Evidence of instability of the wealth effect in the USA is presented through the estimation of a Markov switching model of the long-run aggregate consumption function. The dating of the regimes appears to bear relation to movements in asset prices. A model-based explanation of the findings is suggested, highlighting the importance of the short-run relation between consumption, income and wealth in explaining the estimated long-run coefficients.Parameter instability; Markov switching; Consumption; Wealth effect

Fernando Alexandre

Pedro Bação

Vasco J. Gabriel

English

Research Papers in Economics

råáp=== = = ======råáîÉêëáíó=çÑ=pìêêÉó 
 
  
 
Discussion Papers in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Department of Economics 
University of Surrey 
Guildford 
Surrey GU2 7XH, UK 
Telephone +44 (0)1483 689380 
Facsimile +44 (0)1483 689548 
Web www.econ.surrey.ac.uk 
ON THE STABILITY OF THE WEALTH EFFECT 
 
By 
 
Fernando Alexandre 
(Department of Economics and NIPE,  
University of Minho, Portugal) 
Pedro Bação 
(Faculty of Economics and GEMF,  
University of Coimbra, Portugal) 
& 
Vasco J. Gabriel 
(University of Surrey and NIPE) 
 
DP 14/05 On the Stability of the Wealth Effect∗
Fernando Alexandre†
Department of Economics and NIPE, University of Minho, Portugal
Pedro Bação
Faculty of Economics and GEMF, University of Coimbra, Portugal
Vasco J. Gabriel
Department of Economics, University of Surrey, UK and NIPE
28th June 2005
Abstract
Evidence of instability of the wealth eﬀect in the USA is presented through the estimation
of a Markov switching model of the long-run aggregate consumption function. The dating of
the regimes appears to bear relation to movements in asset prices. A model-based explanation
of the ﬁndings is suggested, highlighting the importance of the short-run relation between
consumption, income and wealth in explaining the estimated long-run coeﬃcients.
Key Words: Parameter instability; Markov switching; Consumption; Wealth eﬀect
JEL Classiﬁcation: E21; E44; G10
∗The authors acknowledge ﬁnancial support from Fundação para a Ciência e a Tecnologia, research grant
POCI/EGE/56054/2004, POCTI.
†Corresponding author. Address: Department of Economics, University of Minho, 4710-057 Braga, Portugal.
Email: falex@eeg.uminho.pt. Tel: + 351 253 604546. Fax: + 351 253 607635.
11I n t r o d u c t i o n
The 1990s witnessed a remarkable increase in stock prices in the USA. Between January 1995 and
September 2000, the S&P 500 stock price index rose approximately by 230 percent. According
to Poterba (2000), this increase accounted for more than 60 percent of the wealth creation in the
USA during that period. Boom/bust periods in stock markets, and their eﬀects on total wealth
and consumption, have raised several questions for economic analysis. In particular, there is an
ongoing debate on the appropriate Federal Reserve response to movements in stock prices.
Several authors have questioned the stability of the wealth eﬀect estimates. For example,
Ludvigson and Steindel (1999) estimate the wealth eﬀect to be the usual 0.04 for their full
sample, 1953-1997. However, their estimate reaches 0.1 in the 1976-1985 sub-sample, and is
only 0.02 after 1986. Furthermore, Mehra (2001) corroborates the view that the estimate seems
to depend on the econometric model, the measures of wealth and consumption, and on the
sample. Poterba (2000) puts forward three reasons that might explain the observed signs of
instability. First, the fact that only a subset of households own equity, which was the main
source of shocks to aggregate wealth in that period. Second, the growing importance of equity
investments that are held in tax-favored retirement accounts. Third, the falling cost of leaving
bequests. In this paper we report additional empirical evidence on, and provide an explanation
of, the instability of the wealth eﬀect.
The paper is structured as follows. In the next section we use a simple cointegrated Markov
switching model, which allows us to distinguish endogenously between periods with diﬀerent
values for the wealth eﬀect estimate. We relate these sub-samples to periods with diﬀerent
levels of volatility in asset prices. We then present in section 3 a model-based explanation of
the econometric results. Section 4 concludes.
2 Empirical Analysis
The standard derivation of the “consumption function” (see, e.g., Lettau and Ludvigson, 2001,
LL henceforth) begins by assuming that consumption tends to a stationary fraction of total
wealth, which allows us to write a cointegrating relation between (the logs of) consumption (ct)
and total wealth (wt):
ct − wt = ut, (1)
2where ut is a stationary process. Such a result may be obtained from the usual micro-founded
model of consumption (e.g., Campbell and Mankiw, 1989) if one assumes that the period utility
is well approximated by a log function of consumption. The derivation then proceeds to separate
total wealth into human and non-human wealth
wt ≈ ωat +( 1− ω)ht, (2)
where at is log non-human wealth, ht is log human wealth and ω is the average weight of non-
human wealth in total wealth. Since human wealth is not observable, LL (2004) show that an
approximation1 may be obtained by using labor income, yt, as a proxy for ht, resulting in the
following log consumption-wealth ratio
ct − ωat − (1 − ω)yt = u∗
t. (3)
These authors show that ct, at and yt share a common trend, with normalized cointegration
vector (1,−β,−δ) and cointegration residual ct −βat −δyt (cayt in brief)2.T h ec o e ﬃcient β is
interpreted as the “wealth eﬀect”, their estimation yielding ˆ β =0 .3 and ˆ δ =0 .6.
LL (2004, section 4) discuss the stability of the long run relationship, resorting to the sup,
mean and Lc tests of Hansen (1992), which produced ambiguous results. Indeed, sequential
tests of this family may not be able to detect certain types of structural change, such as Markov
regime shifts or threshold eﬀects, as shown by Carrasco (2002). Therefore, a simple way of
assessing instabilities in the consumption-wealth ratio is to allow the relationship to undergo
occasional discrete shifts of the Markov switching type, as suggested by Hall, Psaradakis and
Sola (1997). Thus, we specify the cointegration equation as
ct = µst + βstat + δstyt + ηstεt, (4)
where {εt} is a stationary random sequence with mean zero and unit variance, while st is
a discrete-valued random variable, independent of εt−i for all i. This variable indicates the
unobserved regime operative at time t, forming a homogeneous ﬁrst-order Markov chain with
state space {0,1} and transition probabilities p =P r ( st =1 |st−1 =1 )and q =P r ( st =0 |st−1 =
0).
1See LL (2001 and 2004) for a more detailed discussion of the assumptions employed in the approximation.
2Note that the coeﬃcients β and δ need not sum to 1, since non-durable consumption and services are used
as a measure of total consumption.
3Notice that the formulation in (4) is very ﬂexible, in that it allows the data to determine
when and which parameters have shifted, be it the long run coeﬃcients or the variance (see Hall
et al., 1997, for more details on the use of MS models in a cointegration setting). Other papers
(Ludvigson and Steindel, 1999 and Mehra, 2001, for example) have relied on an ad-hoc choice of
break points. This model, however, will be able to distinguish endogenously periods where, for
instance, asset markets and returns may be behaving diﬀerently. This is particularly convenient
to study the implications of the theoretical model developed in the next section.
Table 1 records the results of maximum likelihood estimation of the parameters of model (4)
and respective asymptotic standard errors3. Both the LR test and the model selection criteria
favour the MS model over the linear cointegration speciﬁcation4. The MS model identiﬁes
two distinct regimes: regime 1 is associated with more volatile periods, which roughly coincide
with historical "bull" markets, such as those of the late 1960s and 1990s, while regime 0 is
associated with "calmer" periods. Figure 1 shows the smoothed probabilities produced by
the MS model, plotted against ﬂuctuations in the consumption-wealth ratio as estimated by
LL (2004), conﬁrming the view that the identiﬁed regimes seem to capture the state of asset
markets and of the economy.
In state 1, the coeﬃcient associated with asset wealth is smaller (0.22) than in state 0
(0.29), which is consistent with the empirical evidence surveyed in Poterba (2000) and Mehra
(2001). It is interesting to notice that the diﬀerence between the β’s and the δ’s across regimes
is approximately the same (0.07). Although these ﬂuctuations do not appear to be sizeable,
the cointegration vector estimated by LL (2004) seems to be a “composite” estimate of the
diﬀerent regimes. The next section presents additional possible explanations for the documented
instability.
3 Model-Based Explanation
LL (2001) note that the estimated coeﬃcients (0.3 and 0.6) are what one would expect if the
aggregate production was well represented by a Cobb-Douglas, since they are very close to the
3We resort to the same dataset used in LL (2004), comprising quarterly data on aggregate consumption, asset
wealth and labor income, spanning from 1951:4 to 2003:3.
4A I Ca n dB I Cv a l u e sf o rt h el i n e a rm o d e la r e−1248.2 and −1234.8, respectively.
4usual income shares of capital and labor. In fact, if aggregate output is
Ot = Kα
t L1−α
t , (5)
then labor income is Yt =( 1− α)Ot and the return to capital is Rt = αOt. Now note that a
standard macroeconomic asset-pricing model would imply
Qt = Et
µ
Qt+1 + Dt+1
1+rt
¶
, (6)
where Qt is the asset price, Dt are “dividends” and rt is the discount rate. In steady state, one
obtains
Q =
ξ
1 − ξ
D, (7)
where ξ is a risk-adjusted discount factor. Noting that non-human wealth is A =( Q + D)K
(cum-dividend price times asset volume) and that the product D × K is just the steady-state
return to capital, then
A =( Q + D)K =
D
1 − ξ
K =
α
1 − ξ
O. (8)
If we write consumption as C = O −X,w h e r eX stands for other uses of aggregate output,
then for an arbitrary θ
C = O − X = θ
1 − ξ
α
A +( 1− θ)
1
1 − α
Y − X. (9)
This seems to suggest that, in the long run, consumption is not related to labor income and
non-human wealth alone, but also to other components of aggregate output. One implication of
this is that, to be able to restore the original result, one must expect X to be somehow related
to A and Y , so that it may be substituted out. One way to proceed would be to assume that A
and Y capture the nonstationarity in X. Stability of the wealth eﬀect would then have to rely
on the stability of the relation of the components of X, for instance, government spending, to
A and Y – given the time span covered by the data, this would appear dubious. But assume
that this problem can be solved in the most simple way: set X =0 , which leads to the same
conclusions as assuming that consumption is a constant fraction of output. In this case
C = θ
1 − ξ
α
A +( 1− θ)
1
1 − α
Y, (10)
i.e., if we run the usual regression in levels, the “wealth eﬀect” is indeterminate, since θ is
arbitrary. Nevertheless, it is common to run the regression in logs. In this case the usual ﬁrst
5order Taylor approximation gives (ignoring a “constant”)
c = ρaa + ρyy, (11)
where the coeﬃcients are
ρa =
θ
1−ξ
α A
C
= θ
O
C
= θ, (12)
ρy =
(1 − θ) 1
1−αY
C
=( 1− θ)
O
C
=1− θ. (13)
Taken literally, the Taylor approximation would imply the sum of the coeﬃcients to equal
unity. The fact that it is just an approximation may explain why this is not the case in the data
– Jensen’s inequality could be the reason for this. Nevertheless, our results in section 2 are
consistent with matching symmetric variations in the coeﬃcients, corresponding to changes in
θ. What matters to us here is that again the aggregate long-run wealth eﬀect is indeterminate.
If this was really the case, one would expect the estimated coeﬃcients to depend on short-run
correlations between the variables. This would result in instability of the coeﬃcients as one
varied the sample. To see this, note that the previous equations relate steady state values. Let
the actual values be
˜ ct = c + εc
t, (14)
˜ at = a − εa
t, (15)
˜ yt = y − ε
y
t, (16)
where the added disturbances reﬂect short-run deviations from the steady state. Then
˜ ct = ρa˜ at + ρy˜ yt + ρaεa
t + ρyε
y
t + εc
t. (17)
Minimisation of the variance of the residual, V
£
ρaεa
t + ρyε
y
t + εc
t
¤
with respect to θ leads to
the following estimate of the wealth eﬀect
ρa = θ =
σyy − σya + σyc − σac
σyy + σaa − 2σya
, (18)
where σij = E
³
εi
tε
j
t
´
, with i,j = c,a,y, represent variances and covariances in the short run.
This result and the reasoning leading to it suggest there may be reasons for concern regarding
the relevance of empirical estimates of the wealth eﬀect in the context of this model. The
estimates may reﬂect short-term correlations, possibly mixed with long-run parameters, as in
6the regression in levels. In particular, notice that, ceteris paribus, an increase in the variance of
asset wealth reduces the size of the estimated wealth eﬀect – this is exactly what the data, as
reported in section 2, shows.
We developed our model from the point where LL (2001) stopped. These authors estimated
the consumption equation and concluded it was consistent with the usual estimates of labor and
capital income shares. We have shown that if we accept this conclusion, then the consumption
function is likely to show signs of instability, and in the simple case studied here it would
even lead to indeterminacy. The reader may ask whether the issue is only about relating the
estimates of the consumption-function coeﬃcients to labor and income shares. It is not. If we
accept the assumptions of the standard derivation of the wealth eﬀect – that consumption tends
to a stationary fraction of wealth, that the average weight of human wealth on total wealth is
stationary and that labor income captures the non-stationarity in human wealth – then the
algebra of I(1) variables says we should also conclude that there is cointegration between any
two-element combination of consumption, wealth and labor income. This is in fact another
instance of indeterminacy, but one that apparently is not upheld by the data. On the other
hand, the implications from the model we have just presented, in particular that the wealth
eﬀect should be unstable, do in fact appear to be matched by the data, as reported in section 2.
In view of the indeterminacy/instability result, what is surprising is actually the relatively
small variation across regimes of the estimates reported in section 2. The empirical results
and the theoretical model taken together suggest that there may be some structure in the
short-run correlations between consumption, income and asset wealth – i.e., consumption,
income and asset prices, which drive asset wealth in the short-run. This short-run structure
reduces the magnitude of oscillation in the estimated coeﬃcients (possibly making the traditional
cointegrating relation a useful reference point), but also makes them react in a particular way
to movements in asset prices.
To sum up, the usual derivation of the consumption cointegrating relation itself suggests
that the estimated coeﬃcients may be changing over time. The simple model investigated here
adds to this a possible connection, through short-run correlations, between asset wealth (i.e.,
asset prices) and these coeﬃcients. This warrants a more detailed study of the short-run relation
between these variables, which is the subject of Gabriel, Alexandre and Bação (2005).
74C o n c l u s i o n
This paper documents patterns and sources of instability in the consumption-wealth ratio. We
estimated a simple Markov switching model and found diﬀerent estimates of the wealth eﬀect
associated with two regimes. These regimes seem to correspond to periods of high/low volatility
in asset prices. This suggests that consumption reacts diﬀerently to asset prices depending on
whether these changes are perceived to be permanent or transitory. We then show that the
instability is not surprising given the way the consumption equation is derived. In this way,
we add to the explanations given in Poterba (2000) a model-based reason for instability in the
wealth eﬀect.
The magnitude of parameter instability is relatively small, and linear cointegration may
perhaps be capturing the fundamental path of the relationship. Indeed, in a related paper
(Gabriel et al., 2005), we suggest that short term asymmetries, not long run instability, may be
more important in describing the dynamics in this trivariate system.
References
[1] Campbell, J. Y. and Mankiw, N. G. (1989), Consumption, Income and Interest Rates:
Reinterpreting the Time Series Evidence, in Olivier J. Blanchard and Stanley Fischer, eds.,
NBER macroeconomics annual: 1989, Cambridge, MA, MIT Press, 185-216.
[2] Carrasco, M. (2002), Misspeciﬁed Structural Change, Threshold and Markov Switching
Models, Journal of Econometrics, 109, 239-273.
[3] Hall, S. G., Psaradakis, Z., Sola, M. (1997), Cointegration and Changes in Regime: the
Japanese Consumption Function., Journal of Applied Econometrics, 12, 151-168.
[4] Hansen, B. E. (1992), Tests for Parameter Instability in Regressions with I(1) Processes,
Journal of Business and Economic Statistics, 10, 321-335.
[5] Gabriel, V. J., Alexandre, F. and Bação, P. (2005), The Consumption-Wealth Ratio under
Asymmetric Adjustment, unpublished manuscript.
[6] Lettau, M. and Ludvigson, S. (2001), Consumption, Aggregate Wealth and Expected Stock
Returns, Journal of Finance, 56, 815-849.
8[7] Lettau, M. and Ludvigson, S. (2004), Understanding Trend and Cycle in Asset Values,
American Economic Review, 94, 276-299.
[8] Ludvigson, S. and Steindel, C. (1999), How important is the stock market eﬀect on con-
sumption?, Economic Policy Review, Federal Reserve Bank of New York, 5, 29-51.
[9] Mehra, Y. P. (2001), The Wealth Eﬀect in Empirical Life-Cycle Aggregate Consumption
Equations, Economic Quarterly, Federal Reserve Bank of Richmond, 87, 45-68.
[10] Poterba, J. M. (2000), Stock Market Wealth and Consumption, Journal of Economic Per-
spectives, 14, 99-118.
95 Appendix
Figure 1: Regime 0 probabilities and C/W ﬂuctuations
0.0
0.2
0.4
0.6
0.8
1.0
-0.06
-0.04
-0.02
0.00
0.02
0.04
55 60 65 70 75 80 85 90 95 00
smoothed probabilities C/W fluctuations
10Table 1: Markov switching cointegration estimates
µ0 µ1 β0 β1 δ0 δ1
0.5961
(0.06)
0.7749
(0.049)
0.2944
(0.018)
0.2177
(0.024)
0.6051
(0.017)
0.6761
(0.027)
η0 η1 pq
0.006
(0.0008)
0.0089
(0.0014)
0.927
(0.54)
0.9429
(0.35)
LR: 69.261 AIC: 1374.7 BIC: 1341.4
11

Cointegration and Changes in Regime: the Japanese Consumption Function.,

Consumption, Aggregate Wealth and Expected Stock Returns,

Consumption, Income and Interest Rates: Reinterpreting the Time Series Evidence,

How important is the stock market eﬀect on consumption?, Economic Policy Review, Federal Reserve Bank of

Misspeciﬁed Structural Change, Threshold and Markov Switching Models,

Stock Market Wealth and Consumption,

Tests for Parameter Instability in Regressions with I(1) Processes,

The Consumption-Wealth Ratio under Asymmetric Adjustment,

The Wealth Eﬀect in Empirical Life-Cycle Aggregate Consumption Equations, Economic Quarterly, Federal Reserve Bank of Richmond,

On the Stability of the Wealth Effect

http://www.fahs.surrey.ac.uk/economics/discussion_papers/2005/DP14-05.pdf

On the Stability of the Wealth Effect

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics