This paper presents a modified nonlinear Hicks model of the cycle and a method for deriving estimators based on Nonlinear Least Squares and other relevant criteria. 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. An empirical application for the US economy (1960-2008) demonstrates the almost ideal performance of the modified model and the proposed method

Arapis, Gerasimos

Belegri-Roboli, Athena

Michaelides, Panayotis G.

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Estimating a modified nonlinear Hicks
model: Evidence from the US economy
(1960-2008)
Panayotis G. Michaelides and Athena Belegri-Roboli and
Gerasimos Arapis
National Technical University of Athens
2010
Online at https://mpra.ub.uni-muenchen.de/74461/
MPRA Paper No. 74461, posted 19 October 2016 21:35 UTC
Estimating a modified nonlinear Hicks model: 
Evidence from the US economy (1960-2008)  
 
Abstract: This paper presents a modified nonlinear Hicks model of the cycle and a 
method for deriving estimators based on Nonlinear Least Squares and other relevant 
criteria. 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. An empirical 
application for the US economy (1960-2008) demonstrates the almost ideal 
performance of the modified model and the proposed method.   
 
 
Keywords: Nonlinear, Hicks model, dynamics, US economy.   
  
1. Introduction 
Modern economics is often regarded as a mathematical 
science and draws heavily on the tools of nonlinear mathematics 
(Debreu, 1987). These tools are considered as promising ways 
towards overcoming the problems associated with the traditional 
approaches and have developed through different strands of 
thought and across diverse disciplines (Fontana, 2006; Rosser, 
1999; Samuelson, 1939).  
A seminal contribution in the economics literature on 
Business Cycles was Hicks (1950) where the author developed 
his own endogenous nonlinear model of the cycle. In this paper, 
we introduce a modified nonlinear Hicks model and we propose 
a method for its empirical estimation. Next, we apply it to the 
US economy to test its validity, using real world data for the 
time period 1960-2008.   
The paper is structured as follows: section 2 presents the 
modified Hicks model; section 3 sets out our approach for 
estimating the model; the empirical application of the model for 
the US economy is illustrated in section 4; section 5 concludes 
the paper.  
 
2. The Model  
According to Hicks, Consumption ( tC ) 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 so-called multiplier and 1tY − denotes output 
one period back.  
Meanwhile, Hicks thought that fluctuations in investment 
are caused by (i) nonlinear changes in autonomous investment 
and (ii) the acceleration principle governing induced investment. 
Analytically, Hicks though that autonomous investment 
expenditure may be growing exponentially at a constant rate g: 
               tt gAA )1(0 +=                                
where 0Α >0  is the autonomous investment.  
Also, there is the induced part of investment which 
responds to changes in output. This part of investment is 
assumed to be proportional to the changes in output, or: 
)( 21 −− −= ttt YYuIΝ  
where tIΝ denotes induced investment in time period t, 1tY − and 
2tY − output one and two periods back, respectively, and u (>0) is 
the so-called accelerator. Thus:  
                               )()1( 210 −− −++= tttt YYugAI                    (2) 
where tI denotes total investment in time period t.  
Hicks modeled the growth process of a closed economy, 
within a Keynesian framework. In this context:  
                                        ttt ICY +=                                         (3) 
Consider now a consumption function (1) with constant 
term:  
       ttt vYsCC +−+= −10 )1(                               (4) 
where 0C >0  is the constant term and tv is the random error 
term, independent and identically distributed (i.i.d.). This 
formulation is consistent with economic theory since 0C  
expresses the so-called autonomous consumption.  
 
Also, consider the investment function (2) with constant 
term:  
                ttt
t
t eYYugAΑI +−+++= −− )()1( 2101                   (5) 
where 1Α >0 is the constant term and te  the random error (i.i.d.). 
Τhe constant term expresses the part of investment which does 
not depend on output and does not grow exogenously. 
Meanwhile, both constant terms introduced to the model, namely 
0C and 1Α , are convenient for the econometric implementation 
expressing the intercepts of the regressions.   
By substituting equations (4) and (5) into (3) and 
rearranging we get the following second order difference 
equation  
             
1 2 0 1 0(1 ) (1 )
t
t t tY s u Y u Y A g A C− −− − + + = + + +             (6) 
Also, 1λ  and 2λ are the roots of its characteristic equation  
          [ ]uusus 4)1(1
2
1, 221 −+−±+−=λλ                         (7) 
The complete solution for (6) is 
                            )()()( tYtYtY ec +=                                        (8) 
where )(tYc  is the complementary function and )(tYe  is the 
particular integral. It is easy to show that the particular integral 
is equal to the following “moving equilibrium” expression 
          
2
0 1 0
2
( )( ) ( )
(1 )
t
e p
A m m A Ct Y t
m s u m u s
+Υ = = +− − − +               (9) 
The solution depends on the discriminant 
uus 4)1( 2 −+−=Δ . Analytically:  
(a) When 0Δ > , i.e. 1λ  and 2λ are both real and unequal, the 
complementary function takes the form: 
                          ttc KKt 2211)( λ+λ=Υ                                    (10) 
(b) When 0Δ = , i.e. 1λ  and 2λ  are both real and equal 
( 1λ = 2λ = λ  where λ is real), the complementary function takes 
the form    
                                   1 2( ) ( ) tY t K K t λ= +                               (11) 
(c) When 0<Δ , i.e. 1λ  and 2λ  are both complex ( 1λ = λ and 
2λ = λ , where λ is complex) the complementary function takes 
the form 
                         1 2( ) ( cos sin )tc t r B t B tϑ ϑΥ = +                    (12) 
where  
                          1,2 biλ α= ±                                                    (13) 
 2 2r a b= +                                                  (14) 
                          1tan ( / )b aϑ −=                                               (15) 
Finally, stability conditions are 1iλ <  for i=1,2 in the real root 
case ( 0Δ >  or 0Δ = ) and 1r <  in the complex root case 
( 0<Δ ), impying that the solution is periodically convergent.  
 
3. Econometric Estimation  
It is clear that the proposed model should be confronted 
with real world data, not only to assess the model’s ability to 
replicate the behavior of observed output but also in order to 
allow formal inference for parameters and functions of interest.  
The estimation of the consumption function (4) is 
straightforward using 2 Stages Least Squares (2SLS) relevant for 
the estimation of multiplier–accelerator systems given the 
structure of the problem (Chow, 2007).  
In what follows, the proposed method estimates the 
modified nonlinear investment function. It is the case that the 
Least Squares (LS) estimation principle applies as a method for 
deriving estimators, i.e. the nonlinear least squares (NLS). 
Unlike Ordinary Least Squares (OLS) and 2SLS, NLS estimators 
cannot be obtained analytically as closed form expressions.  
 
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. The algorithm 
begins with some initial guess for the coefficient (starting value), 
and then proceeds by a series of steps. We provide the routine 
with starting values that are good guesses of the coefficient, 
given that we have very good knowledge of the nature of the 
economic problem being studied and this suggests plausible 
coefficient values. Consider the following procedure:  
 
Step 1: Let g )(i ℜ∈βαβα∈ ,],,[ , 1,..,i I=  be drawn from a 
uniform distribution.  
Step 2: For 1=i  and g = g )(i = g )1(  estimate 1Α , 0Α  and u in the 
following (intrinsically) linear equation using 2SLS: 
                       ttt
t
t eYYugAΑI +−+++= −− )()1( 2101             (5)  
Step 3: Compute the Sum of Squared Residuals SSR )(i  for 
equation (5), for 1=i .  
Step 4: Repeat for Ii ,...,2= and select the value g of g )(i  that 
yields the minimum SSR )(i , subject to 1Α , 0Α  and u being 
statistically significant. 
Step 5: Given the value of g estimated in the previous step, keep 
the estimates of 1Α , 0Α  and u.  
Since g expresses the economy’s growth rate in 
(autonomous) investment it should, normally, be positive in the 
long run (i.e. over several decades) and range between 0% and 
20%. This relatively small range of plausible coefficient values 
makes it possible to iterate on each value -with reasonable 
accuracy- and to reach, thus, a global minimum in this range.  
It should be made clear that once the parameter g takes 
on a certain value (even if not the ‘optimal’ value g ), the 
investment function becomes intrinsically linear and its 
estimation is straightforward employing 2SLS.  
 
4. Empirical Analysis: The US Economy (1960-2008)  
 
4.1 Model Estimation  
In order to apply the model to explain US output data, 
we define Consumption ( tC ) to include both private and 
government consumption (Chow, 2007, p. 113). The data for the 
US economy come from AMECO (at 2000 constant prices) and 
cover the time period 1960-2008.  
The estimation of equation (4) employing 2SLS yields:  
1278.36 0.82t t tC Y v−= + +  
                                  (7.64)   (156.63) 2R =0.99, SSE =97.69 
Next, following the procedure described above we find that for 
g =0.04, the estimation of equation (5) yields the minimum 
value of SSR (see Fig. 1).  
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
g
SS
R
 
Fig. 1. Calculated SSR for the values of g  
 
The model could be selected using the 2R  goodness-of-
fit criterion, according to which one should select the value of 
parameter g that maximizes R2.Other criteria that could be used 
for the selection of parameter g include the minimization of SIC 
(Schwartz, 1978) and AIC (Akaike, 1974).  
 
Figs. 2-4 illustrate the value of g that optimizes the 
aforementioned criteria.  
0.750
0.800
0.850
0.900
0.950
1.000
0.00 0.05 0.10 0.15 0.20
g
R
sq
ua
re
d
Rsquared
 
Fig. 2. Calculated 2R  for the values of g  
12.00
12.50
13.00
13.50
14.00
14.50
0.00 0.05 0.10 0.15 0.20
g
A
IC AIC
 
Fig. 3. Calculated  AIC for the values of g  
12.00
12.50
13.00
13.50
14.00
14.50
0.00 0.05 0.10 0.15 0.20
g
SI
C SIC
 
Fig. 4. Calculated  SIC for the values of g  
 
Thus, for g =0.04 all the aforementioned criteria (SSR, 2R , 
AIC, SIC) are optimized. So, for g =0.04, the estimation of the 
intrinsically linear equation (5) using 2SLS yields:  
      ttt
t
t eYYΙ +−+++= −− )(33.0)04.01(57.31686.84 21  
               (2.49)  (27.91)                   (2.67) 2R =0.97, SSE=102.33 
The values in parentheses are t-statistics which imply that 
all estimated parameters in both equations are statistically 
significant. Also, we note that the proposed model provides an 
excellent fit to the data, as expressed through 2R  for both 
equations. Finally, our findings are consistent with economic 
theory given that 0C >0, 1Α >0, 0Α >0, u>0 and 0 <1-s<1. 
        In Fig. 5 we illustrate the estimated values (Iestimated) of 
Investment along with its real values (Ireal). 
0
500
1000
1500
2000
2500
1960 1970 1980 1990 2000 2010
Year
I(t
) Ireal
Iestimated
 
Fig. 5. Real Investment vs estimated Investment for the US economy (1960-2008) 
 
The calculated correlation coefficient ( 98.0=ncorrelatior ) is 
another indication of the almost ideal fit of the model.  
 
4.2 Solutions and Stability 
Substitution in (9) yields the particular integral  
                          10.1991)04.1(63.1568)( += te tΥ                (10) 
Substitution in (7), given that 0Δ > , yields  
1λ = 0.60 and 2λ = 0.54 
By substituting these values into equation (10) we get  
        1 2( ) 0.60 0.54t tc t K KΥ = +                    
and 
10.1991)04.1(63.156854.060.0)( 21 +++= ttt KKtY  
Finally, given the two initial conditions (i.e. actual values for 
Υ(0) and Y(1)) we get the values for the arbitrary constants 
1 7951.57K = −  and 2 6883.61K = . 
 
Conclusively 
( ) 7951.57(0.60) 6883.61(0.54) 1568.63(1.04) 1991.10t t tY t =− + + + . 
This is the analytical solution for ( )Y t  and by substituting the 
values of t we get the estimated values (Yestimated) of ( )Y t  
which are illustrated in Fig. 6 along with the real output values 
(Yreal). 
0
2000
4000
6000
8000
10000
12000
14000
1960 1970 1980 1990 2000 2010
Year
Y(
t) Yreal
Yestimated
 
Fig. 6. Real GDP vs estimated GDP for the US economy (1960-2008) 
 
The calculated correlation coefficient ( 0.99correlationr = ) is 
another indication of the almost ideal fit of the model. Finally, 
since 1 1λ <  and 2 1λ <  the solution is periodically convergent, 
i.e. stable. 
 
5. Conclusions  
In this paper, we proposed a modified non-linear Hicks model of 
the Cycle and a relevant method for estimating it, in a 
Keynesian system of equations. The proposed method yielded 
very satisfactory results fitted to data for the US economy over 
the time period 1960-2008. The results of this paper suggest that 
the proposed 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. Also, preliminary results from other 
countries are extremely encouraging. Clearly, future research 
would be of great interest.  
References 
Akaike, H., 1974. A new look at the statistical model 
identification, IEEE Trans. Automatic Control, Vol. 19, 
No. 6, 716–723.  
Chow, G., 2007. China’s Economic Transformation, Blackwell 
Publishing. 
Debreu, G., 1987. Mathematical Economics. The New Palgrave: 
A Dictionary of Economics, Vol. 3, 401-03,  
Fontana, M., 2006. Computer Simulations, Mathematics, and 
Economics. Rivista Internazionale di Scienze Economiche 
e Commerciali, 53(1), 96-124. 
Hicks, J., 1950. A Contribution to the theory of the Trade Cycle. 
Clarendon Press, Oxford. 
Rosser, B. J. Jr., 1999. On the Complexities of Complex 
Economic Dynamics. Journal of Economic Perspectives, 13 
(4), 169-192.  
Samuelson, P., 1939. Interactions between the multiplier 
analysis and the principle of acceleration. Review of 
Economics and Statistics 21 (2), 75-78. 
Schwartz, G., 1978. Estimating the dimension of a model, 
Annals of Statistics 6 (2), 461-464. 


Estimating a modified nonlinear Hicks model: Evidence from the US economy (1960-2008)

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveEstimating a modified nonlinear Hicksmodel: Evidence from the US economy(1960-2008)Michaelides, Panayotis G. and Belegri-Roboli, Athena andArapis, GerasimosNational Technical University of Athens2010Online at https://mpra.ub.uni-muenchen.de/74461/MPRA Paper No. 74461, posted 19 Oct 2016 21:35 UTCEstimating a modified nonlinear Hicks model: Evidence from the US economy (1960-2008)   Abstract: This paper presents a modified nonlinear Hicks model of the cycle and a method for deriving estimators based on Nonlinear Least Squares and other relevant criteria. 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. An empirical application for the US economy (1960-2008) demonstrates the almost ideal performance of the modified model and the proposed method.     Keywords: Nonlinear, Hicks model, dynamics, US economy.     1. Introduction Modern economics is often regarded as a mathematical science and draws heavily on the tools of nonlinear mathematics (Debreu, 1987). These tools are considered as promising ways towards overcoming the problems associated with the traditional approaches and have developed through different strands of thought and across diverse disciplines (Fontana, 2006; Rosser, 1999; Samuelson, 1939).  A seminal contribution in the economics literature on Business Cycles was Hicks (1950) where the author developed his own endogenous nonlinear model of the cycle. In this paper, we introduce a modified nonlinear Hicks model and we propose a method for its empirical estimation. Next, we apply it to the US economy to test its validity, using real world data for the time period 1960-2008.   The paper is structured as follows: section 2 presents the modified Hicks model; section 3 sets out our approach for estimating the model; the empirical application of the model for the US economy is illustrated in section 4; section 5 concludes the paper.   2. The Model  According to Hicks, Consumption ( tC ) 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 so-called multiplier and 1tY − denotes output one period back.  Meanwhile, Hicks thought that fluctuations in investment are caused by (i) nonlinear changes in autonomous investment and (ii) the acceleration principle governing induced investment. Analytically, Hicks though that autonomous investment expenditure may be growing exponentially at a constant rate g:                tt gAA )1(0 +=                                where 0Α >0  is the autonomous investment.  Also, there is the induced part of investment which responds to changes in output. This part of investment is assumed to be proportional to the changes in output, or: )( 21 −− −= ttt YYuIΝ  where tIΝ denotes induced investment in time period t, 1tY − and 2tY − output one and two periods back, respectively, and u (>0) is the so-called accelerator. Thus:                                 )()1( 210 −− −++= tttt YYugAI                    (2) where tI denotes total investment in time period t.  Hicks modeled the growth process of a closed economy, within a Keynesian framework. In this context:                                          ttt ICY +=                                         (3) Consider now a consumption function (1) with constant term:         ttt vYsCC +−+= −10 )1(                               (4) where 0C >0  is the constant term and tv is the random error term, independent and identically distributed (i.i.d.). This formulation is consistent with economic theory since 0C  expresses the so-called autonomous consumption.   Also, consider the investment function (2) with constant term:                  ttttt eYYugAΑI +−+++= −− )()1( 2101                   (5) where 1Α >0 is the constant term and te  the random error (i.i.d.). Τhe constant term expresses the part of investment which does not depend on output and does not grow exogenously. Meanwhile, both constant terms introduced to the model, namely 0C and 1Α , are convenient for the econometric implementation expressing the intercepts of the regressions.   By substituting equations (4) and (5) into (3) and rearranging we get the following second order difference equation               1 2 0 1 0(1 ) (1 )tt t tY s u Y u Y A g A C− −− − + + = + + +             (6) Also, 1λ  and 2λ are the roots of its characteristic equation            [ ]uusus 4)1(121, 221 −+−±+−=λλ                         (7) The complete solution for (6) is                             )()()( tYtYtY ec +=                                        (8) where )(tYc  is the complementary function and )(tYe  is the particular integral. It is easy to show that the particular integral is equal to the following “moving equilibrium” expression           20 1 02( )( ) ( )(1 )te pA m m A Ct Y tm s u m u s+Υ = = +− − − +              (9) The solution depends on the discriminant uus 4)1( 2 −+−=Δ . Analytically:  (a) When 0Δ > , i.e. 1λ  and 2λ are both real and unequal, the complementary function takes the form:                           ttc KKt 2211)( λ+λ=Υ                                    (10) (b) When 0Δ = , i.e. 1λ  and 2λ  are both real and equal ( 1λ = 2λ = λ  where λ is real), the complementary function takes the form                                       1 2( ) ( )tY t K K t λ= +                               (11) (c) When 0<Δ , i.e. 1λ  and 2λ  are both complex ( 1λ = λ and 2λ = λ , where λ is complex) the complementary function takes the form                          1 2( ) ( cos sin )tc t r B t B tϑ ϑΥ = +                    (12) where                            1,2 biλ α= ±                                                    (13)  2 2r a b= +                                                  (14)                           1tan ( / )b aϑ −=                                               (15) Finally, stability conditions are 1iλ <  for i=1,2 in the real root case ( 0Δ >  or 0Δ = ) and 1r <  in the complex root case ( 0<Δ ), impying that the solution is periodically convergent.   3. Econometric Estimation  It is clear that the proposed model should be confronted with real world data, not only to assess the model’s ability to replicate the behavior of observed output but also in order to allow formal inference for parameters and functions of interest.  The estimation of the consumption function (4) is straightforward using 2 Stages Least Squares (2SLS) relevant for the estimation of multiplier–accelerator systems given the structure of the problem (Chow, 2007).  In what follows, the proposed method estimates the modified nonlinear investment function. It is the case that the Least Squares (LS) estimation principle applies as a method for deriving estimators, i.e. the nonlinear least squares (NLS). Unlike Ordinary Least Squares (OLS) and 2SLS, NLS estimators cannot be obtained analytically as closed form expressions.   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. The algorithm begins with some initial guess for the coefficient (starting value), and then proceeds by a series of steps. We provide the routine with starting values that are good guesses of the coefficient, given that we have very good knowledge of the nature of the economic problem being studied and this suggests plausible coefficient values. Consider the following procedure:   Step 1: Let g )(i ℜ∈βαβα∈ ,],,[ , 1,..,i I=  be drawn from a uniform distribution.  Step 2: For 1=i  and g = g )(i = g )1(  estimate 1Α , 0Α  and u in the following (intrinsically) linear equation using 2SLS:                        ttttt eYYugAΑI +−+++= −− )()1( 2101             (5)  Step 3: Compute the Sum of Squared Residuals SSR )(i  for equation (5), for 1=i .  Step 4: Repeat for Ii ,...,2= and select the value g of g )(i  that yields the minimum SSR )(i , subject to 1Α , 0Α  and u being statistically significant. Step 5: Given the value of g estimated in the previous step, keep the estimates of 1Α , 0Α  and u.  Since g expresses the economy’s growth rate in (autonomous) investment it should, normally, be positive in the long run (i.e. over several decades) and range between 0% and 20%. This relatively small range of plausible coefficient values makes it possible to iterate on each value -with reasonable accuracy- and to reach, thus, a global minimum in this range.  It should be made clear that once the parameter g takes on a certain value (even if not the ‘optimal’ value g ), the investment function becomes intrinsically linear and its estimation is straightforward employing 2SLS.   4. Empirical Analysis: The US Economy (1960-2008)   4.1 Model Estimation  In order to apply the model to explain US output data, we define Consumption ( tC ) to include both private and government consumption (Chow, 2007, p. 113). The data for the US economy come from AMECO (at 2000 constant prices) and cover the time period 1960-2008.  The estimation of equation (4) employing 2SLS yields:  1278.36 0.82t t tC Y v−= + +                                    (7.64)   (156.63) 2R =0.99, SSE =97.69 Next, following the procedure described above we find that for g =0.04, the estimation of equation (5) yields the minimum value of SSR (see Fig. 1).  05000001000000150000020000002500000300000035000000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2gSSR Fig. 1. Calculated SSR for the values of g   The model could be selected using the 2R  goodness-of-fit criterion, according to which one should select the value of parameter g that maximizes R2.Other criteria that could be used for the selection of parameter g include the minimization of SIC (Schwartz, 1978) and AIC (Akaike, 1974).   Figs. 2-4 illustrate the value of g that optimizes the aforementioned criteria.  0.7500.8000.8500.9000.9501.0000.00 0.05 0.10 0.15 0.20gRsquaredRsquared Fig. 2. Calculated 2R  for the values of g  12.0012.5013.0013.5014.0014.500.00 0.05 0.10 0.15 0.20gAIC AIC Fig. 3. Calculated  AIC for the values of g  12.0012.5013.0013.5014.0014.500.00 0.05 0.10 0.15 0.20gSIC SIC Fig. 4. Calculated  SIC for the values of g   Thus, for g =0.04 all the aforementioned criteria (SSR, 2R , AIC, SIC) are optimized. So, for g =0.04, the estimation of the intrinsically linear equation (5) using 2SLS yields:        ttttt eYYΙ +−+++= −− )(33.0)04.01(57.31686.84 21                 (2.49)  (27.91)                   (2.67) 2R =0.97, SSE=102.33 The values in parentheses are t-statistics which imply that all estimated parameters in both equations are statistically significant. Also, we note that the proposed model provides an excellent fit to the data, as expressed through 2R  for both equations. Finally, our findings are consistent with economic theory given that 0C >0, 1Α >0, 0Α >0, u>0 and 0 <1-s<1.         In Fig. 5 we illustrate the estimated values (Iestimated) of Investment along with its real values (Ireal). 050010001500200025001960 1970 1980 1990 2000 2010YearI(t)IrealIestimated Fig. 5. Real Investment vs estimated Investment for the US economy (1960-2008)  The calculated correlation coefficient ( 98.0=ncorrelatior ) is another indication of the almost ideal fit of the model.   4.2 Solutions and Stability Substitution in (9) yields the particular integral                            10.1991)04.1(63.1568)( += te tΥ                (10) Substitution in (7), given that 0Δ > , yields  1λ = 0.60 and 2λ = 0.54 By substituting these values into equation (10) we get          1 2( ) 0.60 0.54t tc t K KΥ = +                    and 10.1991)04.1(63.156854.060.0)( 21 +++=ttt KKtY  Finally, given the two initial conditions (i.e. actual values for Υ(0) and Y(1)) we get the values for the arbitrary constants 1 7951.57K = −  and 2 6883.61K = .  Conclusively ( ) 7951.57(0.60) 6883.61(0.54) 1568.63(1.04) 1991.10t t tY t =− + + + . This is the analytical solution for ( )Y t  and by substituting the values of t we get the estimated values (Yestimated) of ( )Y t  which are illustrated in Fig. 6 along with the real output values (Yreal). 020004000600080001000012000140001960 1970 1980 1990 2000 2010YearY(t) YrealYestimated Fig. 6. Real GDP vs estimated GDP for the US economy (1960-2008)  The calculated correlation coefficient ( 0.99correlationr = ) is another indication of the almost ideal fit of the model. Finally, since 1 1λ <  and 2 1λ <  the solution is periodically convergent, i.e. stable.  5. Conclusions  In this paper, we proposed a modified non-linear Hicks model of the Cycle and a relevant method for estimating it, in a Keynesian system of equations. The proposed method yielded very satisfactory results fitted to data for the US economy over the time period 1960-2008. The results of this paper suggest that the proposed 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. Also, preliminary results from other countries are extremely encouraging. Clearly, future research would be of great interest.  References Akaike, H., 1974. A new look at the statistical model identification, IEEE Trans. Automatic Control, Vol. 19, No. 6, 716–723.  Chow, G., 2007. China’s Economic Transformation, Blackwell Publishing. Debreu, G., 1987. Mathematical Economics. The New Palgrave: A Dictionary of Economics, Vol. 3, 401-03,  Fontana, M., 2006. Computer Simulations, Mathematics, and Economics. Rivista Internazionale di Scienze Economiche e Commerciali, 53(1), 96-124. Hicks, J., 1950. A Contribution to the theory of the Trade Cycle. Clarendon Press, Oxford. Rosser, B. J. Jr., 1999. On the Complexities of Complex Economic Dynamics. Journal of Economic Perspectives, 13 (4), 169-192.  Samuelson, P., 1939. Interactions between the multiplier analysis and the principle of acceleration. Review of Economics and Statistics 21 (2), 75-78. Schwartz, G., 1978. Estimating the dimension of a model, Annals of Statistics 6 (2), 461-464. 

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Estimating a modified nonlinear Hicks model: Evidence from the US economy (1960-2008)

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