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Interest Rates and Business Cycles
Fluctuations: a Focus on Higher Moments
By Andrea Beccarini,
University of L’Aquila, Italy.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Two Stylised Facts:
- Non Linearity and Regimes in the Interest Rates process
- Non Linearity and Regimes in the Business Cycles Variables
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Use the C-Capm to explain the non linearity in the
interest rates by expanding the expected marginal utility
of future consumption for the F.O.C.

Pt ,n 
E[u ' (ct n )];
u ' (ct )
Up to the fourth order:
1
E[u ' (ct n )]  u ' (ct n )  u ' ' (ct n ) E[(ct n  ct n )]  u ' ' ' (ct n ) E[(ct n  ct n )2 ] 
2
1 iv
1 v
3
u (ct n ) E[(ct n  ct n ) ]  u (ct n ) E[(ct n  ct n ) 4 ]
6
24
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
This is consistent with:
- Rationality assumption of the C-Capm:
the representative agent is an optimiser; all information must be exploited
for the agent to be rational.
- Non linearity of the Business Cycles variables:
the alternation of phases of recessions and expansions make non-normal the
distribution of the total output.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
The theoretical moment links interest rates with
moments up to the fourth of the consumption
(Business Cycles variable)
Hence, non linearity in the interest rates may depend
on the higher moments of Business Cycles i.e.
asymmetry and kurtosis.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Furthermore,
By testing the above model one may also verify:
- Rationality of financial markets, implicit in the maximization
problem and in the use of all available information.
- Market risk aversion, represented by higher order utility
function.
- Precautionary Saving Hypothesis, represented by the
significance of the second moment.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Asymmetry of the Business Cycles
D1
Strong Recession
D0
Rec.
Expansion
Strong Expansion
Distribution
Expected Value
Variance
Third Moment
D0
E[x]=0
Var[x]
E[(x-E(x))^3]=0
D1
E[x]=0
Var[x]
E[(x-E(x))^3]>0
Risk aversion implies the preference of a positive third moment.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Statistical models for the Business Cycles variable:
1. Switching in Mean model (SM):
yt  st  et
t  1,2,..., T
et  iid (0,  s2t )
2. Switching in Mean, Autoregressive model (SWAR):
yt   st   ( yt 1   st 1 )  et
et  iid (0,  s2t )
t  1,2,..., T
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Moments of the Business Cycles variable as a function of the
Switching in Mean model’s parameters:
- First moment:
- Second moment:
 t   t 1  (1   t )  2
 t2   t 12  (1   t ) 22   t (1   t )(1   2 ) 2
- Third moment:
E[( yt   t ) 3 ]   t (1   t )(1   2 )[3( 12   22 )  (1  2 t )( 2  12 ) 2 ]
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Estimation Strategy:
Dependent variable:
Euro area
3-months
interest rates
Technique:
Regressors:
Linear model
(OLS).
1°, 2° and 3°
moment of the
Euro area GDP
growth rate,
(Business Cycles
variable).
Regime
Switching
model.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
OLS regression’s results of real interest rates on Gdp moments
(SM model).
Regression 1: 3-Months Quarterly Real Interest Rate, First Difference
Variable
Constant
First moment
Third moment(-1)
R-squared
Adjusted R-squared
S.E. of regression
Coefficient
-0.52
0.70
30.04
0.215
0.177
0.342
Std. Error
0.14
0.23
13.83
t-Statistic
-3.68
3.08
2.17
Sum squared resid
Log likelihood
Durbin-Watson stat
Prob.
0.0007
0.0037
0.0358
4.785
-13.62
2.068
The first and third moments are significant and they have the
expected sign.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
OLS regression’s results of real interest rates on Gdp moments
(SWAR(1) model).
Regression 2: 3-Months Quarterly Real Interest Rate, First Difference
Variable
Constant
First moment
Third moment
R-squared
Adjusted R-squared
S.E. of regression
Coefficient
-0.73
1.28
2.47
0.244
0.206
0.338
Std. Error
0.22
0.42
1.43
t-Statistic
-3.36
3.02
1.72
Sum squared resid
Log likelihood
Durbin-Watson stat
Prob.
0.0017
0.0044
0.0922
4.57
-12.82
2.26
The first and third moments are significant and they have the
expected sign.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
R-S regression’s results of real interest rates on Gdp moments
(SM model).
Regression 3: 3-Months Quarterly Real Interest Rate, First Difference
Constant
First moment
Second moment
Third moment
Standard deviation
R-squared
Sum squared resid
Durbin-Watson stat
REGIME 1
Coefficient
Std. Error
-0.68
0.23
0.46
1.22
-0.33
18.49
7.94
76.95
0.21
0.81
1.18
1.98
Log likelihood
AIC
BIC
REGIME 2
Coefficient
Std. Error
-0.06
0.18
0.62
0.57
-2.71
7.13
17.68
32.12
0.19
-18.73
61.46
82.87
The first, the second and third moments have the expected sign
but they are only jointly significant.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
R-S regression’s results of real interest rates on Gdp moments
(SWAR(1) model).
Regression 4: 3-Months Quarterly Real Interest Rate, First Difference
Constant
First moment
Second moment
Third moment
Standard deviation
R-squared
Sum squared resid.
Durbin-Watson stat
REGIME 1
Coefficient
Std. Error
-0.65
0.34
0.58
1.56
-0.06
4.08
0.57
3.66
0.27
0.77
1.39
2.27
Log likelihood
AIC
BIC
REGIME 2
Coefficient
Std. Error
-0.35
0.31
1.07
0.68
-0.42
1.73
2.07
1.93
0.20
-25.57
55.15
76.56
The first, the second and third moments have the expected sign
but they are only jointly significant.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Main Results:
- The positive relationship between the first moment of Gdp and the
first difference of interest rates, consistently with basic C-Capm.
- The negative relationship between the second moment of Gdp and
the first difference of interest rates, hence Precautionary Saving
motivations.
- The positive relationship between the third moment of Gdp and the
first difference of interest rates, consistently with the presence of
relevant information in higher moments.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Other Results:
- The importance of fitted moments of the Gdp rather than the row
time series.
- The evidence of the rational expectation formation rather than
the adaptive formation.
- The Precautionary Saving hypothesis rather than the Permanent
Income Hypothesis.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
All in all:
The business cycles variable
(Gdp) is not normal but
shows signs of asymmetry;
this is due to the alternation
of recessions and
expansions.
The interest rates
are non-linear and
embed a mixture
of different
stochastic
processes.
The C-Capm implies
the considerations of
high moment of state
variable (see the
Taylor’s expansion up
to the fourth order).
- The theoretical model and the statistical evidence find a robust
connection with interest rates and moments of the B.C up to the
fourth order.
- Rationality of Financial market, Risk Aversions, Precautionary
Saving are sustained by theory and evidence.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Annex 1: Estimated parameters of the SM (switching in mean) model:
ˆ1  .76 (.06) ˆ 2  .23 (.05)
ˆ 12  .06 (.02) ˆ 22  .03 (.015)
Residual sum of Squares: 1.417
Log-Likelihood value:
-20.93
p11  0.8 (.13) p 22  0.71 (.11)
DW-statistic:
Rsq value:
1.49
0.723
The average rate of growth of the Gdp during expansions is 0.76; during
recessions is 0.23.
The probability of remaining in the expansion phase is 0.81. The
probability of remaining in the recession phase is 0.71.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Annex 2: Estimated parameters of the SWAR(1) (switching in mean,
autoregressive) model:
ˆ 1  .62 (.36) ˆ 2  .22 (.30)
  .73 (.25)
ˆ 12  .05 (.004) ˆ 22  .01 (.002)
p11  .67 (.22) p 22  .43 (.32)
Residual sum of Squares: .9199
Log-Likelihood value:
-6.02
DW-statistic:
Rsq value:
2.25
0.816
The average rate of growth of the Gdp during expansions is 0.62; during
recessions is 0.22.
The probability of remaining in the expansion phase is 0.67. The
probability of remaining in the recession phase is 0.43.
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
The end of the presentation titled:
Interest Rates and Business Cycles Fluctuations: a
Focus on Higher Moments
By Andrea Beccarini,
University of L’Aquila, Italy.
E-mail address: [email protected]
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini
Interest Rates and Business Cycles Fluctuations: a Focus on Higher Moments, by Andrea Beccarini