Stephen M. Miller Equation(7)does not separate the effect of the excess supply f money into movements in real income and the price level. Al lowing for these differential effects, Equation(5)becomes Dlny=φ2(1-81nM-hnM) D In P 22(1-8)n MS-In Mp) whereΦ2=Φa1+Φ2,Now, dividing Equations(4,(5a),and⑤5b) byΦ1,Φal,andΦ2;, respectively, and then subtracting twice Equa- tion (4)from the sum of Equations (5a)and (5b) gives In M-In Mt=-(1/1D In T,+(1/2p2 D In y, +(1/2p22)D In P, And finally, substituting from Equation(1)results in In M=ao+aInr,+aIn y, +aaIn P, -(1/p,)D In re +(1/22)Dhy+(1/2a)DhP+∈ Now, first-differencing Equation (1)yields In M,-In Mi-1=a, D In rt-1+ a2D In y,-1 3D In Pt-1+ Et where Equation (3b)defines the adjustments in the interest rate real income, and the price level. Substituting into Equation( 8)from quations(4),(5a), and(5b)generates InMp-In Mp-1=Q(n M,-1-In Mp-1)+eE-1,(9) where =-∝1中181+(221+a32)(1-81) (10) Equation(9)represents, not surprisingly, a demand-adjusting for
Stephen M. Miller Equation (7) does not separate the effect of the excess supply of money into movements in real income and the price level. Allowing for these differential effects, Equation (5) becomes D In qt = cP,,(l - &)(ln Mf - ln Mf) , (54 and D In P, = cP,,(l - S,)(ln Mf - In Mf) , W where apz = a21 + az2. Now, dividing Equations (4), (5a), and (5b) by aI, QS1, and $a, respectively, and then subtracting twice Equation (4) from the sum of Equations (5a) and (5b) gives In Mf - In MF = -(l/al)0 In r, + (l/2@& In yi + (l/2@.& In P, . And finally, substituting from Equation (1) results in In Mf = a0 + a,ln r, + olJn qt + ol,ln P, - (l/@&I In r, + (1/2@&I In qt + (1/2@.&0 In P, + l , . Now, first-differencing Equation (1) yields lnM:- In ME, = alD In t-,-r + ozD In gt-l + c@ In P,-I + E, - l tel , (64 (74 (8) where Equation (3b) defines the adjustments in the interest rate, real income, and the price level. Substituting into Equation (8) horn Equations (4), (Sa), and (5b) generates In Mf' - ln ME, = LR(ln Mf-, - ln ML,) + E, - Q-~, (9) where n = -a,@16, + (a&!1 + c&&)(1 - 6,) . (10) Equation (9) represents, not surprisingly, a demand-adjusting for- 568
Disequilibrium Macroeconomic he tradition of Starleaf (1970), Artis and Lewis(1976), and Coats(1982). Cordon(1984)states that two major problems face monetary economists-the large coefficients of lagged money and the high autocorrelations in post-1973 samples. Lagged money was originally introduced to account for sluggish portfolio adjustment(Chow 1966) but the post- 1973 coefficients of lagged money suggest implausibly slow speeds of portfolio adjustment. Further, high autocorrelation may indicate model misspecification The existing literature has several things to say about these two issues. Goodfriend(1985)argues that the money market can lear each period and that lagged money does not belong theoret ically in money demand. Measurement errors in the exogenous variables can explain the significance of the coefficient of lagged money and the high autocorrelation. Laidler(1985)and Gordon(1984) argue that money demand regression equations represent semi-re duced-form equations. That is, the parameters of the money de- mand regressions combine the parameters from the money demand and other equations of the macroeconomy offer a competing explanation for these problems hased on Equations(1),(7),(7a), and (9 ). The post-1973 money market ex perienced significant disequilibrium. But the dynamic adjustment is of the demand-, rather than the supply-, adjusting type. Equation (9)shows how my formulation of money-market adjustment con- forms with the demand-adjusting view. Now, a well-behaved (that is, white-noise)error structure in Equation(1) implies a well-be- haved error structure in Equations (7)and(7a)but a moving-at erage error structure with a unit root in Equation(9). If, alterna tively, the partial-adjustment equation possesses a well-behaved error structure,then Equations (1),(7), and(7a) exhibit autocorrelated Equations(7a)and(9) are comparable to Starleaf's (1970, 751-52) Equations (3.4)and (3.5)after several adjustments. First, Starleaf assumes that the adjustment quation(that is, [3. 4]does not involve a random error. Equation( 9)includes a gn. seco mand for money and the demand-adjustment equation are in real terms. Thus, the price terms appearing in Equation(7a)disappear in Starleaf's specification. Thir Starleaf assumes that this periods demand for money adjusts to the differe tween this period's money supply and last period's money demand. Equation (9) has last periods money supply instead of this period s. Starleaf's adjustment eq tion results when Equation (3a) is adopted rather than Equation(3b)as the dis equilihrium adjustment specification. Finally, to derive Starleaf s quation(3.5)from Equation (7a), assume that n2=a 1=a2p2i
Disequilibrium Macroeconomics mulation in the tradition of Starleaf (1970), Artis and Lewis (1976), and Coats (1982).’ Gordon (1984) states that two major problems face monetary economists-the large coefficients of lagged money and the high autocorrelations in post-1978 samples. Lagged money was originally introduced to account for sluggish portfolio adjustment (Chow 1966); but the post-1973 coefficients of lagged money suggest implausibly slow speeds of portfolio adjustment. Further, high autocorrelation may indicate model misspecification. The existing literature has several things to say about these two issues. Goodfriend (1985) argues that the money market can clear each period and that lagged money does not belong theoretically in money demand. Measurement errors in the exogenous variables can explain the significance of the coefficient of lagged money and the high autocorrelation. Laidler (1985) and Gordon (1984) argue that money demand regression equations represent semi-reduced-form equations. That is, the parameters of the money demand regressions combine the parameters from the money demand and other equations of the macroeconomy. I offer a competing explanation for these problems based on Equations (l), (7), (7a), and (9). The post-1973 money market experienced significant disequilibrium. But the dynamic adjustment is of the demand-, rather than the supply-, adjusting type. Equation (9) shows how my formulation of money-market adjustment conforms with the demand-adjusting view. Now, a well-behaved (that is, white-noise) error structure in Equation (1) implies a well-behaved error structure in Equations (7) and (7a) but a moving-average error structure with a unit root in Equation (9). If, alternatively, the partial-adjustment equation possesses a well-behaved error structure, then Equations (l), (7), and (7a) exhibit autocorrelated ‘Equations (7a) and (9) are comparable to Starleaf’s (1970, 751-52) Equations (3.4) and (3.5) after several adjustments. First, Starleaf assumes that the adjustment equation (that is, [3.4]) does not involve a random error. Equation (9) includes a random error due to different model design. Second, Starleaf assumes that the demand for money and the demand-adjustment equation are in real terms. Thus, the price terms appearing in Equation (7a) disappear in Starleaf’s specification. Third, Starleaf assumes that this periods demand for money adjusts to the difference between this periods money supply and last period’s money demand. Equation (9) has last periods money supply instead of this periods. Starleaf’s adjustment equation results when Equation (3a) is adopted rather than Equation (3b) as the disequilibrium adjustment specifkation. Finally, to derive Starleaf’s Equation (3.5) from Equation (7a), assume that n = a,@, = up&I. 569
Stephen M. Miller error structures with unit roots. In sum. a well-behaved demand- djusting partial-adjustment model of the money market implies an autoregressive crror structure with a unit root for estimated moncy demand equations, potentially explaining the high autocorrelation in the post-1973 money demand regressions Estimation of Equations (7)and (7a)present several econo- metric problems. First, the equations contain right-side endogenous variables. The rates of change in the interest rate nominal and real income, and the price level, since they are based on Equation(3b), follow, in a timing sense, the other variables in the equations, it cluding the left-hand-side money stock. Thus, two-stage estimation appears appropriate, assuming that the rate of change variables are endogenous. But, such an approach implies an exogenous left-hand side variable. Cointegration and error-correction modeling, consid ered in the next section, provide a possible solution to these prob. ems Cointegration and Error-Correction Econometric method precedes econometric practice, some- times with a substantial lead. Fo or exam ple, the possibility of spu us co-movement between variables has been acknowledged for a long time(for example, Jevons 1884, 3), with Yule(1926)conduct ing the first formal analysis(Hendry 1986 provides more details) Nonetheless. econometricians continued to use standard time-series regressions with little concern for whether the relationships were sUch a dichotomy does not occur with the supply-adjusting model, where the error structures of the partial-adjustment and estimating equations are identical Gordon(1984, 414)introduces the error term into the partial-adjustment, rather han the demand, equation with little effect, since the final error structure of the stimating equation is unaffected. Such is not the case for the demand-adjusting EStimation also assumes constant parameters, inviting the Lucas(1976)criti- cism. The speeds of adjustment( that is,中,中;,φa,andφ2y) are especially open to this criticism, since they measure how the interest rate, nominal and real in (1985)makes this point as it applies to the estimation of standard post-1973 money demand functions. In addition, exogenous oil-price shocks cause temporary pertur- bations in the price- level adjustment process. For example, as the price level larger(smaller)changes in D In P than are indicated by previ equilibria. As a consequence, the estimates of and aa are biased (downward) during the time when the oil-price shock is being transmitted to the domestic price level
Stephen M. Miller error structures with unit roots. In sum, a well-behaved demandadjusting partial-adjustment model of the money market implies an autoregressive error structure with a unit root for estimated money demand equations, potentially explaining the high autocorrelation in the post-1973 money demand regressions.‘j Estimation of Equations (7) and (7a) present several econometric problems. First, the equations contain right-side endogenous variables. The rates of change in the interest rate, nominal and real income, and the price level, since they are based on Equation (3b), follow, in a timing sense, the other variables in the equations, including the left-hand-side money stock. Thus, two-stage estimation appears appropriate, assuming that the rate of change variables are endogenous. But, such an approach implies an exogenous left-handside variable. Cointegration and error-correction modeling, considered in the next section, provide a possible solution to these problems.7 Cointegration and Error-Correction Econometric method precedes econometric practice, sometimes with a substantial lead. For example, the possibility of spurious co-movement between variables has been acknowledged for a long time (for example, Jevons 1884, 3), with Yule (1926) conducting the first formal analysis (Hendry 1986 provides more details). Nonetheless, econometricians continued to use standard time-series regressions with little concern for whether the relationships were real or spurious. Spurious regression can occur when the regression ‘Such a dichotomy does not occur with the supply-adjusting model, where the error structures of the partial-adjustment and estimating equations are identical. Gordon (1984, 414) introduces the error term into the partial-adjustment, rather than the demand, equation with little effect, since the final error structure of the estimating equation is unatfected. Such is not the case for the demand-adjusting framework. ‘Estimation also assumes constant parameters, inviting the Lucas (1976) criticism. The speeds of adjustment (that is, a,, $, Q2,, and a,,) are especially open to this criticism, since they measure how the interest rate, nominal and real income, and the price level respond to disequilibria in the money market. Laidler (1985) makes this point as it applies to the estimation of standard post-1973 money demand functions. In addition, exogenous oil-price shocks cause temporary perturbations in the price-level adjustment process. For example, as the price level rises in response to previous excess supplies of money, oil-price increases (decreases) cause larger (smaller) changes in D In P than are indicated by previous moneymarket disequilibria. As a consequence, the estimates of Qz and @a are biased upward (downward) during the time when the oil-price shock is being transmitted to the domestic price level. 570