Abstract
The buffer stock role of absorbing temporary discrepancies between purchases and sales is assigned to money because money, being the most liquid of all assets, performs the buffer function best. However, as this paper shows, the attempts to model the buffer stock role have led to certain incoherencies. Specifically, this paper shows that the econometric models of buffer stock money published in the literature are incompatible with the theory of buffer stock money and imply two different probability distributions for the same variable, thus resulting in an incoherency.
1. Introduction
A substantial and influential amount of recent money demand literature has stressed the role of money as a buffer asset or shock absorber which temporarily smooths the economy’s response to unexpected changes in the money supply. The genesis of this approach has been attributed to Jonson (1976) and Knoester (1979) although its roots trace back to the inventory-theoretic contributions of Baumol (1952) and Tobin (1956). 2/ Subsequent methodological accounts and extensions of what the buffer stock concept is and how it relates to other approaches of monetary analysis include discussions by Goodhart (1982, 1984), Judd and Scadding (1982a), Judd (1983), Knoester (1984), Laidler (1984, 1985, 1987), Bain and McGregor (1985), Cuthbertson (1985), Cuthbertson and Taylor (1987a), Milbourne (1987), and Muscatelli (1988). Empirical applications of the buffer stock concept include works by Carr and Darby (1981), Laidler (1980), Coats (1982), Kahn and Knight (1982), Judd and Scadding (1982b), Carr, Darby, and Thornton (1985), Kanniainen and Tarkka (1986), Swamy and Tavlas (1987), and Cuthbertson and Taylor (1987b). 3/
Such has been the impact of the buffer stock approach that Bain and McGregor (1985, p. 386), employing Tobin’s (1981) distinction between Monetarism I and II, argue that “buffer-stock Monetarism might well be interpreted as an improved version or more precise statement of the impact interval analysis of Monetarism I.” Additionally, Cobham (1984), emphasizing similarities between the buffer stock approach and disequilibrium money demand studies by economists such as Artis and Lewis (1976), considers the buffer stock approach to be an area of what he convincingly characterises as the emerging convergence between “some Keynesians and some Monetarists.” As an indication of the importance attached to the approach Laidler (1984, p. 32) states that it is the “starting point towards further progress in monetary economics.”
This paper is a critique of the empirical application of the buffer stock approach. In particular, it is argued that: (i) while discussions of the buffer stock concept offer novel and important insights about the workings of the monetary economy, in several major respects the empirical application of buffer stock money is in contradiction with buffer stock money as a concept; (ii) in some instances the empirical application has been logically inconsistent; and (iii) the empirical models of the buffer stock notion which have been developed are as likely to magnify certain problematic attributes of conventional money demand models which the buffer stock concept is designed to remedy. 4/
The paper is divided into five sections including this introduction. Section 2 provides a brief overview of the buffer stock notion, with attention focused on those characteristics which its adherents consider paramount to the approach. Section 3 examines the advantages (or disadvantages) offered by the buffer stock concept in terms of its ability (or inability) to correct certain undesirable properties incorrectly attributed to conventional money demand models. One of these advantages is the apparent ability of the buffer stock model to explain why short-run interest rate overshooting—a presumed characteristic of conventional money demand models—does not exist in the real world. In this section we point out some problems of statistical inference inherent in attempts to invert conventional models in order to test for overshooting. Section 4 offers a critical discussion of the empirical implementation of the buffer stock notion. Concluding remarks are provided in Section 5.
2. Buffer stock money—an overview
Fundamental to the buffer stock concept is the existence of a system of monetary exchange in which there are costs in gathering and processing information. An economic agent desires money not for its own sake, but because he “confidently expects someone else to accept it from him in due course for something else” (Laidler, 1984, p. 18). The buffer stock approach presupposes the need for an asset to absorb shocks during the interval which elapses between the occurrence of a shock and the subsequent response of the economic agent.
Shocks may simply reflect the random timing of receipts and payments, or may appear in the form of unanticipated macroeconomic disturbances. For example, if an unanticipated increase in the money supply is generated, it is initially held by economic agents, thus providing a buffer of purchasing power, even though it forces the holders temporarily off their short-run money demand curves. The approach stresses that decisions to change plans entail costs—as Goodhart (1984, p. 255) observes, the resale price of a durable asset is usually well below its purchase price. Thus “economic agents may rationally and optimally decide to respond to the continuing stream of developments, ‘news and shocks,’ not by a thoroughgoing, continuous reconsideration of their full economic dispositions, but by allowing such shocks to impinge initially upon certain assets/liabilities whose characteristics make them suitable to act as buffers” (Goodhart, 1984, p. 255). Money, being the most liquid of all assets, performs the buffer function best. Hence emphasis is placed on the role of money as a medium of exchange or temporary abode of purchasing power (Laidler, 1984, p. 18; Goodhart, 1984, pp. 255-56). 5/
The buffer stock approach assumes that there is no single-valued demand for money. Rather, as in the Miller-Orr model of money demand, there is a band around which money holdings may vary. In turn, the width of the band is determind by economic variables, such as the scale and variability of cash flows. Once holdings of money balances exceed their upper prescribed limit, they are used to purchase a broad spectrum of assets—financial as well as real (Jonson, 1976 p. 513). Goodhart’s (1984, p. 266) analogy with a reservoir is instructive. With the passage of time an individual’s reservoir of money balances spills over onto canals of spending on a wide spectrum of assets and goods as real balance effects are set into motion. Hence the buffer stock approach is able to account for the broadly based monetary transmission mechanism long advocated by monetarists (e.g., Friedman and Meiselman, 1963). 6/ As Laidler (1984) has demonstrated, the approach is also able to distinguish between the microeconomic money demand experiment, where the individual passively adjusts his money balances to changes in opportunity costs and transactions or wealth variables, and the macroeconomic experiment, where changes in the money supply might be exogenous or policy induced. Accordingly, in situations where monetary authorities have the capability to influence the money supply and do not accommodate all changes in money demand, the buffer stock concept is especially relevant.
Two properties of the buffer stock approach are particularly noteworthy for purposes of this paper. First, its adherents emphasize that the buffer function of money is a short-run, or temporary, phenomenon. In this vein Goodhart (1982, p. 1545) states that an unanticipated change in the money stock is “willingly held temporarily” (original italics). Elsewhere he [Goodhart (1984 p. 257)] argues that “an initial shock to the money supply is likely to be buffered, in the short run.” Similarly, Knoester (1984, p. 255) refers to the “temporary character of monetary buffers since the spillover to the real sphere implies absorbent feedback effects which diminish the initial buffer.” Second, buffer stock adherents argue that the validity of the approach is an empirical matter. In this regard, Laidler (1984, p. 27) states that “the case for the buffer-stock approach ultimately rests as much upon its empirical content as upon a priori reasoning.” Likewise, Judd (1983, p. 41) emphasizes that “the existence of these [buffer stock] effects is primarily an empirical issue.”
3. Why buffer stock money?
The conventional specification 7/ of money demand treats the demand for real money balances in the following form:
where mt is the logarithm of a measure of nominal money stock, pt is the logarithm of a price index, yt is the logarithm of a scale variable such as income or wealth, rt is the logarithm of an opportunity cost variable, and ut is an error term which may or may not be white noise, depending upon how model (1) is generated. 8/ Among the reasons advanced for the presence of the lagged dependent variable in equation (1) are generalized adjustment costs—i.e., the partial adjustment model—and permanent, rather than transitory, values of the explanatory variables. 9/
Several attributes of empirical studies based on the specification of model (1) have been particularly troublesome. First, empirical estimates of the coefficients of model (1) have often yielded highly inaccurate predictions of mt, particularly since the mid-1970s, indicating perhaps that the demand for money function (1) became unstable in the years after the mid-1970s. Second, an implication of model (1) that does cot seem plausible to some buffer stock monetarists is that changes in the money supply are accompanied by interest rate overshooting in the short run. To see why they think interest rate overshooting is a property of model (1), note that when yt and rt are exogenous relative to (mt - pt), and ut is white noise, the short-run (i.e., impact) interest elasticity of the demand for real money balances is a2 (a negative number), and the long-run (after lagged adjustment has occurred) interest elasticity is a2/(1-a3). The values of these two elasticities satisfy the inequality |a2|≤|a2/(1-a3)|, with equality if and only if a3 = 0. 10/ This inequality implies that |1/a2|≥|(1-a3)/a2|. The latter inequality is called the overshooting effect. Thus, from the result that the demand for money function (1) is less interest elastic in the short run than in the long run (due, for example, to costs of adjusting portfolios), it is concluded that a given change in the money supply has to be accompanied by a larger interest rate response in the short run (than in the long run) in order to induce economic agents to absorb the giver change in money. 11/
When prices remain constant, this overshooting effect can be demonstrated with reference to Figure 1. An increase in the real money stock from M0/P to M1/P necessitates a fall in the interest rate from r0 to r1 in order for the increase in the money supply to be matched by an increase in money demand. This is equivalent to a movement from point a, to point b, along the short-run money demand curve
Short-run interest rate overshooting may be even more pronounced in an environment where financial deregulation has taken place, if such deregulation allows for the explicit payment of market-determined yields on demand deposits. The opportunity cost of holding money is the spread between the market rate on a substitute financial asset and the own deposit rate on money. If the adjustment of deposit rates is to follow market rates closely, then the opportunity cost of money may have to vary less than market rates. That is, the achievement of a particular change in relative yields (between substitute financial assets and bank deposit rates) may require a larger change in market yields than was necessary when deposit rates were controlled. Therefore, it is unlikely that the inequality, |a2|≤|a2/(1-a3)|, will be reversed if we replace rt in equation (1) by the spread between a market rate on a substitute financial asset and the own deposit rate on money.
The belief in short-run interest rate overshooting is not shared by buffer stock monetarists. Thus, Laidler (1982) convincingly argues that overshooting effects do not sit well with buffer stock theory. In particular, Laidler shows that overshooting refers to models in which an interest rate is the dependent variable while buffer stock theory refers to the money demand function which cannot be inverted.
a. Money demand inversion: problems of statistical inference
Nevertheless, several studies proceed to invert the money demand function (see, e.g., Artis and Lewis (1976), Goodhart (1984)). In this regard, Goodhart (1984, p. 259) has observed that, when interest rates are themselves regressed on movements in the money stock, “there if rarely any trace of such apparent overshooting.” In addition to Laidler’s concern about the incompatibility of such an equation inversion with buffer stock theory, there are also valid statistical reasons why Goodhart’s (1984) regression may not give the correct values for 1/a2 and (1-a3)/a2. As we have already pointed out, the direct regression of (mt - pt) on yt, rt, and (mt-1 - Pt-1) in CD gives the correct values for a2 and a2/(1-a3) if yt and rt are exogenous relative to (mt - pt) and if ut is white noise. In this case, the reverse regression of rt on yt, (mt- pt), and (mt-1-pt-1) most give wrong values a2 and a2/(1-a3) because the error term of this regression is not mean independent of yt and (mt - pt.) In the alternative case, where yt and (mt - pt) are exogneous relative to rt and the error term of the reverse regression is white noise, it is the reverse regression (but not the direct regression) that gives the correct values for 1/a2 and (1-a3)/a2. These results follow from Goldberger’s (1984) commentary on direct and reverse regressions in a different context. For example, if (mt, rt) is bivariate normal, then
and
These equations show that the coefficient of mt in E(rt|mt) is not the reciprocal of the coefficient of rt in E(mt|rt), unless the correlation coefficient between mt and rt is 1. The situation represented by the above two conditional means does not correspond to the situation depicted by Figure 1.
The point we are making here is that the theoretical and empirical assessments of the short-run and long-run effects of changes in mt on rt should be based on the conditional mean of rt given mt and other relevant variables, and not on the conditional mean of mt, given rt and other variables. It may then turn out that such correctly made assessments will not even permit the formulation of the overshooting hypothesis. For example, an estimate of the mean value of the time-varying coefficient on money growth in an interest rate equation estimated by Swamy, Kolluri, and Singamsetti (1987) is -11. Since this coefficient is time dependent, it is zero whenever the money growth is zero. Therefore, if money growth is zero in the long run, then the mean of its coefficient may take on the same value in both the short run and the long run. Thus, there is no evidence to show that the magnitude of the coefficient on mt in an interest equation in any time period is larger than the magnitude of the corresponding-long-run coefficient. The preceding argument proves that it is statistically incorrect to invert equation (1) to deduce the overshooting effect. Such a deduction raises issues of statistical inference.
It might be tempting to resolve the overshooting issue by any of the instrumental variables techniques. However, while instrumental variables can, in principle, be used to estimate either direct or reverse regression when some of its right-hand side variables are endogenous, these techniques rest on the hypothesis that certain observed variables used as instruments are truly exogenous even at they are thought to have an important influence on the endogenous right-hand variables. These two requirements can, however, be contradictory, as shown by Swamy and von zur Muehlen (1988). This point can be illustrated as follows: Let a structural equation imbedded in a system of equations by
where Yj is a matrix of observations on a set of endogenous variables, βj is a vector of coefficients, Xj is a submatrix of X, the matrix of observations on all the exogenous variables in the system, and uj is an error vector. Further, let the portion of a reduced form that relates to Yj be
Then the matrix X(X’X)-1 X’Yj is a matrix of observations on a legitimate set, of instruments for Yj if IIj is nonnull whenever γj is nonnull and if X is truly exogenous. It is typically the case that in practical situations we are unable to verify whether the assumption of the exogeneity of X contradicts the assumption of nonnull IIj and nonnull γj, It can happen that either the exogeneity assumption about X is incorrect or the exclusion restrictions imposed on the structural equation imply that IIj is null whenever γj is nonnull (for further discussion, see Swamy and von zur Muehlen (1988)). Another difficulty is that the structural coefficients, γj and βj, are not unique in the over-identified case even when the rank condition for the identification of these coefficient is satisfied (for a proof of this statement, see Swamy and von zur Muehlen (1988)). Consequently, the use of selected observed variables as instruments may imply an unwanted imposition of a set of contradictory model restrictions, thereby invalidating any regression results. Therefore, caution should be exercised in interpreting coefficient estimates based on direct and reverse regressions whenever instrumental variables techniques are used.
The buffer stock monetarists’ solution to the problem of interest rate overshooting is to include the unexpected component of the money supply as an additional regressor in equation (1). In terms of Figure 1, an unforeseen increase in the money supply leads to a rightward shift in the short-run money demand function to
Thus, buffer stock adherents formulate the demand for money function in the following form.
where
and
Here Zt is a set of variables that agents assume have an important influence on the money supply,
Equation (2) is not supposed to exhibit short-term interest rate overshooting, at least not to the extent implied in model (1), if the variable (mt -
4. A critique of buffer stock models
In what follows it is also argued that the empirical model used to test the buffer stock concept is faulty in several respects: (i) to the extent that some researchers attempt to model monetary anticipations on the basis of the rational expectations hypothesis (RE), buffer stock models are subject to criticisms dealing with the empirical estimation of rational expectations; (ii) the empirical model runs contrary to a major insight of the buffer stock concept; (iii) it is logically inconsistent; and (iv) it can magnify certain characteristics of the conventional money demand approach which the buffer stock concept seeks to mitigate. Attempts to estimate model. (2) include, works by Carr and Darby (1981); Coats (1982); Carr, Darby, and Thornton (1985); Juld and Scadding (1982b); MacKinnon and Milbourne (1984); Cuthbertson and Taylor (1982b); Cuthbertson (1986); and Kanniainen and Tarkka (1986). 14/
a. Incorporating rational expectations
Given that the buffer stock model has been identified with the rational expectations hypothesis, some criticisms leveled at RE (Swamy, Barth, and Tinsley, 1982; Swamy and von zur Muehlen, 1988) can also be directed at equation (2). As Cuthbertson and Taylor (1987a, p. 103) have argued, “certain buffer stock models are not necessarily inconsistent with RE and, indeed, one can usefully combine the two approaches,” and this has provided justification for the addition of the term a4 (mt-
Equations (2)-(4) satisfy the RE hypothesis, if they give the same distribution for mt if the distribution of mt given by equation (2) has a frequency interpretation, and if the distribution of mt given by equation (4) has a subjective interpretation. Conversely, the RE hypothesis is not true unless the distribution of mt has both frequency and subjective interpretations. It has been shown in the statistics literature that the conditions of a law of large numbers under which probabilities have a frequency interpretation are neither necessary nor sufficient for the Bayesian conditions of coherent behavior under which probabilities have a subjective interpretation. But the distribution of mt can be given joint frequency and subjective interpretations only if both these sets of conditions are true. Bayesians and non-Bayesians, alike, are not willing to adopt these two sets of conditions for the simple reason that their conjunction is more restrictive than either of them alone and may, indeed, be absurd, as the following assertions based on Suppes’ (1974) argument show: “[The conditions of a law of large numbers] do not in any direct sense represent axioms of rationality that should be satisfied by any ideally rational person but, rather, they represent structural assumptions about the environment that may or may not be satisfied in given applications…. Intuitively, a…[structural assumption]…as opposed to a rationality axiom is existential in character…. It is not a part of rationality to require that the decision-maker enlarge his decision space, for example, by adding a coin that may be flipped any finite number of times…the intrinsic theory of rationality should be prepared to deal with a given set of states of nature…, and it is the responsibility of the formal theory of belief…to provide a theory of how to deal with these restricted situations without introducing strong structural assumptions.” Furthermore, equations (2) and (4) may not give the same distribution for mt, as we shall show below.
b. Incompatibility between conceptual and empirical approaches
The buffer stock concept assumes that unforeseen changes in the money supply are initially “absorbed in money balances, and over time are used to purchase a wide spectrum of assets and goods, as real balance effects are set into motion (Zellner, Huang, and Chau, 1965). We shall show that the specification of model (2) is incompatible with this view of the transmission process. It follows from equations (3) and (4) that
because the variance of vt is not time dependent. In other words, plim mt ≠
It may be comforting to buffer stock monetarists to note that the above argument is based on the assumption that (mt -
Another approach is to reparameterize equation (2) as in MacKinnon and Milbourne (1984):
This equation shows that the addition of the term a4(mt-
c. Logical inconsistency
Equations (2) and (4), used to empirically implement the buffer stock approach, can be mutually inconsistent, although this has not been previously recognized. To show this, note that equation (6) which is an alternative form of equation (2), can be written as:
where pt on the left-hand side is combined with the price term on the right-hand side and g’zt. has been substituted for
However, according to equation (4) the conditional expectation of mt, given xt, is
If the conditional expectations (8) and (9) are unequal, then equations (2)-(4) imply two different distributions for the same variable. The implications of such incoherencies that may trap the unwary are discussed by Swamy and von zur Muehlen (1988).
d. The impact of unanticipated money on overshooting
As noted, the buffer stock concept has been formulated, in part, to deal with the interest rate overshooting characteristics of the conventional money demand model. The buffer stock model captures the influence of unexpected increases in the money supply via the coefficient, a4, on the term (mt -
5. Concluding remarks
Buffer stock monetarists assume that unforeseen changes in the money supply are initially absorbed in money balances, and over time are used to purchase a wide spectrum of assets and goods as real balance effects are set into motion. They model this buffer stock role of money by including the unanticipated component of money supply as an additional regressor in the conventional money demand function. Furthermore, buffer stock monetarists believe that, without this additional regressor, the conventional money demand function gives rise to the following anomaly: a given change in the money supply has to be accompanied by a larger interest rate response in the short run than in the long run. This anomaly is deduced from the result that the conventional money demand function is less interest elastic in the short run than in the long run. This paper has argued that neither their model of the buffer stock role of money nor their deduction of the anomaly from a property of the conventional money demand function is necessarily right. Specifically, the anomaly may vanish, if the coefficient on money in an interest rate equation is not the reciprocal of the coefficient on an interest rate variable in the conventional money demand equation. Also, the buffer stock empirical model may not even be compatible with the buffer stock notion which is reasonable. More importantly, the buffer stock model is incoherent if, as is possible, its constituent models of money demand and expectations about money imply two different distributions for money.
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The views expressed in this paper are those of the authors and do not represent the views of the Board of Governors of the Federal Reserve System or its staff, or of the International Monetary Fund. We are grateful to Martin Bailey, Lawrence Klein, David Laidler, Liliana Rojas-Suarez, Hari Vittas, and Peter von zur Muehlen for helpful comments.
The buffer stock notion also relates to Keynes’ (1936) precautionary motive for holding money balances. The terms “buffer” and “shock absorbers” are probably due to Friedman and Schwartz (1963). In their discussion of portfolio adjustment following a monetary innovation, Friedman and Schwartz (1963, p. 63) state that: “It is this interconnection of stocks and flows that stretches the effects of shocks out in time, produces a diffusion over different economic categories, and gives rise to cyclical reaction mechanisms. The stocks serve as buffers or shock absorbers of initial changes in rates of flow by expanding or contracting from their ‘normal’ or ‘natural’ or ‘desired’ state, and they slowly alter other flows as holders try to regain that state.” An early and interesting empirical application of the buffer stock notion in the context of modeling money demand in a developing country was made by Riechel (1979).
Not all empirical studies of the buffer stock approach have been successful; see MacKinnon and Milbourne (1984), Bailey and Tavlas (1986), and Cuthbertson (1986). The predictive performance of buffer stock models may not be as good as those of some other models, as shown by Swamy and Tavlas (1987). The reason for this failure may be the way the buffer stock notion is modeled. The surge of interest in the buffer stock approach is attested to by the workshop on buffer stock money held in London at the Center for Economic Policy Research (CEPR) in June 1986. See the CEPR Bulletin August 1986, for a summary of the proceedings.
The criticisms in this paper of the empirical implementation of buffer stock money are not applicable to empirical work on disequilibrium money, which deals with the gap between desired and actual money stock. See Zellner, Huang, and Chau (1965) and Cuthbertson (1985).
However, Bain and McGregor (1985) argue that the tendency to associate the medium of exchange and financial buffer functions is inappropriate. They extend the notion of financial buffers to general liquid asset holdings and short-term liabilities.
Jonson (1976, p. 513) states that “if, in an uncertain world, people use money as a buffer stock and make their decisions about consumption, the labor they will supply or demand, purchases of durable assets, holdings of bonds and other decisions, and, at the end of the day, look to their money balances to signal the aggregate impact of their economic decisions, money plays a crucial role.”
However, in recent years there has been increased application of the error-correction specifications pioneered by Hendry and his associates. See, for example, Hendry and Ericsson (1985). Some of the claims made about the error-correction models are incorrect, as shown by Swamy and von zur Muehlen (1988).
In particular, while the partial adjustment model leads to an error term that may be spherical, a model which uses the adaptive expectations mechanism to generate permanent values of the regressors results in an error term which may be of the moving average variety. See Zellner and Geisel (1970) and Theil (1971, pp. 261-62).
Laidler (1982) is critical of the short-run long-run demand for money distinction in terms of adjustment costs. However, recognizing the need for the lagged dependent variable in empirical work, he argues that the short-run demand for money should be viewed as a misspecified price level adjustment equation. He goes on to suggest that the short-run money demand function is not strictly a structural relationship but a mixture of structural and reduced form effects of the whole economy.
Model stability requires that a3 be less than unity.
This assumes that adjustment is faster in asset markets than in goods markets. Alternatively, Carr and Dary (1981) also note that model (1) implies price level overshooting.
Laidler (1984, p. 19) refers to this process of adjustment to the long-run demand curve as an adjustment path involving “points on various, essentially Marshallian, short-run demand for money functions.”
Differences exist among individual researchers with regard to the empirical methods used to estimate Mt. For example, Carr and Darby (1981) use an unspecified ARIMA model for Mt while some researchers use a first-order autoregressive model. The difficulty with these methods is that the conditions for the existence of an ARIMA model for Mt may contradict those for the existence of a structural model for Mt (see Swamy and von zur Muehlen (1988)). Therefore, a combination of these two models should not be used if such a combination involves a contradiction. Differences also exist with respect to the zt variables. For example, Carr and Darby include a transitory income while Laidler (1980) is critical of the use of such a variable. It is important to point out that Laidler derives a buffer stock money demand model that is a reduced form of a structural model in which price adjustment is sluggish. Laidler emphasizes the importance of the theory of buffer stock money but, like us, he is critical of the empirical implementation of that theory.
While MacKinnon and Milbourne (1984) reject the buffer stock hypothesis, they do so on the basis of the number of instruments used to implement two-stage estimation. Following Klein’s (1969) results with respect to macroeconometric modeling, they argue that two stage least squares produces better results if the number of instruments is reduced considerably from the 25 used in Carr and Darby (1981) study. This argument as well as Carr and Darby’s instrumental variables technique is incorrect if it is based on contradictory restrictions (see Swamy and von zur Muehlen (1988)). Cuthbertson (1986) also rejects the buffer stock hypothesis, but in line with MacKinnon and Milbourne, he does so on the basis of a model specified in accordance with model (2) in the text. Thus, Cuthbertson’s rejeccion of the buffer stock hypothesis might be due to the particular expectations scheme he uses to generate the anticipated money supply series.
