Finally, Strikholm and Terasvirta (2005) have proposed a testing procedure to determine the
optimal number of regimes within a threshold analysis. In this work it seems reasonable to restrict
the choice between two or, at most, three regimes, since both the spreads and its components are
strictly stationary time series. The idea of the test is to compare the linear model with smoothed
transitions regressions that allow for either two or three regimes13. Test results are supportive of
the two-regime specification against both linearity and three-regime model, as shown in Table 5.
Strikholm - Terasvirta Test
Number of
Regimes
(n, m) 1 vs 2 2 vs 3
|
(120, 3) |
(0.000) |
(0.457) |
|
(60, 3) |
(0.000) |
(0.342) |
|
(36, 3) |
(0.000) |
(0.193) |
|
(24,3) |
(0.000) |
(0.107) |
|
(12, 3) |
(0.000) |
(0.146) |
|
(6, 3) |
(0.996) |
/ |
sample jan64-jun02; p-values
Table 5
5. Empirical Results
In this Section we present some further evidence regarding the instability over time of the linear
equation to test the expectations hypothesis; then we present the threshold estimates. We believe
that when the deviations between the expected and the actual spread are large, which occurs when
also the variance of the term premium shoots exponentially, single equation models fail to support
the expectations theory. We thus propose a non linear model to analyse the informative content of
the term structure. The empirical analysis of the expectations hypothesis is performed within a
threshold framework in which regimes are determined by the level of the term premium. We recall
that the term premium is proxy for monetary policy surprise, since it measures the unexpected
component of the yield spread. According to Campbell and Shiller (1991), the term premium is
13 The testing procedure implies a sequential comparison of the sum of squared residuals obtained by logistic-STR
models with different number of regimes. In particular, if the one-regime specification is rejected against the alternative
two-regime specification, we proceed to test the two-regime model against the three-regime model, and so on. Smooth
transition regressions are based on the logistic function. The test considers the first-order Taylor approximation of the
logistic function around the parameter that governs the transition between regimes. The test has been performed both
with and without the Taylor approximation obtaining similar results. The practical implementation of the test requires
specification of the parameters. In testing two regimes against linearity we set the threshold value in the logistic
function equal to the mean of the series. This value has been chosen to make the test independent of the threshold
estimate. After estimating the threshold model, we also performed the test with the estimated threshold value ^
obtaining similar results. In both cases the non linear framework has been preferred to the null of linearity. Without loss
of generality, the strategy followed for testing two against three regimes has been to set, threshold c1 and threshold c2
respectively equal to the 33% and the 66% quantiles in the three-regime logistic-STR specification. Following
Strikholm and Terasvirta we use an asymptotic F approximation of the ∕2 test statistics.
15