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A note on modelling downturns
Economics Letters 3 (1979) 0 North-Holland Publishing
333-339 Company
A NOTE ON MODELLING DOWNTURNS A Non-linear Model vs. Simple Linear Autoregressive Carry
J. SCHINASI
*
Federal Reserve Board, Washington, DC 20551, Received
18 September
Schemes
USA
1979
This note presents empirical/simulations results which compare a simple Kaldor-type model and comparable linear autoregressive schemes as models of sharp movements observed in macroeconomic time series that exhibit persistent fluctuations.
non-linear often
1. Introduction One of the more fundamental regularities of macroeconomic time series is the persistence of fluctuations in either the levels of economic variables around an exponential or linear trend or growth rates. In addition, one often observes an asymmetry in upturns and downturns: the upturns exhibit gently rising series; the downturns exhibit sharp declines. ’ This note presents selected preliminary empirical/simulations results on the relative abilities of a simple non-linear macromodel (an endogenous cycle model) and comparable linear autogressive models to capture sharp movements - especially downturns - often observed in macroeconomic time series that exhibit persistent fluctuations, or what we might call business cycles.
* The author is Economist in the Special Studies Section of the Division of Research and Statistics of the Board of Governors of the Federal Reserve System. The Board does not necessarily subscribe to the views expressed in the paper. The author acknowledges the many enlightening and useful discussions with his dissertation advisor Prof. John B. Taylor, Columbia University. Discussions with David Folkerts-Landau and Peter von zur Muehlen were also helpful. ’ Hicks (1950) states: ‘One of the most striking characteristics of actual cycles is that output once it has passed the peak, falls off rather rapidly. (This) happened sufficiently often for it to seem likely that a rapid downswing, ought to be a feature of the theoretical cycles, if we are on the right lines.’ Keynes (1964, p. 314) ‘There is however another characteristic of what we call the Trade Cycle which our explanation must cover if it is to be adequate; namely the phenomenon of the crises - the fact that the substitution of a downward for an upward tendency often takes place suddenly and violently, whereas there is as a rule, no such sharp turning point when an upward is substituted for a downward tendency.’ 333
334
G.J. Schinasi /A note on modelling downturns
2. Discussion Economists have traditionally used two modelling techniques in their efforts to describe and explain the persistence of fluctuations in macroeconomic variables. The first technique - which is currently the more conventional modelling technique was originally developed by Frisch and Slutsky (FS) in the late thirties and recently applied in the work of Poole, Lucas, Sargent-Wallace, and Taylor. The technique relies on simple autogressive processes (AR) to describe fluctuations. The other approach which has not received much attention in the past few decades was developed primarily by Kaldor and Goodwin although Hicks and others made contributions as well. This approach in its original version utilized deterministic non-linear difference and differential equations to describe what was believed to be a fundamental characteristic of industrialized mixed economies - the existence of self-generating endogenous business cycles or growth cycles. The AR processes have not performed well in modelling the downturn - simulations exhibit a smoothness not possessed by the observed series - while the second approach has not been tried. Here we estimate a simple Kaldor-type non-linear model and compare simulations of the model to simulations of estimated AR(l) and AR(2) processes.
3. The models We are interested in determining how well these models simulate Net National Product (for example). AR(l) and AR(2) processes are estimated as representations of the FS approach and a Kaldor-type non-linear model is estimated as a representation of the non-linear approach. * The estimated non-linear model is as follows: 3,4
’ In principle, the AR(l) and AR(2) processes can be viewed as reduced form representations of the non-linear model with fist- and second-order (respectively), investment accelerator equations replacing the non-linear investment function. We make the additional assumption that the forcing functions are white noise. Roughly then, the comparison made here is between linear (non-cyclic) systematic components and non-linear (cyclic) systematic components. 3 Note that here we constrain the equation Zr = bl Ytal + bzR, since we set al = f. When a search over the interval al (0, 2) was performed it was found that a1 = 1.2 maximizes the log likelihood function. We also can reject the hypothesis that al = f, and cannot reject the hypothesis that al = 1.0, when we perform a likelihood ratio test with al = f and al = 1.0 as the constrained estimate, respectively, and ar = 1.2, as the unconstrained maximum of the likelihood function. The ‘fit’ for al = 1.2 is of course better. As compared to the R* = 0.836 0.9316 and R* = 0.9274 for the equations with al = 1.2 and al = 1.0 forar=$,wehaveR*= respectively. By reference to fig. 1 however, where the various fits are compared, it can be seen that we are not creating much of an injustice by using the equation with al = f. It is instructive then to explore the dynamic properties of the model with al = 4, since we can then
335
G. J. Schinasi /A note on modelling downturns
Y,=C,+I,+G,-D,, C, = 0.38(YD, (6.6)
(1) - 0.865YD,_,) (18.6)
+ 0.865C,_1
,
D-W = 2.05,
R2 = 0.896, (2a)
YD, = Y, - W, , I, = 3.3qyy3 (5.6)
(2b)
- 0.847Y;i3,) (17.7)
Mt = O.O805(Y, - 0.95Y,_,) (33.3) (5 .O)
- 1.41(R, - 0.847R,_,) + 0.8471,-l, (-2.2) D-W = 1.92, R2 = 0.836,
(3)
- 0.342(R, (-1.5)
(4)
- 0.95R,_1)
+ 0.95il~‘,_~,
D-W=
1.03,
R2 =0.95.
Eqs. (1) through (3) represent the product market. Eq. (1) is the income accounting identity where Y,, Ct, I,, G,, and D, represent Net Domestic Product, Personal Consumption Expenditure, Gross Private Domestic Investment Expenditure, Government Purchases of Goods and Services, and Capital Consumption Allowances (inclusive of capital consumption adjustments). All variables are real, measured in 1972 prices. Y,, C,, I,, and G, were detrended using an exponential trend and D, was calculated as a residual. Eq. (2) is a 1’ mear consumption function with YD, representing detrended Disposable Income. Eq. (3) is the investment function which is non-linear in NNP and linear in an interest rate variable, R,. This specification is consistent with the Kaldor shape. R, is the rate on 4-6 month commercial paper. An advantage of eq. (3) is that we can estimate it using linear techniques. Eq. (2b) is an identity which defines disposable income; we assume that W,, the difference between NNP and YD, is exogenous. Eq. (4) is the equilibrium condition in the money market where M, represents the real money supply detrended (Ml), and the right-hand side is the liquidity preference function which is linear in NNP and R,. We solve for the reduced form in Y, to the extent that the non-linearity permits and assume that the linear combination of the predetermined variable and R, is white noise. This allows us to isolate the cyclical properties of the non-linear mechanism from the cyclical properties of the right-hand side variables. This is represented in eq. (5) (1 - 0.38125)Y,
- (1.41295)Y;‘3
- RAND,.
(5)
The estimated linear AR schemes are as follows: Y, = 0.97067 Y,_ 1 + RAND,, (34.4)
(6)
determine if there is any plausibility of the Kaldor hypothesis as a tool whereby we can understand the cyclical mechanisms which may be operative in the U.S. 4 The equations were estimated using the Cochrane-Orcutt iterative technique, since, OLS estimates of the equations indicated that the residuals were serially correlated.
G.J. Schinasi /A note on modelling downturns
336
-
1
60
40 20 620 40 60 80
IIIIII III 1950
I
1955
I
IIIII II
II
I II
1950
I I I
III 1955
1965
1960
Fig. 1. A comparison
i
I I I
1960
I
I
I
I
I.1
1965
Fig. 2. The non-linear
1
II
1970
of investment
I
Ill
II
1975
specifications.
I
I I I I I I 1970
I
I
-
60
-
40
~
20
-
40
-
60
-
80
I
100
1975
mechanism.
v v
v
100
40 60
t
hi ,l,,,llllllll,l,,;llllIllIIul
1950
1955
1960
1965
Fig. 3. The AR(l)
process.
1970
1975
80 100
331
G.J. Schinasi /A note on modelling downturns
Y, = 1.426Y,_r (17.4)
- 0.485 Yt-z + RAND,. (-5.8)
(7)
We now simulate each of these ‘reduced form’ equations assuming a white noise process of equal variance (RAND,) for each forcing function (here we have chosen a variance equal to that of the interest rate series). Note however that there is no explicit dynamic in eq. (5). Given a drawing of RAND, say RAND,, the solution to (5) is a triplet of roots. At least one of these roots is real for all finite drawings of RAND; an equilibrium always exists. For successive drawings of RAND,+j,i= 1,2, . . .. we require a dynamic story which tells the model how to proceed from one set of equilibria to the next. The rule used in the simulation routine was a minimum distance rule. Hence if the initial condition is one in which there is a unique equilibrium, and the next drawing of RAND implies a triplet of real equilibria, the simulation routine will choose the equilibrium which minimizes the distance from the prior equilibrium. This rule is consistent with
Prediction
1950
1955
fpr al=1
.2
1965
1960
Fig. 4. The AR(2)
1970
1975
1970
1975
process.
r
1955
1960
1965
Fig. 5. Actual
NNP.
60
338
G.J. Schinasi /A note on modelling
downturns
market behavior in markets in which there are costs to adjusting. [For more details see Schinasi (1979, ch. S).] The results are presented in figs. 2,3, and 4, respectively. Fig. 5 presents a plot of actual NNP. Note that we do not examine these series to determine which is a more accurate representation of the actual series. We cannot hope to produce accurate predictions with such simplistic models. Instead we observe the general shape of the series with particular reference to upturns and downturns, and the ability of the alternative models to generate persistent fluctuations of a cyclical nature. With this in mind note how the non-linear mechanism generates persistent cyclical behavior. Also note how the non-linear model captures sharp declines. 5 In transitions through the steady state the sharper movements dominate. 6 The AR( 1) process is much smoother despite the fact that it is ‘shocked’ by the identical forcing function - RAND,. The AR(l) process exhibits a long gradual swing - a long-run serial correlation that does not appear in the non-linear series or the actual series. Although the AR(2) process is an improvement over the AR(l) process, it does not match the ability of the non-linear mechanism to generate persistent fluctuations. The AR(2) process retains a smoothness not possessed by the actual series of NNP or the simulated non-linear mechanism. It too fails to capture the sharp downturns in the actual series. Although the models are not accurate predictors, the non-linear mechanism appears to capture sharp downturns better than the AR processes. These results indicate that the non-linear approach to modelling fluctuations’the endogenous cycle approach - might lead to interesting and useful models as empirical descriptions of persistent fluctuations. The next obvious task is to develop a statistical test on second differences, whereby one can discriminate between the abilities of the non-linear model and AR processes to simulate the actual series. 7 This is however a difficult task and remains an unsolved problem.
References Frisch, R., 1933, Propagation problems and impulse problems in dynamic economics, in: Economic essays in honour of Gustav Cassel (Allen and Unwin, London). Goodwin, R.M., 1951, The nonlinear accelerator and the persistence of business cycles, Econometrica 19, 1-17. Hicks, J.R., 1950, A contribution to the theory of the trade cycle (Clarendon Press, Oxford). 5 Since the model is simple we were not able to model the asymmetry of upturns and downturns. The upturns in the series generated by the non-linear model also exhibit sharpness. 6 In fact, the movements of this series through the steady state - the apparent discontinuities are actually simulations of what is now called in the mathematics literature a ‘catastrophe’. See Schinasi (1979, especially chapter 5) for references and further details. 7 Neftci has done this for selected linear models.
G.J. Schinasi /A note on modelling
downturns
339
Kaldor, N., 1940, A model of the trade cycle, Economic Journal 50, 78-92. Keynes, J.M., 1964, The general theory of employment, interest, and money (Harcourt Brace World, New York). Lucas, R.E. Jr., 1972, Expectations and the neutrality of money, Journal of Economic Theory 4, 103ff. Neftci, S., 1979, The optimal prediction of cyclical downturns, Working paper (Department of Economics, George Washington University). Poole, W., 1970, Optimal choice of monetary policy instruments in a simple stochastic macro model, Quarterly Journal of Economics 84, 197-216. Sargent, T. and N. Wallace, 1975, Rational expectations, the optimal monetary instrument and the optimal money supply rule, Journal of Political Economy 83, 241-254. Schinasi, G.J., 1979, A nonlinear disequilibrium dynamic macroeconomic model of short run fluctuations, Unpublished Ph.D. dissertation (Columbia University, New York). Slutsky, E., 1937, The summation of random causes as the source of cyclic processes, Econometrica 5, 105-146. Taylor, J.B., 1975, Monetary policy during transition to rational expectations, Journal of Political Economy 83, 1009-1021.
333-339 Company
A NOTE ON MODELLING DOWNTURNS A Non-linear Model vs. Simple Linear Autoregressive Carry
J. SCHINASI
*
Federal Reserve Board, Washington, DC 20551, Received
18 September
Schemes
USA
1979
This note presents empirical/simulations results which compare a simple Kaldor-type model and comparable linear autoregressive schemes as models of sharp movements observed in macroeconomic time series that exhibit persistent fluctuations.
non-linear often
1. Introduction One of the more fundamental regularities of macroeconomic time series is the persistence of fluctuations in either the levels of economic variables around an exponential or linear trend or growth rates. In addition, one often observes an asymmetry in upturns and downturns: the upturns exhibit gently rising series; the downturns exhibit sharp declines. ’ This note presents selected preliminary empirical/simulations results on the relative abilities of a simple non-linear macromodel (an endogenous cycle model) and comparable linear autogressive models to capture sharp movements - especially downturns - often observed in macroeconomic time series that exhibit persistent fluctuations, or what we might call business cycles.
* The author is Economist in the Special Studies Section of the Division of Research and Statistics of the Board of Governors of the Federal Reserve System. The Board does not necessarily subscribe to the views expressed in the paper. The author acknowledges the many enlightening and useful discussions with his dissertation advisor Prof. John B. Taylor, Columbia University. Discussions with David Folkerts-Landau and Peter von zur Muehlen were also helpful. ’ Hicks (1950) states: ‘One of the most striking characteristics of actual cycles is that output once it has passed the peak, falls off rather rapidly. (This) happened sufficiently often for it to seem likely that a rapid downswing, ought to be a feature of the theoretical cycles, if we are on the right lines.’ Keynes (1964, p. 314) ‘There is however another characteristic of what we call the Trade Cycle which our explanation must cover if it is to be adequate; namely the phenomenon of the crises - the fact that the substitution of a downward for an upward tendency often takes place suddenly and violently, whereas there is as a rule, no such sharp turning point when an upward is substituted for a downward tendency.’ 333
334
G.J. Schinasi /A note on modelling downturns
2. Discussion Economists have traditionally used two modelling techniques in their efforts to describe and explain the persistence of fluctuations in macroeconomic variables. The first technique - which is currently the more conventional modelling technique was originally developed by Frisch and Slutsky (FS) in the late thirties and recently applied in the work of Poole, Lucas, Sargent-Wallace, and Taylor. The technique relies on simple autogressive processes (AR) to describe fluctuations. The other approach which has not received much attention in the past few decades was developed primarily by Kaldor and Goodwin although Hicks and others made contributions as well. This approach in its original version utilized deterministic non-linear difference and differential equations to describe what was believed to be a fundamental characteristic of industrialized mixed economies - the existence of self-generating endogenous business cycles or growth cycles. The AR processes have not performed well in modelling the downturn - simulations exhibit a smoothness not possessed by the observed series - while the second approach has not been tried. Here we estimate a simple Kaldor-type non-linear model and compare simulations of the model to simulations of estimated AR(l) and AR(2) processes.
3. The models We are interested in determining how well these models simulate Net National Product (for example). AR(l) and AR(2) processes are estimated as representations of the FS approach and a Kaldor-type non-linear model is estimated as a representation of the non-linear approach. * The estimated non-linear model is as follows: 3,4
’ In principle, the AR(l) and AR(2) processes can be viewed as reduced form representations of the non-linear model with fist- and second-order (respectively), investment accelerator equations replacing the non-linear investment function. We make the additional assumption that the forcing functions are white noise. Roughly then, the comparison made here is between linear (non-cyclic) systematic components and non-linear (cyclic) systematic components. 3 Note that here we constrain the equation Zr = bl Ytal + bzR, since we set al = f. When a search over the interval al (0, 2) was performed it was found that a1 = 1.2 maximizes the log likelihood function. We also can reject the hypothesis that al = f, and cannot reject the hypothesis that al = 1.0, when we perform a likelihood ratio test with al = f and al = 1.0 as the constrained estimate, respectively, and ar = 1.2, as the unconstrained maximum of the likelihood function. The ‘fit’ for al = 1.2 is of course better. As compared to the R* = 0.836 0.9316 and R* = 0.9274 for the equations with al = 1.2 and al = 1.0 forar=$,wehaveR*= respectively. By reference to fig. 1 however, where the various fits are compared, it can be seen that we are not creating much of an injustice by using the equation with al = f. It is instructive then to explore the dynamic properties of the model with al = 4, since we can then
335
G. J. Schinasi /A note on modelling downturns
Y,=C,+I,+G,-D,, C, = 0.38(YD, (6.6)
(1) - 0.865YD,_,) (18.6)
+ 0.865C,_1
,
D-W = 2.05,
R2 = 0.896, (2a)
YD, = Y, - W, , I, = 3.3qyy3 (5.6)
(2b)
- 0.847Y;i3,) (17.7)
Mt = O.O805(Y, - 0.95Y,_,) (33.3) (5 .O)
- 1.41(R, - 0.847R,_,) + 0.8471,-l, (-2.2) D-W = 1.92, R2 = 0.836,
(3)
- 0.342(R, (-1.5)
(4)
- 0.95R,_1)
+ 0.95il~‘,_~,
D-W=
1.03,
R2 =0.95.
Eqs. (1) through (3) represent the product market. Eq. (1) is the income accounting identity where Y,, Ct, I,, G,, and D, represent Net Domestic Product, Personal Consumption Expenditure, Gross Private Domestic Investment Expenditure, Government Purchases of Goods and Services, and Capital Consumption Allowances (inclusive of capital consumption adjustments). All variables are real, measured in 1972 prices. Y,, C,, I,, and G, were detrended using an exponential trend and D, was calculated as a residual. Eq. (2) is a 1’ mear consumption function with YD, representing detrended Disposable Income. Eq. (3) is the investment function which is non-linear in NNP and linear in an interest rate variable, R,. This specification is consistent with the Kaldor shape. R, is the rate on 4-6 month commercial paper. An advantage of eq. (3) is that we can estimate it using linear techniques. Eq. (2b) is an identity which defines disposable income; we assume that W,, the difference between NNP and YD, is exogenous. Eq. (4) is the equilibrium condition in the money market where M, represents the real money supply detrended (Ml), and the right-hand side is the liquidity preference function which is linear in NNP and R,. We solve for the reduced form in Y, to the extent that the non-linearity permits and assume that the linear combination of the predetermined variable and R, is white noise. This allows us to isolate the cyclical properties of the non-linear mechanism from the cyclical properties of the right-hand side variables. This is represented in eq. (5) (1 - 0.38125)Y,
- (1.41295)Y;‘3
- RAND,.
(5)
The estimated linear AR schemes are as follows: Y, = 0.97067 Y,_ 1 + RAND,, (34.4)
(6)
determine if there is any plausibility of the Kaldor hypothesis as a tool whereby we can understand the cyclical mechanisms which may be operative in the U.S. 4 The equations were estimated using the Cochrane-Orcutt iterative technique, since, OLS estimates of the equations indicated that the residuals were serially correlated.
G.J. Schinasi /A note on modelling downturns
336
-
1
60
40 20 620 40 60 80
IIIIII III 1950
I
1955
I
IIIII II
II
I II
1950
I I I
III 1955
1965
1960
Fig. 1. A comparison
i
I I I
1960
I
I
I
I
I.1
1965
Fig. 2. The non-linear
1
II
1970
of investment
I
Ill
II
1975
specifications.
I
I I I I I I 1970
I
I
-
60
-
40
~
20
-
40
-
60
-
80
I
100
1975
mechanism.
v v
v
100
40 60
t
hi ,l,,,llllllll,l,,;llllIllIIul
1950
1955
1960
1965
Fig. 3. The AR(l)
process.
1970
1975
80 100
331
G.J. Schinasi /A note on modelling downturns
Y, = 1.426Y,_r (17.4)
- 0.485 Yt-z + RAND,. (-5.8)
(7)
We now simulate each of these ‘reduced form’ equations assuming a white noise process of equal variance (RAND,) for each forcing function (here we have chosen a variance equal to that of the interest rate series). Note however that there is no explicit dynamic in eq. (5). Given a drawing of RAND, say RAND,, the solution to (5) is a triplet of roots. At least one of these roots is real for all finite drawings of RAND; an equilibrium always exists. For successive drawings of RAND,+j,i= 1,2, . . .. we require a dynamic story which tells the model how to proceed from one set of equilibria to the next. The rule used in the simulation routine was a minimum distance rule. Hence if the initial condition is one in which there is a unique equilibrium, and the next drawing of RAND implies a triplet of real equilibria, the simulation routine will choose the equilibrium which minimizes the distance from the prior equilibrium. This rule is consistent with
Prediction
1950
1955
fpr al=1
.2
1965
1960
Fig. 4. The AR(2)
1970
1975
1970
1975
process.
r
1955
1960
1965
Fig. 5. Actual
NNP.
60
338
G.J. Schinasi /A note on modelling
downturns
market behavior in markets in which there are costs to adjusting. [For more details see Schinasi (1979, ch. S).] The results are presented in figs. 2,3, and 4, respectively. Fig. 5 presents a plot of actual NNP. Note that we do not examine these series to determine which is a more accurate representation of the actual series. We cannot hope to produce accurate predictions with such simplistic models. Instead we observe the general shape of the series with particular reference to upturns and downturns, and the ability of the alternative models to generate persistent fluctuations of a cyclical nature. With this in mind note how the non-linear mechanism generates persistent cyclical behavior. Also note how the non-linear model captures sharp declines. 5 In transitions through the steady state the sharper movements dominate. 6 The AR( 1) process is much smoother despite the fact that it is ‘shocked’ by the identical forcing function - RAND,. The AR(l) process exhibits a long gradual swing - a long-run serial correlation that does not appear in the non-linear series or the actual series. Although the AR(2) process is an improvement over the AR(l) process, it does not match the ability of the non-linear mechanism to generate persistent fluctuations. The AR(2) process retains a smoothness not possessed by the actual series of NNP or the simulated non-linear mechanism. It too fails to capture the sharp downturns in the actual series. Although the models are not accurate predictors, the non-linear mechanism appears to capture sharp downturns better than the AR processes. These results indicate that the non-linear approach to modelling fluctuations’the endogenous cycle approach - might lead to interesting and useful models as empirical descriptions of persistent fluctuations. The next obvious task is to develop a statistical test on second differences, whereby one can discriminate between the abilities of the non-linear model and AR processes to simulate the actual series. 7 This is however a difficult task and remains an unsolved problem.
References Frisch, R., 1933, Propagation problems and impulse problems in dynamic economics, in: Economic essays in honour of Gustav Cassel (Allen and Unwin, London). Goodwin, R.M., 1951, The nonlinear accelerator and the persistence of business cycles, Econometrica 19, 1-17. Hicks, J.R., 1950, A contribution to the theory of the trade cycle (Clarendon Press, Oxford). 5 Since the model is simple we were not able to model the asymmetry of upturns and downturns. The upturns in the series generated by the non-linear model also exhibit sharpness. 6 In fact, the movements of this series through the steady state - the apparent discontinuities are actually simulations of what is now called in the mathematics literature a ‘catastrophe’. See Schinasi (1979, especially chapter 5) for references and further details. 7 Neftci has done this for selected linear models.
G.J. Schinasi /A note on modelling
downturns
339
Kaldor, N., 1940, A model of the trade cycle, Economic Journal 50, 78-92. Keynes, J.M., 1964, The general theory of employment, interest, and money (Harcourt Brace World, New York). Lucas, R.E. Jr., 1972, Expectations and the neutrality of money, Journal of Economic Theory 4, 103ff. Neftci, S., 1979, The optimal prediction of cyclical downturns, Working paper (Department of Economics, George Washington University). Poole, W., 1970, Optimal choice of monetary policy instruments in a simple stochastic macro model, Quarterly Journal of Economics 84, 197-216. Sargent, T. and N. Wallace, 1975, Rational expectations, the optimal monetary instrument and the optimal money supply rule, Journal of Political Economy 83, 241-254. Schinasi, G.J., 1979, A nonlinear disequilibrium dynamic macroeconomic model of short run fluctuations, Unpublished Ph.D. dissertation (Columbia University, New York). Slutsky, E., 1937, The summation of random causes as the source of cyclic processes, Econometrica 5, 105-146. Taylor, J.B., 1975, Monetary policy during transition to rational expectations, Journal of Political Economy 83, 1009-1021.