Long Memory Processes and Chronic Inflation: Detecting Homogeneous Components in a Linear Rational Expectations Model

This paper is an empirical study of the links between monetary variables and inflation based on Cagan’s equation and its rational expectations solution, when the forcing variable is a fractionally integrated process. As demonstrated by Hamilton and Whiteman, the existence of bubbles and other extraneous influences can be detected only by verifying the difference in the order of integration between the monetary base and the price level series. This paper shows that a test based on fractional differencing overcomes Evans’ critique and that chronic inflation is essentially a monetary phenomenon caused by fiscal imbalance.

Abstract

This paper is an empirical study of the links between monetary variables and inflation based on Cagan’s equation and its rational expectations solution, when the forcing variable is a fractionally integrated process. As demonstrated by Hamilton and Whiteman, the existence of bubbles and other extraneous influences can be detected only by verifying the difference in the order of integration between the monetary base and the price level series. This paper shows that a test based on fractional differencing overcomes Evans’ critique and that chronic inflation is essentially a monetary phenomenon caused by fiscal imbalance.

The idea that increases in consumer price index may not reflect movements in fundamentals is hardly new. The idea, in essence, was already contained in Cagan (1956) and the problem of the non-uniqueness of equilibrium in a dynamic monetary economy has surfaced since Sargent and Wallace (1973). Brock (1974) undertook a comprehensive treatment of stable and unstable solutions in models of intertemporal optimization with real money balances in the utility function.1 However, it was the seminal article by Flood and Garber (1980b)2 that renewed the interest in Cagan’s model and made the notion of rational bubbles popular, by showing how self-fulfilling expectations might arise in a rational expectations context derived as a particular case of Brock (1974).

Nonfundamental influences are sometimes confused with persistence, but these are conceptually distinct phenomena. In fact, in linear dynamic models, bubbles and other nonfundamental influences are represented by the homogeneous part of the solution to a linear difference equation and logically are not related to the way expectations are formed. Expectations in turn affect the persistence of inflation: adaptive or backward-looking expectations would delay the effect of any change in policy as the agents’ reactions lag.3

This study will test the presence of homogeneous components in the solution to Cagan’s equation.4 Hamilton and Whiteman (1985), generalizing the results by Burmeister, Flood, and Garber (1983), showed that bubbles, sunspots, and related processes are observationally equivalent to “fundamental” equilibria once a fairly general dynamic specification for the driving variables has been postulated. Hence, the only falsifiable hypothesis implied by self-fulfilling expectations is the difference in the order of integration of the relevant variables. In the presence of bubbles, money supply has a lower order of integration than the price level or, stated differently, changes in money supply are followed by larger movements in the price level.5

Evans (1991) criticized Hamilton and Whiteman (1985) by showing in a Monte Carlo study that a specific form of bubble process cannot be detected by conventional unit root tests. This paper argues that Evans’s critique can be overcome by exploiting the properties of a recently developed time series model, the so-called Autoregressive Fractionally Integrated Moving Average (ARFIMA) (see Brockwell and Davis (1991), Hosking (1981), and Granger and Joyeux (1980)).6 This model generalizes the treatment of classic integrated processes by considering noninteger orders of integration, thereby providing an accurate representation of time series with slowly decaying autocovariance structures (also called long memory structures). The advantages of ARFIMA models in testing for the presence of bubbles are as follows:

1. A greater degree of diagnostic precision than the standard stationarity tests (see Diebold and Rudebusch (1991)) because, unlike the Autoregressive Integrated Moving Average (ARIMA) models, the ARFIMA models do not place any restrictions on the long-run characteristics of the series.

2. Separate analysis of the short-and long-term dynamics of a process, which is essential in the study of inflation. The long-term dynamics can be interpreted as changes in macroeconomic policies while short-term effects are the result of measures that do not attack the roots of inflation.

3. Extreme generality in the sense that, unlike other methods, ARFIMA-based tests are not sensitive to the functional form of the homogeneous component.

The existence of bubbles, sunspots, or other nonfundamental effects has far-reaching consequences for economic policy. In particular, if inflation expectations are self-sustaining or depend on causes beyond economic rationale, the price level will not respond to conventional monetary and fiscal measures. The cost of stabilization achieved through monetary and fiscal discipline, hence, will be extremely high.

The paper is organized as follows. In Section I, a solution to Cagan’s equation is formulated on the assumption that the driving variable, in this case money supply, is fractionally integrated. Section II discusses Evans’ critique, its relevance, and the reasons why ARFIMA models offer an appropriate response. The results of the empirical analysis for six countries (Argentina, Bolivia, Brazil, Chile, Peru, and the former Socialist Federal Republic of Yugoslavia) are presented in Section III. Section IV concludes the paper with a discussion on the implications for economic policy.

I. Cagan’s Equation with Fractionally Differenced Variables

Cagan’s money demand function has the form

Mt/Pt=cexp{δπt+1*},(1)

where Mt is nominal money balances at time t, Pt is the price level at time t, and πt+1* is expected inflation at time t + 1, while 8 and c are constants, the first reflecting the impact of expected inflation and the second summarizing all other effects. Indicating the variables in logarithms by lowercase letters and normalizing c to 1, expected inflation can be expressed as

π*=E[Pt+1|Ωt]PtPtE[pt+1|Ωt]pt,

where is E[•|Ωt] denotes mathematical expectation conditional on the information set at time t, Ωt. The money demand equation (1) can then be rewritten as

mtpt=δ(E[pt+1|Ωt]pt).(2)

Adding a money demand disturbance nt, equation (2) can be equivalently expressed as

E[pt+1|Ωt]αpt=xt+nt,(3)

where α = (1 - δ)/δ and xt = mt,/(1 + δ) = (1 - α)mt. Equation (3) can also be obtained as a log-linear approximation to an Overlapping Generation (OLG) model with money. This interpretation, as explained in Chapter 5 of Blanchard and Fisher (1989), does not place any restriction on α, while in Cagan’s formulation α lies between 0 and 1.

Monetary Policy and Fiscal Regime

To get testable implications from equation (3) we need to define the stochastic process governing money supply and the money demand disturbance. Furthermore, to implement the test, the driving variable must be exogenous. A rather general specification was proposed by Hamilton and Whiteman (1985):

(1L)dxt=A(L)ε1,t+B(L)ε2,t(1L)dnt=R(L)ε1,t+S(L)ε2,t,(4)

where the white noise innovations €i, t, i = 1,2 are jointly fundamental for the bivariate process (xt, nt), d is fractional, and A(L), B(L), R(L), and S(L) are polynomial in the lag operator L, with mean square converging terms. One can think of xt as a variable observed by the econometrician, in the sense that time series of past realizations are available, and nt as unobservable by the econometrician (as no data are available), but observable by the agents. For example nt can be interpreted as the effect of variables—other than fundamentals—influencing the agents’ forecasts. The first feedback rule in equation (4) asserts that the monetary authority reacts to unexpected shocks in the economy i, by choosing A(L) and B(L).

The government budget constraint is represented by

Gt+(1+rt1)Bt1Tt=(MtMt1)/Pt+Bt.(5)

The identity (5) asserts that the difference between total government spending (the sum of real expenditures Gt and interest payments (1 + rt-1) Bt-1) and revenues Tt is covered by money printing Mt- Mt-1 (which extracts a seigniorage equal to (Mt-Mt-1)/Pt) or by issuing bonds Bt bearing a real interest rate rt.

A major implication of equation (5) is that inflation and fiscal deficit are not necessarily contemporaneous as long as governments can resort to borrowing. So if a government is committed to finance its debt exclusively by issuing bonds (Ricardian Rule), i.e. Mt- Mt-1 = 0, deficits are not inflationary, provided that the future stream of expenditures equals the future stream of revenues. Stated differently, government debts are not inflationary when they are temporary, so that the budget is balanced in a present value sense. In reality, however, the fiscal authority sets Gt and Tt, and the monetary authority then decides to cover the debt by money creation or by borrowing, or by a combination of the two.

This digression explains, by a twofold rationale, the choice of the functional form (4) for the money supply. First, in periods of chronic high inflation only small increases in Bt are feasible, i.e., Bt-Bt-1 ≈ 0, so persistent deficits are almost fully monetized. Second, fiscal imbalance is the product of historical conditions unlikely to change suddenly, so deficits must be modeled as a highly persistent stochastic process, as in equation (4). Furthermore, the substantial dependence of monetary policy on the fiscal regime determines the exogeneity of money supply with respect to the price level, which is an implicit prerequisite for the Hamilton and Whiteman test.

The Solution to the Expectational Equation and Its Testable Implications

Equation (3) with variables specified as in equation (4) can be solved using the z-transform method. Hamilton and Whiteman (1985) in fact obtain

(1αL)(1L)dpt={ξ0(1L)d+LA(L)+LR(L)}ε1,t+{k0(1L)d+LB(L)+LS(L)}ε2,t.(6)

Depending on the value of α, the constants ξ0 and k0 can be determined by the requirement that the functions (1 - z)dξ(z) and (1 - z)dK(z) must be analytic on the unit circle. For | α | > 1 this leads to

ξ0=(11/α)d{α1A(1/α)+1/αR(1/α)}κ0=(11/α)d{α1B(1/α)+1/αS(1/α)}.(7)

For |α| < 1, these conditions are not required for analyticity, but hold if the parsimony principle is invoked, that is if model specification involves the minimum number of explanatory variables.

The solution (6) - (7) to equation (3), valid for all processes xt and nt representable in terms of square summable operators A(L), B(L) R(L), and S(L), is called the fundamental solution because it depends on the driving variable only.

By contrast, solutions to (3) that depend on other—possibly completely independent—variables are referred to as bubble, sunspot, or nonfundamental solutions. Following Hamilton and Whiteman (1985) and Burmeister, Flood, and Garber (1983), the particular solution as a function of any finite number of white noises ηi, t i = 1,2, … ,m, completely unrelated to (xt, nt), is, for any value Qi in:

y(1αL)(1L)dPt=(1αL)(1L)dpt*+Q1(1L)dη1,t+...+Qm(1L)dηm,t,(8)

where pt* is the fundamental solution. The condition Qt ≠ 0 for some i implies the existence of an extraneous influence.

Hamilton and Whiteman propose to estimate the difference between the order of integration of the variables on the left-hand side and that of those on the right. In fact, by dividing both sides of equation (6) by (1 - αL) we obtain

(1L)dpt=(1L)dpt*+(1αL)1[Q1(1L)dη1,t+...+Qm(1L)ηm,t]+Kαt,(9)

where K is an arbitrary constant. If K = Qi = 0 αi, i.e., in the absence of bubbles or sunspots, both sides of equation (8) have the same order of integration. Otherwise, the nonstationary terms (1 - αL)-1ηi, t, by the algebra of integrated series (see Granger (1980)), would render the right-hand side integrated of an order higher than d.

As argued by Hamilton and Whiteman (1985), this difference in the order of integration is the only testable hypothesis implied by nonfundamental solutions, because the parameters in equations (3) and (4) are observationally equivalent whether or not bubbles and other extraneous influences are present. In addition, this test does not rely on the functional form of the nonstationary component, while previous studies depend crucially on the arbitrary choice of this functional form.

II. Detecting Nonfundamental Influences

This section covers some basic issues in testing for the presence of nonfundamental influences in rational expectations models. The starting point is Evans’s critique of the Hamilton and Whiteman test, which epitomizes the confusion arising in the analysis of nonstationary processes when standard techniques are employed. Evans shows that when a particular bubble process is added to a fundamental solution, conventional tests fail to detect the presence of the bubble. This critique high-lights the inadequacy of the statistical methods based on integer orders of integration.7 The crucial feature of Evans’s process is its periodic collapse. When added to the fundamental solution, this homogeneous component affects only the high-frequency components of the spectral density. The nonstationarity of a series is directly linked to its spectral density, and in particular to its slope near the origin; hence, a “bubble” is not detectable by standard tests unless it alters in a significant way the lower-frequency terms. The Dickey-Fuller test, based on integer orders of integration, concentrates on the low-frequency components of the spectrum and is therefore inadequate to analyze the process suggested by Evans (1991) as well as all the homogeneous terms that alter the high-frequency (short-term) components in the fundamental solution to equation (3). By contrast, fractionally integrated models provide a viable way of detecting the presence of bubbles or extraneous influences.

Using the ARIMA conventional notation, an ARFIMA model can be expressed as

(1L)dA(L)xt=B(L)εt,

where d is a real number and not an integer as in standard time series analysis. An ARFIMA process is covariance stationary when dє (–0.5, 0.5).

A formal discussion on Evans’s critique can be introduced by looking at the spectrum of ARFIMA(p, d, q) processes (see Geweke and Porter-Hudak (1983)):

f(λ)=|1eiλ|2dfu(λ),

where

fu(λ)=(σ2|Θexp{iλ}|2)/(2π|Φexp{iλ}2|)

is the spectral density of the ARMA (p, q) process ut = (1 - L)dXt. Taking the natural logarithm, adding and subtracting log(fu(0)), we obtain

lnf(λ)=lnfu(0)dln|1eiλ|2+ln[fu(λ)/fu(0)].

Therefore, the order of differentiation d is essentially the negative slope of the log spectrum near the origin, which is equivalent to asserting that the order of integration has a pronounced dependence on the low-frequency components.

To verify this theoretical proposition, the order of integration of a number of synthetic processes was estimated with the same parameters used by Evans in his paper. In all cases, the estimated order of integration was about 0.3, which implies, first, that this type of process is covariance stationary, and second, that it can be detected by estimating the difference in the order of integration as Hamilton and Whiteman suggest. This conclusion is obviously not limited to the specific form of process Evans considers, but applies to all stationary and nonstationary processes with a nonzero order of integration.

III. Empirical Analysis

In essence, the empirical analysis consists of estimating the fractional order of integration of the money supply and the consumer price level. If no bubbles or sunspots are present, the orders should be approximately equal.

The estimation was carried out using a FORTRAN program written by Fallaw Sowell at Carnegie-Mellon University and based on Sowell (1989).8 The data were obtained from the IMF’s International Financial Statistics (IFS), which provides a homogeneous definition for both variables across countries and therefore allows a meaningful comparison of the results. The money series in this analysis refers to narrow money (line 34 in the IFS) and the price level is the consumer price level (line 64 in the IFS).

The data span different periods but are always on a monthly basis. For Argentina and Peru, they cover January 1971 to December 1989, while for Bolivia, Brazil, and Chile they extend only through December 1987, February 1986, and June 1985 respectively. The data on the former Socialist Federal Republic of Yugoslavia cover the period January 1975 to June 1990.

The results are shown in Table 1. A full description of the estimates is contained in Scacciavillani (1994), while this version presents only the results for the models identified by the Schwartz Information Criterion (SIC) and the Akaike Information Criterion (AIC)—invariably the ARFIMA(1,d,0)—which in this case are also the best ones according to the log-likelihood test. Specifically, Table 1 reports the estimated value of d (with t-statistics in parentheses) and the 95 percent confidence interval of the estimates; the estimated autoregressive parameter (AR); and the ratio of the variance of the predicted values to the variance of the data (a sort of R2) denoted by σ/var.

Table 1.

Estimates of the ARFIMA Parameters for Price Level and Money Supply in Selected Countries

article image

95 percent confidence interval of the d estimate.

In no case—even for models other than the ARFIMA(1,d,0), not reported in Table 1—do the estimates indicate that the variables have different levels of nonstationarity. Furthermore, only in one case (Brazil) is the point estimate of d for the price level outside the 95 percent confidence interval estimation of d for the money supply. Therefore the evidence against the presence of a homogeneous component is rather strong. The AR parameters are all very significant, ranging from 0.38 to 0.64, with three of them close to 0.50, showing also that the short-term components of the processes are fairly similar.

A detailed account of the results for each country is presented below. Former Socialist Federal Republic of Yugoslavia. The difference in the order of integration between the money supply and price level is negligible. The AIC and SIC values leave no doubt about the identification of the ARFIMA(1, d, 0) model. Further, the other models (not reported here) yield estimates of d that are similar to the ARFIMA(1, d, 0).

Peru. The results for Peru are analogous to those for the former Socialist Federal Republic of Yugoslavia: the estimates of d for both series are extremely close, although the a/var is lower in this case.

Chile. The model ARFIMA(1, d, 0) identified by the AIC and the SIC for both series exhibits an estimated d equal to 2.31 for the money supply and 2.4 for the price level. The result does not change substantially when we exclude from the sample period the years 1971, 1972, and 1973, that is, the period before the military took power.

This difference in the orders of integration is not substantial: the hypothesis that they are different would not be accepted at conventional significance levels. Further, the AIC and, especially, the SIC for the model ARFIMA(2,d,0) (not reported here) are close to those for ARFIMA(1,d,0), so the identification procedure in this case leaves some degree of uncertainty. The estimates of d in the ARFIMA(2,d,0) models (not reported here) are practically identical: 2.48 for price level and 2.47 for money supply.

In conclusion, even for Chile the evidence in favor of the “no bubbles” hypothesis is strong, although not as categorical as in the first two cases.

Argentina. The difference in the estimates of d is more pronounced than in the case of Chile. The ARFIMA(1,d,0) representation, selected by the AIC and the SIC, yields an estimate of d equal to 2.47 for the price level and 2.35 for the money supply. Moreover, the identification methodology does not indicate other plausible alternatives to the ARFIMA(1,d,0) model. Therefore, a slight difference in the order of integration is possible, but the presence of homogeneous components that are lasting enough to induce a difference in levels of nonstationarity can be reasonably refuted.

Bolivia. Strict adherence to the AIC and SIC again points to the ARFIMA(1,d,0) representation. As in the case of Argentina, the estimates of d differ somewhat, but not enough to establish the presence of homogeneous components. Moreover, the AIC and SIC provide evidence that the ARFIMA(2,d,0) model (not reported in Table 1) could be adequate to describe the dynamics of money supply. The estimate of d for the ARFIMA(2,d,0) is 2.46, much closer to the value 2.43 obtained with the ARFIMA(1,d,0), the most reliable model for the price level.

Brazil. The difference in the estimated order of integration for the ARFIMA(1,d,0) is the largest of those in the countries studied. On this basis, the existence of a rapidly collapsing bubble process cannot be excluded, but there is rather weak evidence of difference in the order of stationarity.

However, unlike for Argentina, the strength of this conclusion is mitigated by the fact that for money supply, the values of AIC and SIC relative to the ARFIMA(2,d,0) (equal to 473 and 464 respectively), are comparable to the values relative to the ARFIMA(1,d,0) (439 and 433 respectively). The estimated d parameter in the ARFIMA(2, d, 0) representation is 2.43, close to the value of 2.48 obtained for the price level.

In summary, the econometric analysis supports a strong rejection of the existence of nonfundamental influences altering the degree of stationarity of the solution to Cagan’s equation. However, in the case of Argentina and Brazil one cannot exclude the existence of stationary nonfundamental influences like the rapidly collapsing bubble suggested by Evans.

IV. Summary and Conclusions

This paper has tested the hypothesis that high chronic inflation is caused by nonfundamental influences represented by a nonstationary homogeneous term in the solution to Cagan’s expectational equation (3).

As stressed by Hamilton and Whiteman (1985), the presence of a homogeneous term can only be verified by analyzing the difference in the dynamics of price level and money supply as reflected in the orders of integration. This paper tested the significance of this difference for a number of countries during the 1970s and part of the 1980s, based on recent econometric advances in the theory of fractionally integrated processes.

The results show that the difference in the order of integration is in general small, although in a few cases it is not negligible. The AIC and the SIC always point to an ARFIMA(1,d,0) model for both price levels and money supplies; the estimates of d lie in the interval [2.23, 2.49], indicating that the series are stationary in second differences.

Overall, the empirical evidence against the existence of persistent self-fulfilling expectations and long-lasting nonfundamental influences in general appears to be strong. What cannot be ignored in some cases is the sporadic temporary divergence between the short-cycle dynamics of money supply and price level. This phenomenon could be attributed in some instances to stabilization programs based on isolated actions that do not modify substantially the fiscal regime, and in other cases to backward-looking forms of indexation.

These results are similar in spirit—though based on different methodology—to other findings in the recent literature on hyperinflation, including Phylaktis and Taylor (1993), who use the same data set as does this paper, and Blackburn and Sola (1993), who concluded that for Argentina the presence of a rapidly bursting bubble could not be rejected.

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*

I greatly benefited from conversations with Robert Flood, Leonardo Bartolini, and Peter Clark. In addition, I wish to thank Fallaw Sowell for his estimation software and Mark Taylor for the suggestions and data that he provided. Andrea Malagoli wrote the maximization code used in the estimation software, which is copyrighted by the University of Chicago Department of Astrophysics. All remaining errors are solely my responsibility.

1

A critical review of this literature is contained in Gray (1984).

2

Kingston (1982) proved that Cagan’s equation can be derived as a special case of the general equilibrium model in Brock (1974).

3

See Sargent’s (1986) introduction to Chapter 3, “The End of Four Big Inflations,” for a lucid and concise treatment of the dichotomy between adaptive and rational expectations.

4

Nonfundamental influences have been long debated and have been brought up in several circumstances to explain anomalies in speculative markets or in macroeconomic data. For instance, bubbles may arise when the current value of an asset is determined (at least in part) by the expected rate of market price change. The mere self-fulfilling assessment of a future change can drive the current value to a level unwarranted by economic fundamentals.

5

Even after Hamilton and Whiteman (1985) was published the fundamental importance of this falsifiable hypothesis has not always been perceived in the literature. For example, Casella (1989), in her analysis of the post-World War I German hyperinflation, takes the second difference of the data, therefore assuming, without testing, that the series have the same order of nonstationarity.

6

Two other ways to overcome this problem were proposed by Funke, Hall, and Sola (1994) and Blackburn and Sola (1993), who used the Markov regime-switching model by Hamilton (1989).

7

Hall and Sola (1993) and Blackburn and Sola (l993) have resorted to the Markov regime switching proposed by Hamilton (1989) to overcome Evans’ critique. While this approach is based on premises quite different from the ARFIMA models, it nevertheless is well suited for detecting the presence of Evans’ processes.

8

A different subroutine for calculating the roots of autoregressive polynomials was substituted for the original version of the software written by Fallaw Sowell; this code employs Muller’s method. The estimation was performed on a Sun Spark II at the University of Chicago Social Science Computer Center.