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W: .‘II'III. w“ $18“, WIN!llll“NW\llllllUlllllllllllllllllll 3 1293 01055 3208 This is to certify that the dissertation entitled S trongly Dependent Economic Time Series : Theory and Applications presented by Margie A. Tieslau has been accepted towards fulfillment of the requirements for Ph . D . degree in Economics 77 In . Major professor Date 7. '12 “777 MS U is an Affirmative Action/Sq ual Opportunity Institution 0-12771 3” - v.1 1?" _ bel‘ .-.' ? Bu g :13}. iii-33 6:31.“ 5 r“ ‘ 1 55B i 5 I '- " 1 4 ‘ .fir L: \, ‘3. '8 t 54:. . . l l" ' < '-- 'J 1'33 t ‘1;..‘: .’- 'g 3'1: 5'3“... 9; ’r 1"! i f ‘ k”-H '3'.- " 'gewf.ifi 8‘? *- my if {’n! 3min ‘ J Pei PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 7| ll MSU Is An Affirmative Action/Equal Opportunity Institution chMmS-nt STRONGLY DEPENDENT ECONOMIC TIME SERIES: THEORY AND APPLICATIONS By Margie A. Tieslau A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1992 M?" ..3 “weak .5, l x .' 5a. ABSTRACT STRONGLY DEPENDENT ECONOMIC TIME SERIES: THEORY AND APPLICATIONS By Margie A. Tieslau This dissertation contains three essays which address the issues of persistence and stationarity of macroeconomic time series, estimation of persistent series through the procedure of generalized method of moments, and the existence of stable long-run money demand functions and stable real exchange rates. The first essay investigates the Autoregressive Fractionally Integrated Moving Average ARFIMA(p,d,q) model as a process for describing strongly persistent time series. The essay proposes the ARFIMA(p,d,q)- GARCH(P,Q) model. which, offers the flexibility' of' modelling strongly dependent series that are also characterized by non-homoskedastic error. The ARFIMA-GARCH model is applied to the inflation rate series for both low- and high-inflation economies and the inflation rate is found to be highly persistent but, none the less, mean reverting and thus stationary. In addition, empirical support is found for the relatively high-inflation economies for the Friedman hypothesis which implies a direct association between increased levels of inflation.and increased.inf1ation variability. The second essay presents the theoretical derivation of a generalized method of moments (CMM) estimator for strongly dependent time series. The GMM estimation technique may 'be preferred. to maximum likelihood estimation methods since GMM does not rely on distributional assumptions. The moment conditions exploited by the GMM estimator make use of' theoretical and. estimated. autocorrelation. functions, and. the derivation of these functions is presented here. Numerical results from variance calculations using all available moment conditions as well as groups of moment conditions are presented to examine the relative efficiency of the GMM estimator. The third essay investigates the existence of cointegrating relationships betweenlmacroeconomic variables in Canada and.the U.S. This essay examines the long-run equilibrium relationship between real money balances, real income, and short—term interest rates for Canada and the U.S., and investigates the existence of cross-country effects between these countries. Evidence of stable long-run money demand functions is found for both countries with estimated long run income elasticities near 1.0 and long-run interest elasticities near -0.5. The stationarity of the nominal exchange rate and the relative price levels for Canada and the U.S. is also investigated with some evidence found in support of this hypothesis for the post-Bretton Woods era. This dissertation is dedicated to my mother, Herta Tieslau, and to the memory of my father, Adolf Tieslau. iv ACKNOWLEDGEMENTS The successful completion of this dissertation would not have been possible without the input, advice, assistance and encouragement of many individuals throughout the entire process. I am happy to have this opportunity to express my gratitude to these individuals, although I realize that it may be impossible to express, in these few lines, the depth of my appreciation to the many who are deserving. My greatest appreciation must go to my dissertation committee chairman, Professor Richard T. Baillie, who introduced me to some of the topics of this dissertation and who inspired my work in this area. I have had the great fortune of working under his expert guidance for the past three years, and during this time I have benefitted greatly from his experience and wisdom. He has endured an endless stream of questions on my part and has read repeated drafts of this dissertation, offering many suggestions for improvement. I am indebted to him for the time and effort he has given to me and to my work, and for the kindness and patience that he has shown me throughout the completion of this dissertation. I have also had the good fortune of working closely with the two remaining members of my dissertation committee, Professors Robert H. Rasche and Peter Schmidt, whose guidance and direction have been invaluable in completing this dissertation. I have learned a great deal from them and.have benefitted greatly from their knowledge and advice. In addition, I would like to express my appreciation to Professor Rowena Pecchenino who read several earlier drafts of parts of this dissertation and offered many valuable comments. I am also grateful to her for all of the support and advisement that she offered to me throughout my four years at Michigan State University. I owe a debt of gratitude, too, to all of my colleagues in the Ph.D. program at Michigan State University who shared with me the toils of class work and prelims, and who continuously offered me words of encouragement when necessary (which was often). I am particularly indebted to my office mate and friend, Kel Utendorf, who bravely endured my many moods with a smile and a positive word, even through the darkest hours, and never asked for anything in return. My success throughout graduate school could not have been possible without the love and support of my wonderful family. My mother has continuously offered me her unwavering support, even when she personally may have preferred a different course for my life. My father stood by me from the beginning and left me with an unconditional belief in myself that has carried me through many rough times. I thank the rest of my family, too, for their support and encouragement. My only regret is that my work on this dissertation has meant many missed hours with them. Finally, no account of my gratitude could be complete without a special word of thanks to Michael Redfearn whose undying support and encouragement, both academic and emotional, have been a constant source of strength and inspiration to me these past four years. vi TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER I: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 1 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 CHAPTER II: THE THEORETICAL CHARACTERISTICS AND ESTIMATION PROCEDURES ASSOCIATED WITH LONG-MEMORY TIME SERIES MODELS 1. INTRODUCTION . . . . . 5 2. THEORETICAL CHARACTERISTICS OF LONC MEMORY TIME SERIES AND THE ARFIMA(p, d, q) PROCESS . . . . . . 7 3. A SURVEY OF LONG TERM PERSISTENCE AND BROWNIAN MOTION . . . 12 a. A SURVEY OF LONC- TERM PERSISTENCE AND FRACTIONAL DIFFERENCING . . . . . . . 20 5. THE THEORETICAL PROPERTIES OF THE ARFIMA CARCH PROCESS . . . 34 6. SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . . . . . 40 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 CHAPTER III: LONG-TERM PERSISTENCE, MEAN REVERSION, AND STATIONARITY: A MODEL OF INFLATION AS AN ARFIMA-CARCH PROCESS 1. INTRODUCTION . . . . . . . . 45 2. LONG MEMORY, MEAN REVERSION, AND THE PERSISTENCE OF INFLATION . . . . . . . . . . . . . . . . . . . 47 3. THE VARIABILITY OF INFLATION . . . . . . . . . . . 50 4. INFLATION CONSIDERED AS A LONC- MEMORY PROCESS . . . . . . . 52 5. ESTIMATION OF THE ARFIMA- GARCH MODEL . . . . . . . . . . . . 59 6. SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . . . . . 70 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 7O vii CHAPTER IV: A GENERALIZED METHOD OF MOMENTS ESTIMATOR FOR LONG-MEMORY PROCESSES INTRODUCTION . NH INTEGRATED MODEL . 3. ASYMPTOTIC DISTRIBUTION THEORY . GMM ESTIMATION IN THE CONTEXT OF THE FRACTIONALLY 4. ASYMPTOTIC PERFORMANCE OF THE GMM ESTIMATOR FOR THE FRACTIONALLY INTEGRATED MODEL 5. SUMMARY AND CONCLUSION . REFERENCES CHAPTER V: EQUILIBRIUM MONEY DEMAND FUNCTIONS, AND PPP: AN ANALYSIS OF COINTEGRATING RELATIONSHIPS IN CANADA AND THE US 1. INTRODUCTION . 2. THE MONETARY BALANCE OF PAYMENTS THEORY 2.1 LONG- RUN MONEY DEMAND FUNCTIONS REAL EXCHANGE RATES, 2. 2 REAL EXCHANGE RATES AND THE LAW OF ONE PRICE 2.3 THE LINK BETWEEN THE CANADIAN AND U.S. ECONOMIES 2.4 THE EQUILIBRIUM MONETARY MODEL 3. CANADIAN EQUILIBRIUM MONEY DEMAND FUNCTIONS 4. US EQUILIBRIUM MONEY DEMAND FUNCTIONS 5. JOINT ESTIMATION OF THE CANADIAN AND U. S. MONEY DEMAND FUNCTIONS 6. STATIONARITY OF REAL EXCHANGE RATES 7. SUMMARY AND CONCLUSION . REFERENCES viii 101 102 106 109 115 122 123 126 127 128 129 130 131 139 144 152 157 166 CHAPTER III TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE 10 ll 12 l3 14 LIST OF TABLES Autocorrelations of CPI Inflation Series Autocorrelations of First Differenced Inflation Series Autocorrelations Series, Argentina Autocorrelations Series, Brazil Autocorrelations Series, Canada . Autocorrelations Series, France Autocorrelations Series, Germany Autocorrelations Series, Israel Autocorrelations Series, Italy Autocorrelations Series, Japan of of of of of of of of Filtered Filtered Filtered Filtered Filtered Filtered Filtered Filtered Autocorrelations of Filtered Series, United Kingdom . Autocorrelations of Filtered Series, United States Tests for Order of Integration of Different Countries' Inflation Series Application of the Geweke Porter-Hudak Method to Estimate d ix CPI CPI CPI CPI CPI CPI CPI CPI CPI CPI Inflation Inflation Inflation Inflation Inflation Inflation Inflation Inflation Inflation Inflation 73 74 75 76 77 78 79 8O 81 82 83 84 85 86 TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE CHAPTER IV TABLE TABLE TABLE 15 l6 17 18 19 20 21 22 23 24 25 l 2 3 TABLE 4 Estimated ARFIMA (0, d, 1) Student t-GARCH (1, 1) Model for US CPI Inflation . Estimated ARFIMA-GARCH Model for CPI Inflation: Argentina Estimated ARFIMA-GARCH Model for CPI Inflation: Brazil . Estimated ARFIMA-GARCH Model for CPI Inflation: Canada . Estimated ARFIMA-GARCH Model for CPI Inflation: France Estimated ARFIMA-GARCH Model for CPI Inflation: Germany Estimated ARFIMA-GARCH Model for CPI Inflation: Israel Estimated ARFIMA-GARCH Model for CPI Inflation: Italy Estimated ARFIMA-GARCH Model for CPI Inflation: Japan Estimated ARFIMA-GARCH Model for CPI Inflation: United Kingdom . Likelihood Ratio Tests of Relationship Between Mean and Variability of Inflation, yt Asymptotic Variances of /T (d - d) Using Moments 1, 2, 3, . _ .’ n . . . . . . , , Asymptotic Variances of /T (d - d) Using Moment n only . e o . . . . . . . . Asymptotic Variances of JT (d - d) Using Five Moments ' - - - - - - - - - - . Asymptotic Variances of /T (d d) Using Ten Moments ° ° ° - - - - - - . 87 88 89 90 91 92 93 94 95 96 97 117 119 120 121 CHAPTER V TABLE 1 TABLE 2 TABLE 3 TABLE 4 TABLE 5 TABLE 6 TABLE 7 Phillips and Perron Unit Toot Tests Cointegration Tests, U.S. Data . Cointegration Tests, Canadian Data Cointegration Tests with Velocity Restrictions, Canadian Data Cointegration Tests for the Joint Model, Logarithmic Specification Cointegration Tests for the Joint Model, Semi-Logarithmic Specification . Cointegration Tests for Canadian/U S. Exchange Rate‘. xi 159 160 161 162 163 164 165 I . INTRODUCTION CHAPTER I INTRODUCTION Many economic and financial time series are characterized by strong persistence in their mean and variance such that when expressed in levels the series appear to be nonstationary, or contain a unit root, yet when expressed in first differences the series appear to be overdifferenced. The traditional nonstationary Autoregressive-Integrated Moving Average ARIMA(p,d,q) model of Box and Jenkins (1976), where the parameter of integration can take on only integer values, is not able to account for the long-term persistence which is characteristic of these series. This can be especially relevant in applied analysis since empirical investigation of such series must be done within the framework of a model which is able to account for the substantially high order of correlation present in these series. The first essay of this dissertation, Chapters II and III, considers the long-run characteristics of the CPI inflation rate series which is one macroeconomic variable that exhibits characteristics common to strongly persistent time series. This variable is considered.within the context of the Autoregressive Fractionally Integrated Moving Average ARFIMA(p,d,q) process introduced.by Granger and Joyeux (1980) and Hosking (1981), which allows for the order of integration of a series to be fractional, as well as the Generalized Autoregressive Conditionally Heteroskedastic GARCH(P,Q) process of Engle (1982) and Bollerslev (1986). In this way the model is able to allow for simultaneous modelling of both long-term persistence and time varying conditional variance which have been found to be present in the inflation rate series. The application. to the inflation rate series is particularly appealing since this series does not appear to be well described by either the traditional stationary 1(0) process or the non-stationary I(l) process. In addition, this essay investigates the validity of the hypothesis posited by Friedman in his 1977 Nobel lecture that increased levels of inflation were likely to be associated.with increased levels of the variance of inflation. Empirical support for such a relationship would impLy a significant role in the economy fOr policy directed at maximizing net benefit. Several estimation techniques have been developed in recent years to estimate the degree of persistence of strongly dependent time series utilizing both ordinary least squares and maximum likelihood estimation procedures (Janacek (1982), Geweke and Porter-Hudak (1983), Hosking (1984), Fox and Taqqu (1986), and Sowell (1992)). The second essay, Chapter IV, proposes a generalized method of moments (GMM) estimator to determine the degree of fractional integration of a strongly dependent series. The estimation technique is set within the context of the general ARFIMA(0,d,0) process, but may be extended to include autoregressive and moving average parameters in the model as well. The GM estimation technique provides an attractive alternative estimation procedure for the fractionally integrated process since it does not require the distributional assumptions necessary under maximum likelihood estimation techniques. This essay also examines the asymptotic performance of the generalized method of moments estimator and compares its efficiency relative to that of the maximum likelihood estimator. Attention is 3 restricted, in this analysis, to that range of the parameter of fractional integration for which the moment conditions exploited by the model exhibit the usual JT consistency and normality. The final essay of this dissertation, Chapter V, addresses the issues of stable empirical money demand functions and the stationarity of the nominal exchange rate and relative price levels for the countries of Canada and the U.S. The investigation is set within the framework of the monetary balance of payments theory and utilizes the methodology proposed by Johansen (1988) and Johansen and Juselius (1989) to identify cointegrating relationships among sets of variables. The investigation.of long-run equilibrium relationships within and between the economies of Canada and the U.S. is especially appropriate due to the parallel nature of macroeconomic variables in the two countries. The framework used in this essay allows for an investigation into the existence of cross-country relationships between monetary and international variables in the two countries and allows for an investigation into the degree of similarity in the dynamics between the two countries. Empirical evidence of cross-country effects or similarity in dynamics between the two countries can have significant implications for the conduct of monetary policy and international transactions within and among the two countries. (f) L” l r1 3 4 REFERENCES Bollerslev, T. (1986), "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31, 307-327. Box, G.E.P. and G.M. Jenkins (1976), Time Series Analysis Forecasting and Control, second edition, San Francisco: Holden Day. Engle, R.F. (1982), "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK Inflation", Economatrica, 50, 987- 1008. Fox, R. and M.S. Taqqu (1986), "Large Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time-Series", Annals of Statistics, 14, 517-532. Friedman, M. (1977), "Nobel Lecture: Inflation and Unemployment", Journal of Political Economy, 85, 451-472. Geweke, J. and S. Porter-Hudak (1983), "The Estimation and Application of Long.Memory Time Series Models", Journal of Time SerieafiAnalysis, 4, 221-238. Granger, C.W.J. and.R. Joyeux (1980), "An Introduction to Long Memory Time Series Models and Fractional Differencing", Journal of Time Series Analysis, 1, 15-39. Hosking, J.R.M. (1981), "Fractional Differencing", Biometrika, 68, 165- 176. Hosking, J.R.M. (1984), "Modelling Persistence in Hydrologic Time Series Using Fractional Differencing", Water Resources Research, 20, 1898- 1908. Janacek, G.J. (1982), "Determining the Degree of Differencing for Time Series via the Log Spectrum", Journal of Time Series Analysis, 3, 177-83. Johansen, S. (1988), "Statistical Analysis of Cointegration Vectors", Journal of Economic Dynamica_and Control, 12, 231-254. Johansen, S. and K. Juselius (1989), "The Full Information Maximum Likelihood Procedure for Inference on Cointegration. 'with Applications", Preprint No. 4. Institute of Mathematigal Statistics, University of Copenhagen. Sowell, F.B. (1992), "Maximum Likelihood Estimation of Stationary Univariate Fractionally-Integrated Time-Series Models", Journal of Econometrics, forthcoming. II. THE THEORETICAL CHARACTERISTICS AND ESTIMATION PROCEDURES ASSOCIATED WITH LONG-MEMORY TIME SERIES MODELS CHAPTER II THE THEORETICAL CHARACTERISTICS AND ESTIMATION PROCEDURES ASSOCIATED WITH LONG-MEMORY TIME SERIES MODELS 1. INTRODUCTION The use of time series analysis in the field of economics is instrumental in investigating the long-run properties of various economic series. Often times it is the case when fitting parametric models to a time series that the series under investigation initially requires first differencing in order to achieve stationarity. The standard Box and Jenkins (1976) methodology is to difference a series an integer number of times until it appears stationary. The Autoregressive Moving Average (ARMA) model of Box and Jenkins has been used to model the long-run behavior of certain series by expressing today's realization of a variable as a weighted sum of past observations of that variable and past and present innovations. ARMA models assume that the dependence between observations decays exponentially and therefore is relatively' weak. However, many economic and financial time series, while being stationary, are characterized by the property of relatively significant dependence between observations which occur at distant intervals. Such series, although stationary, have been termed "long memory". Two general classes of models have arisen in modelling series which exhibit longgmemory; The first is the fractional Brownian motion process, which is a generalization of Brownian motion. The use of fractional Brownian motion was motivated by the existence of long-term persistence which was observed by hydrological engineers in streamflow data. 6 Consequently, the fractional Brownian motion process has been widely applied in the field of hydrology. The second model is the fractionally integrated process, which is a generalization of the Autoregressive- Integrated Moving Average (ARIMA) model of Box and.Jenkins (1976). Use of the fractionally integrated process was motivated by the observation that the conventional stationary models of Box and Jenkins were not able to capture the persistent nature of long-memory time series. This class of model, which was proposed independently by Granger and Joyeux (1980) and Hosking (1981), offered. an .alternative to the standard..ARIMA(p,d,q) process by not restricting the degree of differencing, the parameter d, to be limited. to an, integer ‘value 'but rather allowing it to take on fractional values. Fractionally integrated processes are capable of generating extreme persistence represented by the hyperbolic decline of the impulse response weights and the autocorrelation functions. Hence fractionally integrated processes allow for a much slower decay of past observations of the series, relative to the faster exponential decline of the weights of the traditional ARMA model. This chapter will provide a comprehensive survey of long—memory processes, outlining the theoretical properties of series which exhibit strong dependence and discussing the techniques used to model these processes. The next section will discuss both sample and population characteristics of long-memory time series and also will discuss the characteristics of the Autoregressive-Fractionally Integrated. Moving Average process. Section 3 will survey the Brownian motion process and fractional Brownian motion as a method used to model long-term behavior. This section will examine the estimation procedures involving fractional 7 Brownian motion which have been used to estimate the degree of persistence of a series exhibiting long memory. Section 4 will survey the fractionally integrated process as an alternative method of modelling long-term behavior, examining various estimation techniques involving the fractionally integrated process which have been proposed to estimate the degree of persistence (the degree of fractional differencing) of a series as well as any other model parameters. Section 5 provides a description of the theoretical properties of the model which is developed in this dissertation, the ARFIMA-GARCH process. This process offers the flexibility' of' modelling series which. exhibit long-term. persistence compounded with the presence of non-normal or non4homoskedastic error. In Chapter III, this process is applied in a macroeconomic analysis of the aggregate price level. The final section of the current chapter presents a brief summary and conclusion. 2. THEORETICAL CHARACTERISTICS OF LONG-MEMORY TIME SERIES AND THE ARFIMA(p,d,q) PROCESS Consider a stationary time series, {yt}, which can be expressed in ARIMA(p,d,q) form as (1.) mm - L>d (yt - p>= aunt. or alternatively as (In) (1 - L)d (yt - u) = o/¢(L) % — “t- In this expression, L represents the lag operator, the polynomial ¢(L) 8 incorporates the autoregressive parameters of the model and is expressed as ¢(L) - 1 - ¢1L1 - ¢2L2 - . . . - dpLP, the polynomial 0(L) incorporates the moving average parameters of the model and is expressed as 0(L) - 1 - 01L1 - 02L2 - . . . — oqu, all roots of ¢(L) and 0(L) lie outside the unit circle, and ut is iid. In the case where d is zero the model in (l.') simply reduces to the standard ARMA process. However, in the case where d is assumed to be fractional,1 the model expressed in (1.) takes on characteristics unique to long-memory time series. From the expression in (1.), the difference operator (1 - L)d can be expressed in terms of the difference parameter, d, as: 1 2 (2.) (1 - L)d = [1 - dL ..3 + d(d-l)/2LL - d(d-1)(d-2)/3IL g+ . . .1 r". Similarly, this expression can be defined by the Binomial Theorem as: d . . [ ]<-1>JLJ, 0 j or as defined by the Maclaurin series: (1 - L)d = z j= 1 The range of values of d is restricted to -h < d.<:!1. Series containing a value of d within this range are stationary and mean reverting” The value of d may be expressed outside of this range as well, for example a value of d equal to .80, in which case it is assumed that the series requires first differencing in order to achieve a value of d corresponding to the range stated above. That is, a value of d - .80 for a non first differenced series corresponds to a value d - -.20 when the series is appropriately first differenced, etc. d _ ” P(-d+j) j (1 ' L) jfo I F(-d)P(j+1) IL which makes use of the standard gamma function, P(z), defined as m 1 f 32. e-Sds, for z > 0 F(z) - o m , for z — 0 . Using the formulation expressed in (2.) it can beyn seen that the model expressed in (1.) reduces to the standard stationary time series models of Box and Jenkins (1976) for integer values of d. In this case, the expression in (2.) has a finite number of non-zero terms indicating the relatively short memory of the process yt. In the case where d takes on a value of one, the operator generates first differences: (1 - L)d’yt - [1 - (1)L1 + 0]yt - yt - yt-l' In the case where d - 2 the operator generates second differences: (1.- L)dyt = [1 - (2)L1 + L2 - 0]yt ' A(yt ' yt-1)' In the case where d takes on fractional values, however, the expression for (1 - L)d has an infinite number of non:Eero terms such that the current realization of yt is a function of a long history of observations on y. This model exhibits a relatively slow decline of the weights on observations farther back in time. The model with fractional d is characterized by relatively long-term persistence and series which exhibit this type of behavior are called fractionally integrated time series. Formally, a series is fractionally integrated if after applying the fractional difference operator, (1 - L)d, it follows a finite order ARMA(p,q) process. In such case, the series is said to be integrated of Y..9 Lu» 5» am I“ f) 10 order (1, I(d), and may be expressed as an Autoregressive-Fractiona11y Integrated Moving Average, ARFIMA(p,d,q) process where for -B < d < 8 and d#0 the series is sationary and mean reverting. The theoretical characteristics of the ARFIMA process have been studied extensively by Granger (1980), Granger and Joyeux (1980), and Hosking (1981). In the simplest case where yt is an ARFIMA(0,d,0) process, the model may be expressed as d \ ./ (3.) <1 - L) yt - at. \, or alternatively, , -d (3. ) yt (1 - L) ut. Expressions for the one-sided representations that correspond to infinite autoregressive and infinite moving average processes are given, respectively, by (I) (4.) y = 2 «.y _. + u t j=1 JtJ t and m (5.) y = 2 ¢.u _. t j=0 J t J and are derived from the binomial expansions, (2.). In the special case Where ut is iid (0, 02), yt is said to be fractionally integrated white noise and (4.) and (5.) may be interpreted as being infinite autoregressive and infinite moving average representations, respectively. The coefficients on these representations for the fractionally integrated 11 process may be expressed in terms of the gamma function as 4'} = N j r(-d)rF f“ 321' Granger and Joyeux (1980) have shown that as j + m, ijl = Alj'd'l and Pj = Azjd'l, where the A1 i=1,2 are constants to be determined from initial boundary conditions and are expressed as A.l - [l/(d-l)!] and A2 -r[l/(-d-l)!]. Clearly, |1rj| and zbj will decay to zero more slowly than the decay exhibited by the moving average coefficients of the standard stationary and invertible ARMA model. From this property it should be apparent that no finite-order ARMA model could adequately approximate a long-memory time series for large lags. Hosking (1981) derives an.expression for the autocovariance function of the fractionally integrated process, 1 a cov[Xt,Xt_j] for t-O, i1, i2, J , and shows that as the lag j increases the autocovariance function behaves as 7j - 0(j2d-l). Granger and Joyeux (1980) show that the fractionally integrated process is covariance stationary for 0‘<rr ' 12 where -h < d < 8. Using Sheppard's formula,2 for large j the autocorrelation function of the fractionally integrated process may be expressed as where A3 is a constant expressed as [-d!/(d—l)!]. In contrast, the autocorrelation function. of the standard stationary invertible ARMA process may be expressed for large j as ._._ j pj A40 . where |0| < l, which tends to zero exponentially as j n Q. As with the impulse response weights expressed in (4.) and (5.), the rate of decline of the autocorrelations of the stationary ARMA.process is much faster than that of the fractionally integrated process, further indicating the inability of the finite ARMA process to model the type of persistence present in long-memory data. 3. A SURVEY OF LONG-TERM PERSISTENCE AND BROWNIAN MOTION The existence of highly persistent, long-memory data can be found not onLy in the field of hydrology in geophysical data (Hurst (1951, 1956), Mandelbrot and Wallis (1969a), and McLeod and Hipel (1978)) but also in the areas of image processing (Kashyap and lapsa (1984)) and meteorology (Kashyap and Eom (1988)) as well. In searching for a physical ‘2 Sheppard' formula indicates that for large lag, j, the expression F(j+a)/P(j+ ) is well approximated by ja’b. 13 explanation of the cause of strong persistence in data, Hosking (1985) noted that the existence of long-term dependence in a series quite likely ——' could arise in models which require solving stochastic differential, f “NW." - equations. These models generally give rise to the properties of self- / K . similarity and power-law covariance, which are characteristic of long- memory data. Models which utilize stochastic differential equations often arise in hydrological and geophysical applications, as noted by Hosking (1985), since these types of processes offer the potential to provide physically realistic specifications of many variables which are encountered in this field. It should not be surprising, then, that pioneering research in the area of strongly dependent time series originated in the field of hydrology with the work of Hurst (1951, 1956) who noted the long-term dependence in geophysical and hydrological time series in analyzing data on river discharge and reservoir storage capacity.3 In analyzing series which displayed long-term dependence Hurst developed the rescaled range statistic, R/S, which expresses the range of Partial sums of deviations of a time series rescaled by the standard deviation of the series. Consider the time series {xr: t-1,2,...,n}, _ Which Hurst ass ed to be_normal and independentlLdistribgted. The R/S Elli—m an.-.”.__.,..._...._--_._W______l statistic is calculated as R/S - s-1{max(U1,...,Un) - min(U1,...,Un)} \ 3 An excellent survey on modelling long-term persistence in hydrological time series can be found in Lawrance and Kottegoda (1977). 14 j _ _ _1 n where U. = 2 (xt- x) , x - n 2 xt J t-l t—l n and 32 - n-1 E (xt- x)2. t-l The term max(U1,...,Un) represents the maximum, over j, of the partial sums of the first j deviations of x from the sample mean, E. This t quantity will always be nonnegative since the deviation of x from its mean, summed over all observations, will be zero. Similarly, the term min(U1,...,Un) is the minimum, over j, of the partial sums of the first j deviations of xt from the sample mean, and this quantity will always be nonpositive. The range is defined as the difference between these two quantities. The variable 3 is the usual maximum likelihood standard deviation estimator and is used to normalize, or "scale", the range. A.notable development of Hurst's work was the finding of an.apparent discrepancy between what was predicted by theory for the value of the R/S statistic and what was observed in empirical analysis for many hydrological and geophysical time series. In applied work Hurst observed that the value of R/S could be approximated by (n/2)k, where n represents sample size and 0.6 < k < 0.8. However, when the data were generated by a random process which assumed.the dependence between distant observations to be negligible, the value of the rescaled range was asymptotically proportional to n7. It was then hypothesized that Hurst's observed phenomenon should. not hold for relatively large samples; but this hypothesis could not be confirmed.empiricallyu Hurst investigated.data on water flows of several rivers, including the Nile, and various other physical series including rainfall, temperatures, thickness of tree rings, 15 thickness of stratified mud beds, and sunspot numbers. In each case the discrepancy ‘between theoretical expectations and empirical evidence remained” This discrepancy came to be known as the "Hurst phenomenon" and its implication was that the geophysical records which displayed this phenomenon must be considered to posses an infinite span of statistical interdependence. Subsequent research demonstrated that the presence of the Hurst phenomenon in certain data could in fact be explained by the degree of persistence in a series. That is to say, the Hurst phenomenon would be present in those stationary time series which displayed long-term dependence.“ When. applied. to long-memory' data, the rescaled. range statistic would not behave as a function of n8 as it would.when applied to short-memory series. These findings began to indicate the importance of recognizing the existence of long-memory characteristics in data and indicated the need for the specialized type of analysis that these series require. Since the work of Hurst in 1951, the rescaled range has been refined by Mandelbrot (1972, 1975) and others [Mandelbrot and Taqqu (1979), and Mandelbrot and Wallis (1968, 1969a, 1969b), for example], and Mandelbrot has suggested the use of the rescaled range statistic as a method for detecting long-range dependence in a series. Several researchers have shown the benefits of using this method over the use of, for example, the autocorrelation structure of a series or some measure of variance ratios. Mandelbrot (1972, 1975), for example, has shown the advantage of the rescaled range statistic by demonstrating the almost-sure convergence of ‘ See, for example, Mandelbrot and Wallis (1968), Klemes (1974), McLeod and Hipel (1978) and Hosking (1984). 16 this statistic for series with non-finite variance. In addition, Monte Carlo simulations by Mandelbrot and Wallis (1969a) have indicated that the rescaled range statistic can be used to identify long-term dependence in time series which are non-Gaussian and are compounded by some degree of skewness and kurtosis. The usefulness of the rescaled range statistic in detecting long-range dependence is discussed more fully in L0 (1991). The analysis undertaken by Hurst was extended by several researchers in the field including Mandelbrot and Van Ness (1968) who suggested that strong persistence and stationarity were not necessarily mutually exclusive. Mandelbrot and Van Ness (1968) proposed the use of the fractional Brownian motion process to model strongly dependent time series since this process was able to characterize the long correlation structures associated with long-memory series. The attractive feature of fractional Brownian motion which made it useful in modelling long-term dependence was the infinite span of interdependence between the increments of the process. The derivation and statistical properties of the Brownian motion are well documented in the statistical and physical science literature (see Mandelbrot and Van Ness (1968), Hosking (1981), Mandelbrot (1982), and.Jonas (1983), for more information and further references) and only a brief summary is given here. The formal notion of Brownian motion was first brought to the attention of the scientific community in the early 18003 to describe the diffusion processes investigated by botanist Robert Brown. The Brownian motion, represented as B(t), is a continuous-time stochastic process. The increments of the Brownian motion, represented as B(t+u) - B(t), are Gaussian distributed with zero mean and variance u. A generalization of the Brownian motion process is fractional Brownian motion, represented as 17 B (t), which is a continuous time, self-similar process. The characteristic of self-similarity is defined for a continuous-time stochastic process, {xt}, which satisfies the condition that for some positive constants a and h, xt and a-hx t have the same finite joint 1% aired W That is to say, the distribution of a self-similar process \k distributions . will be invariant to the scale under which the process is observed; the V _________._. limb _ . statistical behavior of an observed phenomenon characterized by self- ’W _. .... .. _ -—-. -...H." " similarit should have the same probabilistic structure whether the ‘-—-.u—-—_.H‘ phenomenon is observed daily, weekly, monthly, or otherwise. AL“...— “amt- -. ,_ .. . -.. sis. -.-.- ”J1 -.4 7- H)th fractional ,F‘. ,._ L._ ...-rm" Fractional Brownian motion, .- whichhis ”the (1': w —--—--v .ngm.~—_ _.. derivative of Brownian motion, has stationary Gaussian increments, and .. pr F. I“ I- u 'M Iv-“l’ *N" ‘ 7-,," .- _ma-r “an... ”n..v.1-.o-,—- - ... u-A""' these increments are known as "fractional CaUSSi§Q»DQISQ"- The increments _‘fi._..--~\-‘.- . 0f the fractional Brownian motion have hyperbolically decaying correlation functions which enable the process to account for the long-term behavior 0f Strongly dependent time series. This is due to the relatively slowly varying span of interdependence between increments of fractional Brownian motion which vary according to power law. The autocovariance function of fractional Gaussian noise, 7k - E[xt’xt-k] , is given as ‘ 7k = C{(|kl + 1)2H - 2Ik|2H + (Ikl - 1>2”}. v———-— Where C is a positive constant and the parameter H satisfies 0 <_,_ng___1.. The fractional Brownian motion process may be expressed, as i \ t .. i (6.) BH(t;) = f (t-s)HJS dB(s), where -00 < t < 00. NA ’ The Parameter H incorporates the long-memory element of the model, and is \__ ‘w—v kn Own as the Hurst coefficient. To ensure stationarity of the process the W l8 parameter H must satisfy the conditign_0‘< H < l, and fractional Brownian motion can be divided LLQEQ...-.§h.1:se -..Sl.i.f.f.91?§fl.t......falniliefit...§§.§.§.ntially. f depending on the rangaflgfggalueswgwaJ For H in the range of H < H < l, the fractional Brownian motion process willbe meaningfulflfggwempirigal applications in hydrology; the case where 05 <5 will*not tbe useful in Mmrm‘ FHIELESEE$DS° When H - H, the fractional Brownian motion simply reduces \----rvm.- eh 'm- ,._..,..,.... -.r to the Brownian motion process implying that observations separatedbflyma mfimw Wmu—t—a "x“. relatively large time span are statistically independent. Since the __‘__fl___,,_mw,_ .1 expression in (6.) above is divergent, it is only feasible to look at stationary increments of BH(t), the fractional Gaussian noise process, which can be expressed as t o (7.) BH(t) - BH(O) = f (t:-s)H'L2 dB(s) - f (-s)H'8 dB(s). Estimation of the degree of persistence of the series, then, involves estimation of the parameter H, the Hurst coefficient, from the k \_——r 4“ '— specification given in (7.). Estimation of the Hurst coefficient has been explored in.Hurst (1951, 1956) and in Mandelbrot and Wallis (1969a, 1969b) who suggested constructing :pox—diagrams" as a method of estimating the parameter H. The pox diagram plots various values of the R/S statistic of a series along‘with their average. The scatter of points usually produces a short, convex section and a longer, linear section which will have an average slope equal to H. Thus, the value of H is approximated from the shape of the curve in the pox diagram. Further research utilizing the fractional Brownian motion to model strongly dependent series and to estimate the degree of persistence of a series includes Mandelbrot (1972, 19 1975), Mandelbrot and Taqqu (1979), and Mandelbrot and Wallis (1968), among others. The use of fractional Brownian motion to model strongly dependent, time series involves some disadvantages, however, and hence may not be ideal in modelling all series exhibiting long-term persistence. For example, Mandelbrot and Wallis (1969b) have foundflgreatjvariability in the estimated values of H when the R/S pox diagram technique is used, and.have even noted that in certain cases the researcher must be content to wager an intelligent, though imprecise, guess as to the value of H. Mandelbrot and Wallis (1969b) note that estimation of H through the use of the pox .... a ...-M...»- w w..-“ ”M— --- diagram becomes even more difficult in the presence of strong periodic elements in a series. In addition, Wallis and Matalas (1970) have proven that estimates of the Hurst coefficient have relatively large biases and large sampling variances, and have indicated that not much is known about the large sample efficiency of the estimator. An additional shortcoming ... ”...... of the model is its limited flexibility in modelling series which embody other characteristics in addition to the long-memory element. Such series w would require multiple parametersinwthemodel yet fractional Brownian motion offers onlyone parameter, H, with which to describe the behavior ~y_.,.— a»; of a series. Consequently, this process limits the range of correlation structures which may be represented by fractional Brownian motion. In light of the shortcomings of fractional Brownian motion, anflalternative class of models has been proposed which generalizes the ARIMA(p,d,q) models of Box and Jenkins (1976) to allow for non-integer values of d. This class of model, known as the fractionally dIfferenCedmprOCess, may allow for more flexibility in modelling series which contain long-memory. The next section provides a detailed survey of the fractionally 20 differenced process. 4. A SURVEY OF LONG-TERM PERSISTENCE AND FRACTIONAL DIFFERENCING It is often the case in macroeconomics and financial economics that time series analysis of certain variables involves the problem of extreme persistence in the mean and'variance of these variables. The existence of such long-team dependence in economic data was first noted by Granger (1980) who suggested that the aggregation of certain dynamic economic -r‘... . A ._ f” relationships produced. a class of model that exhibited long-memory characteristics. J“, For example, consider the series x1t and x2t which are generated by the process (8.) xjt=ajxj,t-l+6jt ,j=1, 2 , and elt’ €2t are independent white noise processes with zero mean. As noted in Granger (1980) the sum of the processes, it - x1t + x2t’ will obey an ARMA(2,1) process where the autoregressive part of the model is given by (l-aJL)(l-a2L). Consider such an aggregation for N independent series each following an AR(l) process as expressed in (8.) and each with differing values for the autoregressive parameter, a. The aggregation of these processes will follow an ARMA(N,N-l) _nrocess, provided that no cancellation of roots occurs betwggnmautoregressivewandflmovingfiaveragem parameters in the model. In assuming a specific distribution from which ‘_ -..- .-.. .a-.n -' " .-r..—-.'--- ¢ ~ 21 the autoregressive parameters are drawn,5 Granger (1980) derives an approximation.of the'ktilorder autocovariance function.of the aggregated process. This approximation is derived fromthe_standard Fourier series expansion of the spectrum ofmthe“processmandmismgiyeg as 5k — A5k(1-Q) o. A-‘nmm -‘Ulm-o-Hflv—g—I'flh " where A5 is some constant. Recall from section 2 that the h:th order autocovariance function.of the fractionally integrated.process is of order \.____..___.__. .kSEE-l). The parameter of integration in the fractionally differenced model, d, corresponds to (l-q/2) in.this case such that the aggregation.of w‘- the N dynamic processes, it, is said to be integrated of order (l-q/2). The model exhibiting this form of autocovariance function is characteristic. of' the strongly' persistent, long-memory, fractionally integrated process, as discussed in section 2. Thus for N very large the aggregation of N individual dynamic . .-—..-..~u’- - processes produces series WhiCh display those characteristics typical of ,flahmhflgywwmflw‘afl,. long-term dependence as represented.by the fractionally integrated medel. ~ r-..._ , Hw~nv_ ‘_ .—..- ... ...... “"“‘* cvr‘ Ineconomics, such models are very likely to arise since many economic variables are aggregates of a large number of smaller_series; relevant examples include the aggregate price level or ‘the .inflation Irate, aggregate consumption data,wnational income data, etc. As noted by Kfinsch (1986), the properties of such long-memory data cannot be explained asymptotically by stationary models characterizedwby finite variances and weak dependence. The alternative proposed by Granger and Joyeux (1980) and Hosking (1981) was derived by fractionally 5 For mathematical simplicity Granger assumes a beta distribution for a, considered on the range (0,1). The particular choice of the distribution assumed for a is not crucial for deriving the result of highly persistent series through aggregation of dynamic processes. Granger investigates further generalizations which consider alternative distributions that produce the same result. 22 differencing the random walk process. The fractionally differenced model offers an alternative to the fractional Gaussian. noise processflflgf Mandelbrot and Van.Ness (1968) and.Mandelbrot and Wallis (1969a), although the two models are not completely without similarities. The fractionally differenced process has a covariance structure similar to that of fractional Gaussian noise and is also asymptotically self-similar. In addition, the parameter of fractional differencing in.the model of Granger and Joyeux (1980) and Hosking (1981) is related to the Hurst coefficient as d - (H - H), where H is the parameter described in (6.) and (7.) (see Granger and Joyeux (1980), Hosking (1981), and Geweke and Porter-Hudak (1983), for a more detailed analysis). Recently, much attention has been given to estimation of the degree of fractional differencing of series that display forms of long-term persistence. The earliest contributions in this area.were made by Janacek (1982), who proposed a technique to estimate the degree of fractional differencing of a series based on the log of the power spectrum of the series, and.Geweke and.Porter-Hudak (1983) who proposed.a semi-parametric, two-step estimation procedure which also utilizes the spectrum of a series. These procedures are based in the frequency domain since long- memory time series can be equally well represented in either the time domain, where series can be expressed in terms of stochastic difference equations, or in the frequency domain, where series can be expressed in terms of the relative importance of various cycles that occur within the series. In the context of the frequency domain a long-memory process as expressed in (1.), which exhibits spectral density function, fy(w) - [4sin2(w/2)]quu(w), will be characterized by an infinitely increasing 23 spectral density function for frequency values, w, near zero.6 The estimation technique proposed by Janacek (1982) was designed to determine the order of fractional integration of a series and is grounded in the idea that the low-frequency end of the log-spectrum of a series gives some information about the degree of persistence of the series. Janacek (1982) considered a univariate series {yt: t = l, 2, ... ,N} which when differenced (1 time gives rise to a stationary process, ut, with rational spectrum fu(w) where w represents frequency. Applying the standard procedures of linear filters, an expression for the log of the spectrum of x is given as log fx(w) = -d log[2(l-cos w)] = log fu(w). Introducing a "weighting" function as W(w) = -H log[2(l-cos w)] and using the standard results for Fourier series, an integral expression for the weighted log of the spectrum can be given as 1l’ co 00 l/« f W(9) log fx(e) do = d X 1/R2 + 2 aK/(ZK) 0 K=l K-l where fu(w) - a0 + alcos w + a2cos 2w + . . ., such that the aK, K-1, 2, 3, . . . , are the Fourier coefficients of the spectrwm of ut. This allows for estimation of d without the need for model specification. The O 2 . minimum-mean square error of predict1on, a , is given as 5 This particular characteristic corresponds to hyperbolically decaying correlation functions when the series is expressed in the time domain. Where 24 2 -1’r log a = x I log 2n f(w) dw 0 and the step mean-square error of prediction, ai is given as 02(1 + bi + 2 2 , b2 + . . . + bK ). The bj s can be expressed as (b1 c1), (b2 c2 + 2 . c1 /2!), . . . , where ZCK = aK. The cK will decay exponentially for a short-memory, stationary process where the cK are assumed to be zero for all values of K greater than some finite number, M. For a long-memory series, however, the cK will decay at the much slower hyperbolic rate. Janacek (1982) proposed estimation of d based on an estimate of the cK from the following formulation “ -2 -/ZK 0‘” MK M W A where S - 1/n f W(w) log f(w) dw . 0 Clearly, in this specification, the estimate of the degree of persistence will depend on the value of M. Janacek proposed that for M large enough the estimate of d will be unbiased, and suggested that M be chosen as being equal to two standard deviations of d. Choosing M too large, however, does necessitate a trade off in increased variance of the estimated parameter, and.Janacek (1982) presented small scale simulations to investigate this trade off. The results of the simulations indicate that for M chosen as suggested above, the variance of the estimate is within a 95% confidence interval of theoretical predictions. Similarly, the motivation for the estimation technique proposed by 25 Geweke and Porter-Hudak (1983) is like that proposed by Janacek (1982) in that it utilizes the low-frequency end of the spectrum of a series. The procedure proposed by Geweke and Porter-Hudak is based on the observation that the spectral density function of a long-memory time series when expressed in levels is unbounded at w - 0 but disappears at w - 0 when the series is first differenced” Consequently, Geweke and Porter-Hudak (1983) use the low-frequency component of the spectral density function to estimate the degree of persistence of a series. Consider the simple case of the ARFIMA(0,d,0) process as expressed in (3.) where ut is a white noise process. Again, applying the property of linear filters, the spectral density function of yt, fy(w), can be expressed as -iw|-2d (9.) fy(w ) = II - e fu(w). An equivalent expression may be given as (9.') fy(w) = [4sin2(w/2)]-dfu(w) where fu(w) represents the spectral density' function. of 11. Taking logarithms of both sides of equation (9.') and evaluating at the harmonic ordinates wj - 2nj/T gives . 2 (10.) log[fy(wj)] - log[fu(0)]-dlog[451n (wj/2)] + log[fy(wj)/fu(0)]. 26 The motivation for this estimation procedure is based on the similarity of (10.) to an ordinary least squares regression where the last term on the right hand side of (10.) represents the regression error and the slope parameter, -d, can be estimated with the usual least squares techniques. The estimation procedure of Geweke and Porter-Hudak (1983), hereafter referred to as GPH, requires two steps. The first step involves obtaining an estimate of d from equation (10.) using OLS. This regression is performed over the range of low-frequency ordinates of the spectrum, wl, w2, .. wm, and involves replacing fy(wj) with the periodogram of the series. In this formulation it is critical to estimate the OLS regression using the appropriate number' of’ ordinates, m, since the estimation technique is based on performing the regression over only the low- frequency end of the spectrum” Use of too small a regression sample, that is, inclusion of too few ordinates, can mean that relevant long-memory information contained in the low-frequency end of the spectrum is not being included in estimation. Conversely, use of too many ordinates can lead to erroneous inclusion of medium- and high-frequency ordinates in estimation, which will cause the estimate of d to be "contaminated". Geweke and Porter-Hudak (1983) have suggested that the optimal number of low-frequency ordinates to use in the OLS regression should be a function of the sample size of the series under investigation. They suggest choosing the number of spectral ordinates as being Ta, where a would typically take on values such as .50, .60, and .70. Alternatively, Sowell (1990) has suggested that the optimal number of spectral ordinates to use in the OLS regression.should not depend on sample size but, rather, should be a function of the annual periodicity of the data and the long-run frequency of the series. Consequently, Sowell (1990) has suggested 27 choosing the optimal number of ordinates, m, as a number of years of the data lowest long-run frequency of the series ' m Once a consistent estimate of the parameter (1, the degree of fractional integration, is obtained by the GPH procedure outlined above, this value is then used in the second step of the estimation procedure to transform, or filter, the observed series for the appropriate degree of fractional differencing. Once the series has been properly transformed the remaining model parameters may be estimated by standard procedures of time series analysis performed on this transformed series. This second step of estimation will provide estimates for the autoregressive and moving average components of the series. The consistency of the GPH estimator of d in the presence of weak dependency in u is proven in t Geweke and Porter-Hudak (1983) for -8 < d < 0, and is conjectured for 0 < d < h. The estimator is shown to be asymptotically normal, but is not JT consistent. To date, there appears to be limited applied work utilizing the estimation procedure outlined by Janacek (1982); however, the GPH estimator has been widely applied in many areas since its introduction (Diebold and Rudebusch (1989), Choi and. Wohar (1990), Baillie and Pecchenino (1991), Cheung (1991), Tieslau (1991), and Agiakloglou, Newbold, and Wohar (1992), for example). Some difficulty has been encountered, however, in applying the GPH estimator in practice. Agiakloglou, Newbold and Wohar (1992) , for example, have demonstrated with simulation results that the GPH estimator can have substantial bias when yt is generated by an autoregressive process with a substantially large positive AR parameter, or whenyt is generated by a moving average process 28 with a substantially large positive MA parameter. Agiakloglou, Newbold and.Wohar (1992) discuss the reasons for these results noting that whenyt exhibits a large positive value for its AR parameter, the log spectrum is downward sloping in the low-frequency range, rather than constant. As such, there will be significant bias in estimating the slope parameter. Agiakloglou, Newbold and Wohar (1992) present simulation evidence indicating that the bias of the GPH estimate decreases slowly as sample size increases, but this comes at the expense of a trade-off of increasing the standard error of the estimate. Similarly, the presence of large bias in the GPH estimate when yt exhibits a large positive moving average coefficient is due to fact that for this type of process the spectral density of yt is close to zero in the low-frequency range of the spectrum. Agiakloglou, Newbold and Wohar (1992) show that for series exhibiting large positive MA parameters the bias of the GPH estimate for the model also decreases slowly as sample size increases. In addition, Hosking (1985) noted the limited applicability of the fractionally differenced ARMA process in applied work in hydrology due to the linearity of this model. Hosking noted that many typical hydrological time series, such as daily streamflow data, often embody non-linear characteristics which are unlikely to be well explained by the type of model proposed by Geweke and Porter-Hudak (1983). As a result, the GPH estimation procedure appears to be most appropriate in modelling purely fractionally integrated. white noise, leaving little flexibility for practical applications. Since the seminal work of Geweke and Porter-Hudak (1983) on estimation of fractionally integrated series, several alternative 29 estimation procedures have been proposed for the ARFIMA process described in equation (3.). One such. procedure involves maximum likelihood estimation (MLE) of the parameters of the model given in (3.), and three alternative approaches which utilize maximum likelihood estimation techniques have been proposed. Tflna first technique proposed involves approximate MLE based in the time domain, the second involves approximate MLE based in the frequency domain, and the third estimation technique involves exact MLE based in the time domain. Each of these are further discussed below. The first alternative estimation procedure utilizing maximum likelihood estimation techniques was proposed by Hosking (1984) who, in dealing with many non-linear hydrological and geophysical data series, suggested that methods based on maximizing the likelihood function of a series were the most likely methods for producing efficient parameter estimates for fractionally integrated time series. The method.proposed by Hosking (1984) utilizes approximate MLE techniques in the time domain where the log likelihood function of a series, yt, is expressed as log £' IV1‘1 (y-ul) where A represents the vector of model parameters, p represents the mean of the process, V(A) is the covariance matrix of y, which is assumed to be independent of the mean, and 1 is a vector of ones. Hosking (1984) assumed, for convenience, that the likelihood function followed a normal marginal distribution. Direct maximization of the likelihood above has been.proposed in.applied.work by McLeod and.Hipel (1978) in estimating the parameters of the fractional Gaussian noise process of (7.). However, 30 Hosking (1984) noted the computational expense of this method since it requires the inversion of a (T x T) matrix of covariances at each iteration of the procedure which necessitates a nmnber of arithmetic operations of order T3. The cost of repeated iterations of this process in maximizing the likelihood function, f, can be excessive. As an.alternative, Hosking (1984) suggested replacing the likelihood function with an approximation, 2, which would be more easily evaluated and then.maximizing this approximation. This approximation considers the fractionally integrated parameters of the model separately from the ARMA parameters. Estimation of the model parameters, then, simply involved application of the standard algorithm of Ansley (1979) to the approximation of the likelihood function. The second alternative estimation procedure, proposed by Fox and Taqqu (1986), also utilizes approximate nmximum likelihood estimation techniques but is based in the frequency domain rather than in the time, domain. Within this framework Fox and Taqqu (1986) assume a stationary Gaussian series and construct an asymptotic approximation to the likelihood function of an ARFIMA process. Their work follows the approach suggested by Whittle (1951) which involves maximizing - ex 0 P [ Z'AT(0)Z] 2T02 where Z - (X1 - E ., X - XT)', and X represents the sample average T’ T of the process X. The matrix AT(0) is a (T x T) matrix of covariances which has j,kth element aj_k(0) as given by 31 1 " i'x -1 aj(6) = 2 f e 3 [f(x,6)] dx. (2«) -« The maximum likelihood estimates, ET and ET, are defined as those values of 0 and a that satisfy the above maximization; maximization of the above expression is equivalent to choosing FT to minimize a$(0) - [Z'AT(9)Z]/T and then setting '52 equal to mid-T). Fox and Taqqu (1986) derive T consistent and.asymptotically normal estimates of the model parameters for the case when d is in the range 0 < d < H. Maximization.of the likelihood function is based in the frequency domain and the procedure is invariant to the true, but unknown, mean of the process. The third estimation procedure, proposed by Sowell (1992), utilizes an exact maximum likelihood estimation technique which is based in the time domain. This estimation procedure considers the ARFIMA(p,d,q) process as given in (1.) with ut ~ NID (0, 02) and d < 8. The probability density function is given by Sowell to be f = (210“2 IEI'l/2 exp{-HytE-1yt} where E is the covariance matrix of the ARFIMA.process yt, and.E is of the standard block Toeplitz form. The likelihood function of the process is derived in the usual manner and maximization is achieved with standard computer algorithms and use of the Choleski decomposition in evaluating the inverse of the covariance matrix. The inversion of the (TT><'T) matrix of autocovariances which is required by this time domain exact MLE technique is the main disadvantage of this procedure. Inversion of this covariance matrix is required at 32 each iteration of the estimation process and this becomes particularly cumbersome for fractionally integrated processes since even for fractionally integrated white noise eaCh autocovariance is :1 ratio of gamma functions. This makes arithmetic operations quite difficult. Furthermore, for the more general ARFIMA(p,d,q) process, a typical autocovariance is a nonlinear function of four hypergeometric functions necessitating even. greater computational expense. Clearly, maximum likelihood estimation isldifficult even.for an ARFIMA.process with.NID (0, 02) disturbances. The relative efficiency of the two estimation procedures of Sowell (1992) and Fox and Taqqu (1986) has been analyzed in a recent study by Cheung and Diebold (1991). The exact MLE time-domain procedure proposed by Sowell (1992) is asymptotically equivalent to the approximate MLE frequency-domain procedure proposed by Fox and Taqqu (1986), although the properties from estimation in finite samples will not be equivalent. Cheung and Diebold (1991) perform a detailed simulation study which compares both large and small sample properties of the two estimation techniques. They analyzed the fractionally integrated white noise process with non-zero mean, p, which may be represented as: (1-L)d(yt-#) - at where 6t ~ NID (0, 02). The results of this analysis indicate that for finite sample sizes when the mean of the process is known, an unambiguous ranking of the two estimation procedures is evident. That is, the mean—squared error (MSE) 33 of the parameter estimate of the frequency-domain approximate MLE of Fox and Taqqu (1986) is greater than that of the time-domain exact MLE of Sowell (1992) for the case of known mean. By this criterion the exact MLE approach of Sowell (1992) is superior to the approximate MLE approach of Fox and Taqqu (1986) when the true mean of the process is known. However, in the case of unknown mean in finite sample sizes, Cheung and Diebold (1991) show that the relative efficiency of the approximate MLE increases significantly. This is particularly relevant since most practical applications of these estimation procedures will arise in cases where the true mean of the process is unknown and consequently must be estimated. Such estimation is based on de-meaned data and Cheung and Diebold (1991) have shown that precise estimation of the mean of a long- memory process is often difficult. The frequency-domain estimation procedure of Fox and Taqqu (1986) will be invariant to the true mean of the process; the time-domain estimation procedure of Sowell (1992) will not be. By this criterion, the estimation procedure of Fox and Taqqu (1986) may be preferred to that of Sowell (1992). In addition Cheung and Diebold (1991) indicate that for the case where the mean of the process must be estimated, the efficiency of the exact MLE time-domain approach of Sowell (1992) decreases as the value of d increases. That is, the rate of convergence of the sample mean of the process is a function of the parameter (1 in that as the value of (1 increases the rate of convergence decreases, which further decreases the performance of the exact MLE time-domain approach. This is due to the fact that while the maximum likelihood estimates of the ARMA parameters expressed in (1.) are If consistent, the rate of convergence of the maximum T+8-d likelihood estimate of the population mean is (for further 34 discussion, see Li and McLeod (1986)). The maximum likelihood estimation procedures discussed above, whether approximate or exact.MLE, are each predicated on the assumption.of NID innovations. That is to say, the approximate MLE technique of Fox and Taqqu (1986), for example, cannot be applied to data which exhibit forms of time dependent heteroskedasticity, which may well be encountered in practical applications. To date, no attempt has been made to address the problem of obtaining maximum likelihood estimates for a process which involves more complicated data generating processes where the presence of fractional integration may be compounded with non-normality and non- homoskedastic error. This can prove to be a significant factor in assessing the usefulness of an.estimation procedure since such data series are likely to arise in practice. The next section will present one such estimation procedure which utilizes the estimation technique of maximum likelihood to estimate the model parameters of a series but yet does not restrict the distribution of the series to satisfy the assumption of normal and independently distributed error. 5. THE THEORETICAL PROPERTIES OF THE ARFIMA—GARCH PROCESS The model developed in this section merges the ARFIMA(p,d,q) process of Granger and Joyeux (1980) and Hosking (1981) with the GARCH(P,Q) process of Engle (1982) and Bollerslev (1986) to allow for simultaneous modelling of fractionally integrated behavior and time dependent conditional heteroskedasticity. The property of time varying conditional variance has been investigated by Engle (1982, 1983) who proposed the Autoregressive Conditionally Heteroskedastic, ARCH, process as a suitable model for series exhibiting this characteristic. ihn this model, the 35 conditional variance is assumed to be a linear function of past squared errors; the conditional variance is allowed to change over time as a. function of these past squared errors, leaving the unconditional variance constant. The ARCH model, then, is able to characterize series which tend to exhibit periods of extreme values followed by other periods of extreme values, whether of the same sign or not. Engle (1982) proposed this model for inflation realizing that the uncertainty of inflation tended to change over time. The Autoregressive Conditionally Heteroskedastic model of Engle (1982) was later extended by Bollerslev (1986) who generalized the model to allow for a much more flexible lag structure. The generalization proposed by Bollerslev (1986) allows for an additional parameter in the conditional variance equation; under this specification the conditional variance is assumed to be influenced by the squared residuals of the series as well as lagged values of the conditional variance. This model, known as the Generalized Autoregressive Conditionally Heteroskedastic GARCH(P,Q) model, is used to model series which exhibit non-constant conditional variance over time. The general model proposed in this analysis will be referred to as the ARFIMA(p,d,q)-GARCH(P,Q) model and can be represented with equations (11.) through (14.) below: + 60 + u d . (11.) (1-L) yt = b xlt t t (12.) ¢(L)ut = 6(L)et 0 ~ D(0, 02) (13.) etl t_1 t 36 (14.) fi(L)aE = w + a(L)e§ + y'th. In this specification, d.E (-8, 8) such that yt is fractionally integrated of order d, and x1t and x2t are vectors of predetermined variables. The polynomials ¢(L), 0(L), 6(L), and a(L) are represented as ¢(L) - 1-¢1L - HH¢EP,oao-1w L+ l 1 a2L2 +...+aQL9, and all the roots of ¢(L), 0(L), fi(L) and.a(L) are assumed to lie outside the unit circle. The innovations of the model, 6 q P L -...-0qL , p(L) = l-filL -...-flPL , a(L) - a t’ are assumed to follow a conditional density D, which may be assumed to be either Normal or Student t. Additionally the time dependent heteroskedasticity of 0: follows the Generalized Autoregressive Conditionally Heteroskedastic, or GARCH(P,Q), model of Engle (1982) and Bollerslev (1986). In the special case where 6 - 0 and b - 0, equations (11.) and (12.) describe the ARFIMA process of' Granger and Joyeux (1980). One useful application afforded by the ARFIMA-GARCH model is that the specification will allow for the possibility of testing for the presence of feedback between the standard deviation of y and lagged values of y. In this framework, the parameter 6 is used to test for this effect such that when 6 fl 0 volatility is allowed to influence the mean of yt. The model specification will also allow for a test of whether lagged predetermined variables influence the conditional variance of'yt. This is achieved by allowing for the predetermined variable to enter into the conditional variance equation (14.) by being included in x2t' A positive and significant value for the parameter 7, then, would indicate the existence of an effect of lagged predetermined variables on yt. Such tests will prove valuable in empirical work in subsequent analysis. The likelihood function of the full model represented by equations 37 (11.) through (14.) is given by £(A,d;y) where A is the parameter vector A' - (b'6 ¢'0'w a'fi'). Bollerslev (1987) presents an extension to the ARCH model of Engle (1982) in which the conditional density of the process yt is assumed to be Student t with v degrees of freedom. This specification can by particularly useful for analyzing series which exhibit excessive tail distribution, perhaps due to outliers in the series, since it allows for specific modelling,of unconditional excess kurtosis in an observed series. Following Bollerslev (1987), the log likelihood function for the Student t distribution with T observations can be expressed as (15.) log £(A.d;y) = T[log r<£§l> - log r<§> - % log 1 - T l 2 [log 02 + (u+1) log(1+e2 0-2 2 t=1 t t t (u-2>'11. When u-1 approaches zero, the t distribution approaches a normal distribution; but for u-1 > O the t distribution will be quite different from the normal distribution, exhibiting substantial leptokurtosis or fat tail distribution. Consequently, the likelihood function expressed in (15.) will be appropriate for those series for which the inverse of the degrees of freedom parameter, v'l, takes on values greater than zero. Estimation of the model represented by the system of equations (11.) through (14.) may be achieved through standard maximum likelihood 38 estimation (MLE) techniques. The process of performing exact MLE on the entire system, however, can be rather difficult especially given the allowance for general conditional densities, D, and the presence of time dependent conditional variance as specified in (14.). Attention may be restricted to D being either Normal or Student t to circumvent this difficulty, and approximate maximum likelihood estimates of all model parameters may be obtained via two separate methods which are described below. The first method which may be used to obtain approximate maximum likelihood estimates of A is referred to as Method I and involves direct maximization of equation (15.). This may be achieved numerically through standard maximization of the likelihood function via use of any of the standard computing algorithms such as the Berndt et. a1. (1974) algorithm. This method provides approximate ML estimates of all model parameters simultaneously'anduwill'provide appropriate asymptotic standard errors for the parameter vector A. 'Unfortunately, this method entails the problem of setting starting values for the process since the presence of d, the fractional differencing parameter, has the effect of making initialization conditions persist for a longer period than would be the case with the standard ARMA process. The second method which may be used to obtain approximate maximum likelihood estimates of A is referred to as Method II and considers the conditional likelihood with respect to the fractionally integrated parameter, d. The likelihood function for this method is given by {*(A;y) and is related to the full likelihood as: £j. Consequently, the Jacobian of the transformation, J(d), is unity so that * £ (My) = £(»\.d;y). Once this trivial adjustment is realized, full maximum likelihood estimates of the parameter vector A may be obtained from the standard first-order conditions which are represented as a£ _ 0 6A 6A ' This will give the maximum likelihood estimates, A(d,y). The concentrated likelihood with respect to A is then defined as £° - £[A. when the series has been regressed on an intercept and also possibly a time trend. The estimate of the disturbance variance, 32(k), is computed in the same manner in which its equivalent in the Phillips and Perron (1988) test is computed; a Bartlett window adjustment based on the first k sample autocovariances is used, as suggested by Newey and West (1987). The test statistic ; represents the statistic when the residuals are computed from a regression with only an intercept. The test statistic hf represents that for' which. a time trend is included in the initial regression. Both hp and ;T are shown to be asymptotic functions of a Brownian bridge under the null of stationarity. The critical values, which are produced in KPSS (1992), for 9p and ;f are .739 and .216 at the .01 level of' significance, and. .463 and .146 at the .05 level of significance, respectively. For fractionally integrated series, neither the hypothesis of nonstationarity nor that of stationarity describes the processes well. Cormequently, the combined application of both a conventional unit root test such as the Phillips and Perron (1988) test which has a null Ianothesis of nonstationarity, and the KPSS (1992) test which has a null hypothesis of stationarity, can be used to detect the presence of frfictionally integrated series. Application of the Phillips and Perron testl, denoted PP, and the Kwiatkowski, Phillips, Schmidt, and Shin test Pr°<1uces four possible outcomes which are summarized as: 57 Reject Ho Do Not Reject Ho PP test: Ho: I(l) PP test: H0: I(l) Reject Ho (i) Inflation not well (ii) Strong evidence of a KPSS test: represented as unit root exists in Ho: 1(0) 1(1) or I(O). the data. POSSIBLE EVIDENCE FOR I(l) FRACTIONAL INTEGRATION Do Not Reject Ho (iii) Strong evidence of (iv) The data is insuf- KPSS test: stationarity exists ficiently informa- Ho: I(0) in the data. tive on the long 1(0) run characteristics of the series. Rejection of both null hypotheses for a given series, scenario (i) above, indicates that the series is not well described by either an I(l) nonstationary or an I(0) stationary process, and consequently may be evidence of fractionally integrated behavior. Table 13 presents the results of applying the above tests to the inflation rate series for each of the ten countries represented in Table 1. Scenario (i), in which both null hypotheses are rejected, arises for (right out of ten countries: Argentina, Brazil, Canada, France, Italy, Israel, the U.K. and the U.S. The implication, then, is that the thflation rate series for each of these countries are not well described as being either stationary or nonstationary, which suggests that the Series may be fractionally integrated. The results for Germany and Japan, however, seem to indicate that tflle inflation rate series for these two countries are stationary. This result is not surprising given the extent to which officials in these conntries have intervened, historically, in the operation of their economies to maintain a steady and stable rate of inflation. These results of a stationary inflation rate for Germany and Japan are further 58 confirmed in subsequent estimation. That is, for Japan the estimated value of the parameter of integration, d, is found to be of the same magnitude as that of the moving average parameter such that cancellation of the two result in a value of d near zero. Similarly for Germany, the estimated value of d is found to be close to zero. These results support the hypothesis of a stationarity inflation rate series for these two countries. Table 14 presents the results of applying the Geweke and Porter- Hudak (1983) estimation technique, which is described in detail in chapter II, to the inflation series of the ten countries: Argentina, Brazil, Canada, France, Germany, Israel, Italy, Japan, the U.K. and the U.S. The parameter of integration, d, has been estimated over the range of low- frequency ordinates used in the spectral regression, as suggested by Geweke and Porter-Hudak, for the values Ta, where a a .50, .525, .55, .575, and .60. The results indicate the extreme sensitivity of the parameter estimate to the number of ordinates used in the spectral regression. The estimated value of d for each country's inflation series varies within a substantial range, depending upon the number of ordinates used during estimation. Alternatively, Table 14 also provides the results of estimating the parameter of integration, d, for each country's inflation rate using the number of ordinates for the spectral regression as suggested by Sowell (1990). Since each of the low-inflation economies span approximately 41 years of data, and each of the high-inflation economies span approximately 31 years of data, and assuming a low-frequency period of five years for the inflation rate series, the values of m implied by this technique for the low- and high-inflation economies are 6 and 8, respectively. This .57.. . _ -1 ii 59 approach further confirms the extreme sensitivity of the GPH estimation technique to the range over which the spectral regression is estimated. The results presented in Table 14 of estimating d for the inflation rate series using the Geweke and Porter-Hudak (1983) estimation technique clearly indicate the sensitivity of this procedure to the data used in the analysis. These results indicate the importance of considering alternative estimation procedures when dealing with fractionally integrated series. Consequently, the ARFIMA-GARCH model developed in Chapter II has been applied to the inflation rate series for the G-7 countries, Argentina, Brazil, and Israel to estimate both the parameter of fractional differencing for these series as well as all other model parameters of the series. By applying an appropriate modelling procedure which can account for not only the long-memory characteristics of the series but the time varying homoskedasticity as well, it will be possible to examine the true long-run characteristics of the inflation rate which should aid in our understanding of the macroeconomy. 5. ESTIMATION OF THE ARFIMA-GARCH MODEL Section 5 of Chapter 11 detailed the ARFIMA(0,d,l) - GARCH(1,1) model as represented by equations (11.) through (14.) of that chapter and discussed the two estimation procedures termed Method I and Method II. These estimation procedures were applied to the CPI inflation rate series for the G-7 countries as well as Argentina, Brazil, and Israel. The estimation procedure for each of these countries and the results of estimation are outlined below. The ARFIMA(0,d,1) - GARCH(1,1) model as applied to the U.S. CPI inflation rate can be expressed by the following equations: 60 d (2.) 100(1-L) A log CPIt - b + 6t + oet_1 + sat 2 -1 (3.) et|nt_1 ~ c(o, at, u ) (4) 02-w+a62 +1302 +1Alo CPI ' t t-l t-l g t’ In this specification, yt = 100Alog CPIt is the consumer price index measure of inflation. The random error, 6t’ is assumed to follow a conditional density D, assumed to be either normal or Student t, with mean zero and variance of. The error at is conditional on the information set at time (t-l), 0 The model parameters to be estimated include: b, the t-l' mean of inflation; 6, the effect of the volatility of inflation on the mean level of inflation; 0, the moving average parameter; w, the intercept in the conditional variance; a, the effect of lagged squared residual on the conditional variance (the ARCH effect); 3, the effect of lagged conditional variance on the current variance (the GARCH effect); 1, the effect of lagged inflation on volatility; and, due to the presence of excess kurtosis in the series, the degrees of freedom from the Student t distribution, u, must also be estimated. The specification of the ARFIMA-GARCH model will allow for empirical tests of the Friedman hypothesis as well as tests for the presence of feedback between lagged inflation and the degree of volatility of inflation in the current period. The validity of the Friedman hypothesis is investigated through the parameter 6; when 6 fl 0, volatility is allowed to influence the mean of inflation in that a positive and significant It.” 61 value for this parameter indicates that higher levels of inflation are associated with higher volatility of the series. Such a finding would give positive empirical support to the Friedman hypothesis. The model specification will also allow for a test of whether lagged inflation, which is predetermined, enters the conditional variance equation (4.). A positive and significant value for the parameter 1 will lend support to the hypothesis that last period's inflation rate is directly correlated with a higher value of the current period's inflation variance. The results of estimation of 6 and 1 are presented in Table 25 and are discussed in more detail later in this section. In estimation of each country's inflation rate the statistics m4 and m3, which are measures of the sample kurtosis and skewness, respectively, are estimated numerically by the Berndt et. a1. (1974) algorithm. In the case where the true distribution of a series is normal, m and m3 will 4 have asymptotic distributions given by N(0,24/T) and N(0,6/T), respectively. The estimated value of m4 is used as a diagnostic to determine whether the distribution of the model should be assumed normal or Student t, since the presence of significant kurtosis in the residuals indicates the inappropriateness of the assumption of normality of the unconditional distribution of 100Alog CPIt. When evidence of significant kurtosis is present in a series as evidenced by the estimated.value of ma, a Student t distribution, rather than the normal, will be assumed for the model. Use of the Student t distribution will necessitate estimation of the degrees of freedom parameter, v. The appropriateness of the Student t distribution can be examined by comparing the estimated value of the sample kurtosis, m4, to the level kurtosis implied by the estimated degrees of freedom parameter. That is, in using the Student t 62 distribution, the estimated value of u implies a conditional kurtosis of 3(; - 2)/(; - 4), and this value may be directly compared to the estimated value of ma. The estimation procedure employed.in this analysis also produces the standard Ljung and Box (1978) test statistic, Q(k), which tests for ktfll order serial correlation in the estimated residuals. In addition, the statistic Q2(k) is also calculated and this statistic is used to test for kth order serial correlation in the squared residuals. The Q2(k) statistic will be used in this analysis to provide an LM test of the ARCH(k) specification where a null hypothesis of no serial correlation in the squared residuals of the inflation series is tested against the alternative hypothesis of kth order correlation. Under the null hypothesis of conditional homoskedasticity, these statistics will be asymptotically distributed as chi-squared with k degrees of freedom. In this particular analysis k will be set equal to 10 so that the Lung-Box statistics Q(lO) and Q2(10) are calculated in estimation of the model for each country. It is worth noting the problems of interpreting the Ljung-Box statistics in the case of the model for the inflation rate, however, since the power of these statistics can be influenced by the presence of significant ARCH effects in the data generating process. As pointed out by Cumby and Huizinga (1988), when testing residuals for autocorrelation, the presence of heteroskedasticity in a series (as is the case with the inflation rate) will tend to 'bias the Ljung-Box statistics towards rejecting the null hypothesis. The ARFIMA-GARCH.model described above is used to estimate all model parameters given in equations (2.) through (4.) of this section, for the 63 inflation rate series of eath of the ten countries of this analysis. Table 15 presents the results of estimating the ARFIMA(0,d,l)-GARCH(1,1) process for the U.S. The first column of the table presents the full maximum likelihood estimates of the parameters of the model, obtained by Method I which was described in some detail in Chapter II. The MLE of d by this method is .36, which implies that the series is highly persistent but none the less mean reverting. In this way, the inflation series for the U.S. can be considered to be a stationary process since mean reversion implies that shocks to the series will not persist indefinitely into the future but, rather, will die out over time. The results of estimation of the model by Method II, as described in Chapter II, are also presented in Table 15. This involves estimation over the transformed, or filtered, series for ten values of d, 0.00 through 0.90. That is to say, the maximum likelihood estimation procedure is performed ten times for the inflation rate series and the resulting parameter vector which produces the maximized log likelihood is taken to be the true MLE. In the case of the U.S. model, in observing the values of the maximized log likelihood function produced at each value of d, it appears that the log likelihood is relatively flat in the range of d from .30 to .50. The maximized value occurs when d - .40, which is consistent with the ML estimate of d obtained by Method 1, and also with the value predicted by observing the autocorrelation functions of the filtered U.S. inflation series presented in Table 12. The value of the Ljung-Box statistics at the MLE indicates that after fitting the GARCH(1,1) model to the inflation rate series the null hypotheses of uncorrelated residuals and uncorrelated squared residuals cannot be rejected. In addition, the estimated value of u at the MLE implies a conditional kurtosis of 4.28 - .-.- in 64 which is relatively close to the estimated value of 4.42, indicating the appropriateness of the use of the Student t distribution for the U.S. model. It is interesting to note that a clear trade off exists for the maximized conditional log likelihood between the estimated moving average parameter, 3, and the fractional differencing parameter, 3, as can be observed in Table 15. As the value of the fractional differencing parameter increases the estimated value of the moving average parameter decreases. A similar trade off can be observed for the maximized conditional log likelihood between the estimated mean parameter, b, and the fractional differencing parameter; the MLE of S does appear to change conditional on d. The ARFIMA-GARCH model was applied to the inflation rate series of the remaining nine countries: .Argentina, Brazil, Canada, France, Germany, Israel, Italy, Japan, and the United Kingdom, and the results of estimation are similar to that of the U.S. For each of these countries, however, some degree of seasonality in the conditional mean of the inflation rate series was apparent. This seasonality can be observed by noting the pattern of decay of the autocorrelation functions of each of the countries presented in Table 1. For each country except the U.S., the correlation decreases steadily to zero as the lag increases, but picks up somewhat around lag twelve and then continues to decline again after this point. Consequently, an ARFIMA(O , d, 13) -GARCH(1 , 1) process was estimated for each of these countries to account for the seasonality in the data. This model includes moving average terms of lags 1, 12, and 13 to account for this seasonalityu The multiplicative seasonal restriction of 013 - 01012, 65 however, was not imposed in estimation” The model for these countries may be represented by the following equations: + 0 0 6a ‘ 13‘t-13 + c d 2 (6.) et|0t_1 ~ N(O, at) (7) 02-w+a62 +1302 +1Alo CPI ' t t-l t-l g t’ This model specification is very similar to that for the U.S. except that the above specification includes the two additional parameters, 012 and 0 which account for the seasonality of the model. In addition, the 13’ above model assumes a zero mean for the inflation rate. In the cases of France and Israel, the inflation rate series were found to exhibit a substantial degree of excess kurtosis and so the models for these countries were estimated using the Student t density rather than the normal. Recall that the estimated value of the degrees of freedom parameter estimated.under the Student t distribution implies a conditional kurtosis of 30: - 2)/(; - 4). In the case of the French model, for example, the estimated value of 3 implies a conditional kurtosis of 10.22 which is relatively close to the estimated sample kurtosis for France, m4 = 9.22. This will indicate the appropriateness of the use of the Student t distribution over that of the normal in estimating the models for these countries. Tables 16 through 24 present the results of estimating the ARFIMA(0,d,l3)-GARCH(1,1) process via Method II for the remaining nine countries; results for Method I are not reported due to the excessive 66 computational difficulty of applying direct maximization to the seasonal moving average model. As with the U.S. model, the value of the Ljung-Box statistics for the models of each of the remaining countries indicates that at the MLE there is no case in which the null hypothesis of no serial correlation in the squared residuals can be rejected against the 2 For Canada, alternative hypothesis of tenth-order serial correlation. France, Germany, Italy, and the U.K., which are all considered to be relatively low-inflation economies, the log likelihood function is maximized at a value of d which is greater than zero but less than or equal to 8. This implies that the inflation series for these countries, like that for the U.S., are highly persistent but none the less mean reverting. In this way these series may be viewed. as stationary processes, contrary to what many researchers have found in examining the inflation rate within the framework of models other than that of the fractionally integrated model. The results for the relatively high-inflation economies of Argentina, Brazil, and Israel also indicate the highly persistent and mean reverting behavior of the inflation series for these countries. The log likelihood functions for these countries were maximized when the value of d was less than one, but also greater than B. This indicates that the \nnconditional variance of the inflation rate for these relatively high- inflation economies is infinite. The fact that all three high-inflation economies exhibit this characteristic should not be surprising due to the great variability in the inflation rate experienced by these countries 2 In each case, the null hypothesis cannot be rejected at the 10% leveal of significance and in some cases the null hypothesis also cannot be rejeected at the 5% level of significance. 67 during the time period under investigation. Perhaps somewhat more surprising is the result that for France and Italy the MLE of d is found to be exactly equal to 8. This implies an infinite unconditional variance for these countries as well. The results of applying Method 11 to the remaining countries' inflation series appear to be quite similar to the results for the U.S. The same trade off may be observed for the countries represented in Tables 16 through 23 as was observed for the U.S. That is, as the value of the fractional differencing parameter increases, the value of the estimated A moving, average parameter, 01, decreases. In addition, all of the estimated models with the exception of the U.K. exhibit strong persistence in their variances as evidenced by the sum of the estimated.ARCH and.GARCH parameters, a and 6, being close to one. Again, this confirms the highly persistent nature of the inflation rate series for these countries. For each of the ten countries examined in this analysis, likelihood ratio tests were performed to investigate whether the value of d produced by Method II was significantly different from zero or from one. That is, a test of the null hypothesis that d =- 0.00 versus the alternative hypothesis that d was equal to the MLE produced.by Method 11 was performed for those inflation series for which 0 < d < 8. Similarly, a test of the null hypothesis that d - 1.00 versus the alternative hypothesis that d = dMLE was performed for those series for which.8 < d < 1. In each case the \ results indicate that the null hypothesis could be strongly rejected against the MLE value of d. The results of these tests for the U.S. are presented in Table 15. Table 25 presents the results of likelihood ratio tests which were designed to test the validity of the Friedman hypothesis and also to test 68 for the presence of a feedback relationship between lagged inflation and the conditional variance of inflation in the current period. The first row of Table 25 provides the results of testing for whether lagged volatility Granger causes the mean of inflation. As discussed in section 4, this test allows the volatility of inflation (the standard deviation) to influence the mean of inflation through the parameter 6 in equation (2.) for the U.S. model and equation (5.) for the models of the remaining countries. The results indicate that for the low-inflation economies of Canada, France, Italy, Japan, and the U.S., there is no evidence of volatility causing the mean of inflation. The results for the U.S. are consistent with the findings of previous studies by Engle (1983) and Cosimano and Jansen (1988). For the high-inflation economies, and also surprisingly for the U.K., there is strong evidence of joint feedback between the conditional mean and variance of inflation. The second row of Table 25 provides the results of testing whether lagged inflation Granger causes inflation volatility. This test allows lagged inflation to influence the volatility of inflation through the parameter 1 in the conditional variance equation, equation (4.) for the U.S and equation (7.) for the remaining countries. This can be interpreted as a direct test of the Friedman hypothesis which states that the volatility or uncertainty of inflation increases in high inflation regimes. The results indicate that for the low-inflation economies of Canada, France, Germany, Italy, and.Japan, there is no support for a valid Friedman hypothesis. These findings are consistent with those of Gordon (1971), Logue and Willett (1976), and Fischer (1981) who failed to find evidence of a positive correlation between the level and variability of inflation for relatively'highly industrialized economies. The results for It: 69 the high-inflation economies and the U.K., on the other hand, indicate that there is strong evidence in support of the Friedman hypothesis. For these countries, the parameter 1 is found to be positive and significant. These results indicate that for the high-inflation economies, and again also for the U.K., periods of increased inflation should be expected to be associated with periods of increased inflation variability. This is consistent with the findings of Logue and Willett (1976) and Fischer (1981) who noted.that for economies experiencing relative instability, for example in the form of hyperinflation or political unrest, there existed a significant positive correlation between increased levels of inflation and increased inflation variability. The finding of empirical support for the Friedman hypothesis for only the relatively high-inflation economies should not be surprising if one considers that the hypothesized link between inflation and its variance ‘was most likely directed. at ‘high inflation. economies (see Friedman 1977). Logue and Willett (1976), who found no link between inflation and its variance for economies experiencing relatively low levels of inflation, suggested that there existed some "threshold" level of inflation below which the Friedman hypothesis was not valid. That is to say, Logue and Willett hypothesized that there was some minimum level of inflation3 below which an increase in the level of inflation would not lead to a subsequent increase in inflation variability; the higher the average rate of inflation, the more likely there was to be a positive association between the level and variability of inflation. As a result, 3 Based on calculations performed in their analysis, Logue and Willett (19 76) propose that this threshold level of inflation should lie somewhere between two to four percent. In:- . 7O evidence of the Friedman hypothesis should be weakest for relatively low- inflation economies, such as those of the G-7, and strongest for relatively high inflation countries, such as Argentina, Brazil, and Israel. The apparent inconsistency of the results for the U.K. are somewhat surprising, however. In one respect the inflation rate for the U.K. behaves like the inflation rate series for the relatively low-inflation economies of the G-7 countries, having a value of d which falls within.the range of 0 < d < 8. This indicates the highly persistent but mean reverting nature of the U.K. inflation series and implies a finite variance. Yet in another respect the inflation rate for the U.K. behaves like the inflation rate series for the relatively high-inflation economies of Argentina, Brazil, and Israel in that empirical support is found for a valid Friedman hypothesis and the existence of a feedback mechanism between the mean of inflation and its variance. These results seem to separate the U.K. inflation rate from the norm of either a relatively low- inflation economy or a relatively high-inflation economy, indicating some sort of atypical behavior on the part of the U.K. 6. SUMMARY AND CONCLUSION This chapter examines the long-run characteristics of the inflation rate series, which is clearly one of the key variables in understanding the macroeconomy, in the context of the long-memory process. The model applied here allows for a more precise investigation into the true long- run characteristics of the inflation rate series since the series is (Hnnsidered, for the first time, within the framework of the fractionally intxegrated ARMA model, the ARFIMA process. The additional characteristic 71 of the inflation.rate in the form of homoskedastic error is also accounted for in the model by use of the ARFIMA-GARCH process. Two methods of obtaining approximate maximum likelihood estimates are applied to the inflation rate series for the Group of Seven countries, which are considered to be relatively low-inflation economies, and to Argentina, Brazil, and Israel, which are considered.to be relatively'high- inflation economies. A distinct difference is observed in the estimated parameters of the two types of economies, implying fundamentally different long-run characteristics for high- and low-inflation economies. The results of this analysis indicate that the inflation rate series, regardless of whether they represent high-inflation regimes or low-inflation regimes, are all highly persistent but none the less mean reverting and stationary. The inflation rate series for Argentina, Brazil, Israel, and the U.K., however, were found to exhibit infinite variances, which may not be unexpected in each case other than the U.K. due to the volatile nature of the series for these countries during the time period under which this investigation takes place. For each of the relatively high-inflation economies of Argentina, lSrazil, Israel, and also for the U.K., there is strong empirical support :Eor the Friedman hypothesis that high inflation should be expected to be zissociated with increased inflation volatility. This relationship does ruat appear to hold for any of the relatively low-inflation economies, with tile exception of the U.K. which seemed to exhibit atypical behavior; This finding should be consistent with theoretical expectations if one Considers that the Friedman hypothesis was directed at high-inflation r("figimes only. One issue of interest in pursuing further the findings of this 72 analysis is to consider the effect of exogenous influences on the long-run characteristics of the inflation rate. For example, Alesina (1989) has considered the impact of political stability on an economy's performance and discusses the degree of autonomy in performing monetary policy of the central banks of several countries. The extent to which a country's central bank is divorced from the fiscal activity of the economy should have some effect on the stability of the inflation rate for that country; that is, central banks which are tied directly to government policy can often increase the volatility of inflation by their monetary policy actions. In addition, Bernake (1992) provides a useful discussion of central bank behavior and the degree of autonomy of the central banks of six industrialized countries. Fischer (1981) also discusses the effect of the use of monetary policy on the part of central banks, noting that the validity of a Friedman-type hypothesis which links the level and variability of inflation may depend heavily on the degree of accommodation in a country's monetary policy. Two additional areas which might be of interest in examining the effect of exogenous influences on the inflation rate include the degree to which a country's wages are indexed to the inflation rate, and the degree to which central banks engage in interest rate smoothing in maintaining their policy objectives (see for example Gray (1976) and Goodfriend (1987)). These analyses may provide further insight into the questions which economists pose about the characteristics of inflation and its variability and the subsequent implications these issues have for the macroeconomy. 73 TABLE 1 Autocorrelations of CPI Inflation Series Country Lag Argentina Brazil Canada France Germany Israel Italy Japan U.K. U.S 1 .758 .886 .434 .428 .362 .736 .253 .121 .267 .467 2 .561 .789 .369 .169 .275 .653 .280 .092 .232 .423 3 .368 .698 .409 .120 .210 .671 .316 .180 .197 .399 4 .272 .645 .380 .088 .011 .635 .240 .036 .203 .360 5 .373 .611 .362 -.102 .063 .658 .368 .098 .223 .316 6 .410 .588 .288 -.077 -.026 .658 .204 .082 .313 .305 7 .464 .557 .311 -.078 -.024 .602 .310 .025 .160 .312 8 .510 .531 .349 -.022 .047 .614 .326 .166 .146 .359 9 .420 .510 .317 -.025 .000 .615 .194 .112 .210 .386 10 .355 .506 .272 .057 .044 .600 .220 .021 .173 .349 11 .293 .531 .311 .155 069 .575 .194 .031 .201 .320 12 .281 .549 .419 .222 .055 .624 .370 .151 .403 .278 13 .228 .552 .286 .193 .070 .507 .229 .076 .156 .234 14 .219 .550 .216 .113 -.016 .470 .193 .135 .144 .180 15 .192 .512 .235 .140 .095 .456 .246 .083 .168 .211 16 .173 .498 .225 .102 .021 .445 .188 .019 .106 .229 17 .164 .773 .195 -.035 .184 .461 .296 .088 .138 .160 18 .163 .449 .144 -.043 -.135 .487 .199 .013 .192 .129 T 408 409 512 511 507 410 510 511 512 525 Key: The inflation series are defined as 100 Alog CPI 74 TABLE 2 Autocorrelations of First Differenced Inflation Series Country Lag Argentina Brazil Canada France Germany Israel Italy Japan U.K. U.S 1 -.092 “.074 -.437 -.191 -.419 -.343 .505 .467 .471 .499 2 -.009 -.024 -.094 -.214 -.017 -.191 .018 .075 .003 .112 3 -.201 -.169 .057 -.030 .063 .102 .072 .130 .024 .086 4 -.304 -.087 - 004 .150 -.082 -.114 .138 .108 .018 .032 5 -.077 -.015 .038 -.189 .069 .047 .185 .028 .033 .050 6 .072 -.019 -.075 -.100 -.025 104 .153 .038 .160 .013 7 .013 .002 -.018 -.018 -.054 ‘.130 .057 .119 .095 .024 8 .281 -.024 .068 .108 .093 .021 .087 .095 .053 .028 9 -.050 -.066 .017 -.033 -.074 .030 .098 .058 .065 .042 10 -.008 -.151 -.070 -.050 -.012 .020 .030 .069 .041 .029 11 -.104 -.094 -.072 .041 .035 - 140 .132 .105 .120 .024 12 .086 .092 .222 .163 .003 .314 .209 .226 .309 .037 13 -.091 .013 -.052 .051 .089 -.151 .070 .094 .161 .034 14 .039 .204 -.084 -.151 -.035 -.044 .052 .065 .025 .089 15 -.016 -.102 .025 .062 -.134 -.005 .067 .005 .050 .073 16 -.021 .069 .018 .082 .178 -.051 .116 .069 .056 .008 17 -.018 -.004 .008 -.110 -.127 -.019 .130 .100 .016 .084 18 -.033 -.001 -.034 ~.051 .050 .159 .064 .117 .078 .036 Key The series are defined as 100 Azlog CPIt. ‘75 TYiBJLEZ 3 Autocorrelations of Filtered CPI Inflation Series .‘3‘1 .159->- -“ Argentina Value of d: Lag 00 0 1o 0 20 0 30 0 40 0.50 0 60 0 70 0 80 0 9o .00 1 .758 .644 .534 .435 .346 .263 .116 .112 .041 -.026 .091 2 .561 .395 .261 .165 .099 .054 .024 .006 —.005 -.008 .007 3 .366 .147 -.018 — 127 - 191 -.225 -.240 -.241 -.234 -.221 .204 4 .272 .031 — 144 -.254 -.315 —.345 -.335 -.352 -.341 - 326 .307 5 .323 .121 -.o19 -.099 - 136 -.146 -.144 -.132 -.116 -.097 .078 6 .410 .253 .146 .087 .060 .051 .051 .056 .062 .069 .075 7 .464 .326 .229 .169 .133 .109 .089 .071 .052 .033 .014 8 .510 .404 .336 .302 .288 .264 .263 .284 .264 .283 .262 9 .420 .261 .181 .118 .078 .050 .027 .006 -.014 -.033 -.052 10 .355 .207 .104 .045 .015 .000 -.007 -.010 - 010 -.008 - 006 11 .293 .140 .035 -.026 -.059 -.077 -.087 -.094 - 099 -.103 -.107 12 .281 .150 .066 .031 .020 .024 .034 .047 .060 .074 .067 13 .228 .090 .002 - 044 - 066 - 075 -.060 -.082 —.065 -.066 -.092 14 .219 .097 .026 -.003 —.009 -.004 .004 .014 .023 .032 .039 15 .192 .068 -.004 -.034 -.041 -.036 -.032 -.026 -.021 - 017 —.015 Key: The inflation series is 100(l-L)d'Alog CPIt - (1-L)d'yt. The series begins in January of 1957 and runs through December of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 76 TABLE 4 Autocorrelations of Filtered CPI Inflation Series Brazil Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .886 .811 .706 .591 .479 .375 .197 .190 .107 .030 “.042 2 .789 .662 .512 .368 .248 .158 .091 .042 .007 -.016 -.030 3 .698 .528 .340 .170 .040 -.049 ”.107 -.144 -.166 -.l78 -.183 4 .645 .464 .274 .114 .005 -.059 -.091 -.104 '.104 -.098 ”.088 5 .611 .429 .245 .099 .008 -.037 -.053 -.052 -.044 -.033 -.020 6 .582 .397 .216 .077 -.004 -.040 -.048 -.043 -.032 -.021 -.011 7 .557 .368 .186 .051 -.025 “.055 -.058 -.048 -.034 “.021 -.010 8 .531 .337 .152 .016 -.059 -.085 -.083 -.068 -.050 -.031 -.015 9 .510 .309 .115 -.030 ".113 '.144 “.146 -.132 ‘.113 -.093 -.073 10 .506 .305 .106 -.050 -.l46 -.191 -.206 -.204 -.196 -.184 -.172 11 .531 .359 .185 .046 -.040 -.082 -.096 -.097 -.091 -.084 -.077 12 .549 .412 .275 .168 .103 .072 .062 .063 .067 .072 .076 13 .552 .438 .320 .223 .157 .117 .093 .076 .061 .046 .031 14 .550 .456 .362 .288 .241 .217 .206 .203 .203 .203 .205 15 .512 .405 .289 .187 .108 .051 .009 -.025 -.054 “.080 '.103 Key: The inflation series is 100(1-1.)d Alog CPIt =- (1-L)d yt. The series begins in January of 1957 and runs through January of 1991. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 77 TABLE 5 Autocorrelations of Filtered CPI Inflation Series Canada Value of d: 1 .446 .299 .172 .037 -.092 -.200 ‘.280 -.338 -.381 -.416 '.445 2 .384 .259 .162 .066 -.018 -.076 “.106 '.116 ‘.113 ”.103 *.090 3 .424 .323 .249 .176 .114 .073 .053 .047 .049 .054 .060 4 .395 .302 .230 .154 .086 .036 .006 ".009 -.016 -.020 -.022 5 .392 .308 .243 .177 .120 .081 .061 .054 .055 .059 .063 6 .320 .219 .140 .060 -.010 -.058 -.084 -.096 -.099 -.098 -.097 7 .355 .272 .205 .135 .072 .029 .004 -.007 -.011 -.011 ”.010 8 .401 .337 .280 .219 .163 .122 .099 .087 .082 .080 .078 9 .360 .293 .229 .159 .094 .047 .019 .005 -.001 -.003 “.004 10 .323 .252 .183 .105 .035 -.015 -.044 -.056 -.060 -.059 -.056 11 .349 .283 .212 .132 .059 .003 -.033 -.053 -.065 -.073 -.079 12 .461 .435 .393 .341 .291 .254 .231 .217 .209 .204 .200 13 .351 .307 .248 .176 .109 .057 .024 .004 -.007 -.015 -.021 14 .265 .216 .153 .077 .008 -.042 -.071 -.084 -.089 -.089 -.087 15 .275 .243 .193 .129 .070 .029 .007 -.002 “.004 -.003 -.001 Key: Each series is 100(1-L)d Alog CPIt - (l-L)d yt. The series begins in January of 1948 and runs through August of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 78 TABLE 6 Autocorrelations of Filtered CPI Inflation Series France Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .428 .380 .251 .139 .042 “.042 -.116 ".181 “.239 '.290 “.335 2 .169 .227 .115 .033 ‘.025 -.063 -.087 -.101 “.106 ‘.104 -.098 3 .120 .184 .095 .037 .005 ‘.010 -.014 -.011 -.003 .007 .018 4 .088 .105 .020 -.031 -.058 -.069 -.070 -.066 -.060 -.053 -.045 5 -.102 .073 -.008 -.055 -.018 -.093 “.097 -.097 “.095 -.094 ‘.092 6 -.077 .155 .097 .068 .057 .057 .062 .070 .077 .085 .091 7 -.078 .116 .051 .014 ~.005 ‘.013 “.016 '.016 ".016 -.014 -.013 8 -.022 .092 .023 -.020 “.046 -.062 -.072 “.080 ‘.086 -.092 ‘.098 9 -.025 .197 .155 .138 .135 .140 .149 .161 .172 .185 .195 10 .057 .047 -.033 -.088 -.125 -.151 ‘.170 -.185 -.196 -.205 “.212 11 .155 .166 .115 .086 .071 .063 .059 .058 .057 .057 .057 12 .222 .215 .177 .157 .148 .143 .141 .139 .138 .136 .134 13 .193 .102 .049 .018 -.002 -.013 -.021 -.026 -.030 '.032 -.034 14 .113 .032 -.029 -.068 -.094 -.112 -.126 -.137 -.145 -.152 '.157 15 .140 .163 .136 .125 .121 .122 .124 .126 .129 .131 .133 Key: The inflation series is 100(1-L)d Alog CPIt- (l-L)d yt. The series begins in January of 1948 and runs through July of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 79 TABLE 7 Autocorrelations of Filtered CPI Inflation Series Germany Value of d: 1 .362 .183 .076 -.015 -.092 '.158 -.216 -.266 ’.311 -.350 '.386 2 .275 .044 -.027 -.073 -.101 -.117 -.123 -.122 -.117 -.109 -.097 3 .210 .027 -.016 “.035 -.039 “.035 -.025 -.013 -.001 .011 .021 4 .011 -.022 -.056 -.067 -.066 -.058 '.048 -.038 -.029 '.021 -.015 5 .063 -.039 -.067 ”.074 “.070 -.061 ”.051 ”.041 -.032 '.025 -.019 6 -.026 -.050 -.079 -.087 -.084 -.076 -.066 -.057 -.047 -.039 -.032 7 -.024 -.004 -.036 -.050 -.056 -.058 -.059 -.060 -.061 -.064 ’.067 8 .041 .154 .139 .139 .144 .152 .160 .167 .173 .178 .182 9 .000 .010 -.034 -.062 -.081 '.094 -.105 -.114 -.122 -.129 -.135 10 .044 .063 .026 .007 -.003 -.006 “.006 -.004 .000 .004 .010 11 .069 .108 .068 .042 .024 .010 -.001 -.010 '.019 ‘.026 -.033 12 .055 .231 .206 .191 .181 .172 .164 .156 .148 .140 .133 13 .070 .148 .122 .107 .097 .090 .083 .077 .070 .065 .059 14 -.016 ".022 -.054 -.069 -.077 -.082 -.084 -.084 -.084 -.083 -.082 15 “.095 '.047 -.069 “.076 '.077 -.075 '.072 “.068 '.065 ".062 ".060 Key: The inflation series is 100(1-L)d'Alog CPIt - (l-L)d’yt. The series begins in January of 1948 and runs through March of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. A. 80 TABLE 8 Autocorrelations of Filtered CPI Inflation Series Israel Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .736 .570 .374 .199 .059 -.046 -.126 -.129 “.240 -.282 -.319 2 .653 .444 .226 .050 -.070 ‘.144 -.186 -.108 -.216 -.216 -.211 3 .671 .491 .318 .190 .114 .077 .065 .068 .078 .091 .106 4 .635 .433 .238 .090 -.007 ’.063 -.094 -.111 -.119 -.124 -.127 5 .658 .486 .323 .204 .130 .090 .069 .060 .056 .055 .055 6 .658 .492 .341 .232 .167 .132 .116 .109 .108 .108 .109 7 .602 .404 .219 .080 -.009 -.062 “.093 -.110 -.120 -.127 -.131 8 .614 .431 .267 .149 .077 .037 .017 .009 .006 .007 .009 9 .615 .446 .294 .186 .119 .080 .060 .049 .043 .040 .038 10 .600 .431 .279 .169 .100 .058 .035 .021 .014 .010 .008 11 .575 .407 .252 .136 .058 .008 '.027 -.051 ‘.071 -.087 -.100 12 .624 .483 .381 .318 .287 .274 .270 .270 -.273 .275 .278 13 .507 .316 .155 .043 -.025 -.065 '.089 '.103 -.113 -.121 -.128 14 .470 .266 .106 .001 -.055 -.080 -.088 -.087 -.082 -.076 -.069 15 .456 .269 .123 .035 -.008 -.022 '.022 '.015 -.006 .003 .012 K_ey: The inflation series is 100(1-L)d Alog CPIt - (1.1.)d y. The series begins in January of 1957 and runs through February of 1991. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. fl) 81 TABLE 9 Autocorrelations of Filtered CPI Inflation Series Italy Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .253 .126 “.063 “.191 “.280 “.346 “.398 “.440 “.475 “.505 “.532 2 .280 .227 .104 .041 .012 .003 .005 .013 .023 .036 .050 3 .316 .205 .090 .033 .009 .001 .000 .002 .006 .010 .014 4 .240 .160 .042 “.014 “.039 “.048 “.050 “.049 “.047 “.044 “.042 5 .368 .183 .075 .026 .006 .001 .002 .005 .009 .014 .018 6 .204 .171 .059 .003 “.023 “.035 “.041 “.044 “.046 “.047 “.048 7 .310 .249 .153 .107 .087 .078 .073 .071 .069 .066 .064 8 .326 .212 .112 .065 .044 .035 .031 .029 .027 .026 .025 9 .194 .140 .030 “.021 “.043 “.051 “.054 “.054 “.053 “.052 “.051 10 .220 .147 .043 “.004 “.023 “.030 “.031 “.030 “.029 “.027 “.025 11 .194 .194 .101 .060 .045 .039 .038 .038 .039 .039 .040 12 .370 .172 .075 .031 .012 .004 “.001 “.003 “.005 “.006 “.008 13 .229 .165 .071 .029 .012 .005 .002 .001 .000 .000 “.001 14 .193 .162 .072 .035 .021 .018 .018 .020 .021 .023 .024 15 .246 .114 .017 “.025 “.041 “.046 “.047 “.046 “.045 “.043 “.042 Key: The inflation series is 100(1-L)d’Alog CPIt - (l-L)d’yt. The series begins in January of 1948 and runs through June of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 82 TABLE 10 Autocorrelations of Filtered CPI Inflation Series Japan Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 l .121 “.008 “.097 “.169 “.228 “.279 “.322 “.361 “.395 “.425 “.452 2 .092 “.048 “.090 “.111 “.118 “.117 “.110 “.100 “.086 “.071 “.055 3 .180 .010 “.011 “.017 “.014 “.008 .001 .009 .017 .024 .030 4 .036 “.035 “.058 “.066 “.065 “.061 “.055 “.048 “.041 “.034 “.027 5 .098 “.045 “.077 “.095 “.105 “.112 “.116 “.119 “.122 “.124 “.127 6 .082 .190 .177 .173 .175 .178 .183 .187 .191 .195 .198 7 .025 .035 .004 “.016 “.029 “.039 “.046 “.053 “.059 “.065 “.070 8 .166 .041 .017 .005 “.001 “.003 “.004 “.004 “.004 “.003 “.003 9 .112 .065 .050 .047 .049 .053 .058 .063 .068 .072 .075 10 “.021 “.062 “.086 “.098 “.102 “.103 “.101 “.099 “.096 “.093 “.089 11 “.031 “.019 “.049 “.069 “.084 “.095 “.106 “.115 “.124 “.133 “.140 12 .151 .328 .334 .343 .352 .359 .364 .368 .370 .370 .370 13 “.076 “.108 “.138 “.157 “.170 “.181 “.190 “.197 “.204 “.210 “.215 14 “.135 “.086 “.097 “.097 “.092 “.085 “.077 “.068 “.059 “.051 “.042 15 “.083 .013 .010 .016 .024 .033 “.041 .047 .053 .057 .061 Qy: The inflation series is 100(1-L)d Alog CPIt - (1-L)d yt. The series begins in January of 1948 and runs through July of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. M “”‘fl 83 TABLE 11 Autocorrelations of Filtered CPI Inflation Series United Kingdom Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .267 .118 “.016 “.115 “.190 “.251 “.302 “.346 “.384 “.418 “.448 2 .232 .075 “.014 “.061 “.083 “.089 “.087 “.079 “.068 “.054 “.038 3 .197 .074 .001 “.032 “.044 “.044 “.038 “.029 “.020 “.011 “.002 4 .203 .063 “.009 “.043 “.057 “.060 “.059 “.055 “.051 “.047 “.043 5 .223 .118 .051 .017 .000 “.009 “.015 “.020 “.023 “.027 “.032 6 .313 .236 .192 .175 .171 .173 .176 .179 .181 .183 .185 7 .160 .028 “.043 “.078 “.096 “.105 “.110 “.114 “.116 “.118 “.120 8 .146 .036 “.027 “.054 “.064 “.065 “.063 “.059 “.054 “.050 “.045 9 .210 .115 .064 .044 .039 .040 .044 .049 .054 .058 .061 10 .173 .085 .022 “.008 “.023 “.027 “.028 “.027 “.025 “.022 “.019 11 .201 .077 “.001 “.047 “.078 “.101 “.119 “.135 “.149 “.161 “.172 12 .403 .391 .367 .361 .362 .365 “.369 .372 .375 .378 .380 13 .156 .030 “.047 “.092 “.122 “.144 “.162 “.177 “.190 “.202 “.213 14 .144 .063 .010 “.011 “.018 “.018 “.014 “.009 “.003 .003 .009 15 .168 .068 .023 .008 .066 .009 .014 .019 .024 .028 .032 Key: The inflation series is 100(1-1.)Cl Alog CPIt - (1-L)d yt. The series begins in January of 1948 and runs through August of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. 84 TABLE 12 Autocorrelations of Filtered CPI Inflation Series United States Value of d: Lag 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1 .467 .313 .121 “.028 “.142 “.231 “.302 “.360 “.409 “.451 “.499 2 .423 .353 .220 .137 .090 .067 .058 .058 .064 .073 .112 3 .399 .225 .084 .000 “.045 “.067 “.076 “.078 “.076 “.073 “.086 4 .360 .221 .095 .028 “.003 “.015 “.015 “.011 “.006 .001 .032 5 .316 .172 .046 “.020 “.050 “.060 “.061 “.057 “.051 “.046 “.050 6 .305 .185 .069 .009 “.016 “.024 “.024 “.020 “.014 “.009 “.013 7 .312 .207 .094 .034 .005 “.009 “.014 “.016 “.016 “.016 “.024 8 .359 .257 .154 .097 .006 .049 .039 .032 .027 .023 .028 9 .386 .275 .179 .125 .096 .079 .069 .061 .056 .052 .042 10 .349 .235 .132 .073 .038 .017 .003 “.008 “.016 “.022 “.029 11 .320 .242 .156 .101 .076 .062 .054 .049 .045 .042 .024 12 .278 .213 .122 .076 .054 .043 .037 .034 .031 .030 .037 13 .234 .135 .035 “.016 “.038 “.048 “.050 “.050 “.048 “.045 “.034 14 .180 .103 .001 “.050 “.074 “.083 “.086 “.086 “.084 “.082 “.089 15 .211 .187 .110 .076 .063 .060 .060 .061 .062 .063 .073 Key: The inflation series is 100(1-L)d Alog CPIt - (l-L)d'yt. The series begins in January of 1948 and runs through August of 1990. The first 30 observations were omitted before the autocorrelations were computed for each filtered series. ‘"*_”7] 85 TABLE 13 Tests for Order of Integration of Different Countries' Inflation Series HO: 1(1) H : 1(0) Country Z(ta*) Z(ta) 3, Argentina -4.76** -3.67* .56** 0.21* Brazil - 3 46*"r - 2.50 .44Mr o .44“ Canada - 1o . 61** - 6 . 23” .13“ 0 .26’“k France - 12 . 76M - 1o . 46’” . 20 0 .2” Germany -15.11** -13.12** .24 0.14 Italy - 15 .19M - 9 . 06** .70” 0 .46“ Israel -4.51** -3.46* .16** 0.3?* Japan -18.38** -15.76** .33 0.17 UK - 14. 35“ -8 . 76** .88** 0 .26” us - 9 . 64** - 5 . 66** .80** 0 .36M Key: Z(ta*) and 2(t5) are the Phillips Perron adjusted t statistics of the lagged dependent variable in a regression with intercept only, and The critical values for 2(ta*) and 2(t5) are -2.86 and -3.41 at the .05 level of significance and -3.43 and -3.96 at the .01 level of significance. 3 and 37 are the KPS test statistics and are based on residuals p from regressions with an intercept, intercept and time trend included respectively. and intercept and time trend and n1 are 0.463 and 0.146 respectively; the .01 critical values are 0.739 and 0.216 respectively. respectively. The .05 critical values for up All test statistics reported in this table are based on Newey and West (1987) adjustments using 8 lags. Two asterisks denote calculated test statistics which are significant at the .01 level; one asterisk corresponds to significance at the .05 level. ova M1~m t 2 e (2 «- bi () (7 cl t-l ( ' c) a w + ac + fiaz -1 t-l d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 91 .180 .014 -.120 —.249 -.368 -.475 -.827 -.737 -.838 -.918 “ ( 049) ( 048) ( 048) (.048) (.044) (.040) ( 035) ( 029) ( 027) (.023) “" .289 .248 .224 .208 .220 .225 .221 .220 .227 .273 12 ( 048) ( 047) (.047) (.045) (.045) ( 047) (.048) (.038) (.048) (.042) 13 .230 .148 .078 .023 -.033 - 078 -.124 -.144 -.189 —.218 (.048) ( 049) (.049) (.048) (.047) (.047) (.048) ( 039) (.049) ( 044) w .0038 .0038 .0049 .0091 .0211 .0204 .0148 .0059 - 0009 .0200 (.0025) (.0028) (.0038) (.0088) (.0079) (.0056) (.0053) (.0024) (.0004) (.0017) a .054 .048 .048 .058 .084 .094 .074 .092 .047 .078 ( 019) (.018) ( 021) (.028) ( 029) ( 022) (.023) ( 021) (.013) ( 021) p .918 .920 .908 .859 .723 .720 .792 .841 .958 .712 ( 031) ( 034) (.048) ( 081) (.092) (.065) ( 082) (.038) ( 008) ( 013) Q(10) 24.04 13.28 10.08 10.38 11.52 13.40 14.15 13.80 10.98 9.08 02(10) 7.97 8.82 9.70 9.43 8.37 8.14 9.77 10.06 14.28 8.34 ms 0.388 0.383 0.372 0.381 0.358 0.359 0.329 0.399 0.254 0.365 m“ 3.899 3.915 3.932 3.942 3.935 3.937 3.870 3.999 3.888 3.929 2 -197.05 -171.43 -160.17 -158.03 -153.58 —154.85 -154 12 -155 57 -158 21 -171.55 Key: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung-Box tests based on the residuals and squared. residuals, m3 and. 1114 are the sample skewness and. kurtosis statistics based on the standardized residuals, and £ is the maximized value of the log likelihood. 91. 'IAdiLEZIl9 Estimated ARFIMA-GARCH Model for CPI Inflation: France d. :12 i13 100 l-L A 10 CPI - 1 + 0 L + 0 L + 6 L e ( ) g c ( 1 12 13 ) c 2 -1 e (2 - t: 0 0' u tl t'l ( ’ t, ) 2 2 2 a - w + a6 + 0 t t-l fl t—l d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 91 .338 .194 .081 —.082 - 177 -.288 - 401 -.518 -.845 —.788 (.043) (.043) (.042) (.042) (.041) (.040) (.038) (.035) (.032) (.030) 012 .247 .210 .194 .187 .182 .179 .178 .173 .187 .151 (.045) ( 037) (.037) ( 037) (.038) ( 038) ( 039) ( 039) (.039) (.040) 913 .293 .194 .103 .038 -.012 -.049 —.080 -.110 -.141 -.157 ( 041) (.042) (.041) (.041) ( 040) (.040) ( 040) (.040) (.040) (.041) w .0012 .0008 .0004 .0004 .0004 .0004 .0004 .0005 .0005 .0010 ( 0010) (.0008) ( 0005) (.0005) (.0005) (.0005) (.0005) (.0005) (.0006) (.0008) .138 .071 .048 .042 .041 .042 .043 .045 .048 .078 ( 038) ( 023) ( 018) (.015) (.018) ( 015) ( 015) (.018) (.018) (.024) p .883 .962 .948 .951 .951 .951 .949 .947 .945 .912 (.033) (.022) ( 015) (.014) (.014) (.014) (.015) ( 015) (.018) (.024) 0'1 .172 .198 .215 .211 .207 .204 .201 .198 .198 .030 (.016) (.000) (.085) (.017) (.017) (.015) (.003) (.001) (.019) (.000) Q(10) 42.10 34.84 27.58 22.91 22.82 24.88 27.33 29.51 30.72 27.39 02(10) 2.15 3.35 4.82 5.43 5.98 8.53 7.13 7.78 8.39 8.18 m3 0.840 0.878 1.022 1.082 1.093 1.077 1.045 0.998 0.950 0.918 (:14 7.524 8.384 8.905 9.149 9.218 9.133 8.913 8.551 8.117 7.451 £ -281.35 -220 82 -202.92 -198.77 -198 20 -198.08 -200.78 -203.18 -204.48 -205.53 Key; All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung-Box tests based on the residuals and squared residuals, m3 and. 1114 are the sample skewness and ‘kurtosis statistics based on the standardized residuals, and f is the maximized value of the log likelihood. IT (I) V €92 'TAd3lJ3 :20 Estimated ARFIMA-GARCH Model for CPI Inflation: Germany d. .12 113 100 l-L A 10 CPI = 1 + 0 L + 0 L + 9 L e ( ) g t ( 1 12 13 ) c 2 e 0 ~ N O a tl c-1 ( ' t) 2 2 = w + as + 0 t t-l 5 c-1 d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 91 .214 .100 -.004 —.104 -.207 -.315 -.444 - 825 -.848 -.895 (.047) (.047) (.045) (.045) ( 045) (.044) (.041) (.034) (.024) (.021) 12 .228 .201 .191 .190 .189 .193 .202 .215 .232 .234 (.042) ( 041) (.041) (.041) (.041) (.041) (.041) (.040) (.038) (.040) 13 .208 .184 .134 .108 .082 .052 .014 -.058 -.178 -.188 (.046) ( 048) (.047) (.048) (.048) (.048) (.047) (.045) (.041) ( 041) w .0004 .0003 .0002 .0002 .0002 .0002 .0003 .0002 .0003 .0040 (.0004) (.0003) ( 0003) (.0004) ( 0004) (.0004) (.0004) (.0004) (.0004) (.0004) a .030 .028 .028 .028 .028 .029 .029 .028 .028 .031 ( 007) (.006) ( 008) (.006) (.007) (.007) (.007) (.007) (.007) (.008) p .983 .988 .987 .987 .988 .988 .966 .987 .987 .982 (.009) ( 008) (.008) ( 008) (.008) (.008) (.009) (.008) (.008) (.010) Q(10) 17.22 18.95 21.95 28.52 34.44 42.34 50.93 85.47 84.09 43.99 02(10) 8.55 10.51 12.28 13.87 15.45 18.72 17.78 17.29 14.32 13.93 m3 0.553 0.587 0.584 0.587 0.545 0.521 0.494 0.450 0.344 0.289 m“ 4.875 4.784 4.853 4.547 4.448 4.351 4.288 4.199 4.198 4.185 s -170.90 -158.07 -155.10 -157 32 -181 01 -185.22 -189.02 -171 07 -187 15 -183.34 Key: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung Box tests based on the residuals and squared. residuals, m3 and. 1114 are the sample skewness and. kurtosis statistics based on the standardized residuals, and f is the maximized value of the log likelihood. . .-4... “...—... _H 923 TYKBIJE 221 Estimated ARFIMA-GARCH Model for CPI Inflation: Israel d. 112 113 100 1—L A 1o CPI - 1 + 0 L + 0 L + 0 L e ( ) g t ( 1 12 13 ) t 2 -1. e (2 - t: 0 a' u tl t-l ( 9 t, ) 2 2 2 a‘ .. (a -+ (28 -+ <7 t t-l fl t-l d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 91 .317 .177 .083 - 087 - 201 —.353 -.475 -.813 -.732 -.782 (.058) (.054) (.056) (.052) ( 050) (.052) ( 049) ( 040) (.038) ( 038) 12 .221 .228 .212 .223 .208 .235 .220 .203 .220 .187 (.039) ( 040) ( 035) ( 039) (.039) (.039) (.040) (.038) (.038) (.039) 13 .108 .075 .040 .001 -.033 -.078 -.115 -.138 —.178 -.143 (.039) (.042) (.034) ( 043) (.042) (.043) (.042) (.040) (.038) (.041) .181 .105 .403 .104 .133 .141 .180 .180 .153 .341 (.089) (.049) (.117) (.050) ( 058) (.087) ( 084) (.087) (.060) (.133) a .308 .174 .371 .141 .138 .184 .209 .128 .178 .612 (.074) ( 049) (.091) (.044) (.044) (.082) ( 078) (.042) (.055) (.166) B .871 .783 .490 .807 .793 .774 .772 .787 .750 .509 ( 054) (.044) ( 072) (.045) (.050) ( 055) ( 080) (.055) (.080) ( 078) 0'1 .030 .132 .100 .138 .115 .170 .240 .128 .090 .084 (.000) (.000) (.000) (.000) (.000) (.000) (.010) ( 000) ( 000) (.000) Q(10) 38.44 37.37 38.81 22.14 21.32 22.09 23.20 23.43 20.80 23.10 02(10) 8.22 10.43 13.88 11.49 10.28 8.95 8.04 7.27 7.10 5.97 m3 1.283 1.375 1.895 1.508 1.524 1.481 1.432 1.427 1.414 1.583 m4 7.708 7.593 8.890 8.135 8.387 9.253 8.222 8.542 8.352 9.877 £ -777.49 -759.30 -759 25 -748.15 -744.53 -740.80 -737.11 -740.51 -742.01 —745.27 Key: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(lO) are the Ljung-Box tests based on the residuals and squared. residuals, m3 and. 1114 are the sample skewness and. kurtosis statistics based on the standardized residuals, and £ is the maximized value of the log likelihood. in.“ : -— ..A. [ .... .D 94 TABLE 22 Estimated ARFIMA-GARCH Model for CPI Inflation: Italy d 12 13 100 l-L A 10 CPI = 1 + 0 L + 0 L + 0 L E ( ) g c < 1 12 13 )c 2 e 0 ~ N O a tl t-l ( ' t) 2 2 2 a - 00 +06 + 0 t t-l fl t-l d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 91 .158 .019 -.108 -.228 -.343 -.458 -.577 -.894 -.789 -.888 (.046) (.046) (.045) (.045) (.043) (.041) (.038) (.034) (.030) (.024) a 2 .187 .129 .099 .092 .075 .070 .065 .057 .073 .080 1 (.047) (.047) (.047) ( 047) (.047) (.047) (.047) (.048) ( 045) ( 039) 13 .223 .144 .090 .057 .027 .002 -.024 -.050 -.072 -.080 (.042) (.043) (.044) (.045) (.046) (.046) (.046) (.047) (.044) (.038) w .003 .002 .002 .002 .002 .002 .002 .002 .004 .007 (.002) (.001) (.001) (.002) (.002) (.002) (.002) (.002) (.003) (.003) a .112 .108 .110 .117 .114 .114 .115 .115 .130 .168 (.018) (.016) (.016) (.018) (.018) (.018) (.018) (.018) (.023) (.023) B .886 .893 .891 .882 .886 .885 .885 .884 .864 .819 (.015) (.013) (.013) (.015) (.015) (.015) (.015) (.015) (.019) (.022) Q(10) 22.94 13.84 11.37 11.51 12.53 13.79 14.12 12.86 10.66 8.96 02(10) 7.67 8.35 8.15 7.89 6.99 6.35 5.88 5.61 6.36 9.41 ms 0.536 0.665 0.726 0.756 0.759 0.759 0.762 0.772 0.792 0.829 1114 3.779 3.864 3.944 4.025 4.059 4.091 4.127 4.161 4.251 4.409 £ -426.0 -392.3 '378.9 ~374.9 “374.4 -375.3 '376.2 '376.2 -376.0 -377.4 Key: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung-Box statistics based on the residuals and squared residuals, m3 and ma are the sample skewness and kurtosis statistics based on the standardized residuals, and f is the maximized value of the log likelihood. 955 IHXBIJS 235 Estimated ARFIMA-GARCH Model for CPI Inflation: Japan d. 212 113 100 l-L A 10 CPI = l + 9 L + 0 L + 9 L e < ) g t ( 1 12 13 ) t 2 e (2 - 11 () <7 tl t-l ( ' t) 2 2 2 0 - w + (26 + 0 t t-l p t-l d .10 .20 .30 .40 .50 .80 .70 .80 .90 1.00 01 -.019 -.181 —.348 -.499 -.817 - 709 -.788 -.858 - 901 -.927 ( 403) (.045) (.042) ( 038) ( 034) (.031) (.028) (.025) (.022) (.021) 12 .321 .298 .284 .287 .258 .248 .215 .259 .274 .294 (.043) (.044) (.044) (.044) (.044) (.045) ( 045) (.045) (.044) (.043) 13 -.045 -.142 ~.225 -.278 -.301 -.305 —.301 -.291 *.287 -.289 (.455) ( 045) ( 043) (.041) ( 041) (.042) ( 044) (.044) ( 044) (.043) 0 .0100 .0108 .0132 .0151 .0145 .0120 .0088 .0085 .0057 .0083 (.0042) (.0050) (.0082) (.0070) (.0089) (.0420) (.0048) (.0036) (.0032) (.0032) a .080 .082 .090 .099 .098 .087 .073 .082 .057 .080 (.018) ( 018) (.017) (.017) (.018) (.014) (.013) (.012) (.013) (.013) 8 .908 .905 .893 .883 .885 .898 .914 .927 .933 .930 (.178) (.018) (.019) (.021) ( 020) (.017) (.015) ( 014) (.015) (.015) Q(10) 17.70 22.18 27.07 29.52 29.88 28.78 28.95 24.48 22.47 21.03 02(10) 9.41 10.88 11.77 12.94 14.24 15.33 18.01 14.78 11.25 8.81 m3 0.413 0.443 0.481 0.483 0.452 0.431 0 399 0.348 0 289 0.198 m4 3.973 4.009 4.015 4.035 4.098 4.211 4.380 4.525 4.857 4.882 £ -813.4 -810.1 -810.2 -810 4 -810 2 —809 8 -808 7 -808.0 -809 1 -814.0 Key: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung-Box statistics based on the residuals and squared residuals, m3 and m4 are the sample skewness and kurtosis statistics based on the standardized residuals, and £ is the maximized value of the log likelihood. 96 TABLE 24 Estimated ARFIMA-GARCH Model for CPI Inflation: United Kingdom d 12 13 100 l-L A 10 CPI - 1 + o L + o L + o L e ( ) g t ( 1 12 13 ) t 2 e 0 ~ N 0 0 cl c-1 ( ' c) 2 2 2 0 ‘3 Q)‘+ (16 +‘ a c t-l 5 t-l d .10 .20 .30 .40 .50 .60 .70 .80 .90 1.00 91 .177 .053 ”.055 ‘.164 '.282 ”.412 ‘.556 '.673 ”.762 '.866 (.059) (.061) (.062) (.060) (.058) (.055) (.050) (.044) (.039) (.030) 2 .336 .304 .287 .278 .271 .265 .254 .239 .229 .227 1 (.037) (.037) (.037) (.037) (.037) (.037) (.037) (.037) (.038) (.037) 13 .017 ‘.068 “.126 “.172 '.211 “.251 ”.292 '.310 ‘.306 '.275 (.045) (.047) (.048) (.049) (.049) (.049) (.048) (.045) (.044) (.042) .093 .104 .118 .135 .137 .144 .142 .141 .154 .107 (.026) (.028) (.034) (.041) (.044) (.047) (.047) (.047) (.050) (.032) a .232 .226 .207 .204 .196 .200 .189 .182 .192 .104 (.044) (.044) (.046) (.050) (.052) (.054) (.052) (.050) (.053) (.041) 8 .590 .556 .529 .486 .491 .474 .488 .494 .456 .610 (.075) (.081) (.097) (.119) (.128) (.135) (.135) (.135) (.143) (.103) Q(10) 24.29 20.03 20.70 24.16 28.20 31.08 31.46 30.00 27.68 22.14 Q2(10) 18.79 19.93 18.76 17.15 14.51 13.33 12.37 11.87 12.16 11.14 m3 0.682 0.815 0.870 0.876 0.865 0.855 0.864 0.886 0.899 0.954 m4 4.186 4.506 4.727 4.807 4.850 4.886 5.001 5.082 5.093 5.398 2 '481.76 '466.34 “461.74 “462.21 “453.64 '465.60 “466.66 ‘466.62 ’466.08 ‘465.03 £21: All models are estimated by approximate MLE having concentrated out d. Q(10) and Q2(10) are the Ljung-Box tests based on the residuals and squared. residuals, m3 and. 1114 are the sample skewness and. kurtosis statistics based on the standardized residuals, and f is the maximized value of the log likelihood. 97 TABLE 25 Likelihood Ratio Tests of Relationship Between Mean and Variability of Inflation, yt d 12 (l-L) yt = (1+61L+012 13 2 -l etlnt-l ~ D(0, at, u ) 2 2 2 at = w + aet_1 + fiat_1 + 7yt_1 LR Tests Argentina Brazil France Germany Israel Italy 13 L +0 L )et+60t Japan UK US 6=0 8.90* 4.26* 0.00 2.40 5.46* 0.72 1=0 12.92** 7.14** 1.54 0.80 10.88** 1.12 1.44 3.88* 0.40 0.44 9.12** 1.28 Key: yt is 100 Alog CPIt, the conditional density D is student t for France and Israel and is Normal otherwise. Under the null hypothesis all test statistics are distributed as asymptotic X3! random variables. Two asterisks denotes significance at the .01 level and one asterisk denotes significance at the .05 level. 98 REFERENCES Alesina, A. 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Schwert (1977), "Short-Term Interest Rates as Predictors of Inflation: 0n Testing the Hypothesis that the Real Rate of Interest is Constant", American Economic Review, 67, 478- 486. Newey, W.K., and K.D. West (1987), "A Simple, Positive, Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix", Econometrica, 55, 703-708. Okun, A.M. (1971), "The Mirage of Steady Inflation", Brookings Papera on Economic Activity, 2, 485-498. Pagan, A.R., Hall, A.D., and P.K. Trivedi (1983), "Assessing the Variability of Inflation", Review of Economic Studies, 50, 585-596. Parks, R.W. (1978), "Inflation.and.Relative Price Variability", Journal of Political Economy, 86, 79-96. Phillips, P.C.B. and P. Perron (1988), "Testing for a Unit Root in Time Series Regression", Biometrika, 75, 335-346. Rose, A. (1988), "Is the Real Interest Rate Stable?", Journal of Finance, 43, 1095-1112. Sowell, F.B. (1990), "Modeling Long Run Behavior with the Fractional ARIMA Model", GSIA Carnegie Mellon University working paper. Sweeney, R.J. (1987), "Some Macro Implications of Risk", Journal of'MoneyI Credit and Banking, 19, 222-234. Taylor, J.B. (1981), "On the Relation Between the Variability of Inflation and. the .Average Inflation. Rate", Carnegie-Rochester Conference Series on Public Policy, 15, 57-85. L“ W IV. A GENERALIZED METHOD OF MOMENTS ESTIMATOR FOR LONG-MEMORY PROCESSES 131-.”‘1‘ ‘T‘LI CHAPTER IV A GENERALIZED METHOD OF MOMENTS ESTIMATOR FOR LONG-MEMORY PROCESSES 1. INTRODUCTION As discussed in Chapters II and III, there currently exist two general types of estimation procedures which have been used to estimate the degree of persistence of a fractionally integrated, long-memory process. The frequency-domain based, two-step estimation procedure of Geweke and Porter-Hudak (1983) utilizes the spectrum of the series in estimating the parameter of fractional integration. Alternatively, the maximum likelihood.estimation procedures of'Hosking (1984b), Fox and.Taqqu (1986), and Sowell (1992), which have been based in both the time and frequency domains, utilize standard first-order conditions in maximizing the log of the likelihood function of the fractionally integrated process. The generalized method of moments (GMM) estimation technique is an attractive alternative framework in which to estimate the parameter of fractional integration of a long-memory process since it does not require the distributional assumptions necessary under maximum likelihood estimation techniques and consequently offers the advantage of robustness in parameter estimation. In addition, as evidenced in Chapter III, approximate MLE can often involve numerically cumbersome techniques which may be avoided, in some part, with the technique of generalized methods of moments. For the fractionally integrated process, the GMM estimation technique exploits the set of moment conditions that equate the expected value of the sample autocorrelations to the corresponding population autocorrelations, evaluated at the true parameter values. In this way a 101 ‘ i 102 consistent estimate of the parameters can be obtained. This chapter provides the derivation of a generalized method of moments estimator for the degree of persistence, d, of a long-memory time series and examines its efficiency relative to those estimation procedures discussed in Chapters II and III. The following section discusses the GMM estimation technique in the context of the fractionally integrated model and presents the motivation for the use of GMM in this context. Section 3 presents the derivation of the asymptotic distribution of the estimated autocorrelations under specified assumptions. This section also presents the derivation of the asymptotic variance of the GMM estimator. Section 4 provides an investigation of the estimation procedure by examining the asymptotic efficiency of the estimator for a range of values of the parameter d using various moment conditions as well as subsets of moment conditions. The chapter ends with a brief summary and concluding section. 2. GMM ESTIMATION IN THE CONTEXT OF THE FRACTIONALLY INTEGRATED MODEL The estimation technique of generalized method of moments makes use of a set of orthogonality conditions that are implied by the model to be estimated such that the expected value of the orthogonality condition is equated to zero at the true parameter value. For the case of the fractionally integrated model, consider the non-zero mean, stationary time series {yt} expressed in.ARFIMA(p,d,q) form as introduced in Chapter II as (1.) <1 - I.)d (yt - 4) = 4(L)/¢(L) ct = II: where for -5 < d < 5, yt is said to be fractionally integrated of order d, the polynomials 6(L) and ¢(L) are as defined in Chapter II, and ut is a 103 stationary and invertible error process. Following the framework of Hansen's (1982) GMM estimator, estimation of a (p x l) parameter vector A via the GM estimation technique involves the use of m orthogonality restrictions where m is at least as great as p. Defining the (m x 1) vector of orthogonality conditions as some function g(yt,A), the GM estimator of A is given as that value of the parameter vector which satisfies (2-) min §(y.A)’ W 20$). A 1 where §(y,A) is the standard expression for the orthogonality condition of the GMM estimator written in the form of an average as T 2 - 1/T Z g. t=l In. this formulation” W' is an 0n >< no positive definite, symmetric weighting matrix defined as that matrix which has the characteristic of minimizing the sample orthogonality conditions. The minimized value of the criterion function (2.) will be asymptotically distributed as Chi- square with (m - p) degrees of freedom. Within the context of the GMM estimation procedure, the expression g(y,A) should converge to zero for 1 In the context of the fractionally integrated process expressed in (1.), use of an orthogonality condition of the form g(y,A) is not directly applicable due to the difficulty in expressing the orthogonality conditions in the form of an average. This problem arises because the typical orthogonality condition of the fractionally integrated.process is a function of an infinite number of terms. Therefore, as an alternative, the function g(-) is expressed in the form of a moment condition for the fractionally integrated process, as discussed later in this section. -' - - ..-‘q 104 the true parameter vector and not for any other element of the parameter space. Additionally, the optimal weighting matrix, W, is given as -l W - [cov g(yt,A)] . Under weak regularity conditions, Hansen (1982) shows that the GMM estimator of the parameter vector A satisfies ‘ , -1 -1 JT (Am, - A) ~N(0.[D c D] ) 1 is the optimal weighting matrix. In this representation, D is where C- defined as the (m x p) matrix of partial derivatives of the moment conditions with respect to the parameter vector; that is, 0 _ 6 8(Y§.A). a A’ The GM estimation procedure may be applied to many standard econometric models, each of which exploits its own unique set of moment conditions and asymptotically optimal weighting matrix. For the case of the fractionally integrated model, the moment conditions exploited make use of the theoretical and estimated autocorrelation functions of the model. Consider, for simplicity, the zero-mean ARFIMA(O,d,O) process (3-) (1 ' L)d yt . Ut where ut is a stationary error process, d.e (~8,8), and pj = corr(yt,yt_j) h is defined as the jt autocorrelation function of the process. The simple 105 model expressed in (3.) is a single parameter model such that A consists of a single element, d.2 Recall that for the model given by (3.), pj may be expressed simply as a function of d, as given earlier in Chapter II, as _ F(1-d)P(j+d) - a (d+i-1) pj F(d)F(j+d-1) ,EI'TETE)‘ The moment condition exploited by the fractionally integrated model, considering the first k moments, may be expressed as E[S - p(d)] - 0 where p = [p1, . . . .pkl’ and p(d) - [p1(d). . . . .pk] and the asymptotically optimal weighting matrix, W, is given as W - [cov13 - p)]‘1. In considering the efficiency of the GM estimator, it should be the 2 This estimation procedure may be applied to the more general, multi- parameter ARFIMA representation given by (1.) in which case A would be a vector and would include the parameters of the autoregressive and moving average polynomials. 106 case that any estimator based on all available moment conditions should.be relatively more efficient than that based on only a subset of these moment conditions. However, in the case of the fractionally integrated process there will be some advantage to considering the GMM estimator based on a subset of moment conditions, especially in the case where a stationary ARMA component exists in the series. In such a model, the autocorrelation functions for the lower-order moments of the process will be a function of the autoregressive and moving average parameters of the model as well the parameter d. As such, the autocorrelation functions for the lower-order moments will be quite different from those autocorrelations that exist at higher-order moments, which are simply a function of the parameter d. In this sense the autocorrelation functions for the lower-order moments may be thought of as being "contaminated" when a stationary ARMA component exists in the series. As a result, it would be of interest in this context to determine whether the efficiency of the GM estimator is maintained.when.using some subset of the moment conditions, say moments (r + 1) through {(r + l) + k), such that the first r moments may be discarded. Simple asymptotic variance calculations may be employed to determine these relative efficiencies, and these operations are discussed further in section 4. 3. ASYMPTOTIC DISTRIBUTION THEORY In order to determine the asymptotic distribution and optimal weighting matrix of the generalized method of moments estimator for the fractionally integrated process, it is necessary to derive the asymptotic distribution of the moment condition, [0 - p(d)]. Recall that agglsolves the operation 68(d)/88 = 0 as given in equation (4.). This expression may 107 be written in the form of its Taylor-series expansion as 88 d 2 . 88(8) ‘_ ( ) _+ 8 S(d) “1_ d), 88 88 adg Where d* lies between d and d. Equating the above expansion to zero and solving for (d - d) gives A 2 -1 (d _ d) = _ [a Séd)] 88(8) 6d* 6d where 88(8)/88 = -2 D’W[B - p(d)], 628(d) 2 . /88 - 2 D wn + op(l), and D is as defined previously. It follows that the asymptotic distribution of the GMM estimator of d satisfies 2 . _ 8 S(d) -1 88(8) JT (8 - 8) - 17;§§_1 J1 ad A -1 A - (D'wn) D'Wfflp - p(d)]. The asymptotic distribution of'.fT[S - p(d)] for the fractionally integrated, long-memory process is given in Hosking (1984a). Hosking considers the fractionally integratedHARIMA(p,d,q) process as expressed in (1.) where e is an independent and identically, but not necessarily t normally, distributed white noise error process with mean zero and variance 0% 6t has a finite fourth moment, and yt has mean u. The sample __._._~__‘_Il 108 autocorrelation function is defined as T1 A §1(Yt ' 37) (Yt+j ' S") pj a: t — if“ — 7)2 where 7 = 1/Tti1yt is the sample mean of the process. For the standard, stationary, short-memory time series process where d takes on integer values, there are standard results for the asymptotic distribution of the sample autocovariance function. However, in the case of the fractionally integrated, long-memory time series process where -H <2<13< 8, Hosking (1984a) shows that these standard results hold for d e [-8,%) but not for d z 18. This discrepancy may be attributed to the treatment of the estimation of the mean of the fractionally integrated process. That is, for k s d < 8 the effect of replacing p with 7 is not negligible, even asymptotically, and large bias (of the same order of magnitude as that of the standard deviation) is introduced into the estimate of the autocorrelation function. Consequently, the remaining analysis of this chapter will restrict attention to the range of values of the parameter vector for which d E [-k,k). Within this range3 the estimated autocorrelation functions will be distributed asymptotically normal with variance of order l/T. Following Hosking (1984a), the estimated autocorrelation function, A has covariance matrix C which has ihjth element as given by 3 For d - k, asymptotic normality is retained but the variance of the estimated autocorrelation function is of order l/T(log T). For d.e (8,8), asymptotic normality is not retained.and the variance is of order Tfiuerk A w~_M‘.._—-i u 109 - 2 pips)(ps+. + p . - 2 9.108)} (5.) c- - = 1/T {5219.48 + ps-i J s-J J 1,] and C - {cij}' In applying the GMM estimation procedure to the single parameter fractionally integrated process, then, the asymptotic distribution of [3 - p(d)] will be given by Jfi§(; - p(d)) ~ N(0,C) where the dimension of C will be defined by the number of moments used in estimation, and the asymptotic distribution of the GMM estimator will be given by (6.) fi(8 - 8) ~ N[0,(D’C'1D)'1]." 4. ASYMPTOTIC PERFORMANCE OF THE GMM ESTIMATOR FOR THE FRACTIONALLY INTEGRATED MODEL The asymptotic performance of the generalized method of moments estimator for the fractionally integrated process is examined by calculating the large sample variance of 8 as given in (6.). To perform this calculation it is necessary to compute [D’C'lD]-1 where D and C are functions of the parameter (1 and the number of moments, k, used in estimation. In the general case, the efficiency of the GMM estimator should be greatest when calculations are performed utilizing all available moment conditions. However, in the case of the fractionally integrated process, which uses the estimated autocorrelations in calculation, the possible number of available moment conditions is infinite. Relative efficiency, then, should continue to increase as 81 greater number of ‘ In.this representation the optimal weightinglnatrix, defined in (2.) as W, is given by C'l. 110 moments are used in estimation such that more moments will always be preferred. As such, the use of any subset of moments in estimation should provide lower levels of efficiency relative to that in which a greater number of moments are employed. The calculation of the vector of partial derivatives, D, and the covariance matrix of the estimated autocorrelation functions, C, is as follows. Recall that the (k X 1) vector p(d) is given by _ -9— - l-d .8. 9:1 l-d 2-d p(d) = d d+1 d+2 d+(k-1) _ l-d 2-d 3-d ' ' ' k-d It follows then that D is given by - 1 - (d-l)2 -2(-l-2d+2d2) (d-l)2 (d-2)2 3(4+128-982-883+384) n — (d-1)2 . (d-1)2>() or q > 0, the lower-order autocorrelations may be substantially different than those for the (0,d,0) part of the process. Since it appears that these lower-order autocorrelations cannot be dropped from estimation without sacrificing efficiency, it is reasonable to consider GMM estimation of the ARFIMA(p,d,q) model in the context in which d is estimated.jointly with the autoregressive and moving average parameters of the process. This is an important topic for further research. I - . -_—-——- .- 117 emmn. nmee. enae. moae. mwwe. mm8e. muse. 8mom. menu. nmen. m8oe. shoe. seem. n8om. ce8m. emeo. cal: cues. whee. enae. aaae. amNe. ea8e. mane. muan. mn8n. C8en. 8amm. mmoa. «mam. mama. Gene. mauo.a ml: 8ahm. eeee. emae. ouae. mnwe. ~8me. mnee. mmwm. 8aem. mace. mu8e. moam. eeua. euem. m8ao.a mmmo.a ml: «New. mane. enae. mmae. name. maee. 8nae. Numn. mans. emwe. omee. amam. mmem. mmoo.a maeo.a umoa.a Blah m8mn. O8ne. nmae. mmae. meme. none. moan. menu. ewoe. 8mm». emom. memo. «cao.a oaeo.a mmma.a oaea.a Ola eece. meme. enae. 8eae. 888e. mmee. 8oun. 8oen. e8me. came. oamm. muao.a memo.a onma.a meo~.a aae~.a m Ia..— meme. muse. eeae. nmme. aeme. 8NON. owns. unae. 8aee. em8¢. 8o~o.a m8mo.a 805a.a Ne8~.a memm.a 8nmm.a 8|: ~m8e. meme. name. a8me. «use. 58mm. mace. muse. swam. 8m8o.a amma.a 8n-.a emNm.a aeu8.a nmme.a maNe.a ml: 8mme. meme. meme. meme. 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OH.H du.lc ON.IU ma.lv OH.IU mo.lv oo.lv no.-lc OH.-IU ma.-lv o~.-lu mm.-lfi om.-lv mm.-lfl oe.-lv we.-lv a8..ne 122 REFERENCES Fox, R. and M.S. Taqqu (1986), "Large Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time-Series", Annals of Statistics, 14, 517-532. Geweke, J. and S. Porter-Hudak (1983), "The Estimation and Application of Long Memory Time Series Models", Journal of Time Series Analysis, 4, 221-238. Hosking, J.R.M. (1984a), "Asymptotic Distributions of the Sample Mean, Autocovariances and Autocorrelations of Long—Memory Time Series", Mathematics Research Center Technical Summary Report #2752, University of Wisconsin-Madison. Hosking, J.R.M. (1984b), "Modelling Persistence in Hydrologic Time Series Using Fractional Differencing", Water Resources Research, 20, 1898- 1908. Li, W.K. and A.I. McLeod (1986), "Fractional Time Series Modelling", Biometrika, 73, 217-21. Sowell, F.B. (1992), "Maximum Likelihood Estimation Univariate Fractionally-Integrated Time-Series Models", Journa Econometrics, forthcoming. of Stationary of .-- -0 .— .. . A. V. EQUILIBRIUM MONEY DEMAND FUNCTIONS, REAL EXCHANGE RATES, AND PPP: AN ANALYSIS OF COINTEGRATING RELATIONSHIPS IN CANADA AND THE U.S. I. A4 l--AAA CHAPTER V EQUILIBRIUM HONEY DEMAND FUNCTIONS, REAL EXCHANGE RATES, AND PPP: AN ANALYSIS OF COINTEGRATING RELATIONSHIPS IN CANADA AND THE U.S. 1. INTRODUCTION Investigation of empirical money demand functions has long been of interest to macroeconomists seeking to identify the existence of a stable long-run relationship between real money balances and measures of economic activity in an economy. Empirical evidence of a stable money demand function can have far-reaching implications for, among other things, the conduct of monetary disinflationary policy within an economy. In addition, estimation of long-run money demand functions can provide researchers with measures of income and interest elasticities of the demand for real balances which can provide insight into the behavior of individuals in the economy. Traditional modelling of money demand functions has been grounded in the theory that money demand should be linked, in a predictable manner, to the behavior of some scale variable and to some measure of the opportunity cost of holding money. This tradition is now augmented by recent advances in econometrics, in the form of cointegration tests, which allow for a more rigorous analysis of long-run economic relationships. These advances allow researchers to identify stable long-run relationships among a set of non-stationary variables, providing for nearly an ideal framework with which to examine the stability of money demand functions. In addition, the recent advances in the area of cointegration provide a useful methodology in which to analyze the stability of the real 123 124 exchange rate between two countries. These techniques can be used to test for the existence of such relationships as the interest rate parity condition or the validity of a long-run Purchasing Power Parity (PPP) between two countries, which are intuitively appealing relationships from an economic standpoint. The issue of stable exchange rates and valid PPP is by no means clear cut among many nations' currencies, and insight into this area can provide valuable information in the areas of international policy and trade. This chapter investigates the existence of stable money demand functions and exchange rates for Canada and the U.S., utilizing the technique of cointegration as developed by Engle and Granger (1987). The analysis is made within the context of the monetary balance of payments theory which provides a useful method for analyzing issues of the balance of payments and exchange rates that stress the interaction of the supply and demand for money. This theory is grounded on several key assumptions and the validity of these assumptions may be investigated empirically through the cointegration methodology. The contribution of this chapter is two-fold. First, the long-run equilibrium relationship between real money balances, real output, and short-term interest rates in Canada and the U.S. is examined using the cointegration techniques of Johansen (1988) and Johansen and Juselius (1989). This approach will allow for a unique investigation into the existence of cross-country relationships in the monetary model for these two countries. Second, the stability of the Canadian/U.S. exchange rate is examined, also using cointegration techniques, and an investigation is made into the validity of long-run PPP and interest rate parity for these two countries. The use of the monetary' balance of payments framework is of 1:3 I). ' .' [A 125 particular interest in the case of Canada and the U.S. due to the relationship that exists between these two countries with respect to their position as trading partners and the nature of their trade operations. The usual criteria which often cause the theory of the monetary balance of payments to break down do not exist in this case given the unique relationship between Canada and the U.S. or, at least, exist in rather weak form. In this sense, the theory of the monetary balance of payments provides a useful framework in which to examine the issues of interest in this chapter. The plan of the rest of the chapter is as follows. Section 2 provides a survey of the theoretical underpinnings of the monetary balance of payments framework and the equilibrium monetary model. Section 3 reviews the Johansen and Juselius (1988) methodology for detecting the presence of cointegration and presents the estimation of the equilibrium money demand function for Canada. Particular attention is given, in this section, to the apparent collapse of the money demand relationship in the Canadian data after 1980. Section 4 examines the existence of a stable equilibrium money demand function for the U.S. and confirms its existence for the sample period of this analysis. Section 5 presents the joint estimation of the Canadian and U.S. money demand functions to examine the similarity of the dynamics in the two countries and to investigate the existence of cross-country effects in the model. An alternative to the double-logarithmic specification of the money demand equation for the joint. model is also considered. in this section to investigate the robustness of the equilibriuml money demand function to alternative functional forms. Section.6 examines the stationarity of nominal exchange rates and relative prices and addresses the issue of long-run purchasing 126 power parity. A brief concluding section follows. 2. THE MONETARY BALANCE OF PAYMENTS THEORY The monetary approach to the balance of payments concentrates on the direct relationship between the money market and the balance of payments. This analytical framework deals exclusively with long-run equilibrium relationships which focus on the connection between prices, output, interest rates, and the balance of payments, and is based on a few central assumptions. Under these assumptions, the balance of payments will reflect any disequilibrium that emerges in the money market.1 The primary assumption of the monetary approach to the balance of payments is that the demand for money is a stable function of a given set of 'variables. This implies that there exists a stable, long-run, equilibrium money demand function for each country in the model. Another assumption is that there is perfect mobility of goods and financial assets 'between countries so that there is perfect substitutability of these goods and assets. The implication of this assumption is that the market should ensure a single price for each commodity and a single rate of interest. That is, changes in relative goods prices should be proportional to Changes in the nominal exchange rate so that the law of one price holds in the long run. In addition, the expected return on interest bearing securities and assets which are denominated in different currencies should 1 The origin of this theory began in the 18th century with the work of David Hume. Contemporary revival of the theory came about with the work of James Meade in the early 19505 and further development is attributed to Polak and his associates at the International Monetary Fund in the late 19505. Interest in this area was greatly expanded with the ‘workLOf'Mundell (1971) and Johnson (1972). An excellent survey of many of the contributions of the theory can be found in Kreinin and Officer (1978). 127 be the same. In addition, it is assumed that output and employment tend to full-employment levels, at least in the long run. The specific nature of each of these assumptions will be discussed more fully below. 2.1 LONG-RUN MONEY DEMAND FUNCTIONS The basic premise of the monetary approach to the balance of payments is that in the long run there exists a stable demand for the stock of money balances as a function of a given set of variables. In particular, the demand for real money balances is posited to be a function of real income and nominal interest rates. To derive this relationship, consider that the demand for nominal money balances, M% is a function of nominal income, Q, short-term interest rates, R, and the price level, P. This relationship can be expressed as: Md - f(Q,R,P). Assuming that individuals in the economy are not subject to money illusion, the money demand function can be written as homogeneous of degree one in prices, and can be re-written as the demand for real money balances: hf&P - f(Q/P,R). In this representation, the demand for real money balances, PN??, is a function of real income, Q/P, and interest rates, R. An increase in real income, ceteris paribus, would be expected to increase the demand for money balances since this increases the amount of transactions one is able to finance. An increase in the interest rate, on the other hand, is expected to decrease the demand for money balances since this increases the opportunity cost of holding money. If one assumes that the income velocity of money, k, varies with the interest rate, and assumes further that the interest elasticity of money demand is unity, then the above money demand relationship can.be expressed in the form of the familiar Cambridge cash balances equation as Md/P - ... .‘___-— . u 128 k(r)Q/P. Expressed in this manner, the demand for real money balances is homogeneous of degree one in real income. 2.2 REAL EXCHANGE RATES AND THE LAW OF ONE PRICE As formulated in the monetary approach to the balance of payments, real exchange rates are simply relative prices of the foreign and domestic countries' currencies. That is, the real exchange rate is the foreign currency price of a unit of domestic currency. The determination of this price occurs via the equilibrium supply and demand for the stock of foreign and domestic monies. Under this approach, exchange rates are considered to be predominately a monetary phenomenon, but "real" factors, operating through monetary channels, are also assumed to influence the equilibrium exchange rate. Given the assumption under the monetary approach to the balance of payments of perfect substitutability across countries in both the goods and capital markets, a natural outcome of the model is that the law of one price should hold for all countries in the model. This implies the existence of a single integrated market for all traded goods and capital where the actions of the market ensure a single price for each commodity and a unique interest rate. That is, the monetary approach to the balance of payments is based on the assumption that interest rate parity holds. Due to efficient arbitrage in assets with similar characteristics, the interest rate differential between two countries will be reflected in the forward premium of the exchange rate so that securities that share the same characteristics will yield the same return in equilibrium. One further implication of the framework of the monetary balance of payments is that in the long run a unit of domestic currency is expected «Ina-4-5.- ..- Q 129 to command. the same purchasing 'power in the foreign. country, when converted into the foreign currency, as it would command in the domestic country. This is the so called Purchasing Power Parity (PPP) theory. The implication of PPP is that the nominal exchange rate, 5, which is expressed in terms of units of foreign currency per unit of domestic currency, will be proportional to the ratio of the foreign price level, P*, and the domestic price level, P: s = P*/P. This relationship suggests that, adjusted for changes in exchange rates, the price levels of the domestic and foreign countries move in step with one another. The resulting implication, assuming that s, P, and P* are all non stationary, is that a linear combination of the logarithms of the nominal exchange rate and relative prices defines a stationary time series. 2.3 THE LINK BETWEEN THE CANADIAN AND U.S. ECONOMIES The economies of Canada and the U.S. are so closely linked to one another that fluctuations in one country's macroeconomic variables often closely parallel fluctuations in the macroeconomic variables of the other country. Poole (1967) investigated the strength of the relationship between Canadian and U.S. macroeconomic variables from 1950 through 1962 and identified a strong linear relationship between. many of these variables. In addition, Poole noted that the magnitude of movements in the two economies were very similar. These similarities make the Canadian and U.S. economies quite interesting from the perspective of the monetary approach to the balance of payments. The U.S. is by far Canada's largest trading partner with approximately eighty percent of all Canadian exports going to the U.S. and approximately twenty-five percent of all U.S. exports going to Canada. In fact, the trade relationship that exists 130 between Canada and the U.S. is the largest of any two trading partners in the world. In addition, the flow of foreign direct investment between Canada and the U.S. is the largest two-way flow of foreign direct investment anywhere in the world. (See, for example, Rugman (1990) and Hill and Whalley (1985).) The international trade relationship that exists between Canada and the U.S. can no doubt be attributed partly to the close geographic location of the two countries and partly to the large size of the U.S. relative to Canada in the international economy. The close proximity of the two' countries allows for relatively easy’ mobility’ of' goods and financial assets across borders. In addition, trade relationships between Canada and the U.S. have been steadily improving over the past decade with the advent of actions such as the General Agreement on Tariffs and Trade (GATT) and the Canada/U.S. Free Trade Agreement, such that tariff and.non- tariff barriers to trade between the two countries have been substantially lowered. With the end of the Tokyo Round of the GATT negotiations in 1987, Canadian and U.S. tariffs had been reduced to such a level that approximateLy 80% of Canadian exports entered the U3S. duty free and approximately another 15% entered at tariff rates of less than 5%. For Canada, approximately 65% of U.S. exports entered Canada duty free by 1987 with another 25% entering at rates of less than 5%. 2.4 THE EQUILIBRIUM MONETARY MODEL Given the conditions set out in the monetary approach to the balance of payments theory, the equilibrium monetary model of money demand and exchange rates may be formalized as shown in equations (1) through (3). ‘0. a -..K :0 ‘4‘ 131 (1) log(Mt/Pt) = 60+ 6llog(Qt/Pt) - 6210g(Rt) + elt ** *3? * (2) 10g(Mt/Pt) = (o + (110g(Qt/Pt) - C210g(Rt) + ‘2: * (3) 10g - 92(Rt) + 64t 5 1 M* 9* 1 * 9* R* ( > og< t/ t) — wo + w, 08(Qt/ t> - ¢2( t> + ‘5: Recall that estimation of the double-logarithmic specification of the stacked model clearly produces two cointegrating vectors which may be interpreted as stable long-run velocity functions for Canada and the U.S. -.-—--9__—~_. - ._ I. . o ' 1' 151 Table 6 presents the trace test statistics and cointegrating vector from estimation of the semi-logarithmic specification of the money demand functions for the joint Canadian/U.S. model with velocity restrictions imposed. These results indicate the existence of three cointegrating vectors, implying the presence of an additional long-run equilibrium relationship in the model. The presence of this third cointegrating vector in the joint, semi- 1ogarithmic specification of the model may be accounted for by the interest rate parity condition that results from profit-seeking arbitrage activities. More specifically, consider the joint model written in terms of the velocity restrictions as (6) VC + 02Rt = €4t * * (7) Vt + ¢2Rt - 65t * (8) Rt - a + ¢Rt + 66 t' In this representation, V embodies the velocity restriction imposed on the model and equations (6) through (8) represent three unique equilibrium relationships which correspond to those identified by the Johansen and Juselius procedure for the semi-logarithmic specification of the joint model. The parameters 0 and p represent the interest elasticities of the U.S. and Canada, respectively; the parameter ¢ embodies the uncovered interest parity condition. According to the theory of uncovered interest parity, the difference between.the domestic and foreign interest rates of two countries shouldee 152 stationary. This implies a value for ¢ in equation (8) of unity, which corresponds to the finding of a third cointegrating vector for the model estimated via the semi-logarithmic specification. That is, under uncovered interest parity, (Rt - R3) - a + €6t should be stationary. If uncovered interest parity holds, and therefor accounts for the presence of the third cointegrating vector identified in estimation, then the coefficient on the foreign interest rate should be the negative of that on the domestic interest rate. In order to test the hypothesis that the third cointegrating vector identified by the Johansen and Juselius procedure is indeed attributable to the condition of uncovered interest parity, a chi-square test of the restriction that 3 is equal to -l was performed. The calculated test statistic indicates that the null hypothesis cannot be rejected; that is, uncovered interest rate parity holds. As such, the third cointegrating vector identified in the semi-logarithmic specification of the joint model has the interpretation of being a stationary difference between the Canadian and U.S. interest rates. This combined with the existence of two stable long-run velocity functions for Canada and the U.S. accounts for three unique, long-run equilibrium relationships among these key macroeconomic variables in the joint Canadian/U.S. model. 6. STATIONARITY OF REAL EXCHANGE RATES Although much of the recent empirical literature on exchange rates tends to support the notion that exchange rates behave as a random walk, 153 the possibility still remains that exchange rates may belong to a larger equilibrium system in which short-run deviations from equilibrium are possible. Such deviations are discussed by Dornbusch (1976) who noted that exchange rates often times were experiencing temporary overshooting. Examination of the Canadian/U.S. exchange rate in the context of a long-run equilibrium relationship is a natural setting in which to apply the techniques of cointegration. Use of this methodology enables one to investigate the possible existence a stable long-run equilibrium relationship for the real exchange rate between any two countries. The possible existence of such a relationship between the economies of Canada and the U.S. is particularly interesting given the parallel nature of the two economies. Before proceeding with the investigation of a cointegrating relationship for the Canadian/U.S. exchange rate it is necessary to determine whether the nominal Canadian/U.S. exchange rate and the price levels of the two countries are non-stationary 1(1) variables. Table 1 indicates that each of the variables is indeed 1(1) and as such it is possible to test whether a particular linear combination of these variables, the real exchange rate, is stationary. Evidence of a single cointegrating vector among these variables would indicate the existence of a stable long-run relationship between the nominal exchange rate and the relative price levels. Recall from equation (3) the relationship between the nominal Canadian/U.S. exchange rate and the Canadian and U.S. price levels: 71’ 71' (3) log - 71103(Pt) - v2log + e3t .’-..1 "f 'N". 4 D 154 If it is found that 71 - 1 a 1 then equation (3) may be interpreted as 2 the real Canadian/U.S. exchange rate. Evidence of a stable long-run relationship in this formulation may be interpreted as the existence of a stationary real Canadian/U.S. exchange rate. The vector process, Xt, for the analysis of the real exchange rate may be represented as _ log 5 X ‘ [log P/P*] This representation uses the log of the ratio of the price levels of the U.S. and Canada rather than the individual price levels so that the hypothesis of long-run Purchasing Power Parity, PPP, may be examined later. In estimating the real Canadian/U.S. exchange rate the period considered in this investigation corresponds to the post-Bretton Woods period in whidh most of the Group of Seven countries abandoned their systems of fixed exchange rates. The data run from January of 1974 through September of 1990. The rmmdnal exchange rate was taken from International Monetary Fund data which contains end—of-month observations. The nominal exchange rate is expressed in terms of Canadian dollars per U.S. dollar. Results of estimation of the exchange rate model are provided in Table 7. The results of the Johansen trace tests for cointegration clearly indicate the presence of one unique cointegrating relationship between.the nominal exchange rate and the ratio of the U.S. and Canadian price levels over the period of 1974 to 1990. This is consistent with finding a stationary real Canadian/U.S. exchange rate for this period. The results 155 of this analysis indicate that the estimated parameter on the ratio of the price levels is 1.40. The model was also estimated over a sample containing the period in which Canada initially abandoned their system of fixed exchange rates, the period beginning in June of 1970, but the results over this period were inconclusive on the long-run properties of the data. The trace test statistics of the analysis over this sample period do not indicate a stationary long-run real Canadian/U.S. real exchange rate for the period beginning in 1970. This apparent tenuousness of the model indicates the sensitivity of the stability of the Canadian/U.S. real exchange rate to the time period under consideration” This may not be a surprising result, however, if one considers that the period immediately following the collapse of the Bretton.Woods system (June of 1970) was one in which there was still substantial adjustment occurring in many of the exchange rates which began to float at this time. An interesting,implication.of the 1974-1990 estimation result canfbe made with regard to the validity of the long-run PPP relationship. If an estimated parameter of unity is found for the ratio of real prices in the model, this indicates that changes in relative goods prices between the two countries will be proportional to changes in the nominal exchange rate. This is consistent with the assumptions laid out in the monetary approach to the balance of payments. The existence of a valid PPP relationship may not be a reasonable expectation for all pairs of countries in the world, but is particularly appealing for the U.S. and Canada. These two countries seem to satisfy, most nearly, the conditions described by the monetary approach to the balance of payments for which PPP should be expected to hold. Canada is 156 relatively small, compared to the U.S., and the two countries are the largest trading partners of any two countries in the world. The close proximity of the two countries allows for relatively easy mobility of goods and capital across borders with very limited restrictions to trade between the two countries. In addition, evidence of stable, long-run money demand functions for both countries has been found, as indicated in sections 3 through 5. Given these conditions, one would expect that the nominal Canadian/U.S. exchange rate and the relative price levels of the two countries, although non-stationary by themselves, should not move too far from one another in the long run and as such should have a stable linear representation. That is, it should be reasonable to expect that the variables comprising the Canadian/U.S. real exchange rate should move together over time and that the relative goods prices between the two countries, adjusted for changes in the exchange rate, should move in step with one another. The validity of the PPP hypothesis can be tested within the framework of the Johansen model by imposing the restriction of a unitary coefficient on the ratio of the price levels in the real exchange rate equation. The results of this investigation are also presented in Table 7. The value of the likelihood ratio test statistic for the hypothesis that the coefficient on the ratio of the price levels is the opposite of that on the nominal exchange rate fails to reject the maintained hypothesis at the 1% level, although the null hypothesis is rejected at the 5% level. This may be interpreted as evidence in favor of the long- run PPP relationship between Canada and the U.S., although this evidence is somewhat weak. It may be appropriate, therefore, to conclude that a 157 valid long-run PPP relationship exists between the U.S. and Canada, consistent with theoretical expectations. 7. SUMMARY AND CONCLUSION This chapter has investigated key elements of the theory of the monetary balance of payments for the economies of Canada and the U.S., using the methodology proposed by Johansen (1988) and Johansen and Juselius (1989). The primary element of the monetary balance of payments theory, the existence of a stable long-run demand for real money balances, is found to exist for both Canada and the U.S. for the period of the late 19505 through 1989. With respect to the monetary model, the dynamics between the two countries are found to be very similar in that the estimated interest elasticities of 'velocity are found. to 'be insignificantly different from .50 for both countries. In addition, there appear to be no long- or short-run cross-country effects between the two countries with regard to the formulation of the demand for real money balances. A test of the null hypothesis that all cross-country effects in the model are zero could not be rejected, despite the finding of some long-run influence in the U.S. money demand equation with respect to Canadian real income. Zhn general, then, it may not be reasonable to expect that influences from U.S. monetary conditions would be useful in predicting the demand for the stock of real money balances in Canada, and vice versa. A further component of the theory of the monetary balance of payments, the existence of a stable Canadian/U.S. real exchange rate, is found in this analysis as well. Empirical evidence of a stable real exchange rate is found for the period after the breakdown of the Bretton 158 Woods system of fixed exchange rates, which spans the period from January of 1974 to the present. This indicates that the Canadian/U.S. nominal exchange rate and the relative price levels of the two countries tend to move together in the long run, even though individually they may tend to wander without bound. In addition, strong support is also found for the model of Canada and the U.S. for the validity of the law of one price, especially with regard to the capital market. That is, uncovered interest rate parity is found to hold for these two countries for the period from the late 19505 through 1989. With respect to the goods market, some evidence is found for the validity of long-run purchasing power parity between Canada and the U.S., although the sensitivity of this relationship to the time period under investigation must be noted. The null hypothesis of a valid PPP relationship between Canada and the U.S. is maintained at the 1% level of significance but not at the 5% level, and the only time period for which the relationship holds is between January of 1974 and September of 1990. Despite the tenuousness of the long-run purchasing power parity relationship for the Canadian and U.S. economies, it does appear that each of the key components of the theory of the monetary balance of payments are well-grounded for the case of Canada and the U.S. ..‘.Iw-u“‘—‘_I 159 TABLE 1 Phillips and Perron Unit Root Tests test statistic: 2(tu8) Z(t;) lag length: 4 12 4 12 series: P 1.46 4.78 8.92 7.42 R -2.07 -2.65 -2.18 -l.95 M/P 1.77 1.25 -3.05 -2.22 Q/P -2.32 -2.18 1.43 1.43 P* -2.80 -2.14 6.89 4.32 R* -2.22 -2.08 -1.83 -1.71 M*/P* -2.36 -2.28 -l.67 -l.46 Qf/P* -1.90 -l.73 1.56 1.65 s -0.38 -0.69 -1.02 —1.10 Key: Z(ta*) and Z(t&) are the Phillips and Perron adjusted t statistics used to test the parameter on the lagged dependent variable in. a regression with an intercept, and with an intercept and a time trend, respectively. The test statistic Z(ta*) tests the null hypothesis of a unit root, HO: a* - 1, in the regression yt - u* + a*yt_1 + 5:. The test statistic Z(t;) tests the null hypothesis of a unit root, HO: 5 - 1, in the regression yt - u + fl[(t—n)/2] + Eyt_1 + Ft. The critical values for these test statistics at the .05 level are -2.86 and -3.41, respectively, and at the .10 level are -3.43 and -3.91, respectively. 1 } “m. “'tfi1 ‘ . . __.( !? " 160 TABLE 2 Cointegration Tests, U.S. Data Real M1, Real GNP, and Treasury Bill Rate (double-logarithmic specification) Johansen Trace Test Statistics NOrmalized Cointegrating Vector sample r50 r51 r52 M/P O/P .4; 1959.1 - 90.3 37.41 15.77 5.60 1.0 -0.76 0.40 (with dummy) (0.12) (0.07) with velocity restriction: Johansen Trace Test LR test of Estimated Statistics Velocity Interest r50 r51 Restriction Elasticity 27.84 7.51 1.34 0.52 (0.06) Key: The standard errors are given beneath the parameter estimates in parentheses. Critical values for trace tests an: 5% level are 31.53, 17.95, and 8.18 for r50, rsl, and.r52 cointegrating vectors, respectively. Results are based on k=4 lag specification. The critical value for the likelihood ratio test, x2(1), at 5% level is equal to 3.84. 161 TABLE 3 Cointegration Tests, Canadian Data Real M1, Real GNP, and Treasury Bill Rate (double-logarithmic specification) Johansen'Trace.Test Statistics Normalized Cointegrating Vector sample r50 r51 r52 M/P O/P 3 1956.1 - 89.3 40.41 8.04 0.71 1.0 -15.01 16.98 (81.16) (94.09) 1956.1 - 89.3 42.97 8.15 0.41 1.0 -1.06 0.59 (with dummy) (0.26) (0.27) 1956.1 - 79.4 30.89 11.74 3.83 1.0 -1.05 0.49 (0.26) (0.33) 1956.1 - 80.4 28.31 8.70 1.51 1.0 -l.01 0.46 (0.21) (0.31) 1956.1 - 81.1 31.11 7.70 0.26 1.0 —l.15 0.63 (0.31) (0.39) 1956.1 - 81.2 32.21 7.57 0.00 1.0 -1.17 0.66 (0.49) (0.41) 1956.1 - 81.3 33.49 7.56 0.05 1.0 -l.43 1.03 (0.84) (1.11) 1956.1 - 81.4 37.07 8.67 0.05 1.0 2.14 -3.65 (12.62) (16.27) 1956.1 - 82.4 37.59 7.59 0.99 1.0 3.20 -2.89 (4.33) (4.83) 1956.1 - 83.4 34.34 7.07 0.38 1.0 87.57 99.63 (137.1) (157.2) Key: The standard errors are given beneath the parameter estimates in parentheses. Critical values for trace tests at 5% level are 31.53, 17.95, and 8.18 for r50, r51, and r52, respectively. Results are based on k=4 lag specification. 162 TABLE 4 Cointegration Tests with Velocity Restrictions, Canadian Data Real Ml, Real GNP, and Treasury Bill Rate (double-logarithmic specification) Johansen Trace Test LR test of Estimated Statistics Velocity Interest sample r50 r51 Restriction Elasticity “ 1956.1 - 89.3 26.57 1.03 0.06 1.02 i (0.22) 1 1956.1 - 89.3 29.89 1.89 0.03 0.57 '7 (with dummy) (0.12) 1956.1 - 79.4 24.02 4.89 0.39 0.43 (0.08) 1956.1 - 80.4 24.96 5.36 0.16 0.45 (0.08) 1956.1 - 81.4 26.53 1.87 1.12 0.63 (0.20) 1956.1 - 82.4 24.86 3.57 0.15 1.09 (0.87) 1956.1 - 83.4 22.54 0.86 1.12 0.72 (0.22) Key: The standard errors are given beneath the parameter estimates in parentheses. Critical values for trace tests at 5% level are 17.95 and 8.18 for r.<_0 and r51, respectively. Results are based on k-4 lag specification. The critical value for the Likelihood Ratio test, x2(1)’ at 5% level of significance is 3.84. 163 TABLE 5 Cointegration Tests for the Joint Model double-logarithmic specification Real Ml, Real GNP, and Treasury Bill Rate Johansen Trace Test Statistics sample r50 .r51 r52 r53 r54 r55 1956.1 - 89.3 143.12 67.25 32.47 15.01 3.60 0.29 (with dummies) Cointegrating Vector M/P M*/P* Q/P 0* /P* R R' 1.0 1.0 -1.16 -0.73 0.59 0.36 With Velocity Restrictions: Johansen Trace Test Statistics LR test of Estimated Interest Velocity Elasticities for r50 r51 r52 r53 Restrictions Canada U.S. 118.61 55.30 10.12 4.12 4.05 -0.48 -0.49 (0.16) (0.19) Key: The standard errors are given beneath the parameter estimates in parentheses. Critical values for trace tests at 5% level are r:Q r21 r=2 r23 r23 r25 95.18 70.60 48.28 31.53 17.95 8.18 when testing without imposing velocity restrictions, and £29 £21 r22 :53 48.28 31.53 17.95 8.18 when imposing velocity restrictions. Results are based on k—4 lag specification. The critical value for the Likelihood Ratio test, x2(2), at 5% level of significance is 5.99. 4 1M1_1'__ \ :1. 164 TABLE 6 Cointegration Tests for the Joint Model Semi-Logarithmic Specification Johansen Trace Test Statistics sample r50 r51 r52 r53 1956.1 - 89.3 72.56 32.26 17.43 3.35 (with dummies) Cointegrating Vector 0 0 ¢ .0813 .0457 -.1727 (.0017) (.0013) (.4253) Likelihood Ratio Statistic: 1.61 Key: The standard errors are given beneath the parameter estimates in parentheses. Critical values for trace tests at 5% and 10% levels are: r59 r51 r52 r53 5% 48.28 31.53 17.95 8.18 10% 45.23 28.71 15.66 6.50 Results are based on k-4 lag specification. The critical value for the Likelihood Ratio test, x2(1) at 5% level of significance is 3.85. 165 TABLE 7 Cointegration Tests for Canadian/U.S. Real Exchange Rate Johansen Trace Test sample Statistics Estimated period r50 r51 Coefficient 1970.6 - 1984.12 4.40 0.15 1.91 (0.45) 1970.6 - 1985.12 4.79 0.01 2.12 (0.51) 1970.6 - 1986.12 5.21 0.10 1.89 (0.31) 1970.6 - 1987.12 4.16 0.01 1.63 (0.26) 1970.6 - 1988.12 1.82 0.11 1.14 (0.19) 1970.6 - 1989.12 10.44 1.50 1.40 (0.17) 1974.1 - 1987.12 5.89 2.19 0.88 (0.95) 1974.1 - 1988.12 7.04 1.05 11.16 (51.00) 1974.1 - 1989.12 13.09 3.83 1.45 (0.25) LR test of HO: 71/72='1 1974.1 - 1990.09 18.51 5.52 5.18 1.40 (0.24) Key: The standard errors are given in parentheses beneath the parameter estimates. Critical vaules for the trace tests at the 5% level are 17.95 and 8.17 for r50 and r51, respectively. 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