Inflation, Money Growth, and I(2) Analysis

DISCUSSION PAPERS Institute of Economics University of Copenhagen 04-31 Inflation, Money Growth, and I(2) Analysis Katarina Juselius Studiestræde 6, DK-1455 Copenhagen K., Denmark Tel. +45 35 32 30 82 - Fax +45 35 32 30 00 http://www.econ.ku.dk In ation, Money Growth, and I(2) Analysis Katarina Juselius Studiestr de 6, 1455 Copenhagen K, Denmark Abstract The paper discusses the dynamics of in ation and money growth in a stochastic framework, allowing for double unit roots in the nominal variables. It gives some examples of typical I(2) 'symptoms' in empirical I(1) models and provides both a nontechnical and a technical discussion of the basic di erences between the I(1) and the I(2) model. The notion of long-run and medium-run price homogeneity is discussed in terms of testable restrictions on the I(2) model. The Brazilian high in ation period of 1977:1-1985:5 illustrates the applicability of the I(2) model and its usefulness to address questions related to in ation dynamics. JEL classi cation: C32, E41, E31. Keywords: Cointegrated VAR, Price Homogeneity, Cagan Model, Hyper In ation 1 Introduction1 The purpose of this paper is to give an intuitive account of the cointegrated VAR model for I(2) data and to demonstrate that the rich structure of the I(2) model is particularly relevant for the empirical analyses of economic data characterized by highly persistent shocks to the growth rates. Such data are usually found in applications of economic models explaining the determination of nominal magnitudes. For example, the explicit assumption of a nonstationary error term in some models of money demand during periods of high or hyper in ation (Cagan, 1956, Sargent, 1977), implies that nominal money and prices are I(2). Thus, 1 Useful comments from Michael Goldberg, S ren Johansen, and Mikael Juselius are gratefully acknowledged. The article was produced with nacial support from the Danish Social Sciences Research Council. 1 the empirical analysis of such models would only make sense in the I(2) model framework. However, as argued in Juselius and Vuojesevic (2003), prices in hyperin ationary episodes should not be modelled as an I(2) but rather as an explosive root process. Though such episodes are (almost by de nition) short they are usually preceded by periods of high in ation rates for which the I(2) analysis is more adequate. Even though in ationary shocks in such periods are usually large, it is worth stressing that the (double) unit root property, as such, is not related to the magnitude but the permanence of shocks. Therefore, we may equally well nd double unit roots in prices during periods of low in ation rates, like the nineties, and not just in periods of high in ation rates like the seventies. But, while the persistence of shocks determine whether price in ation is I(1) or I(0), the magnitude of in ationary shocks is probably much more indicative of a risk for hyper in ation. High in ation periods are, therefore, particularly interesting as they are likely to contain valuable information about the mechanisms which subsequently might lead to hyper-in ation. The empirical application to the Brazilian high-in ation period of 1977-1985 o ers a good illustration of the potential advantages of using the I(2) model and demonstrates how it can be used to study important aspects of the in ationary mechanism in periods preceding hyper in ation. The Cagan hyper in ation model is rst translated into set of testable empirical hypotheses on the pulling and pushing forces described by the cointegrated I(2) model in AR and MA form. The paper nds strong empirical support for one of the hypothetical pulling forces, the Cagan money demand relation with the opportunity cost of holding money measured by a combination of CPI in ation and currency depreciation in the black market. The Cagan's coe cient, de ning the average in ation rate at which government can gain maximum seignorage, is estimated to be approximately 40-50% which is usually considered to describe hyper in ation. Thus, it seems likely that the seed to the subsequent Brazilian hyper in ation episode can be found in the present data. This is further supported by the nding that (1) there is a small explosive root in the VAR model, (2) the condition for long-run price homogeneity was strongly violated, and (3) the CPI price in ation showed lack of equilibrium correction behavior. The latter is associated with the widespread use of wage and price indexation, which prohibited market forces to adjust back to equilibrium after a price distortion. As a consequence domestic price in ation gained momentum as a result of increasing in ationary expectations in the foreign exchange market. 2 The organization of the paper is as follows: Section 2 discusses money growth and in ation in a Cagan type of high / hyper in ation model framework. Section 3 reformulates the high in ation problem in a stochastic framework allowing for double unit roots in the nominal variables. Section 4 discusses typical 'symptoms' in the VAR analysis when incorrectly assuming that the data are I(1) instead of I(2) and gives a rst intuitive account of the basic di erence between the I(1) and the I(2) analysis. Section 5 de nes formally the I(2) model in the AR and the MA form, discusses the role of deterministic components in the I(2) model and introduces the two-step procedure for determining the two cointegration rank indices. Section 6 gives an interpretation of the various components in the I(2) model and illustrates with the Brazilian data. Section 7 discusses long-run and medium-run price homogeneity and how these can formulated as testable restrictions on the I(2) model. Section 8 presents the empirical model for money growth, currency depreciation and price in ation in Brazil. Section 9 concludes. 2 Money growth and in ation It is widely believed that the growth in money supply in excess of real productive growth is the cause of in ation, at least in the long run. The economic intuition behind this is that other factors are limited in scope, whereas money in principle is unlimited in supply (Romer, 1996). Generally, the reasoning is based on equilibrium in the money market so that money supply equals money demand: M=P = L(R; Y r ); (1) where M is the money stock, P the price level, Y r real income, R an interest rate, and L( ) the demand for real money balances. In a high (and accelerating) in ation period, the Cagan model for hyper in ation predicts that aggregate money demand is more appropriately described by : M=P = L( e ; Y r ); L e < 0; LY r > 0 (2) where e is expected in ation. The latter model (2) is chosen as the baseline model in the subsequent empirical analysis of the Brazilian high in ation experience in the seventies until the mid eighties. The data consists of money stock measured as M3, the CPI price index, the black market spot exchange rate, and the real industrial production and covers the period 1977:1,...,1985:5. 3 -10 Lm3 .15 DiLfm3 .1 -12.5 .05 0 -15 1980 -17.5 1985 Lcpi 1985 1980 1985 1980 1985 DifLcpi .1 -20 .05 1980 -20 1980 1985 DifLexch Lexch .3 .2 -22.5 .1 0 -25 1980 1985 Figure 1. Nominal M3, CPI, and exchange rates in levels and di erences. The graphs of the data in levels and di erences (after taking logs) gives a rst indication of the order of integration. The growth rates of all three nominal variables in Figure 1 exhibit typical I(1) behavior, implying that the levels of the variables are I(2). In contrast the graphs of the log of the industrial production in levels and di erences in Figure 2 do not suggest I(2) behavior: The smooth behavior typical of I(2) variables is not present in the level of industrial production and the di erenced process looks signi cantly mean-reverting. 4 Industrial production 0.2 The change in industrial production 0.1 4.6 0.0 4.4 -0.1 1980 1985 The log of real M3 measured by CPI 7.4 1980 -2.75 1985 The black market excahnge rate relative to CPI -3.00 -3.25 7.2 -3.50 1980 -20.0 1985 1980 1985 The depreciation rate in the official and the black market The official and black market exchange rate 0.2 -22.5 -25.0 0.0 1980 1985 1980 1985 Figure 2. The graphs of industrial production in levels and di erences (upper part), M3 and exchange rate both de ated with CPI (middle panel), and the black and white market exchange rate in levels and di erences (lower panel). The middle part of Figure 2 demonstrates how real money stock (lnM3 - lnCPI ) and real exchange rates (lnLexch - lnCPI ) have evolved in a nonstationary manner and increasingly so after 1981. Figure 2, lower panel compares the levels and the di erences of the o cial and black market exchange rate. While the o cial rate seems to have stayed below the black market rate for some periods the graphs show that the two major devaluations brought the two series back to the same level. Thus, it seems likely that the black market exchange rate is a good proxy for the 'true' value of the Brazilian currency in this period. When data are nonstationary, the Cagan model can be formulated as a cointegrating relation, i.e.: (M=P )t L( et ; Yt ) = vt (3) where vt is a stationary process measuring the deviation from the steadystate position at time t. The stationarity of vt implies that whenever the system has been shocked it will adjust back to equilibrium and is, therefore, essential for the interpretation of (3) as a steady-state relation. If vt is nonstationary as explicitly assumed in Sargent (1977) money supply has deviated from the steady-state value of money demand. As this case generally 5 implies a double unit root in the data, the choice of the I(2) model for the econometric analysis seems natural. Therefore, when addressing empirical questions related to the mechanisms behind in ation and money growth in a high or hyper in ation regime we need to understand and interpret the I(2) model . 3 Formulating the economic problem in a stochastic framework2 Cointegration and stochastic trends are two sides of the same coin: if there are cointegration relations there are also common stochastic trends. Therefore, to be able to address the transmission mechanism of monetary policy in a stochastic framework it is useful rst to consider a conventional decomposition into trend, T ; cycle, C; and irregular component, I; of a typical macroeconomic variable. X=T C I and allow the trend to be both deterministic, Td ; and stochastic, Ts ; i.e. T = Ts Td ; and the cyclical component to be of long duration, say 6-10 years, Cl , and of shorter duration, say 3-5 years, Cs ; i.e. C = Cl Cs : The reason for distinguishing between short and long cycles is that a long/short cycle can either be treated as nonstationary or stationary depending on the time perspective of the study. For example, the graph of the trend-adjusted industrial production in Figure 5, lower panel, illustrates long cycles in the data that were found nonstationary by the statistical analysis. An additive formulation is obtained by taking logarithms: x = (ts + td ) + (cl + cs ) + i (4) where lower case letters indicate a logarithmic transformation. Even if the stochastic trends are of primary interest for the subsequent analyses, a linear time trend is needed to account for average linear growth rates typical of most economic data. 3.1 Stochastic and deterministic trends As an illustration of a trend-cycle decomposition we consider the following vector of variables xt = [m; p; sb ; y r ]t ; t = 1977:1,...,1985:5; where m is the log of M3, p is the log CPI, sb is the log of black market exchange rate, and y r is the log of industrial production. All variables are treated 2 This section draws heavily on Section 4 in Juselius (1999a) 6 as stochastic and will be modelled, independently of whether they are considered endogenous or exogenous in the economic model. A stochastic trend describes the cumulated impact of all previous permanent shocks on a variable, i.e. it summarizes all the shocks with a long lasting e ect. This is contrary to a transitory shock, the e ect of which cancels either during the next period or over the next few periods. For example, the income level of a household can be thought of as the cumulation of all previous permanent income changes (shocks), whereas the e ect of temporary shocks, like lottery prizes, will not cumulate as it is only a temporary change in income. If in ation rate is found to be I(1), then the present level of in ation can be thought of as the sum of all previous shocks to in ation, i.e. t = t P "i + 0: (5) i=1 Because the e ect of transitory shocks disappears in the cumulation a stochastic trend, ts ; P is de ned as the cumulative sum of previous permanent shocks, ts;t = ti=1 "i : The di erence between a linear stochastic and a linear deterministic trend is that the increments of a stochastic trend change randomly, whereas those of a deterministic trend are constant over time. Figure 3 illustrates three di erent stochastic trends measured as the once cumulated residuals from the money, price and exchange rate equations. A representation of prices is obtained by integrating (5) once, i.e. pt = t P s=1 s = t P s P "i + 0t + p0 : (6) s=1 i=1 Thus, if in ation is I(1) with a nonzero mean (as most studies nd), prices are I(2) with a linear trend. Figure 4 illustrates the twice and once cumulated residuals from the CPI price equation of the VAR model de ned in the next section. 7 .1 Sm 3 0 1977 1978 1979 1980 1981 1982 1983 1984 1985 1978 1979 1980 1981 1982 1983 1984 1985 1979 1980 1981 1982 1983 1984 1985 Sp 0 1977 .25 Sbm 0 -.25 1977 1978 Figure 3. The graphs of the cumulated residuals from the money, price, and exchange rate equations of the estimated VAR. 0 SSp -.5 -1 1977 .025 1978 1979 1980 1981 1982 1983 1984 1985 1978 1979 1980 1981 1982 1983 1984 1985 Sp 0 .025 1977 Figure 4. The graphs of the twice and once cumlated residuals from the price equation. 3.2 A trend-cycle scenario Given the set of variables discussed above, one would expect (at least) two autonomous shocks u1;t and u2;t , of which u1;t is a nominal shock and u2;t is a real shock. If there are second order stochastic trends in the 8 data it seems plausible that they have been generated from the nominal shocks. We will, therefore, tentatively assume that the second order long-run stochastic Pt trend Ps ts in (4) is described by the twice cumulated nominal shocks, s=1 i=1 u1i : The long cyclical components cl in the data will then be Ptdescribed by a combination of the once cumulated P nominal shocks, i=1 u1i ; and the once cumulated real shocks, ti=1 u2i : This allows us to distinguish between the long-run stochastic Pt empirically Ps trend in nominal levels, P s=1 i=1 u1i ; the medium-run stochastic trend t in nominal growth P rates, i=1 u1i ; and the medium-run stochastic trend in real activity, ti=1 u2i : Figure 5 illustrates. trad m3 0 -.0 5 19 77 .0 05 19 78 19 79 19 80 19 81 19 82 19 83 19 84 19 85 19 79 19 80 19 81 19 82 19 83 19 84 19 85 19 79 19 80 19 81 19 82 19 83 19 84 19 85 D trad m3 0 .0 05 19 77 .2 19 78 tren d ad Y 0 -.2 19 77 19 78 Figure 5. The graphs of trend-adjusted M3 in levels and di erences (upper and lower panel) and trend-adjusted industrial production (lower panel). The trend-cycle formulation below illustrates the ideas: 3 2 3 2 2 3 3 d11 d12 mt c1 g1 Pt t P s 6 d21 d22 7 6 pt 7 6 c2 7 P 6 7 6 6 b7 = 6 7 7 Pti=1 u1i +6 g2 7 [t]+stat.comp. + u 1i 4 d31 d32 5 4 st 5 4 c3 5 s=1 i=1 4 g3 5 i=1 u2i r yt d41 d42 0 g4 (7) The deterministic trend component, td = t; is needed to account for linear growth trends present in the levels of the variables. If g4 = 0 and Pt d41 = 0 in (7), then i=1 u2;i is likely to describe the long-run trend 2 9 in Ptindustrial production. In this case it may be possible to interpret i=1 u2;i as a "structural" unit root process (cf. the discussion in King, Plosser, Stock and Watson (1991) on stochastic versus deterministic real growth models). If, on the other hand, g4 6= 0; then it seems plausible that the longrun real trend Pcan be approximated by a linear deterministic time trend. In this case ti=1 u2;i is likely to describe medium-run deviations from the linear trend, i.e. the long business cycle. The graph of the trendadjusted industrial production in the lower panel of Figure 5 illustrates such a long cycle starting from the long upturn from 1977-1980:6 and ending with the downturn 1980:6-1984. Note also the shorter cycles of approximately a year's duration imbedded in the long cycle. Pt Therefore, the possibility of interpreting the second stochastic trend, i=1 u2;i ; as a long-run structural trend depends crucially on whether one includes a linear trend in (7) or not. The trend components of mt ; pt ; st ; and yt in (7) can now be represented by: P P PP mt = c1 P P u1i +d11 P u1i +d12 P u2i +g1 t + stat: comp: pt = c2 P P u1i +d21 P u1i +d22 P u2i +g2 t + stat: comp u1i +d31 P u1i +d32 P u2i +g3 t + stat: comp st = c 3 yt = +d41 u1i +d42 u2i +g4 t + stat: comp If (c1 ; c2 ; c3 ) 6= 0; then fmt ; pt ; st g then (8) I(2): If, in addition, c1 = c2 = c3 P P pt = (d11 d21 ) P u1i +(d12 d22 ) P u2i +(g1 g2 )t +stat:comp: st = (d21 d31 ) P u1i +(d22 d32 ) P u2i +(g2 g3 )t +stat:comp: st = (d11 d31 ) P u1i +(d12 d32 ) P u2i +(g1 g3 )t +stat:comp: +g4 t +stat:comp: +d42 u2i yt = +d41 u1i (9) The real variables are at most I(1) but, unless (g1 = g2 ); (g2 = g3 ); and (g1 = g3 ); they are I(1) around a linear trend. Figure 5 illustrates the trend-adjusted behavior of real M3 and industrial production. Long-run price homogeneity among all the variables implies that both the long-run stochastic I(2) trends and the linear deterministic trends should cancel in (9). But, even if overall long-run homogeneity is rejected, some of the individual components of fmt pt ; pt st ; mt st g can, nevertheless, exhibit long-run price homogeneity. For example, the case (mt pt ) I(1) is a testable hypothesis which implies that money stock and prices are moving together in the long-run, though not necessarily in the medium-run (over the business cycle). mt pt mt 10 The condition for long-run and medium-run price homogeneity is fc11 = c21 ; and d11 = d21 g; i.e. that the nominal shocks u1t a ect nominal money and prices in the same way both in theP long run and in the medium run. Because the real stochastic trend u2i is likely to enter mt but not necessarily pt ; testing long-run and medium-run price homogeneity jointly is not equivalent to testing (mt pt ) I(0): Testing the composite hypothesis is more involved than the long-run price homogeneity alone. It is important to note that (mt pt ) I(1) implies ( mt pt ) I(0); i.e. long-run price homogeneity implies a stationary spread between price in ation and money growth. In this case the stochastic trend in in ation is the same as the stochastic trend in money growth. The econometric formulation of long-run and medium-run price-homogeneity in the I(2) model will be discussed in Section 7. When overall long-run price homogeneity holds it is convenient to transform the nominal system (8) to a system consisting of real variables and a nominal growth rate, for example: 2 mt 6 st 6 4 3 2 d11 pt 6 d21 pt 7 7=6 pt 5 4 yt d21 d12 d31 d22 c21 d41 2 3 g1 d22 Pt 6 u d32 7 7 Pti=1 1;i + 6 g2 40 05 i=1 u2;i g4 d42 3 g2 g3 7 7 [t] + ::: 5 (10) Given long-run price homogeneity all variables are at most I(1) in (10). The nominal growth rate (measured by pt ;P mt ; or st ) is only t a ected by the once cumulated nominal trend, i=1 u1;i ; but all the other variables can (but need not) be a ected by both stochastic trends, Pt Pt i=1 u1;i and i=1 u2;i . The case (mt pt yt ) I(0); i.e. the inverse velocity of circulation is a stationary variable, requires that d11 d21 d41 = 0; d12 d22 d42 = 0 and g1 g2 g4 = 0: If d11 = d21 (i.e. medium run price homogeneity), d22 = 0 (real stochastic growth does not a ect prices), d41 = 0 (mediumrun in ationary movements do not a ect real income), and d12 = d42 ; then mt pt yt I(0). In this case real money stock and realPaggregate income share one common trend, the real stochastic trend u2i : The stationarity of money velocity, implying common movements in money, prices, and income, would then be consistent with the conventional monetarist assumption as stated by Friedman (1970) that "in ation always and everywhere is a monetary problem". This case would correspond to model (1) in Section 2. The case (mt pt yt ) I(1); implies that the two common stochastic trends a ect the level of real money stock and real income di erently. 11 Cagan's model of money demand in a high (hyper) in ation period suggests that the nonstationarity of the liquidity ratio is related to the expected rate of in ation Et ( pt+1 ): The latter is generally not observable, but as long as Et ( pt+1 ) pt is a stationary disturbance, one can replace the unobserved expected in ation with actual in ation without loosing cointegration. The condition that fEt ( pt+1 ) pt g I(0) seems plausible considering that f pt+1 pt g I(0) when pt I(2): It amounts to assuming that fEt ( pt+1 ) pt+1 g I(0); i.e. agents' in ationary expectations do not systematically deviate from actual ination. Therefore, from a cointegration point of view we can replace the expected in ation with the actual in ation: mt pt y t + a1 p t (mt pt y t ) + a2 s t I(0); (11) or, equivalently: I(0): where under the Cagan model a1 > 0; a2 > 0: 4 Diagnosing I(2) VAR models are widely used in empirical macroeconomics based on the assumption that data are I(1) without rst testing for I(2) or checking whether a near unit root remains in the model after the cointegration rank has been imposed. Unfortunately, when the data contains a double unit root essentially all inference in the I(1) model is a ected. To avoid making wrong inference it is, therefore, important to be able to diagnose typical I(2) symptoms in the I(1) VAR model. For the Brazilian data, the unrestricted VAR model was speci ed as: xt = 1 xt 1 + xt 2 + 1 t + 0 + p Dp83:8t + "t Np (0; ); t = 1; :::; T s Qst + "t ; (12) where xt = [mt ; pt ; sbt ; ytr ]; mt = ln(M 3); pt = ln(CP It ); sbt = ln(Exchbt ); ytr = 0 ln(industrial production); t =1977:1,...,1985:5, = ; 1= 1; 01 = ; and ( ; ; ; ; ) are unrestricted. The estimates have been 1 p s 01 0 calculated using CATS for RATS, Hansen and Juselius (1994). Misspeci cation tests are reported in the Appendix. The data are distinctly trending and we need to allow for linear trends both in the data and in the cointegration relations when testing for cointegration rank (Nielsen and Rahbek, 2000). The industrial production, ytr ; exhibits strong seasonal variation and we include 11 seasonal dummies, Qst ; and a constant, 0 in the VAR model. Finally, the 12 graphs of the di erenced black market exchange rate and nominal M3 money stock exhibited an extraordinary large shock at 1983:8, which was accounted for by an unrestricted impulse dummy Dp83:8t = 1 for t = 1983:8 and 0 otherwise. A permanent shock to the changes corresponds to a level shift in the variables, which may or may not cancel in the cointegration relations. To account for the latter possibility the shift dummy, Ds83:8t = 0 for t = 1983:8 and 1 otherwise, was restricted to be in the cointegration relations. It was found to be insigni cant (p-value 0.88) and was left out. The I(1) estimation procedure is based on the so called R-model in which the short-run e ects have rst been concentrated out: R0t = 0 R1t + "t : (13) where R0t and R1t are de ned by: ^11 xt 1 + const + B13 Dpt + R0t x =B |{z} |{z}t | {z } (14) ^21 xt 1 + const + B23 Dpt + R1t : xt 1 = B |{z} | {z } |{z} (15) I(1) and I(2) I(0) I(1) I(2) I(1) Dpt is a catch-all for all the dummy variables. If xt I(2) then xt I(1) and (14) is a regression of an I(1) process on its own lag. Thus, the regressand and the regressor contain the same common trend which will cancel in regression. This implies that R0t I(0); even if xt I(2): On the other hand equation (15) is a regression of an I(2) variable, xt 1 ; on an I(1) variable, xt 1 : Because an I(2) trend cannot be canceled by regressing on an I(1) trend, it follows that R1t I(2): Therefore, when xt I(2) (13) is a regression of an I(0) variable (R0t ) on an I(2) variable (R1t ): Under the (testable) assumption that "t I(0); either 0 R1t = 0 or 0 R1t I(0) for the equation (13) to 0 hold. Because the linear combination R1t transforms the process from I(2) to I(0), the estimate ^ is super-super consistent (Johansen, 1992). Even though is precisely estimated in the I(1) model when data are I(2), the interpretation of 0 xt as a stationary long-run relation has to be modi ed as will be demonstrated below. It is easy to demonstrate the connection between 0 x~t 2 and 0 R1t by inserting (15) into (13) : 13 0 R0t = R1t + "t (~ xt 1 B2 xt 1 ) + "t 0 0 = ( x~t 1 B2 xt 1 ) + "t = ( 0 x~t 1 ! 0 xt 1 ) + "t | {z } | {z } 0 I(1) | I(1) {z I(0) (16) } where ! = 0 B2 : It appears that the stationary relations 0 R1t consists of two components 0 x~t 1 and ! 0 xt 1 both of which are generally I(1). The stationarity of 0 R1t is, therefore, a consequence of cointegration between 0 x~t 1 I(1) and ! 0 xt 1 I(1): Thus, when data are I(2), 0i x~t I(1); while 0i R1t I(0) for at least one i; i = 1; :::; r: It is, therefore, a clear sign of double unit roots (or, alternatively, a unit root and an explosive root) in the model when the graphs of 0i x~t exhibits nonstationary behavior whereas 0i R1t looks stationary. As an illustration we have reported the graphs of all four cointegration relations (of which 01 R1t and 02 R1t are stationary) in Figures 6-9. The upper panels contain the relations, 0i x~t ; and the lower panels the cointegration relations corrected for short-run dynamics, 0i R1t : V1 ` * Zk(t) 68.4 67.2 66.0 64.8 63.6 62.4 61.2 60.0 58.8 1977 1978 1979 1980 1981 1982 1983 1984 1985 1984 1985 V1 ` * Rk (t) 4 3 2 1 0 -1 -2 -3 1977 1978 Figure 6. The graphs of 1979 0 1 xt 1980 1981 1982 1983 (upper panel) and 14 0 1 R1t (lower panel). V2 ` * Zk(t) -39.6 -40.8 -42.0 -43.2 -44.4 -45.6 -46.8 -48.0 -49.2 1977 1978 1979 1980 1981 1982 1983 1984 1985 1982 1983 1984 1985 V2 ` * Rk (t) 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 1977 1978 Figure 7. The graphs of 1979 0 2 xt 1980 1981 (upper panel) and 0 2 R1t (lower panel). V 3` * Zk (t) 121 120 119 118 117 116 115 114 1977 1978 1979 1980 1981 1982 1983 1984 1985 1982 1983 1984 1985 V 3` * Rk (t) 3.2 2.4 1.6 0.8 0.0 -0.8 -1.6 -2.4 1977 1978 Figure 8. The graphs of 1979 0 3 xt 1980 1981 (upper panel) and 15 0 3 R1t (lower panel). V 4` * Zk (t) -35.2 -36.0 -36.8 -37.6 -38.4 -39.2 -40.0 -40.8 -41.6 1977 1978 1979 1980 1981 1982 1983 1984 1985 1984 1985 V 4` * Rk (t) 2 1 0 -1 -2 -3 1977 1978 1979 Figure 9. The graphs of 1980 0 4 xt 1981 1982 1983 (upper panel) and 0 4 R1t (lower panel). Among the graphs in Figures 6 and 7 01 x~t and 02 x~t exhibit distinctly nonstationary behavior whereas the graphs of the corresponding 0i R1;t look reasonably stationary. This is strong evidence of double roots in the data. As all the remaining graphs seem de nitely nonstationary, this suggests r = 2 and at least one I(2) trend in the data. Another way of diagnosing I(2) behavior is to calculate the characteristic roots of the VAR model for di erent choices of the cointegration rank r. When xt I(2) the number of unit roots in the characteristic polynomial of the VAR model is s1 +2s2 ; where s1 and s2 are the number of autonomous I(1) and I(2) trends respectively and s1 + s2 = p r. The characteristic roots contain information on unit roots associated with both and ; whereas the standard I(1) trace test is only related to the number of unit roots in the matrix. If the data are I(1) the number of unit roots (or near unit roots) should be p r; otherwise p r + s2 : Therefore, if for any reasonable choice of r there are still (near) unit roots in the model; it is a clear sign of I(2) behavior in at least some of the variables. Because the additional unit root(s) are related to xt 1 ; i.e. belong to the matrix = I 1 , lowering the value of r does not remove the s2 additional unit root associated with the I(2) behavior. In the Brazilian nominal model there are altogether p k = 4 2 = 8 eigenvalue roots in the characteristic polynomial which are reported below for r = 1; :::; 4. Unrestricted near unit roots are indicated with 16 bold face. V AR(p) 1:002 0:97 0:90 0:90 0:38 r=3 1:0 1:002 0:91 0:91 0:38 r=2 1:0 1:0 0:99 0:86 0:38 r=1 1:0 1:0 1:0 1:001 0:61 0:33 0:33 0:32 0:33 0:06 0:06 0:09 0:09 0:06 0:06 0:07 0:00 In the unrestricted model two of the roots are very close to the unit circle, one is larger than unity possibly indicating explosive behavior, the other is a stable near unit root. In addition there is a complex pair of two fairly large roots. The presence of an unstable root can be seen in the graph of the rst cointegration relation 01 x ~t in Figure 6: The equilibrium error in the `steady-state' relation in levels grows in an unstable manner at the end of the period, but is `compensated' by a similar increase in the in ation rate, so 01 R1;t looks stationary. This suggests that the seed to the Brazilian hyper in ation in the subsequent period can already be found in the present data. However, the explosive part of the root is very small and might not be statistically signi cant. In such a case we would expect the unstable root to disappear when restricting the rank. We notice that for r = 3 and r = 1 the explosive root is still left in the model, whereas for r = 2 it has disappeared. Independently of the choice of r; a near unit root remains in the model consistent with I(2), or moderately explosive, behavior. Therefore, we continue with r = 2 and disregard the possibility of an explosive root in the econometric analysis. Subsequently we will use the empirical results to demonstrate where in the model the seed to the subsequent hyper in ationary behavior can be found. In most cases a graphical inspection of the data is su cient to detect I(2) behavior and it might seem meaningless to estimate the I(1) model when xt is in fact I(2): However, a variety of hypotheses can be adequately tested using the I(1) procedure with the caveat that the interpretation of the cointegration results should be in terms of CI(2; 1) relations, i.e. relations which cointegrated from I(2) to I(1), and not from I(1) to I(0). One of the more important hypotheses which can be tested is the long-run price homogeneity of to be discussed in Section 7. 5 De ning the I(2) model It is useful to reformulate the VAR model de ned in the previous section in acceleration rates, changes and levels: 2 xt = xt 1 + xt 1 + "t Np (0; p Dp;t + s Qs;t + ); t = 1; :::; T 17 0 + 1t + "t ; (17) where = (I Section 5.3). 5.1 1) and 1 = 1:0 is restricted to lie in sp( ) (cf. The AR formulation The hypothesis that xt is I(2) is formulated in Johansen (1992) as two reduced rank hypotheses: 0 = , where ; are p r are (p r) (18) and 0 ? ? 0 = ; where ; s1 : (19) The rst condition is the usual I(1) reduced rank condition associated with the variables in levels, whereas the second condition is associated with the variables in di erences. The intuition is that the di erenced process also contains unit roots when data are I(2). Note, however, that (19) is formulated as a reduced rank condition on the transformed : The intuition behind this can be seen by pre-multiplying (17) with ? 0 (and post-multiplying by ? ): This makes the levels component xt 2 disappear and reduces the model to a ((p r) (p r))-dimensional system of equations in rst- and second order di erences. In this system the hypothesis of reduced rank of the matrix 0? ? is tested in the usual way. Thus, the second reduced rank condition is similar to the rst except that the reduced rank regression is on the p r common driving trends. Using (19) it is possible to decompose ? and ? into the I(1) and I(2) directions: ? =f ?1 ; ?2 g and ? =f ?1 ; ?2 g; (20) and ?;1 = ? ( 0? ? ) 1 is p s1 ; ?2 = s2 ; and ? ; ? are the orthogonal com? ? and ?2 = ? ? is p plements of and ; respectively. Note that the matrices ?1 ; ?2 ; ?1 ; and ?2 are called 1 ; 2 ; 1 and 2 in the many papers on I(2) by Johansen and coauthors. The reason why we deviate here from the simpler notation is that we need to distinguish between di erent and vectors in the empirical analysis and, hence, use the latter notation for this purpose. While the I(1) model is only based on the distinction between r cointegrating relations and p r non-cointegrating relations, the I(2) model makes an additional distinction between s1 I(1) trends and s2 I(2) trends. Furthermore, when r > s2 ; the r cointegrating relations can be divided into r0 = r s2 directly stationary CI(2; 2) relations (cointegrating from I(2) to I(0)) and s2 polynomially cointegrating relations. This distinction will be illustrated in Section 6 based on the Brazilian data. where ?1 = 0 ?( ? ?) 1 18 5.2 The moving average representation The moving average representation of (17) describes the variables as a function of stochastic and deterministic trends, stationary components, initial values and deterministic dummy variables. It is given by: xt = C2 +C1 t P t P s P s=1 i=1 "i + C2 21 "s + C1 p t P 2 0 t + C2 Dps + C2 p s +Yt + A + Bt; t = 1; :::; T Dpi + C2 s s t P P Qsi s=1 i=1 s=1 i=1 t P s=1 s=1 s=1 s t P P Qss + (C1 + 21 C2 ) 0 t + 1t (21) where Yt de nes the stationary part of the process, A and B are functions of the initial values x0 ; x 1 ; :::; x k+1 ; and the coe cient matrices satisfy: C2 = 0 ?2 ( ?2 ?2 ) 1 0 0 ?2 ; 0 C1 = C2 ; 0 ?1 C1 0 +I where = 1 and the shorthand notation used: See Johansen (1992, 1995). We denote e?2 = ?2 ( 0?2 ?2 ) 1 so that C2 = e?2 0 ?2 = 0 ?1 (I = ( 0 C2 ) (22) ) 1 is (23) i:e: the C2 matrix has a similar reduced rank representation as C1 in the 0 I(1) model. It is, therefore, natural to interpret ?2 "i as the second order stochastic trend that has a ected the variables xt with weights e?2 : However, the C1 matrix cannot be decomposed similarly. It is a more complex function of the AR parameters of the model and the C2 matrix and the interpretation of the parameters ?1 and ?1 is less intuitive. The MA representation (22) together with (23) can be used to obtain ML estimates of the stochastic and deterministic trends and cycles and their loadings in the intuitive scenario (8) of Section 3. This will be illustrated in Section 6. 5.3 Deterministic components in the I(2) model It appears from (21) that an unrestricted constant in the model is consistent with linear and quadratic trends in the data. Johansen (1992) suggested the decomposition of the constant term 0 into: 0 = 0 + where 19 1 + 2; 0 is a constant term in the stationary cointegration relations, 1 is the slope coe cient of linear trends in the variables, and 2 is the slope coe cient of quadratic trends in the variables. Quadratic trends in the levels of the variables is consistent with linear trends in the growth rates, i.e. in in ation rates, which generally does not seem plausible (not even as a local approximation). Therefore, the empirical model will be based on the assumption that the data contain linear but no quadratic trends, i.e. that 2 = 0: Similar arguments can be given for the dummy variables. An unrestricted shift dummy, such as Ds83:8t ; in the model is consistent with a broken quadratic trend in the data, whereas an unrestricted blip dummy, such as Dp83:8t = Ds83:8t ; is consistent with a broken linear trend in the data. Thus, a correct speci cation of dummies is important as they are likely to strongly a ect both the model estimates and the asymptotic distribution of the rank test. In many cases it is important to allow for trend-stationary relations in the I(2) model (Rahbek, Kongsted, and J rgensen, 1999). In this case 1 t 6= 0 and the vector 1 needs to be decomposed in a similar way as the constant term: 1 = 0 0 + 1 + 2; where is the slope coe cient of a linear trend in the cointegration relations, 0 1 is the slope coe cient of quadratic trends in the variables, and 2 is the slope coe cient of cubic trends in the variables. Since the presence of deterministic quadratic or cubic trends are not very plausible we will assume that 1 = 2 = 0. 5.4 The determination of the two rank indices The cointegration rank r can be determined either by the two-step estimation procedure in Johansen (1995) based on the polynomial cointegration property of 0 xt , or by the F IM L procedure in Johansen (1997) based on the CI(2; 1) property of 0 xt and 0?1 xt : The idea of the twostep procedure is as follows: The rst step determines r = r based on the trace test in the standard I(1) model and the estimates ^ and ^ : The 20 Table 1: Testing the two rank indices in the I(2) model p-r r FIML test procedure: Q(s1 ; r) Q(r) i 4 0 323:87 220:09 149:68 99:01 95:91 0.43 [0:00] 3 1 2 2 1 3 s2 [0:00] [0:00] [0:00] [0:00] 141:95 73:89 51:69 44:16 0.24 48:92 24:14 19:67 0.12 13:40 7:29 0.05 1 0 [0:00] [0:02] [0:05] [0:08] [0:47] [0:34] 4 3 2 [0:04] [0:25] [0:32] second step determines s1 = s1 by solving the reduced rank problem for the matrix (^ 0? ^ ? ): The practical procedure is to calculate the trace test for all possible combinations of r and s1 so that the joint hypothesis (r; s1 ) can be tested using the procedure in Paruolo (1996). Based on a broad simulation study Nielsen and Rahbek (2003) show that the F IM L procedure has better size properties than the two-step procedure. The estimates here are, therefore, based on the F IM L procedure using the new version 2.0 of CATS for RATS developed by Jonathan Dennis. Table 1 reports the test of the joint hypothesis (r; s1 ) with the 95% quantiles of the simulated distribution given in brackets. They are derived for a model with a linear trend restricted to be in the cointegration space. The test procedure starts with the most restricted model (r = 0; s1 = 0; s2 = 4) in the upper left hand corner, continues to the end of the rst row (r = 0; s1 = 4; s2 = 0), and proceeds similarly rowwise from left to right until the rst acceptance. The rst acceptance is at (r = 1; s1 = 1; s2 = 1) with a p-value of 0.08. However, the case (r = 2; s1 = 1; s2 = 1) is accepted with a much higher p-value 0.47 and will be our preferred choice. As a matter of fact, the subsequent results will demonstrate that the second relation plays a crucial role in the price mechanisms which led to hyper in ation. To improve the small sample properties of the test procedures, a Bartlett correction can be employed (Johansen, 2000). Even though it signi cantly improves the size of the cointegration rank, the power of the tests is generally very low for I(2) or near I(2) data. The Paruolo procedure delivers a correct size asymptotically, but does not solve the problem of low power. Because economic theory is often consistent with few rather than many common trends, a reversed order of testing might be preferable from an economic point of view. However, in that case the test will no longer deliver a correct asymptotic 21 size. Furthermore, when the I(2) model contains intervention dummies that cumulate to trends in the DGP , standard asymptotic tables are no longer valid. For example, an unrestricted impulse dummy, like Dp83.8 t , will cumulate to a broken linear trend in the data. The asymptotic distributions for the I(2) model do not account for this feature. Since the null of a unit root is not necessarily reasonable from an economic point of view, the low power and the impact of the dummies on the distributions can be a serious problem. This can sometimes be a strong argument for basing the choice of r and s1 on prior information given by the economic insight as well as the statistical information in the data. As demonstrated in Section 4 such information can be a graphical inspection and the number of (near) unit roots in the characteristic polynomial of the VAR. For the present choice of rank (r = 2; s1 = 1; s2 = 1) the characteristic roots of the V AR model became 1:0 1:0 1:0 0:89 0:39 0:06 0:09 0:32 leaving a fairly large root in the model. Therefore, another possibility would have been to choose r = 2 ; s1 = 0; s2 = 2: 6 Interpreting the I(2) structure It is no easy task to give the intuition for the di erent levels of integration and cointegration in the I(2) model and how they can be translated into economically relevant relationships. Table 2 illustrates the I(2) decomposition of the Brazilian data, which is based on the following assumptions (anticipating the subsequent results): mt I(2); pt I(2); sbt I(2); ytr I(1) and r| {z = 2} ; and r0 =1;r1 =1 p r =2 | {z } s1 =1;s2 =1 The left hand side of Table 2 illustrates the decomposition of xt into two and two ? directions corresponding to r = 2 and p r = 2. This decomposition de nes two stationary polynomially cointegrating relations, 01 xt + ! 01 xt and 02 xt + ! 02 xt ; and two nonstationary relations, 0 I(1) and 0?2 xt I(2): Note that 0?1 xt is cointegrating from ?1 xt I(2) to I(1), and can become I(0) by di erencing once, whereas 0?2 xt is not cointegrating at all and, thus, can only become I(0) by di erencing twice. 22 Table 2: Decomposing the data vector using the I(2) model The ; ? decomposition of xt The ; ? decomposition r=2 [ 0:1 xt + ! 0:1 xt ] I(0) 1 :short-run adjustment coe cients | {z } | {z } I(1) I(1) [ 0:2 xt + ! 0:2 xt ] | {z } | {z } I(1) I(0) 2: short-run adjustment coe cients Pt I(1) s1 = 1 0 ?1 xt I(1) 0 ?1 s2 = 1 0 ?2 xt I(2) 0 ?2 i=1 Pt s=1 "i : I(1) stochastic trend Ps i=1 "i : I(2) stochastic trend When r > s2 the polynomially cointegrating relations can be further decomposed into r0 = r s2 = 1 directly cointegrating relations, 00 xt ; and r1 = r r0 = s2 = 1 polynomially cointegrating relations, 01 xt + 0 xt ; where is a p s2 matrix proportional ?2 : The right hand side of Table 2 illustrates the corresponding decomposition into the and the ? directions, where 1 and 2 measure the short-run adjustment coe cients associated with the polynomially cointegrating relations, whereas ?1 and ?2 measure the loadings to the rst and second order stochastic trends. Both 0 xt and 0?1 xt are CI(2; 1) but they di er in the sense that the former can become stationary by polynomial cointegration, whereas the latter can only become stationary by di erencing. Thus, even in the I(2) model the interpretation of the reduced rank of the matrix is that there are r relations that can become stationary either by cointegration or by multi-cointegration, and p r relations that only become stationary by di erencing. Thus, the I(2) model can distinguish between the CI(2; 1) relations between levels f 0 xt ; 0?1 xt g; the CI(1; 1) relations between levels and di erences f 0 xt 1 + ! 0 xt g; and nally the CI(1; 1) relations between di erences f 0?1 xt g: As a consequence, when discussing the economic interpretation of these components, we need to modify the generic concept of "long-run" steady-state relations accordingly. We will here use the interpretation of 0 0 xt as a static long-run equilibrium relation; 0 0 xt as a dynamic long-run equilibrium relation, 1 xt + 23 Table 3: Unrestricted estimates of the I(0), I(1), and I(2) directions of and m p sb yr The stationary cointegrating relations ^0 1.00 -0.07 -0.91 -1.22 ^1 -0.67 1.00 -0.06 0.37 -4.87 -3.78 -5.48 -0.30 The adjustment coe cients ^0 0.04 -0.03 0.17 0.01 ^1 0.10 0.04 0.11 -0.01 The nonstationary relations ^ 4.54 0.46 -3.99 6.69 ?1 ^ 0.52 0.40 0.58 -0.03 ?2 The common stochastic trends ^ ?1 0.038 -0.007 -0.012 0.122 ^ ?2 -0.053 -0.078 -0.016 -0.023 ^" 0.016 0.010 0.054 0.028 0 ?1 xt as a medium-run equilibrium relation: As mentioned above the parameters of Table 2 can be estimated either by the two-step procedure or by the F IM L procedure. Paruolo (2000) showed that the two-step procedure gives asymptotically e cient M L estimates. The F IM L procedure solves just one reduced rank problem in which the eigenvectors determine the space spanned by ( ; ?1 ); i.e. the p s2 I(1) directions of the process: Independently of the estimation procedure, the crucial estimates are f ^ ; ^ ?1 g; because for given values of these it is possible to derive the estimates of f ; ?1 ; ?2 ; ?2 g and, if r > s2 ; to further decompose and into = f 0 ; 1 g and = f 0 ; 1 g: The parameter estimates in Table 3 are based on the two-step procedure for r = 2, s1 = 1; and s2 = 1. We have imposed identifying restrictions on two cointegration relations by distinguishing between the directly stationary relation, 00 xt ; and the polynomially cointegrated rext ; where is proportional to ?2 : Note, however, that lation, 01 xt + this is just one of many identi cation schemes which happen to be possible because r s2 = 1: In Section 8 we will present another identi ed structure where both relations are polynomially cointegrating. 0 The b?1 xt relation is a CI(2; 1) cointegrating relation which only can become stationary by di erencing. We interpret such a relation as 24 a medium long-run steady-state relation. The estimated coe cients of b suggest a rst tentative interpretation: ?1 ytr = 0:60 sbt 0:68 mt i.e. real industrial production has increased in the medium run with the currency depreciation relative to the growth of money stock. The estimate of ?2 determines the stochastic I(2) trend ^ 0?2 ^"i = u b2i ; where ^"i is the vector of estimated residuals from (17) and u b2t = ^ 0?2^"t : Permanent shocks to money stock relative to price shocks, to black market exchange rates and to industrial production seem to have generated the I(2) trend in this period. The standard deviations of the VAR residuals are reported in the bottom row of the table. P The estimate of describes the second I(1) stochastic trend, u b2i = ?1 P b 0?1 ^"i . The coe cient to real industrial production has by far the largest weight in b ?1 suggesting that it measures an autonomous real shock. This is consistent with the hypothetical scenario (7) of Section 3. I2 tren d .5 19 77 .0 2 19 78 19 79 19 80 19 81 19 82 19 83 19 84 19 85 19 78 19 79 19 80 19 81 19 82 19 83 19 84 19 85 19 78 19 79 19 80 19 81 19 82 19 83 19 84 19 85 I1 n tren d .0 1 0 -.0 1 19 77 .2 Sy .1 0 -.1 19 77 Figure 10. The graphs of the estimated I(2) trend in the upper panel, the nominal I(1) trend (i.e. the di erenced I(2) trend) in the middel panel and the real I(1) trend in the lower panel. Figure P P 10, upper panel, shows the graph of the I(2) stochastic trend, ^"i ; where ^ ?2 is from Table 3. The graph in the middle panel is the di erenced I(2) trend Pand the graph in the lower panel is the real stochastic trend given by ^"y;i : ^ 0?2 25 The vector ^ ?2 describes the weights ci ; i = 1; ::; 4 of the I(2) trend in the scenario (7) of Section 3 for the Brazilian variables. Nominal money, prices and exchange rates have large coe cients of approximately the same size, whereas the coe cient to real income is very small. This suggests that only the nominal variables are I(2) consistent with the assumption behind the scenario in (7). 7 Nominal growth in the long run and the medium run3 The notion of price homogeneity plays an important role for the analysis of price adjustment in the long run and the medium run. Both in the I(1) and the I(2) model, long-run price homogeneity can be de ned as a zero sum restriction on : Under the assumption that industrial production is not a ected by the I(2) trend, long-run price homogeneity for the Brazilian data can be expressed as: 0 i 0 ?1 0 ?2 = [ai ; ! i ai ; (1 ! i )ai ; ; ]; i = 1; :::; 2; = [b; ! 3 b; (1 ! 3 )b; ]; = [c; c; c; 0]: (24) where and ?1 de ne CI(2; 1) relations and ?2 de ne the variables which are a ected by the I(2) trends. Overall price homogeneity is testable either as a joint hypothesis of the rst two conditions or as a single hypothesis of the last condition in (24) (see, Kongsted, 2004). The rst condition in (24) describes price homogeneity between the levels of the nominal variables. It can be easily tested in the standard I(1) model as a linear hypothesis on either expressed as R0 i = 0; i = 1; 2; :::; r, where for the Brazilian data R0 = [1; 1; 1; 0; 0] or, equivalently, as = H' where ' is a (p1 1) r matrix of free coe cients and 2 3 1 000 6 1 1 0 07 6 7 7: 0 1 0 0 H=6 6 7 4 0 0 1 05 0 001 The hypothesis of price homogeneity was strongly rejected based on a LR test statistic of 41.9, asymptotically distributed as 2 (2): We note that the rst three coe cients of ^ 1 in Table 3 do not even approximately sum to zero, whereas those of ^ 0 are much closer to zero. The ^ ?2 estimates in Table 3 suggest that nominal money stock and black market exchange rate have been similarly a ected by the I(2) 3 This Section draws heavvily on Section 2 in Juselius (1999b) 26 trend, whereas the CPI price index has a smaller weight. Furthermore, the coe cient to industrial production is close to zero, consistent with the hypothesis that the latter has not been a ected by the I(2) trend. This can be formally tested based on the LR procedure (Johansen, 2004) as an hypothesis that industrial production is I(1). The test, distributed as 2 (1), was accepted based a test statistic of 1.30 and a p-value of 0.25. In the I(2) model, there is the additional possibility of mediumrun price homogeneity de ned as homogeneity between nominal growth rates. This is, in general, associated with real variables being I(1). For example, if (m p) I(1) and (sb p) I(1); then ( m p) I(0) and ( sb p) I(0) and there is medium-run price homogeneity in the sense of nominal growth rates being pairwise cointegrated (1, -1). Hence, a rejection of long-run price homogeneity implies a rejection of homogeneity between the nominal growth rates. We note that the rst three coe cients of ^ ?1 in Table 3 do not even roughly sum to zero consistent with the rejection of long-run price homogeneity. The previous section demonstrated that the levels component, xt 2 and the di erences component, xt 1 in (17) are closely tied together by polynomial cointegration. In addition xt 1 contains information 0 about ? xt 1 ; i.e. about the medium long-run relation between growth rates. Relying on results in Johansen (1995) the levels and di erence components of model (17) can be decomposed as: xt 1 + xt 1 =( ) 0 | I(0) 0 +( x {z t }1 ?1 + ?1 ) 0 x | ?1 {z t }1 I(0) 0 +( + + 1 0 ?2 ) 0 x | 1{zt }1 | 0 ?2 x {z t }1 (25) I(1) I(1) 0 0 xt 1 | {z } I(0) where = ( 0 ) 1 and is similarly de ned. The matrix is decomposed into three parts describing di erent dynamic e ects from the growth rates, and the matrix into two parts describing the e ects from the stationary relation, 00 xt 1 ; and the nonstationary relation, 01 xt 1 . The matrices in brackets correspond to the adjustment coe cients. The interpretation of the rst component in (25), ( ) 0 xt 1 ; is that prices not just adjusting to the equilibrium error between the price 27 levels, 0 xt 2 ; but also to the change in the equilibrium error, 0 xt 1 . Under long-run price homogeneity it would have represented a homogeneous e ect in in ation rates. The second component, ( 0 ?1 + ?1 ) 0?1 xt 1 ; corresponds to a stationary medium long-run relation between growth rates of nominal magnitudes. Because of the rejection of long-run price homogeneity, this represents a non-homogeneous e ect in nominal growth rates. The third component, ( 0 ?2 ) 0?2 xt 1 ; and the fourth component, 1 01 xt ; are both I(1) relations which combine to a stationary I(0); where polynomial cointegration relation, 1 ( 01 xt 1 + 0 xt 1 ) 0 0 0 =( 1 ?2 ) ?2 . The long-run matrix is the sum of the two levels components measured by: = 0 0 0 + 0 1 1: Hypothetically, the matrix is likely to satisfy the condition for long-run price homogeneity in a regime where in ation is under control. Thus, the lack of price homogeneity is likely to be the rst sign of in ation running out of control. The growth-rates matrix is the sum of the three di erent components measured by =( ) 0 +( 0 ?1 + ?1 ) 0 ?1 +( 0 ?2 ) 0 ?2 : The matrix is, however, not likely to exhibit medium-run price homogeneity, even under the case of long-run price homogeneity. This is because R0 = 0 implies R0 ?2 6= 0: The intuition is as follows: When 0 xt I(0); a non-homogeneous reaction in nominal growth rates is needed to achieve an adjustment towards a stationary long-run equilibrium position. Therefore, medium-run price homogeneity interpreted as a zero sum restriction of rows of would in general be inconsistent with overall long-run price homogeneity. 0 Table 4 reports the estimates of = (I 1) = ? and ? 0 = : We notice that the coe cients of each row do not sum to zero. Next section will show that the di erence is statistically signi cant. The diagonal elements of the matrix are particularly interesting as they provide information of equilibrium correction behavior, or the lack of it, of the variables in this system. We notice a signi cant positive coe cient in the diagonal element of the domestic prices, which in a single equation model would imply accelerating prices. In a VAR model absence of equilibrium correction in one variable can be compensated by a su ciently strong counteracting reaction from the other variables 28 in the system. It is noticeable that the only truely market determined variable, the black market exchange rate, is signi cantly equilibriumcorrecting variable, whereas money stock is only borderline so. Section 3 demonstrated that the unrestricted characteristic roots of the VAR model contained a small explosive root, which disappeared when two unit roots were imposed. Nevertheless, the positive diagonal element of prices suggest that the spiral of price increases which subsequently became hyper in ation had already started at the end of this sample. 8 Money growth, currency depreciation, and price in ation in Brazil Long-run price homogeneity is an important property of a nominal system and rejecting it is likely to have serious implications both for the interpretation of the results and for the validity of the nominal to real transformation. The empirical analysis of Durevall (1998) was based on a nominal to real transformation without rst testing its validity. We will here use the I(2) model for the empirical investigation of the money-price spiral without having to impose invalid long-run price homogeneity. 8.1 Identifying the relations The estimates of 0 ; 1 and in Table 3 are uniquely identi ed by the CI(2; 2) property of 00 xt : However, other linear combinations of 0 and 1 may be more relevant from an economic point of view, but these will be I(1) and will, therefore, have to be combined with the di erenced I(2) variables to become stationary. To obtain more interpretable results three overidentifying restrictions have been imposed on the two relations (see johansen and Juselius, 1994). The LR test of overidentifying restrictions, distributed as 2 (3) became 1:41 and the restrictions were accepted based on a p-value of 0.70. The estimates of the two identi ed relations became: c0 1;t xt = mt c0 2;t xt = pt 1 1 stb 1 ytr 0:64 (mt (18:3) 1 1 0:005trend ( 2:5) r yt 1 ) 0:008trend (26) ( 2:5) The rst relation is essentially describing a trend-adjusted liquidity ratio, except that the black market exchange rate is used instead of the CPI as a measure of the price level. The liquidity ratio with CPI instead of the exchange rate was strongly rejected. This suggests that in ationary expectations were strongly a ected by the expansion of money stock and that these expectations in uenced the rise of the black market nominal exchange rate. 29 Both relations need a linear deterministic trend. The estimated trend coe cient of the rst relation in suggests that 'the liquidity ratio' grew on average with 6% (0.005 12 100) per year in this period. The second relation shows that prices grew less than proportionally with the expansion of M3 money stock relative to industrial production after having accounted for an average price increase of approximately 9% (0.008 12 100) per year. 3.0 Lm3-Lcpi-LY 2.8 2.6 2.4 1977 1978 1979 1980 1981 1982 1983 1984 1985 1982 1983 1984 1985 Lm3-Lexc-LY 6.25 6.00 5.75 5.50 1977 1978 1979 1980 1981 Figure 11. The graphs of inverse velocity with CPI as a price variable (upper panel) and with nominal exchange rate (lower panel). The graphs in Figure 11 of the liquidity ratio based on the nominal exchange rate and on the CPI index, respectively, may explain why nominal exchange rates instead of domestic prices were empirically more relevant in the rst relation. It is interesting to note that the graphs are very similar until the end of 1980, whereafter the black market exchange rate started to grow faster than CPI prices. Thus, the results suggest that money stock grew faster than prices in the crucial years before the rst hyper in ation episode, but also that the depreciation rate of the black market currency was more closely related to money stock expansion. This period coincided with the Mexican moratorium, the repercussions of which were strongly and painfully felt in the Brazilian economy. The recession and the major decline of Brazilian exports caused the government to abandon its previous more orthodox policy of ghting in ation by maintaining a revalued currency and, instead, engage in a much looser monetary policy. For a comprehensive review of 30 the Brazilian exchange rate policy over the last four decades, see Bobomo and Terra (1999). Under the assumption that the black market exchange rate is a fairly good proxy for the `true' value of the Brazilian currency, the following scenario seems plausible: The expansion of money stock needed to nance the recession and devaluations in the rst case increased in ationary expectations in the black market, which then gradually spread to the whole domestic economy. Because of the widespread use of wage and price indexation in this period there were no e ective mechanisms to prevent the accelerating price in ation. 8.2 Dynamic equilibrium relations This scenario can be further investigated by polynomial cointegration. In the I(2) model 0 xt I(1) has to be combined with the nominal growth rates to yield a stationary dynamic equilibrium relation. The two identi ed relations, 01;1 xt and 01;2 xt in (26) need to be combined with nominal growth rates to become stationary. Table 5 reports various versions of the estimated dynamic equilibrium relations. The rst dynamic steady-state relation corresponds essentially to Cagan's money demand relation in periods of hyper in ation. However, the price level is measured by the black market nominal exchange rate and the opportunity cost of holding money is measured both by the CPI in ation and by the currency depreciation. The coe cient to in ation corresponds to Cagan's coe cient which de nes the average in ation rate (1= ) at which the government can obtain maximum seignorage. The present estimate suggests average in ation rates of an order of magnitude of 0.40-0.50 which corresponds to the usual de nition of hyper in ation periods. The second relation is more di cult to interpret from a theoretical point of view but seems crucial for the mechanisms behind the increasingly high in ation of this period and the hyper in ation of the subsequent periods. Eq. (3) shows that the `gap' between prices and `excess' money as measured by 01;2 xt is cointegrated with changes in money stock and prices, but not with currency depreciation. Eq. (4) combines 0 m; and price in ation, p; Eq. (5) with 1;2 xt with money growth, p and Eq. (6) with m: Although both nominal growth rates are individually cointegrating with 01;2 xt , there is an important di erence between them: The relationship between money growth and the relation 01;2 xt suggests error-correcting behavior in money stock, whereas the one between price in ation and 01;2 xt indicates lack error-correcting behavior in prices. The latter would typically describe a price mechanism leading ultimately to hyper in ation unless counterbalanced by 31 Table 4: The unrestricted parameter estimates The estimated = 0? ? matrix mt pt s bt y rt 2 mt : -1.07 -0.06 0.00 0.02 2 pt : -0.02 -0.55 0.01 -0.01 2 b -0.42 0.42 -0.92 0.03 st : 2 r yt : -0.13 0.20 0.04 -1.32 0 The estimated = matrix b mt 1 pt 1 st 1 ytr 1 trend 2 mt : 0:03 0:11 0:05 0:00 -0:001 ( 1:7) 2 pt : 2 b st : 2 r yt : 0:05 (7:5) ( 3:3) (0:4) ( 6:6) 0:03 0:03 0:05 0:00 ( 5:1) (3:3) 0:11 0:15 (1:9) 0:02 0:7 (3:1) 0:19 (2:8) ( 3:9) -0:01 0:01 0:3 0:5 (4:8) (0:1) 0:15 0:002 (4:2) ( 2:6) -0:02 0:00 0:6 ( 0:1) Table 5: Estimates of the polynomially cointegrated relations The dynamic equilibrium relations 0 xt + ! 0 xt ^ 0 xt ! 1;1 m t ! 1;2 p t ! 1;3 s b 1;1 t (1) 1.0 0:62 2:52 0:59 (1:1) (3:4) (2:7) (2) 1.0 - 2:02 0:53 (3) ^ 0 xt 1;2 1.0 ! 2;1 m t 5:80 ! 2;2 p t 11:32 ! 2;3 s bt 0:34 (67) (9:9) (1:0) (4) 1.0 6:02 11:38 (3:4) (7:1) (5) (2:5) - (10:0) 1.0 16:57 - (15:4) (6) 1.0 11:42 (12:4) 32 - - other compensating measures, such as currency control. 8.3 The short-run dynamic adjustment structure The in ationary mechanisms will now be further investigated based on the estimated short-run dynamic adjustment structure. Current as well as lagged changes of industrial production were insigni cant in the system and were, therefore, left out. Thus, real growth rates do not seem to have had any signi cant e ect on the short-run adjustment of nominal growth rates which is usually assumed to be the case in a high in ation regime. Furthermore, based on a F-test the lagged depreciation rate was also found insigni cant in the system and was similarly left out. Table 6 reports the estimated short-run structure of the simpli ed model. Most of the signi cant coe cients describe feed-back e ects from the dynamic steady-state relations de ned by Eq. (2) and Eq. (4) in Table 5 and the medium-run steady-state relation between growth rates, 0?1 xt de ned in Table 3. It is notable that the residual correlations are altogether very small, so that interpretation of the results should be robust to linear transformations of the system. The short-run adjustment results generally con rm the previous ndings. Price in ation has not been equilibrium correcting in the second steady-state relation, whereas the growth in money stock has been so in both of the two dynamic steady-state relations. The depreciation of the black market exchange rate has been equilibrium correcting to the rst steady-state relation measuring the liquidity ratio relation and has been strongly a ected by the second price 'gap' relation. Furthermore, it has reacted strongly to changes in money stock con rming the above interpretation of the important role of in ationary expectations (measured by changes in money stock) for the currency depreciation rate. After the initial expansion of money stock at around 1981 (which might have been fatal in terms of the subsequent hyper in ation experience) money supply seems primarily to have accommodated the increasing price in ation. The lack of equilibrium correction behavior in the latter was probably related to the widespread use of wage and price indexation in this period. Thus, the lack of market mechanism to correct for excessive price changes allowed domestic price in ation to gain momentum as a result of high in ationary expectations in the foreign exchange market. 9 Concluding remarks The purpose of this paper was partly to give an intuitive account of the cointegrated I(2) model and its rich (but also complicated) statistical structure, partly to illustrate how this model can be used to address 33 important questions related to in ationary mechanisms in high in ation periods. The empirical analysis was based on data from the Brazilian high in ation period, 1977:1-1985:5. An additional advantage of this period was that it was succeeded by almost a decade of hyper-in ationary episodes. The paper demonstrates empirically that it is possible to uncover certain features in the data and the model which at an early stage may suggest a lack of control in the price mechanism. Thus, a violation of two distinct properties, price homogeneity and equilibrium correction, usually prevalent in periods of controlled in ation, seemed to have a high signal value as a means to detect an increasing risk for a full-blown hyper in ation. The paper demonstrates that: 1. prices started to grow in a non-homogeneous manner at the beginning of the eighties when the repercussions of the Mexican moratorium strongly and painfully hit the Brazilian economy. The expansion of money stock needed to nance the recession and the subsequent devaluations increased in ationary expectations in the black market, which then spread to the whole domestic economy. 2. the widespread use of wage and price indexation in this period switched o the natural equilibrium correction behavior of the price mechanism. Without other compensating control measures which might have dampened in ationary expectations, it was not possible to prevent price in ation to accelerate. Acknowledgement 1 Useful comments from Michael Goldberg are gratefully acknowledged. The paper was produced with nancial support from the Danish Social Sciences Research Councel. 10 Appendix A: Misspeci cation diagnostics The univariate normality test in Table A.1 is a Jarque-Bera test, distributed as 2 (2): The multivariate normality test is described in Doornik and Hansen (1995) distributed as 2 (8). The AR-test is the F-test described in Doornik (1996), page 4. P-values are in brackets. 34 Table 6: Dynamic adjustment and feed-back e ects in the nominal system Ref. Regressors: mt 1 pt Table 5 (2) Table 5 (4) Eq.: 0:59 0:76 (5:6) w^1;1 x)t 1 ( ^ 1;2 x w^1;2 x)t 1 Table 3 pt 0:11 (4:2) 1 ( ^ 1;1 x ^0 ?1 mt 0:33 0:03 AR(1) 0:19 (0:66) (0:43) 0:00 (0:95) 0:03 0:06 0:02 0:06 (4:3) 0:005 ( 2:2) (2:1) 1.0 -0.02 0.08 1.0 -0.12 (0:84) 0:03 (0:87) p 1:67 (0:43) 1:27 (0:26) Skewness -0.13 0.21 0.09 -0.21 Kurtosis 3.06 3.29 2.62 3.27 Multivariate misspeci cation tests Normality, 2 (8) 4.43 (0.82) AR(1) 5.59 (0.99) AR(4) 62.21 (0.54) 35 0:08 (1:9) Table 7: Misspeci cation tests Univariate misspeci cation tests yr sb m 2 Normality, (2) 0:66 1:71 0:36 (0:72) (12:1) ( 2:9) +0:008 Residual correlations: (2:7) ( 2:3) (6:4) xt (2:4) s bt 0:91 (2:0) - 1.0 LY LY LUSDbm Lcpi 1 1 0 0 0 0 1 5 -1 10 1 5 10 -1 1 5 -1 10 1 1 1 1 0 0 0 0 -1 Lm3 Lm3 1 -1 1 5 -1 10 1 5 10 -1 1 5 -1 10 1 1 1 1 0 0 0 0 -1 Lcpi LUSDbm 1 1 5 -1 10 1 5 10 -1 1 5 -1 10 1 1 1 1 0 0 0 0 -1 1 5 -1 10 1 5 10 -1 1 5 -1 10 1 5 10 1 5 10 1 5 10 1 5 10 Figure A.1: Residual autocorrelograms and crosscorrelograms with 95% con dence bands. LY LU SD bm 25 25 20 20 15 15 10 10 5 5 0 -4 -2 0 2 0 -4 4 -2 Lm 3 25 20 20 15 15 10 10 5 5 -2 0 2 4 2 4 Lc pi 25 0 -4 0 2 0 -4 4 -2 0 Figure A.2: Residual histograms for the four equations. Figure A.1 shows the residual auto-correlograms and cross-correlograms of order 10 for all four equations. Figure A.2 shows the residual histograms compared to the normal distribution for all equations. Both gures have been produced with the program Me2, described in Omtzigt (2003) 36 11 References Bonomo, M. and Terra, C., (1999). The political economy of exchange rate policy in Brazil: 1964-1997, Graduate School of Economics, Getulio Vargas Foundation, Rio de Janeiro, Brazil. Cagan, P., (1956). The monetary dynamic of hyperin ation, in: M. Friedman (Ed.), Studies in the Quantity Theory of Money, University Press, Chicago. Doornik, J.A. (1996), \Testing vector error autocorrelation and heteroscedasticity", Available at http://www.nu .ox.ac.uk/users/doornik Doornik, J.A. and H. 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