The present paper discusses the transition from linear modelling to the first nonlinear models in economic analysis. In this vein, an important contribution was J.

Hicks’s A Contribution to the Theory of the Trade Cycle where he developed his own endogenous model of the cycle. Hicks thought that fluctuations in investment,– caused by nonlinear changes in autonomous investment and the acceleration principle governing induced investment – led to an adjustment process taking place throughout many periods. In this paper we introduce some modifications regarding the econometric estimation of Hicks’s nonlinear model and an empirical application for Greece (1960-2007) takes place demonstrating the almost ideal fit of the model

Arapis, Gerasimos

Belegri-Roboli, Athena

Michaelides, Panayotis G.

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Early Nonlinear Modelling in Economic
Analysis: The Hicks Model for Greece
Revisited
Panayotis G. Michaelides and Athena Belegri-Roboli and
Gerasimos Arapis
National Technical University of Athens
2009
Online at https://mpra.ub.uni-muenchen.de/67112/
MPRA Paper No. 67112, posted 8. October 2015 02:17 UTC
Early Nonlinear Modelling in Economic Analysis:  
The Hicks Model for Greece Revisited
 
Panayotis G. Michaelides, Athena Belegri-Roboli and Gerasimos Arapis 
 
National Technical University of Athens, Greece  
School of Applied Mathematics and Physics 
Department of Humanities, Social Sciences and Law 
Laboratory of Theoretical and Applied Economics 
Email: pmichael@central.ntua.gr  
 
 
Abstract: The present paper discusses the transition from linear modelling to the first 
nonlinear models in economic analysis. In this vein, an important contribution was J. 
Hicks’s A Contribution to the Theory of the Trade Cycle where he developed his own 
endogenous model of the cycle. Hicks thought that fluctuations in investment, – 
caused by nonlinear changes in autonomous investment and the acceleration principle 
governing induced investment – led to an adjustment process taking place throughout 
many periods. In this paper we introduce some modifications regarding the 
econometric estimation of Hicks’s nonlinear model and an empirical application for 
Greece (1960-2007) takes place demonstrating the almost ideal fit of the model.   
 
Keywords: Linear, Nonlinear, dynamics, Hicks, economics.   
  
1. Introduction 
Modern economic analysis draws heavily on the tools of mathematics ([7], 
[9]). In fact, the tools of nonlinear analysis are regarded as promising ways 
towards overcoming the problems associated with the traditional approaches 
and have developed through different strands of thought and across diverse 
disciplines ([5], [8], [20], [21], [22]).  
An important contribution was Hicks’s A Contribution to the Theory 
of the Trade Cycle where he developed his own endogenous model of the 
cycle. In this paper we introduce a novel approach regarding the econometric 
estimation of Hicks’s nonlinear model. Next, an empirical application for the 
Greek economy using real world data for the time period 1960-2007 takes 
place and shows the ideal fit of the model.  
The paper is structured as follows: section 2 deals with the transition 
from early linear to the first nonlinear model in economics; section 3 presents 
the Hicks model; section 4 describes the proposed methodological framework 
for its empirical estimation; section 5 presents the empirical analysis for the 
case of Greece; section 6 concludes the paper.  
  
         Michaelides et al.  
2. From linear to nonlinear models in Economics: A Review 
This section, which draws heavily on [17], deals with the development of 
early nonlinear models in economics. Early empirical efforts, focused on time 
series models whose ad hoc nature was (and still is) criticised by many 
researchers, who insisted on the importance of adopting ‘structural’ models 
[12]. It should be stressed that before the linear (typically Keynesian) 
approach dominated economic theory, numerous economists were involved 
in the development of nonlinear models ([3], [6], [25]). This trend was 
associated with the study of business fluctuations, a very hot topic on most 
economists’ agenda [18]. Consequently, the study of business fluctuations 
came up because several economies were unstable.  
One of the seminal papers was Frisch’s 1933 Propagation problems 
and impulse problems in dynamics economics. According to it, the study of 
cycles was dynamic, and could be decomposed into two aspects: propagation 
and impulse. Frisch tried to think of several ways to make his system more 
realistic. One idea was to produce oscillations through the introduction of 
time-lags in the investment function [10]. In this context, Frisch constructed a 
mixed differential-difference equation system which was capable of 
producing cycles. However, the model’s oscillations died out in a finite time. 
This situation made him add an exogenous shock to his model [10]. 
However, several other papers criticised it for not including the Keynesian 
multiplier, an omission that prevented it from being a purely endogenous 
model of the cycle [23].  
Another seminal contribution in the development of nonlinear 
dynamic modelling in economic analysis is Samuelson’s famous model [19]. 
Actually, while Frisch constructed a dynamic system based on investment 
lags, Samuelson used time-lags in his model in a different way. However, the 
dynamic behaviour expressed by his model was, practically, devoid of 
meaning [19]. As a result, the model would only reproduce damped 
oscillations and thus one had to also introduce exogenous shocks [2].  
In this line of thought, Kalecki ([15], [16]) constructed a model 
focusing on the time lag between investment decisions and the production of 
capital goods. Kalecki’s model was not based on the acceleration principle to 
generate cycles. Rather, the thrust of his model’s theoretical idea was that the 
rate of investment decisions was a positive function of the gap between the 
rate of profit and the rate of interest, influenced by lenders’ confidence as 
well as the money market. Both the prospective rate of profit and the rate of 
interest were in turn a function of investment and the stock of capital. [16]. 
In 1940, Kaldor [14] employed nonlinearities to account for 
economic fluctuations. The model was different from the previous ones since 
the author did not make it dynamic by utilising a time-lag structure on the 
investment function. Rather, Kaldor followed a different strategy: his model 
was dynamic in nature because ex ante Savings differed from Investment, 
which together with the multiplier-accelerator, yielded a model that produced 
endogenous cyclical fluctuations.  
The Hicks Model for Greece Revisited 3 
Harrod stressed the fact that oscillations were produced by the lag 
itself rather than by mechanisms endogenous to the system. In Harrod’s 
model dynamics were not caused by the lags, but rather depended on the 
internal structure of the model [11]. Harrod proposed a very simple model, 
based on the multiplier-accelerator mechanism and derived his well-known 
equation. He also came to show that for a static equilibrium, the process was 
unstable. Moreover, he showed that instability was, at least partly, 
responsible for generating cycles.  
Harrod’s model was criticised for being simplistic by Baumol [3] 
and Alexander [1]. A fundamental contribution in this vein was Hicks’s A 
Contribution to the Theory of the Trade Cycle [13] where he defended a 
combination of both approaches. Hicks thought that fluctuations in 
investment led to a process taking place throughout many periods, because of 
lags in the multiplier. Of course, Hicks came to the standard solutions of a 
higher order linear system. Also, another crucial aspect of the model was the 
introduction of nonlinearities in the investment function.  
3. The Hicks Model  
According to Hicks the consumption function is a linear function of 1tY !  
                                                           1)1( !!" tt YsC                                (1) 
where 0 < 1-s< 1 is the so-called marginal propensity to consume, 
1/ s is the multiplier and 1tY ! denotes output one period back. In the 
Samuelson business cycle model, investment was determined by the growth 
in output, through the so-called acceleration principle. Analytically, 
investment was assumed to be proportional to the change in output, or: 
)( 21 !! !" ttt YYuI  
where tI denotes investment in time period t, 1tY ! and 2tY ! output 
one and two periods back, respectively, and u (>0) is the accelerator. 
However, Hicks thought that fluctuations in investment, caused by nonlinear 
changes in autonomous investment and the acceleration principle governing 
induced investment, led to an adjustment process taking place throughout 
many periods. In this spirit, Hicks introduced autonomous expenditures 
which may be growing exponentially. Investment has two components: 
autonomous investment growing at a constant rate g: 
               
t
t gAA )1(0 #"                                
where 0$  is the autonomous investment. Of course, there is also the induced 
part of the investment which responds to changes in output. Thus:  
)()1( 210 !! !##" tt
t
t YYugAI                    (2) 
Hicks modeled the growth process of a closed economy without government 
sector, within the Keynesian framework. In this context:  
                                                        ttt ICY #"                                         (3) 
         Michaelides et al.  
In what follows, we introduce a novel approach for the econometric 
estimation of Hicks’s nonlinear model and apply it for the Greek economy in 
the time period 1960-2007 using data from the AMECO database.  
4. Econometric Estimation  
The estimation of the consumption function (1) is straightforward using the 2 
Stages Least Squares (2SLS) method. This method is typically used for the 
estimation of multiplier–accelerator models [4].  
For the estimation of the consumption function we use 2SLS:  
ttt vYsCC #!#" !10 )1(                               (4) 
where 0C  is the intercept of the regression, and tv is the random error term, 
independent and identically distributed (i.i.d.).  
In what follows, the proposed methodology estimates the Hicks 
model, based on nonlinear investments. It is the case that the LS estimation 
principle applies for deriving estimators, ones that are called Nonlinear Least 
Squares (NLS). Unlike Ordinary Least Squares (O.L.S.) estimators, NLS 
estimators cannot be obtained analytically. However, the minimization of the 
Sum of Squared Residuals (SSR) is a well-defined optimization problem that 
can be solved numerically by iterating on a solution:   
 
Step 1: Let g %&'('(& ,],,[ : ')(  be drawn from a uniform 
distribution.  
Step 2: Estimate 0$ >0 and u>0 in the following equation using O.L.S.: 
                               ttt
t
t eYYugA!I #!###" !! )()1( 2101             (5) 
where 1!  is the intercept of the estimation and te is the random error (i.i.d.).  
Step 3: Compute the Sum of Squared Residuals SSR(i) for the equation.  
Step 4: Repeat for 1,..,i I" and select the value g that yields the minimum 
SSR(i), subject to 0$ >0 and u>0 being statistically significant. 
Step 5: Given the value of g estimated in the previous step, use 2SLS to 
estimate the linearized equation (5) and keep the values of the parameters. 
Given that g is the economy’s growth rate in autonomous investment 
it should, normally, be positive over time and range between 0% and 15%. 
Also, other criteria for model selection include the SIC (Schwartz 
Information Criterion) and the AIC (Akaike Information Criterion).  
 
 
The Hicks Model for Greece Revisited 5 
5. Empirical Analysis: The Greek Economy (1960-2007)
5.1 Estimation Results 
The estimation of equation (4) by means of 2SLS yields:  
14.46 0.83t t tC Y v!" ! # #  
                              (-2.80) (49.79)  
2R =0.98, S.S.E.=3.99 
The equation (5) by means of the procedure developed above, yields:  
1 211.54 2.69(1 0.05) 0.49( )
t
t t t tI Y Y e! !" # # # ! #
                          (7.45)    (8.37)                      (1.99) 
2R =0.68, S.S.E.=5.20 
Values in parentheses are t-statistics which imply that all estimated 
parameters are statistically significant. It should be stressed that for g =0.05 
the estimation of equation (5) yields the minimum value of SSR, under which 
0$  and u are statistically significant. Thus s=0.17, 0C = -4.46, 
u=0.49, 0A =2.69, 1A =11.54 and gm #"1 =1.05. 
 
5.2 Solutions and Stability 
By substituting equations (4) and (5) into (3) and rearranging we get   
                   
1 2 0 1 0(1 ) (1 )
t
t t tY s u Y uY A g A C! !! ! # # " # # #             (6) 
Also, 1*  and 2* are the roots of the characteristic equation  
                 + ,uusus 4)1(1
2
1
, 221 !#!-#!"**
                        (7) 
The complete solution for (6) is: 
                                           )()()( tYtYtY ec #"                                        (8) 
where )(tYc is the complementarity function and )(tYe is the particular 
integral. We can show that the particular integral is equal to 
                  
2
0 1 0
2
( )
( ) ( )
(1 )
t
e p
A m m A C
t Y t
m s u m u s
#
. " " #
! ! ! #
              (9) 
So, the solution which clearly implies a “moving equilibrium” output is 
 
                                  ( ) 14.36 (1.05) 41.65teY t " / #                             (10) 
As is well known, the solution depends on the discriminant 
uus 4)1( 2 !#!"0 . Given that in our case the discriminant takes a 
negative value ( 0)0 ), the complementarity function is 
                                   
1 2( ) ( cos sin )
t
c t r B t B t1 1. " #                   (11) 
where:  
                                          
2 2r a b" #                                                  (12) 
                                        
1tan ( / )b a1 !"                                               (13) 
         Michaelides et al.  
                                          1,2 bi* (" -                                                    (14) 
Substitution in (7), given that 0)0 , yields  
1,2 0.66 0.23i* " -                                    
Which implies 
0.7r "  and 0.341 " . 
By substituting these values into equation (11) we get  
        
1 2( ) 0.7 ( cos 0.34 sin 0.34 )
t
c t B t B t. " #                    
So 
1 2( ) 0.7 ( cos0.34 sin0.34 ) 14.36 (1.05) 41.65
t tt B t B t. " #! # / # . 
Finally, given the two initial conditions (i.e. actual values for !(0) and Y(1)) 
we get the values for the arbitrary constants 
1 31 .00B " !  and 2 33 .5 7B " ! . 
Conslusively  
( ) 0.7 ( 31cos0.34 33.57sin0.34 ) 14.36 (1.05) 41.65t tt t t. " ! #! # / # .                  
This is the analytical solution for ( )Y t , i.e. Gross Domestic Product (GDP) 
and by substituting the values of t we get the estimated values of ( )Y t . The 
estimated GDP is illustrated in the same plot with the real GDP values and 
the calculated correlation coefficient ( 0.99correlationr " ) implies an ideal fit.      
0
50
100
150
200
60 65 70 75 80 85 90 95 00 05 10
GDPREAL GDPESTIMATED
Figure 1. Estimated GDP vs real GDP for the Greek economy (1960-2007) 
 
Finally, stability depends on the discriminant uus 4)1( 2 !#!"0 . Since  
0)0 , tr  (with 1r ) ) will dampen the fluctuations caused by the 
1 2( cos 0.34 sin 0.34 )B t B t# . In another formulation, since 1)1(
2 ))! us  
or 0.35<0.49<1, the solution is periodically convergent, i.e. stable [24]. 
The Hicks Model for Greece Revisited 7 
 
6. Conclusions  
In this paper, after discussing the transition from early linear modelling to the 
first nonlinear models in economics, we proposed a novel approach for 
estimating, in a Keynesian system of equations, the non-linear Trade 
(Business) Cycle model developed by Hicks. The proposed methodology 
yields very satisfactory results when fitted to GDP data for the Greek 
economy over the time period 1960-2007. We believe that the results of this 
paper suggest that the modified Hicks model with its generality, conformity 
with theory and simplicity of structure is an appropriate vehicle for testing, 
expanding and improving conventional business cycle theory in empirical 
applications. Meanwhile, preliminary results of the proposed approach from 
other countries are extremely encouraging. Clearly, future and more extended 
research on the subject would be of great interest.  
 
References 
[1] Alexander, S. Mr. Harrod’s dynamic model. Economic Journal 60 (240), 
724-39, 1950. 
[2] Baumol, W., Benhabib, J. Chaos: Significance, Mechanism, and 
Economic Implications. Journal of Economic Perspectives, Vol. 3, No. 1, 77-
105, 1989. 
[3] Baumol, W.: Notes on some dynamic models. Economic Journal 58 
(232), 506-21, 1948. 
[4] Chow, G. China’s Economic Transformation, Blackwell Publishing, 
2007. 
[5] Colander, D., Pluralism and Heterodox Economics: Suggestions for an 
“Inside the Mainstream” Heterodoxy. Middlebury College Economics 
Discussion Paper No. 07-24, October, 2007.  
[6] Day R. H.: Nonlinear dynamics and evolutionary economics. In Day, 
R.H., Chen, P.  (eds.): Nonlinear Dynamics and Evolutionary Economics. 
Oxford University Press, New York and Oxford, 1993. 
[7] Debreu, G.: Mathematical Economics. The New Palgrave: A Dictionary 
of Economics, Vol. 3, 401-03, 1987. 
[8] Fontana, M.: Computer Simulations, Mathematics, and Economics. 
Rivista Internazionale di Scienze Economiche e Commerciali, 53(1), 96-124, 
2006. 
[9] Friedman, M., Essays in Positive Economics. University of Chicago 
Press, Chicago, 1953. 
[10] Frisch, R., Propagation problems and impulse problems in dynamic 
economics (1933) In Bjerkholt, O.: Foundations of Modern Econometrics: 
The Selected Essays of Ragnar Frisch. Edward Elgar, Cheltenham and 
Northampton, 1995. 
[11] Harrod, R. An essay in dynamic theory. Economic Journal 49 (193), 14-
33 (1939). 
         Michaelides et al.  
[12] Hendry D., Morgan, M. The Foundations of Econometric Analysis. 
Cambridge University Press, Cambridge, 1995. 
[13] Hicks, J. A Contribution to the theory of the Trade Cycle. Clarendon 
Press, Oxford, 1950. 
[14] Kaldor, N. A model of the trade cycle. Economic Journal 50 (197), 78-
92, 1940. 
[15] Kalecki, M. A macrodynamic theory of business cycles. Econometrica 
3(3), 327-44, 1935. 
[16] Kalecki, M. A theory of the business cycle. Review of Economic Studies, 
4 (2), 77-97, 1937. 
[17] Michaelides, P. The Development of Non Linear Dynamics in Economic 
Analysis, in Kyamakya, K., Halang, W., Rulkov, R., Chedjou, J.C., Li, Z., 
Bouchachia, A. and Unger, H. (eds) : Recent Advances in Nonlinear 
Dynamics and Synchronization– Theory and Applications, Springer, 
Germany, Chapter 13, 2009.  
[18] Nagatani K. Macroeconomic Dynamics. Cambridge University Press, 
Cambridge, 1981. 
[19] Puu, T. Nonlinear Economic Dynamics. Springer-Verlag, Heidelberg 
and New York, 1993. 
[20] Rappaport, S. Abstraction and Unrealistic Assumptions in Economics. 
Journal of Economic Methodology, 3(2), 215–236, 1996.  
[21] Rosser, B. J. Jr. On the Complexities of Complex Economic Dynamics. 
Journal of Economic Perspectives, 13 (4), 169-192, 1999. 
[22] Samuelson, P. Interactions between the multiplier analysis and the 
principle of acceleration. Review of Economics and Statistics 21 (2), 75-78, 
1939. 
[23] Thalberg, B. A reconsideration of Frisch’s original cycle model. (1990) 
In Velupillai, K.: Nonlinear and Multisectoral Macrodynamics: Essays in 
Honour of Richard Goodwin. Macmillan, Basingstoke (1990) 
[24] Tu, P. N.V., Dynamical Systems. Springer – Verlag, 1994.  
[25] Tvede L. Business Cycles - The Business Cycle Problem from John Law 
to Chaos Theory. Harwood Academic Publishers, Amsterdam, 1998. 
 


Early Nonlinear Modelling in Economic Analysis: The Hicks Model for Greece Revisited

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveEarly Nonlinear Modelling in EconomicAnalysis: The Hicks Model for GreeceRevisitedMichaelides, Panayotis G. and Belegri-Roboli, Athena andArapis, GerasimosNational Technical University of Athens2009Online at https://mpra.ub.uni-muenchen.de/67112/MPRA Paper No. 67112, posted 08 Oct 2015 02:17 UTCEarly Nonlinear Modelling in Economic Analysis:  The Hicks Model for Greece Revisited Panayotis G. Michaelides, Athena Belegri-Roboli and Gerasimos Arapis  National Technical University of Athens, Greece  School of Applied Mathematics and Physics Department of Humanities, Social Sciences and Law Laboratory of Theoretical and Applied Economics Email: pmichael@central.ntua.gr    Abstract: The present paper discusses the transition from linear modelling to the first nonlinear models in economic analysis. In this vein, an important contribution was J. Hicks’s A Contribution to the Theory of the Trade Cycle where he developed his own endogenous model of the cycle. Hicks thought that fluctuations in investment, – caused by nonlinear changes in autonomous investment and the acceleration principle governing induced investment – led to an adjustment process taking place throughout many periods. In this paper we introduce some modifications regarding the econometric estimation of Hicks’s nonlinear model and an empirical application for Greece (1960-2007) takes place demonstrating the almost ideal fit of the model.    Keywords: Linear, Nonlinear, dynamics, Hicks, economics.     1. Introduction Modern economic analysis draws heavily on the tools of mathematics ([7], [9]). In fact, the tools of nonlinear analysis are regarded as promising ways towards overcoming the problems associated with the traditional approaches and have developed through different strands of thought and across diverse disciplines ([5], [8], [20], [21], [22]).  An important contribution was Hicks’s A Contribution to the Theory of the Trade Cycle where he developed his own endogenous model of the cycle. In this paper we introduce a novel approach regarding the econometric estimation of Hicks’s nonlinear model. Next, an empirical application for the Greek economy using real world data for the time period 1960-2007 takes place and shows the ideal fit of the model.  The paper is structured as follows: section 2 deals with the transition from early linear to the first nonlinear model in economics; section 3 presents the Hicks model; section 4 describes the proposed methodological framework for its empirical estimation; section 5 presents the empirical analysis for the case of Greece; section 6 concludes the paper.             Michaelides et al.  2. From linear to nonlinear models in Economics: A Review This section, which draws heavily on [17], deals with the development of early nonlinear models in economics. Early empirical efforts, focused on time series models whose ad hoc nature was (and still is) criticised by many researchers, who insisted on the importance of adopting ‘structural’ models [12]. It should be stressed that before the linear (typically Keynesian) approach dominated economic theory, numerous economists were involved in the development of nonlinear models ([3], [6], [25]). This trend was associated with the study of business fluctuations, a very hot topic on most economists’ agenda [18]. Consequently, the study of business fluctuations came up because several economies were unstable.  One of the seminal papers was Frisch’s 1933 Propagation problems and impulse problems in dynamics economics. According to it, the study of cycles was dynamic, and could be decomposed into two aspects: propagation and impulse. Frisch tried to think of several ways to make his system more realistic. One idea was to produce oscillations through the introduction of time-lags in the investment function [10]. In this context, Frisch constructed a mixed differential-difference equation system which was capable of producing cycles. However, the model’s oscillations died out in a finite time. This situation made him add an exogenous shock to his model [10]. However, several other papers criticised it for not including the Keynesian multiplier, an omission that prevented it from being a purely endogenous model of the cycle [23].  Another seminal contribution in the development of nonlinear dynamic modelling in economic analysis is Samuelson’s famous model [19]. Actually, while Frisch constructed a dynamic system based on investment lags, Samuelson used time-lags in his model in a different way. However, the dynamic behaviour expressed by his model was, practically, devoid of meaning [19]. As a result, the model would only reproduce damped oscillations and thus one had to also introduce exogenous shocks [2].  In this line of thought, Kalecki ([15], [16]) constructed a model focusing on the time lag between investment decisions and the production of capital goods. Kalecki’s model was not based on the acceleration principle to generate cycles. Rather, the thrust of his model’s theoretical idea was that the rate of investment decisions was a positive function of the gap between the rate of profit and the rate of interest, influenced by lenders’ confidence as well as the money market. Both the prospective rate of profit and the rate of interest were in turn a function of investment and the stock of capital. [16]. In 1940, Kaldor [14] employed nonlinearities to account for economic fluctuations. The model was different from the previous ones since the author did not make it dynamic by utilising a time-lag structure on the investment function. Rather, Kaldor followed a different strategy: his model was dynamic in nature because ex ante Savings differed from Investment, which together with the multiplier-accelerator, yielded a model that produced endogenous cyclical fluctuations.  The Hicks Model for Greece Revisited 3 Harrod stressed the fact that oscillations were produced by the lag itself rather than by mechanisms endogenous to the system. In Harrod’s model dynamics were not caused by the lags, but rather depended on the internal structure of the model [11]. Harrod proposed a very simple model, based on the multiplier-accelerator mechanism and derived his well-known equation. He also came to show that for a static equilibrium, the process was unstable. Moreover, he showed that instability was, at least partly, responsible for generating cycles.  Harrod’s model was criticised for being simplistic by Baumol [3] and Alexander [1]. A fundamental contribution in this vein was Hicks’s A Contribution to the Theory of the Trade Cycle [13] where he defended a combination of both approaches. Hicks thought that fluctuations in investment led to a process taking place throughout many periods, because of lags in the multiplier. Of course, Hicks came to the standard solutions of a higher order linear system. Also, another crucial aspect of the model was the introduction of nonlinearities in the investment function.  3. The Hicks Model  According to Hicks the consumption function is a linear function of 1tY !                                                             1)1( !!" tt YsC                                (1) where 0 < 1-s< 1 is the so-called marginal propensity to consume, 1/ s is the multiplier and 1tY ! denotes output one period back. In the Samuelson business cycle model, investment was determined by the growth in output, through the so-called acceleration principle. Analytically, investment was assumed to be proportional to the change in output, or: )( 21 !! !" ttt YYuI  where tI denotes investment in time period t, 1tY ! and 2tY ! output one and two periods back, respectively, and u (>0) is the accelerator. However, Hicks thought that fluctuations in investment, caused by nonlinear changes in autonomous investment and the acceleration principle governing induced investment, led to an adjustment process taking place throughout many periods. In this spirit, Hicks introduced autonomous expenditures which may be growing exponentially. Investment has two components: autonomous investment growing at a constant rate g:                tt gAA )1(0 #"                                where 0$  is the autonomous investment. Of course, there is also the induced part of the investment which responds to changes in output. Thus:  )()1( 210 !! !##" tttt YYugAI                    (2) Hicks modeled the growth process of a closed economy without government sector, within the Keynesian framework. In this context:                                                          ttt ICY #"                                         (3)          Michaelides et al.  In what follows, we introduce a novel approach for the econometric estimation of Hicks’s nonlinear model and apply it for the Greek economy in the time period 1960-2007 using data from the AMECO database.  4. Econometric Estimation  The estimation of the consumption function (1) is straightforward using the 2 Stages Least Squares (2SLS) method. This method is typically used for the estimation of multiplier–accelerator models [4].  For the estimation of the consumption function we use 2SLS:  ttt vYsCC #!#" !10 )1(                               (4) where 0C  is the intercept of the regression, and tv is the random error term, independent and identically distributed (i.i.d.).  In what follows, the proposed methodology estimates the Hicks model, based on nonlinear investments. It is the case that the LS estimation principle applies for deriving estimators, ones that are called Nonlinear Least Squares (NLS). Unlike Ordinary Least Squares (O.L.S.) estimators, NLS estimators cannot be obtained analytically. However, the minimization of the Sum of Squared Residuals (SSR) is a well-defined optimization problem that can be solved numerically by iterating on a solution:    Step 1: Let g %&'('(& ,],,[ : ')(  be drawn from a uniform distribution.  Step 2: Estimate 0$ >0 and u>0 in the following equation using O.L.S.:                                ttttt eYYugA!I #!###" !! )()1( 2101             (5) where 1!  is the intercept of the estimation and te is the random error (i.i.d.).  Step 3: Compute the Sum of Squared Residuals SSR(i) for the equation.  Step 4: Repeat for 1,..,i I" and select the value g that yields the minimum SSR(i), subject to 0$ >0 and u>0 being statistically significant. Step 5: Given the value of g estimated in the previous step, use 2SLS to estimate the linearized equation (5) and keep the values of the parameters. Given that g is the economy’s growth rate in autonomous investment it should, normally, be positive over time and range between 0% and 15%. Also, other criteria for model selection include the SIC (Schwartz Information Criterion) and the AIC (Akaike Information Criterion).    The Hicks Model for Greece Revisited 5 5. Empirical Analysis: The Greek Economy (1960-2007)5.1 Estimation Results The estimation of equation (4) by means of 2SLS yields:  14.46 0.83t t tC Y v!" ! # #                                (-2.80) (49.79)  2R =0.98, S.S.E.=3.99 The equation (5) by means of the procedure developed above, yields:  1 211.54 2.69(1 0.05) 0.49( )tt t t tI Y Y e! !" # # # ! #                          (7.45)    (8.37)                      (1.99) 2R =0.68, S.S.E.=5.20 Values in parentheses are t-statistics which imply that all estimated parameters are statistically significant. It should be stressed that for g =0.05 the estimation of equation (5) yields the minimum value of SSR, under which 0$  and u are statistically significant. Thus s=0.17, 0C = -4.46, u=0.49, 0A =2.69, 1A =11.54 and gm #"1 =1.05.  5.2 Solutions and Stability By substituting equations (4) and (5) into (3) and rearranging we get                      1 2 0 1 0(1 ) (1 )tt t tY s u Y uY A g A C! !! ! # # " # # #             (6) Also, 1*  and 2* are the roots of the characteristic equation                   + ,uusus 4)1(121, 221 !#!-#!"**                        (7) The complete solution for (6) is:                                            )()()( tYtYtY ec #"                                        (8) where )(tYc is the complementarity function and )(tYe is the particular integral. We can show that the particular integral is equal to                   20 1 02( )( ) ( )(1 )te pA m m A Ct Y tm s u m u s#. " " #! ! ! #              (9) So, the solution which clearly implies a “moving equilibrium” output is                                    ( ) 14.36 (1.05) 41.65teY t " / #                             (10) As is well known, the solution depends on the discriminant uus 4)1( 2 !#!"0 . Given that in our case the discriminant takes a negative value ( 0)0 ), the complementarity function is                                    1 2( ) ( cos sin )tc t r B t B t1 1. " #                   (11) where:                                            2 2r a b" #                                                  (12)                                         1tan ( / )b a1 !"                                               (13)          Michaelides et al.                                            1,2 bi* (" -                                                    (14) Substitution in (7), given that 0)0 , yields  1,2 0.66 0.23i* " -                                    Which implies 0.7r "  and 0.341 " . By substituting these values into equation (11) we get          1 2( ) 0.7 ( cos 0.34 sin 0.34 )tc t B t B t. " #                    So 1 2( ) 0.7 ( cos0.34 sin0.34 ) 14.36 (1.05) 41.65t tt B t B t. " #! # / # . Finally, given the two initial conditions (i.e. actual values for !(0) and Y(1)) we get the values for the arbitrary constants 1 31 .00B " !  and 2 33 .5 7B " ! . Conslusively  ( ) 0.7 ( 31cos0.34 33.57sin0.34 ) 14.36 (1.05) 41.65t tt t t. " ! #! # / # .                  This is the analytical solution for ( )Y t , i.e. Gross Domestic Product (GDP) and by substituting the values of t we get the estimated values of ( )Y t . The estimated GDP is illustrated in the same plot with the real GDP values and the calculated correlation coefficient ( 0.99correlationr " ) implies an ideal fit.      05010015020060 65 70 75 80 85 90 95 00 05 10GDPREAL GDPESTIMATEDFigure 1. Estimated GDP vs real GDP for the Greek economy (1960-2007)  Finally, stability depends on the discriminant uus 4)1( 2 !#!"0 . Since  0)0 , tr  (with 1r ) ) will dampen the fluctuations caused by the 1 2( cos 0.34 sin 0.34 )B t B t# . In another formulation, since 1)1(2 ))! us  or 0.35<0.49<1, the solution is periodically convergent, i.e. stable [24]. The Hicks Model for Greece Revisited 7  6. Conclusions  In this paper, after discussing the transition from early linear modelling to the first nonlinear models in economics, we proposed a novel approach for estimating, in a Keynesian system of equations, the non-linear Trade (Business) Cycle model developed by Hicks. The proposed methodology yields very satisfactory results when fitted to GDP data for the Greek economy over the time period 1960-2007. We believe that the results of this paper suggest that the modified Hicks model with its generality, conformity with theory and simplicity of structure is an appropriate vehicle for testing, expanding and improving conventional business cycle theory in empirical applications. Meanwhile, preliminary results of the proposed approach from other countries are extremely encouraging. Clearly, future and more extended research on the subject would be of great interest.   References [1] Alexander, S. Mr. Harrod’s dynamic model. Economic Journal 60 (240), 724-39, 1950. [2] Baumol, W., Benhabib, J. Chaos: Significance, Mechanism, and Economic Implications. Journal of Economic Perspectives, Vol. 3, No. 1, 77-105, 1989. [3] Baumol, W.: Notes on some dynamic models. Economic Journal 58 (232), 506-21, 1948. [4] Chow, G. China’s Economic Transformation, Blackwell Publishing, 2007. [5] Colander, D., Pluralism and Heterodox Economics: Suggestions for an “Inside the Mainstream” Heterodoxy. 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Early Nonlinear Modelling in Economic Analysis: The Hicks Model for Greece Revisited

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