2.1 Surplus Consumption ratio
advertisement
Return Predictability and State Variables in ConsumptionBased CAPMs: International Perspectives
Bin Li
Maosen Zhong
UQ Business School, the University of Queensland, Brisbane, QLD, 4072, Australia
1 March 2006
_____________________________________________________________________
Corresponding Author.
Email Address: m.zhong@business.uq.edu.au (M. Zhong);
Fax: (+61) 7 3365 6788.
Return Predictability and State Variables in ConsumptionBased CAPMs: International Perspectives
Abstract
Asset pricing literature contends that aggregate stock returns can be predicted by
business cycle state variables. In this study, we compare the return predictability of
three state variables in the context of consumption-based capital asset pricing model
(CCAPMs): consumption growth, the consumption-wealth ratio, and the consumption
in surplus of habit ratio in the eight major equity markets in the world: Australia,
Canada, France, Germany, Italy, Japan, the U.K. and the U.S.. We also explore the
time-series relationship between expected excess market returns and the conditional
risk aversion associated with the world as well as local consumption in these eight
countries. Our results reveal that the consumption-wealth ratio has much more
predictive power for returns in the international markets than the surplus consumption
ratio and consumption growth rate.
- JEL Classification: G12; G15
Keywords:
Consumption-based CAPM, Consumption-wealth Ratio, Habit
Formation, Predictability, Asset Pricing
1. Introduction
Recent literature has related the aggregate stock market return predictability to
consumption variables that reflect the business cycle. Starting from Rubinstein (1976),
Lucas (1978) and Breeden (1979), many authors define equilibrium in the capital
markets using consumption variables. Under a number of assumptions, the asset
returns should be linearly related to the growth rate in aggregate consumption as long
as the parameters of the linear relationship remain constant over time. Despite the
theoretical soundness and simplicity, the consumption-based capital asset pricing
model (CCAPM) is usually not easy to be explicitly estimated for the highly nonlinear nature and non-separability of consumption utility among periods. The
canonical CCAPM does not performed well empirically both on the U.S. data (see for
example, Hansen and Singleton 1982; Mankiw and Shapiro 1986; Breeden, Gibbons
and Litzenberger 1989) and international data (Wheatley 1988).
In response to this, researchers have modified the consumption-based CAPM
hoping to account for the time-varying risk aversion left out in the canonical model.
Campbell and Cochrane (1999) (hereafter as CC) introduce a habit formation variable
– the surplus consumption variable, which is the time-varying consumption in surplus
to habit, to modify the optimal choices of consumption over time to explain the
cyclical variation in expected returns and volatility. The habit formation of CC is nonlinear, infinite-horizon, slow-moving and external in response to the history of
consumption. Investors tend to consume more relative to their habits in good times as
a result of low risk aversion and consume less relative to habit in recessions. CC find
that due to the fact that risk aversion is inversely related to surplus consumption, a
high level of consumption exceeding habit should forecast low expected stock market
returns. Li (2001) proposes an alternative habit formation based on a finite-horizon
linear model. Li’s (2001) model preserves the inverse relation between expected
returns and surplus consumption, and it performs almost as well as CC’s nonlinear
habit model for the U.S. data.
The modified consumption-based asset pricing model not only can explain
U.S. stock market returns, but also has predictive power for industry portfolios of the
U.S. market and the international stock market returns as well (See Li and Zhong
2005; Li, Lu and Zhong 2004; Jacobs and Wang 2004).
1
Furthermore, the recent studies by Lettau and Ludvigson (2001a) explain time
series variation of US portfolios returns with a log consumption-wealth ratio called
cay t , which is constructed as the residual from the shared trend among log
consumption (ct), log asset holding (at) and log labor income (yt). Under the
assumption of a representative agent’s binding intertemporal budget constraints, they
find there is cointegration relationship between log consumption, log asset holding
and log labor income, and their constructed log consumption-wealth ratio ( cay t ) is a
strong and positive predictor for U.S. stock market excess returns at short and
intermediate horizons. cay t can help explain about 9% of one quarter ahead excess
market returns, 18% for annual excess returns, and 15% for the 3-year excess returns.
The economic intuition for these results is that investors want to maintain a smooth
consumption path and insulate future consumption from fluctuations in expected
returns. Thus, they increase consumption out of current asset wealth and labor income
(allowing consumption to rise above the shared trend) when expected return is high
and decrease consumption out of current asset wealth and labor income (allowing
consumption to fall below the shared trend) when expected return is low.
There are several potential contributions of our study. First, though the
performance scrt and cay t have been well documented in the U.S. market, the
international evidence of return predictability is quite limited, partly because of the
unavailability of reasonably good data to construct scrt and cay t . We fill the gap by
providing international evidence of the predictability of the consumption state
variables. Second, to our knowledge, little study has compared the relative
performance of the predictive power of three state variables in the context of
CCAPMs: the consumption growth, scrt , and cay t .
We show that the consumption-wealth ratio can predict the variation of excess
stock market returns much better than the surplus consumption ratio scrt and the
consumption growth in the eight major markets: Australia, Canada, France, Germany,
Italy, Japan, the U.K. and the U.S.. Moreover, each of the three state variables
explains the return variations that are unexplained by the other. This indicates the two
variables represent different states of aggregate markets. Our interpretation is that scrt
captures the risk aversion of the representative agent, whereas cay t represents
2
conditioning variable on the consumption risk factor that induces the time-varying
risk premium. Both state variables can affect the risk premia of the aggregate stock
market but in different dimensions.
The rest of the paper is organized as follows: Section 2 presents the theoretical
development of the surplus consumption ratio ( scrt ), the consumption-wealth ratio
( cay t ) and the corresponding econometric models for testing. Section 3 presents the
empirical comparison results of the international markets using consumption growth,
scrt and cay t . Section 4 concludes this paper.
2. The Model
This section first presents the theoretical development of the surplus
consumption ratio ( scrt ) and the aggregate consumption-wealth ratio ( cay t ) in the
context of CCAPM.
2.1 Surplus Consumption ratio
Campbell and Cochrane (1999) present CCAPM with external habit formation
where a representative agent derives utility from the difference between consumption
and a time-varying habit level, and the model can capture much of excess stock
returns in the long horizon. We provide a concise introduction of this model.
Assume a representative agent take the utility function as
Et
j
(Ct j X t j )1 1
j 0
,
1
(1)
where Ct is the per capita real consumption, Xt is the habit level, δ is the subjective
discount factor, and γ is the utility curvature parameter. Abel (1990) and Campbell
and Cochrane (1999) suggest the level of habit Xt to be external. Xt is also assumed to
depend upon a long history of aggregate consumption and follows an infinite-horizon
nonlinear formation process (CC 1999). The surplus consumption ratio SCRt , is
defined by
SCRt
Ct X t
1
Ct
.
(2)
3
SCRt indicates the state of the economy. When SCRt declines, consumption Ct drops
to the habit level Xt, and the economy is reaching recession and investor risk aversion
rises. On the other side, rising SCRt means that current consumption exceeds Xt, and
the economy is expanding, and investor risk aversion decreases.
Hereafter, we use uppercase letters to denote variables at their original scale,
and lowercase letters to denote the logs of the corresponding uppercase letters.
Campbell and Cochrane (1999) suggest a heteroscedastic AR(1) process for the log
surplus consumption ratio, scrt log(SCRt ) :
scrt 1 (1 ) scr scrt ( scrt ) c ,t 1 ,
(3)
where φ is the habit persistent parameter, and (scrt ) is the sensitivity function, and
c ,t 1 is the innovation in consumption growth. Campbell and Cochrane (1999)
suggest that we should use a large value of the persistence parameter in order to
obtain the predictive power of the log surplus consumption ratio. As in Li and Zhong
(2005), we use φ = 0.90 in this paper1. (scrt ) represents the conditional sensitivity of
SCRt to Ct, and it is inversely related to the surplus consumption ratio SCRt . When
SCRt falls, the sensitivity function (scrt ) and expected excess returns rise.
Consumption growth is assumed independently and identically distributed (i.i.d.).
Imposing three conditions that the risk-free rate is fixed, habit is
predetermined at the steady state, and habit moves non-negatively with
contemporaneous consumption elsewhere, we can specify the sensitivity function
(scrt ) max{0, (1/ SCR) 1 2( scrt scr ) 1} ,
(4)
where the steady state is SCR c /(1 ) , and c is the standard deviation of the
unexpected consumption growth c ,t 1 . The resulting sensitivity function (scrt ) is
inversely related to the log surplus consumption ratio, scrt . Following Campbell and
Cochrane (1999) and Li, Lu and Zhong (2005), we choose curvature parameter 2
to compute the steady-state value of scrt .
Under the external habit formation, (1) implies the intertemporal marginal rate
of substitution of the representative investor at time t 1 is
Alternative habit persistence values ( 0.85, 0.95 ) do not alter any of the conclusions reached in
this study.
1
4
M t 1 (
Ct 1St 1
) .
Ct St
(5)
Let Ri ,t 1 denote one plus the rate of return on asset i from time t to t 1 , then
Ri ,t 1 satisfies the Euler equation of the following form:
Et [ M t 1 Ri ,t 1 ] 1 ,
(6)
where Et is the expectation conditional on the information set as of time t.
Under the assumption of the jointly lognormally distribution of asset returns
and consumption growth, the Euler equation (6) implies that expected excess returns
on any asset is
1
Et [rie,t 1 ] it2 [1 ( scrt )]covt (ri ,t 1 , ct 1 )
2
In (7),
2
.
(7)
1 2
it is the Jensen’s alpha, and the risk premiums on asset i are the price of
2
risk, [1 ( scrt )] , times the conditional covariance of the asset’s returns with
consumption growth. The price of risk depends on the utility curvature parameter
and the sensitivity function (scrt ) . Given that the expected excess returns are
inversely related to scrt , (scrt ) is inversely related to the surplus consumption
ratio, scrt (Eq. (4)). Thus investors require higher expected returns on assets when
consumption falls toward habit.
Under the assumption that the conditional covariance of returns with
consumption growth is constant, linear approximations to the sensitivity functions
(scrt ) imply that the expected excess return on asset i can be written as
Et [rte,t 1 ] 1scrt
,
(8)
where scrt is the log surplus consumption ratio and the slope coefficient 1 is constant.
Sensitivity function (scrt ) , according to (7), is positively related to the expected
excess market returns, and scrt is inversely related to (scrt ) , so the coefficient 1 is
expected to be negative.
The model for expected one-period returns in (8) can be easily extended to a
model
2
for
expected
returns
over
multiple
periods.
Let
See Li (2001) for detailed derivation.
5
K
r
k 1
e
i ,t k
rie,t 1 rie,t 2
rie,t k denote the cumulative excess stock market returns with
continuous compounding over K periods. If scrt are highly persistent, they should be
able to predict multi-period returns. We write expected K-period excess returns as
K
Et [ rie,t k ] K 1K scrt
,
(9)
k 1
where 1K are constant slope coefficients3.
Assuming the growth rates of consumption and dividends are i.i.d., expected
excess stock returns are determined by a single state variable – the surplus
consumption ratio. Campbell and Cochrane (1999) and Li (2001) show that expected
excess returns should be negatively related to the state variable because high (low)
surplus consumption at the business cycle troughs (peaks) are associated low (high)
investor risk aversion, thus reducing (increasing) the required rate of returns.
2.2 Consumption-wealth ratio
Here we theoretically derive the aggregate consumption-wealth ratio cay t that
provides useful conditioning information for asset returns. Consider a representative
investor who invests his wealth in a single asset with a time-varying risky return Rt4.
We denote Wt as aggregate wealth including human capital and household asset
holding at time t, Ct as consumption and Rt+1 is the net return on the market portfolio
of all invested wealth (including financial and non-financial wealth). For a completemarket model where wealth includes human capital as well as financial assets, under
the intertemporal budget constraint, the period-to-period aggregate wealth is
Wt 1 (1 Rt 1 )(Wt Ct ) .
(10)
Campbell and Mankiw (1989) derive the log consumption ratio from (10). In
order to make the budget constraint function linear, we approximate it by taking first
order Taylor expansion (see Campbell and Mankiw 1989 for details), resulting in
wt 1 wt 1 wt k rt 1 (1 1/ )(ct wt ) ,
3
Similar to Campbell et al. (1997), if
coefficients
(11)
scrt follows an AR(1) process with an autocorrelation
, then the law of iterated expectations implies 1K 1 (1 K ) /(1 ) .
4
Individuals, usually in empirical research, are assumed to be aggregated into a single representative
agent economy (see Campbell, Lo and MacKinlay (1997), p.304).
6
where the parameter ρ is the average ratio of invested wealth, W-C, to total wealth, W,
and k is a constant.
The wealth growth rate wt 1 can be arranged in terms of consumption growth
rate and the change of log consumption-wealth ratio. Solving it forward recursively,
we can get a log-linear version of the infinite-horizon budget constraint
ct wt Et i (rt 1 ct 1 )
.
(12)
i 1
(12) links the log consumption-wealth ratio with future market return and the future
consumption growth, suggesting that a higher log consumption-wealth ratio at this
period must forecast either higher returns on the market portfolio at future periods or
low future consumption growth rates. Moreover, economic intuition indicates that,
when investors expected low future returns on assets, they will drop today’s
consumption temporarily below the long term relationship among consumption, asset
and wealth to secure future higher consumption. Therefore, trend deviation cay t and
excess stock returns should have positive covariance.
Given that aggregate wealth Wt is the sum of financial asset At and human
capital Ht, the log linear approximation of Wt is a convex combination of the loglinear approximation of At and Ht,
wt at (1 )ht
,
(13)
where ω equals the average share of asset holdings in total wealth, A/W. If aggregate
labor income Yt can describe the non-stationary component of human capital Ht (see
Lettau and Ludvigson (2001a) for details), then we can obtain the following
relationship between log consumption-aggregate wealth ratio
ct at (1 ) yt Et i ra ,t i (1 )rh ,t i ct i (1 ) zt
(14)
i 1
Assuming all terms on the right-hand side of (14) are stationary, then the left-hand
side must also be stationary, implying ct, at, yt must have cointegration relation with a
cointegrating vector (1, -ω, ω-1). We now denote cay t as the left side of (14),
ct at (1 ) yt . This equation suggests that as long as expected future returns on
human capital rh,t+i and consumption growth ∆ct+i are not too volatile, or if they have
high correlation with expected future returns on assets, then cay t should help forecast
the expected future asset returns.
7
(14) suggests that cay t has positive linear relation with the excess stock
market return. To examine the implication of the single-state-variable model for the
predictability of returns on the market portfolio, the regression equation is as follows:
Et [rt e,t 1 ] 2 cay t
,
(15)
where Et [ri ,t 1 ] is one-quarter ahead expected excess stock market returns; is a
e
constant, and 2 is the slope coefficient.
Accordingly, the model for expected one-period returns in (15) can be easily
extended to a model for expected returns over multiple periods. If cay t are highly
persistent, they should be able to predict multi-period returns. We write expected Kperiod excess returns as
K
Et [ rie,t k ] K 2K cay t ,
(16)
k 1
where 2K are constant slope coefficients.
2.3 The relationship between surplus consumption ratio and
consumption-wealth ratio
Analogous to Campbell and Cochrane (1999), who shows that the priceconsumption ratio is a function of the surplus consumption ratio, Li (2005)
demonstrates that the wealth-consumption ratio dependents upon the surplus
consumption ratio. Rather than define the represent investor’s budget constraint as
(10), he defines it as
Wt 1 Ct 1 Rt 1Wt ,
(17)
Rt 1 (Wt 1 Ct 1 ) / Wt .
(18)
which is equivalent to
Substituting (18) into (6) yields
Wt
C
W
Et [ M t 1 t 1 (1 t 1 )] .
(19)
Ct
Ct
Ct 1
Equation (19) implies that the wealth-consumption ratio is a function of marginal rate
of substitution M t 1 , which is a function of the surplus consumption ratio as specified
in equation (5). As the consumption-wealth ratio is the inverse of the wealthconsumption ratio, the consumption-wealth ratio should also depend upon the surplus
consumption ratio and the consumption growth rate. Therefore, the consumption-
8
wealth ratio and the surplus consumption ratio should be related in the predictability
tests.
3. Data
We compile the data of aggregate market returns and macroeconomic data for
eight major countries: Australia, Canada, France, Germany, Italy, Japan, the U.K. and
the U.S.. As discussed above, the key state variables for return predictability test are
the log surplus consumption ratio and the log consumption-wealth ratio. For
Australian market, our macroeconomic data include household consumption (nondurable goods and service), after-tax labor income and net household wealth in
Australia. Our financial data include All Ordinary stock market returns, and the shortterm bond rate. Appendix A provides detailed descriptions of data source and data
construction in Australian market.
For Canada, France, Germany, Italy, Japan and the U.K., we obtain the
macroeconomic data and financial data from the Datastream International. The
consumption data is consumption expenditure from household and non-profit
organization serving household. We use household disposable income as the income
data. Both the consumption data and income data are quarterly in frequency. However,
we are unable to get the household asset holding data at quarterly level. The
household wealth data are in the form of the percentages of household income in
Datastream. We multiply these percentages with their corresponding household
income data to get the household wealth series. They are only available in annual
frequency, so we use ARIMA(1,1) model to extrapolate the annual data to the
quarterly frequency. These data series are from the first quarter of 1985 to the last
quarter of 2004.
We also retrieve the consumer price index and the population data from
Datastream. Then we use the first quarter of 2000 as the base date, and compute the
real per capita household consumption, household income, and household wealth. We
take logs of these three variables, and use the dynamic least square estimation to
obtain the log consumption-wealth ratio for these six markets (four lags are used in
the dynamic least square estimation5).
5
Other lags are also tried, and the results have not been substantially changed.
9
The market indices of six countries are: S&P/TSX Composite for Canadian
market, CAC 40 for French market, DAX 30 for German market, Milan COMIT
performance index for Italian market, NIKKEI 225 Stock Average for Japanese
market, and FTSE 100 for the U.K. market. All of the indices data are from
Datastream International. We use Treasury Bill rate to proxy for risk-free rate for
these countries except for Japan, for which we use the deposit rate instead because
Treasury Bill rate of Japan is not available. The risk-free rates are from the
International Financial Statistics.
The U.S. consumption-wealth ratio data come from the website of Martin
Lettau6, which provides quarterly data from the fourth quarter of 1951 to the fourth
quarter of 2004. The variables which are used to construct the U.S. consumptionwealth ratio - consumption, labor income, household net wealth – are log real per
capita data. We also use the consumption data to construct the surplus consumption
ratio. S&P500 stock market index are from Datastream. The risk-free rate is proxied
by the U.S. three-month Treasury bill rate.
4. Empirical Results
4.1 Summary Statistics
We construct the surplus consumption ratio and the consumption wealth ratio
according to models discussed in section 2. Specifically, for the consumption wealth
ratio, we employed augmented Dickey-Fuller (1979) and Philips-Perron (1988) Unit
Root test in the series of household consumption, labor income and net household
wealth. We find there is only one unit root in each of these three series. Next we move
to test whether there is any cointegration relationship among the log per-capita real
consumption, wealth, and labor income. To do this, we conduct two kinds of
cointegration test: one is Phillips-Ouliaris (1990) residual-based cointegration test,
which is to discern whether there is at least one cointegration vector among these
three variables; the other test is a more popular one: Johansen’s (1988, 1991) L-Max
test and trace statistics test, which will tell us the number of cointegration vectors of
the long-term relationship. Appendix B displays the cointegration test results. The
6
Quarterly cay t data for the U.S. can be retrieved from Martin Lettau’s Webpage:
http://pages.stern.nyu.edu/~mlettau/.
10
three variables are cointegrated and hence share a common trend in the long run.
Therefore, we follow Lettau and Ludvigson (2001a) and construct the consumption
wealth ratio using the Dynamic Least Square regression.
Table 1 presents the summary statistics of the variables in eight countries:
Australia, Canada, France, Germany, Italy, Japan, the U.K., and the U.S. For every
country, there are two sub-panels: the correlation matrix and univariate summary
statistics. The quarterly variables include log excess return on the market index rt e1 ,
the surplus consumption ratio scrt , and the consumption-wealth ratio cay t and the
consumption growth rate ct 1 . The sample period of each country is given at the top
of the each panel.
<< Insert Table 1 Here>>
Sub-panel A in each panel demonstrates that excess return on the stock market
is negatively related to the surplus consumption ratio scrt in most markets, but
positively correlated with the local consumption-wealth ratio. The sign of the
correlation is consistent with the theory we discussed in Section 2.
Sub-panel B in each panel presents the first-order autocorrelation of the
variables at lag 1. They show that in most countries, excess stock market return and
the consumption growth rate exhibit low autocorrelation, but the surplus consumption
ratio and the consumption-wealth ratio exhibit high autocorrelation. The high
persistence of these two variables may be helpful for forecasting excess market
returns over long horizons.
4.2 Long-horizon Forecasting
Now, we move to examine the predictive power of the surplus consumption
ratio scrt, the consumption-wealth ratio cay t and the consumption growth rate ct 1
for the aggregated stock market returns in the long horizons in the world’s eight major
equity markets. Table 2 reports the linear regression estimation results using single
state variable: the log surplus consumption ratio, scrt, and the log consumption-wealth
ratio, cay t , and the consumption growth rate ct 1 as predictive variables. We run the
following least square regressions using full sample over the horizons spanning from
11
K
1 to 16. The dependent variable is cumulative K-period excess market returns:
r
k 1
e
t k
.
Table 2 shows the estimated coefficients of the regression with the state variable at
horizons from quarter 1 to quarter 16. It also reports the adjusted R2, and Newey-West
(1987) adjusted t-statistics to account for any residual serial correlation. Newey-West
adjusted t-statistics are placed in parentheses under the estimated coefficients, and the
adjusted R2s of the regressions are placed in the square brackets below the t-statistics.
<< Insert Table 2 Here>>
The log surplus consumption variable, scrt, which is derived from modified
Campbell and Cochrane (1999) model. Here, we use the parameter of the curvature of
risk aversion γ=2.0, as suggested by Campbell and Cochrane (1999); the habit
persistence parameter φ=0.95.
In Australia, the first regression shows that scrt can only predict at the very
short horizon. At quarter 1, the estimated coefficient is -0.05, Newey-West adjusted tstatistics is -2.51 and scrt can explain 3% of the variation of the excess stock returns.
However, scrt hardly has any predictive power of for excess market returns over
intermediate and long horizons. The adjusted R2 at intermediate horizon is negligible
over long horizons. The estimated coefficient is not significant at the 10% level.
Our finding that the estimated coefficient of scrt at short horizons is negative
is consistent with the theory. Under the assumption of i.i.d of the growth rates of
consumption and dividends, Campbell and Cochrane (1999) and Li (2001) point out
that expected excess stock returns are determined by a single state variable – the
surplus consumption ratio, and expected excess returns should be inversely related to
the state variable because high surplus consumption at the business cycle troughs are
associated with low investor risk aversion, thus lowering the required rate of returns.
The second regression shows the forecastability of log excess stock market
returns over long horizon using cay t as a single state variable. The predictive power of
12
cay t is quite strong in the intermediate horizons (from 1 year to 3 years). The
adjusted R2 has jumped monotonically from quarter 1 to quarter 8, partly because the
returns are overlapping returns. At quarter 8 (year 2), the predictive effect of cay t on
accumulated excess returns is quite large; the point estimate of the coefficient on cay t
is about 5.6. Newey-West adjusted t-statistics is 3.3 and cay t can explain 32% of the
variation of the accumulated excess stock returns. At short horizons, cay t has little
explanatory power, the R 2 is close to zero. cay t ’s low forecasting power at short
horizons may be subsumed by its marginal predicting power of surplus consumption
ratio at short horizons.
As suggested by Eq. (12), a higher log consumption wealth ratio at this period
must forecast either higher returns on the market portfolio at future periods or lower
future consumption growth rate. cay t almost has no predicting power of consumption
growth (result is not reported), thus, a higher log consumption-wealth ratio at this
period must forecast higher returns on the market portfolio at future periods.
Therefore, consumption trend deviation should covary positively with excess stock
returns. As expected, the coefficient of cay t turns out to be positive in all regressions.
The third regression examines the performance of the canonical CCAPM,
where the consumption growth rate is the only factor. No coefficients are significant
at any horizon, therefore the canonical CCAPM performs poorly in explaining
Australian equity return.
Interestingly, when all three variables enter the multiple regression, the same
picture emerges: the significant coefficients enter the same positions as in the single
regressions. The adjusted R2s have improved over those in the single regressions.
These results indicate that the three state variables are not substitutes of each other in
the predictability tests. Rather, they may well represent different aspects of the timevarying risk premia that drive the expected returns.
In Canadian market, the surplus consumption ratio scrt has predictive power
from quarter 4 to quarter 16, where the t-statistics and the adjusted R 2 rise
monotonically. The coefficient of scrt is 0.28 at quarter 16, it is not only statistically
significant, but also economically significant. One percentage in the surplus
13
consumption ratio can explain the 28 basis point changes in the stock market returns.
The model can explain more than half of the variation of the stock returns. The sign of
the coefficient is consistent with the theory, as high surplus consumption would lower
the investor’s risk aversion, thus decrease the investor’s required return. In contrast,
the consumption-wealth ratio plays no role in predicting Canadian stock market return.
Although there are a couple of coefficients are significant for the consumption growth,
2
the explanatory power of the consumption growth rate is poor as the adjusted R s
range below 5%. Like Australia, Canadian multiple regressions with all three state
variables as regressors preserve their significant coefficients in the same positions as
in the single regressions.
Contrary to Canada, the surplus consumption ratio has no predictive power of
the stock market return, but the consumption-wealth ratio performs quite well in
French market. The coefficients are all significant at 5% level except at quarter 16,
which has t-stats of 1.80. At quarter 4, the coefficient is 4.87%, which means for one
percentage change in the consumption-wealth ratio, there will be 487 basis point
2
changes in the excess stock market return subsequently. The adjusted R reaches
above 30%, which means about one third of stock market variation can be explained
by the consumption-wealth ratio. In the multiple regressions, the surplus consumption
ratio and the consumption growth enter significantly at the long horizons, whereas the
consumption-wealth ratio maintain its significance. The sign of the coefficient is
consist with the theory. In contrast, the performance of the consumption growth rate is
in accordance with the poor empirical performance this ratio documented in previous
literature.
For German market, the single and multiple regressions results indicate that
the surplus consumption ratio is only significant at the longest horizon, but the sign is
positive, which is contradictory with the theory. The wrong sign of the coefficient of
the surplus consumption ratio probably is due to the poor measurement of the
consumption data, which warrants further investigation. The consumption-wealth
ratio is statistically and economically significant up to one-year horizon. However the
explanatory power the consumption-wealth ratio is quite limited, as the adjusted R 2 is
below 10%. As in other countries, the canonical CCAPM perform poorly, the
coefficient at the short horizons are negative, which is contrast with the theory.
14
For Italian market, the surplus consumption ratio has explanatory power in the
long horizons and the consumption-wealth ratio only plays a significant role at the
short horizons, and the consumption growth rate does not have any effect on the stock
market return.
For Japanese market, none of the coefficient in the surplus consumption ratio
or the consumption growth rate is significant. In contrast, the consumption-wealth
ratio performs well, economically and statistically significant, and the model can
explain almost half of the variation of the stock returns for quarters 8-16.
In the U.K., both the surplus consumption ratio and the consumption-wealth
ratio are important factors in determining the stock market returns in the long
horizons. The two variables have correct signs of the coefficients, whereas the
consumption-wealth ratio has wrong signs in the multiple regressions.
For the U.S. market, the results are quite similar to what the previous
researchers has done. The surplus consumption ratio and the consumption-wealth
ratio capture the stock market variation very well. As the regression horizons increase,
the adjusted R 2 build up monotonically. They both are significant at every quarter in
the regression except the first quarter. One point need mentioning is that the
consumption growth rate, though, economically and statistically significant in the first
two quarters( high consumption growth are associated with high stock market return),
the explanatory power of this model is quite small, less than 5%. And the coefficients
are no longer significant in the longer horizons. Interestingly, the consumption growth
variable becomes highly significant in the multiple regressions with the surplus
consumption ratio and the consumption wealth ratio.
In short, the consumption-wealth ratio predicts the future returns quite well in
seven out of eight developed markets, and the signs are in consistent with the theory:
high surplus consumption-wealth ratio is associated with higher subsequent stock
market return. The surplus consumption ratio performs well in the short horizon in
some countries (Canada, U.K. and U.S.), and long horizons in other countries
(Australia, France, Germany, Italy, and Japan). Consumption growth generally
performs badly in the single forecasting regressions.
15
4.3 Relative Explanatory Power of Three State Variable
It is worthwhile to investigate the relative explanatory power of the three state
variables (the surplus consumption ratio scrt , the consumption-wealth ratio cay t and
the consumption growth rate ct 1 ) in explaining the long horizon excess stock
market returns in these eight major markets.
Table 3 presents the results for explanatory power of cay t relative to the other
state variables in the long-horizon regressions for each of the world’s eight major
markets. We first run the regression, Rt k a b scrt et k , where Rt k is the
K
cumulative excess market returns over K period: Rt k rt e k , scrt is the log surplus
k 1
consumption ratio, which is derived from modified Campbell and Cochrane (1999)
model, where habit persistent 0.90 and investors’ risk aversion 2 , a and b are
coefficients, and et k is the regression residual; then we regress et k and the predicted
component Rt k et k separately on the consumption-wealth ratio. For the second
sub-panel, we run the similar procedure, except that the first step is a multiple
regression with both scrt and ct 1 : Rt k a b scrt c ct 1 et k .
First consider the regression results in the first sub-panel where the dependent
variable is et k . As the coefficients of cay t in most country panel is significant at 5%
level, most return variation which has not been explained by scrt can be explained by
cay t . When the dependent variable is the predicted component of returns, Rt k et k ,
the significance of the coefficients of cay t in most country panel has dropped
dramatically. When we add ct 1 in the multiple regression, the results do not change
drastically. This finding indicates that the consumption-wealth ratio and the surplus
consumption ratio are not substitute. Rather, they can explain the different portions of
the return variation.
Table 4 presents the results for the relative explanatory power of scrt . The
table takes the same form as Table 3. In the first sub-panel of each panel, first we run
the regression, Rt k a b cay t et k , then we regress et k and Rt k et k separately
16
on the surplus-consumption ratio, scrt . For the second sub-panel, we run the similar
procedure, except that in the first step is ct 1 enters the regression. We find the
similar pattern with Table 3, which reinforce our understanding that the consumptionwealth ratio and the surplus consumption ratio should be used together rather than
alone in explaining the stock market returns.
Table 5 displays the results for the relative explanatory power of the
consumption growth ct 1 . Except for Canada and U.S., the consumption growth does
not explain any return variation that is unexplained by scrt and cay t . Nor does the
consumption growth explain the predicted component of returns predicted by scrt
and cay t for France, U.K. and the U.S.. This corroborates the earlier observation that
the consumption growth generally explains little of the return variation.
5. Conclusion
In this study, we use the consumption growth, the log surplus consumption
ratio scrt of Campbell and Cochrane’s (1999) and the log consumption wealth
variable cay t of Lettau and Ludvigson (2001a) to examine the predictability of stock
market returns in the world major eight equity markets. We find that the surplus
consumption ratio, scrt , can predict excess stock returns in most countries, but some
at short horizons and others at long horizons; on the other hand, the consumptionwealth ratio cay t can predict the variation of excess stock market return in seven out
of eight markets (Australia, Canada, France, Germany, Italy, Japan, the U.K. and the
U.S.). The estimated coefficient of scrt is negative, which is also in accordance with
the theory: because high surplus consumption at the business cycle troughs is
associated with low investor risk aversion, thus lowering the expected required return.
cay t is positively related to excess stock returns, which is consistent with economic
intuition: when investors expect low future returns on assets, they will drop today’s
consumption temporarily below the long term relationship among consumption, asset
and wealth to secure future higher consumption.
Our study compares the relative predictive power of scrt and cay t for excess
market returns. One interesting point we find that the consumption-wealth ratio have
17
relatively better performance than the surplus consumption ratio, which is consistent
with the findings from Li, Lu and Zhong (2004). The lower predictive power of the
surplus consumption ratio may be attributed to serious data measurement errors of
aggregate consumption data as discussed in Campbell and Cochrane (1999). Yet,
Lettau and Ludvigson (2001b) argue that the construction of the consumption-wealth
ratio based on cointegration estimation may have overcome the problems of
unobservable variables. Our study enriches the understanding of long-horizon return
predictability in the international setting.
18
REFERENCES
Abel, A.B., 1990, Asset prices under habit formation and catching up with the Joneses,
American Economic Review Papers and Proceedings 80, 38-42.
Breeden, D., 1979, An intertemporal asset pricing model with stochastic consumption
and investment opportunities, Journal of Financial Economics 7, 265-296.
Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger, 1989,
Empirical tests of the consumption-based CAPM, Journal of Finance 44, 231262
Campbell, John Y., and John H. Cochrane, 1999, By force of habit: a consumptionbased explanation of aggregate stock market behavior, Journal of Political
Economy 107, 205-251.
Campbell, John Y., Andrew W Lo and A. Craig MacKinlay, 1997, The Econometrics
of Financial Markets (Princeton University: Princeton, NJ).
Campbell, John Y., and Gregory Mankiw, 1989, Consumption, income and interest
rates: Reinterpreting the time series evidence; in Olivier J. Blanchard and
Stanley Fischer, eds.: NBER Macroeconomics Annual (MIT Press, Cambridge,
MA).
Dickey, D., and W.A. Fuller, 1979, Distribution of the estimators for time series
regressions with a unit root, Journal of the American Statistical Association 74,
427-431.
Hansen, Lars Peter, and Kenneth J. Singleton, 1982, Generalized instrumental
variables estimation of nonlinear rational expectations models, Econometrica
50, 1269-1288.
Harvey, Campbell R., 1989, Time-varying conditional covariance in tests of asset
pricing models, Journal of Financial Economics 24, 289-317.
Johansen, Soren, 1988, Statistical analysis of cointegrating vectors, Journal of
Economic Dynamics and Control 12, 231-254.
Johansen, Soren, 1991, Estimation and hypothesis testing of cointegration vectors in
Gaussian vector autoregressive models, Econometrica 59, 1551-1580.
Lettau, Martin, and Sydney Ludvigson, 2001a, Consumption, aggregate wealth, and
expected stock returns, The Journal of Finance 56, 815-849.
Lettau, Martin, and Sydney Ludvigson, 2001b, Resurrecting the (C)CAPM: a crosssectional test when risk premia are time varying, Journal of Political Economy
109, 1238-1287.
Li, Yuming, 2001, Expected returns and habit persistence, Review of Financial
Studies 14, 861-899.
Li, Yuming, 2005, The Wealth-consumption ratio and the consumption-habit ratio,
Journal of Business & Economic Statistics 23, 226-241.
Li, Yuming, and Maosen Zhong, 2005, Consumption habit and international stock
returns, Journal of Banking & Finance 29, 579-601.
Li, Yuming, Weili Lu, and Maosen Zhong, 2004, The predictability of industry
portfolio returns, the University of Queensland working paper.
Lucas, Robert E., 1978, Asset prices in an exchange economy, Econometrica 76,
1429-1445.
Mankiw, N. Gregory, and Matthew D. Shapiro, 1986, Risk and return: Consumption
versus market beta, Review of Economics and Statistics 68, 452-459.
19
Newey, Whitney K., and Kenneth D. West, 1987, A simple, positive semidefinite,
heteroscedasticity and autocorrelation consistent covariance matrix,
Econometrica 55, 703-708.
Osterwald-Lenum, Michael, 1992, A note with quartiles of the asymptotic distribution
of the maximum likelihood cointegration rank test statistic, Oxford Bulletin of
Economics and Statistics 54, 461-472.
Phillips, P.C.B., and P. Perron, 1988, Testing for a unit root in time series regressions,
Biometrika 65, 335-346.
Phillips, Peter, and Sam Ouliaris, 1990, Asymptotic properties of residual based tests
for cointegration, Econometrica 58, 165-193.
Rubinstein, Mark, 1976, The valuation of uncertain income streams and the pricing of
options, Bell Journal of economics and Management Science 7, 407-425.
Stock, James H., and Mark Watson, 1993, A simple estimator of cointegrating vectors
in higher order integrated systems, Econometrica 6, 783-820.
Tan, Alvkin, and Graham Voss, 2003, Consumption and wealth in Australia, The
Economic Record 79, 39-56.
Wheatley, Simon, 1988, Some tests of international equity integration, Journal of
Financial Economics 21, 177-212.
20
Table 1. Summary Statistics for the US Market
e
Notes: rt 1 is quarterly log excess return on the market index; scrt is the surplus consumption ratio;
cay t ct a at y y t , where ct is log consumption, at is the log of household net asset wealth, and yt is log
after-tax labor income; ct 1 is the consumption growth rate at time t+1. Autocorrelation coefficient is computed
as the first-order autocorrelation.
Australia
sample period: 1979Q1-2004Q2
e
scrt
rt 1
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
-0.184
1
cay t
0.114
0.013
1
ct 1
-0.009
-0.185
-0.167
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
-0.003
-3.264
-2.197
0.005
0.085
0.279
0.019
0.007
0.050
0.813
0.880
-0.195
Canada
sample period: 1985Q1-2004Q4
e
scrt
rt 1
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
-0.069
1
cay t
0.053
0.559
1
ct 1
0.251
0.059
0.174
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
4.0E-04
-3.541
-1.141
0.004
0.087
0.730
0.014
0.008
-0.029
0.938
0.755
-0.017
France
sample period: 1985Q1-2004Q4
e
rt 1
scrt
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
0.053
1
cay t
0.258
0.259
1
ct 1
-0.227
-0.241
-0.332
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
-0.001
-2.295
-3.132
0.004
0.130
0.266
0.032
0.018
-0.056
0.788
0.759
-0.415
21
Germany
sample period: 1985Q1-2004Q4
e
scrt
rt 1
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
0.046
1
cay t
0.228
-0.091
1
ct 1
-0.257
-0.086
-0.046
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
0.009
-2.877
-2.800
0.005
0.139
0.635
0.035
0.014
-0.080
0.950
0.841
-0.099
Italy
sample period: 1985Q1-2004Q4
e
scrt
rt 1
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
-0.054
1
cay t
0.299
-0.338
1
ct 1
0.143
0.083
-0.166
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
0.008
-2.213
-25.722
0.001
0.133
0.600
0.236
0.022
0.103
0.882
0.900
0.336
Japan
sample period: 1985Q1-2004Q4
e
rt 1
scrt
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
0.050
1
cay t
0.143
-0.478
1
ct 1
0.181
0.228
-0.095
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
-0.003
-3.597
-4.581
0.004
0.120
0.988
0.070
0.010
0.001
0.984
0.917
-0.200
22
U.K.
sample period: 1985Q1-2004Q4
e
scrt
rt 1
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
-0.012
1
cay t
-0.188
0.360
1
ct 1
0.113
0.299
-0.151
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
-0.001
-2.761
-1.689
0.009
0.092
0.352
0.013
0.008
-0.049
0.953
0.733
0.330
U.S.
sample period: 1964Q1-2004Q4
e
rt 1
scrt
cay t
ct 1
A: Correlation Matrix
e
rt 1
1
scrt
-0.083
1
cay t
-0.074
-0.176
1
ct 1
0.263
0.081
-0.275
Mean
Standard Error
Autocorrelation
1
B: Univariate Summary Statistics
0.008
-5.669
-0.129
0.001
0.084
0.922
0.005
0.001
-0.031
0.957
0.887
0.359
23
Table 2. Long-horizon Forecasting Regressions for the International Stock Markets
Notes: This table presents the results from the regression of the long horizon excess market returns in world’s
eight major markets: Australia, Canada, France, Germany, Italy, Japan, U.K., and U.S. K denotes the return
horizon in quarters, spanning from 1 to 16. The dependent variable is the cumulative excess market returns over K
K
period:
r
e
tk
. In each panel, the three explanatory variables enter the regression independently in the first sub-
k 1
panel, which are as the followings: the log surplus consumption ratio, scrt, which is derived from modified
Campbell and Cochrane (1999) model, where habit persistent 0.90 and investors’ risk aversion 2 ; the log
consumption-wealth ratio cay t ct a at y yt , where ct is log consumption, at is the log of household net
asset wealth, and yt is log after-tax labor income, a and y are estimated by dynamic least square method, and
scrt and the consumption-wealth ratio cay t are lagged
one quarter period; another variable is the consumption growth rate, ct 1 ct 1 ct . The second sub-panel,
six lags are used; both the log surplus consumption ratio
scrt , cay t and ct 1 enter the regression simultaneously as the explanatory variables. Significant coefficients at
the 5% significance level are highlighted in bold face. Newey-West adjusted t-statistics are placed in parenthesis
under the estimated coefficients. The adjusted R2 of the regression is placed in the square bracket below the tstatistics.
Country
Australia
Regression
1
2
3
Regression Horizons
1
2
4
8
12
sample period: 1979Q1-2004Q2
Regression with single state variable
Regressor
scrt
cay t
ct 1
-0.05
-0.06
-0.03
0.04
0.05
0.11
(-2.51)
[0.03]
(-1.36)
[0.01]
(-0.35)
[-0.01]
(0.30)
[-0.01]
(0.32)
[-0.01]
(0.47)
[0.00]
0.45
1.32
3.28
5.56
5.03
2.98
(0.87)
[0.00]
(1.44)
[0.05]
(2.36)
[0.17]
(3.32)
[0.32]
(2.07)
[0.20]
(1.07)
[0.05]
-0.10
-2.37
-3.71
-2.73
-3.73
-5.33
(-0.09)
[-0.01]
4
scrt
cay t
ct 1
16
(-1.36)
(-1.37)
(-1.10)
(-1.39)
[0.01]
[0.01]
[0.00]
[0.00]
Regression with multiple state variables
(-1.76)
[0.01]
-0.05
-0.07
-0.02
0.12
0.17
0.18
(-2.50)
(-1.69)
(-0.27)
(1.04)
(1.15)
(0.73)
0.39
1.12
3.13
5.91
5.57
3.42
(0.73)
(1.19)
(2.29)
(3.78)
(2.66)
(1.71)
-0.30
-2.23
-2.05
1.67
0.96
-1.64
(-0.26)
[0.02]
(-1.20)
[0.07]
(-0.86)
[0.16]
(0.79)
[0.33]
(0.47)
[0.21]
(-0.57)
[0.06]
24
Country
Canada
Regression
1
2
3
4
1
2
12
sample period: 1985Q1-2004Q4
Regression with single state variable
scrt
cay t
ct 1
scrt
cay t
ct 1
Country
France
Regression
1
2
3
Regression Horizons
4
8
Regressor
-0.02
-0.03
-0.08
-0.13
-0.18
-0.28
(-1.42)
[0.00]
(-1.64)
[0.02]
(-2.39)
[0.07]
(-3.36)
[0.17]
(-4.42)
[0.31]
(-7.86)
[0.56]
0.33
0.78
1.34
0.37
-0.53
-2.83
(0.43)
[-0.01]
(0.59)
[-0.01]
(0.57)
[0.00]
(0.14)
[-0.01]
(-0.17)
[-0.01]
(-0.87)
[0.01]
2.62
2.63
5.46
4.38
4.42
1.35
(3.07)
[0.05]
(1.71)
(2.50)
(2.02)
(1.86)
[0.02]
[0.06]
[0.02]
[0.02]
Regression with multiple state variables
-0.02
-0.03
-0.08
-0.17
-0.25
-0.30
(-1.23)
(-1.54)
(-2.21)
(-3.11)
(-5.19)
(-6.14)
0.52
1.53
3.22
4.10
4.54
0.93
(0.59)
(1.09)
(1.33)
(1.42)
(1.34)
(0.32)
2.55
2.36
4.93
4.07
1.35
2.53
(2.87)
[0.04]
(1.52)
[0.03]
(2.31)
[0.12]
(1.65)
[0.25]
(0.56)
[0.41]
(0.91)
[0.66]
Regression Horizons
1
2
4
8
12
sample period: 1985Q1-2004Q4
Regression with single state variable
Regressor
scrt
cay t
ct 1
scrt
cay t
ct 1
(0.43)
[-0.01]
16
0.04
-0.03
-0.09
-0.18
-0.35
-0.66
(0.31)
[-0.01]
(-0.15)
[-0.02]
(-0.29)
[-0.01]
(-0.44)
[0.00]
(-0.70)
[0.02]
(-1.50)
[0.09]
1.14
2.50
4.87
5.95
6.07
4.83
(1.98)
[0.05]
(3.40)
[0.16]
(4.67)
[0.34]
(4.57)
[0.32]
(4.21)
[0.24]
(1.80)
[0.08]
-2.12
-1.03
-0.94
-1.60
-2.38
-4.30
(-1.18)
[0.03]
4
16
(-0.72)
(-0.65)
(-0.83)
(-1.18)
[-0.01]
[-0.01]
[-0.01]
[-0.01]
Regression with multiple state variables
(-1.79)
[0.00]
-0.03
-0.15
-0.40
-0.99
-1.53
-1.43
(-0.28)
(-1.05)
(-1.69)
(-3.88)
(-4.44)
(-3.36)
0.95
2.80
5.78
7.58
7.03
5.61
(1.45)
(3.40)
(5.15)
(6.00)
(3.37)
(2.30)
-1.55
0.50
1.91
-0.04
-3.76
-5.69
(-0.80)
[0.04]
(0.36)
[0.15]
(1.58)
[0.42]
(-0.03)
[0.45]
(-2.39)
[0.42]
(-2.02)
[0.27]
25
Country
Germany
Regression
1
2
3
Regression Horizons
4
8
Regressor
1
2
12
sample period: 1985Q1-2004Q4
Regression with single state variable
scrt
cay t
ct 1
0.00
0.02
0.03
0.07
0.19
0.42
(0.14)
[-0.02]
(0.40)
[-0.01]
(0.39)
[-0.01]
(0.56)
[-0.01]
(1.03)
[0.03]
(3.54)
[0.20]
0.97
1.71
2.26
3.57
4.27
3.73
(2.58)
[0.03]
(2.77)
[0.07]
(2.74)
[0.06]
(1.66)
[0.09]
(1.43)
[0.08]
(1.16)
[0.05]
-2.40
-3.03
-2.44
-2.43
-3.24
-3.45
(-3.28)
[0.05]
4
scrt
cay t
ct 1
Country
Italy
Regression
1
2
3
0.04
0.07
0.15
0.33
0.58
(0.35)
(0.64)
(0.72)
(1.22)
(1.74)
(4.31)
0.94
1.71
2.32
3.81
4.85
4.83
(2.48)
(2.87)
(3.21)
(2.17)
(2.30)
(2.68)
-2.27
-2.73
-1.99
-1.53
-1.57
-0.29
(-3.31)
[0.06]
(-2.43)
[0.10]
(-1.20)
[0.06]
(-0.77)
[0.11]
(-0.52)
[0.18]
(-0.08)
[0.37]
Regression Horizons
1
2
4
8
12
sample period: 1985Q1-2004Q4
Regression with single state variable
cay t
ct 1
scrt
cay t
ct 1
(-0.85)
[-0.01]
0.01
16
-0.01
-0.02
-0.03
-0.03
-0.17
-0.31
(-0.58)
[-0.01]
(-0.67)
[-0.01]
(-0.41)
[-0.01]
(-0.42)
[-0.01]
(-1.93)
[0.06]
(-3.20)
[0.18]
-0.06
-0.10
-0.17
-0.18
-0.29
-0.48
(-2.25)
[0.04]
(-2.07)
[0.07]
(-1.77)
[0.09]
(-2.07)
[0.05]
(-3.38)
[0.12]
(-4.81)
[0.29]
0.84
1.09
0.40
0.28
1.16
-0.44
(1.26)
[0.01]
4
(-2.75)
(-1.49)
(-1.01)
(-1.01)
[0.04]
[0.00]
[-0.01]
[-0.01]
Regression with multiple state variables
Regressor
scrt
16
(1.23)
(0.26)
(0.10)
(0.40)
[0.00]
[-0.01]
[-0.01]
[-0.01]
Regression with multiple state variables
(-0.15)
[-0.02]
0.01
0.01
0.02
-0.05
-0.25
-0.39
(0.63)
(0.28)
(0.32)
(-0.55)
(-2.99)
(-7.14)
0.21
0.33
0.46
0.11
-0.14
0.04
(2.79)
(2.55)
(1.87)
(0.41)
(-0.40)
(0.08)
1.16
1.61
1.13
0.56
1.49
0.52
(1.79)
[0.09]
(1.76)
[0.11]
(0.76)
[0.08]
(0.20)
[-0.03]
(0.64)
[0.07]
(0.29)
[0.21]
26
Country
Japan
Regression
1
2
3
Regression Horizons
4
8
Regressor
1
2
12
sample period: 1985Q1-2004Q4
Regression with single state variable
scrt
cay t
ct 1
0.01
0.01
0.01
0.03
0.05
0.06
(0.40)
[-0.01]
(0.34)
[-0.01]
(0.16)
[-0.01]
(0.29)
[-0.01]
(0.44)
[0.00]
(0.43)
[-0.01]
0.25
0.58
1.49
2.95
3.96
4.44
(1.40)
[0.01]
(1.91)
[0.04]
(3.56)
[0.16]
(4.69)
[0.35]
(4.34)
[0.48]
(3.97)
[0.54]
2.07
1.62
3.38
3.51
2.19
3.74
(1.62)
[0.02]
4
scrt
cay t
ct 1
Country
U.K.
Regression
1
2
3
0.02
0.04
0.09
0.13
0.11
(0.45)
(0.83)
(0.76)
(1.28)
(1.61)
(1.21)
0.27
0.60
1.39
2.62
3.29
3.38
(2.14)
(2.94)
(5.47)
(8.27)
(8.81)
(7.53)
1.82
0.94
2.14
0.82
-1.46
0.08
(1.43)
[0.03]
(0.77)
[0.07]
(1.19)
[0.24]
(0.53)
[0.48]
(-0.99)
[0.59]
(0.04)
[0.59]
Regression Horizons
4
8
1
2
12
sample period: 1985Q1-2004Q4
Regression with single state variable
cay t
ct 1
scrt
cay t
ct 1
(0.84)
[-0.01]
0.01
16
-0.02
-0.07
-0.16
-0.41
-0.62
-0.79
(-0.60)
[-0.01]
(-1.10)
[0.00]
(-1.78)
[0.05]
(-2.74)
[0.22]
(-2.67)
[0.33]
(-2.95)
[0.45]
-1.09
-1.85
-0.86
3.56
8.83
14.05
(-1.56)
[0.01]
(-1.63)
[0.02]
(-0.38)
[-0.01]
(1.15)
[0.03]
(2.29)
[0.15]
(2.53)
[0.31]
1.41
3.19
1.01
3.08
-4.06
-5.43
(0.81)
[0.00]
4
(1.14)
(1.28)
(0.90)
(0.42)
[0.00]
[0.01]
[0.00]
[-0.01]
Regression with multiple state variables
Regressor
scrt
16
(1.24)
(0.38)
(0.82)
(-1.75)
[0.02]
[-0.01]
[0.00]
[0.00]
Regression with multiple state variables
(-2.04)
[0.00]
0.03
0.03
0.02
-0.15
-0.20
-0.38
(1.06)
(0.61)
(0.38)
(-1.73)
(-1.42)
(-2.03)
-1.71
-2.79
-3.88
-2.27
-2.86
1.60
(-2.23)
(-2.70)
(-2.01)
(-1.12)
(-1.07)
(0.40)
0.40
1.95
-0.49
4.91
-0.99
1.66
(0.23)
[0.02]
(0.83)
[0.08]
(-0.19)
[0.07]
(1.39)
[0.11]
(-0.41)
[0.12]
(0.56)
[0.19]
27
Country
U.S.
Regression
1
2
3
4
Regression Horizons
1
2
4
8
12
sample period: 1965Q1-2004Q4
Regression with single state variable
Regressor
scrt
cay t
ct 1
scrt
cay t
ct 1
16
-0.01
-0.03
-0.06
-0.12
-0.17
-0.21
(-1.96)
[0.01]
(-2.34)
[0.04]
(-2.90)
[0.10]
(-3.34)
[0.20]
(-3.58)
[0.29]
(-3.78)
[0.38]
-0.68
3.52
10.74
24.90
35.85
41.28
(-0.63)
[0.00]
(1.99)
[0.02]
(2.79)
[0.11]
(4.68)
[0.33]
(6.44)
[0.49]
(5.69)
[0.52]
10.77
15.72
15.07
-1.00
-16.39
-21.14
(3.19)
[0.04]
(2.73)
(1.23)
(-0.08)
(-0.85)
[0.04]
[0.02]
[-0.01]
[0.00]
Regression with multiple state variables
(-1.02)
[0.01]
-0.01
-0.02
-0.03
-0.06
-0.08
-0.12
(-1.65)
(-1.42)
(-1.65)
(-1.91)
(-2.36)
(-3.69)
-0.64
3.87
10.59
22.58
31.37
34.54
(-0.55)
(1.94)
(2.55)
(4.16)
(6.61)
(5.72)
12.25
22.20
29.95
27.68
23.39
28.61
(3.60)
[0.05]
(3.89)
[0.09]
(2.60)
[0.18]
(3.31)
[0.38]
(1.61)
[0.54]
(2.47)
[0.62]
28
Table 3. Relative Explanatory Power of cayt
Notes: This table presents the results of the long horizon regressions in world’s eight major markets: Australia,
Canada, France, Germany, Italy, Japan, U.K., and U.S. K denotes the return horizon in quarters, spanning from 1
to 16. In the first sub-panel of each panel, first we run the regression, Rt k a b scrt et k , where Rt k is
K
the cumulative excess market returns over K period:
Rt k rt e k , scrt is the log surplus consumption ratio,
k 1
which is derived from modified Campbell and Cochrane (1999) model, where habit persistent 0.90 and
investors’ risk aversion 2 , a and b as coefficient, and
et k is the regression residual; then we run et k and
Rt k et k separately on the consumption-wealth ratio, cay t ct a at y yt , where ct is log consumption, at
is the log of household net asset wealth, and yt is log after-tax labor income, a and y are estimated by dynamic
least square method, and six lags are used; both the log surplus consumption ratio
scrt and the consumption-
wealth ratio cay t are lagged one quarter period. For the second sub-panel, we run the similar procedure, except
that the first step is
Rt k a b scrt c ct 1 et k , where ct 1 ct 1 ct is the consumption growth
rate. The results of the second step are reported in the table. Significant coefficients at the 5% significance level
are highlighted in bold face. Newey-West adjusted t-statistics are placed in parenthesis under the estimated
coefficients. The adjusted R2 of the regression is placed in the square bracket below the t-statistics.
Country
Australia
Regression Horizons
4
8
Regressand
1
2
sample period: 1979Q1-2004Q2
12
16
Rt k a b scrt et k
et k
Rt k et k
0.41
1.27
3.25
5.64
5.17
3.34
(0.78)
[0.00]
(1.36)
[0.04]
(2.32)
[0.17]
(3.41)
[0.33]
(2.15)
[0.21]
(1.23)
[0.07]
0.04
0.06
0.03
-0.07
-0.14
-0.35
(0.43)
[-0.01]
(0.43)
[-0.01]
(0.50)
[0.00]
(-1.22)
[0.01]
(-2.73)
[0.05]
(-3.24)
[0.06]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
0.37
1.07
2.98
5.45
4.88
2.92
(0.72)
[0.00]
(1.19)
[0.03]
(2.18)
[0.14]
(3.34)
[0.31]
(2.07)
[0.19]
(1.11)
[0.05]
0.08
0.25
0.30
0.12
0.16
0.07
(0.82)
[0.00]
(1.72)
[0.03]
(2.11)
[0.04]
(1.22)
[0.00]
(1.14)
[0.00]
(0.30)
[-0.01]
29
Country
Canada
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
12
16
Rt k a b scrt et k
et k
Rt k et k
0.57
1.25
2.68
3.24
3.31
0.88
(0.77)
[0.00]
(0.98)
[0.01]
(1.22)
[0.03]
(1.29)
[0.04]
(1.19)
[0.04]
(0.37)
[-0.01]
-0.24
-0.47
-1.34
-3.76
-6.14
-9.68
(-5.21)
[0.30]
(-4.64)
[0.30]
(-4.03)
[0.30]
(-3.84)
[0.30]
(-4.08)
[0.29]
(-4.43)
[0.33]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
Country
France
0.35
1.02
2.15
2.75
3.04
0.59
(0.48)
[-0.01]
(0.83)
[0.00]
(1.02)
[0.02]
(1.14)
[0.03]
(1.11)
[0.03]
(0.25)
[-0.01]
-0.01
-0.24
-0.82
-3.27
-5.87
-9.39
(-0.08)
[-0.01]
(-1.31)
[0.00]
(-1.68)
[0.02]
(-3.10)
[0.18]
(-3.83)
[0.26]
(-4.24)
[0.31]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b scrt et k
et k
Rt k et k
1.08
2.55
5.14
7.07
7.03
5.81
(1.93)
[0.04]
(3.43)
[0.16]
(4.93)
[0.39]
(6.28)
[0.38]
(3.81)
[0.24]
(2.18)
[0.11]
0.06
-0.05
-0.28
-1.00
-2.20
-2.62
(1.33)
[0.05]
(-1.17)
[0.05]
(-1.01)
[0.05]
(-0.98)
[0.05]
(-1.34)
[0.07]
(-1.58)
[0.13]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
0.81
2.39
4.94
6.48
5.70
4.20
(1.48)
[0.02]
(3.15)
[0.14]
(4.61)
[0.36]
(6.30)
[0.33]
(3.84)
[0.17]
(1.73)
[0.06]
0.33
0.10
-0.07
-0.41
-0.87
-1.01
(2.19)
[0.09]
(1.05)
[0.02]
(-0.24)
[-0.01]
(-0.37)
[-0.01]
(-0.46)
[-0.01]
(-0.56)
[-0.01]
30
Country
Germany
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
12
16
Rt k a b scrt et k
et k
Rt k et k
0.98
1.76
2.35
3.81
4.82
4.76
(2.56)
[0.04]
(2.75)
[0.08]
(2.88)
[0.07]
(1.99)
[0.11]
(2.01)
[0.12]
(2.26)
[0.14]
-0.02
-0.05
-0.09
-0.23
-0.55
-1.03
(-0.66)
[-0.01]
(-0.57)
[-0.01]
(-0.48)
[-0.01]
(-0.55)
[-0.01]
(-0.64)
[-0.01]
(-0.71)
[-0.01]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
Country
Italy
0.93
1.69
2.29
3.75
4.75
4.73
(2.48)
[0.03]
(2.84)
[0.08]
(2.98)
[0.07]
(1.98)
[0.11]
(2.01)
[0.12]
(2.26)
[0.14]
0.04
0.02
-0.03
-0.18
-0.49
-1.00
(0.23)
[-0.02]
(0.08)
[-0.02]
(-0.13)
[-0.02]
(-0.40)
[-0.01]
(-0.55)
[-0.01]
(-0.68)
[-0.01]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b scrt et k
et k
Rt k et k
0.17
0.27
0.40
0.09
-0.14
0.03
(2.41)
[0.07]
(2.07)
[0.08]
(1.56)
[0.08]
(0.31)
[-0.01]
(-0.37)
[-0.01]
(0.05)
[-0.02]
0.01
0.02
0.03
0.05
0.21
0.36
(3.00)
[0.10]
(2.54)
[0.10]
(2.10)
[0.10]
(1.79)
[0.10]
(1.68)
[0.10]
(1.62)
[0.09]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
0.18
0.28
0.40
0.09
-0.12
0.04
(2.69)
[0.08]
(2.27)
[0.09]
(1.60)
[0.08]
(0.33)
[-0.01]
(-0.31)
[-0.01]
(0.06)
[-0.02]
0.00
0.01
0.03
0.05
0.19
0.35
(-0.18)
[-0.01]
(0.78)
[-0.01]
(1.88)
[0.04]
(1.75)
[0.07]
(1.66)
[0.07]
(1.61)
[0.08]
31
Country
Japan
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
12
16
Rt k a b scrt et k
et k
Rt k et k
0.30
0.65
1.57
3.15
4.24
4.68
(1.65)
[0.01]
(2.21)
[0.06]
(3.91)
[0.18]
(5.82)
[0.41]
(5.58)
[0.56]
(4.88)
[0.61]
-0.04
-0.08
-0.08
-0.20
-0.29
-0.24
(-2.80)
[0.22]
(-2.35)
[0.22]
(-1.93)
[0.22]
(-1.59)
[0.20]
(-1.44)
[0.17]
(-1.35)
[0.12]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
Country
U.K.
0.29
0.65
1.56
3.14
4.24
4.67
(1.69)
[0.01]
(2.23)
[0.06]
(3.95)
[0.18]
(5.89)
[0.41]
(5.60)
[0.56]
(4.98)
[0.61]
-0.04
-0.07
-0.07
-0.19
-0.28
-0.23
(-1.04)
[0.00]
(-1.69)
[0.06]
(-0.97)
[0.00]
(-1.26)
[0.08]
(-1.33)
[0.15]
(-1.10)
[0.06]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b scrt et k
et k
Rt k et k
-1.40
-2.37
-1.80
1.68
5.96
11.02
(-1.99)
[0.02]
(-2.19)
[0.04]
(-0.90)
[0.01]
(0.69)
[0.00]
(3.09)
[0.06]
(3.23)
[0.20]
-0.03
-0.25
-0.75
-2.17
-3.90
-5.26
(-2.52)
[0.12]
(-2.14)
[0.12]
(-1.79)
[0.12]
(-1.59)
[0.12]
(-1.56)
[0.11]
(-1.49)
[0.10]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
-1.14
-1.76
-1.45
2.76
6.11
11.28
(-1.64)
[0.01]
(-1.67)
[0.02]
(-0.72)
[0.00]
(1.15)
[0.02]
(3.18)
[0.06]
(3.37)
[0.21]
-0.28
-0.86
-1.10
-3.25
-4.05
-5.52
(-3.05)
[0.08]
(-3.46)
[0.12]
(-2.47)
[0.19]
(-2.19)
[0.19]
(-1.60)
[0.12]
(-1.53)
[0.11]
32
Country
U.S.
Regression Horizons
4
8
Regressand
1
2
sample period: 1965Q1-2004Q4
12
16
Rt k a b scrt et k
et k
Rt k et k
-1.26
2.16
7.60
18.44
26.56
29.23
(-1.21)
[0.00]
(1.25)
[0.00]
(2.00)
[0.05]
(3.61)
[0.21]
(5.06)
[0.36]
(4.70)
[0.40]
0.58
1.36
3.14
6.46
9.30
12.05
(4.77)
[0.12]
(4.07)
[0.12]
(3.38)
[0.12]
(2.78)
[0.12]
(2.53)
[0.12]
(2.42)
[0.12]
Rt k a b scrt c ct 1 et k
et k
Rt k et k
-0.48
3.37
9.01
19.23
26.73
29.50
(-0.51)
[-0.01]
(2.18)
[0.02]
(2.66)
[0.08]
(3.96)
[0.24]
(5.13)
[0.36]
(4.80)
[0.41]
-0.20
0.15
1.73
5.68
9.13
11.78
(-0.48)
[0.00]
(0.19)
[-0.01]
(1.32)
[0.02]
(2.33)
[0.09]
(2.48)
[0.12]
(2.36)
[0.12]
33
Table 4. Relative Explanatory Power of scrt
Notes: This table presents the results of the long horizon regressions in world’s eight major markets: Australia,
Canada, France, Germany, Italy, Japan, U.K., and U.S. K denotes the return horizon in quarters, spanning from 1
to 16. In the first sub-panel of each panel, first we run the regression,
Rt k a b cay t et k , where Rt k is
K
Rt k rt e k , cay t ct a at y yt is the consumption-
the cumulative excess market returns over K period:
k 1
wealth ratio, where ct is log consumption, at is the log of household net asset wealth, and yt is log after-tax labor
income, a and y are estimated by dynamic least square method, and six lags are used, a and b are slope
coefficients, and
et k is the regression residual; then we run et k and Rt k et k separately on scrt , the log
surplus consumption ratio, which is derived from modified Campbell and Cochrane (1999) model, where habit
persistent 0.90 and investors’ risk aversion 2 ; both the log surplus consumption ratio scrt and the
consumption-wealth ratio cay t are lagged one quarter period. For the second sub-panel, we run the similar
procedure, except that the first step is
Rt k a b cay t c ct 1 et k , where ct 1 ct 1 ct is the
consumption growth rate. The results of the second step are reported in the table. Significant coefficients at the
5% significance level are highlighted in bold face. Newey-West adjusted t-statistics are placed in parenthesis
under the estimated coefficients. The adjusted R2 of the regression is placed in the square bracket below the tstatistics.
Country
Australia
Regressand
1
2
sample period: 1979Q1-2004Q2
Regression Horizons
4
8
12
16
Rt k a b cay t et k
et k
Rt k et k
-0.06
-0.06
-0.02
0.11
0.14
0.17
(-2.49)
[0.02]
(-1.27)
[0.01]
(-0.21)
[-0.01]
(0.67)
[0.01]
(0.68)
[0.01]
(0.58)
[0.01]
4.6E-04
-6.2E-05
-0.01
-0.05
-0.07
-0.01
(0.09)
[-0.01]
(0.00)
[-0.01]
(-0.17)
[-0.01]
(-0.86)
[0.02]
(-1.88)
[0.07]
(-2.14)
[0.10]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
-0.06
-0.07
-0.03
0.10
0.12
0.12
(-2.47)
[0.02]
(-1.47)
[0.02]
(-0.33)
[-0.01]
(0.64)
[0.00]
(0.59)
[0.00]
(0.41)
[0.00]
0.00
0.01
0.01
-0.05
-0.05
0.04
(-0.02)
[-0.01]
(0.62)
[0.00]
(0.16)
[-0.01]
(-0.79)
[0.01]
(-1.39)
[0.02]
(2.17)
[0.04]
34
Country
Canada
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
12
16
Rt k a b cay t et k
et k
Rt k et k
-0.01
-0.02
-0.06
-0.12
-0.17
-0.20
(-1.11)
[0.00]
(-1.26)
[0.01]
(-1.84)
[0.05]
(-2.61)
[0.16]
(-4.35)
[0.28]
(-5.62)
[0.38]
0.00
0.01
0.01
-0.01
-0.03
-0.09
(5.84)
[0.30]
(5.42)
[0.30]
(4.97)
[0.30]
(-5.50)
[0.30]
(-5.93)
[0.29]
(-8.19)
[0.33]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
Country
France
-0.01
-0.02
-0.06
-0.12
-0.17
-0.20
(-1.04)
[0.00]
(-1.22)
[0.01]
(-1.83)
[0.05]
(-2.61)
[0.16]
(-4.37)
[0.28]
(-5.63)
[0.38]
0.00
0.01
0.01
-0.01
-0.03
-0.09
(0.97)
[-0.01]
(2.47)
[0.04]
(1.76)
[0.02]
(-1.71)
[0.01]
(-6.56)
[0.25]
(-8.51)
[0.32]
12
16
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
Rt k a b cay t et k
et k
Rt k et k
-0.01
-0.14
-0.39
-0.92
-1.35
-1.18
(-0.09)
[-0.02]
(-0.98)
[0.01]
(-1.52)
[0.11]
(-3.31)
[0.26]
(-4.96)
[0.33]
(-3.34)
[0.21]
0.05
0.11
0.21
0.27
0.22
0.18
(1.47)
[0.05]
(1.30)
[0.05]
(1.15)
[0.05]
(1.17)
[0.05]
(1.59)
[0.07]
(2.34)
[0.13]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
-0.03
-0.13
-0.36
-0.90
-1.36
-1.21
(-0.27)
[-0.02]
(-0.91)
[0.00]
(-1.38)
[0.09]
(-3.12)
[0.25]
(-5.03)
[0.34]
(-3.51)
[0.22]
0.07
0.10
0.18
0.24
0.23
0.20
(1.66)
[0.08]
(1.17)
[0.04]
(0.94)
[0.03]
(1.03)
[0.04]
(1.68)
[0.08]
(3.03)
[0.12]
35
Country
Germany
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
12
16
Rt k a b cay t et k
et k
Rt k et k
0.02
0.04
0.07
0.15
0.33
0.57
(0.51)
[-0.01]
(0.76)
[0.00]
(0.78)
[0.01]
(1.21)
[0.04]
(1.73)
[0.12]
(4.15)
[0.35]
-0.01
-0.01
-0.01
-0.02
-0.04
-0.03
(-0.62)
[-0.01]
(-0.53)
[-0.01]
(-0.45)
[-0.01]
(-0.50)
[-0.01]
(-0.58)
[-0.01]
(-0.65)
[-0.01]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
Country
Italy
0.01
0.04
0.06
0.15
0.32
0.56
(0.35)
[-0.02]
(0.64)
[0.00]
(0.73)
[0.00]
(1.17)
[0.04]
(1.67)
[0.12]
(4.11)
[0.34]
0.00
0.00
-0.01
-0.02
-0.03
-0.02
(0.04)
[-0.02]
(-0.14)
[-0.02]
(-0.28)
[-0.02]
(-0.42)
[-0.01]
(-0.47)
[-0.01]
(-0.44)
[-0.02]
12
16
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
Rt k a b cay t et k
et k
Rt k et k
0.01
0.01
0.02
-0.04
-0.22
-0.35
(0.68)
[-0.01]
(0.30)
[-0.01]
(0.28)
[-0.01]
(-0.47)
[-0.01]
(-2.20)
[0.08]
(-3.65)
[0.19]
-0.02
-0.04
-0.05
-0.02
-0.01
-0.04
(-2.51)
[0.10]
(-2.21)
[0.10]
(-2.00)
[0.10]
(-2.12)
[0.10]
(-2.16)
[0.10]
(-1.92)
[0.09]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
0.01
0.01
0.02
-0.04
-0.22
-0.35
(0.66)
[-0.01]
(0.25)
[-0.01]
(0.27)
[-0.01]
(-0.48)
[-0.01]
(-2.22)
[0.08]
(-3.65)
[0.19]
-0.02
-0.03
-0.05
-0.02
-0.01
-0.04
(-2.17)
[0.06]
(-2.11)
[0.07]
(-2.05)
[0.09]
(-2.20)
[0.08]
(-1.51)
[0.00]
(-1.92)
[0.09]
36
Country
Japan
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
12
16
Rt k a b cay t et k
et k
Rt k et k
0.02
0.03
0.07
0.14
0.20
0.22
(1.00)
[0.00]
(1.16)
[0.02]
(1.34)
[0.06]
(2.10)
[0.20]
(3.02)
[0.32]
(3.21)
[0.30]
-0.01
-0.02
-0.05
-0.11
-0.15
-0.16
(-4.90)
[0.22]
(-4.06)
[0.22]
(-3.26)
[0.22]
(-2.52)
[0.20]
(-2.09)
[0.17]
(-1.67)
[0.12]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
Country
U.K.
0.01
0.03
0.06
0.13
0.19
0.21
(0.71)
[-0.01]
(1.01)
[0.01]
(1.15)
[0.04]
(1.94)
[0.17]
(2.88)
[0.30]
(2.97)
[0.28]
0.00
-0.02
-0.04
-0.10
-0.14
-0.14
(-1.44)
[0.00]
(-2.89)
[0.10]
(-2.47)
[0.12]
(-2.16)
[0.15]
(-1.92)
[0.14]
(-1.52)
[0.10]
12
16
Regressand
1
2
sample period: 1985Q1-2004Q4
Regression Horizons
4
8
Rt k a b cay t et k
et k
Rt k et k
0.01
0.01
-0.04
-0.20
-0.34
-0.49
(0.57)
[-0.01]
(0.22)
[-0.01]
(-0.66)
[-0.01]
(-2.12)
[0.10]
(-2.39)
[0.22]
(-3.16)
[0.38]
-0.02
-0.03
-0.03
-0.01
0.02
0.05
(-3.18)
[0.12]
(-2.72)
[0.12]
(-2.38)
[0.12]
(-2.36)
[0.12]
(2.52)
[0.11]
(2.16)
[0.10]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
0.01
-0.01
-0.04
-0.22
-0.31
-0.45
(0.25)
[-0.01]
(-0.24)
[-0.01]
(-0.70)
[0.00]
(-2.35)
[0.13]
(-2.19)
[0.18]
(-2.90)
[0.33]
-0.01
-0.01
-0.03
0.02
-0.01
0.02
(-1.40)
[0.02]
(-0.85)
[0.00]
(-2.09)
[0.09]
(2.10)
[0.05]
(-0.56)
[-0.01]
(0.74)
[-0.01]
37
Country
U.S.
Regressand
1
2
sample period: 1965Q1-2004Q4
Regression Horizons
4
8
12
16
Rt k a b cay t et k
et k
Rt k et k
-0.01
-0.01
-0.03
-0.05
-0.07
-0.11
(-1.42)
[0.01]
(-1.10)
[0.01]
(-1.32)
[0.02]
(-1.74)
[0.06]
(-2.18)
[0.12]
(-3.72)
[0.22]
1.4E-03
-0.01
-0.02
-0.05
-0.07
-0.08
(3.28)
[0.12]
(-2.76)
[0.12]
(-2.28)
[0.12]
(-1.89)
[0.12]
(-1.72)
[0.12]
(-1.62)
[0.12]
Rt k a b cay t c ct 1 et k
et k
Rt k et k
-0.01
-0.02
-0.03
-0.06
-0.08
-0.11
(-1.89)
[0.02]
(-1.65)
[0.02]
(-1.80)
[0.04]
(-1.99)
[0.08]
(-2.35)
[0.14]
(-4.00)
[0.25]
4.1E-03
-2.1E-03
-0.02
-0.05
-0.07
-0.08
(2.51)
[0.04]
(-0.48)
[0.00]
(-1.36)
[0.03]
(-1.63)
[0.09]
(-1.60)
[0.11]
(-1.52)
[0.11]
38
Table 5. Relative Explanatory Power of ct 1
Notes: This table presents the results of the long horizon regressions in world’s eight major markets: Australia,
Canada, France, Germany, Italy, Japan, U.K., and U.S. K denotes the return horizon in quarters, spanning from 1
to 16. In the first sub-panel of each panel, first we run the regression,
Rt k a b cay t c scrt et k ,
K
where
Rt k is the cumulative excess market returns over K period: Rt k rt e k , cay t ct a at y yt is the
k 1
consumption-wealth ratio, where ct is log consumption, at is the log of household net asset wealth, and yt is log
after-tax labor income, a and y are estimated by dynamic least square method, and six lags are used, scrt is the
log surplus consumption ratio, which is derived from modified Campbell and Cochrane (1999) model, where
habit persistent 0.90 and investors’ risk aversion 2 , a, b and c are slope coefficients, and et k is the
scrt and the consumption-wealth ratio cay t are
et k separately on ct 1 , where ct 1 ct 1 ct is the
regression residual; both the log surplus consumption ratio
lagged one quarter period. Then we run
et k and Rt k
consumption growth rate. The results of the second step are reported in the table. Significant coefficients at the
5% significance level are highlighted in bold face. Newey-West adjusted t-statistics are placed in parenthesis
under the estimated coefficients. The adjusted R2 of the regression is placed in the square bracket below the tstatistics.
Country
Australia
Regression Horizons
4
8
Regressand
1
2
sample period: 1979Q1-2004Q2
12
16
Rt k a b cay t c scrt et k
et k
Rt k et k
Country
Canada
-0.27
(-0.27)
[-0.01]
-2.24
(-1.30)
[0.01]
-2.72
(-1.04)
[0.00]
0.02
(0.01)
[-0.01]
-1.03
(-0.42)
[-0.01]
-3.28
(-1.05)
[0.00]
0.17
(0.73)
[-0.01]
-0.13
(-0.34)
[-0.01]
-0.99
(-1.65)
[0.01]
-2.75
(-3.52)
[0.04]
-2.70
(-4.70)
[0.08]
-2.05
(-5.24)
[0.12]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b cay t c scrt et k
et k
Rt k et k
2.46
(2.95)
[0.04]
2.28
(1.51)
[0.01]
4.76
(2.29)
[0.05]
3.92
(1.71)
[0.02]
1.29
(0.59)
[-0.01]
2.41
(0.91)
[0.00]
0.16
(0.89)
[0.00]
0.35
(0.91)
[0.00]
0.70
(0.78)
[0.00]
0.50
(0.31)
[-0.01]
0.06
(0.03)
[-0.02]
-1.59
(-0.54)
[-0.01]
39
Country
France
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
12
16
Rt k a b cay t c scrt et k
et k
Rt k et k
Country
Germany
-1.34
(-0.71)
[0.00]
0.43
(0.29)
[-0.02]
1.65
(1.27)
[0.00]
-0.03
(-0.02)
[-0.02]
-3.14
(-2.49)
[0.00]
-4.68
(-2.48)
[0.01]
-0.78
(-2.74)
[0.09]
-1.46
(-2.19)
[0.05]
-2.60
(-1.88)
[0.03]
-2.35
(-1.18)
[0.00]
-1.16
(-0.61)
[-0.02]
-0.85
(-0.61)
[-0.02]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b cay t c scrt et k
et k
Rt k et k
Country
Italy
-2.24
(-3.36)
[0.04]
-2.70
(-2.51)
[0.03]
-1.97
(-1.31)
[0.00]
-1.52
(-0.72)
[-0.02]
-1.54
(-0.54)
[-0.02]
-0.28
(-0.08)
[-0.02]
-0.16
(-0.48)
[-0.01]
-0.32
(-0.54)
[-0.01]
-0.47
(-0.61)
[-0.01]
-0.91
(-0.89)
[-0.01]
-1.70
(-1.67)
[-0.01]
-3.16
(-2.52)
[0.00]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b cay t c scrt et k
et k
Rt k et k
1.12
(1.76)
[0.03]
1.56
(1.70)
[0.02]
1.10
(0.68)
[-0.01]
0.55
(0.19)
[-0.01]
1.45
(0.57)
[-0.01]
0.51
(0.20)
[-0.02]
-0.28
(-1.72)
[0.01]
-0.48
(-1.71)
[0.01]
-0.70
(-1.64)
[0.01]
-0.26
(-1.46)
[0.01]
-0.29
(-0.51)
[-0.01]
-0.94
(-1.20)
[-0.01]
40
Country
Japan
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
12
16
Rt k a b cay t c scrt et k
et k
Rt k et k
Country
U.K.
1.90
(1.62)
[0.02]
1.30
(1.02)
[-0.01]
2.95
(1.47)
[0.01]
2.51
(1.34)
[0.00]
0.77
(0.42)
[-0.01]
2.36
(0.87)
[0.00]
0.17
(0.70)
[-0.01]
0.32
(0.59)
[-0.01]
0.43
(0.31)
[-0.01]
1.00
(0.33)
[-0.01]
1.42
(0.37)
[-0.01]
1.39
(0.40)
[-0.01]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1985Q1-2004Q4
Rt k a b cay t c scrt et k
et k
Rt k et k
Country
U.S.
0.78
(0.45)
[-0.01]
2.37
(0.94)
[0.01]
1.15
(0.44)
[-0.01]
6.84
(2.19)
[0.05]
3.04
(1.73)
[0.00]
5.33
(2.28)
[0.02]
0.63
(2.65)
[0.05]
0.82
(1.74)
[0.02]
-0.14
(-0.24)
[-0.01]
-3.76
(-3.70)
[0.12]
-7.09
(-4.39)
[0.14]
-10.77
(-4.14)
[0.16]
12
16
Regression Horizons
4
8
Regressand
1
2
sample period: 1965Q1-2004Q4
Rt k a b cay t c scrt et k
et k
Rt k et k
11.10
(3.34)
[0.04]
20.10
(3.68)
[0.07]
27.01
(2.63)
[0.08]
25.01
(3.38)
[0.05]
21.14
(1.46)
[0.03]
24.96
(2.16)
[0.05]
-0.32
(-0.56)
[0.00]
-4.37
(-3.16)
[0.09]
-11.94
(-2.75)
[0.09]
-26.01
(-2.44)
[0.09]
-37.53
(-2.27)
[0.09]
-46.10
(-2.22)
[0.09]
41
Appendix A: Description of Australian Dataset
Macroeconomic data usually only exist in lower frequency such as quarterly or annually. In Australia,
Australia Bureau of Statistics (ABS) maintains a good record of macroeconomic data. Our consumption, wealth
and labor income data are constructed from the time series spreadsheets from AusStats Database 7. These variables
are only available in quarterly or annually. For this paper, we use the data dating back to the fourth quarter of
1976 (1976Q4) and until the second quarter of 2004 (2004Q2), which yields 111 observations. The data used here
are quarterly, seasonally adjusted, real per capita data, measured in 1989-90 dollar.
The consumption data are for non-durables goods and services. Using non-durables goods and services
for consumption data has been used by many researchers (e.g. Lettau and Ludvigson, 2001a; Li, Lu and Zhong,
2004) It is calculated as Total Household Final Consumption Expenditure less Clothing and Footwear,
Furnishings and Household Equipment, and Purchase of Vehicles ($m, seasonally adjusted in current prices),
which is taken from series 5206058.1 of AusStats Time Series Spreadsheets.
CPI (Consumer Price Index), from Series 640101b of AusStats Time Series Spreadsheets, is used to
deflate all nominal variables in this paper. It is weighted average of all groups index of eight capital cities, with
base index 1989-90 = 100.
Aggregate wealth, particularly human capital, is not directly observable. In order to use it for forecasting
asset returns, we must find a proxy for human capital. To overcome this obstacle, we use after tax labor income to
proxy human capital. We use a number of series from AusStats Time Series Spreadsheets to construct after-tax
labor income.
After-tax labor income is defined as wages and salaries plus transfer payment minus labor income tax.
INCOME WAGES TRANSFERS * TAX
(20)
Wages is quarterly non-farmer wage & salary earners’ average earnings. It is constructed from Non-farmer Wage
& Salary Earner’s average weekly earnings and measures of employment. Average weekly earnings (AWE) are
from AusStats Series 1364019 (seasonally adjusted, in $). WSE is the total number of non-farm civilian wage
and salary earners, available from AusStats Series 1364010. ω is the number of weeks in one quarter, calculated
as (1/7)*(365/4).
(21)
WAGES * AWE *WSE
Transfers are constructed as Total Secondary Income Receivable less Social Contributions for Workers
Compensation, available from AusStats Series 5206036. γ is the proportion of labor income in the total
household income, which is calculated as WAGES/(Total Primary Income). Tax is calculated as the sum of
Income Tax Payable and Other Current Tax on income, wealth etc. All above series are available in AusStats
Series 5206036.
The labor income we constructed above is aggregate labor income. We deflate this series with Australian
Population (available from AusStats Series 1364010) and CPI to get per capita real after-tax labor income.
Quarterly net household wealth data for the period from 1976Q4 to 1999Q3 are taken from Tan & Voss
(2003)8, which includes financial wealth and non-financial wealth. Using annual net wealth of household balance
sheet from AusStats Series 5204050, we extend household wealth data from 1999Q4 to 2004Q2 by interpolation.
We employed augmented Dickey-Fuller (1979) and Philips-Perron (1988) Unit Root test in the series of
household consumption, labor income and net household wealth. We find there is only one unit root in each of
these three series. Next we move to test whether there is any cointegration relationship among these three
variables. To do this, we conduct two kinds of cointegration test: one is Phillips-Ouliaris (1990) residual-based
cointegration test, which is to discern whether there is at least one cointegration vector among these three
variables; the other test is a more popular one: Johansen’s (1988, 1991) L-Max test and trace statistics test, which
will tell us the number of cointegration vectors of the long-term relationship. Both tests suggest there is only one
cointegration vector among the three variables. The results of Phillips-Ouliaris cointegration test and Johansen
cointegration test are given in Panel A and Panel B of Appendix B, respectively.
Cointegration tests suggest that there is a shared trend in consumption, labor income and net asset wealth.
In order to examine the predictability of stock market returns using cay t , we should estimate the parameters of
the cointegration relation. Due to endogenously determined nature of ct, at and yt series, we use the single
equation procedure suggested in Stock and Watson (1993) and use the dynamic least squares (DLS) estimates
with Newey and West (1987) to correct for any residual serial correlation,
k
k
i 1
i 1
ct a at y yt bai at i byi yt i t .
Ordinary Least Square (OLS) estimate of (22) produces the super-consistent estimate of
(22)
a
and
y of
cointegration parameters (Lettau and Ludvigson 2001). Adding leads and lags of the first difference of net asset
wealth and labor income will capture the effects of regressor endogeneity in the linear regression of consumption
7
8
These series can be retrieved at http://www.abs.gov.au/ausstats/abs@.nsf/w2.3.1?OpenView.
Real per capita net household wealth data can be downloaded at http://web.uvic.ca/~gvoss/.
42
on asset wealth and labor income. From DLS estimation of (22), we can obtain the estimated trend deviation
cayt ct a at y yt , where ‘hat’ means estimated parameter.
Using quarterly data from 1976Q4 to 2004Q2, we obtain the estimated coefficients of the trend deviation
(Corresponding t-statistics are given below in the parenthesis) of constant, net asset wealth and labor income:
ct 1.476 0.405at 0.239 yt
(3.06) (11.85)
Thus
(23)
(2.29)
cayt ct 0.405at 0.239 yt .
For Australian stock market index, there are two commonly used indices: AllOrd Index and ASX/S&P
200 Index. AllOrd Index is from the Centre for Research in Finance (CRIF) database of the Australian Graduate
School of Management (AGSM), which is dating back to December 1979 with base index of 500 in that month.
ASX/S&P 200 index is collected from the bulletin statistics of Reserve Bank of Australia (RBA) with base index
of 500 at December 1979. They are monthly data, so we convert them into quarterly data using the average value
of the three months in each quarter. AllOrd Index is the value weighted price index for the companies traded in
Australian Stock Exchange (ASX) which should provide a better proxy for nonhuman components of household
asset wealth than ASX/S&P 200 index. We try stock market returns using both AllOrd Index and ASX/S&P 200
index, and the empirical results are not sensitive to the type of index used. We deflate the stock market index by
CPI to get the real stock market return. Stock market returns are log returns:
rt = log(AllOrdt) – log(AllOrdt-1) ,
(24)
where AllOrd is the series of All Ord Index.
For risk-free rate, we use the short-term 2-year government bond as a proxy. Quarterly and monthly 2year government short-term bond yields are also available in IFS until June 2004. Let us denote rmt as the log real
return of the stock market index, and rft as the log real return on the risk-free rate, then log real excess return on
the stock market is
rte rmt rft .
Appendix B: Cointegration Tests of Consumption, Household Income and Household Wealth in Eight
Countries
Notes: Panel A examines the presence of unit root from the cointegrating regression of consumption on after-tax
labor income and net household wealth using Philips-Ouliaris (1990) residual-based cointegration test for
Australia. Augmented Dickey-Fuller test is applied to the residuals from the regression of consumption (ct) on
wealth (at) and household income (yt). The optimal lag is chosen by the AIC criterion. The critical values are
assuming trending series. Significant t-test statistics at the 5% significance level are highlighted in bold face.
Panel B reports Johansen L-Max and Trace Statistics for cointegration tests among consumption, income and
asset wealth. A constant is included in the cointegration space. The column labeled “L-max90” and “L-max95”
denote the 90% and 95% confidence level of L-max statistics, respectively; the “Trace90” and “Trace95” gives
the 90% and 95% confidence level of trace statistics, respectively. “r” is the number of the cointegration relation.
Optimal lag is chosen by AIC criterion. Significant test statistics at 95% (90%) confidence level are highlighted in
bold (italics) face. The critical values of the Johansen cointegration tests are obtained from Osterwald-Lenum
(1992, Table 1).
Australia
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.94
-1.62
-2.88
H 0: r
0
1
2
L-max Statistics
34.58
9.77
6.45
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
50.81
15.67
16.23
9.24
6.45
Trace90
32.00
17.85
7.52
43
Trace95
34.91
19.96
9.24
Canada
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-0.63
-1.94
-1.62
H 0: r
0
1
2
L-max Statistics
27.02
14.05
2.12
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
43.19
15.67
16.17
9.24
2.12
Trace90
32.00
17.85
7.52
Trace95
34.91
19.96
9.24
France
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.16
-1.94
-1.62
H 0: r
0
1
2
L-max Statistics
21.19
8.24
3.49
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
32.93
15.67
11.73
9.24
3.49
Trace90
32.00
17.85
7.52
Trace95
34.91
19.96
9.24
Germany
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.27
-1.94
-1.62
H 0: r
0
1
2
L-max Statistics
19.51
11.51
2.44
Panel B. Johansen Cointegration Test
L-max90
L-max95
Trace Statistics
19.77
22.00
33.46
13.75
15.67
13.95
7.52
9.24
2.44
Trace90
32.00
17.85
7.52
Trace95
34.91
19.96
9.24
Italy
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.94
-1.62
-2.04
H 0: r
0
1
2
L-max Statistics
26.98
9.03
3.17
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
39.19
15.67
12.21
9.24
3.17
Trace90
32.00
17.85
7.52
44
Trace95
34.91
19.96
9.24
Japan
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.94
-1.62
-1.83
H 0: r
0
1
2
L-max Statistics
35.20
24.83
6.22
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
66.24
15.67
31.05
9.24
6.22
Trace90
32.00
17.85
7.52
Trace95
34.91
19.96
9.24
U.K.
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.16
-1.94
-1.62
H 0: r
0
1
2
L-max Statistics
30.83
19.64
3.51
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
53.97
15.67
23.15
9.24
3.51
Trace90
32.00
17.85
7.52
Trace95
34.91
19.96
9.24
U.S.
Panel A. Phillips-Ouliaris Test for Cointegration Relationship
Critical Value
Augmented Dickey-Fuller T-statistics
5% Critical Value
10% Critical Value
-1.94
-1.62
-2.04
H 0: r
0
1
2
L-max Statistics
50.91
10.31
7.09
L-max90
19.77
13.75
7.52
Panel B. Johansen Cointegration Test
L-max95
Trace Statistics
22.00
68.31
15.67
17.40
9.24
7.09
Trace90
32.00
17.85
7.52
45
Trace95
34.91
19.96
9.24
