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Long Memory Processes and Chronic Inflation

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Long Memory Processes and Chronic Inflation IMF Staff Papers Vol. 41, No. 3 (September 1994) © 1994 International Monetary Fund Long Memory Processes and Chronic Inflation Detecting Homogeneous Components in a Linear Rational Expectations Model FABIO SCACCIAVILLANI * 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 differ- ence in the order of integration between the monetary base and the price level series. This paper shows that a test based on fractional differenc- ing overcomes Evans' critique and that chronic inflation is essentially a monetary phenomenon caused by fiscal imbalance. [JEL C22, E31, E58] T^HE IDEA that increases in consumer price index may not reflect JL 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 comprehen- sive treatment of stable and unstable solutions in models of intertemporal optimization with real money balances in the utility function.1 However, *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 pro- vided. 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. 1A critical review of this literature is contained in Gray (1984). 488 ©International Monetary Fund. Not for Redistribution LONG MEMORY PROCESSES AND INFLATION 489 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 pop- ular, 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. Expec- tations 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), general- izing 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 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 Infla- tions," for a lucid and concise treatment of the dichotomy between adaptive and rational expectations. 4Nonifundamental 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 assum- ing, without testing, that the series have the same order of nonstationarity. ©International Monetary Fund. Not for Redistribution 490 FABIO SCACCIAVILLANI 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 represen- tation 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 station- arity tests (see Diebold and Rudebusch (1991)) because, unlike the Auto- regressive 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 coun- tries (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. 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). ©International Monetary Fund. Not for Redistribution LONG MEMORY PROCESSES AND INFLATION 491 I. Cagan's Equation with Fractionally Differenced Variables Cagan's money demand function has the form (1) where Mt is nominal money balances at time t, Pt is the price level at time t, and ir£-i 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 where E [•|ft,] denotes mathematical expectation conditional on the information set at time t, flt. The money demand equation (1) can then be rewritten as (2) Adding a money demand disturbance nt, equation (2) can be equivalently expressed as (3) where and Equation (3) can also be obtained as a log-linear approximation to an Overlapping Gener- ation (OLG) model with money. This interpretation, as explained in Chapter 5 of Blanchard and Fisher (1989), does not place any restriction on a, while in Cagan's formulation a 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 dis- turbance. Furthermore, to implement the test, the driving variable must be exogenous. A rather general specification was proposed by Hamilton and Whiteman (1985): (4) where the white noise innovations e,,, i = 1,2 are jointly fundamental for the bivariate process (*„ nt), d is fractional, and A(L\ B(L\ R(L), and 5(L) are polynomial in the lag operator L, with mean square converging ©International Monetary Fund. Not for Redistribution 492 FABIO SCACCIAVILLANI 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 ob- servable by the agents. For example nt can be interpreted as the effect of variables—other than fundamentals—influencing the agents' fore- casts. The first feedback rule in equation (4) asserts that the monetary authority reacts to unexpected shocks in the economy €/, by choosing A(L) and B(L). The government budget constraint is represented by (5) The identity (5) asserts that the difference between total government spending (the sum of real expenditures Gt and interest payments (1 + r,_i)£,_i) and revenues Tt is covered by money printing Mt - Af,_i (which extracts a seigniorage equal to i ',) 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.
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