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Terminal conditions as a means of ensuring unique solutions for rational expectations models with forward expectations
Economics Letters 4 (1979) 117-120 0 North-Holland Publishing Company
TERMINAL CONDITIONS AS A MEANS OF ENSURING UNIQUE SO&UTIONS FOR RATIONAL EXPECTATIONS MODELS WITH FORWARD EXPECTATIONS Patrick MINFORD, Kent MATTHEWS and Satwant MARWAHA University of Liverpool, Liverpool, L69 3BX, UK
Received 3 March 1980
Terminal conditions solution superior
are imposed on rational expectations models as a means of finding a unique among a continuum of potential solutions. It is argued that using terminal conditions is to invoking stability. The method can be applied to non-linear as well as linear systems.
1. Introduction A general problem with rational expectations models in which there are current or lagged expectations of future variables (REFV), is that there is an infinity of solution paths [e.g., Shiller (1978)]; m most REFV models some of these paths are also divergent. This problem has typically been solved [e.g., Muth (1961, p. 325), Sargent aI,d Wallace (1973), Minford (1978)] by imposing the additional condition on the model that the solution be convergent. Sargent and Wallace have referred to this as ‘ruling out speculative bubbles’. This condition sets the coefficient on divergent roots within the general solution to zero. The coefficients on the remaining roots have then been uniquely determined within these models. In effect the convergence condition provides sufficient additional restrictions on the solution to ensure uniqueness. However, Taylor (1977) has given an example of an REFV model where this condition is insufficient to ensure a unique solution. In this example there is only one root and it is stable; hence the whole infinite set of solutions is convergent. The convergence condition therefore cannot ‘bite’. In principle, such a problem can apply to models of any degree of complexity. For example if there are future expectations for these variables, the model will need to have unstable roots to ensure uniqueness; there is no apparent necessity for this to be the case. This problem has been discussed within the context of linear models. With nonlinear models there is superimposed on it the more general problem that, even if a unique solution can be determined for a linear approximation to the model, we cannot be sure that this is the only solution to the model in its original form. Our proposal - described in Minford and Matthews (1978) is that the imposi117
118
P. Minford et al. / Terminal conditions
tion of terminal equilibrium conditions on REFV models is warranted on economic grounds and that it ensures unique solutions for these models whether linear or nonlinear. These solutions are also identical with the solutions obtained by imposing the convergence condition, for those linear REFV models where this condition is applicable.
2. An example of the terminal condition
in a simple linear model
The following is a simple illustrative macro model with a dichotomised ‘monetary’ sector: y
=k-ap+Ay-,,
fi
‘-YY
+P-Pq+,
-_ElPh
Y-Y*=o(P-_ElP)+uP-g_El(Y-Y*)+,,
‘real’ and
(IS curve),
(1)
(LM curve),
(2)
(supply curve),
(3)
where G = exogenous money supply, p = price level, y = output, y* = trend output, E_ix+i = expectation at time i of x+i, E = white noise. All variables are in natural logs except for p which is the real rate of interest expressed as a fraction per period of time. The supply curve incorporates intertemporal effects with expected future output terms subsuming a Lucas-Rapping (1969)type supply of labour. The LM curve incorporates the costs of inflation expectations through a Cagan (1956) type mechanism. It turns out that with future expectations on y and p we obtain a thirdorder difference equation in the endogenous terms in which we require 2 unstable roots for the convergence condition to give uniqueness. This is best seen from the reduced form for p, where rii = y * = 0 for simplicity,
(4) Consider the solution for p = C&, Tie-i, this generates a series of identities
in the
ns, -(atfJ)?r,
=(artu)+ar/ane
-(otu)(l
t/?)rrr
+ {p((Y+u)--(Ylp(l
for
(5)
E ,
t/3)}n,
t~crp~,
x u01
tukr,=-
fore-r, (6)
(Y@p7Ti+* + @(CX+ U) for E-i,
i>2.
-
a@(1 + /3)}7r[+r - {(o +
U)
(1 + 3) + /3UX}7Ti
+
Uh(1 + $)?T_l
=O (7)
119
P. Minford et al. / Terminal conditions
It can be seen that we have 2 restrictions too few to determine all the 7~s.Applying the terminal condition that beyond some date t + N the endogenous variables reach equilibrium, this implies that rN+r = rrN+2 = 0; while from the general solution we have rri = AS’:’ + B6F’ f C‘S’,+‘. If both F 2 and F 3 are unstable, the terminal conditions sets B = C 2 0 for large N as would the convergence condition. If only 6 a is unstable (i.e., the convergence condition fails) the terminal chooses a unique path, setting the coefficient on the larger of the 2 stable roots to approximately zero. For example, for values of a: = 1.8,p = 1 S, u = h = 4 = 0.5, eq. (7) has roots 2.977, 1.471, 0.106. It can easily be verified that the use of a terminal condition, PN+r = pN+a = 0 would generate a path for p described by PN = (0.106)Np,.
3. Numerical solution Application of a terminal condition requires the arbitrary choice of terminal date. Analytically we would proceed by letting N + 00. However, analytic solution is impracticable for linear models of any size and for non-linear models. The question arises whether the numerical solution is sensitive to the terminal date. We investigate this within our model by introducing a simple form of non-linearity, we make the demand for money respond to the variance of expected inflation (proxied by its square). Hence rewrite (2) as fi =-Yy +P -P(_EIP+l
- _EIP)’ +e.
For values of fi = lO,y*
= 9 and y = 1, the terminal condition
Table 1 Solution for the price level, differences Period
N-3
N-2
1 2 3 4 5 6 7 8 9 10 11 12 13
-0.0004 -0.0003 0.0006 -0.0012 0.0012 0.0008 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
-0.0001 -0.0004 0.0000 0.0008 0.0010 0.0007 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
a Column
shows base run solution
from Base run N = 10.
N-l
-0.000 1 -0.0002 -0.0002 0.0002 0.0005 0.0005 0.0002 0.000 1 0.0000 0.0000 0.0000 0.0000 0.0000 value.
PN+~ = ?%N+r -
ov)” 1.1730 1.0065 0.999 0.9985 0.9987 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000
N+l -0.0001 -0.0001 0.0002 0.0002 -0.0002 -0.0005 -0.0005 -0.0002 -0.0001 0.0000 0.0000 0.0000 0.0000
N+2
N+3
-0.0001 -0.0001 0.0002 0.0004 0.0000 -0.0007 0.0000 -0.0007 -0.0003 0.0001 0.0000 0.0000 0.0000
-0.0001 -0.0001 0.0001 0.0004 0.0002 -0.0005 -~0.0012 -0.0012 -0.0008 -0.0003 -0.0001 0.0000 0.0000
120
P. Minford et al. / Terminal conditions
y;L’+r = 1 .O. As regards the numerical sensitivity of the path, we solve the model ’ for N = 10, u = 2 and a starting value of y,, = 5 .O. The terminal data was then varied on either side of N. Table 1 shows that the solution path for p is quite insensitive to this.
4. Economic rationale of terminal conditions We argue that the convergence condition is justified for reasons such as those previously given by Sargent and Wallace (1973), but that it is economically sensible to suppose that a convergent system will at some point in the future after a shock come to total rest. Hence the appropriate terminal condition is equilibrium. The sensitivity
of the solution
some ‘feel’ for the terminal is highly insensitive,
hence
Ijath with respect
date. However
to the terminal
the example
the cost in imposing
(table
date gives us
1) shows that the path
the ‘wrong’ terminal
date is likely to
be small. 2
References Cagan, P., 1956, The monetary dynamics of hyperinflation, in: M. Friedman, ed., Studies in the quantity theory of money (University of Chigaco Press, Chigaco, IL). Lucas, R. and L.A. Rapping, 1969, Real wages, employment and inflation, Journal of Political Economy 70,452-470. Matthews, K.G.P. and S. Marwaha, 1979, Numerical properties of the LITP model, SSRC Project Working Paper 7903 (University of Liverpool). Minford, A.P.L., 1978, Substitution effects, speculation and exchange rate stability (NorthHolland, Amsterdam). Minford, A.P.L. and K.G.P. Matthews, 1978, Terminal conditions, uniqueness and the solution of rational expectations models, SSRC Project Working Paper 7805 (University of Liverpool). Muth, J.F., 1961, Rational expectations and the theory of price movements, Econometrica 29, 315-33s. Sargent, T.J. and N. Wallace, 1973, Rational expectations and the dynamics of hyperinflation, International Economic Review 14, 328-350. ShiIler, R.J., 1978, Rational expectations and the dynamic structure of macroeconomic models: A critical review, Journal of Monetary Economics 4, l-44. Taylor, J.B., 1977, On conditions for unique solutions in stochastic macroeconomic models with price expectations, Econometrica 45, 1377-1387.
’ The numerical method involves solving the model with starting values for future expectations via a Gauss-Seidel algorithm; the expectations are then altered towards the forecasts thus produced via a Jacobi algorithm; solution occurs when expectations = forecast. * This is confiimed by Matthews and Marwaha’s (1979) report of simulations with a non-linear model of 15 equations.
TERMINAL CONDITIONS AS A MEANS OF ENSURING UNIQUE SO&UTIONS FOR RATIONAL EXPECTATIONS MODELS WITH FORWARD EXPECTATIONS Patrick MINFORD, Kent MATTHEWS and Satwant MARWAHA University of Liverpool, Liverpool, L69 3BX, UK
Received 3 March 1980
Terminal conditions solution superior
are imposed on rational expectations models as a means of finding a unique among a continuum of potential solutions. It is argued that using terminal conditions is to invoking stability. The method can be applied to non-linear as well as linear systems.
1. Introduction A general problem with rational expectations models in which there are current or lagged expectations of future variables (REFV), is that there is an infinity of solution paths [e.g., Shiller (1978)]; m most REFV models some of these paths are also divergent. This problem has typically been solved [e.g., Muth (1961, p. 325), Sargent aI,d Wallace (1973), Minford (1978)] by imposing the additional condition on the model that the solution be convergent. Sargent and Wallace have referred to this as ‘ruling out speculative bubbles’. This condition sets the coefficient on divergent roots within the general solution to zero. The coefficients on the remaining roots have then been uniquely determined within these models. In effect the convergence condition provides sufficient additional restrictions on the solution to ensure uniqueness. However, Taylor (1977) has given an example of an REFV model where this condition is insufficient to ensure a unique solution. In this example there is only one root and it is stable; hence the whole infinite set of solutions is convergent. The convergence condition therefore cannot ‘bite’. In principle, such a problem can apply to models of any degree of complexity. For example if there are future expectations for these variables, the model will need to have unstable roots to ensure uniqueness; there is no apparent necessity for this to be the case. This problem has been discussed within the context of linear models. With nonlinear models there is superimposed on it the more general problem that, even if a unique solution can be determined for a linear approximation to the model, we cannot be sure that this is the only solution to the model in its original form. Our proposal - described in Minford and Matthews (1978) is that the imposi117
118
P. Minford et al. / Terminal conditions
tion of terminal equilibrium conditions on REFV models is warranted on economic grounds and that it ensures unique solutions for these models whether linear or nonlinear. These solutions are also identical with the solutions obtained by imposing the convergence condition, for those linear REFV models where this condition is applicable.
2. An example of the terminal condition
in a simple linear model
The following is a simple illustrative macro model with a dichotomised ‘monetary’ sector: y
=k-ap+Ay-,,
fi
‘-YY
+P-Pq+,
-_ElPh
Y-Y*=o(P-_ElP)+uP-g_El(Y-Y*)+,,
‘real’ and
(IS curve),
(1)
(LM curve),
(2)
(supply curve),
(3)
where G = exogenous money supply, p = price level, y = output, y* = trend output, E_ix+i = expectation at time i of x+i, E = white noise. All variables are in natural logs except for p which is the real rate of interest expressed as a fraction per period of time. The supply curve incorporates intertemporal effects with expected future output terms subsuming a Lucas-Rapping (1969)type supply of labour. The LM curve incorporates the costs of inflation expectations through a Cagan (1956) type mechanism. It turns out that with future expectations on y and p we obtain a thirdorder difference equation in the endogenous terms in which we require 2 unstable roots for the convergence condition to give uniqueness. This is best seen from the reduced form for p, where rii = y * = 0 for simplicity,
(4) Consider the solution for p = C&, Tie-i, this generates a series of identities
in the
ns, -(atfJ)?r,
=(artu)+ar/ane
-(otu)(l
t/?)rrr
+ {p((Y+u)--(Ylp(l
for
(5)
E ,
t/3)}n,
t~crp~,
x u01
tukr,=-
fore-r, (6)
(Y@p7Ti+* + @(CX+ U) for E-i,
i>2.
-
a@(1 + /3)}7r[+r - {(o +
U)
(1 + 3) + /3UX}7Ti
+
Uh(1 + $)?T_l
=O (7)
119
P. Minford et al. / Terminal conditions
It can be seen that we have 2 restrictions too few to determine all the 7~s.Applying the terminal condition that beyond some date t + N the endogenous variables reach equilibrium, this implies that rN+r = rrN+2 = 0; while from the general solution we have rri = AS’:’ + B6F’ f C‘S’,+‘. If both F 2 and F 3 are unstable, the terminal conditions sets B = C 2 0 for large N as would the convergence condition. If only 6 a is unstable (i.e., the convergence condition fails) the terminal chooses a unique path, setting the coefficient on the larger of the 2 stable roots to approximately zero. For example, for values of a: = 1.8,p = 1 S, u = h = 4 = 0.5, eq. (7) has roots 2.977, 1.471, 0.106. It can easily be verified that the use of a terminal condition, PN+r = pN+a = 0 would generate a path for p described by PN = (0.106)Np,.
3. Numerical solution Application of a terminal condition requires the arbitrary choice of terminal date. Analytically we would proceed by letting N + 00. However, analytic solution is impracticable for linear models of any size and for non-linear models. The question arises whether the numerical solution is sensitive to the terminal date. We investigate this within our model by introducing a simple form of non-linearity, we make the demand for money respond to the variance of expected inflation (proxied by its square). Hence rewrite (2) as fi =-Yy +P -P(_EIP+l
- _EIP)’ +e.
For values of fi = lO,y*
= 9 and y = 1, the terminal condition
Table 1 Solution for the price level, differences Period
N-3
N-2
1 2 3 4 5 6 7 8 9 10 11 12 13
-0.0004 -0.0003 0.0006 -0.0012 0.0012 0.0008 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
-0.0001 -0.0004 0.0000 0.0008 0.0010 0.0007 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
a Column
shows base run solution
from Base run N = 10.
N-l
-0.000 1 -0.0002 -0.0002 0.0002 0.0005 0.0005 0.0002 0.000 1 0.0000 0.0000 0.0000 0.0000 0.0000 value.
PN+~ = ?%N+r -
ov)” 1.1730 1.0065 0.999 0.9985 0.9987 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000
N+l -0.0001 -0.0001 0.0002 0.0002 -0.0002 -0.0005 -0.0005 -0.0002 -0.0001 0.0000 0.0000 0.0000 0.0000
N+2
N+3
-0.0001 -0.0001 0.0002 0.0004 0.0000 -0.0007 0.0000 -0.0007 -0.0003 0.0001 0.0000 0.0000 0.0000
-0.0001 -0.0001 0.0001 0.0004 0.0002 -0.0005 -~0.0012 -0.0012 -0.0008 -0.0003 -0.0001 0.0000 0.0000
120
P. Minford et al. / Terminal conditions
y;L’+r = 1 .O. As regards the numerical sensitivity of the path, we solve the model ’ for N = 10, u = 2 and a starting value of y,, = 5 .O. The terminal data was then varied on either side of N. Table 1 shows that the solution path for p is quite insensitive to this.
4. Economic rationale of terminal conditions We argue that the convergence condition is justified for reasons such as those previously given by Sargent and Wallace (1973), but that it is economically sensible to suppose that a convergent system will at some point in the future after a shock come to total rest. Hence the appropriate terminal condition is equilibrium. The sensitivity
of the solution
some ‘feel’ for the terminal is highly insensitive,
hence
Ijath with respect
date. However
to the terminal
the example
the cost in imposing
(table
date gives us
1) shows that the path
the ‘wrong’ terminal
date is likely to
be small. 2
References Cagan, P., 1956, The monetary dynamics of hyperinflation, in: M. Friedman, ed., Studies in the quantity theory of money (University of Chigaco Press, Chigaco, IL). Lucas, R. and L.A. Rapping, 1969, Real wages, employment and inflation, Journal of Political Economy 70,452-470. Matthews, K.G.P. and S. Marwaha, 1979, Numerical properties of the LITP model, SSRC Project Working Paper 7903 (University of Liverpool). Minford, A.P.L., 1978, Substitution effects, speculation and exchange rate stability (NorthHolland, Amsterdam). Minford, A.P.L. and K.G.P. Matthews, 1978, Terminal conditions, uniqueness and the solution of rational expectations models, SSRC Project Working Paper 7805 (University of Liverpool). Muth, J.F., 1961, Rational expectations and the theory of price movements, Econometrica 29, 315-33s. Sargent, T.J. and N. Wallace, 1973, Rational expectations and the dynamics of hyperinflation, International Economic Review 14, 328-350. ShiIler, R.J., 1978, Rational expectations and the dynamic structure of macroeconomic models: A critical review, Journal of Monetary Economics 4, l-44. Taylor, J.B., 1977, On conditions for unique solutions in stochastic macroeconomic models with price expectations, Econometrica 45, 1377-1387.
’ The numerical method involves solving the model with starting values for future expectations via a Gauss-Seidel algorithm; the expectations are then altered towards the forecasts thus produced via a Jacobi algorithm; solution occurs when expectations = forecast. * This is confiimed by Matthews and Marwaha’s (1979) report of simulations with a non-linear model of 15 equations.