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Working Paper 89-2 Cointegration and a Test of the Quantity Theory of Money Yash P. Mehra Federal Reserve Bank of Richmond P.O. Box 27622 Richmond, VA 23261 804-697-8247 April 1989 thanks William Cullison, Michael *Research Officer and Economist. The author helpful comments. The views Dotsey, Tom Humphrey and Roy Webb for manynot necessarily reflect the views of expressed are those of the author and do Federal Reserve System. the Federal Reserve Bank of Richmond or the Abstract The main implication of the Quantity Theory of Money is that long-run movements in the price level are determined primarily by long-run movements in the excess of money over real output. This implication is related to the concept of cointegration discussed in Granger (1986), which states cointegrated multiple time series share common long-run movements. It is shown that the general price level is cointegrated with money, real output, and the nominal rate of interest. These economic variables enter a price Furthermore, the appearance of equation based on the Equation of Exchange. this cointegration in the data seems consistent with the presence of Grangercausality from money and real output to the price level. It is also shown that an inflation equation that incorporates the abovestated implication of the Quantity Theory of Money predicts quite well the actual behavior of inflation during the past decade or so. These results however hold for M2, not Ml, measure of money. One of the most influential economic doctrines is the quantity theory of money (hereafter denoted QTM), which in its simplest form states that long-run movements in the general price level are determined primarily by long-term movements in the excess of money over real output.1 The theory allows the price level to deviate in the short run from this long-run relationship. However, it is postulated that such deviations would trigger forces which would cause the actual price level to move towards the long-run value implied by the QTM. Such short-run price deviations, according to this theory, are transitory. This article draws on recent developments in the theory of cointegrated processes to test whether the QTM holds as a long-run equilibrium relation. A price equation summarizing the main determinants of the price level suggested by the QTM is derived. If long-run movements in the price level are related to long-run movements in the variables included in the price equation, then the price level is cointegrated with such variables.2 If such cointegration exists, then short-term deviations in the price level from its long-run value are stationary, and actual changes in the price level could satisfy an error correction mechanism described in Engle and Granger (1987). In that case, cointegration implies Granger-causation. This article shows that for the M2 measure of money the abovestated implications of the QTM are consistent with the data.3 In particular, long-run movements in the price 'The theory allows in addition the influence of velocity on the price level. However, it is generally assumed that velocity in the long run is determined by institutional factors and thus does not change or changes slowly over time. For an excellent review of the Quantity theory of money, see Humphrey (1984). Interpreting cointegration as a long run equilibrium relation is proposed by Engle and Granger (1987). 2 3 The results do not hold if Ml measure of money is used. - 2 level appear to be determined primarily by long-run movements in the excess of money over real output. This article is organized as follows. regression equation consistent with the QTM. Section 1 presents a It also discusses how recent developments in the theory of cointegrated processes could be used to test the main implication of the QTM. Section 2 contains the empirical results and Section 3 the concluding remarks. I A Price Equation Consistent with the Ouantitv Theory of MoneY A price equation that summarizes the main determinants of the price level suggested by the QTM can be written as It = + 2 Rt + In Mt Iln yt + nt (1) where p is the general price level; R, a market rate of interest; M, the quantity of money in circulation; yt, real output; and nt, a random error term. In is the natural logarithm. One could derive this equation from the equation of exchange in which the level of velocity depends upon the opportunity cost variable measured by a market rate of interest.4 The price equation (1) implies the price level is proportional to the quantity of money, given levels of real output and the market rate of interest. 4 This can be seen as follows. The usual equation of exchange expressed in log form is lnp = In M - In y + In V. Moore, Porter and Small (1988) and Hetzel and Mehra (1989), among others, have shown that the level of M2 velocity depends upon the opportunity cost variable measured as the differential between the market rate of interest (Rj and the own rate of return on money. Since several deposit components of M2 now pay market-determinedyields, the own rate of return on money in the long run is likely to move with the market rate. Hence, the long-run value of velocity is likely to depend on the market rate of interest. - 3 The variables in (1) are the long term determinants of the price level. In the short run the actual price level could differ from the value suggested by such determinants. This is implied by the presence of the error term nt in (1). However, if equation (1) is true, then nt is a stationary zero mean process,5 though it could be serially correlated, heteroscedastic, and even correlated with some of the righthand side explanatory variables. Testing the Quantity Theory of Money: The Issue of Cointegration If levels of the variables included in the price equation (1) above have stochastic trends, the proposition that this price equation describes the long-run relationship among the variables can be interpreted to mean that the stochastic trend in the price level is related to stochastic trends in money, real output and the nominal rate of interest. This implication is related to the concept of cointegration discussed in Granger (1986), which states cointegrated multiple time series share common stochastic trends.6 Hence, the long-run implication of the QTM can be examined using the test of cointegration proposed in Engle and Granger (1987). If the QTM is true, then the price level should be cointegrated with money, real output and the market rate of interest. This test for cointegration consists of two steps. The first step tests whether each variable in equation (1) has a stochastic trend. That is 5 The stationarity of nt is required by the QTM, which regards any short term deviations from the price level implied by values of R, M and y as transitory. 6 Let xt be a vector of N component time series, each first difference stationary. Then xt is said to be cointegrated if there exists a vector a such that Zt = a Xt is stationary. The intuition behind this definition is that even if each element of xt is nonstationary, there might exist linear combinations of such multiple time series that are stationary. In that case, multiple time We can series are cointegrated and share some common stochastic trends. interpret the presence of cointegration to imply that the stochastic trends (long-run movements) in these multiple time series are related to each other. -4 investigated by performing the unit root tests on the variables. The second step tests whether stochastic trends in these variables are related to each other. In particular, the question of interest here is whether the stochastic component in the price level is related to stochastic components in money, real output and the nominal rate of interest. This can be examined by estimating the cointegrating regression of the form (2) lnpt to Rt + Y2 n Mt + '(3 in yt + et y 1 (2) and then testing whether the residual et in (2) has a unit root or not. If et in (2) does not appear to have a unit root while the lefthand and righthand variables had each a unit root, then the variables are said to be cointegrated.7 In that case, ordinary least squares estimates of the parameters of (2) are consistent.8 Furthermore, as shown in West (1988), these ordinary least squares estimators have even asympototic normal distributions if, in addition to sharing a common unit root, the unconditional mean of first differences of the nonstationary variables is non-zero. In that case standard inference procedures based on t and F values can proceed in the usual test for cointegration proposed in Engle and Granger (1987) thus consists of finding linear combinations of nonstationary multiple time series that are stationary. This test is particularly suited to test the QTM, because the particular linear combination of interest is the one in which the price level is the dependent variable and money, real output and the nominal interest rates as the righthand side variables. Alternative test procedures for cointegration among multiple time series look for the number of common stochastic trends and are based on transformations of the original variables (see, for example, Stock and Watson (1988), Johansen (1988) and Bossaerts (1988)). These alternative test procedures are useful if the objective is to find the number of common trends. 7The If levels of the nonstationary variables included in equation (2) are not cointegrated, then ordinary least squares estimators of this equation would not possess any desirable asymptotic properties. As shown in Phillips (1986), ordinary least squares estimates of the regression parameters would not converge to constants, and the usual t and F-ratio test statistics would not possess limiting distributions but diverge as the sample size T goes to infinity. 8 way.9 The implication is that estimated coefficients of the cointegrating regression can be used to test hypotheses about the long-run impact of money, real output and the nominal rate of interest on the price level. Short-Run Dynamics: CointeQration and Granger-Causalitv Granger (1988) also points out that if a pair of series are cointegrated, then there must be causation in at least one direction. ° Suppose one finds that the residual et in (2) is stationary. This result implies the presence of cointegration. The question of interest is whether the presence of cointegration implies changes in the price level are jointly Granger-caused by lagged levels of money, real output and the nominal rate of interest, as suggested in Granger (1988).1" I present below some evidence consistent with the presence of such causality. Furthermore, I also show that an inflation equation incorporating the equilibrium relationship exhibited in 9 The standard errors of the estimated coefficients need to be adjusted to allow for the presence of time dependent and heteroscedastic error term et in (2). 10To illustrate, assume we have two series x and y. Assume that x and y are both integrated of order one (i.e., both series are stationary after first differencing) and that they are cointegrated. Then, as shown in Granger (1988), they satisfy an error-correction model of the form Axt = lagged Axt, Ayt + I1 Zt1 Ayt = lagged Axt, &y, + 72 Zt-1 where Zt is the residual from the cointegrating regression and where one of y1, 72 # 0. Since Zt-1 depends upon lagged levels of x and y the model above implies that either Ax or Ay (or both) must be caused by lagged levels of the variables. The intuition behind this result is that for a pair of series to have attainable equilibrium, there must be causation between them to provide the necessary dynamics. "1 As is quite clear from the illustration given in ft. note 10, the presence of cointegration does not necessarily imply that money, real output and the nominal rate of interest would jointly Granger-cause the rate of inflation. -6 (1) does reasonably well in predicting the actual behavior of inflation during the last decade or so. II This section presents the empirical results. quarterly and cover the sample period 1952Q1 to 1988Q4. The data used are The general price level (p) is measured by the implicit GNP deflator; real output (y), by real GNP; nominal money (M), by M2 measure of money; and opportunity cost (R), by 4-6 month commercial paper rate.12 I first present unit root test results and show that levels of variables included in the price equation (1) have each a However, levels of these variables appear to be cointegrated in unit root. the sense that the residual from the 'cointegrating regression' of the form (2) is stationary. Test Results for Cointearation The test used to detect a unit root in a given time series Xt is the Augmented Dickey-Fuller (ADF) test and is performed estimating the following regression n AXt = a + s-1 b5 AXt 5 + c T + d X + t(3) where et is the i.i.d. disturbance and n is the number of lagged values of first differences that are included to allow for serially correlated errors. If there is a unit root in Xt, then the estimated coefficient d above should not be different from zero. The results of estimating (3) for price level, opportunity cost of holding M2 is measured by the market rate of interest. This reflects the assumption made here that in the long run the own rate of return on M2 depends on the market rate of interest. 12The - 7 real output, M2, and the nominal interest rate data are presented in Table 1. These test results are consistent with the presence of a unit root in each of the relevant variables.1 3 Table 2 presents results of regressing the price level on levels of money, real output and the nominal rate of interest as in equation (2). The estimates of this regression are presented in the upper panel of Table 2. The middle panel of this Table presents the first ten autocorrelations of the residual from the cointegrating regression. These autocorrelations appear to decline quite rapidly, suggesting that the residual is stationary. The lower panel presents results of applying the formal ADF test for detecting a unit root in the residual series. Since there does not appear to be any serial correlation in first differenced residual series, I also present results without including own lags. The estimated coefficient that appears on the lagged level of the residual in the relevant regression range between -.2 (n=0) to -.3 (n=4) and is significantly different from zero at the .05 level.1 4 This result implies that the residual et is stationary.115 13I get similar results using the unit root tests proposed in Phillips and Perron (1988). Their procedure accounts for non-independent and non-identically distributed errors using non-parametric adjustment to the standard Dickey-Fuller procedure. The adjusted Dickey-Fuller t values denoted as Z (to) in Phillips and Perron (1988) for lnp, In y, In M2, and RCP are, respectively, -2.0, -2.1, -2.3 and 3.0. The 5% critical value (Fuller (1976), Table 8.5.2) is -3.45. None of the t values is significant, implying the existence of a unit root each in the data. 14 The unit root test proposed in Phillips (1987) when applied to the residual et yielded the following regression et = .76 et-1 + Et. The t statistic for the hypothesis that the estimated coefficient on et-1 in the regression above is unity (denoted as Zt in Phillips (1987)) is -4.4. The 5% critical value (Engle and Yoo (1987), Table 3) is 4.0. This result indicates that the residual et does not have a unit root. 15I also examined the stationarity of the residual et if the opportunity cost variable in the regression (2) was measured as the differential between the market rate of interest and the own rate of return on money. These test results continue to support the conclusion that the residual et is stationary. - 8 The results presented above imply that levels of the variables entering the price equation are nonstationary but cointegrated.1 6 The parameter estimates of (2) presented in Table 2 are therefore consistent. The coefficients that are estimated on In M2 and In yt are close to unity, as implied by the quantity theory. Since the residual et is serially correlated and possibly heteroscedastic, the least squares standard errors were corrected as suggested in West (1988). The parameter estimates shown in Table 2 still exceed their standard errors by a substantial margin1 7 '6 The Engle and Granger test as done here suggests the presence of at least one cointeprating vector. I also computed the likelihood ratio test, -2lnQr, proposed in Johansen (1988), for the no of linearly independent cointegrating vectors r, or equivalently the number of common unit roots (4-r) in a VAR (4) for the set of four time series included in equation (3) of the text. The likelihood ratio test values for their being 3, 2, 1, and 0 cointegrating vectors are 1.2, 10.5, 30.3 and 58.5, respectively. The ninety five percent quantiles for r = 3, 2, 1, and 0 are 4.2, 12.0, 23.8 and 38.6, respectively (Johansen (1988), Table 1). This evidence is consistent with the presence of at least two linearly independent cointegrating vectors in the four-variable system. 1 7 It is not clear if standard t and F values could be used to test hypotheses based on the estimated price equation (3). This is so because certain conditions stated in West (1988) are not satisfied. For example, even though the unconditional mean of lnM and lny is nonzero, that of R is not. Moreover, the formal test done for the number of common unit roots suggests the presence of more than a single unit root (see footnote 16). However, as shown in West (1988), the price equation (1) can be reformulated as an equation with a single nonstationary righthand explanatory variable as shown below in (B) lnpt = y. +-f1 ln(Mt/y) + 1 2 (Rt - RMJ) + Vt (B) where RM is the own rate of return on money. The opportunity cost variable is measured as the differential between the market rate and the own rate of return on money. The unit root tests suggest that (R-RM) is stationary, but ln(MJyt) is not. The unconditional mean of ln(MJyt) is nonzero. The least squares estimation of (B) yielded lnpt = 5.04 + .98 ln(MJyt) + .014 (Rt - RMJ) + Vt The coefficient that appears on money in (B) is not different from its theoretical unitary value and is statistically different from zero (t value is 87.3). It should also be pointed out that the coefficient that appears on (RRM) above is .014, which is not too different from the value .011 found on R in equation (2) of the text. See also footnote 15. - 9 - Short-Run DYnamics: Evidence on Causality and Forecasting Performance I now present some evidence on the extent to which short-run movements in the rate of inflation are influenced by deviations of the price level from the long-run equilibrium relation (2) estimated here. The evidence consists of estimating the following regression 2 AlPt = f + n1 1 fis A 2 n2 n3 lnpt-s +J21 fsAlnMt-s +J21 fsAlnyt- n4 + 21 f4 5 ARt 5 + A et 1 + C (4) and then testing whether X is statistically different from zero.'8 I also estimate these regressions for a shorter sample period, 1952Q1-1979Q4.19 Table 3 presents estimates of the coefficient that appears on the error-correction variable et 1 in regression (4). As can be seen, the estimated coefficient X is generally negative as expected and is statistically different from zero.2 0 This result implies that lagged levels, as opposed to first differences, of money, real output and the nominal rate of interest are relevant in modeling short-run dynamics of the rate of inflation. The earlier empirical work which tested causality using only first differenced variables might be suspect, because they might have missed detecting causality that 18In equation (4) the price level is in second differences whereas money, real output and the nominal rate of interest in first. This specification is suggested by unit root tests results. In particular, unit root tests indicated the presence of two unit roots in the price level, implying that this regressor needs to be differenced twice. 19 The residual series et used in the shorter-sample period is constructed using parameters of the 'cointegrating regression' (2) estimated over the whole sample period 1952Q1 to 1988Q4. I get similar results if the 'cointegration regression' is estimated over the shorter period and its residual series is used. 2 0 The coefficient that is estimated on the error correction variable remains negative and is statistically significant if the price level regressor in (4) is in first differences. - 10 - enters through the error-correction variable (Granger (1988)). How well does the price equation incorporating the error-correction mechanism explain the long-run behavior of inflation? I present below some evidence on this issue by examining the out-of-sample forecasting performance of a price regression of the form (4) over the period 1977 to 1988. In particular, the price regression that underlies this exercise is A2 lnpt =fo + S21 f1 S A2 lnpt-s + f A Rt + ) (In pt-l - ln Pt 1) + ft (5) where In Pt-1 is simply set to In (Mt ,/Ytl). That is, the long-run value of the price level (lnp*) is determined primarily by the excess of the log of money over real output.2 1 The regression is estimated first over the period 1952Q1 to 1976Q4 and dynamically simulated out-of-sample over the 4-quarter and 8-quarter periods, 1977Q1 to 1977Q4 and 1977Q1 to 1978Q4. The errors that occur predicting the inflation rate are calculated for the 4-quarter and 8-quarter horizons. The end of the estimation period is then advanced by four quarters, and estimation and out-of-sample simulation is repeated using this new period 1952Q1 to 1977Q4. That procedure is repeated until the price equation is reestimated and simulations prepared based on data ending in each fourth quarter through 1987Q4 (in case of 4-quarter horizon) and 1986Q4 (in case of 8-quarter horizon). Table 4 reports the errors from 2 1 Lag lengths of the right hand side variables in (5) were selected by the 'final prediction error' criterion. First differences of money and real output did not enter the regression. This inflation regression appears to pass several tests of model adequacy. In particular, the regression passes the Chow test of parameter stability over the period 1952Q1 to 1988Q4. The Godfrey test for serial correlation and heteroscedascity did not indicate the presence of serial correlation or heteroscedasticity in the residuals of (5). See also Hallman, Porter, and Small (1989) for another price regression that is similar in spirit to (5). - 11 - this exercise. As can be seen in the Table, this equation does a reasonable job in predicting the rate of inflation. The bias is small and the root mean squared error (RMSE) is 1.11 percentage points for the 4-quarter horizon and 1.44 percentage points for the 8-quarter horizon. On balance, these results imply that forecasts of the rate of inflation from the price equation estimated under the restriction implied by the QTM are not out of line with the actual behavior of inflation during the last decade or so.2 2 III Concluding Remarks The quantity theory of money does appear to hold as a long run equilibrium relation in the sense that short-term deviations in the general price level from the value implied by the excess of money over real output are stationary. The proposition that the rate of inflation in the long run is determined primarily by the rate of growth of money in excess of real output is consistent with the data. However, these results hold only for the M2 measure of money and reflect perhaps the underlying stability of the M2 demand behavior recently noted in Rasche (1987), Moore, Porter, and Small (1988) and Hetzel and Mehra (1989). Other analysts (see for example, Reichenstein and Elliott (1987) and Mehra (1988)) have also produced evidence consistent with the view that money measured by M2 remains relevant in predicting the long-term behavior of inflation. 2 2 This is not to suggest that the price equation (6) is the best forecasting equation but rather it captures quite well the long-term behavior of the price level. 12 - References Akaike, H. "Fitting autoregressive models for Prediction." Annals of International Statistics and Mathematics, 1969, 21, 243-247. Bossaerts, Peter. "Common Nonstationary Components of Asset Prices." Journal of Economic Dynamics and Control, June 1988, 347-364. Engle, Robert F. and Byung Sam Yoo. "Forecasting and Testing in a CoIntegrated Systems." Journal of Econometrics, 35, 1987, 143-159. Engle, Robert F. and C.W.J. Granger. "Cointegration and Error Correction: Representation, Estimation, and Testing." Econometrica, Vol. 55, No. 2, March 1987, 251-276. Fuller, W.A. Introduction to Statistical Time Series, 1976, Wiley, New York. Granger, C.W.J. "Developments in the Study of Cointegrated Economic Variables". Oxford Bulletin of Economics and Statistics, 48, 3, 1986, 213-228. . "Some Recent Developments in a Concept of Causality." Journal of Econometrics, 39, 1988, 199-211. Hallman, Jeffrey, J., Richard D. Porter, and David H. Small. "M2 Per Unit of Potential GNP as an Anchor for the Price Level." Staff Study #157, Board of Governors of the Federal Reserve System, April 1989. Hetzel, Robert L., and Yash P. Mehra. "The Behavior of Money Demand in the 1980s." Federal Reserve Bank of Richmond, March 1989. Photocopy. Forthcoming in the Journal of Money. Credit and Banking. Johansen, Soren. "Statistical Analysis of Cointegrating Vectors." Journal of Economic Dynamics and Control, June 1988, 231-54. Mehra, Yash P. "The Forecast Performance of Alternative Models of Inflation." Federal Reserve Bank of Richmond, Economic Review, September/October 1988, pp. 10-18. Moore, George R., Richard D. Porter, and David H. Small. "Modeling the Disaggregated Demands for M2 and Ml in the 1980s: The U.S. Experience." Board of Governors of the Federal Reserve System, May 1988. Photocopy. Phillips, P.C.B. "Time Series Regression with a Unit Root." Vol. 52, March 1987, 277-301. Econometrica . "Understanding Spurious Regressions in Econometrics." Journal of Econometrics, 33, 1986, 311-340. Phillips, P.C.B and Pierre Perron. "Testing for a Unit Root in Time Series Regression." Biometrika, 1988, 75, 2, 335-46. - 13 Rasche, Robert H. "Mi-Velocity and Money-Demand Functions: Do Stable Relationships Exist?" In Empirical Studies of Velocity, Real Exchange Rates, Unemployment and Productivity. Carnegie-Rochester Conference Series on Public Policy, Vol. 27, ed. by Karl Brunner and Allan H. Meltzer. Amsterdam: North Holland, Autumn 1987, pp. 9-88. Reichenstein, William, and J. Walter Elliott. "A Comparison of Models of Long-Term Inflationary Expectations." Journal of Monetary Economics 19 (May 1987): 405-25. Stock, James H. and Mark W. Watson. "Testing for Common Trends," Working Paper #E-87-2, Hoover Institution, Stanford University. West, Kenneth D. "Asymptotic Normality When Regressors Have a Unit Root." Econometrica, Vol. 56, no. 6, Nov. 1988, 1397-1417. Table 1 Augmented Dickey-Fuller Test Results A 4 A A A Augmented Dickey-Fuller Regression: AXt = a + si l b AXt s + c T + d Xt- 1 A + E Time Series xt Notes: a (t value) F Q(sl) npt -.006 (-1.96) 32.4* 26.7(.87) 1nM2t -.013 (-2.3) 19.9* 41.8(.23) 1nyt -.047 (-2.1) 6.1* 21.9(.97) -.13 8.0* 38.7(.35) (-3.2) The Augmented Dickey-Fuller regression is estimated over the period, 1952Q1-1988Q4. p is the price level; y, real GNP; M2, M2 measure of money; and R, the 4-6 month commercial paper rate. in is natural logarithm and A, the first difference operator. d is the estimated coefficient that appears on the lagged level of the variable in question and the associated parenthesis contain the t value; the 5 percent critical value of the t statistic is 3.45 (Fuller (1976), Table 8.5.2). F is a test of the null hypothesis that four lagged values of AX do not enter the Dickey-Fuller regression stated above. Q(sl) is the Ljung-Box Q-statistic based on 36 autocorrelations of the residual and sl is the significance level. * significant at .05 level. Table 2 Cointegration Test Results A. Estimates of a Cointegrating Regression; 1952Q1-1988Q4 lnpt = y0 +If, lnMt + 2 1nyt + Rt + 3 et 71i12 i3 1.0 .011 -1.2 (011, .024) B. (A) (.001, .002) (.028, .063) Autocorrelations (from 1 to 10) of the residuals from the Cointegrating Regression (A) 1 2 3 4 5 6 7 8 9 10 .76 .56 .46 .36 .23 .15 .13 .12 .09 .05 C. Augmented Dickey-Fuller Test of Residuals n Aet 3 =sX a (t Notes: bs &et-. + a value) n=4 -.26(-4.0 n=0 -.23(4.4) F * .90 et-l + Et Q(sl) 28.7(.68) 35.7(.48) See notes in Table 1 for definition of variables. In the top panel above, two values in the parenthesis below the estimated coefficients are the standard errors. The first value is the least squares value, whereas the second value allows for et to be serially correlated and heteroscedastic and is calculated as in West (1988). F and Q(sl) are defined as in Table 1. *The 5% and 10% critical values are, respectively, 4.02 and 3.71 (Engle and Yoo (1987), Table.3). mn n oD 0 LO tD COqa. N .4- I,- CO 00 4- -4 '4- 4-) oC S.. C) -. ~0 CO E =<: 04S.O 0 '4>C CO CD0 -4 -0 LA. 0 0. S'-i . ._-> =_Q , -a) =. a) 4-) 5 1L) 4. 5 I- E'V -a 4-) 0 MDt 4 si r c 4) -4 -4 08 4-) co 0 0 C -4 + 4-4-) C 0 0*i.-cm Ci S0 4. 5 0V) C 0C .4- C a 1-_-4 Il _-_ C c'S) AIr qd NW 0C) 0~~~~ V) 4L- 0) 0I Ll aC 4-=4.) cU) -4 C0 0 4-. 04 Ci *_ tD CL -4 aL) F- 4- 0 S ~ Li *'w 1-. o-- c 2) 0 5-4 o Ci 0}_ 4._ 0 0 Li a ci 0 Lo-) 4-b S 5._to CL 4.Cci (I_ S.C~~ aa C_- C 0 + 4 40CA 44A Q 5._ .) 4-) 0 pUL. 4- - a) _- s -4 -4 a)'~ CD C4- *c C LJ 0C i 4-C -sw 4-) -4 I" CC to 4) C U. -~ ca) oV 0cn0>,=: cia U')-4 '4- 4i U)0a)t U)0 0 a)i >, c) C - '- i 4 .4-4 _ CS lie 0 = LUJ CD-S ctia)C 4 Ci 4-) C 0)~~~ 14 CC+ 4-N N _II U-) tv *4=.s1 0co CM C o. Cii- 0 C 0. C E)a a0a) a) >4- Cj S-I C a) .C(AWC C =n 00 4- to< -. LUI ,0 0 0 )4-) 4- o Un .4. c0 + _ C C as C4 * i U) *_ .,S*- E0 cn v U) U 4.*_ 4, -- IN CJ C 0)C C C, -C _- *_ U) > 8 4.' 00 .4n4 a) -SC C S S.. )a )0 i0 ic V 4..) 0) C ci -J 0 4. J00 Ca) . C Table 4 Out-of-Sample Forecasting Performance of the Restricted Price Equation,; 1977-1988 Estimation Period Ends In 4-guarter Ahead Period Actual Predicted Error 8-quarter Ahead Period Actual Predicted Error 1976Q4 6.6 6.7 -.09 7.1 7.1 .01 1977Q4 7.7 7.2 .43 8.1 7.2 .84 1978Q4 8.5 7.7 .74 8.9 7.4 1.53 1979Q4 9.4 7.6 1.79 8.8 7.9 .98 1980Q4 8.3 9.6 -1.29 6.7 8.9 -2.19 1981Q4 5.0 7.1 -2.05 4.3 6.9 -2.56 1982Q4 3.6 5.0 -1.4 3.5 5.0 -1.53 1983Q4 3.3 4.1 -.71 3.1 3.9 -.78 1984Q4 2.9 3.1 -.20 2.8 3.3 -.52 1985Q4 2.7 3.4 -.67 2.9 4.0 -1.13 1986Q4 3.0 4.3 -1.28 3.5 4.5 -.96 1987Q4 4.0 3.5 .51 Mean Error .34 -.69 Mean Absolute Error .93 1.25 1.11 1.44 Root Mean Squared Error Notes: The error-correction variable in the price equation (see equation (5) in the text) is computed using the restriction that the equilibrium price level (in logs) equals ln(M2dy,). The numbers in the Table are the annualized rate of growth of the price level (4Q to 4Q) over the out-of-estimation forecast horizons.
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