where ξt ^N (0,σξ2),η^ N (η, σfy and ɪɪai < 1. However, modeling the interest rate for the
i=1
very long run through constant coefficient models is likely to be an unrealistic assumption, since
a number of factors, such as the economic cycle, oil crises, stock market crises, productivity and
technology shocks may account for a time-varying behavior in the data generation process of
the interest rate. In this respect, we introduce two models that are time-heterogeneous in the
sense that they account for the possibility of time varying parameters and regime changes. Our
Regime-Switching (RS) model is one with two regimes as follows:
p
rt = ηk + et, et = aiket-i + ξt (4)
i=1
where ξt ^ IIDN(0, σ^), k = 1, 2 for the first and second regime, respectively. Each regime
incorporates a different speed of mean-reversion, along with a different permanent component,
ηk , and error variance. The probability of being in each regime at time t is specified as a Markov
1 process, i.e. it depends only on the regime at time t - 1, with the matrix of the transition
probabilities assumed to be constant.3 A more convenient way to account for infinite regimes
in the interest rate process is through a time varying coefficient model, such as an AR(1) model
with an AR(p) coefficient, namely a State Space (SS) model, given by the following system of
equations:
rt = η + αtrt-1 + et,
p
αt = ηiαt-i + ut ,
i=1
(ut)-N(∣0∣∙∣i й|) (5)
3 Empirical Results
3.1 Data and Estimation Results
Our dataset consists of nominal interest rates transformed to real interest rates by subtracting
the annual change in the Consumer Price Index for the period 1800 to 2001.4 To smooth very
short-term fluctuations, a 3-year moving average of the real interest rate series is employed and
in order to avoid negative interest rates, we use the natural logarithms of the series.
A variety of unit root tests confirmed that the UK real interest rate is a stationary process.5
Our AR(4) model displays relatively rapid reversion to the implied unconditional mean of 3.32%
(see Table 1, Panel A). However, our estimates for the RS model (see Table 1, Panel B) indicate
the presence of two distinct regimes (modelled as AR(2) processes). The unconditional means
of each are 2.14% and 3.70% and mean reversion is faster in the latter. The first regime has
an estimated duration of 4 years, while the second one is more persistent with a duration of 15
years. Overall, the estimates of this model suggest that low interest rate periods are quickly
mean-reverting, surrounded by greater uncertainty and transit more often to high interest rates
periods which are more persistent and less uncertain. Turning to our SS model, the parameter
estimates (see Table 1, Panel C) suggest that the state process is highly persistent, almost
a random-walk process, as indicated by the estimate of the autoregressive coefficient. The
constant of our model suggests a minimum of 1.31% for the interest rate process.
3 The matrix of probabilities is as follows:
Pr ob(Rt =1| Rt-1 =1)=P, Pr ob(Rt =2| Rt-1 =2)=Q
Pr ob(Rt =2| Rt-1 =1)=1-P, Pr ob(Rt =1| Rt-1 =2)=1-Q
where Rt refers to the regime at time t.
4The nominal interest rate is the United Kingdom 2 1/2% Consol Yield. Data provided by the Global Financial
Data, Inc, available at http://www.globalfindata.com.
5 Unit root tests are not reported for brevity but are available upon request.