Interest Rates and Business Cycles
Fluctuations: a Focus on Higher Moments
Andrea Beccarini,
University of L’aquila, Italy1.
Abstract
This work aims at analysing the relationships between market interest rates
and the business fluctuations. Asymmetries in the business cycle affect
saving decisions of agents and interest rates. The relationships between
interest rates and the expected value, the variance, the skewness and the
kurtosis of the business cycle are demonstrated. The process for the
business cycle variable is estimated by a Markov-switching model which
allows explicitly to consider the alternation of the business cycles phases.
Afterwards, conditional, time-varying moments of the business cycles are
calculated. Then, these conditional moments are used as regressors for
interest rates.
JEL Classification: D91, E32, E43.
Key words: Real Interest Rates, Business Fluctuations, Asymmetry,
Markov-switching models.
1
Address for correspondence: Andrea Beccarini, Department of Economic Systems and
Institutions, University of L'Aquila, Piazza del Santuario, 19, I-67040 Roio Poggio (AQ),
Italy. Tel. Number: +39 0746 483600. E-mail: fpbec@tin.it.
1
Introduction
A large literature has documented the presence of non-linearities in the
market interest rates. One kind of non-linearities seems to be particularly
important and emerges from the presence of the so-called regimes or states
which may alternate, for example, according to a Markov-chain process.
On the light of the empirical evidences and the theoretical ground which
both support the presence of regimes in the interest rates, this paper has a
twofold objective. Firstly, it provides a further theoretical foundation for
the phenomenon in question which is based on an intertemporal
maximisation problem of the representative agent dealing with aggregate
consumption (C-Capm). The assumption of rationality embedded in that
model suggests to take into account not only the first two moments of
consumption but to look at higher moments as well, consistently with the
consideration of all relevant information which is implicit in the rationality
property. The first-order condition of the above maximization problem is
expanded and it has been found that higher moments, with a decreasing
weight, affect saving and consumptions decisions, hence interest rates. It
has been argued that moments, higher than the second,
may be a
precious information instrument for rational expectation agents. Secondly,
the focus is on how to measure these conditional moments, especially the
third moment of a time-series which is a priori, highly suspected to embed
switching parameters. In fact, consumption or other related variables may
follow the business cycles fluctuations.
After having assumed an appropriate Markov-switching model (MSM) for
the business cycle variable, consistently with Timmerman (2000), the
time-series of the relevant moments are constructed, by the means of the
estimated parameters. As stressed in Timmerman (2000), in some models,
the variance of time-series which embeds some regimes may not simply
be a weighted average of the variances proper to regimes but there is also
another term which captures the so-called “jump effect” which occurs
when switching among regimes. The focus is also on the importance of
both the risk aversion and rationality property of the representative agent
who prefers a positive asymmetry of his/her consumption. In the
developed context one is able to test the validity of some well-known
theories dealing with consumption, that is the Precautionary Saving theory
of Leland (1968) and the Permanent Income Hypothesis of Friedman
(1957). The related literature is divided in two main strands. The first one
focuses on the statistical detection of regimes in interest rates, in particular
on the short term interest rates. One may quote Ang and Bekaert (2002a,
2000b), Bansal et all (2002) respectively for short-term interest rates and
for the entire term structure; Garcia et all (1996) for the real interest rates.
They often detect two or three regimes in the interest rates with a transition
probability among regimes which can be constant across time or varying
with the level of the lagged interest rate. The second strand provides an
economic justification of this phenomenon, searching the causes in other
2
macro-variables. It has been argued that either monetary policy regimes or
business cycles phases influence either inflation expectations or real
interest rates. A micro-foundation of this phenomenon may rely on the
different price flexibility or on transmission channels of the monetary
policy in a context of asymmetric information; see, for example, Cover
(1992), Garcia et all (1995), Morgan (1993), Ravn et all (1996) and Weise
(1999). As regards the literature of higher moments, see Guedhami et all
(2005) and the quoted references, in particular for the relationship between
the conditional skewness and financial variables.
This paper is organized as follows. The first paragraph provides a
microeconomic foundation for the presence of asymmetry and kurtosis of
the business cycles affecting interest rates. In the second paragraph, a
way for constructing the time series of the relevant conditional moments is
described. In the third one, empirical evidences for the Euro area are
reported. Finally, after an appendix, some conclusions follow.2
1. Technical notes
Consider the first-order condition of an optimisation problem of a
representative agent who faces a two-periods maximisation problem:
 u ' (c t n )

E 
(1  rt ,t n )  1
 u ' ( ct )

Where,u'(.)ctnr, are,respectively,thediscountfactor,themarginalutilityoftherepresentativeagent,consumptionattimetandt+nandtheholdingperiodreturnofa
security held n times. Considering a risk-free investment and taking out
of the expected value operator what is fixed, then one has:

u ' ( ct )
E[ u' (ct  n )]  Pt , n
Now, assuming that the utility function u(.) be indefinitely differentiable,
consider the Taylor expansion of u ' (ct  n ) around the conditional expected
value of consumption ct  n :
u ' (c t  n )  u' (c t  n )  u ' ' (c t  n )( ct  n  ct  n ) 
1
u' ' ' ( ct n )(c t  n  ct  n ) 2 
2
This paper is based on a chapter of the author’s PhD dissertation at the university
LUISS Guido Carli (Rome).
2
3
1 iv
1 v
u ( ct  n )(c t n  c t  n ) 3 
u (c t n )( c t  n  ct  n ) 4
6
24
(3)
which yields:
E[u'(ctn)]
u ' ' ( ct  n ) E[(c t n  c t  n )] 
1
u' ' ' ( ct n ) E[(c t  n  ct  n ) 2 ] 
2
(4)
1 iv
1
u ( ct  n ) E[(c t  n  c t  n ) 3 ]  u v (c t  n ) E[( ct  n  ct  n ) 4 ]
6
24
Now notice that, in equation (4), the term which multiplies the second
derivative is zero and the terms multiplying the remaining derivatives are,
respectively, the variance of consumption in t+n and higher moments.
Equation (4), basically states that the stochastic discount factor is also a
function of the variance, the skewness and the kurtosis of future
consumption, provided an appropriate shape of the utility function.
Substituting into the first-order condition, one obtains:
Pt, n 

u ' ( ct )

1


u
'
(
c
)

u
'
'
(
c
)
E
[(
c

c
)]

u ' ' ' (c t  n ) E[( c t  n  ct  n ) 2 ]  
t

n
t

n
t

n
t

n

2




1

1
 u iv ( ct  n ) E[(c t n  c t  n ) 3 ]  u v (c t  n ) E[( ct  n  ct  n ) 4 ]

6
24

(5)
For every maturity n, one can model the relative price as a function of the
central moments of consumption at t+n, provided that the partial
derivatives of the utility function, up to the fifth order, are different from
zero.
Equation (5) embeds the precautionary saving theory; in fact, assuming the
third partial derivative of u(.) be positive, an increase in the variance of
consumption, induces the agent to save more pushing up the bond’s price.
This equation states more. The agent may care not only about the expected
consumption and the uncertainty (volatility) around it, but also whether the
mass probability concentrates more on higher levels of consumption rather
4
than lower levels. When the agent expects an economic recession more
likely than an expansion, there is a reason more to save and independently
of the perception of the uncertainty of the future, represented by a
volatility. Considering the third moment as an indicator of the asymmetry
of the probability distribution of the future consumption, the bond’s price
must be affected by this measure.
Resuming, by testing equation (5) one can implicitly verify the following
argumentations:
-
Rationality of financial markets implicit in the maximization problem
and in the use of all available information.
-
Market risk-aversion which is represented by higher order utility
function derivatives, different from zero.
-
Precautionary Saving Hypothesis which is described by the
significance of the second moment.
Consider, finally the relationship between equation (5) and the Permanent
Income Hypothesis. One of the implications of this theory is that marginal
utility of consumption is constant along time3. As can be seen below, this
property may be verified in the empirical section by looking at the
stability of the parameters in question; in fact, if prices or interest rates are
regressed (by the OLS) on conditional moments, then by virtue of
equation (5) the derivatives of the utility function correspond to the
estimated parameters. Alternatively, a model with time-varying
parameters (like a MSM) may be used and the validity of this model may
be considered as a proof against the Permanent Income Hypothesis.
The meaning of the business cycle variable’s asymmetry
As can be seen in equation (5), the third moment is weighted by the
derivative of the fourth order of the utility function which is negative if the
representative agent is risk-adverse. Hence, according to that equation an
increase of the third moment causes a fall in security prices. In order to
clarify this relationship, one can think the risk-adverse agent as the one
whose sensitivity for strong recessions is higher than for the other states
of the economy. Besides, an increase of the third moment implies a
redistribution of the mass probability from very low values of the state
variable to very high values and from small high values to small low
values of the state variable. A graphical representation of this concept is
provided in graph 1. Now, risk-aversion implies a preference for a positive
third moment of the state variable’s distribution because it implies a
preference of swapping between a strong recession in favour of a strong
3
This property implies that higher order derivatives are also constant along time.
5
expansion even though with the renouncement of a low expansion for a
low recession.
Graph 1. A symmetric and an asymmetric distribution.
D1
Strong Recession
Distribution
D0
D1
D0
Rec.
Expans.
Expected Value
E[x]=0
E[x]=0
Strong Expansion
Variance
Var[x]
Var[x]
Third Moment
E[(x-E(x))^3]=0
E[(x-E(x))^3]>0
In sum, an increase of the third moment implies a better situation for the
risk-adverse agent whose propensity to save decreases, letting prices to fall
and interest rates to go up. Notice further, that rationality implies to use of
all possible information the agents have at their disposal. Under a
statistical point of view, the complete information about a random variable
is provided only by its distribution and not only by the first and the second
moments, unless in the case of a Gaussian distribution. Nevertheless, the
knowledge of the entire set of moments is equivalent to the knowledge of
the distribution function. However, considering the presence of alternating
phases in the economy, it is implausible to think of the distribution of the
state variable as symmetric, hence Gaussian.
2. Statistical models for the business cycle variable
One can model the third and the forth moments by an appropriate regimeswitching specification. Following Timmermann (2000),4 one could
specify a Markov-switching process for consumption or for other macro4
Notice that, differently from Timmermann (2000), here, the moments are considered as
conditional.
6
factors that are eligible to enter the utility function. Two MSMs are
performed, namely the model with switches in mean (SM) and the
autoregressive (of order 1) model with switches in mean, labelled as
SWAR(1).
The SM model has the following features:
y t   st  et
t  1,2,..., T
et  iid ( 0,  s2t )
with the following specifications:
 St   1 (1  st )   2 st
 S2t   12 (1  s t )   22 s t
(7)
st  0,1 Re gime 1,2




P
r
s

0
|
s

0

p
P
r
s

1
|
s

1

p
t
t

1
1
1
t
t

1
2
2
where y t is the dependent variable; st is the dichotomous latent variable
that allocates observations in the two specified regimes; 1 ,  2 are the
conditional means of regimes 1 and 2, respectively;
 12 ,  22 are the
conditional variances of regimes 1 and 2, respectively; p11 , p22 are the
transition probabilities of remaining in regime 1 and 2 at each time, having
assumed the regime 1 and 2 for the previous period, respectively.
Furthermore,  t will indicate the inferred probability for the process to
stay in regime 1, for the time t5.
The SWAR(1) model has the following specification:
y t   st   ( y t1   st 1 )  et t  1, 2,..., T
(8)
et  iid ( 0,  )
2
st
where  is the autoregressive parameter, independent of switches; system
(7) also describes the means, the variances and the transition probabilities.
5
In other terms, this is the probability of the latent state variable s t to occur.
7
It has been already stated that, for the two-regimes SM model6 of
equations (6) and (7), the unconditional moments (unconditional with
respect to regimes and conditional with respect to the remaining
information) are the following:
t1(1)2
(9)





(
1


)



(
1


)
(



)
2
2
t
t1
2
t 2 t
t
2
1 2
(10)
where the third addend captures the so-called “jump effect”; the third
central moment (for y) is:









3
2
2
2
2
E
[
(
y

)
]

(
1

)
(

)
[
3
(

)

(
1

2
)
(

)
]
(11)
t
t
t
t
1
2
1
2
t
2
1
Recalling that the approximation by the Taylor series expansion is
conceived around the expected value of consumption at time t+n, this
quantity is substituted out by observing that:

 

c

E
[
c
]


(
1

)2
t

n
t
t

n
t1
t
The next step is to substitute the above moments7 into equation (5) in
order to obtain:
Pt, n 

u ' (c t )

u ' ( t  1  (1   t )  2 ) 





1

2
2
2
 u' ' ' ( t  1  (1   t )  2 )[1  (1   t ) 2   t (1   t )(  1   2 ) ]  (12)
 2



1

 u iv ( t 1  (1   t )  2 ) 

6





2
2
2 2
[ t (1   t )( 1   2 )[3( 1   2 )  (1  2 t )(  2   1 ) ]]

6
7
For the moments of the SWAR(1) model, the reader may refer to Timmermann (2000).
Bearing in mind that they are only valid for the SM model.
8
Which is only a function of the regime-switching parameters and the
present consumption. It has been already noticed that, in principle, the
derivatives of every order of the utility function are regime-dependent,
hence time-varying. So, the model reveals itself to be highly non-linear.
However, one may attempt to regress prices or interest rates (by the means
of the OLS) on the conditional moments (equations (9), (10) and (11)) and
on a constant relying on the assumption that the following ratios (13) are
constant:

u ' ( ct )
u ' ( t  1  (1   t )  2 )
1 
u' ' ' ( t 1  (1   t )  2 )
2 u ' (c t )
(13)
1 
u iv ( t 1  (1   t )  2 )
6 u' (c t )
A second approach is based on the observation that the ratios of (13) are
all regime dependent, hence one may attempt to regress prices or interest
rates on the conditional moments but by a MSM model as well. If these
assumptions turn out to be implausible then the Kalman filter may be an
alternative methodology to overcome the problem. In this work, the first
two approaches are followed. For the empirical part, the real rate of
growth of Gdp is assumed to be the variable entering the utility function
u(.). This is consistent with its high correlation with consumption and the
praxis of substituting out consumption, like in I-CAPM models, in favour
of other macroeconomic factors that better represent the economy as a
whole. Also, the literature for the regime-switching process of the GDP is
huge. A two-regime specification is then consistent with the recession and
expansion phases of the economy.
Resuming, one can proceed, firstly, by constructing the time-varying
moments that derive from the regime-switching specification of the chosen
macroeconomic factor; then, these variables are considered as regressors
for the interest rates (or bond prices). One potential problem to be faced is
the high non-linear relationship between interests rates and the conditional
moments; this may cause the instability of the parameters of the interest
rate (OLS) regression. To overcome this problem a regime-switching
specification also for the first difference of interest rate is used.
9
3.
Empirical evidence
The Euro area quantities are considered8. The first difference of real
interest rates is regressed on the relevant moments of the rate of growth of
the real Gdp (quarterly change). Both the OLS and the MSM, for this
regression, are attempted. It is important to stress that a reliable
verification of the theoretical model depends on a good estimate of the
conditional moments since they are used as regressors. That is the reason
why two MSMs (SM ad SWAR(1)) are also attempted to estimate the
conditional moments of the Gdp. This paragraph is divided in two parts.
First, by the OLS, interest rates are regressed on conditional moments
obtained by the SM model and then they are regressed on the SWAR(1)
moments. The second part also consists of two regressions, but performed
by MSMs, of interest rates on conditional moments, assuming, now, a tworegimes process for their parameters9. The following table 1. should clarify
this framework.
Table 1. Regressions for the Euro area real interest rates.
OLS
MSM
Dependent Interest Rates Interest Rates Interest Rates Interest Rates
Variable
(First Diff.) (First Diff.) (First Diff.)
(First Diff.)
Regressors
SM
SWAR(1)
SM
SWAR(1)
Moments
Moments
Moments
Moments
OLS Interest rates regression on SM and SWAR(1) moments
Table 2. and table 3. show the estimates of the first difference of the 3months quarterly real interest rate on moments up to the third order, of
both SM and SWAR(1) regressions, after having selected the appropriate
lags. The second and the fourth moments and any leads or lags of them
turned out to be not significant.
Table 2. OLS regression of interest rates on SM cond. moments of Gdp.
Regression 1: 3-Months Quarterly Real Interest Rate, First Difference
Variable
Coefficient Std. Error
t-Statistic
Prob.
Constant
-0.52
0.14
-3.68
0.0007
First moment
0.70
0.23
3.08
0.0037
Third moment(-1)
30.04
13.83
2.17
0.0358
R-squared
Adjusted R-squared
S.E. of regression
0.215
0.177
0.342
Sum squared resid
Log likelihood
Durbin-Watson stat
8
4.785
-13.62
2.068
The source of the data is the ECB. Quarterly data are considered and they range from
1994:2 to 2005:1.
9 The reader may refer to Hamilton et all (1996) for the relationship between financial
and real economy regimes.
10
Table 3.OLS regression of interest rates on Gdp SWAR(1) cond. moments.
Regression 2: 3-Months Quarterly Real Interest Rate, First Difference
Variable
Coefficient Std. Error
t-Statistic
Prob.
Constant
-0.73
0.22
-3.36
0.0017
First moment
1.28
0.42
3.02
0.0044
Third moment
2.47
1.43
1.72
0.0922
R-squared
Adjusted R-squared
S.E. of regression
0.244
0.206
0.338
Sum squared resid
Log likelihood
Durbin-Watson stat
4.57
-12.82
2.26
For both regressions, usual diagnostic tests show no sign of
misspecification; the appendix provides the details of the SM and
SWAR(1) regressions for the real Gdp. According to the estimated
parameters of regression 1, an increase of an expected percentage point of
the Gdp rate of growth between quarter t and t-1, implies an increase of
0,7 (and 1,28 for regression 2) of the 3-month real interest rate, over the
same period. This is consistent with the first order condition (equation (2));
prices fall and interest rates go up when the ratio of marginal utilities
decreases, hence consumption growth increases. This feature is also
consistent with the LM relationship: people use short-term notes to
finance the increasing purchases. For regression 2, the third moment is
significant10 (and the first lag of it for regression 1); their signs are
consistent with the market risk aversion (the derivative of even order of the
utility function is negative). Once, in time t, agents perceive a strong
expansion more plausible than a strong recession11 (implicit in a positive
third moment), interest rates between time t and t+1 increase. Again, this
should occur because of the more favourable economy’s scenario, hence
agents save less and consume more. Further, the first and the second lags
of the first difference of interest rates (dependent variable), in both
regressions, showed themselves not to be significant while the significance
of the other regressor remained unchanged. This confirms the rationality
assumption of market participants against the hypothesis of adaptive
expectations. Again, the significance of the third moment may be
considered as positive response for a market risk-aversion test. The
precautionary saving theory seems not to be supported by data since the
second moment is not significant in both regressions. The same conclusion
has been reached when the standard deviation instead of the variance, were
included among regressors. A possible explanation may rely on the small
magnitude of the variances within the regimes along with the fact that
regimes are very well separated (they mostly occur with probability 0 or
10
11
The third moment-regressor is significant only at the level of significance of 10%.
But also a low recession more probable than a low expansion.
11
1). According to equation (10) the jump effect on the unconditional
variance (unconditional with respect to regimes) is negligible and the
unconditional variance essentially coincides either with the variance or
regime 1 , which is 0.057 or with the variance of regime 2: 0.027.
However, both variances are very low, suggesting that these are not good
indicators as measures of uncertainty of the business cycles. In a certain
sense, the significance of the third moment can also be reconducted to
precautionary saving motivations. The inclusion of the series of the rate of
growth of Gdp neither provides any further useful information nor makes
the other regressors not significant12; this suggests that, indeed, decisions
of saving and consequently on interest rates are made on the basis of
expectations and uncertainty perceptions.
Markov–switching Interest rates regression on SM and on
SWAR(1) moments
Table 4. and table 5. present the estimates obtained by regressing the first
difference of interest rates, respectively on the SM and on the SWAR(1)
moments, in a two-regimes regression.
Table 4. MSM regression of interest rates on SM conditional moments
Regression 3: 3-Months Quarterly Real Interest Rate, First Difference
REGIME 1
REGIME 2
Coefficient Std. Error Coefficient Std. Error
Constant
-0.68
0.23
-0.06
0.18
First moment
0.46
1.22
0.62
0.57
Second moment
-0.33
18.49
-2.71
7.13
Third moment
7.94
76.95
17.68
32.12
Standard deviation
0.21
0.19
R-squared
Sum squared resid
Durbin-Watson stat
0.81
1.18
1.98
Log likelihood
AIC
BIC
-18.73
61.46
82.87
Table 5. MSM regression of interest rates on SWAR(1) cond. moments
Regression 4: 3-Months Quarterly Real Interest Rate, First Difference
REGIME 1
REGIME 2
Coefficient Std. Error Coefficient Std. Error
Constant
-0.65
0.34
-0.35
0.31
First moment
0.58
1.56
1.07
0.68
Second moment
-0.06
4.08
-0.42
1.73
Third moment
0.57
3.66
2.07
1.93
Standard deviation
0.27
0.20
12
Although the multicollinearity with its expected value is detected.
12
R-squared
Sum squared resid.
Durbin-Watson stat
0.77
1.39
2.27
Log likelihood
AIC
BIC
-25.57
55.15
76.56
Both regressions 3. and 4. present a very high R-squared, also in
comparison with their one-regime counterparts. All estimated parameters
have the expected signs but, according to their standard errors they are not
significant at the usual levels of significance; this is in contrast with the
high R-squared. One possible reason may rely on the high correlation
(multi-collinearity) within regimes of the regressors, this phenomenon
may produce the rejection of regressors when considered singularly even
though a test for the joint significance does not lead to a rejection. Leaving
some regressors, always produces a significant decrease in the R-squared
and a change of the parameters signs, hence the presented results seem to
be the most reliable. On the basis of the provided results, it is possible to
conclude both for the joint significance of the conditional moments as
regressors for interest rates and for a non linear relationship between these
moments and interest rates as the theoretical model of equations (5) or (12)
predicts. So, the conclusions for regression 1. and 2. are here, basically
confirmed:
-
A positive relationship between the first moment of Gdp and the first
difference of interest rates, suggesting a LM relationship.
-
A negative relationship between the second moment of Gdp and the
first difference of interest rates, hence precautionary saving
motivations.
-
A positive relationship between the third moment of Gdp and the first
difference of interest rates, hence market-risk aversion.
-
The importance of the moments (expectations,..) of the Gdp rather
than the row time-series.
-
The evidence of the rational expectation formation rather than the
adaptive formation.
Furthermore, conditional high moments matter for the determination of
interest rates, not only as regressors but also for the determination of the
(time-varying) parameters. Notice that, in principle, the time-varying
property of the parameters may be due not only to the variability of the
arguments the enter the considered derivatives of the utility function13 but
also to the changes along business cycle of the specific parameters of
13
That is the expected value of the real Gdp and its present value.
13
u(.).14 If the first hypothesis is accepted, this consists of an evidence
against the Permanent Income Hypothesis since this theory implies that
marginal utilities (hence their ratios) be constant across time. Graphs 2.
and 3. show the Euro area quarterly real interest rates (first difference)
along with the inferred smoothed probability obtained, respectively as a
by-product of regressions 3. and 4..
Graph 2. Euro area interest rates and the probability of regime1, based on
regression 3.
0.8
0.4
0.0
-0.4
-0.8
1.0
-1.2
0.8
0.6
0.4
0.2
0.0
1994
1996
1997
1998
1999
2000
2001
2002
2003
2004
Euro Area real interes t rates (firs t diff.)
Probability Regim e 1
14
In both cases there should be a correlation between the probability inferred from the
Gdp regressions (which should describe the business cycle fluctuations) and the
probability inferred from the interest rates regression. However, financial markets may
predict the Business Cycles turning points hence, the regimes embedded in the interest
rates may precede the regimes inherent to real quantities. The correlation between the two
inferred probabilities is .2; instead when considering the first lag of the probability
regarding the business cycle (in order to capture the anticipating property of financial
markets) the correlation is .14 .
14
Graph 3. Euro area interest rates and the probability of regime1, based on
regression 4.
0.8
0.4
0.0
-0.4
-0.8
1.0
-1.2
0.8
0.6
0.4
0.2
0.0
1994
1996
1997
1998
1999
2000
2001
2002
2003
2004
Euro area real interes t rates (firs t diff.)
Probability Regim e 1
Conclusions
The proposed model, is based on the observation that, a rational and riskadverse agent considers all the available information of his/her future
consumption when deciding about the present amount of saving.
Statistically, the complete information regarding a random variable is
provided by the density function and not only by the first and the second
moments. However, the entire set of moments provides the same
information of the density function. If the distribution of the random
variable in question is not Gaussian, then the rationality assumption
requires, to include in the information set, some moments higher than the
second one. The third moment, for example, provides a measure of the
asymmetry of the distribution. Asymmetries seem to be present in the
conditional distribution of the business cycles variables hence, they are
considered by rational agents when choosing consumption and
determining saving. In the proposed model, the importance of the riskaversion of agents is expressed by the form of the utility function. The
effect of the third moment derives from the existence (and different from
15
zero) of the derivative till the fourth order. All these principles have been
used in order to find a relationship between market interest rates and the
conditional moments of the distribution of the business cycle. This
relationship generalizes the Precautionary Saving theory, in the sense that
it extends the uncertainty sources from the volatility to the asymmetry
measures. When the third moment decreases, holding the other moments
fixed, the probability of a strong recession increases along with the
probability of mild expansion; at the same time, the probability of strong
expansion decreases along with the probability of mild recession.
Assuming market risk-aversion, the increase of the chances of a strong
recession is not compensated by an increase of the chances of a mild
expansion; hence, this phenomenon consists of a further reason to save,
besides the one based on the volatility of the business cycle. This work
also proposes an alternative way to estimate conditional moments of the
population and to use them to test the precautionary saving and its impact
on interest rates. Since a very used specification for business cycle
quantities refers to the Markov-switching models which explicitly account
for the presence of regimes in the stochastic process, it is reasonable to
estimate the relevant conditional moments as a function of the parameters
and the probabilities of the inferred regimes to occur. The strategy
followed here, consisted of estimating some Markov-switching models for
the rate of growth of the real Gdp. Then, the conditional moments are
calculated and, their ability of explaining the first difference of the 3months real interest rate is tested in a twofold way. Four regressions for
the interest rates are performed. First, two OLS regressions are performed,
they consider as regressors the conditional moments derived from the SM
and the SWAR(1) models. Then, on the basis of the observation that the
ratios of derivatives could be time-varying a two-regime version of the
above regressions is attempted. Generally, from the OLS, the significance
of the first and the third moments clearly emerges, hence, the importance
of the skewness of the business cycle is confirmed. From the regimeswitching regressions it is not possible to ascertain the significance of the
single regressors but they yield very high R-squared (.77 and .82), also in
comparison to their “one-regime” versions. So, the set of conditional
moments, as a whole, is highly important for the determination of interest
rates. Also, in all regressions, the parameters have the sign that the
theoretical model predicts. A positive relationship is detected between the
first difference of interest rates and the first moment; this is consistent with
the first order conditions of the maximization problem and with the LM
relationship. The second and the third moments have, respectively a
negative and a positive coefficient, consistently with the market riskaversion. The usefulness of all the three moments, in both MSM
regressions, confirms the rationality assumption: firstly, because it
confirms that, indeed, market behave as they solve a maximization
problem, in order to decide consumption and saving; secondly, because it
consists of an evidence for the importance of a wider set of information
16
that includes higher moments as well. The variability of parameters is an
evidence against the Permanent Income Hypothesis since this theory
implies that marginal utilities (hence their ratios) be constant across time.
So, precautionary saving motivations are confirmed in a twofold way:
firstly because some higher moments are significant for saving, hence
interest rates; secondly because the alternative PIH is not accepted by the
evidence. Furthermore, both the theoretical model and the regimeswitching regressions provide some hints on the relationship between
“financial regimes” and “business cycle regimes”: it seems to be that they
are slightly correlated.
Appendix
Model with switches in mean
In this part, the estimates of the switching in mean-model (equation (6),
(7)) are presented. Recall its main features:
gdpt   st  et
t  1, 2,..., T
et  iid ( 0, s2t )
with the following specification:
17
 St   1 (1  st )   2 st
 S2t   12 (1  s t )   22 s t
st  0,1 Re gime 1,2




P
r
s

0
|
s

0

p
P
r
s

1
|
s

1

p
t
t

1
1
1
t
t

1
2
2
The estimates results are resumed in table 6, with standard errors in
brackets:
Table 6: SM model for the Euro-area quarterly rate of growth of real Gdp.
ˆ 1  .76 (.06) ˆ 2  .23 (.05)
ˆ 12  .06 (.02) ˆ 22  .03 (.015)
Residual sum of Squares:
Log-Likelihood value:
1.417
-20.93
p11  0.8 (.13)
p 22  0.71 (.11)
DW-statistic:
Rsq value:
1.49
0.723
Graph 4. reports the inferred smoothed probability for the economy of
been in regime 1, the expansion regime. Shaded area indicates periods of
troughs.
Graph 4. Smoothed probability of the SM model and the Euro area Gdp.
1.6
1.2
0.8
0.4
1.0
0.0
0.8
-0.4
0.6
0.4
0.2
0.0
94
95
96
97
98
99
Euro Area GDP
18
00
01
02
Probability Regime 1
03
04
Descriptive statistics (table 7) of the estimated conditional moments15
along with their graphs are also reported.
Table 7.. Statistics of conditional moments of the SM model.
Mean
Median
Maximum
Minimum
Std. Dev.
Moment 1
0.537
0.648
0.762
0.229
0.238
Moment 2
0.058
0.057
0.114
0.027
0.025
Moment 3
0.002
1.50E-05
0.015
-0.0005
0.004
The following graphs show the estimated conditional moments along with
the Gdp series.
Graph 5. Conditional moments of the SM model
15
These moments are conditional with respect to all information available at the time
they refer to but they are unconditional with respect to regimes.
19
1.6
1.2
0.8
0.4
0.0
-0.4
94
95
96
97
98
99
00
01
02
03
04
Euro Area Gdp
First Moment (Expected Value)
1.6
1.2
0.8
0.4
.12
0.0
.10
-0.4
.08
.06
.04
.02
94
95
96
97
98
99
Euro Are Gdp
00
01
02
03
04
Second Moment (Variance)
1.6
1.2
0.8
0.4
.016
0.0
.012
-0.4
.008
.004
.000
-.004
94
95
96
97
98
99
Euro Area Gdp
20
00
01
02
Third Moment
03
04
Autoregressive model with switching in mean, SWAR(1)
In this part, the estimates of the autoregressive model, AR(1), with
switches in mean, are presented. Recall its main features:
gdpt   st   ( gdpt 1   st1 )  et
t  1,2,...,T
et  iid ( 0, s2t )
with the following specification:
 St   1 (1  st )   2 st
 S2t   12 (1  s t )   22 s t
st  0,1 Re gime 1,2




P
r
s

0
|
s

0

p
P
r
s

1
|
s

1

p
t
t

1
1
1
t
t

1
2
2
The estimates are resumed in table 8, with standard errors in brackets:
Table 8: SWAR(1) model for the Euro-area quarterly rate of growth of
real Gdp.
ˆ 1  .62 (.36) ˆ 2  .22 (.30)   .73 (.25)
ˆ 12  .05 (.004) ˆ 22  .01 (.002)
Residual sum of Squares:
.9199
Log-Likelihood value:
-6.02
p11  .67 (.22)
DW-statistic:
Rsq value:
p 22  .43 (. 32)
2.25
0.816
Graph 6. reports the inferred smoothed probability for the economy of
been in regime 1, the expansion regime. Shaded area indicates periods of
troughs.
21
Graph 6. Smoothed probability of the SWAR(1) model and the Euro area
Gdp.
1.6
1.2
0.8
0.4
0.0
1.0
-0.4
0.8
0.6
0.4
0.2
0.0
1994
1996
1997
1998
1999
Euro Area GDP
2000
2001
2002
2003
2004
Probability Regime 1
Descriptive statistics (table 9) of the estimated conditional moments16
along with their graphs are also reported.
Table 9. Statistics of conditional moments of the SWAR(1) model.
Mean
Median
Maximum
Minimum
Std. Dev.
Moment 1
0.525
0.507
1.04
0.04
0.256
Moment 2
0.184
0.186
0.357
0.078
0.067
Moment 3
-0.010
0.006
0.125
-0.236
0.076
The following graphs show the estimated conditional moments along with
the Gdp series.
16
These moments are conditional with respect to all information available at the time
they refer to but they are unconditional with respect to regimes.
22
Graph 7. Conditional moments of the SWAR(1) model
1.6
1.2
0.8
0.4
0.0
-0.4
1994
1996
1997
1998
1999
Euro Area Gdp
2000
2001
2002
2003
2004
First Moment (Expcted Value)
1.6
1.2
0.8
.4
0.4
.3
0.0
.2
-0.4
.1
.0
1994
1996
1997
1998
1999
Euro Area Gdp
2000
2001
2002
2003
2004
Second Moment (Variance)
1.6
1.2
0.8
0.4
.2
0.0
.1
-0.4
.0
-.1
-.2
-.3
1994
1996
1997
1998
1999
2000
Euro Area Gdp
2001
2002
Third Moment
23
2003
2004
References
-
Ahrens, R., 2002, Predicting recessions with interest rate spreads: a
multicountry regime-switching analysis, Journal of international
Money and Finance, vol. 21, pp. 519-537.
-
Andreou, E., Pelloni, A. and Sensier, M., 2003, The effect of nominal
shock uncertainty on output growt’, Centre for Growth and Business
Cycle Research Discussion Paper Series, University of Manchester,
No. 40.
-
Ang, A. Bekaert, G., 2002, Regime switches in interest rates, Journal
of Business and Economic Statistics, 20, 2, 163-182.
-
Ang A. Bekaert G., 2002, Short rate non-linearities and regime
switches, Journal of Economic Dynamics and Control, 26, 7-8, 12431274.
-
Bansal, R., Zhou, H., 2002, Term structure of interest rates with regime
shifts, The Journal of Finance, 57, 1997-2043.
-
Berk, J.M., 1998, The information content of the yield curve for
monetary policy: a survey, De Economist 146, 303-320.
-
Campbell, J., Y., Luis, M. Viceira, 2001, Who should buy long-term
bonds?, American Economic Review, Vol. 91 (1) pp. 99-127.
American Economic Association.
-
Campbell, J., 1998, Asset prices, consumption, and the business
cycle," NBER Working Papers 6485, National Bureau of Economic
Research.
24
-
Campbell, J. Andrew, W., Lo, A. Craig MacKinlay, 1994, Models of
the term structure of interest rates, Working Papers 94-10, Federal
Reserve Bank of Philadelphia.
-
Campbell, J. Clarida, R. H. 1986, The term structure of Euro-market
interest rates: an empirical investigation, Cowles Foundation
Discussion Papers 772R, Cowles Foundation, Yale University.
-
Campbell, J., Mankiw, N.G., 1989, Consumption, income and interest
rates: reinterpreting the time series evidence, in O.J. Blanchard e S.
Fisher eds. NBER Macroeocnomics Annual 4, 185-216.
-
Campbell, J., Mankiw, N.G., 1991, The response of consumption to
income: a cross-country investigation, European Economic Review 35,
723-767.
-
Campbell, J., 1993, Intertemporal asset pricing without consumption
data" American Economic Review. 83, 487-512.
-
Chan, K., Karolyi, G., Longstaff, F., Sanders, A., 1992, An empirical
comparison of alternative models of the short-term interest rate, The
Journal of Finance, 47, 1209-1227.
-
Chauvet, M., Potter, S., 2001, Predicting a recession: evidence from
the yield curve in the presence of structural breaks, University of
California, Riverside - Department of Economics and Government of
the United States of America - Federal Reserve Bank of New York.
-
Clements, M., Krolzig, H.-M., 2004, Can regime-switching models
reproduce the business cycle features of US aggregate consumption,
investment and output?, International Journal of Finance & Economics
2004 9 (1)1-14.
25
-
Cover, J.P., (1992), Asymmetric effects of positive and negative
supply shocks, Quarterly Journal of Economics, vol. 107, 1261-82.
-
Dynan, K, 1993, How prudent are consumers?, Journal of Political
Economy, 101, December, 1104-1113.
-
Estrella, A., Mishkin, F.S., 1995, The term structure of interest rates
and its role in monetary policy for the European Central Bank, Federal
Reserve of New York, Research Paper 9526.
-
Evans, B. D., 2000, Regimes shifts, risk and the term structure,
Working Paper, Georgetown University.
-
Frankel, J. 1995, Financial markets and monetary policy, MIT Press.
-
Franses, P. H, Van Dijk, V., 2000, Non-linear time series models in
empirical finance, Cambridge University Press.
-
Friedman, M., 1957, A theory of the consumption function. Princeton
NJ: Princeton University Press.
-
Garcia, R., Schaller, H., 1995, Are the effects of Monetary policy
asymmetric?, Mimeo Carleton University, Canada.
-
Garcia, R., Perron, P., 1996, An analysis of real interest under regime
shift, Review of Economics and Statistics, 78, 111-125.
-
Gray, S., 1996, Modelling the conditional distribution of interest rates
as a regime-switching process, Journal of Financial Economics, 42 2762.
-
Guedhami, O., Sy, Oumar, 2005, Does conditional market skewness
resolve the puzzling market risk-return relationship? The Quarterly
Review of Economics and Finance, 45, 582-598.
26
-
Hamilton, J.D., 1988, Rational expectations econometric analysis of
changes in regime: An investigation of the term structure of interest
rates, Journal of Economics, Dynamic and Control, 12, 385-423.
-
Hamilton, J.D., 1989, A new approach to the economic analysis of
non-stationary
time
series
and
the
business
cycle,
march,
Econometrica.
-
Hamilton, J.D., 1990, Analysis of time series subject to changes in
regime, Journal of Econometrics, vol. 9 pp. 27-39.
-
Hamilton, J.D., 1995, Rational expectations and the economic
consequences of changes in regime, in K. Hoover, Macroeconometrics:
Developments, Tensions and Prospects, Dordrecht, Kluwer.
-
Hamilton, J.D., Kim, D.H., 2002, A re-examination of the
predictability of the yield spread for real economic activity, Journal of
Money, Credit, and Banking, vol. 34, pp. 340-360.
-
Hamilton, J.D., Lin, G. 1996, Stock Market volatility and the business
cycle, Journal of Applied Econometrics, Vol. 11, 573-593.
-
Harvey, Campbell, H. R., 1988, The real term structure and
consumption growth, Journal of Financial Economics, 22 305-333.
-
Leland, H., 1968, Saving and uncertainty: the precautionary demand
for saving, Quarterly Journal of Economics 82, August: 465-473.
-
Morgan, D.P., 1993, Asymmetric effects of monetary policy, Federal
Reserve Bank of Kansas City Economic Review, vol. 78(2), 21-33.
-
Naik, V. M., Lee, H., 1997, Yield curve dynamics with discrete shifts
in economics regimes: theory and estimation, Working Paper,
University of British Columbia.
27
-
Osborn, D., 2002, The prediction of business cycle phases: financial
variables and international linkages, The University of Manchester,
Discussion Paper Series, No 15.
-
Psaradakis, Z., Sola, M., Spagnolo, F.,
2001, Risk premia with
Markov regimes and the term structure of interest rates, School of
Economics, Mathematics, Statistics, Birkbeck College, London.
-
Peel, D., A., Ioannidis, C., 2003, Empirical evidence on the
relationship between the term structure of interest rates and future real
output changes when there are changes in policy regimes, Economics
Letters, 2003, vol. 78, issue 2, pages 147-152.
-
Ravn, M.O., Sola, M., 1996, Asymmetric effects of monetary policy in
the US: positive vs. negative or big vs. small?, mimeo, University of
Southampton.
-
Rothman, P., 1999, Non-linear time series analysis of economic and
financial data. Klewer Academic Publishers.
-
Timmermann, A., 2000, Moments of
Markov-switching models,
Journal of Econometrics 96, 75-111.
Weise, C.L., 1999, The asymmetric effects of monetary policy: a
non-linear vector autoregression approach, Journal of Money, Credit
and Banking, vol. 31, 85-108.
-
28