In this paper, we develop a new model that explicitly considers two endogenous consumption items and investigates its applicability to consumption-capital asset pricing model (C-CAPM) by testing it with various sets of instruments. We found that our model is not rejected with reasonable values for both risk aversion and time preference parameters.C-CAPM, multiple consumption commodities, inter-temporal and intra-temporal choice

Atsushi Maki

Kenji Wada

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 2 
  
Estimation of consumption-capital asset pricing model (C-CAPM) with two 
clusters of consumption expenditures 
  
 
 
Atsushi Maki  
Department of Economics, Tokyo International 
University 
Kenji Wada  
Faculty of Business and Commerce, Keio University
Abstract 
In this paper, we develop a new model that explicitly considers two endogenous consumption items and investigates its 
applicability to consumption-capital asset pricing model (C-CAPM) by testing it with various sets of instruments. We 
found that our model is not rejected with reasonable values for both risk aversion and time preference parameters.
We gratefully acknowledge the helpful comments from the seminar participants at the Department of Economics, University of Wisconsin 
Milwaukee. This research is supported by the JSPS research grant-in-aid no. 15330063. 
Citation: Atsushi Maki and Kenji Wada, (2011) ''Estimation of consumption-capital asset pricing model (C-CAPM) with two clusters of 
consumption expenditures'', Economics Bulletin, Vol. 31 no.2 pp. 1183-1187. 
Submitted: Jan 04 2011.   Published: April 17, 2011. 
 
     1.  Introduction 
 
In this paper, we extend the standard Consumption-Capital Asset Pricing Model (C-
CAPM) for a single consumption item by explicitly modeling two endogenous consumption 
items, namely non-durables and services. The introduction of two consumption items in the 
inter-temporal consumption-investment problem makes it possible to investigate the role of 
distinct consumption items and their prices in both inter-temporal and intra-temporal 
optimizations. This paper fills a gap noted by Pennacchi (2008, p7) between microeconomics 
that primarily analyzes consumers’ optimal choice among multiple goods and financial 
economics that mainly focuses on the optimal choice between a single consumption item and 
financial assets at different points in time under varying conditions and states of nature.  
Our model is close to Mankiw, Rotemberg and Summers (1985) and Eichenbaum, 
Hansen and Singleton (1988) that introduced two endogenous variables, namely consumption 
and leisure, in the utility function of agents.  
We also consider the effect of a difference of an instrument set on the estimation and 
testing of the model. If we test a model with only a few sets of instruments and reject it, the 
rejection may be due to the particular choice of instruments. Thus, we form a large set of 
instruments and see whether or not our model is rejected. We found that our model is not 
rejected with reasonable values for both the risk aversion and time preference parameters 
involving many sets of instruments.  
 
2.  Models with two consumption items 
 
  In our model, we explicitly classify total consumption into two items, namely non-
durables and services. Both variables are endogenous. We consider three types of financial 
assets: stocks, bonds and risk-free assets. The consumer in the economy develops a consumption 
and portfolio plan so as to maximize the expected utility subject to the budget constraint. There 
is both an intra-temporal choice between non-durables and services and an inter-temporal choice 
between consumption and investment. The lifetime utility function and the budget constraint at 
time t are given by: 
 
  
,,
,, 0
{,} , {,,}
[( , ) ] max
FBS
nt st t t t
k
tn t k s t k k
cc NNN
EU c c β
∞
++ = ∑            ( 1 )  
 
subject to:  
 
   111 1* ( ) ( )
FB B BS S S n n s s F F B B S S
tt t t t t t t t t t t t t t t t t YNP D NP D N c p c p P N P N P N −−− ++ + + + = + +++      (2) 
 
whereβ  is the time preference parameter,  t E  is the expectation conditioned on the information 
set at time t,  t Y  is nominal disposable income, 1 is the payoff of the riskless asset which is a 
discount bond, 
F
t N is the number of shares of the riskless asset, 
B
t P and 
B
t D  are the nominal price 
and coupon payment of bonds, respectively, 
B
t N is the number of shares of  bonds, 
S
t P and 
S
t D  
are the nominal price and dividend payment of stocks, respectively, and 
S
t N  is the number of 
shares of  stocks. Also,  , nt p and  , s t p   are the price deflators for non-durables and services,   1
respectively, and  ,, , nt st cc  are per capita real consumption for non-durables and services, 
respectively. The time separable utility function ,, (,) nt st Uc c  is specified as: 
 
  
11
,,
,,
1
(,)
(1 )(1 )
nt st
nt st
cc
Uc c
γδ
γ δ
−− −
=
−−
            ( 3 )  
 
where γ and δ are the risk aversion parameters for non-durables and services, respectively. Both 
parameters take positive values from a theoretical restriction.   
Note that we have two ways to deflate nominal series into real series in our model. One is 
to employ the price deflator for non-durables and the other is to use that for services. Thus we 
have two pairs of real rates of return for each nominal rate of return from financial assets. We 
denote the nominal rates of return from time t to t+1 for risk-free assets, bonds and stocks by 
,1 f t R + ,  ,1 bt R +  and  ,1 s t R + . Then the real rates of returns for these three financial assets are given by: 
 
   ,1 ,1 ,1 , 1( 1 ) / ( / )
n
j tj t n t n t rR p p ++ + += +    (  j = f, b, s)      (4) 
 
   ,1 ,1 ,1 , 1( 1 ) / ( / )
s
j tj t s t s t rR p p ++ + += +    (  j = f, b, s)      (5) 
 
The first order conditions for the inter-temporal optimization between consumption and 
investment are given by: 
 
  
,1 ,11
,1
,,
[( )( ) ( 1 ) ]1
nt st n
tj t
nt st
cc
Er
cc
γδ β
++ −−
+ +=  (  j = f, b, s)      (6) 
 
  
,1 ,1 1
,1
,,
[( ) ( )( 1 ) ]1
nt st s
tj t
nt st
cc
Er
cc
γδ β
++ −−
+ +=  ( j = f, b, s)      (7) 
 
The first order condition for the intra-temporal optimization between non-durables and services 
is as follows: 
 
  
1
[] 1
1
ss
tt
t nn
tt
cp
E
cp
γ
δ
−
=
−
              ( 8 )  
 
Following Mankiw, Rotemberg and Summers (1985), we estimate and test all the seven 
equations simultaneously.  
 
3.  Data 
 
For the consumption series, we employ monthly household-level consumption data 
derived from the Family Income and Expenditure Survey (FIES) from January 1980 through 
April 2002 compiled by the Statistics Bureau of Japan. The total number of observations is 279. 
For the financial returns, we employ nominal stocks, bonds and risk free returns from Stocks,   2
Bonds, Bills and Inflation (SBBI) compiled by Ibbotson Associates Japan. The mean real returns 
on the stock index, long-term government bond and money market instruments calculated by the 
price deflator of non-durables are 1.0035, 1.0057 and 1.0024 respectively, and those evaluated 
by the price deflator of services are 1.0025, 1.0047 and 1.0014, respectively.  
We apply seasonal adjustment to the seasonally unadjusted consumption series published 
by the Statistics Bureau because seasonal fluctuations such as an outlay for Christmas shopping 
are not specified in our model. We follow Ghysels and Osborn (2001) who argue that seasonal 
adjustment should be applied at the last stage. Note that both seasonally unadjusted consumption 
series and the price deflators for households are available from FIES compiled by the Statistics 
Bureau. We divide the nominal seasonally unadjusted consumption item by the corresponding 
seasonally unadjusted price deflator. Then we divide the real consumption item by the number of 
household members. Finally, we apply the X-12ARIMA method developed by the Bureau of 
Labor Statistics to obtain the seasonally adjusted real per capita consumption series. The average 
growth rates between t and t+1 period on non-durables and that on services are 1.00084 and 
1.00145, respectively. 
 
4.  Instruments and GMM estimation results 
 
  In order to estimate the model: (6), (7) and (8) simultaneously, we employ the GMM 
method. We use the Newey-West HAC estimator with a Bartlett kernel (MA=4). We selected the 
following set of instruments including constant, consumption for non-durables, services; returns 
of stocks, bonds and risk-free assets evaluated by the price deflator for non-durables or services:  
 
   ,1 ,1 , , , , , , ( 1 , , ,,,,,,)
nsnsn s
nt st st st bt bt f t f t c c rrrrrr −−          ( 7 )  
 
We take every combination of instrument sets from the above variables and have a total of 255 
sets of instruments. When we test the model, we impose the following criteria on the magnitude 
of parameters and test statistics: 
 
(1) P-value of the model > 0.05 
(2) t-values for β, γ and  δ  > 1.96 
(3) β < 1 
(4) γ, δ > 0 
 
The first criterion is on the magnitude of the J-test statistics at the significance level of five 
percent, the second is on the t-statistics for β, γ and δ at the significance level of five percent, the 
third and fourth are the theoretical restrictions regarding preference parameters.  
Out of 255 instrument sets, we obtain 221 sets that converge for the distinct estimation 
and testing results. Out of these 221 results, the model is not rejected in 195 cases by J-test. In 
these 195 cases, 156 cases are not rejected statistically judging from the t-statistics. The 
theoretical restrictions indicated in (3) and (4) are fully satisfied by the above 156 cases. Thus, 
156 out of 255 combinations satisfy the above four conditions.  
The magnitude of γ  is between 0.250 and 0.490 and that of δ  is between 0.332 and 
0.559 in 152 cases. In the remaining four cases, the maximum magnitude of both γ  and δ  
coefficients are l6.023 and 5.60 respectively. The time preference parameter is between 0.9907   3
and 0.9991 in all 156 cases. Table I indicates the distribution of the number of parameter 
estimates for γ and δ. 
 
 
 
Table I: The distribution of the number of parameter estimates for γ and δ  
 
Range     γ     δ 
-------------------------------------------------------------- 
  0.25  –  0.50   152   117 
  0.50 – 0.75         0     35 
  0.75 – 1.0         0      0 
  1.0 – 2.0         0      0 
  2.0 – 5.0         1      3 
  5.0 –  7.0         3      1 
 ------------------------------------------------------------- 
Total    156   156 
 
Note: the minimum values for γ and δ are 0.2501 and 0.3322, respectively and the 
maximum values for γ and δ are 6.023 and 5.602, respectively. 
 
 
5.  Conclusion 
 
  In this paper, we proposed an extension of the standard one-commodity C-CAPM model 
into a multiple-commodity C-CAPM by explicitly modeling non-durables and services 
separately. As a working hypothesis, we set the null hypothesis that our multiple-commodity 
model is correct. Based on this hypothesis, we estimated the parameters of the time preference, 
and the two sets of the risk aversion parameters.  As the present model is robust, we have a 
possibility to extend the scope for SDF (stochastic discount factor). 
 
 
 
References 
 
Eichenbaum, M., Hansen L. and Singleton, K. (1988) “A time series analysis of representative 
agent models of consumption and leisure choices under uncertainty” Quarterly Journal of 
Economics, 103, 51-78. 
 
Ghysels. E. and Osborn, D. (2001) The Econometric Analysis of Seasonal Time Series, 
Cambridge UP: Cambridge. 
 
Mankiw, N., Rotemberg J. and Summers L. (1985) “Intertemporal Substitution in 
Macroeconomics” Quarterly Journal of Economics, 100, 225-251. 
  
Pennacchi, G., (2008) Theory of Asset Pricing, Pearson/Addison-Wesley: Boston, Mass. 

A time series analysis of representative agent models of consumption and leisure choices under uncertainty”

Intertemporal Substitution in

The Econometric Analysis of Seasonal Time Series,

Theory of Asset Pricing,

Estimation of consumption-capital asset pricing model (C-CAPM) with two clusters of consumption expenditures

http://www.accessecon.com/Pubs/EB/2011/Volume31/EB-11-V31-I2-P110.pdf

Estimation of consumption-capital asset pricing model (C-CAPM) with two clusters of consumption expenditures

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics