This paper presents both an extended nonlinear Hicks model of the cycle for a semi-open economy and a method for deriving estimators based on Nonlinear Least Squares as a Numerical Optimization Problem. 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 validity of the model and its almost ideal fit to real world data

Arapis, G

Belegri-Roboli, A

Michaelides, P

Michaelides, Panayotis G.

Belegri-Roboli, Athena

Arapis, Gerasimos

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
An Extended Nonlinear Hicks Model of
the Cycle for a Semi-Open Economy:
Application to the USA (1960-2008)
Panayotis G. Michaelides and Athena Belegri-Roboli and
Gerasimos Arapis
National Technical University of Athens
2009
Online at https://mpra.ub.uni-muenchen.de/74451/
MPRA Paper No. 74451, posted 16 October 2016 05:47 UTC
An Extended Nonlinear Hicks Model of the Cycle 
for a Semi-Open Economy: Application to the 
USA (1960-2008) 
Panayotis Michaelides, Athena Belegri-Roboli and Gerasimos Arapis 
Department of Applied Mathematics and Physics 
National Technical University of Athens  
Athens, Greece 
pmichael@central.ntua.gr 
 
 
Abstract—This paper presents both an extended nonlinear 
Hicks model of the cycle for a semi-open economy and a method 
for deriving estimators based on Nonlinear Least Squares as a 
Numerical Optimization Problem. 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 validity of the model and 
its almost ideal fit to real world data.   
Keywords—Nonlinear; Hicks model; dynamics; US economy  
I. INTRODUCTION  
Modern economics is often regarded as a mathematical 
science and draws heavily on the tools of nonlinear 
mathematics [1]. 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 ([2]-[5]).  
A seminal contribution in the economics literature on 
Business Cycles (BC) was Hicks’s famous A Contribution to 
the theory of the Trade Cycle (1950), where the author 
developed his own endogenous nonlinear model of the cycle. 
Although, there is a plethora of models on BC, the proposed 
model with its generality, conformity with theory and 
simplicity of structure is an appropriate vehicle for testing, 
expanding and improving conventional BC theory in empirical 
applications. However, so far, it has found no applications in 
the relevant literature because of the lack of an appropriate 
method for its empirical estimation. In other words, first, we 
propose a modified nonlinear Hicks model for a semi-open 
economy and, second, we propose a method for its empirical 
application. In this context, we apply it to the USA to test its 
validity, using real world data for the time period 1960-2008.   
 
II. 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 worked in a Keynesian framework. In this context:  
                                ttt ICY += .                                  (3) 
Consider now a consumption function 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 following investment function:                                 
       ttt
t
t eYYugAΑI +−+++= −− )()1( 2101        (5) 
where 1Α >0 is the constant term and te  the random error 
(i.i.d.). Once again, the constant term expresses that part of 
investment which does not depend on output and does not 
grow exogenously. Both constant terms introduced, namely 
0C and 1Α , are convenient for the econometric estimation.    
     By substituting (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 uY A g A C− −− − + + = + + + . (6)      
Also, 1λ  and 2λ are the roots of its characteristic equation  [ ]uusus 4)1(15.0, 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 2
0 1 0( ) ( ) [( ) ]/[ (1 ) ] [( )/ ]
t
e pY t Y t Am m m s u m u A C s= = − − − + + + . (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 )
t
c 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 ), implying that the solution is periodically 
convergent, i.e. stable.  
III. 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 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 [6].  
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 very good 
guesses of the coefficient, given that we have 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.  
It should be made clear that once the parameter g takes on 
a certain value (even if not the ‘optimal’ value of g ), the 
investment function becomes intrinsically linear and its 
estimation is straightforward employing 2SLS.  
IV. EMPIRICAL ANALYSIS 
In order to apply the model to explain US output data, we 
define Consumption ( tC ) to include private, government as 
well as consumption from abroad. The data for the US 
economy come from AMECO and cover the time period 
1960-2008, at constant prices.  
The estimation of (4) employing 2SLS yields 
 
1430.71 0.76t tC Y −= +  
                                  (10.3)    (113.0)  
                            2R =0.996, SSE=102.02. 
      
Next, following the procedure described above we find that 
for g =0.04, the estimation of (5) yields the minimum value 
of SSR (see Fig. 1).  
0500000
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
 
Figure 1. Calculated SSR for the values of g  
The model could also 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 [7] and AIC [8].  
      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
 
Figure 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
 
Figure 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
 
Figure 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:      
1 272.32 322.39(1 0.04) 0.38( )
t
t t t tΙ Y Y e− −= + + + − +
               (1.99)    (2.35)                            (30.13)    
2R =0.96, SSE=104.03 
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 below we illustrate the estimated values 
(Iestimated) of Investment along with its real values (Ireal). 
        The calculated correlation coefficient ( 98.0=ncorrelatior ) 
is another indication of the almost ideal fit of the model.  
V. SOLUTIONS AND STABILITY 
         Substitution in (9) yields the particular integral  
            ( ) 1282.87(1.04) 2132.35teΥ t = +                
Substitution in (7), given that 0Δ < , yields  
1 0.571 0.228iλ = + and 2 0.571 0.228iλ = −  
Also, based on (14)-(15), equation (12) yields:  
        1 2( ) 0.61 ( cos 0.38 sin 0.38 )
t
c t B t B tΥ = +                    
and 
1 2( ) 0.61( cos0.38 sin0.38 ) 1282.87(1.04) 2132.35
t t
c t B t B tΥ = + + + .
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 1026.7341B = −  and 2 1906.65B = − . 
Conclusively, the analytical solution for ( )Y t is: 
 
( ) 0.61[ 1026.73cos(0.38 ) 1906.65sin(0.38 )] 1282.87(1.04) 2132.35t tY t t t= − − + +  
0
500
1000
1500
2000
2500
1960 1970 1980 1990 2000 2010
Year
I(t
) Ireal
Iestimated
 
Figure 5. Real VS estimated Investment for the US economy (1960-2008) 
 
02000
4000
6000
8000
10000
12000
14000
1960 1970 1980 1990 2000 2010
Year
Y(
t) Yreal
Yestimated
 
Figure 6. Real VS estimated GDP for the US economy (1960-2008) 
 
       By substituting the values of t we get the estimated values 
(Yestimated) of ( )Y t  which are illustrated in Fig. 6 above 
along with the real output values (Yreal).  
       The calculated correlation coefficient ( 0.99correlationr = ) 
is another indication of the almost ideal fit of the model. Also, 
since 1r < , given that 0.61<1, the solution of the 
complementary function is periodically convergent, i.e. stable. 
VI. CONCLUSIONS 
The proposed approach for a semi-open economy yielded 
very satisfactory results fitted to data for the US economy 
(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 BC theory and 
predictions. Clearly, future research would be of great interest.    
REFERENCES 
[1]    Debreu, G., “Mathmatical Economics,” in The New Palgrave: A 
Dictionary of Economics, Vol. 3, pp. 401-03, 1987. 
[2]    Colander, D., “Pluralism and Heterodox Economics: Suggestions for an 
‘Inside the Mainstream’ Heterrodoxy,”.Middlebury College Economics 
Discussion Paper No. 07-24, October, 2007.  
[3]    Fontana, M., “Computer Simulations, Mathematics, and Economics,”    
Rivista Internazionale di Scienze Economiche e Commerciali, 53 (1), pp. 
96-124, 2006. 
[4]    Rosser, B. J. Jr., “On the Complexities of Complex Economic 
Dynamics,” Journal of Economic Perpectives, 13 (4), pp. 169-192, 
1999. 
[5]    Samuelson, P., “Interactions between the multiplier analysis and the 
principle of acceleration,” Review of Economics and Statistics 21(2), pp. 
75-78, 1939. 
[6]    Chow, G., China’s Economic Tranformation, Blackwell Publising, 2007. 
[7]    Schwartz, G., “Estimating the dimension of a model,” Annals of 
Statistics 6 (2), pp. 461-464, 1978.  
[8]    Akaike, H., “A new look at the statistical model identification,” IEEE 
Trans Automatic Control, Vol. 19, No.6, pp. 716-723, 1974. 
 


An Extended Nonlinear Hicks Model of the Cycle for a Semi-Open Economy: Application to the USA (1960-2008)

DSpace at NTUA

An extended nonlinear Hicks model of the cycle for a semi-open economy: Application to the USA (1960-2008)

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveAn Extended Nonlinear Hicks Model ofthe Cycle for a Semi-Open Economy:Application to the USA (1960-2008)Michaelides, Panayotis G. and Belegri-Roboli, Athena andArapis, GerasimosNational Technical University of Athens2009Online at https://mpra.ub.uni-muenchen.de/74451/MPRA Paper No. 74451, posted 16 Oct 2016 05:47 UTCAn Extended Nonlinear Hicks Model of the Cycle for a Semi-Open Economy: Application to the USA (1960-2008) Panayotis Michaelides, Athena Belegri-Roboli and Gerasimos Arapis Department of Applied Mathematics and Physics National Technical University of Athens  Athens, Greece pmichael@central.ntua.gr   Abstract—This paper presents both an extended nonlinear Hicks model of the cycle for a semi-open economy and a method for deriving estimators based on Nonlinear Least Squares as a Numerical Optimization Problem. 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 validity of the model and its almost ideal fit to real world data.   Keywords—Nonlinear; Hicks model; dynamics; US economy  I. INTRODUCTION  Modern economics is often regarded as a mathematical science and draws heavily on the tools of nonlinear mathematics [1]. 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 ([2]-[5]).  A seminal contribution in the economics literature on Business Cycles (BC) was Hicks’s famous A Contribution to the theory of the Trade Cycle (1950), where the author developed his own endogenous nonlinear model of the cycle. Although, there is a plethora of models on BC, the proposed model with its generality, conformity with theory and simplicity of structure is an appropriate vehicle for testing, expanding and improving conventional BC theory in empirical applications. However, so far, it has found no applications in the relevant literature because of the lack of an appropriate method for its empirical estimation. In other words, first, we propose a modified nonlinear Hicks model for a semi-open economy and, second, we propose a method for its empirical application. In this context, we apply it to the USA to test its validity, using real world data for the time period 1960-2008.    II. 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 worked in a Keynesian framework. In this context:                                  ttt ICY += .                                  (3) Consider now a consumption function 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 following investment function:                                        ttttt eYYugAΑI +−+++= −− )()1( 2101        (5) where 1Α >0 is the constant term and te  the random error (i.i.d.). Once again, the constant term expresses that part of investment which does not depend on output and does not grow exogenously. Both constant terms introduced, namely 0C and 1Α , are convenient for the econometric estimation.         By substituting (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 uY A g A C− −− − + + = + + + . (6)      Also, 1λ  and 2λ are the roots of its characteristic equation  [ ]uusus 4)1(15.0, 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 20 1 0( ) ( ) [( ) ]/[ (1 ) ] [( )/ ]te pY t Y t Am m m s u m u A C s= = − − − + + + . (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 ), implying that the solution is periodically convergent, i.e. stable.  III. 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 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 [6].  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 very good guesses of the coefficient, given that we have 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.  It should be made clear that once the parameter g takes on a certain value (even if not the ‘optimal’ value of g ), the investment function becomes intrinsically linear and its estimation is straightforward employing 2SLS.  IV. EMPIRICAL ANALYSIS In order to apply the model to explain US output data, we define Consumption ( tC ) to include private, government as well as consumption from abroad. The data for the US economy come from AMECO and cover the time period 1960-2008, at constant prices.  The estimation of (4) employing 2SLS yields  1430.71 0.76t tC Y −= +                                    (10.3)    (113.0)                              2R =0.996, SSE=102.02.       Next, following the procedure described above we find that for g =0.04, the estimation of (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 Figure 1. Calculated SSR for the values of g  The model could also 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 [7] and AIC [8].        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 Figure 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 Figure 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 Figure 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:      1 272.32 322.39(1 0.04) 0.38( )tt t t tΙ Y Y e− −= + + + − +               (1.99)    (2.35)                            (30.13)    2R =0.96, SSE=104.03 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 below we illustrate the estimated values (Iestimated) of Investment along with its real values (Ireal).         The calculated correlation coefficient ( 98.0=ncorrelatior ) is another indication of the almost ideal fit of the model.  V. SOLUTIONS AND STABILITY          Substitution in (9) yields the particular integral              ( ) 1282.87(1.04) 2132.35teΥ t = +                Substitution in (7), given that 0Δ < , yields  1 0.571 0.228iλ = + and 2 0.571 0.228iλ = −  Also, based on (14)-(15), equation (12) yields:          1 2( ) 0.61 ( cos 0.38 sin 0.38 )tc t B t B tΥ = +                    and 1 2( ) 0.61( cos0.38 sin0.38 ) 1282.87(1.04) 2132.35t tc t B t B tΥ = + + + .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 1026.7341B = −  and 2 1906.65B = − . Conclusively, the analytical solution for ( )Y t is:  ( ) 0.61[ 1026.73cos(0.38 ) 1906.65sin(0.38 )] 1282.87(1.04) 2132.35t tY t t t= − − + +  050010001500200025001960 1970 1980 1990 2000 2010YearI(t)IrealIestimated Figure 5. Real VS estimated Investment for the US economy (1960-2008)  020004000600080001000012000140001960 1970 1980 1990 2000 2010YearY(t) YrealYestimated Figure 6. Real VS estimated GDP for the US economy (1960-2008)         By substituting the values of t we get the estimated values (Yestimated) of ( )Y t  which are illustrated in Fig. 6 above along with the real output values (Yreal).         The calculated correlation coefficient ( 0.99correlationr = ) is another indication of the almost ideal fit of the model. Also, since 1r < , given that 0.61<1, the solution of the complementary function is periodically convergent, i.e. stable. VI. CONCLUSIONS The proposed approach for a semi-open economy yielded very satisfactory results fitted to data for the US economy (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 BC theory and predictions. Clearly, future research would be of great interest.    REFERENCES [1]    Debreu, G., “Mathmatical Economics,” in The New Palgrave: A Dictionary of Economics, Vol. 3, pp. 401-03, 1987. [2]    Colander, D., “Pluralism and Heterodox Economics: Suggestions for an ‘Inside the Mainstream’ Heterrodoxy,”.Middlebury College Economics Discussion Paper No. 07-24, October, 2007.  [3]    Fontana, M., “Computer Simulations, Mathematics, and Economics,”    Rivista Internazionale di Scienze Economiche e Commerciali, 53 (1), pp. 96-124, 2006. [4]    Rosser, B. J. Jr., “On the Complexities of Complex Economic Dynamics,” Journal of Economic Perpectives, 13 (4), pp. 169-192, 1999. [5]    Samuelson, P., “Interactions between the multiplier analysis and the principle of acceleration,” Review of Economics and Statistics 21(2), pp. 75-78, 1939. [6]    Chow, G., China’s Economic Tranformation, Blackwell Publising, 2007. [7]    Schwartz, G., “Estimating the dimension of a model,” Annals of Statistics 6 (2), pp. 461-464, 1978.  [8]    Akaike, H., “A new look at the statistical model identification,” IEEE Trans Automatic Control, Vol. 19, No.6, pp. 716-723, 1974.  

https://mpra.ub.uni-muenchen.de/74451/1/MPRA_paper_74451.pdf

An extended nonlinear Hicks model of the cycle for a semi-open economy: Application to the USA (1960-2008)

Abstract

Similar works

Full text

Available Versions

Munich Personal RePEc Archive

DSpace at NTUA

Munich RePEc Personal Archive