. While there was a close relationship for the period as a whole, the 1951–53 (first half) line may be accepted as the basis for determining the relative shares in security and nonsecurity purchases, in the belief that the tendency shown during this period is likely to continue in the near future. The equation of the line is:

Chart 5.Government Purchases of Goods and Services, Total and National Security, 1929–53

(In billions of U.S. dollars)

Chart 6.U.S. Government Purchases of Goods and Services, Total and Nonsecurity, 1947–53

(In billions of U.S. dollars)

where G and N represent, respectively, total government and nonsecurity purchases of goods and services. Substituting this equation in equation (10) through the relationship D = G — N and writing G in place of K,24 we secure the following equation representing the effect on GNP, other things being equal, of a change in government purchases from the 1953 (first half) level of $83 billion:

From this equation, it is now possible to calculate the probable declines in GNP corresponding to given reductions in government purchases of goods and services from the 1953 (first half) level. These estimates are given in Table 4, the figures given after the plus and minus signs being the respective standard errors of these estimates.

Table 4.Effects on Gross National Product of Reductions in Government Purchases(In billions of U.S. dollars)
Reductions in Government Purchases of Goods and Services (ΔG)Resultant Declines in GNP (ΔGNP)Magnitude of Implied Multiplier (ΔGNPG)
11.42 ± 0.181.42
1014.68 ± 1.791.47
2030.38 ± 3.591.52

Within the range of reduction considered in Table 4, the nonlinear effect on the magnitude of the multiplier is not great. Every reduction of $1 billion in government purchases will bring about a decline of approximately $1.5 billion in GNP.

The next problem is to estimate the effects of a simultaneous reduction in government purchases and taxes on the gross product. For instance, it would be of interest to see how the gross product would change if the reductions of $1 billion, $10 billion, and $20 billion in government purchases discussed above were accompanied by reductions in tax revenues of various amounts.25

It has been explained that the predetermined variable t, defined as the ratio of total tax revenue to the gross product (T/GNP), represents an average rate of taxation, and that important categories of actual tax rates have shown a general tendency to change with this average rate.26 It is, in fact, quite necessary from the methodological point of view to use some average of tax rates, instead of tax revenue, as a predetermined variable. Tax revenue is, in all likelihood, a function of both tax rates and many other endogenous variables (such as GNP itself) and is, therefore, an endogenous variable itself. While the Government may be aiming at, say, a reduction of $10 billion in taxes, all that it can do is to reduce certain tax rates; the actual reduction in revenue will probably be different from $10 billion on account of the resultant change in the other endogenous variables (e.g., in the present study, GNP). In order to overcome the same difficulty involved in applying equation (10) to an analysis of the effects of tax reduction, the procedure outlined below will be followed.

First, the magnitude of the variable t corresponding to a desired reduction in revenue is computed on the basis of the existing level of GNP. For instance, for the first half of 1953, GNP is equal to $367.2 billion, and total tax revenue (T) is $90.7 billion. Hence, the average rate of taxation, t, is equal to 24.7 per cent. For this level of GNP, the values of t corresponding to stated declines in revenue are given in Table 5. The reductions in revenue in this tabulation are indicated as “primary reductions” to distinguish them from the final reductions which will ultimately come about as the changes in tax rates (and such changes in government expenditure as may be introduced at the same time) work themselves out in a changed value of GNP.

Table 5.Primary Reduction in Tax Revenue and the Magnitude oft
Primary Reductions in Tax Revenue

from T = $90.7 Billion

(billion dollars)
Magnitudes of t for GNP = $367.2 Billion

(per cent)
1(90.7 - 1)/367.2 = 24.4
5(90.7 - 5)/367.2 = 23.3
10(90.7 - 10)/367.2 = 22.0
15(90.7 - 15)/367.2 = 20.6
20(90.7 - 20)/367.2 = 19.3

These alternative tax rates are then used, together with the figures for D2 and K, in equation (10), to estimate the resulting change in GNP. The final losses in taxes can then be computed from the resultant figures for GNP, on the basis of the magnitude for t that is used.

Various alternative estimates of the effects of a simultaneous reduction in government purchases and taxes on GNP are given in Table 6, together with the standard errors of estimates.

Table 6.Effects of Reductions in Government Purchases and Taxes on Gross National Product
Reduction in

Government

Purchases

ΔG
Primary Reduction

in Tax Revenue

ΔT
Resultant

Reduction in GNP

ΔGNP
Final Losses

of Tax Revenue

ΔT
0.51.05 ± 0.110.8
110.7 ± 0.151.2
114.0 ± 1.64.4
10511.0 ± 1.17.7
107.5 ± 1.511.6
129.7 ± 3.48.2
526.7 ± 2.711.4
201023.2 ± 2.215.0
1519.5 ± 2.419.1
2016.0 ± 3.122.9

When the reduction in government purchases is accompanied by an equal amount of primary reduction in tax revenue, GNP is seen to decline by about 0.7 to 0.8 times the decline in government purchases. In these cases, however, the final decline in tax receipts is about 15 to 20 per cent higher than the primary decline. If tax rates are cut in such a way that final tax revenue falls by an amount equal to the reduction in government purchases, the decline in GNP and the implied multiplier would be as indicated in Table 7 (interpolated from Table 6). It is shown in Table 7 that the implied multipliers are close to unity, as might be expected on theoretical grounds.27

Table 7.Multiplier of an Equal Reduction in Government Purchases and Tax Revenue
ΔG = Final Reduction in Tax RevenueΔGNPImplied Multiplier
10.860.86
108.900.89
209.400.94

As shown in Table 4, a reduction in government purchases that is not accompanied by any change in tax rates would reduce GNP by about 1.5 times the amount of the reduction in purchases. Table 6 shows that GNP changes by intermediate amounts if taxes are reduced by less than the cut in government expenditure.

It will be recalled, however, that the declines in GNP expected from possible reductions in government purchases in 1954 would at least in part be counteracted by the “growth” trend, discussed earlier, implied in the GNP-1 term in equation (10).

Appendix: I. A Discussion of Certain Aspects of the Model

Following upon the consideration which has been given to the individual equations in the exploratory structural and forecasting models, there are also certain aspects of the system as a whole, and certain criticisms which might be made against it, that require further attention. Most of these comments are also applicable to other statistical studies of a similar type.

High correlations and collinearity: their bearing on structural estimation and forecasting scheme

One criticism that is likely to come immediately to the mind of a reader is that the correlation coefficients of the equations in the model are “too high.” With the exception of equations (4a), (4b), and (12), none of the structural or forecasting equations has a correlation coefficient lower than 0.99, and even the exceptions have correlation coefficients higher than 0.98. One may legitimately ask whether there can be such regularity in the economic system that three, four, or five explanatory variables would explain more than 98 per cent of the variations28 in a given variable. Should not such excessively high correlations be a cause for doubt and reservation rather than a source of satisfaction?

It is impossible for any one—and particularly for the social scientist who is constantly harassed by, and hardly can be unaware of, the extreme complexity of the problems he has to deal with—to say that any human behavior can be so simply and completely explained. It cannot literally be true, for instance, that consumption expenditures are almost completely determined by, and only by, the level of disposable income, the amount of cash holdings, and the restrictive influences of defense expenditures, which is what equations (1a) and (1b) in effect purport to say.

It is nevertheless true that such a high correlation has in fact been obtained in all the equations presented in this paper and in many other published and unpublished works. The chances are very small that the high correlations obtained are merely the results of sheer accident. An explanation must be sought for the seemingly contradictory positions: On the one hand, it is impossible to believe that the changes in the magnitude of any given economic variable can be explained almost completely by those of only a few other variables; on the other hand, extremely high correlations have been obtained in all the equations here and elsewhere, which cannot be attributed to chance alone.

The answer lies in the collinearity tendencies, briefly discussed in the early part of this paper, of many economic variables to move more or less together. For instance, there must be scores, or even hundreds, of variables that have a bearing on consumption expenditures; but they tend to move in a certain manner with disposable income, or cash holdings, or defense expenditures, or some combinations of these variables, so that their influences have been attributed to the three variables through the mechanism of statistical calculation. As a result, the three variables successfully explained practically all the variations in consumption. Those who are not familiar with the numerical relationship between the magnitude of the correlation coefficient and the actual sizes of some of the residuals may, indeed, challenge this explanation by saying that even this universal collinearity among economic variables could not account for correlation coefficients as high as 0.99. Some of the variables must, at times, have got out of step with the rest, and produced substantial effects on, for instance, consumption, which could not be attributed to the three explanatory variables included in equations (1a) and (1b). Furthermore, what about the purely random factors which may “shock” the economic system accidentally, but nevertheless quite strongly, at times? Can all of these be reasonably accounted for in the 2 per cent variation left unexplained if the R2 is as high as 0.98?

The answer is in the affirmative, and can best be explained through an example. The R2 of the reduced-form forecasting equation (10) for GNP is as high as (0.998)2 = 0.996. Yet the residuals (actual values minus computed values) obtained from the equation for some of the years are quite high. Out of a total of 23 years in one period, the residuals for 9 years are greater than 5 per cent of the actual value, the highest being 8.6 per cent in 1929 (Table 8).

Table 8.Values of Gross National Product(In billions of U.S. dollars, except for last column)
YearActualComputed from

Equation (10)
Residual (actual

minus computed)
Residual/Actual

(per cent)
1929103.894.98.98.6
193175.981.6−5.77.5
193355.852.73.15.6
193464.960.84.16.3
193884.791.5−6.88.0
1940101.496.25.25.1
1941126.4116.79.77.7
1942161.6170.3−8.75.4
1946211.1225.6−14.56.9

The relatively large residuals in these 9 years, however, do not change the fact that the standard error of estimate for the period of 23 years as a whole is as small as $5.8 billion. It is therefore seen that, where economic variables show a high degree of collinearity, it is possible to chart the movement of any one of them on the basis of the others. This is a blessing from the point of view of forecasting, but at the same time it becomes an almost impossible task to segregate the influence of one exploratory variable from those of other variables in a structural equation. This point is of such fundamental importance that we shall illustrate it with a few examples. In connection with the investment functions, equations (2a) and (2b), it has been explained that GNP and GNP-1 are introduced as explanatory variables on the ground that they reflect the levels of demand for final goods to be produced from the capital assets for which investment is to be made. These considerations, together with D2 and t, succeed in explaining 98 per cent of the variation in investment. But how about the effects of profits on investment? Except for the negative effects of t, profits have not been included as an explanatory variable. But certainly profits should be the main incentive for making investment, and their influence on investment has been found significant in many empirical studies. Is, therefore, the influence of profits on investment lost in equations (2a) and (2b)? The answer is that it is not lost, but is inseparably included in the GNP and GNP-1 variables. Profits are, roughly speaking, the difference between GNP and the sum of certain taxes, total wage bills, and depreciation allowances. The last two items are highly correlated with GNP. Hence, by including GNP and GNP-1 in the equation, the influences of current and past profits are taken into account. There is very little room left in equations (2a) and (2b) to bring current and lagged profits in as additional explanatory variables. The standard errors of the regression coefficients would then be so high that the influences of these variables would not be clearly revealed.29

A similar case is found in the absence of the M variable from equations (2a) and (2b). If private cash balances had an important bearing on consumption, it is reasonable to assume that they would also affect investment decisions. This symmetry argument is probably quite sound. The effects of M on I, however, are probably already included in the GNP and GNP-1 terms in equations (2a) and (2b). As has been shown in equations (4a) and (4b), M is closely tied up with these two variables.

The moral of this discussion is of course quite obvious. Given the limited number of observations that are available, the economic reality is too complicated to establish the kind of structural equation that would permit the measurement of any specific cause-and-effect relationship between one variable and another. If the investment functions, equations (2a) and (2b), are taken as an illustration, it may be said that the GNP and GNP-1 coefficients tend to confirm (or rather do not contradict) each of the three following theories of investment at the same time: (1) the final demand theory, (2) the profit theory, and (3) the liquidity theory. The relative influences of these theories, however, are hopelessly knit together in the two coefficients, incapable of being revealed separately. To replace the GNP and GNP-1 variables with some other variables (say, profits) would present the same difficulty. Yet, the limited number of observations could not accommodate all the variables that have an important bearing on the problem in a structural equation and, at the same time, yield meaningful and significant regression coefficients. This is one of the basic reasons why the current procedure of using the solutions of the structural relationships as the forecasting equations should be rejected.

The same collinearity tendencies are admittedly present in the forecasting equations used in this paper.30 Indeed, as explained earlier, we are dependent upon these collinearity tendencies to “catch” the important forces acting upon a given endogenous variable so that forecasting may be successful. On the other hand, no commitment is made on any of the structural coefficients (such as the propensity to consume) which, although they cannot be estimated accurately, are the bases for forecasting in the procedure suggested in most of the current literature in this field. It should be noted, however, that the regression coefficient of a given predetermined variable in the forecasting equation represents, in addition to whatever influence that variable may have on the endogenous variable, a mixture of many other influences which usually go together with the predetermined variable concerned. It is, therefore, quite possible that in some cases the collinearity tendencies among the predetermined variables are such that the separate mixtures of influences of the different predetermined variables cannot be ascertained with accuracy.

Degree of aggregation and collinearity

A problem closely related to collinearity is the degree of aggregation in the model. There is, of course, general agreement on the desirability of disaggregating the variables as far as possible. With respect to consumption, for instance, it obviously would be an important gain if separate functions could be obtained for durable goods, nondurable goods, and services, instead of only one function for consumption as a whole. Similarly, it would be very desirable to classify disposable personal income either by types (profits, wages, etc.) or by sizes (below $3,000, etc.). The list of desirable disaggregations can be extended almost indefinitely; but all attempts at disaggregation involve the building of larger and larger models. More and more predetermined variables must be brought into the system. The number of predetermined variables, and hence the size of the model, however, is severely limited by the small number of observations available in the data and the collinearity involved in the variables. An example is found in the introduction of the S variable (i.e., the sum of Federal Reserve holdings of government securities and monetary gold stock) into equation (7), it being recalled that equation (6) has the same set of variables except for the omission of S. It has been seen that the S variable appears with an unreasonable sign and with a very large standard error.31 When there are only some 20 observations, collinearity among the economic variables is such that it would not be possible to develop models with more than four or five predetermined variables.

Collinearity among economic variables usually makes it almost impossible to establish meaningful structural relationships.32 It also tends to make it exceedingly difficult to develop large statistical models—and this is a serious limitation, for the larger the model, the greater would be the probability that it would approximate reality. Even for the variables actually capable of being built into the model, it is often difficult to separate accurately the specific influence of one variable on another from other influences. On the other hand, these are obstacles to accurate, quantitative evaluations against which methods other than econometrics are equally or even more hopeless. Moreover, while collinearity would often hinder a conclusive verification of specific economic hypotheses, it usually makes it possible to forecast the future developments of important economic variables on the basis of a few carefully selected key predetermined variables. While the accuracy of these forecasts obviously depends upon continuation of the same collinearities implied in the model, these forecasts can usually be made with a quite high degree of confidence.33

Real terms vs. money values

The existing models are preponderantly in real terms. The models presented in this paper, however, are composed of variables in current values. There do not seem to be sufficient a priori considerations to justify rejection of the hypothesis of a system in current money values as an alternative to a real system. Take the consumption function as an example. The conventional form is to assume real consumption as a function of, among others, real disposable income.34 Much of the savings of a great majority of people consists of pension and life insurance payments. The former is usually a fixed percentage of money income and the latter is a fixed amount of regular periodic payments, both being in current money terms.

The impression created when variables are measured in current money values, instead of in real terms, is often said to illustrate the importance of the “money illusion.” There is a simple way to test whether in any given situation the money illusion has a significant effect. For example, the following simple consumption function in money terms

may be divided throughout by P, an appropriate price level, to obtain the following equation, in which c and yd are consumption and disposable income in real terms:

If the statistical estimate of b (say, 5.0) has a very large statistical error (say, 3.0), then b may be ruled out as statistically not significantly different from zero.

In other words, the term bP drops from equation (18) and real consumption is a function of real income only. Money illusion, then, does not exist to any noticeable extent.

This simple technique may be applied to equation (10), which is the basis of our forecasts for GNP. The constant term in that equation is 52.48, which has a standard error equal to 14.13; it is therefore a quite significant, positive constant term. There are, however, other variables in the equation which are not expressed in current money values. The result of dividing equation (10) throughout by Q, the GNP deflator, is as follows (gnp, k, and p being the real versions of GNP, K, and P, respectively):

This shows that Q plays an important role in the equation.

Weakness in the predetermined variable K

A second possible criticism of the present attempt is directed against the inclusion of the change in business inventories and net foreign investment in K, a predetermined variable. Imports of goods and services probably can be included in the system as an endogenous variable without much difficulty, but this has not been attempted here. If this were done, net foreign investment would be replaced by the value of exports of goods and services as a part in the predetermined variable K. For a study of the U.S. economy, exports may be reasonably considered as a predetermined variable.

To consider changes in business inventories as a part of a predetermined variable is admittedly untenable. There can be little doubt that fluctuations in business inventories are determined largely by forces generated by the economic system itself. It is believed, however, that inventory fluctuations, more than the fluctuations of any other variable, must be explained by models with a unit period much shorter than a year. Models built on annual data are extremely unlikely to be capable of throwing any light on the problem. The inclusion of changes in inventories in K merely evades the issue; it is a violation of the concept of a predetermined variable, though very little can be done to remedy this defect.

In fact, all the predetermined variables (except GNP-1 which is a lagged variable)35 used in this paper can be criticized on similar grounds. Obviously, government expenditures and tax rates are not determined completely independently of the working of the economic system; even less so are the unemployment payments included in the variable P. If the requirements of the concept of a predetermined variable are strictly enforced, there are probably no variables left that can be considered to be predetermined, with the exception of weather conditions and natural calamities (and hints have already been given of the possibility of man-made rainfall). For instance, neither population nor consumers’ tastes and preferences nor changes in production technique, etc., are truly predetermined variables. In economic studies, therefore, there is relative predeterminateness but not much absolute predeterminateness. With the exception of inventory changes and foreign investment, the predetermined variables used in this paper are probably much nearer the concept of predetermined variables than the endogenous variables included in the system.

II. A Comparison of the Forecasting Equations with “Naive Models”

A comparison of the results of the method used above with those of some other approaches that do not involve elaborate methods and extensive work is of interest. An econometric model can be justified only when its results are better than those that can be obtained by simpler and less costly methods. Two such simple standards of comparison exist, and are often referred to as “naive models” I and II.36 As Professor Friedman has put it, any econometric model implies a theory of change, summarizing the essential forces responsible for the changes.37 The theory implied in the econometric model may be tested against the following two alternatives. Naive model I, essentially a stationary hypothesis, says that the value of a variable next year will be the same as the value this year. With allowance for a particular kind of secular change, naive model II assumes that the change in the value of a variable from this year to the next will be in the same direction and by the same amount as that from last year to this year. Thus, if the actual figures of a variable are those given in the first column of Table 9, the computed values of naive models I and II are those given in the second and third columns, respectively.

Table 9.Derivations of Observations for the Naive Models
YearActual ValuesComputed Values
Naive Model INaive Model II
1941110
1942116110
1943124116116 + (116 - 110) = 122
1944121124124 + (124 - 116) = 132

The residuals obtained from equations (10), (11), and (12) are compared with their respective naive models in Tables 10A, 10B, and 10C. Since there are 24 observations, there are 23 possible comparisons with naive model I and 22 with naive model II. The number of years in which the naive models are superior (i.e., the residuals produced by them are smaller) is as follows:

Naive Model INaive Model II
For GNP55
For Consumption18
For Investment77
Table 10A.Comparison of Equation (10) and Naive Models I and II
Residuals from
YearActual GNPEquation (10)Naive Model INaive Model II
1929103.88.9
193090.9−4.1−12.9
193175.9−5.7−15.0−2.1
193258.3−1.9−17.6−2.6
193355.83.1−2.515.1
193464.94.19.111.6
193572.2−3.27.3−1.8
193682.5−0.110.33.0
193790.2−3.27.7−2.6
193884.7−6.8−5.5−13.2
193991.33.06.612.1
1940101.45.210.13.5
1941126.49.725.014.9
1942161.6−8.735.210.2
1943194.3−0.932.7−2.5
1944213.75.519.4−13.3
1945215.2−4.21.5−17.9
1946211.1−14.5−4.1−5.6
1947233.39.522.226.3
1948259.00.325.73.5
1949258.2−3.6−0.8−26.5
1950286.22.628.028.8
1951329.81.943.615.6
1952348.03.118.2−25.4
Table 10B.Comparison of Equation (11) and Naive Models I and II
Residuals from
YearActual ConsumptionEquation (11)Naive Model INaive Model II
192978.87.1
193070.8−2.8−8.0
193161.2−3.5−9.6−1.6
193249.2−2.2−12.0−2.4
193346.30.9−2.99.1
193451.92.95.68.5
193556.2−2.64.3−1.3
193662.5−0.26.32.0
193767.1−3.04.6−1.7
193864.5−5.3−2.6−7.2
193967.52.13.05.6
194072.12.94.61.6
194182.35.810.25.6
194291.2−3.48.9−1.3
1943102.2−1.011.02.1
1944111.63.29.4−1.6
1945123.1−1.611.52.1
1946146.9−7.523.812.3
1947165.68.918.7−5.1
1948177.90.512.3−6.4
1949180.6−0.82.7−9.6
1950194.60.214.011.3
1951208.1−0.513.5−0.5
1952218.10.010.0−3.5
Table 10C.Comparison of Equation (12) and Naive Models I and II
Residuals from
YearActual InvestmentEquation (12)Naive Model INaive Model II
192914.21.8
193010.5−1.4−3.7
19316.8−2.1−3.70.0
19323.50.4−3.30.4
19332.92.1−0.62.7
19343.91.31.01.6
19355.3−0.61.40.4
19367.3−0.12.00.6
19379.1−0.41.8−0.2
19387.3−1.6−1.8−3.6
19399.51.12.24.0
194011.72.42.20.0
194114.53.92.80.6
19428.9−5.2−5.6−8.4
19436.60.1−2.33.3
19448.52.31.94.2
194511.4−2.62.91.0
194622.6−7.011.28.3
194731.00.58.4−2.8
194837.6−0.26.6−1.8
194935.9−2.9−1.7−8.3
195045.02.99.110.8
195147.72.32.7−6.4
195248.83.01.1−1.6

While the naive models, especially model II, are superior in a substantial number of years, the mean square errors of equations (10), (11), and (12) are a great deal smaller than those of the respective naive models, as shown in Table 11. The result of the comparison is therefore relatively favorable to the forecasting scheme proposed here. In general, the method proposed in this paper may be regarded as a far more reliable forecasting device than either of the two naive models.

Table 11.Mean Square Errors
VariableForecasting EquationsNaive Model INaive Model II
GNP33.89364.51214.12
Consumption14.03110.1734.22
Investment6.7119.6520.41
III. Reduced-Form Solutions and Reduced-Form Least Squares Equations: A Preliminary Discussion

In this paper, “least squares reduced-form equations”38 have been used for making forecasts. This method is quite different from that recommended in the current literature, where forecasts are based on the corresponding39 “reduced-form solutions.”40 For a system consisting partly or entirely of “overidentified” structural relationships, the two methods do not give identical results. The reasons for preferring least squares reduced-form equations may be briefly summarized as follows.

The joint probability distribution of the current values of the endogenous variables, given the predetermined variables, can always be expressed in such a way that the parameters in the distribution are the coefficients, and also the elements of the covariance matrix of the disturbances, in the reduced-form functions. The likelihood function of the sample can therefore be expressed as a function of these parameters.41 For an overidentified system, the coefficients in the reduced-form functions are estimated by the procedure recommended in the current literature at those values which maximize this likelihood function, subject to the so-called a priori restrictions (economic hypotheses) implied in the forms of the assumed structural relations.42 In other words, the maximization process is used to attain a restricted maximum (and, for an overidentified system, lower than the absolute or unrestricted maximum) in such a manner that a “backward solution” of the coefficients so obtained for the reduced-form functions will yield unique estimates for the overidentified coefficients in the assumed structural relations. The reduced-form functions thus obtained are what have been referred to above as the “reduced-form solutions” of the identified system. Since the maximum reached is restricted, the probability of obtaining the sample actually observed is smaller if the true values of the coefficients in the reduced-form functions are equal to those in the reduced-form solutions so obtained than if they are equal to those in what has been referred to as the “least squares reduced-form equations.”

The maximum likelihood principle is “compromised” in the reduced-form solutions approach43 on the ground that the so-called a priori restrictions (i.e., the imposed condition that certain variables are absent from a given structural equation) have to be satisfied. These a priori restrictions are really mostly oversimplifications of economic reality. The complexity of the modern economy makes it certain that there will always be many variables (of both the jointly dependent and the predetermined type), besides those actually included, which have an influence on the variable being explained44 in any of the overidentified statistical equations in existence. The realization of this important fact leads to the following conclusions. First, all structural equations are likely to be underidentified (rather than over- or exactly identified). One may start with any overidentified statistical equation in existence (say, any of the estimated U.S. consumption functions or investment functions), and then include in it variables that were absent from it but are clearly relevant.45 The inclusion of these variables (whether of the jointly dependent or of the predetermined type, and whether already included elsewhere in the model or not) will soon make the equation an underidentified one. Second, the a priori restrictions do not constitute any constraint on the likelihood function of the sample, either in an underidentified or in an exactly identified structural equation. The least squares reduced-form equations are thus identical with the corresponding reduced-form solutions in the underidentified, as well as in the exactly identified, cases.46 Since all structural relations are likely to be underidentified, the least squares reduced-form equations should, therefore, always be used as the forecasting equations.47

It has been explained above (pp. 436–38) that some structural estimation is still required for the purpose of selecting the predetermined variables.48 Since this “exploratory” estimation would include only the small number of endogenous variables in whose future magnitudes we are interested, the “exploratory” structural system would usually consist of overidentified structural relationships. After the predetermined variables have been so decided upon and the least squares reduced-form equations have been established, it is possible to go one step further and to throw a little more light on the structural relationships than the overidentified (hence, oversimplified) exploratory model has been able to provide. For the same set of predetermined variables, an underidentified structural system will have the same reduced forms as an exactly identified system. Starting with the overidentified exploratory model, additional endogenous variables, which, next to those already included in the equation, are believed to be the most relevant and important, may be introduced into any given structural equation, until the equation is exactly identified.49 It is understood, of course, that on account of the working of collinearity tendencies the structural coefficients so obtained also possibly represent the influences of many other variables whose movements are more or less similar. These estimated structural relationships, however, have revealed the identities (if not the real magnitudes of their influence) of a number of variables that had a bearing on the dependent variables concerned, the number being the largest that the available data are capable of revealing.

Mr. Liu, economist in the Special Studies Division of the Research Department, is a graduate of National Chiao-Tung University, and received graduate training at Cornell University. He was formerly Assistant Commercial Counselor, Chinese Embassy, Washington, D.C., and Professor of Economics, National Tsing-Hua University, Peiping. He is the author of National Income of China, 1931–36 (The Brookings Institution, 1946) and of several articles in economic journals.

The preparation of this paper was completed in November 1953, before the gross national product and related figures for that year, upon which many of the forecasts in this paper are based, were released by the U.S. Department of Commerce. The figures in Table 2 are therefore forecasts for 1953 obtained from the forecasting equations (9)-(12) on the basis of estimated values of the predetermined variables. The actual values of the predetermined variables became available early in 1954. See Table 2, footnote 1.

The standard errors of the estimated declines in GNP for various reductions in government purchases and taxes range from 10 to 20 per cent of the respective estimates.

For a discussion of the methodological issues, see Appendix III.

The equations used in this method are the solutions for the endogenous variables obtained by solving the structural equations. These equations may be called the “reduced-form solutions of the structural system.”

That is, the structural coefficients cannot be uniquely determined from the available data.

For a detailed discussion of this point, see Appendix I.

These forecasting equations may be called “least squares reduced-form equations.” It can be shown that the least squares reduced-form equations for an underidentified as well as for an exactly identified system are identical with the maximum likelihood estimates of the reduced-form solutions referred to in footnote 4. In these cases, the so-called a priori restrictions do not constitute any constraint on the likelihood function. In the overidentified cases, however, the least squares reduced-form equations are not identical with the maximum likelihood estimates of the reduced-form solutions under the constraint of the a priori restrictions.

This is for the purpose of limiting the “exploratory’’ model to a manageable size. As a result, the exploratory structural relationships tend to be overidentified. The “true” structural relationships are likely to be underidentified.

While the estimated structural coefficients do not actually represent the true influences, it is nevertheless true that insignificant structural coefficients or coefficients with inappropriate signs in the structural equations would mean either that the predetermined variable concerned failed to play the theoretical role expected of it, or that it is “inferior” to others in catching the collinearity tendencies.

The first two components of this item do not properly fulfill the requirements of the concept of a predetermined variable. See p. 461.

It is now well known that, for a simultaneous system of economic relationships, the regression coefficients obtained by fitting the equations separately by the least squares procedure do not yield “consistent” estimates of the parameters. The Cowles Commission method is designed to remedy this shortcoming. The technique of estimation is quite simple if the structural equations are exactly identified. All the functional relationships given on page 440, however, are over-identified. These equations are estimated by the single-equation, limited-information, maximum likelihood methods originally worked out by T. W. Anderson and Herman Rubin (see the two articles by these authors, “Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations” and “The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of Stochastic Equations,” Annals of Mathematical Statistics, March 1949 and December 1950, respectively). The computation procedure used is the one worked out by Jean Bronfenbronner and Herman Chernoff in “Computational Methods Used in Limited Information Treatment of a Set of Linear Stochastic Difference Equations,” Cowles Commission Discussion Papers: Statistics: No. 328. A detailed discussion of this method is also given in L. R. Klein, A Textbook for Econometrics (New York, 1953), and William C. Hood and T. C. Koopmans, eds., Studies in Econometric Method (New York and London, 1953).

While the S variable is present only in equation (4b), it is in the nature of the Cowles Commission method of simultaneous estimation that all equations in system B are different from those in system A.

As defined above, M is the total privately held cash balance, whereas in a consumption function it is only the part of M held by households that is actually relevant as an explanatory variable. For forecasting purposes, however, data on the relevant part need to be available on a current basis; this is unfortunately not possible for the relevant part of M. Hence the total of M is used. A study of past data shows that the total and the relevant part moved very closely together.

It might be thought that a variable such as (DGNP)2 would be preferable to D2, as the proportion of defense output in the total product is a better indication of the strain on the economy than the absolute magnitude of defense expenditures. This, however, would be an unnecessary refinement. Since we have Yd (disposable income which varied closely with GNP and “acts for” GNP in this connection) in equations (1a) and (1b), and Yd and D2 enter into these equations, respectively, positively and negatively, the idea of the ratio is already implied in the equations.

Well-defined relationships have been found to exist between t and (1) personal income tax rates for single persons with net income of $3,000 and no dependents, (2) personal income tax rates for single persons with net income of $10,000, and (3) the ratios of revenue from total corporation income tax (including excess profits tax) to total net income of corporations having a net income, the ratios being roughly adjusted to correct for the changes in the distribution of corporation net income among different income classes.

To include both demand and supply factors in one structural relationship is of course unsatisfactory; the money supply function is therefore more in the nature of an empirical equation than a structural equation. However, no great harm is done, as S is omitted in the subsequent analysis.

The reason for the wrong sign of the S term is, of course, that the nature of the available data is such that the influence of the S variable on GNP cannot be observed (see page 459 and footnote 33).

As indicated by the ratios of the mean square successive difference to the variance δ2S2, given under the equations.

(368.6 - 348.0) 0.39 = 8.0. (364.8 - 348.0) 0.39 = 6.6. See Table 2.

There are many conceivable ways of indicating this degree of stress and strain. For instance, an index of the portion of the labor force devoted to national security purposes (including those producing defense supplies) may be used. The variable D would then be replaced by this index in equation (10), and the multiplier obtainable from the equation would be for government purchases as a whole. This example is given to illustrate the nature of the role played by the D variable. It also indicates that D is a very convenient variable to use for this purpose, as other alternatives (such as the index of the portion of labor force devoted to national security purposes discussed above) would be very difficult to construct.

By use of the following formula derived from equation (10) the multiplier for government purchases of goods and services as a whole may be calculated (G being the amount of government purchases included in K):

The magnitude of this multiplier is relatively insensitive to variations in the ratio of security purchases to other purchases. This can best be illustrated with a concrete example. From 1951 to 1952, total government purchases increased by $14.6 billion. Of this amount, $11.4 billion (or about 78 per cent) was accounted for by increases in security purchases (from $37.0 billion in 1951 to $48.4 billion in 1952). The multiplier of government purchases is

If, however, only 60 per cent of the total increase in government purchases had been on account of security purchases, security expenditure would have risen from $37.0 billion to $37.0 billion + 0.60 ($14.6 billion) = $45.8 billion, with

The change in the magnitude of the multiplier, brought about by this change in the ratio of security purchases to government purchases as a whole, is only about 0.15, or 10 per cent of the original magnitude.

It has been shown, however, that the effects calculated from equation (10) are not very sensitive to variations in these shares within reasonable limits. See footnote 22.

See the definition of K, p. 438.

It is, of course, also of interest to speculate on the effects of tax reductions alone, without a simultaneous change in government purchases. The standard error of the regression coefficient of the t variable in equation (10) is so large, however, relative to the coefficient itself, that an estimate of the separate effect of a reduction in taxes alone cannot be made with any degree of confidence. This standard error is large because of the high degree of collinearity tendency involved in the historical data on taxes and government purchases. In spite of this difficulty, however, the effects of a simultaneous reduction in government purchases and taxes can be estimated with reasonable accuracy, because the co-variances between D2, K, and t coefficients are all negative. This is the reason why the standard errors of the estimated effects of simultaneous reductions in taxes and government purchases (given in Table 6) are relatively small.

See page 442.

See the following papers: T. Haavelmo, “Multiplier Effects of a Balanced Budget,” Econometrica, Vol. 13 (1945), pp. 311–18; G. Haberler, “Multiplier Effects of a Balanced Budget: Some Monetary Implications of Mr. Haavelmo’s Paper,” Econometrica, Vol. 14 (1946), pp. 148–49; R. M. Goodwin, “Multiplier Effects of a Balanced Budget: The Implication of a Lag for Mr. Haavelmo’s Analysis,” Econometrica, Vol. 14 (1946), pp. 150–51; Everette E. Hagen, “Multiplier Effects of a Balanced Budget: Further Analysis,” Econometrica, Vol. 14 (1946), pp. 152–55; T. Haavelmo, “Multiplier Effects of a Balanced Budget: Reply,” Econometrica, Vol. 14 (1946), pp. 156–58.

As measured by the part of the “variance” of the variable in question explained by the equation. This is equal to the square of the correlation coefficients: (0.99)2 = 0.98.

In fact, it is already quite difficult to separate the influences of GNP and GNP-1 even without the profit terms. See equations (2a) and (2b).

The least squares reduced-form equations, each representing a direct least squares regression of an endogenous variable on all predetermined variables.

Another way of looking at the result is that the correlation coefficient of equation (7) is not increased noticeably by the introduction of the S variable, compared with that of equation (6). The S variable has not brought with it any useful information that is not already supplied by the other predetermined variables.

This is in addition to the difficulty of “underidentification”; see pages 436–37 and Appendix III.

The S variable (sum of Federal Reserve holdings of government securities and monetary gold stock) in connection with equations (6) and (7) may again serve as an example. The inclusion of S in equation (7) did not noticeably increase the correlation coefficient, compared with that of equation (6), and the standard error of the S coefficient is large relative to the coefficient itself. This, of course, means that S is so highly correlated with the other five predetermined variables in the equation that its influence on GNP cannot be clearly segregated from the effects of the other five variables. In fact, if S is correlated with the other five variables as explanatory variables, a correlation coefficient in the neighborhood of 0.99 is likely. For forecasting purposes, therefore, the result obtained from equation (6) will be satisfactory (insofar as the effects of S are concerned) if in the future the same collinearity relationship holds between S and the other five variables. The forecast, of course, will not be as good if S moves in such a way that it is out of step with the other variables. It should be noted, however, that the collinearity relationship between S and the other variables implied in equation (7) is based on the experience of 23 years.

This real form is often claimed to have a firm foundation in micro-economic theory. Actually, one usually has to make quite a number of heroic efforts in trying to connect a crude, aggregate function of any kind with the familiar kind of micro-economic theory, containing almost equally crude assumptions different from (and sometimes contradictory to) those underlying the aggregate functions. Even if the gaps between macro- and micro-theories can somehow be assumed as closed, it can easily be shown that the kind of consumption function that would correspond with the maximization of utility functions under budget restraints would be one which has real consumption as a function of, among other things, money (not real) income and prices. A function of real consumption in terms of real income would seem to deviate more from the traditional micro-economic theory than a function of money consumption in terms of money income, since the former leaves no possibility for the price level to play any role in connection with consumption.

Even the use of a lagged endogenous variable as a predetermined variable would weaken certain desirable statistical properties of the least squares estimates.

See Carl Christ, “A Test of an Econometric Model for the United States, 1921–47,” Cowles Commission Papers, New Series, No. 49 (1952), pp. 56–58.

Discussion by Milton Friedman of Christ’s paper, ibid., pp. 109–11.

See footnote 7.

Corresponding in the sense of having the same set of predetermined variables.

See footnote 4. This position is clearly stated by L. R. Klein, op. cit., especially pp. 251–55. See also Carl Christ, op. cit., p. 44; W. C. Hood and T. C. Koopmans, eds., op. cit., especially pp. 176–77; and the many discussions of the bearing of a priori restrictions on the problem of estimation in T. C. Koopmans, ed., Statistical Inference in Dynamic Economic Models (New York, 1950).

See, for instance, equation (5.6) in T. W. Anderson and Herman Rubin, “Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations,” op. cit., p. 53.

The general nature of the procedure is the same, whether the full-information or the limited-information maximum likelihood method is used.

The “compromise” occurs only in the overidentified cases, as the reduced-form solutions and the least squares reduced-form equations are otherwise identical. Practically, however, all the statistically estimated structural equations in existence are overidentified. Underidentified cases are seldom discussed in the literature, although they are in fact the only important cases.

Whether, on account of the collinearity tendencies, these influences are statistically discernible or not is irrelevant to the point at issue.

For instance, any variable which has been found by some other investigator to be a significant explanatory variable for the dependent variable which is under examination. One could always, in this manner, easily compile a long list of “verified” explanatory variables for any given problem.

This is true, even though the structural coefficients in the underidentified cases cannot actually be uniquely determined.

The conclusion that all structural equations are likely to be underidentified can be established by an interesting application of the maximum likelihood principle. The least squares reduced-form equations always explain the movement, during the sample period as a whole, better than the corresponding reduced-form solutions derived from an overidentified system. This is, of course, inherent in the least squares method. Since the least squares reduced-form equations are identical with the corresponding reduced-form solutions only when the structural equations are exactly identified or underidentified, and since a structural relation is unlikely to be exactly identified unless by accident, we conclude that it is most probable that the structural equations are underidentified.

It should be intuitively clear that there is no basis for the belief that the reduced-form solutions derived from an overidentified system, which during the sample period do not “forecast” so well as the corresponding least squares reduced-form equations, would give better results for any future period.

The process of selection naturally implies experiments with different predetermined variables. The view has often been expressed that the significance of a relationship, established only after experimental selection of the explanatory variables, cannot be judged by the usual standards of test of significance. This view has been criticized in a note by S. C. Tsiang, “Experimental Selection of Explanatory Variables and the Significance of Correlation,” Econometrica, Vol. 23 (1955), pp. 330–31. Even if it were accepted, it would not, however, mean that experimental selection of variables is not the best way to develop a forecasting equation; it merely rejects the application of the usual tests of significance to the result of such selections.

The reduced-form solutions of such a system of exactly identified structural relations are the same as the least squares reduced-form equations obtained.

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