3.5 Methodology
3.5.2 Testing for structural breaks with cointegrated variables
There are different avenues one can pursue in testing the time series for cointegration. The Engle and Granger (1987) method is a two-step standard cointegration test which assumes that the cointegrating vector does not change over time, meaning that the cointegrating relationship has a stationary distribution in the long run. The method is based on the Granger representation theorem15 and fundamentally is a residual based method in comparison to other tests where the raw data is examined. The test identifies whether the cointegrating vector had shifted at an unknown point in time. Basically, the method assesses whether the residuals have a unit root, such that the lagged residuals are
15The conjecture asserts that if the data is I(1) and a dynamic linear model with stationary errors is
found, subsequently the variables must be I(1) (Brooks,2008). Besides, the lagged value of the error term
acts as a vector error correction term. The Vector Error Correction Model gives support for short-term relationships, whereas the cointegration represents the long-run relationships.
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Chapter 3 Modelling the Long-run Relationship of Short-term Interest Rate Spreads
regressed on the differenced residuals. If the residuals are proved to be stationary, then the two series are cointegrated. The criticism of this method is that the cointegrating relationships cannot be identified. For this reason, Johansen’s (1995) cointegration test is implemented, and restrictions are placed on the equations. The techniques can be implemented solely if the time series have the same level of integration. The VAR(4) is re-written as a Vector Error Correction Model following the notation used in StataCorp (2011): ∆yt=v+ Πyt−1+ p−1 X i=1 Γi∆yt−1+t (3.10)
whereΠ=Pjj==1pAj−IkandΓi =−Pjj==pi+1Aj. Thevis a (3 x 1) vector of coefficients,
andtis a (3 x 1) vector of normally distributed error terms, with zero mean and constant
variance. Similarly, for the VAR(4) model, t = 1,2, . . . , T and i = 1,2, . . . , I. Earlier, visual inspection of the series revealed that there might be deterministic trends present in the relationships.
The following is the generic representation of the VECM, which allows for a linear trend and a constant in the model:
∆yt=αβ0yt−1+ p−1 X
i=1
Γi∆yt−1+v+δt+t (3.11)
where matrix β consists of the parameters in the cointegrating relationships and α
comprises the error correction terms also called adjustment parameters. α is a (k x r) rank matrix with the deterministic components:
v=αµ+γ (3.12)
and
δt=αρt+τt (3.13)
whereµandρare (r x 1) parameter vectors, andγ andτ are (kx 1) parameter vectors.
γis independent ofαµandτ is independent ofαρ. Thus, the VECM can be re-written as:
Chapter 3 Modelling the Long-run Relationship of Short-term Interest Rate Spreads 45 ∆yt=α(β0yt−1+µ+ρt) + p−1 X i=1 Γi∆yt−1+γ+τt+t (3.14)
Johansen’s (1988) maximum likelihood framework based on the trace test is used to find the cointegrating vectors and relationships (Maddala and Kim,1998), and is given by:
−2 log Lmax ∝T n X
i=1
ln(1−λi) (3.15)
whereλi are the roots of the base equation and the likelihood ratio test statistic is given
by: λtrace=−T n X i=r+1 ln(1−ˆλi) (3.16)
where ˆλr+1, . . . ,λˆn are the eigenvalues of the base equation with the smallest values
(Maddala and Kim,1998).
As shown below, an unrestricted cointegration test along with four restricted cointe- gration tests are implemented. The number of lags to be included in the equations is determined in advance. The decision on the optimal lag order is based on the AIC, the Schwarz’s Bayesian information criterion (BIC), and the Hanna-Quinn information criterion (HQIC) - which are calculated using the likelihood ratio test16. For all five cointegration tests, the null hypothesis states that there are at most r cointegrating vectors in the system of equations.
1. Unrestricted trend
This implies that there are quadratic terms in the levels of the LIBOR-OIS, EU- SWEC and GerUS3M spreads. Moreover, the cointegrating relationships are I(0).
2. Restricted trend, τ = 0
The trends in the spreads are linear and not quadratic, thus the cointegrating relationships are trend stationary.
3. Unrestricted constant, withτ = 0 andρ= 0.
16
The Stata software uses the ‘varsoc’ command to select the optimal number of lags. The null
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Chapter 3 Modelling the Long-run Relationship of Short-term Interest Rate Spreads
In this case there are no quadratic trends but linear trends in the differenced series. Also, the relationships are limited to being stationary with a constant mean.
4. Restricted constant,τ = 0,ρ= 0 and γ = 0.
There are no linear trends in the levels of the differenced LIBOR-OIS, GerUS3M and EUSWEC series. Cointegrating relationships will have a constant mean, how- ever there won’t be trends or constants in the equations.
5. No trend,τ = 0,ρ= 0, γ= 0 and µ= 0.
There are no trends in the relationships, which are stationary with zero mean. Similarly, differences and levels of the series have zero means.
After the short-run and long-run relationships are establised, the Gregory-Hansen (1996) cointegration method is implemented to test for the presence of structural breaks or regime switches. If the cointegrating vector changes at a single unknown time, the series may return to equilibrium and a linear combination of the series becomes stationary. This test allows for serial correlation among the innovations. A dummy is included in the Engle-Granger system of regressions, which helps identify a one-time regime shift in the intercept and slope coefficients. The conventional augmented DF test would not suffice in view of the cointegrating vector shifting at an unknown point in time. The test statistic is similar to a typical Chow test, and is centred on comparing the sum of squares of residuals, a method which measures the amount of variance in the data sets. In normal circumstances, this is not accounted for by the regression model. The null hypothesis of no cointegration is tested against the alternative hypothesis of cointegration with level shift/structural break/regime shift at a single unknown time. The three models of structural breaks follow the Gregory and Hansen (1996) notation, and are as follows:
A Structural break in level
yt1=µ1+µ2 φtτ +αTyt2+γTyt3+t, (3.17)
whereµ1represents the intercept previous to a shift,µ2represents the change in the
intercept at the moment of the shift. αrefers to the cointegrating slope coefficient foryt2 (which is the LIBOR-OIS spread) andγ represents the cointegrating slope
coefficient foryt3 (which represents the EUSWEC rate);t= 1, . . . , nandτ ∈(0,1)
is the unidentified parameter and represents the relative timing of the break point; it can only take integer values. The error term satisfiest∼N(0, σ2). The dummy
Chapter 3 Modelling the Long-run Relationship of Short-term Interest Rate Spreads 47 φtτ = ( 0, ift≤[nτ], 1, ift >[nτ]. (3.18)
The dummy variable has the role of accounting for fluctuations in the constant term and slope coefficients.
B Level shift with trend
yt1 =µ1+µ2 φtτ +βt+αTyt2+γTyt3+t (3.19)
Beside the change in the intercept, a shift in the slope vectorβ is allowed. αrefers to the cointegrating slope coefficient for the LIBOR-OIS spread and γ represents the cointegrating slope coefficient for the EUSWEC spread;t= 1, . . . , nand t∼
N(0, σ2).
C Regime shift
yt1 =µ1+µ2 φtτ +αT1 yt2+αT2 yt2 φtτ
+γ1T yt3+γ2T yt3 φtτ +t, (3.20)
where α1 refers to the cointegrating slope coefficients for the LIBOR-OIS spread
before the regime shift andα2 represents the change in slope coefficients. γ1 refers
to the cointegrating slope coefficients for the EUSWEC spread before the regime shift andγ2 represents the change in the EUSWEC rate after a regime change has
occured. t= 1, . . . , nand t∼N(0, σ2).