COWLES FOUNDATION For Research in Economics at Yale University Studies in
The Cowles Foundation for Research in Economics at Yale Uni 'l:et·sity, established as an activity of the Department of Economics Econometric Method in 1955, has as its purpose the conduct and encouragement of research in economics, finance, commet·ce, industry, and tech nology, including problems of the organization of these activities. The Cowles Foundation seeks to foster the development of logical, mathematical, and statistical methods of analysis for application Edited by in economics and t·elated social sciences. The professional research Wm. C. Hood and staff are, as a rule, faculty membe1·s with appointments and teach Tjalling C. Koopmans ing responsibilities in the Department of Economics and other departments.
~~~3) The Cowles Foundation continues the work of the Cowles Com ( mission for Research in Economics founded in 1932 by Alfred Cowles at Colorado Springs, Colorado. The Commission moved to Chicago in 1939 and was ajjUiated with the University of Chicago until 1955. In 1955 the professional 1·esea1·ch staff of the Com mission accepted appointments at Yale and, along with other members of the Yale Department of Economics, formed the ?'e seat·ch staff of the newly established Cowles Foundation.
A list of Cowles Foundation Monogmphs appears at the end of this volume. r:. -,··-.
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New Haven and London, Yale University Press CHAPTER I
ECONOMIC MEASUREMENTS FOR POLICY AND PREDICTION
BY .JACOB MARSCHAK
Page 1. Useful Knowledge ...... 1 2. Structure...... 3 3. Maintained Structure and Change of Structure ... . 4 4. Controlled and Uncontrolled Changes ...... 8 5. Some Definitions Extended ...... 8 6. The Technician and the Policy-Maker .. 10 7. Random Shocks and Errors ...... 12 8. The Need for Structural Estimation ...... 15 9. The Time Path of Economic Variables; Dynamic Structures...... 17 10. "Steering Wheel" and Automatisms...... 24 11. Mathematics and Prediction...... 25 12. Conclusion...... 26
1. USEFUL KNOWLEDGE \..Knowledge is useful if it helps to make the best decisions. To illustrate useful knowledge we shall take an example from the century-old elementary economics of the firm and of taxation. Such examples are admittedly crude (or, if the reader prefers, neat) compared with the complex actual world since their very purpose is to isolate the essentials of a problem by "idealizing reality." Later sections (beginning with Section 5) will deal with ways of eliminating at least some of the legitimate realistic objections. What kinds of knowledge are useful (A) to guide a monopolistic firm in its choice of the most profitable output level and (B) to guide the government in its choice of the rate of excise tax on the firm's product? Let q represent quantity produced and sold per unit of time; p, price including tax; e, tax per unit of product; 'Y, total cost of producing and selling q units. To fix ideas, suppose that the demand for the product of the firm is known to be (approximately) a linear function of the price and that all costs are known to consist of fixed charges. (This is almost the case with hydroelectric plants.) Write for the demand curve
(1) p =a- {3q ({3 > 0). The firm's profit (net revenue) per unit of time is
(2) r = (p - e)q - 'Y, SEC. 2] MEASUREMENTSFOR POLICY AND PREDICTION 3 2 JACOB MARSCHAK [CHAP.1 interested in high output, it will choose the smaller of the two real roots, or, using (1), say 01 < 02 • If T* exceeds a certain level To , the roots will be not real 2 (3) r = (a - 0 - {Jq)q - 'Y = -{Jq + (a - O)q - 'Y· (i.e., a tax revenue T* > To is unattainable). We thus conclude that if the government knows a and {3 it can choose the best value of 0 for any CASE A: If the firm knows a, {3, and 0, it can use equation (3) to com desired and attainable level T* of tax revenue. pute the difference between the profits that would be attained at any two alternative output levels. To choose the most profitable output of We can sum up as follows: all, it therefore suffices to know a, {3, and 0. It happens in our example, CAsE A: Desired: maximum r. Decision variable: q. Useful knowledge: as in most discussions of classical economies, that the functions inn>lved the form of relations (1) and (2) and the values of the parameters a, are differentiable/ so that the best output level, say q = q, ean be found {3, 0. by putting drI dq = 0. Hence CASE B1. Desired: maximum T. Decision variable: 0. Useful knowledge: the fact that profits are maximized, the form of (1) and (2), and the (4) q = (a - 0)/2{3. value of a.
CASE B2 • Desired: maximum q for given T = T*. Decision variable: CASE B: Assume that the government knows that the firm maximizes 0. Useful knowledge: same as in Case B 2 , plus the knowledge of {3. its profit. What other knowledge is useful to the government? This de pends on its aims: 2. STRUCTURE
CASE B1 : Suppose, first, that the government, which collects from the In all of our examples so far, useful knowledge pertains to certain firm the ta.x revenue T, economic relations. In Case A the firm has to know something about relations (1) and (2). Relation (1), the demand equation, describes the (5) T = Oq, behavior of buyers. The form and the coefficients (a, {3) of this relation wants to maximize this revenue by the proper choice of the excise-tax depend on social and psychological facts, such as the frequency distri rate 8. Then, by equations (4) and (5), bution of consumers by tastes, family size, income, etc. Relation (2), the profit equation, registers the institutional fact that the tax rate is (6) T = O(a - 0)/"2 :i. fixed at 0, and the fact (reflecting the technology of the firm as well as the price and durability of its plant and the interests and rents stipulated Therefore, if the government knows a, it can compute the ratio between in its contracts) that the total cost consists of given fixed charges, 'Y· the tax revenues resulting from fixing any two alternative excise rates. With respect to the decision problem of Case A, relations (1) and (2) This ratio is independent of {3. Hence, to make the best decision (i.e., are called structural relations and are said to constitute the structure; to choose the value of 0 that will bring in the highest tax revenue) it they involve constants (a, {3, 0, 'Y) called structural parameters. In Case is sufficient for the government to know a. In fact, the best value of B the assumed structure includes, in addition to (1) and (2), the as 8 is b = a./2. sumption of profit maximization, which results in relation (4); and defi nition (5) may also be counted as part of the structure. If (1) or (2) or CASE ~: Suppose, on the other hand, that the government wants to both had included a definite pattern of change-say, a linear trend goad the monopolist into maximum production, provided that a fixed this would also be a part of the structure. tax revenue T = T* can be collected. The best tax rate is found by solv In each of the problems studied the form of the structural relations ing the (quadratic) equation (6) for 0 with T* substituted for 1'. The and the values of some (not necessarily all) of their parameters prove to equation will have two real roots, say 8 and 8 (which, in a limiting case, 1 2 constitute useful knowledge. However, we shall presently see that under may coincide), provided that T* is not too large. Since, by equation (4), certain conditions other kinds of knowledge, possibly more easily at q is larger the smaller 8 is, and since the government was assumed to be tained, are sufficient to make the choice of the best decision possible. 1 But see Section 5. 4 JACOB MARSCHAK (CHAP.I SEC. 3] MEASUREMENTSFOR POLICY AND PREDICTION 5
3. MAINTAINED STRUCTUREAND CHANGE OF STRUCTURE predicting r) is a quadratic equation,
We shall show that the knowledge of structure is not necessary if the (7) r = Xl + M + "' structure is not expected to have changed by the time the decision takes its effect.2 Again consider Case A. Assume that the form of the struc say, whose coefficients are related to the coefficients of the structural equations as follows: tural relations (1) and (2) and the values of coefficients a, {3, 'Yare known to have been unchanged in the past and to continue unchanged in the (8) X = -{3, IJ. =a- 0, "= -"(. future, and make three alternative assumptions about the tax rate 0: If the structural relations (I) and (2) are assumed to retain in the fu ture the same (linear) form and the same values of parameters as in the CASE A': 0 has not changed in the past and is not expected to change. observed past, the firm can predict r for a given q by fitting a quadratic equation (7) to past observations on output and profit. It can thus CAsE A": 0 has not changed in the past but is expected to change in determine empirically the parameters X, IJ., " of the reduced form with a known way. out having to pay any attention to the manner [described by equations (8)] in which these parameters are related to the demand and cost con CASE A"': 0 has changed in the past. ditions. In fact, as already mentioned, the firm may display an even stronger disregard for "theory." If the number of observations is large Suppose that in the firm's past experience, of which it has records, it while the firm's confidence in the linearity of the relations (I) and (2) bad tried out varying levels of output q and obtained varying profits r. and hence in the quadratic nature of (7) is small, it may prefer to rely In Case A' it can tabulate the observations of q and r in the form of a altogether on some purely empirical fit. schedule, or fit an empirical curve, and use the table or the curve to Case A" is different. Although the same schedule as in Case A' will predict future profit r for any given output q. It can therefore choose its describe the past relation between output and profit, this schedule will most profitable output without knowing any of the structural parame not help in choosing the most profitable output under the new tax rate. ters a, {3, 'Y, 0. If the firm could conduct a series of experiments under the new tax True, knowledge of the form (not the parameters) of relation (3) may rate, varying the outputs and observing the profits, it could discard the help in filling the gaps in the empirical schedule (if the observations are old schedule and construct a new one to be used in decision-making. few) by suggesting that a quadratic rather than some other relation be But such experiments take time.4 In our case these experiments are not q. fitted to the data on rand Remember that output q was assumed to ' Strictly speaking, if the form of the new schedule is known, one needs only be controlled by the firm independently of any other variables and to as many observations as there are unknown parameters of the schedule. Thus, determine, for given values of the structural parameters (a, {3, 'Y, 0), both three observations, and therefore a delay of three accounting periods, will suffice the profit rand the price p. Accordingly, rand pare said to be "jointly to determine the new quadratic schedule that replaces (7) when the tax rate is dependent" on q, an "independent" variable. Independent variables are changed. If the form of the new schedule is not known, the output that results in maximum profit under the changed schedule can be found by trial and error, the also called "exogenous" ("autonomous," "external"); and the jointly 3 number of necessary trials depending on the firm's skill in hittin~from the begin· dependent variables, "endogenous" ("induced," "internal"). There are ning an output level near the optimal one and in varying the output level by as many jointly dependent variables as there are structural relations amounts not too large and not too small. This skill is equivalent to some approxi in our case, two. Solving the structural relations (1) and (2) for the two mate knowledge of the properties of the new schedule-equation (9) of the text jointly dependent variables we obtain the "reduced form" of the system: in the neighborhood of the optimal point and is therefore enhanced if the firm has approximately the kind of knowledge to be discussed presently (viz., some two relations predicting, respectively, p and r from q. In our case the knowledge of old structural relations and of the change they have undergone). relation predicting p happens to coincide with one of the structural rela However, the full significance of the delay that occurs when, without knowing lations [viz., (I)]. The other equation of the reduced form (viz., the one the structure, one estimates empirically a new reduced-form schedule (such as the relation between the dependent variable r and the independent variable q after 2 See Chapter II, Section 8, of this volume and Hurwicz [1950b). the tax rate 9 has changed) cannot be gauged by the reader as long n.s we deal with • A slight change in definition will be convenient later, when dynamic systems the artificial assumption of exact economic relations such as constitute the usual with lagged endo~~;enousvariables are introduced. See Section 9. economic theory. When, beginning with Section 7, random disturbances of rela- 6 JACOB MARSCHAK (CHAP. I SEC. 3) MEASUREMENTS FOR POLICY AND PREDICTION 7
necessary if the firm knows, in addition to the old observations, the related to q and 0 by an equation of the form form of relations (1) and (2) and both the old and the new tax rates, say 0 and 0*. Then the old schedule will be the reduced-form equation (10) r = - Oq + A.l + 1rq + P, (7). The firm obtains the coefficients of (7) empirically from old observa whose parameters are related to the structural parameters as follows: tions. It knows them to be related to the structural parameters, by (11) equations (8). Under the new tax rate O* the coefficient I" will be replaced A = {3, 7r =a, Jl = --y. by I"* = a - O*, while A and P will not be affected. Hence the new rela If the firm has confidence in the form of the structural equations (1) and tion between profits and outputs will be (2), it will be helped by the knowledge that equation (10) involves a 2 2 product term (- Oq) in the two exogenous variables and a term (A.q) (9) r A.q (!" 0 - O*) q P. = + + + quadratic in q. Thus, Case A"' is analogous to A' except that the re The new schedule can thus be obtained by the firm from the old one by duced form now involves two exogenous variables (q, 0) instead of one (q). inserting the known tax change in a well-defined way. We see that, in the case of a foreseen change in structure, the purely Suppose, however, that a change in the social and psychological con empirical projection of observed past regularities into the future cannot ditions is expected to change the demand equation (1). Suppose, for be used in decision-making. But knowledge of past regularities becomes example, that the slope of the demand curve, which had maintained a useful if supplemented by some knowledge (not necessarily complete constant value {3 during the past observations, is expected to obtain a knowledge) of the past structure and of the way it is expected to change. new value, {3*, while the tax rate 0 and the output q had both under In our case we can replace the old, empirically obtained schedule (7) gone observed variations during the observation period. With the de by the new, not observed schedule (9) if we know (a) the mathematical mand curve thus changed, the coefficient A.in equations (10) and (11) form (viz., quadratic) of these schedules and the role played in them will be replaced by A.*= A + (/3 - {3*). Therefore, the old reduced-form by the tax rate [thls knowledge is derived from the knowledge of the equation (10) cannot be used to predict profits r from given values of form (not the coefficients) of the structural relations (1) and (2)], and tax rate 8 and output q and to decide upon the best output level q un less one knows, in addition, the amount by which the demand param (b) the amount of change of tax rate, O* - 0. Having thus obtained eter {3 is going change. This case is analogous to Case A", with {3 now (9), and ma.ximizing r, we can determine the best output, q = q. In to terms of the tax change and of the coefficients of the old, empirical profit playing the role that was played in Case A" by 0, while q and 0 play schedule (8), the role previously played by q alone. To sum up: (a) for purposes of decision-making it is always necessary q = (O* - 0 - !J)/2A.. to know past and future values of all exogenous variables (i.e., of vari ables that determine the outcome in question and that were observed We now come to Case A"', in whlch the tax rate 0 was observed to to change in the past); (b) if conditions that have not changed in the vary independently in the past, 0 being similar in this respect to the past are expected to change in the future, some knowledge of such con output q. In thls case, both q and 0 are exogenous variables, while a, {3, ditions (called "structure") and of the nature of their change is necessary 'Y are, as before, structural parameters and r is endogenous. From past for decision-making. observations on q, 8, and r, the firm can derive a double-entry table The choice of the best decision presupposes that two or more alterna or fit an empirical surface to predict the profit r for any specified output tive future values are tentatively assigned to a decision variable. If the q and tax rate 0. As in Case A', it is not necessary to know the structural decision variable has varied in the past, it is called an exogenous variable; parameters, although knowledge of the form of the structural relations if it has not, it is usually called a structural parameter. In Cases A', helps to interpolate gaps in the empirical table. Specifically, profit r is A", and A"', q, an exogenous variable, was such a decision variable. In Case B of Section 1 the tax rate 0 was a decision variable, the govern tions and errors in the measurement of variables are introduced, the time ment being the decision-maker. If 8 has varied in the past, and is thus saving aspect of the knowledge of structural relations will appear in a more realis an exogenous variable, the government has to know these variations in tic light. See Section 8. order to choose the best decision on the basis of past relations between SEC. 5) MEASUREMENTS FOR POLICY AND PREDICTION 9 8 JACOB MARSCHAK (CHAP. I
8 and the quantity that it tries to maximize. If 8 bas not varied in the tinuous) variable. Note, however, that in Case B2 the choice had to be past (for example, if 8 was zero) and the government now tries to fix it made between only two values (81 and 02). In every case the decision at its best value, a structural change is planned. To determine the effect maker compares the outcome of alternative decisions, and these may or of such a change the government has to know something about the past may not form a continuous set. It is obviously not essential whether the structure. This knowledge may require more than the knowledge of the alternatives are identified as quantities (as in the examples of the pre past tax rate itself. For example, it is seen from equation (6) that if the vious sections), or by city names (as in the case of location choice), or tax is to be introduced for the first time, the choice of the tax rate that by the words "yes" or "no" (as in the choice between maintaining and will maximize the tax revenue will require knowledge of a, a parameter abolishing rent control). In every case the choice goes to the decision promises the best outcome. of the demand equation. that The extension applies, in fact, to all the variables (including the struc tural parameters), which we had previously introduced as continuous 4. CoNTROLLED AND UNCONTROLLED CHANGES quantities. It has been claimed, for example, that in the interwar period We have noted that a decision variable can be either a structural businessmen's willingness to invest in plant and equipment depended, parameter or an exogenous variable. Structural parameters and exoge other things being equal, on whether the national administration hap nous variables that are decision variables can be called "controlled" pened to be Democratic or Republican. Should an economist take this variables, as distinct from "uncontrolled" variables (both exogenous hypothesis seriously, there is nothing against his regarding the party and endogenous) and parameters. For example, the legally fixed quan label of the administration as a two-valued variable and trying to ex tity 8 is uncontrolled from the point of view of the firm, though con plain certain ''shifts" in the investment schedule as a function of that trolled from the point of view of the government. The psychological and variable. social factors determining a and {3 and the technological and economic Similarly, fluctuations in the supply of a commodity according to the factors determining 'Y were here considered uncontrolled, though a dif four seasons of the year can be conveniently treated by introducing into ferent hypothesis (e.g., involving the effects of an advertising campaign the supply schedule a four-valued exogenous variable called season. designed to change buyers' tastes) might have been discussed instead. This is a more rational approach than the usual mechanical "seasonal
In predicting the effect of its decisions (policies) the government thus ;._'' adjustment" of individual time series, which does not use available !'":.•.. · has to take account of exogenous variables, whether controlled by it . knowledge as to which particular structural relations (such as the tech (the decisions themselves, if they are exogenous variables) or uncontrolled nological supply schedule for crops or buildings or the demand schedule (e.g., weather), and of structural changes, whether controlled by it (the • for winter clothes) are affected by seasons.
!~· decisions themselves, if they change the structure) or uncontrolled (e.g., Finally, consider a structural change that (unlike the changes dis I sudden changes in people's attitudes, in technology, etc.). An analogous I cussed in previous sections) consists, not in changing a certain continu statement would apply to the firm except that, for it, government deci ous parameter, such as the coefficient a of the demand equation (1), sions belong to the category of uncontrolled variables. but in scrapping one equation and replacing it by another. Let the two equations be, respectively, F = 0 and F* = 0, where F and F* are func 5. SoME DEFINITIONS ExTENDED tions involving, in general, several endogenous and exogenous variables and certain parameters. Form the equation oF+ (1 - o)F* = 0, where \ We shall now proceed, as promised in Section 1, to generalize our ois a new structural parameter with the following values: o = 1 before i examples to meet realistic objections. One such objection is that in prac !.. the change, o = 0 afterwards. Then structural change is expressed by a :W 6 tice the decision is frequently qualitative, not quantitative. For ex :~1 change in the value of o. ample, the firm may have to decide in which of a limited number of These examples show that our previous description of structures and :!' eligible locations--each of them near a fuel source, say-it should build decisions in terms of variables (including parameters) is general enough :1:
a plant; the government bas to decide whether to abolish or continue ~: rent control; etc. Such cases look superficially different from Cases A • For example, the introduction of price control, which will be discussed in Section 6, consists in scrapping the equation q• - qd = 0 in (13) and replacing it and B1 , treated in Section 1, where the decision-maker had to choose :~ by the equation p - p = 0. among a large (possibly infinite) number of values of a (possibly con- 10 JACOB MARSCHAK [CHAP. I SEC. 6] MEASUREMENTS FOR POLICY AND PREDICTION 11 if the concepts are properly interpreted. The corresponding generaliza Section 5, that the government ranks the possible results-here the tion of mathematical operations involved is, in principle, feasible. possible pairs of values of T and §-according to its preferences. We Some readers may find it more convenient to give the set of exogenous find that the best value of 8, in this sense, is {) = (a - w)/2. variables and structural parameters a more general name: "conditions." We can imagine a division of labor between the government (or some Similarly, the set of jointly dependent variables can be renamed "re other decision-maker) and the technician. The latter is relieved of the sult." Conditions that undergo changes during the period of observation responsibility of knowing the "utility function" such as (12). The tech correspond to "exogenous variables." Conditions that remain constant nician is merely asked to evaluate the effects of alternative decisions throughout the observation period but may or may not change in the (tax rates 8) separately upon q and T, as in equations (4) and (5). future constitute the "structure." Conditions that can be controlled Clearly, knowledge of the structural coefficients a, 13 is useful for this are called "decisions." Given the conditions, the result is determined. purpose. This knowledge is even necessary if the tax is introduced for I The decision-maker ranks the various achievable results according to the first time (or if a, {3, 8 had all been constant throughout the observed his preferences: some results are more desirable than others. The best past). The technician will thus try to estimate a and {3. The decision I decision consists in fixing controlled conditions so as to obtain the most maker, on the other hand, need not formulate his own utility function I ! desirable of all results consistent with given noncontrolled conditions. U(T, q), say-completely and in advance. It suffices for him to make ! For the economy as a whole, endogenous variables can be roughly the choice only between the particular pairs of values of (T, q) that the identified with what are often called "economic variables." These are technician tells him will result from setting the tax at various considered 1: usually the quantities (stocks or flows) and prices of goods and services, levels. II or their aggregates and averages, such as national income, total invest !i An additional example will illustrate this role of the technician as !! ment, price level, wage level, and so on. The exogenous variables and separated from the decision-maker. The government (or the legislator) the structural parameters are, roughly, "noneconomic variables" (also considers the possibility of guaranteeing some fixed price for a farm l called "data" in the economic literature) and may include the weather product. The technician is asked how many bushels will have to be :J and technological, psychological, and sociological conditions as well as purchased for storage at public expense at any given guaranteed price. :I 'l legal rules and political decisions. But the boundary is movable. Should Suppose that the technician knows the supply and demand functions ! political science ever succeed in explaining political situations (and hence which have so far determined the price in a free market: legislation itself) by economic causes, institutional variables like tax 6 q' = a' + 13'p, rates would have to be counted as endogenous. (13) l =ad- 13ap, 6. THE TECHNICIAN AND THE PoLICY-MAKER q' -l = 0,
Outcomes of alternative decisions are ranked according to their de where q' is the quantity supplied and l is the quantity demanded by .:; sirability by the policy-maker, not by the technician. private people, and where p is the (varying) price at which demand and
Returning to Case B of Section 1, suppose, for example, that the gov supply were equalized in previous years. Under the intended legislation ~I ernment desires both a high tax revenue and a high level of production this system would be replaced by
of the ta:xed commodity. The endogenous variable that is being maxi ]II·~ q' = a' + 13'p, mized is thus neither the tax revenue (as in Case B1) nor the output (as in Case Bt) but a function of the two; for example, this function may be (14) l = ad- 13ap,
• d (12) U = T + wq, q - q = g,
where w, a positive number, indicates the "weight" attached to the pro where q' and qa are, as before, the supply and demand of private people, duction aim relative to the aim of collecting revenue. The statement that and where g is the amount to be purchased by the government when the the government maximizes U is a special case of the statement, made in price is fixed at p. Hence 'i· • See Koopmans [1950c[. (15) g = (a' - ad) ({3' 13d)p. + + J !
'~~i ·' 12 JACOB MARSCHAK [CHAP. I SEC. 7] MBASUREMENTS FOR POLICY AND PREDICTION 13
H the technician can estimate the parameters (a', {t, ad, {f) of the supply diet (viz., the endogenous variables) are therefore random variables. and demand equations, he can tell what alternative pairs of values of Prediction consists in stating the probability distribution of these vari g and pare available for the policy-maker's choice. We can say that the ables.8 latter maximizes some utility function U(g, p) over the set of those As an example, replace the supply and the demand equations in (13) available pairs of values. But this function is of no concern to the tech- and (H) by equations involving shocks (random "shifts," in the econo ruCian.• • 7 mist's language) u' and ud but not errors of observation. In particular, equations (14) become 7. RANDOM SHOCKS AND ERRORS q' = a' + (3'p + u', Exact structural relations such as equations (1) and (2) are admittedly d d ,d- d unrealistic. Even if, in describing the behavior of buyers, we had included, (16) q=a-pp + u, in addition to the price and to the quantity demanded, a few more vari q• - qd = g; ables deemed relevant (such as the national income, the prices of substitutes, etc.), an unexplained residual would remain. It is called "dis accordingly, equation (I5) must be replaced by turbance," or "shock," and can be regarded as the joint effect of numer (17) g = (a' - ad) + ({3' + {3a)z1 + (u' - ud). ous separately insignificant variables that we are unable or unwilling to specify but presume to be independent of observable exogenous vari Suppose that the shocks are known to have the following joint distribu ables. Similarly, numerous separately insignificant variables add up to tion (as already remarked, it must be independent of the observable produce errors in the measurement of each observable variable ( observa exogenous variables; that is, in our case, independent of p): tion errors). Shocks and errors can be regarded as random variables. the probability that u' = 1 and ua = I is 3/6, That is, certain sizes of shocks and observation errors are more probable than others. Their joint probability distribution (i.e., the schedule or (I8) the probability that u' = 1 and ua = -5 is 1/6, formula. giving the probability of a joint occurrence of given sizes of the probability that u• = -2 and ua = I is 2/6. shocks and errors) may be regarded as another characteristic of a given economic structure, along with the structural relations and parameters Then (u' ua) is distributed as follows: we have treated so far. If a.t least some of the variables are subject to observation errors it (u' - ua) = 0 with prubability 3/6, is impossible to predict exactly what the observed value of each of the (19) (u' - ua) 6 with probability 1/6, endogenous variables will be when the observed values of exogenous (u' - ua) variables, together with the structure, are given. But it is possible to -3 with probability 2/6. make a prediction in the form of a. probability statement. The probabil That is, to predict the amount g which the government will have to ity that the observation on a certain endogenous variable will take a purchase if it fixes the price at p, the technician will use the same func certain value, or will fall within a certain range of values, can be stated, tion of pas in equation (I5), plus a random quantity which takes values provided that the probability distribution of observation errors of the 0, 6, or -3, with respective probabilities 3/6, 1/6, 2/6. Our example variables is known. Similarly, no exact predictions, but, in general, only shows how, given the values of exogenous variables (pin our case) and probability statements, can be made if at least one of the structural given the structure [which now includes the probability distribution of relations is subject to random disturbances (shocks), even if all observa shocks u', ua along with the structural relations (16) and their param tions are exact. Few economic observations are free of errors; few eco eters], the technician can state the probability distribution of each en nomic relations are free of shocks. The quantities that we want to pre- dogenous variable (g in our case). He can state with what probability 7 In the above case of "protecting the farm income," g is nonnegative and each endogenous variable will take any specified value, or a value that ji is chosen to be at least equal to the price p that satisfies equations (13) of the will belong to any specified set of numbers or any specified interval. free market. Equations (13) and (14) ca.n also describe the introduction of rent control, with p -" p and with government-financed housing being denoted by -g. 8 See Hurwicz [1950b) and Haavelmo [1944, Chapter VII. 14 JACOB MARSCHAK [CHAP. I SEC. 8] MEASUREMENTS FOR POLICY AND PREDICTION 15
Instead of a discrete probability distribution of ua and u', such as consequence of a certain structural characteristic of the economy, and (18), we might have assumed a continuous probability distribution. the technician would merely have recorded it faithfully. For example, let ud and u' be jointly normally distributed, with zero Note that any function of endogenous variables, and therefore also means, with a correlation coefficient p = 0.6, and with respective stand the utility of a given policy [such as U in equation (12)], now becomes ard deviations ud = 3 and u, = 5 crop units. Then the term u' - ud a random variable. Its distribution depends on the structural relations, in (17) has a normal distribution with zero mean and with variance on the distribution of disturbances and errors, and on the values of exogenous variables, the structural relations and exogenous variables equal to u~+ u: - 2pudu, = 16 and standard deviation equal to VI6 = 4. Hence the odds are approximately 1:2 that the necessary being partly controlled by the policy-maker himself. He will prefer cer government purchase g will have to exceed or fall short of the value tain probability distributions of utility to others and will choose the 9 given in (15) by more than 4 units. The values of a', ad, (3', t, u, , ud , best decision accordingly. In particular, he can choose that decision which p constitute the structure, assuming that the structural equations maximizes the long-run average (the mathematical expectation) of util (16) are linear and that the distribution of u' and ud is normal. The ity. This may result in his preferring policies with a narrow range of knowledge of the structure permits the prediction of the endogenous possible outcomes to policies with a wide range of possible outcomes; variable g, given the exogenous variable p. that is, he may "play for safety." Such is the nature of statistical prediction. It is perhat>S not too well understood in parts of economic literature. Too often economic theory 8. THE NEED FOR STRUCTURAL EsTIMATION is formulated in terms of exact relations (similar to alleged laws of natural The results of Section 3 extend themselves with added force to the science), with the frustrating consequence is always contradicted that it now generalized probabilistic (stochastic, statistical) concept of economic by facts. the numerous causes cannot be, accounted for separately If that structure. The determination of relevant unknowns will now be called are appropriately accounted for in their joint effect as random dis 10 "estimation." Generalizing the example used in Section 3, replace the turbances or as measurement errors, statistical prediction in a well demand and profit equations (I) and (2) and the resulting reduced-form defined sense becomes possible. equation (7) by, respectively, is This not to say that the interval within which a variable is predicted '!i to fall with a given probability may not be large. If it is so large that r· (1') p = a- (3q + u, widely differing policies appear to yield equally desirable results, the (2') r = (p - O)q - 'Y + v, prediction becomes useless as a means of choosing the best decision. (7') r = xl + J.t.q+ " + w, However, provided the technician has used the best available data and ,; the most plausible assumptions, he cannot be blamed for the disturb where u, v, ware random shocks and where, corresponding to equations ances inherent in comple.x processes such as human behavior, weather, (8) of Section 3, ·~
crops, new inventions, and for the errors that have occurred in measuring (8') X = -(3, J.1. = a - 0, " = --y, w = uq + v. ' their manifestations. It is quite possible that some of the structural rela The shock variables represent, respectively, random shifts in tions of our economy are, by their very nature, subject to strong random u, v J; demand behavior and in the total cost and are independent of exogenous fluctuations. Should it be true, for example, that the investment deci ;:~ variables such as q and 0. As an example, u and v may depend partly on ~~ ·' sions of entrepreneurs are essentially made in imitation of the decisions •i ; random fluctuations of the general price level (so that u and v are cor of a very few leaders who, in turn, are affected by conditions of their ij' related) and partly on numerous other causes specific to the demand or personal lives as much as by economic considerations, then the predic to the cost formation. Let u, v be normally distributed with zero means, tion of aggregate investment could be made only within a very large and call their variances and u~and their correlation coefficient p. prediction interval, unless one is content with assigning a very small u: Then w, the random term in the reduced-form equation (7'), will be, by probability to the success of the prediction. This fact would merely be a .-, (8'), normally distributed with zero mean and with variance ~~ • When the sample is small, this calculation must be modified somewhat to ,f :·(, (20) u! = lu~ u~ 2qpu.,u account for errors of estimation in Many economists have been dissatisfied with this picture of economic Po as follows. By equation (22), = 1 - fqo = f. Then, by (21), q1 = changes. Many if not most business cycle theories imply that economic 2po = j; by (22), P1 = 1 - iq1 = i; by (21), q2 = 2pl = t; and so on. fluctuations would·take place even if external conditions remained con In our numerical case q, and p, happen to oscillate around ! and i, stant and no random shocks existed. respectively, approaching these constants as time goes on ("damped This is consistent with the observation, neglected in all of our previous oscillations"): examples, that relations describing human behavior, technology, or legal rules must often involve not only a set of contemporary variables but 0 2 3 ... ~ 00 also their rates of change (time derivatives, or differences between suc cessive values of a variable) or their cumulated values (integrals or sums (23) q, 48/32 40/32 41/32 ... ~ 4/3 over time). For example, net investment may be related to the rate of 24/32 change in annual consumption and also to the existing capacity (i.e., Pt 20/32 22/32 21/32 . . . ~ 2/3. to the cumulated past net investment). To give another example, build ing construction lags behind building plans, and both may play a role in (Note that if, at some time t = T, the price and quantity were to be a system of structural relations. Even supposing that the exogenous artificially set at f and t, respectively, then the demand equation (22) variables and the structure are constant and that random disturbances would be satisfied. Also, by putting t = T, T + 1, · · · in (21) and are absent, such a system would generate variations of endogenous vari (22), we see that qr = qr+J = qr+2 = · · ·, and Pr = Pr+J = Pr+2 = ables through time. The paths of these variables will depend on their · · · . That is, the values f and t, if attained, are maintained. They are initial values and in general will not be parallel to the time axis, except the equilibrium price and quantity.) .I 1 possibly for a particular set of initial values (called "equilibrium values") The time schedule (23) expresses each endogenous variable as a func ·; which, if attained, are maintained. We call a structure that would ad tion of time, a discrete function in our case. We have been able to pre j mit variations of observed endogenous variables, even if exogenous vari dict the values of endogenous variables at each point of time from their l ables did remain constant and if there existed no random disturbances, values in the preceding point of time, using the fact that in equation a "dynamic structure." (21) an endogenous variable was related to the lagged value of another endogenous variable. Another method for obtaining the time schedule A13 an example, we may modify the market system (13) into the (23) is to transform (21) and (22) into equations expressing each endoge "cobweb" case familiar to economists and often used to illustrate the '1 13 nous variable in terms of its own previous value, predicting q from q , ]! so-called "period analysis" of business cycles. Suppose that the sup 1 0 q2 from q1 , etc. These equations are ~ pliers of grain determine output in response to the price that prevails 1• j one year before the harvest, and suppose that the demanders set the (24) ;j q, = ~ + Eqt-1 , price at which they are willing to absorb the whole (perishable) crop ~· immediately after harvest. Thus, transactions take place only once a (25) p, = i' + EPt-1 , year, and the prices and quantities obey the following relations: where E = ~{3·,'Y = ad+ ~a·,o = a'+ f3'ad. Note, moreover, that if (21) q, = a' + ftp,_, (behavior of suppliers), we replace t by t - 1, equation (24) becomes q,_, = o + Eq,_2. Hence, I) J = E) (22) Pe {lq, (behavior of demanders), substituting into the original (24), we have q, o(1 + + /q,_2; =ad+ 2 3 I and, by repeating the procedure, q, = o(1 + E + E ) + E q,_a , and, in where the subscript indicates time. Let a' = 0, /3' = 2, ad = 1, {3d = -l. general, J ·l and suppose that the initial crop q0 = 1. By previous definitions (Sec (26) tion 3), these five quantities can be regarded as structural parameters q, = 0(! + E + •' • + En-!) + t"qt-n (n = 1, 2, '' ')· or, equally well, as exogenous variables that happen to remain constant I Thus, the current value of an endogenous variable can be predicted during the whole period in question. The two endogenous variable:::, 1: from any of its preceding values. In more general cases it can be pre p 1 and q, , will trace certain paths, or time schedules, that we can obtain r dicted from combinations of these values. The form (24), (25) into which "See Leontief [1934], Lundberg [1937], and Samuelson [1947, Chapter XI]. f we have put the dynamic structural system (21), (22) exemplifies a set f' ; ' ~. ; ,.f ' 20 JACOB MARSCHAK [CHAP. I ,;Ec. n] 1\!EM.q;rn:J\IE:\'TS F'OH POLICY .\ !\'D f'HED!f'T!OX 21 of "final equations," in Tinbergen's terminology. 14 Each final equation 1lw pn;;t. Yalurs of the exogenous variables (o and therdore o/) if one is a difference equation (or, in other cases, a differential, or possibly a uses the rcdur·Pd-funn eqwt.tion (30). Alternat.i\·ely, one ean take int.o mixed difference-differential-integral equation), possibly of high order, account only ;;ome of t.hn;;e pa:;t ,·alucs hut mu:;t. then employ as addi in a single variable, with a corresponding time schedule or path, such tional predic:t.ors the pa::;t. values of t l1c prcdiet!'d endogenous variable as those in (23), as its solution. [as in (~0)]or of other cndo~cnousvariables [as in (28) when <"ombined AJ3still another, and the most direct, way to obtain the time schedule with (21)]. :i q, p,) 16 (23), we can express (or in a form involving only the initial values Changes in exogenous variables \Yill, of course, affect the time schedules of endogenous variables. (26), = t: the In put n of the endogenous variables. This fact is generally recognized in the case 1 1 - E 1 of annual seasons. But it is not always recognized with sufficient clarity (27) q 1 = 0 ' - -- + E qo· 1 - E by those who try to discover longer wave-like (so-called cyclical) regu larities of the paths of economic variables without first eliminating the Using (27), q1 is predicted from the following quantities, considered as 17 given: the stmctural parameters, which determine o, E; the initial value effects of noneconomic variables, or try to predict future waves from 16 the past ones without regard for possible changes in the noneconomic q0 ; and time. In fact, (27) is the equation of the time schedule for q, in (23); it is the solution of the "final equation" (24). Equation (27), conditions. together with the analogous one for price p, , corresponds to the re Let us now replace observable exogenous influences by nonobservable duced form previously defined for static systems, since equation (27), random disturbances. Modify our example (21), (22) by letting the like (7) or (13), relates an endogenous (or jointly dependent) variable supply equation undergo random shifts. That is, replace a' in (21) by to the independent quantities only. However, we have already mentioned a' + u', where 1t' is a nonobservable random variable which we shall, other forms that can be used in a dynamic system to predict the future to begin with, assume to have an unchanging probability distribution. value of an endogenous variable. As the reader will remember, those For example, u~may measure the effect of weather on crops in the year t, other forms included as given the lagged values of the endogenous vari and we assume that weather in one year is independent of that in any able to be predicted or of other endogenous variables. of the preceding years but has the same probability distribution; this We can now readmit changes of exogenous variables in our example. is a situation similar to that in which lots are drawn from a sequence of Suppose that the intercept c/ of the demand curve is not a constant but urns, lots of a given kind being present in each urn in the same pro a variable depending on the size of the population, and therefore deter portion. To fix ideas, let this distribution be normal, with zero mean mined outside of our system (21), (22). Suppose that c/ takes successive and variance cr~.The distribution is now a part of the structure, which is described, in addition, by the following equations, with fixed values values ag, a~, · · ·. The demand equation (22) is replaced by attached to each parameter denoted by a Greek letter and also to the (28) Pt = a~+ {3"q, . initial values qo, Po : The "final equations" change accordingly; (24) becomes (31) qt = c/ + {3'p1-1 + u; (behavior of suppliers), (29) q, = Ot + Eql-1 , (32) PI = ad + {3dql (behavior of demanders). ,B'a~_• where o, a' + 1 The reduced-form equation (27) becomes 1 The "final equations" 18 (difference equations in single endogenous vari (30) q, = Ot + EOt-1 + · · · + El-IOt + E qo . ables) (24) and (25) now become "stochastic" (i.e., involve random is now, of course, impossible to predict the endogenous variable q It variables). In particular, equation (24) is replaced by from the constants (E and q0) only; one has to take account also of all 14 Also called the "separated form" [1\'Iarscbak, 1950, Sections 2.4.5 and 2.5.3]. (33) qt = o + Eqc-1 + u;. 15 Also called the '·resolved form" [Marschak, 1950, Sections 2A.5 anrl 2.5.3]. Each successive value of q is a random variable whose distribution de •• The equilibrium values are obtai ncr!, for -1 < < < 0, by puttin~t = "' Thus, in our numerical case(<= f3d·fJ' = -!, 8 = 2), we obtain q~= 1, whi<"h confirms pends on the value af'tually taken by qat the preceding point. (or, gen (23). If<= -1, the oscillation has constant amplitude; if • <-I, the oscillation erally, points) of time. The path of q (and also of p), instead of being a is "explosive." If • > 0, the path is nonoscillatory but converges to a constant if 17 See Marschak [1949]. • < 1. Thus, equilibrium values for q exist only if 1<1< 1. 1 ~This is the "separated form" of footnote 14. l t 1 i 22 JACOB MARSCHAK (CHAP. I SEC. 9) MEASUREMENTS FOR POLICY AND PREDICTION 23 sequence of constants as in (23), has become a "stochastic process." are, accordingly, called "predetermined," while the current endogenous The equation of the path [viz., the reduced-form equation (27)] now variables are called "jointly dependent. " 20 becomes a "stochastic equation." The reader will easily obtain, by the This similarity between exogenous and lagged endogenous variables same recurrent procedures as before, ceases to exist, however, if we drop our assumption that successive ran 1 dom shocks (the random supply shifts u~, due to weather in our 1- E 1 u;, ... (34) q, = o · + E qo + We, --1- E example) are independent. If, for example, we consider not annual but daily weather reeords, the independence of successive shocks may have where to be ruled out. Instead, these shocks u: , u; , ... themselves may con a stochastic process, each shock depending on one or more of (35) Wt = u; + EU;-1 + · · · + EI-IU:, stitute its predecessors. Since the lagged endogenous variable q1-1 depends on the Since all have zero means, so has, by (35), the random u~_, u~ uL u;, · · · w, shock 1 and this is correlated with , q1-1 is not independent of component of q,. Since u:,u;, · · · were assumed independent, the vari u; . Therefore qe-I is not predetermined. It is determined, jointly with ance of w, (and therefore of q,) is the sum of the variances of u;, EU;_1, q1 , ... , q,_2, q,, qt+1, . .. , by the exogenous variables, the coefficients of ... and is therefore equal to the structural equations [such as (31) and (32)], and the joint distribu 21 21 tion of successive shocks entering all of the structural equations. When (36) u!, = u~(l- E )/(1 - l). ever we use weekly or even quarterly instead of annual time series, we As t increases, this variance approaches a constant, provided that the must be wary of predictions that use lagged endogenous variables as though they were exogenous. absolute value of E is smaller than one. (In our numerical example, E = The conclusions of the previous sections can now be generalized to -!and u!, approaches tu~.) But q, itself does not approach any equi librium value. In this and other respects the path actually described by the case of structures that are both stochastic and dynamic. Policy con each endogenous variable will differ from the path (23) generated by the sists in changing those elements of the structure and those exogenous corresponding nonstochastic structure (21), (22). In fact, a stochastic variables that are under the policy-maker's control. Given the values structure may generate explosive oscillations even though the corre of the uncontrollable features of the structure and of the uncontrollable exogenous variables, the technician's task is to predict which stochastic sponding nonstochastic structure [such as (21), (22), with f3'{l = E = 1] 19 processes will be generated by the various proposed policies. vari produces oscillations with a. constant amplitude. The However, the prediction procedure is similar to that of the nonsto ables that are thus predicted are the potentially observable (and hence chastic case if the concept of prediction is appropriately modified (as possibly erroneous, because of measurement errors) values of some eco in Section 7) in the sense of stating the probability that, at a given time, nomic quantities of interest to the policy-maker. To make his best de the endogenous variable in question will fall within a given interval. cision, the policy-maker ranks these alternative outcomes according to Analogously to the nonstochastic case, predictions can be made either his preferences. For example, his objectives may include high income from the structural quantities only [as in (34) and (36), where the pre averaged over time, but also small intensity of variations in time, and, in addition, a high degree of predictability. (small prediction intervals dictors are o, E, qo , and u~];or from the past values of the endogenous variable that is being predicted [as in (33)]; or, more generally, from the for a given probability level). These objectives may conflict, so he will past values of all endogenous variables. rank the various combinations of average income, stability, predicta bility, etc. [for example, by ascribing to them weights analogous to 1 and If we now reintroduce changes in exogenous variables (such as aa in a w in (12)]. previous example), these will have to enter the equations used for pre diction. In fact, under the conditions stated so far, the past values of As in the cases treated earlier, knowledge of past structure is necessary endogenous variables play the same role as exogenous variables in that if the policies under consideration and the expected changes of uncon- they are independent of present random shocks. In this case both the '0 Compare footnote 3. See Koopmans [1950c, Table, p. 406]. exogenous and the lagged endogenous variables determine the current 21 In our example this joint distribution involves only -u;, · · · , u;, · · · . The values of the endogenous variables but are independent of them. They properties of this distribution, such as, for example, the ("serial") correlation coefficient between successive pairs u;_1 , u;, must be considered part of the struc 11 See Frisch (1933a] and Hurwicz [1945). ture. (CHAP. I ~4 JACOB MARSCII:\K SEC. 10] MEASUREME:\TTR FOR POLICY AND PRF.DICTIO:\ 25 trolled conditions involve not only changes in exogenous variables but Our previous conclusions (Sections 3 and 8) can be applied. There is a also changes in the structure itself. difference between changing the exogenous variables and c:hanging the structure. If a certain rule of fiscal or monetary action in response to 10. "STEERING WHEEL" AND AuTOMATISMS changes in national income or in price level has been tried out long enough, in various doses and with various delays, such experience can To the extent to which economic fluctuations are regarded as an evil, 'I policies can be suggested that will dampen such fluctuations. Through indeed suffice to determine when and how intensely the measure should '! an appropriate change of controlled exogenous variables or controlled be applied. If income tax has been operating in various situations and ' structural parameters, the jerky path described by national income and at various tax rates, it is possible to estimate the tax rate that would other economic aggregates in the last hundred years or so may be re best fulfill the task of damping fluctuations of national income. In such placed by a smoother one in the future. In particular, jumps due to cases we have merely to fit an empirical relation between, say, bank sudden changes of exogenous variables or to rare, but nonetheless possi reserve ratios or income-tax rates, on the one hand, and some measure ble, large random disturbances may be counteracted by the construc of the violence of price or national income fluctuations on the other. tion of appropriate "shock-absorbers." If the existing structure is known, The case is then indeed analogous to that of Lerner's motorist, or, for one can attempt to find the extent to which a given and feasible change that matter, to the case of the firm that (as in Section 3) collected ex in the institutional characteristics of the structure would affect certain perience on the effect of output upon profit without ever bothering to properties of the oscillatory path of an important economic variable, explain this effect by the existing behavior of buyers, the cost structure, such as the wave frequency, or the so-called damping ratio between the and the rate of the excise tax on the firm's product. amplitudes of two successive waves in the absence of new impulses. In Suppose, however, that the institution in question is to be introduced this way Tinbergen [1939, p. 169] tried to measure the effect of increas for the first time. To fix in advance the rule of monetary action that will ing or decreasing the (properly defined) rigidity of wages or of prices stabilize prices and national income most quickly and effectively, even upon the shape of the business fluctuations. within a large margin of error, it is necessary to knmv, for example, the Economic history knows remarkable examples of stabilizing institu lags and elasticities involved in the relation describing consumers' re tions. Possibly the best known is the unwritten law that is said to have sponse to changes in national income, prices, cash balances, etc., and the ruled the conduct of the Bank of England during the nineteenth century. lags and elasticities involved in other structural relations at a time when Any serious change in the balance of payments, as indicated by the out the institution was not in force. To experiment with the institution would flow or inflow of gold, was counteracted by changes in the discount rate. require too much trial and error. 22 More recently, in the discussion of the stabilization of employment and of the price level, institutional rules were proposed that would ob 11. MATHEMATICS AND PREDICTION ligate the monetary or fiscal authorities to take specified measures that All of the foregoing was concerned with the logic of economic knowl would nip deflations and inflations in the bud. edge and of its uses. This logic is the same whether or not mathematical It has been argued that the formulation of such rules need not pre symbols are used. However, mathematical presentation is of great help suppose any knowledge of economic structure or, in particular, of its in testing whether a set of structural relations proposed by a theorist is numerical characteristics. As Lerner [1941] put it, the motorist, ignorant internally consistent and whether it can be determined numerically of the car mechanism, steers his wheel quite successfully, responding from observations. Mathematical presentation is hardly avoidable when instantaneously to changes in the surface and the direction of the road. appropriate statistical metl1ods are to be applied to observations in order Other economists have even suggested what we may call "pilotless" devices. Thus, income tax receipts, at a tax rate fixed once and for all, 22 James Angell [1947, p. 291] sees here "the familiar problem of taking the right will rise and fall with money income, thus oounteracting inflation or de compensatory action promptly enough and in the right degree .... How much change in what indices should be the signal for how big a change in what fiscal flation. (Such automatisms have been called, e.g., by Hart [1945], and monetary operations, to offset or reverse a process of undesired general change "built-inflexibilities".) Again, it has been argued that the knowledge of which is already under way? Not only the nature of the actually current movement economic structure is not necessary if one wants to stabilize the economy but the effects of the compensation measures themselves ... must be gauged ... by such devices. if the result is not to be merely the imposition of a new set of 'artificial' or 'in duced' fluctuations on those already operating." 26 JACOB MARSCHAK [CHAP. 1 to estimate the structure or (if no structural change is envisaged) to CHAPTER II estimate its reduced form. As stated in Section 7, the technician cannot be blamed if a certain IDENTIFICATION PROBLEMS IN ECONOMIC MODEL ' kind of data results in a predicted range of values that is so wide, or has CONSTRUCTION 'l ' such a small probability attached to it, as to be useless. The mathematical 1 method and result will merely reveal what otherwise might remain con BY TJALLING C. KooPMANS cealed. Mathematics does not suppress any information available for Page other methods, and it makes clearer when and how additional informa 1. Introduction...... 27 tion must be used-for example, to extend the time series, to supple- 2. Concepts and Examples...... 29 • ment them by cross-section data such as attitude surveys, or to insert 3. The Identification of Structural Parameters...... 35 additional knowledge on technology and institutions. 4. Identifiability Criteria in Linear Models...... 37 5. The Statistical Test of A Priori Uncertain Identifiability...... 39 6. Identification through Disaggregation and Introduction of Specific Ex- 12. CoNCLUSION 23 planatory Variables...... 40 7. Implications of the Choice of the Model...... 44 This chapter has been concerned with the type of knowledge useful 8. For What Purposes Is Identificatio:{l Necessary?...... 46 or necessary for determining the best policy. In particular, the circum r~ ( stances were stated under which the choice of best policy requires the I' I. INTRODUCTION knowledge of "structure." Structure was defined as a set of conditions which did not change while observations were made but which might t The construction of dynamic economic models has become an im change in the future. If a specified change of structure is expected or in I portant tool for the analysis of economic fluctuations and for related tended, prediction of variables of interest to the policy-maker requires [' problems of policy. In these models, macro-econmnic variables are some knowledge of past structure. It follows that if among the policies thought of as determined by a complete system of equations. The meaning considered there are some that involve structural changes, then the choice of the term "complete" is discussed more fully below. At present it may of the policy best calculated to achieve given ends presupposes knowl I suffice to describe a complete system as one in which there are as many edge of the structure that has prevailed before. equations as endogenous variables, that is, variables whose formation is In economics, the conditions that constitute a structure are (1) a set !: to be "explained" by the equations. The equations are usually of, at of relations describing human behavior and institutions as well as tech most, four kinds: equations of economic behavior, institutional rules, nological laws and involving, in general, nonobservable random dis technological laws of transformation, and identities. We shall use the turbances and nonobservable random errors of measurement; (2) the term structural equations to comprise all four types of equations. joint probability distribution of these random quantities. Systems of structural equations may be composed entirely on the basis of economic "theory." By this term we shall understand the combination .,, Economic theories try to explain observed facts by postulating plausi i ble human behavior under given institutional and technological condi of (!'!.)principles of economic behavior derived from general observation partly introspective, partly through interview or experience--of the tions. To be consistent with facts, they should also introduce random ,1 j disturbances and errors. Thus every economic theory susceptible to motives of economic decisions, (b) knowledge of legal and institutional factual tests must describe a structure or a class of structures. rules restricting individual behavior (tax schedules, price controls, reserve requirements, etc.), (c) technological knowledge, and (d) care It follows that a theory may appear unnecessary for policy decisions fully constructed definitions of variables. Alternatively, a structural until a certain structural change is expected or intended. It becomes I 1 necessary then. Since it is difficult to specify in advance what structural I am indebted to present and former Cowles Commission staff members and :! to my students for valuable critical comments regarding contents and presenta changes may be visualized later, it is almost certain that a broad analysis .;·· of economic structure, later to be filled out in detail according to needs, tion of this chapter. An earlier version of this paper was presented before the I, Chicago Meeting of the Econometric Society in December 1947. This chapter is is not a wasted effort. reprinted, w.ith minor revisions and the addition of the sixth example in Section Thus, practice requires theory. 2, from Econometrica, Vol. 17, April, 1949, pp. 125-144. Boldface numbers in brack ets refer to the list of references at the end of the chapter. :!.1 uSee Marschak [1947b]. ~i! 27 !!J ::i ii' II •J