264 Claudio Pizzi
researchers’ interest has mainly been tuned to the problem of estimating the cointe-
gration relationship (it is worth mentioning, amongst others, Johansen [8], Saikko-
nen [18], Stock and Watson [21] Johansen [9] and Strachan and Inder [22]) and
building statistical tests to verify the presence of such a relationship.
The first test suggested by Engle and Granger [5] was followed by the tests pro-
posed by Stock and Watson [20] to identify common trends in the time series assessed.
After them, Phillips and Ouliaris [15] developed a test based on the principal com-
ponents method, followed by a test on regression model residuals [14]. Johansen [8]
instead proposed a test based on the likelihood ratio.
The idea of linear cointegration then has been extended to consider some kind of
nonlinearity. Several research strains can be identified against this background. One
suggests that the response mechanism to the departure from the steady state follows a
threshold autoregressive process (see for example the work by Balke and Fomby [1]).
With regard to the statistical tests to assess the presence of threshold cointegration,
see for example Hansen and Byeongseon [7].
The second strain considers the fractional cointegration: amongst the numerous
contributions, we would like to recall Cheung and Lai [3], Robinson and Marin-
ucci [16], Robinson and Hualde [17] and Caporale and Gil-Alana [2].
Finally, Granger and Yoon [6] introduced the concept of hidden cointegration that
envisages an asymmetrical system answer, i.e., the mechanism that guides the system
to the steady state is only active in the presence of either positive or negative shocks,
but not of both. Schorderet’s [19] work follows up this idea and suggests a procedure
to verify the presence of hidden cointegration.
From a more general standpoint, Park and Phillips [12] considered non-linear
regression with integrated processes, while Lee et al. [11] highlighted the existence of
a spurious nonlinear relationship. In the meantime further developments contemplated
the equilibrium adjustment mechanism guided by a non-observable weak force, on
which further reading is available, by Pellizzari et al. [13].
This work is part of the latter research strain and suggests the recourse to local
linear models (LLM) to build a test for nonlinear cointegration. Indeed, the use of
local models has the advantage of not requiring the a priori definition of the func-
tional form of the cointegration relationship, enabling the construction of a dynamic
adjustment mechanism. In other words, a different model can be considered for each
instant (in the simplest of cases, it is linear) to guide the system towards a new equi-
librium. The residuals of the local model can thus be employed to define a nonlinear
cointegration test. The use of local linear models also enables the construction of a
Local Error Correction Model (LECM) that considers a correction mechanism that
changes in time. The paper is organised as follows. The next section introduces the
idea of nonlinear cointegration, presenting the LECM and the unrestricted Local Er-
ror Correction Model (uLECM). Section 3 presents an application to real data, to
test the nonlinear cointegration assumption. The time series pairs for which the null
hypothesis of no cointegration is rejected will be used to estimate both the uLECM
and the speed of convergence to equilibrium. The paper will end with some closing
remarks.