Testing linearity in cointegrating smooth transition regressions - Feweb

Econometrics Journal (2004), volume 7, pp. 341–365.
Testing linearity in cointegrating smooth transition regressions
IN CHOI ∗ AND PENTTI SAIKKONEN †
∗ Department of Economics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
E-mail: inchoi@ust.hk
† Department of Mathematics and Statistics, P.O. Box 54 (Unioninkatu 37),
FIN-00014 University of Helsinki, Finland
E-mail: pentti.saikkonen@helsinki.fi
Received: November 2003
Summary This paper develops statistical tests that can be used to test linearity in
cointegrating smooth transition regression models. These tests extend previous similar tests
by considering I(1) regressors instead of stationary or mixing regressors and they also allow
for more general transition mechanisms than in previous studies. As is typical in cointegrating
regressions, the regressors and errors of the model can be serially and contemporaneously
correlated. In order to allow for this feature, an endogeneity correction based on a leads-and-lags
approach is employed. The proposed tests are very simple to use because ordinary least squares
techniques and standard chi-square limiting distributions apply. Simulation experiments
indicate that the tests have reasonable finite sample properties. Empirical applications to a
U.K. money demand function illustrate the practical usefulness of the tests.
Keywords: Linearity test, Cointegration, Smooth transition regression, Money demand.
1. INTRODUCTION
Since its introduction in the 1980s, the theory of cointegration has undergone considerable
development and become a central part of modern time series econometrics. Various econometric
and statistical methods have been developed for the analysis of cointegrated time series and
many of them are now routinely applied in empirical studies. However, most of the methods
presently available assume that the cointegrating relations are linear, which according to economic
theory, need not be the case. For instance, economic theories for money demand do not require
the money demand function be linear (see, e.g. Baumol 1952; Tobin 1956; Baba et al. 1992),
although linearity has been assumed in most empirical studies. Being able to investigate the
possibility of a non-linear cointegrating relation is therefore of interest, especially if an estimated
linear cointegrating relation is found to be far from that predicted by economic theory.
The purpose of this paper is to develop test procedures that can be used to test the linearity of
a cointegrating relation in the context of a non-linear cointegrating smooth transition regression
(STR) model. Our test procedures are based on the approach previously used for stationary
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342 In Choi and Pentti Saikkonen
models by Luukkonen et al. (1988) and Granger and Teräsvirta (1993) amongst others (for further
references, see Teräsvirta (1998) and van Dijk et al. (2002)). An advantage of this approach is
that the resulting tests are very simple to use. Computations can be carried out by ordinary least
squares (OLS) and a standard chi-square criterion can be used to obtain asymptotically valid
tests.
Although our tests have the same basis as previous linearity tests developed for STR models,
the fact that we are working with a cointegrating regression brings about notable new features
to the testing problem. Because of the cointegration, we have to relax previous assumptions of
stationary or mixing regressors and allow for nonstationary I(1) regressors. Unlike in the previous
literature, we will not assume that the regressors are exogenous in any sense. Instead, we follow
the common practice in cointegrating regressions and permit both serial and contemporaneous
correlations between the regressors and the error term of the model. In order to allow for this
feature, we use the leads-and-lags approach proposed by Saikkonen (1991), Phillips and Loretan
(1991) and Stock and Watson (1993) for linear cointegrating regressions and Saikkonen and
Choi (2004) for somewhat more general cointegrating STR regressions than those considered
in this paper. As with cointegrating regressions in general, the error term of our model can be
serially correlated. A further point that makes our model more general than previous stationary
STR models is that it may contain several transition functions and more than a single transition
variable.
The rest of the paper is organized as follows. Section 2 introduces the model, hypotheses
and assumptions. Section 3 develops the linearity tests. Section 4 extends the tests of Section
3 to a more general model. Section 5 reports simulation results of the finite sample properties
of the developed tests. Section 6 applies the tests to a U.K. money demand function. Section 7
concludes.
2. THE MODEL, HYPOTHESES AND ASSUMPTIONS
Consider the cointegrating STR model
yt = µ + νg(zst) + α ′ xt + β ′ xt g (zst) + ut, t = 1, 2,...,T , (1)
where xt = [x 1t, ..., xpt] ′ is a p-dimensional I(1) process, ut is a zero-mean stationary error term,
and
zst = γ (xst − c), γ = 0, s ∈{1,...,p}. (2)
Furthermore, g(zst) is a smooth, real-valued transition function of the process xst and the scalar
parameters γ and c. Of the other parameters of the model, µ and ν are scalars and α and β are
p × 1 vectors. In model (1), there is a single non-linear component g(zst) that is affected by the
single transition variable xst. It is straightforward to extend linearity tests for this model to a model
with multiple transition functions and transition variables. We will consider the extension after
the tests for model (1) are fully developed.
Model (1) is a non-linear extension of Engle and Granger’s (1987) linear cointegrating
regression. This model could be extended further by including additional I(1) regressors that
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Testing linearity in cointegrating smooth transition regressions 343
are not affected by the transition functions. For simplicity, this extension is not made explicit in
this paper. 1 The non-linear nature of the model is determined by the transition function g(zst). 2
We are interested in testing the null hypothesis that model (1) reduces to a conventional linear
cointegrating regression. Thus, the null hypothesis of interest is
H0 : ν = 0 and β = 0. (3)
The alternative simply states that the null hypothesis is not true.
STR models have been used to describe economic relations that change smoothly depending
on the location of some economic variables. In model (1), an appropriate choice of the transition
function g(zst) allows the relationship between xt and yt to change depending on where xst are
located relative to the parameter c. The smoothness of the change is determined by the parameter
γ . Examples will be given later. Further examples and discussions on the STR model can be found
in Granger and Teräsvirta (1993), Teräsvirta (1998) and van Dijk et al. (2002), although these
authors do not explicitly consider the case of I(1) processes.
As for the transition function g(z), we assume
Assumption 1.
(i) g(0) = 0.
(ii) The function g(zst) is three times differentiable in an open ball in R with centre 0 and
radius r(r > 0).
= 0.
(iii) ∂g(z)
∂z |z=0
(iv) ∂3 g(z)
∂z 3 |z=0
= 0.
The most commonly used transition functions satisfy Assumption 1, as illustrated by the
following example and the examples given in the aforementioned references. More examples will
be given when model (1) is generalized.
Example 1. (One transition function, one transition variable and two regimes)
1
g (zst) =
, γ > 0.
1 + e 2
In this example, the transition function is a logistic function that makes the regression coefficient
for xt vary smoothly between α − 1
1
β and α + 2 2β. When the value of the regressor x1t is
sufficiently far below the value of the parameter c the regression coefficient takes a value close to
α − 1
2β; and when the value of the regressor xt increases and exceeds the value of the parameter
c the value of the regression coefficient changes and approaches α + 1
β. The intercept term
2
undergoes similar changes.
−γ (xst−c) − 1
1 When there are additional I(1) variables, we add the variables in levels and their differences in leads-and-lags to the
auxiliary regression models on which our test statistics are based (see, e.g. equations (12), (14), (18) and (20)). Asymptotic
distributions of our tests are not affected by this.
2 Variables other than x 1t , ..., xpt may enter the transition function. Extending our tests to this case is straightforward. In
addition, differences of the regressors may be considered as transition variables zit. But then, the endogeneity correction
of this paper does not work.
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344 In Choi and Pentti Saikkonen
We will now discuss assumptions required for model (1). As already mentioned, we assume
Assumption 2.
xt = xt−1 + vt, t = 1, 2,..., (4)
where vt is a zero-mean stationary process and the initial value x0 may be any random vector
with the property Ex 04 < ∞.
Moreover, it will be convenient to assume that the (p + 1) -dimensional process wt = [ut v ′ t ]′
satisfies the following assumption employed by Hansen (1992) in a somewhat weaker form.
Assumption 3. For some r > 4, wt = [ut v ′ t ]′ is a stationary, zero-mean, strong mixing sequence
with mixing coefficients of size −4r/(r − 4) and Ewt r < ∞.
Assumption 3 is fairly general. It covers a variety of weakly dependent processes and implies
that an invariance principle applies to partial sums formed of the process wt (see Hansen 1992
the proof of Theorem 3.1). It also permits the cointegrated system defined by (1) and (4) to
have non-linear short-run dynamics, which is convenient when the cointegrating regression is
non-linear.
As discussed in Saikkonen and Choi (2004), Assumption 3 implies that the process wt has a
continuous spectral density matrix f ww(λ) that we assume to satisfy.
Assumption 4. The spectral density matrix f ww(λ) is bounded away from zero or the matrix
f ww(λ) − εI p+1 is positive semi-definite for some ε>0.
Assumption 4 specialized to the case λ = 0 implies that the components of the I(1) process xt
are not cointegrated. Conformably to the partition of the process wt, we write f ww(λ) = [fab(λ)]
where a, b ∈ {u, v}. The long-run covariance matrix of the process wt is defined by = 2π f ww(0)
and partitioned conformably with the other partitions as
2 ωu ωuv
=
.
ωvu vv
Model (1) and subsequent discussions assume no drifts in the I(1) regressors. In some
applications this may be restrictive and therefore we need to consider a model that allows for
this feature. For this, we assume Assumption 5 instead of Assumption 2.
Assumption 5. Assumption 2 holds with xt generated by
xt = µx + xt−1 + vt, t = 1, 2,..., (5)
where µx = 0.
Assumption 5 implies that the generating mechanism of the regressors can be written as
xt = x0 + µxt +
t
v j, t = 1, 2,..., (6)
j=1
where the deterministic time trend does not vanish. Its presence implies that it is reasonable to
modify the definition of the transition variables zst. If the previous definition in (2) is used in the
case of the logistic transition function, for example, the existence of the time trend implies that
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Testing linearity in cointegrating smooth transition regressions 345
the transition function tends to decrease or increase monotonically as the sample size grows. A
more reasonable specification for the transition variables therefore appears to be
zst = γ (xst − µsxt − c), γ = 0, (7)
where µsx is the sth component of the vector µx. Thus, we assume that detrended regressors
are used as transition variables. Except this the transition mechanism is similar to that assumed
before.
3. TEST PROCEDURES
This section develops tests for the linearity hypothesis (3) against the general cointegrating STR
model (1). We consider the cases of both no drift and drift in regressors. This testing problem is
non-standard because the nuisance parameters γ and c are not identified under the null hypothesis.
This can be seen from equation (1) because, under the null hypothesis, the transition functions
g(zst) and, consequently, the parameters γ and c can take any values without any effect on the
model specification.
There has recently been a great interest in obtaining test procedures when a nuisance parameter
is not identified under the null hypothesis and significant progress has been made by Andrews
(1993), Andrews and Ploberger (1994) and Hansen (1996). However, instead of following the
approach of these authors, we proceed as in Luukkonen et al. (1988) and obtain very simple tests by
replacing the transition functions g(zst) with Taylor series approximations. This approach, which
leads to standard tests, has recently been used by several authors (see Granger and Teräsvirta
1993; Teräsvirta 1998; van Dijk et al. 2002, and the references therein). However, as pointed out
in the Introduction, there are important differences between our model and those employed by
previous authors.
As was done in Luukkonen et al. (1988), we consider two tests referred to as the first-order
and third-order test. Both of these tests are similar to Lagrange Multiplier (LM) tests in that
they require estimating the model only under the null hypothesis of linearity. The simplicity of
these tests stems from this fact because OLS can be used for parameter estimation under the null
hypothesis.
3.1. First-order test
Suppose that on prior grounds the regressors can be assumed to satisfy Assumption 2 and,
consequently, that the transition variables are given by (2). The first-order Taylor series approximation
of the function g(zst) around the origin is given by
g(zst) ≈ bγ (xst − c), (8)
where b = ∂g(z)
∂z |z=0 . Substituting this approximation for g(zst) in model (1) yields
yt = µ + νbγ (xst − c) + α ′ xt + β ′ xtbγ (xst − c) + ηt
p
θkxktxst + ηt, (9)
= φ + ρ ′ xt +
k=1
where the parameters φ (scalar), ρ = [ρ 1, ..., ρ p] ′ and θ k (scalar) are defined implicitly.
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346 In Choi and Pentti Saikkonen
The error term ηt in model (9) is the sum of ut in model (1) and the approximation error.
Under the null hypothesis (3), the approximation error vanishes and, consequently, ηt = ut. The
idea is to test the original null hypothesis (3) by testing the null hypothesis
H ′ 0 : θk = 0(k = 1,...,p) (10)
in the auxiliary regression model (9).
In order to motivate our test procedure, suppose that the error term ut in model (1) is Gaussian
white noise and that the regressor vector xt is strictly exogenous. Then, instead of the null
hypothesis (3), the linearity of the STR model (1) can be tested by testing the null hypothesis
H ′′
0 : γ = 0. Under this null hypothesis the nuisance parameters ν and β are not identified. However,
a test can be obtained in the same way as in Granger and Teräsvirta (1993, pp. 71–72). The first step
is to derive an LM test for the null hypothesis H ′′
0 by assuming that the values of the unidentified
nuisance parameters ν and β are fixed. This yields a test statistic that depends on the values
given for the unidentified nuisance parameters. A standard approach in a case like this is to take
the supremum of the obtained test statistic over the values of the nuisance parameters. It is easy
to see that the test statistics obtained in this way is exactly the same as that obtained by using
the standard LM test for testing the null hypothesis (10) in the auxiliary regression model (9)
(cf. Granger and Teräsvirta (1993, pp. 71–72)). Thus, in this special case the auxiliary regression
model (9) and the related null hypothesis (10) can be used to obtain a ‘supLM’ test for the original
testing problem.
Motivated by the above discussion, we consider the LM test for the null hypothesis (10) in
the auxiliary regression model (9) even in the general case where the simplified assumptions used
above are not satisfied. However, an endogeneity correction is needed because the regressors are
allowed to be correlated with error term. Our endogeneity correction is based on the leads-andlags
approach recently studied by Saikkonen and Choi (2004) in the context of a cointegrating
STR regression model more general than (1). Under Assumptions 3 and 4, we can express the
error term ut in (1) as
∞
ut = π ′ jvt− j + et, (11)
j=−∞
where et is a zero-mean stationary process such that Eetv ′ t− j
∞
j=−∞
π j < ∞
= 0 for all j = 0, ±1, ..., and
(see Saikkonen 1991; Saikkonen and Choi 2004). Expressions for the spectral density function
and long-run variance of the process et can be obtained from the well-known formulas fee(λ) =
fuu(λ) − f uv(λ) f −1
vv (λ) f vu(λ) and ω2 e = ω2 −1
u − ωuv vv ωvu, respectively.
Now, recall that the error term ηt in (9) is the sum of ut and the approximation error due to
the use of the Taylor series approximation (8). Using equations (4) and (11), we can thus write
the auxiliary regression model (9) as
yt = φ + ρ ′ xt +
p
θkxktxst +
k=1
= φ + ρ ′ xt + ζ ′ nt +
K
j=−K
K
j=−K
π ′ j xt− j + ηKt
π ′ j xt− j + ηKt, t = K + 1,...,T − K , (12)
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Testing linearity in cointegrating smooth transition regressions 347
where signifies the difference operator, ζ = [θ 1, ..., θ p] ′ , nt = [x 1t xst, ..., xptxst] ′ , and the
error term ηKt is the sum of the above-mentioned Taylor series approximation error and
et +
| j|>K
π ′ j vt− j
def
= eKt.
The augmented auxiliary regression model (12) will be used to test the null hypothesis (10) or,
equivalently, ζ = 0. In order to eliminate errors caused by truncating the infinite sum in (11), we
have to consider asymptotics in which the integer K tends to infinity with T at a suitable rate. To
this end, we follow Saikkonen (1991) and Saikkonen and Choi (2004) and assume the following:
Assumption 6. K = o(T 3 1/2 ) and T | j|>K π j →0 as T →∞.
Let M be the moment matrix for the auxiliary regression model (12) and (M −1 )nn the block
of the matrix M−1 corresponding to nt. Then the LM test for the null hypothesis (10) is defined
as
T1 = ˆζ ′ ˜ω 2 e (M−1 −1 )nn ˆζ,
where ˆζ is the OLS estimator of ζ in (12) and ˜ω 2 e is a standard long-run variance estimator based
on the residuals of the corresponding restricted OLS estimation. In other words, ˜ω 2 e is obtained
from ũt = yt − ˜φ − ˜ρ ′ xt − K j=−K ˜π ′ jxt− j where ˜φ, ˜ρ and ˜π j are the OLS estimators of the
coefficients φ, ρ and π j, respectively, in the auxiliary regression model (12) constrained by
ζ = 0.
Since the null hypothesis implies that ηKt = eKt in (12) the limiting null distribution of test
statistic T1 can be inferred from the results of Saikkonen and Choi (2004), which apply when
Assumptions 2, 3, 4 and 6 are satisfied. To demonstrate this, note that Saikkonen and Choi (2004)
study a non-linear cointegrating regression model that contains model (12) (with ηKt = eKt) asa
special case. The asymptotic results obtained in that paper were based on the so-called triangular
array asymptotics. However, when the non-linearity is defined by a homogeneous function, as is
the case in (12), the same results are obtained when conventional asymptotics is applied instead of
the triangular array asymptotics. Therefore, the results of Saikkonen and Choi (2004) apply in the
present context and imply that T ˆζ has a mixture normal limiting distribution with (conditional)
variance–covariance matrix given by the weak limit of ˜ω 2 e ((M/T 2 ) −1 )nn. Standard methods used
to construct the consistent long-run variance estimator ˜ω 2 e (e.g. Andrews 1991) also work. These
facts imply that, under the null hypothesis, ˜ω 2 e (M−1 −1/2 )nn ˆζ has a standard normal limiting
distribution and, consequently,
T1
d
−→ χ 2 (p) as T →∞.
Thus, we have shown that a standard chi-square criterion can be used to construct an asymptotic
test for the linearity hypothesis. For this result and other similar results reported below, note that
the assumption of a stationary error term ut is crucial. Methods for testing this assumption are
not yet available.
The above test is based on the assumption that there are no drifts in the regressors. Now suppose
that this assumption is not appropriate so that Assumption 5 and equation (7) are assumed instead
of Assumption 2 and equation (2), respectively. The transition variables defined by equation
(7) contain the unknown drift parameter µx, which has to be replaced by an estimator to make
the situation similar to that above. One possibility is to estimate µx by the sample mean of
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348 In Choi and Pentti Saikkonen
xt(t = 2, ..., T ) (see equation (5)). Another possibility is to use the OLS estimator in a linear
regression of xt on constant and t (see equation (6)). We adopt the former approach and denote
this estimator by ˆµx and its ith component by ˆµix.
The arguments used to obtain the auxiliary regression model (12) can be repeated with xst
replaced by xst − t ˆµsx. This yields the auxiliary regression model
yt = φ + ρ ′ xt + ζ ′ ˆnt +
K
j=−K
π ′ j xt− j + ηKt, t = K + 1,...,T − K , (13)
where ˆnt = [(x1t − t ˆµ1x)(xst − t ˆµsx),...,(x pt − t ˆµpx)(xst − t ˆµsx)] ′ and the notation for the
error term is the same as in (12) because the two error terms are identical under the assumed
null hypothesis. In (13), xt contains both deterministic and stochastic trends that need to be
separated for our test. This can be done by using the model augmented by a time trend
yt = φ + τt + ρ ′ xt + ζ ′ K
ˆnt + π ′ jxt− j + ηKt, t = K + 1,...,T − K . (14)
j=−K
If model (13) is used, the moment matrix is singular in the limit that is expected to bring lower
power of the tests introduced below. Indeed, unreported simulation results confirm serious loss
of power in finite samples when (13) is used.
Instead of T1 we now have the test statistic
−1 ˆζµ,
T1µ = ˆζ ′ 2
µ ˜ω eµ (M −1 ) ˆn ˆn
where ˆζµ is the OLS estimator of ζ in (14), (M −1 ) ˆn ˆn is defined in the same way as (M −1 )nn except
that ˆnt is used in place of nt, and ˜ω 2 eµ is a long run variance estimator obtained from the OLS
residuals of (14). When Assumptions 3–6 are satisfied the limiting null distribution of test statistic
T1µ is the same as that of T1, that is,
T1µ
d
−→ χ 2 (p) as T →∞. (15)
This result can be justified as follows. First, note that, by equation (6), model (14) can be
transformed to
yt = φ † + τ † t + ρ ′
t
v j
+ ζ ′ ˆnt +
K
j=−K
j=1
π ′ j xt− j + ηKt, t = K + 1,...,T − K , (16)
where φ † = φ + ρ ′ x 0 and τ † = ρ ′ µx. By the property of linear projection, this transformation
leaves the OLS estimator of the parameter ζ in (14) and our linearity test invariant. Thus, the
limiting distribution of test statistic T1µ can be derived by using this (infeasible) model instead
of (14). Let Bv(r)(0 ≤ r ≤ 1) be a p-dimensional Brownian motion. By standard arguments we
have T 1/2 ( ˆµx − µx) d → Bv (1) and
T −1/2 x[Tr] − ˆµx[Tr] = T −1/2
[Tr]
v j − T 1/2 ( ˆµx − µx) [Tr]
T + T −1/2 x0
d
→ ¯Bv (r) ,
j=1
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Testing linearity in cointegrating smooth transition regressions 349
where ¯Bv (r) = Bv (r) − rBv (1) is a Brownian bridge. This result implies that the weak limit
of T −1/2 ˆn[Tr] differs from that obtained for T −1/2n[Tr] only in that it is defined in terms of
the Brownian bridge ¯Bv (r) instead of the Brownian motion Bv(r). Second, the presence of the
linear time trend regressor in model (16) does not change the fact that the regressors therein
are asymptotically independent of the regression errors. Therefore, ˜ω 2 e (M−1 −1/2 ) ˆζµ ˆn ˆn has a
standard normal limiting distribution which gives result (15).
3.2. Third-order test
In equations (9) and (12), the parameters θ k are not functions of the parameter ν, which only
affects the parameters φ and ρ. Because the parameter ν cannot be separated from φ and ρ, test
statistic T1 may have low power when the value of ν is large and the elements of β take small
absolute values. To remedy this problem, we follow Luukkonen et al. (1988) and consider a test
based on the third-order Taylor series approximation of g(zst).
Assuming that the transition variables are given by (2), the third-order Taylor series
approximation of the function g(zst) around the origin can be written as
g(zst) ≈ bγ (xst − c) + dγ 2 (xst − c) 2 + hγ 3 (xst − c) 3 , (17)
where b, d and h are constants determined by partial derivatives of g(zst) at the origin. 3 Because
the motivation for using the third-order approximation is to improve the power of test statistic T1
under ν = 0, we use the approximate relation (17) only for the transition of the intercept term and
continue to use the relation (8) for the transition involving the regressors. The resulting auxiliary
regression model that corresponds to model (12) is then
yt = µ + ν bγ (xst − c) + dγ 2 (xst − c) 2 + hγ 3 (xst − c) 3
+ α ′ xt + β ′ xtbγ (xst − c) +
= ψ + ξ ′ xt +
K
j=−K
p
ϕkxktxst + λx 3 st +
k=1
= ψ + ξ ′ xt + ς ′ ht +
K
j=−K
π ′ j xt− j + η ∗ Kt
K
j=−K
π ′ j xt− j + η ∗ Kt
π ′ jxt− j + η ∗ Kt , t = K + 1,...,T − K . (18)
Here ς = [ϕ 1, ..., ϕ p, λ] ′ and ht = [x 1t xst, ..., xptxst, x 3 st ]′ . Moreover, the p × 1 parameter vector
ξ and the scalar parameters ϕ k, λ and ψ are defined implicitly.
Instead of the original null hypothesis (3), the idea is now to test the null hypothesis
H ′′′
0
: ς = 0 (19)
in the auxiliary regression model (18) using the LM test. In order to introduce the test statistic,
let ¯M be the moment matrix for the auxiliary regression model (18) and ( ¯M −1 )hh the block of the
matrix ¯M −1 corresponding to ht. Furthermore, let ˆς be the OLS estimator of ς in (18) and ˜ω 2 e a
3 The second-order derivative of the logistic transition function in Example 1 takes the value zero at the origin but this
may not be the case for all other transition functions (cf. Example 3 in Section 4).
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350 In Choi and Pentti Saikkonen
standard long-run variance estimator based on the residuals of the corresponding OLS estimation
constrained by ς = 0. Then our test statistic is defined as
T2 = ˆς ′ ˜ω 2 e ( ¯M −1 −1 )hh ˆς.
In the same way as in the case of test statistic T1, we can use the results of Saikkonen and Choi
(2004) to show that, when Assumptions 2, 3, 4 and 6 hold, the limiting null distribution of test
statistic T2 is given by
T2
d
−→ χ 2 (p + 1) as T →∞.
Thus, compared to the limiting distribution of test statistic T1, the number of degrees of freedom
has increased by the number of additional restrictions included in the auxiliary null hypothesis.
Arguments similar to those in the previous section can be used to modify the above test to
allow for drifts in the regressors. Thus, suppose that Assumption 2 and equation (2) are replaced
by Assumption 5 and equation (7), respectively. Instead of the auxiliary regression model (18)
we then base our test on
yt = ψ + τt + ξ ′ xt + ς ′ ˆht +
K
j=−K
π ′ jxt− j + η ∗ Kt , t = K + 1,...,T − K , (20)
where ˆht is defined by replacing xst in the definition of ht with xst − t ˆµsx. Testing null hypothesis
(19) in this auxiliary regression model leads to test statistic
T2µ = ˆς ′ 2
µ ˜ω eµ ( ¯M −1 −1 ) ˆh ˆh ˆςµ,
where ˆςµ is the OLS estimator of ς in (20), ( ¯M −1 ) ˆh ˆh is an analog of ¯M −1 )hh based on ˆht instead
of ht, and ˜ω 2 eµ is a long run variance estimator obtained from the OLS residuals of (20) with the
constraint ζ = 0. In the same way as in the previous section we can show that, when Assumptions
3–6 are satisfied, the limiting null distribution of test statistic T2µ is the same as that of test statistic
T2.
If it is a priori known that the coefficients for regressors are not subject to smooth transition
(i.e. β = 0), our test can be based on the regression equation
yt = µ + ν bγ (xst − c) + dγ 2 (xst − c) 2 + hγ 3 (xst − c) 3
+ α ′ xt +
K
j=−K
π ′ j xt− j + η ∗ Kt
= ψ + ξ ′ xt + ϕx 2 st + λx 3 st +
= ψ + ξ ′ xt + ς ′ ht +
K
j=−K
K
j=−K
π ′ j xt− j + η ∗ Kt
π ′ jxt− j + η ∗ Kt , t = K + 1,...,T − K , (21)
where ς = [ϕ, λ] ′ and ht = [x 2 st , x 3 st ]′ . The test for the null hypothesis ς = 0, denoted by T3, is
similarly defined and
T3
d
−→ χ 2 (2) as T →∞.
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Testing linearity in cointegrating smooth transition regressions 351
This test may have higher power than T2 when β = 0. In some cases (e.g. Example 1), d = 0 and
the auxiliary regression model reduces to
yt = ψ + ξ ′ xt + λx 3 st +
K
j=−K
In this case, T3 tests the null hypothesis λ = 0 and
T3
π ′ jxt− j + η ∗ Kt , t = K + 1,...,T − K .
d
−→ χ 2 (1) as T →∞.
When regressors contain drifts, we add a linear time trend in model (21) and use the detrended
regressor xst − t ˆµsx instead of xst. The resulting test will be denoted by T3µ.
4. EXTENSIONS TO A MORE GENERAL MODEL
The test procedures developed in the previous section can straightforwardly be extended to the
cointegrating STR model
yt = µ +
q
ν j g j(z jt) + α ′ xt +
j=1
q
j=1
β ′ j xt g j
where the components of the vector z jt = [z j1t,...,z jmj t] ′ are given by
z jt + ut, t = 1, 2,...,T , (22)
z jit = γ ji(xit − c ji),γji = 0, i = 1,...,m j, (23)
and gj(zjt) is a smooth, real-valued transition function of the process xt and the scalar parameters
γ j1, ..., γ jmj , c j1, ..., cjm j . In equations (22) and (23), parameter q denotes the number of
transition functions and mj the number of transition variables for the transition function gj(·).
Otherwise the notation is the same as in model (1).
In model (22) different forms of smooth transitions may occur depending on more than a
single component of the vector xt. The counterpart of the linearity hypothesis (3) is now
H0 : ν j = 0 and β j = 0 for all j = 1,...,q.
The alternative again states that the null hypothesis is not true.
Model (1) with the logistic transition function defined in Example 1 is an obvious special case
of model (22). More general cases are provided by the following examples.
Example 2. (Two transition functions, one transition variable for each transition function and
three regimes; p = 1, q = 2, m 1 = m 2 = 1)
g1 (z1t) =
g2 (z2t) =
1
1 + e −γ11(x1t −c11)
1
1 + e −γ21(x1t −c21)
− 1
2 , γ11 > 0
− 1
2 , γ21 > 0.
If we here assume that c21 > c11, then the coefficient for x1t changes slowly from α1 − 1
2β11 − 1
2β21 via α1 + 1
2β11 − 1
2β21 to α1 + 1
2β11 + 1
2β21 for increasing values of x1t. Thus, there are basically
three data regimes in this example. A similar interpretation applies when c11 > c21.
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352 In Choi and Pentti Saikkonen
Example 3. (One transition function, two transition variables and two regimes; p = 2, q = 1,
m 1 = 2)
g1 (z1t) =
1
1 + e −γ11(x1t −c11) ×
1
1 + e −γ12(x2t −c12)
− 1
4 , γ11 > 0, γ12 > 0.
This example differs from the previous ones in that there are two transition variables x1t and
x2t. Only when the values of both x1t and x2t are greater than c11 and c12, respectively, does the
coefficient vector of xt become α + 3
4β1. Otherwise, it is α − 1
4β1. So there are two data regimes
characterized by x1t and x2t.
Example 4. (One transition function, two transition variables and three regimes; p = 2, q = 1,
m 1 = 2)
1
g1 (z1t) =
1 + e−γ11(x1t −c11) −
1
1 + e−γ12(x2t −c12) , γ11 > 0,γ12 > 0.
There are three data regimes in this example. When the values of both x1t and x2t are either large
or small simultaneously, the transition function takes the value 0. Otherwise, it takes the value
either 1 or −1.
In place of Assumption 1 we now have to use a similar assumption that allows for the fact
that the argument of the function g is a vector. Specifically, we assume:
Assumption 7.
(i) gj(0) = 0 for all j.
(ii) For all j, the function gj(zj) is three times differentiable in an open ball in Rm j with centre
0 and radius rj(rj > 0).
(iii) For all j, ∂g j (z j )
= 0 for at least one i.
∂z ji |z=0
∂
(iv) For all j,
3g j (z j )
= 0 for at least one (i, k, l).
∂z ji∂z jk∂z jl |z=0
It is straightforward to check that the transition functions of Examples 2–4 satisfy this
assumption.
In the present context, the first-order Taylor series approximation (8) is replaced by g j(z j) ≈
m j
i=1 b jiγ ji(xit − c ji) where b ji = ∂g j (z)
. Following the development in Section 3.1 it can be
∂z ji |z=0
seen here that the counterpart of the auxiliary regression model (12) is
yt = φ + ρ ′ xt +
p
k=i
m
θkixktxit +
i=1
= φ + ρ ′ xt + ζ ′ n t +
K
j=−K
K
j=−K
π ′ j xt− j + ηKt
π ′ j xt− j + ηKt, t = K + 1,...,T − K , (24)
where ζ = [θ11,...,θpm] ′ , nt = [x 2 1t ,...,x ptxmt] ′ and m = max{m 1, ..., mq}. The null hypothesis
tested in this auxiliary regression model is H ′ 0 : ζ = 0. Analogously to test statistic T1 one
obtains
T 1 = ˆζ ′ 2
˜ω e (M −1 −1 )nn ˆζ ,
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Testing linearity in cointegrating smooth transition regressions 353
where the notation differs from that used in test statistic T1 only in that the auxiliary regression
model (24) is used instead of (12). Arguments similar to those used for test statistic T1 show that,
under the null hypothesis,
d
T 1 −→ χ 2
(2p − m + 1)m
as T →∞.
2
This result applies when Assumptions 2, 3, 4, 6 and 7 hold.
If Assumption 5 is used instead of Assumption 2 the regressor nt in the auxiliary regression
model (24) is replaced by ˆn t = [(x1t − t ˆµ1x) 2 ,...,(x pt − t ˆµpx)(xmt − t ˆµmx)] ′ and a time trend is
added to the regressor set. The resulting test statistic T 1µ , with the same limiting null distribution
as T 1 , is obtained in an obvious way.
The third-order test procedure of Section 3.2 can readily be extended to the present context.
The relevant Taylor series approximation is
m
j
m
j m
j
g j(z j) ≈ b jiγ ji(xit − c ji) + d jkiγ jiγ jk(xit − c ji)(xkt − c jk)
i=1
k=i
i=1
m
j m
j m
j
+
h jlkiγ jiγ jkγ jl(xit − c ji)(xkt − c jk)(xlt − c jl).
l=k
k=i
i=1
Because the motivation is to improve the power of test statistic T 1 under ν j = 0 we again use
the third-order approximation only for the transition of the intercept term and continue to use
the first-order approximation for the transition involving the regressors. Inserting the preceding
approximation into model (22) and combining terms in the resulting auxiliary regression model
yields
yt = ψ + ξ ′ xt +
p
k=i
m
ϕikxktxit +
i=1
= ψ + ξ ′ xt + ς ′ h t +
K
j=−K
m
l=k
m
k=i
m
λlkixitxktxlt +
i=1
K
j=−K
π ′ j xt− j + η ∗ Kt
π ′ jxt− j + η ∗ Kt , t = K + 1,...,T − K , (25)
where ς = [ϕ11,...,ϕmp,λ111, ...,λmmm] ′ , ht = [x 2 1t ,...,x ptxmt, x 3 1t ,...,x 3 mt ]′ hypothesis to be tested is H
and the null
′′′
0 : ς = 0. The test statistic becomes
T 2 = ˆς ′ ˜ω 2 e (M−1 −1 )hh ˆς,
where the notation is as before except for being defined in terms of the auxiliary regression model
(25). Under the conditions used for test statistic T 1 ,
d
T 2 −→ χ 2
(2p − m + 1)m
+
2
m(m + 1)(m + 2)
6
as T →∞.
If Assumption 5 is used instead of Assumption 2 the auxiliary regression model (25) is
augmented by adding a time trend to the regressor set and the regressor ht is replaced by ˆh t
defined by replacing xit in the definition of ht by xit − t ˆµix. The resulting test statistic T 2µ has
the same limiting null distribution as T 2 .
Extension of the simplified third-order test T3 of Section 3.2, denoted by T 3 , should be based
on the same model as for T 2 unless the second-order derivatives djki are zero for all i, j and k,
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354 In Choi and Pentti Saikkonen
and has the same limiting distribution as T 2 . In this case, T 3 is no different from T 2 .Ifthe
second-order derivatives are zero for all i, j and k, the second-order terms in model (25) vanish
(i.e. ς = [λ111,...,λmmm] ′ ) and the limiting distribution of T 3 will be χ 2 ( m(m+1)(m+2)
). Extension
6
of T 3 to regressors with drifts can be made as described previously.
We close this section with a remark that is relevant to all the tests we have developed. Because
the auxiliary regression models used to obtain our tests can be considered as approximations of
not only the entertained STR models but also of other non-linear models, a rejection of the null
hypothesis by any of these tests may be due to various non-linearities. Therefore, a rejection of the
linearity hypothesis need not imply that the entertained STR model is the correct or appropriate
alternative. To put this in another way, our tests can also be seen as general tests against non-linear
cointegration. Of course the previous similar tests obtained for stationary models are not different
in this respect.
5. SIMULATION
This section reports simulation results for the empirical size and power of the linearity tests studied
in previous sections. When the tests for regressors with drifts were used, µx was estimated by the
sample mean of xt. We considered three data-generating processes. The first one is the logistic
STR model with one transition function, one transition variable and two regimes. That is, data
were generated by
yt = µ + νg(zt) + αxt + βxt g (zt) + ut;
1
g (zt) =
1 + e−γ (xt
1
−
−c) 2 ;
xt
ω ω
= εt + Bεt−1; B = ;
ut
0 ω
1 0.5
εt ∼ iidN 0,
.
0.5 1
(26)
In this data generation, the coefficient for xt moves slowly between α − 1
2β1 and α + 1
2β1 and
the intercept term shows similar movements. In addition, the regressor xt and the error term ut
are serially and contemporaneously correlated. The degree of correlation is controlled by the
parameter ω. Larger values of ω imply that the regressors and errors are more correlated both
serially and contemporaneously. Also, {xt} were generated such that c is located between the 15th
and 85th percentiles of {xt}. The purpose of this scheme is to make the data-generating process
non-linear under the alternative of non-linearity without any exception. The parameter values for
the experiment using (26) are (ν, β) = (0, 0), (0, 0.5), (0.5, 0), (0.5, 0.5), µ = α = c = 0 and
γ = 1. 4 The parameter values (ν, β) = (0, 0) correspond to the null of linearity, and the rest to the
alternative of non-linearity. We tried two sample sizes T = 100 and T = 200; and the numbers
of leads and lags (K) used were 1, 2 and 3. Nominal size was set at 5% and 5,000 iterations
were performed to calculate the empirical size and power of test statistics T1, T2, T3, T1µ, T2µ and
T3µ. The results are reported in Tables 1 and 3. The added product terms for T1, T2 and T3 are
x2 t ,(x2t , x3t ) and x3t , respectively. Corresponding detrended terms were used for T1µ, T2µ and T3µ.
4In a previous version of this paper, we also considered the cases γ = 5, ∞. Since these provide similar results, we do
not report them here.
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Testing linearity in cointegrating smooth transition regressions 355
Table 1. Empirical size and power of linearity tests.
One transition function, one transition variable, two regimes and no drift.
T1 T2 T3
ω T (ν, β) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0) 0.064 0.071 0.076 0.072 0.076 0.083 0.067 0.076 0.080
(0.0, 0.5) 0.853 0.844 0.835 0.867 0.855 0.852 0.355 0.366 0.370
(0.5, 0.0) 0.120 0.125 0.128 0.118 0.122 0.131 0.116 0.124 0.124
(0.5, 0.5) 0.836 0.830 0.824 0.859 0.854 0.842 0.358 0.367 0.373
200 (0.0, 0.0) 0.057 0.061 0.062 0.058 0.060 0.064 0.061 0.061 0.060
(0.0, 0.5) 0.968 0.967 0.967 0.988 0.985 0.984 0.558 0.563 0.559
(0.5, 0.0) 0.176 0.176 0.176 0.167 0.176 0.174 0.213 0.218 0.217
(0.5, 0.5) 0.957 0.955 0.955 0.977 0.976 0.976 0.529 0.535 0.537
0.5 100 (0.0, 0.0) 0.070 0.078 0.082 0.074 0.080 0.085 0.071 0.073 0.080
(0.0, 0.5) 0.889 0.888 0.886 0.916 0.912 0.910 0.404 0.413 0.417
(0.5, 0.0) 0.116 0.119 0.122 0.107 0.122 0.124 0.119 0.145 0.134
(0.5, 0.5) 0.880 0.876 0.874 0.904 0.906 0.898 0.393 0.402 0.409
200 (0.0, 0.0) 0.060 0.062 0.066 0.062 0.062 0.064 0.064 0.067 0.067
(0.0, 0.5) 0.984 0.983 0.983 0.995 0.995 0.995 0.589 0.587 0.593
(0.5, 0.0) 0.162 0.163 0.166 0.154 0.166 0.159 0.158 0.185 0.168
(0.5, 0.5) 0.972 0.972 0.973 0.986 0.987 0.987 0.572 0.573 0.581
Notes:
(i) Data were generated by (26).
(ii) The parameter values (ν, β) = (0, 0) correspond to the null of linearity, and the rest to the alternative of non-linearity.
(iii) Larger ω implies that the regressors and errors are more correlated both serially and contemporaneously.
(iv) The number of iterations is 5000, and the nominal size 5%.
(v) The long-run variance was estimated using Andrews’ (1991) methods with an AR(4) approximation for the prefilter.
The second data generation we considered is the three-regime, logistic smooth transition
model with one transition variable and two transition functions.
yt = µ +
2
ν j g j(z jt) + αxt +
j=1
2
β j xt g j
j=1
z jt + ut;
1
g1 (z1t) =
1 + e−γ11(xt 1
−
−c11) 2 ; g2
1
(z2t) =
1 + e−γ21(xt −c21)
xt
ω ω
= εt + Bεt−1; B = ;
ut
0 ω
1 0.5
εt ∼ iidN 0,
0.5 1
− 1
2 ;
. (27)
When c21 > c11, the coefficient for xt changes slowly from α − 1
2 β1 − 1
2 β2 via α + 1
2 β1 − 1
2 β2 to
α + 1
2 β1 + 1
2 β2 for increasing values of xt. The coefficient for the intercept term changes similarly.
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356 In Choi and Pentti Saikkonen
Table 2. Empirical size and power of linearity tests.
Two transition functions, one transition variable for each transition function, three regimes and no drift.
T 1 T 2 T 3
ω T (ν 1, ν 2, β 1, β 2) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0, 0.0, 0.0) 0.062 0.066 0.071 0.067 0.071 0.081 0.068 0.070 0.074
(0.0, 0.0, 0.5, 1.0) 0.982 0.984 0.984 0.971 0.967 0.972 0.890 0.897 0.904
(0.5, 1.0, 0.0, 0.0) 0.144 0.151 0.156 0.144 0.157 0.159 0.146 0.151 0.154
(0.5, 1.0, 0.5, 1.0) 0.975 0.977 0.978 0.963 0.964 0.968 0.861 0.867 0.871
200 (0.0, 0.0, 0.0, 0.0) 0.063 0.066 0.067 0.062 0.060 0.062 0.058 0.058 0.061
(0.0, 0.0, 0.5, 1.0) 0.985 0.985 0.986 0.982 0.982 0.982 0.823 0.824 0.831
(0.5, 1.0, 0.0, 0.0) 0.306 0.303 0.305 0.380 0.390 0.379 0.231 0.233 0.229
(0.5, 1.0, 0.5, 1.0) 0.978 0.977 0.978 0.983 0.984 0.984 0.794 0.797 0.804
0.5 100 (0.0, 0.0, 0.0, 0.0) 0.074 0.078 0.087 0.075 0.083 0.092 0.069 0.075 0.081
(0.0, 0.0, 0.5, 1.0) 0.983 0.982 0.982 0.982 0.978 0.979 0.827 0.825 0.836
(0.5, 1.0, 0.0, 0.0) 0.178 0.178 0.178 0.170 0.201 0.191 0.138 0.142 0.140
(0.5, 1.0, 0.5, 1.0) 0.967 0.966 0.966 0.984 0.979 0.980 0.792 0.788 0.800
200 (0.0, 0.0, 0.0, 0.0) 0.059 0.062 0.065 0.059 0.058 0.058 0.060 0.062 0.065
(0.0, 0.0, 0.5, 1.0) 0.993 0.993 0.992 0.996 0.995 0.995 0.746 0.740 0.748
(0.5, 1.0, 0.0, 0.0) 0.327 0.325 0.322 0.395 0.448 0.417 0.233 0.249 0.237
(0.5, 1.0, 0.5, 1.0) 0.983 0.982 0.983 0.995 0.996 0.995 0.696 0.697 0.705
Notes:
(i) Data were generated by (27).
(ii) The parameter values (ν 1, ν 2, β 1, β 2) = (0, 0, 0, 0) correspond to the null of linearity, and the rest to the alternative
of non-linearity.
(iii) Larger ω implies that the regressors and errors are more correlated both serially and contemporaneously.
(iv) The number of iterations is 5000, and the nominal size 5%.
(v) The long-run variance was estimated using Andrews’ (1991) methods with an AR(4) approximation for the prefilter.
As for the data-generating process (26), the regressor and error are serially and contemporaneously
correlated in (27). Also, {xt} were generated such that c11 is greater than the 15th percentile of
{xt} and c21 less than the 85th percentile of {xt}. Thus, the data-generating process (27) is always
non-linear under the alternative of non-linearity. We set µ = α = c11 = 0, c21 = 10, and considered
(ν 1, ν 2, β 1, β 2) = (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.5, 1.0), (0.5, 1.0, 0.0, 0.0), (0.5, 1.0, 0.5, 1.0)
and (γ 11, γ 21) = (1, 1). 5 The null hypothesis of linearity corresponds to (ν 1, ν 2, β 1, β 2) =
(0.0, 0.0, 0.0, 0.0). The employed sample sizes, numbers of leads and lags, nominal size and
number of iterations are the same as for Table 1. The results for the data-generating process (27)
are reported in Tables 2 and 4. The added product terms for T 1 , T 2 and T 3 are x2 t ,(x2t , x3t ) and
x3 t , respectively, and those for T 1µ , T 2µ and T 3µ are corresponding detrended terms.
5 In a previous version of this paper, we also considered the cases (γ 11, γ 21) = (5, 5), (∞, ∞). Since these provide
similar results, we do not report them here.
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Testing linearity in cointegrating smooth transition regressions 357
Table 3. Empirical size and power of linearity tests.
One transition function, two transition variables, two regimes and no drift.
T 1
ω T (ν, β 1, β 2) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0, 0.0) 0.109 0.130 0.157 0.141 0.174 0.227
(0.0, 0.5, 1.0) 0.778 0.802 0.829 0.372 0.394 0.427
(0.5, 0.0, 0.0) 0.134 0.154 0.180 0.158 0.191 0.234
(0.5, 0.5, 1.0) 0.764 0.796 0.821 0.374 0.392 0.427
200 (0.0, 0.0, 0.0) 0.070 0.076 0.081 0.074 0.083 0.090
(0.0, 0.5, 1.0) 0.833 0.835 0.849 0.479 0.481 0.499
(0.5, 0.0, 0.0) 0.136 0.147 0.153 0.119 0.135 0.144
(0.5, 0.5, 1.0) 0.826 0.835 0.849 0.480 0.485 0.494
0.5 100 (0.0, 0.0, 0.0) 0.101 0.126 0.163 0.125 0.153 0.199
(0.0, 0.5, 1.0) 0.824 0.834 0.857 0.443 0.449 0.486
(0.5, 0.0, 0.0) 0.131 0.148 0.190 0.128 0.164 0.209
(0.5, 0.5, 1.0) 0.818 0.830 0.850 0.441 0.470 0.489
200 (0.0, 0.0, 0.0) 0.073 0.074 0.086 0.073 0.081 0.092
(0.0, 0.5, 1.0) 0.894 0.892 0.897 0.575 0.552 0.568
(0.5, 0.0, 0.0) 0.131 0.135 0.147 0.111 0.132 0.144
(0.5, 0.5, 1.0) 0.879 0.876 0.882 0.547 0.536 0.551
Notes:
(i) Data were generated by (28).
(ii) T 2 and T 3 are the same for this data-generating process.
(iii) The parameter values (ν, β 1, β 2) = (0, 0, 0) correspond to the null of linearity, and the rest to the alternative of
non-linearity.
(iv) Larger ω implies that the regressors and errors are more correlated both serially and contemporaneously.
(v) The number of iterations is 5000, and the nominal size 5%.
(vi) The long-run variance was estimated using Andrews’ (1991) methods with an AR(4) approximation for the prefilter.
The third data-generating process we considered is
C○ Royal Economic Society 2004
yt = µ + νg1(z1t) +
2
α j x jt +
j=1
2
β j x jtg1 (z1t) + ut;
1
g1 (z1t) =
1 + e−γ11(x1t −c11) ×
1
1 + e−γ12(x2t 1
−
−c12) 4 ;
⎡ ⎤
ω ω ω
xt
⎢ ⎥
= εt + Bεt−1; B = ⎣ 0 ω ω⎦
;
ut
0 0 ω
⎛ ⎡
1
⎜ ⎢
εt ∼ iidN ⎝0, ⎣ 0.5
0.5
1
⎤⎞
0.5
⎥⎟
0.5 ⎦⎠
. (28)
0.5 0.5 1
j=1
T 2

358 In Choi and Pentti Saikkonen
In this data generation, there are two regressors and two transition variables. Only when the values
of both x1t and x2t are greater than c11 and c12, respectively, the coefficient vector of xjt becomes
α j + 3
4β j. Otherwise, it is α j − 1
4β j. So there are two data regimes characterized by x1t and x2t.
Also, {x 1t} and {x 2t} were generated such that c11 is greater than the maximum of the 15th
percentiles of {x 1t} and {x 2t} and c12 less than the minimum of the 85th percentiles of {x 1t}
and {x 2t}. Wesetµ = α1 = α2 = c11 = 0, c12 = 10, and considered (ν, β 1, β 2) = (0.0, 0.0,
0.0), (0.0, 0.5, 1.0), (0.5, 0.0, 0.0), (0.5, 0.5, 1.0) and (γ 11, γ 12) = (1, 1). The null hypothesis
of linearity corresponds to (ν, β 1, β 2) = (0.0, 0.0, 0.0). The employed sample sizes, numbers of
leads and lags, nominal size and number of iterations are the same as for Table 1. The results for
the data-generating process (28) are reported in Tables 3 and 6. The added product terms for T 1
and T 2 are (x 2 1t , x 1t x 2t, x 2 2t ) and (x 2 1t , x 1t x 2t, x 2 2t , x 3 1t , x 2 1t x 2t, x 1t x 2 2t , x 3 2t ), respectively, and those
for T 1µ and T 2µ are corresponding detrended terms. For the data-generating process (28), T 3 and
T 3µ are the same as T 2 and T 3µ , respectively, because the second derivatives of the transition
function are not identically zero at the origin.
We may summarize the simulation results in Tables 1–3 as follows:
The tests keep the nominal size reasonably well except in Table 3. Size distortions in Table 3
seem to result from more product terms in the auxiliary regression model than in Tables 1
and 2. Comparing the test statistics, T2 (T 2 ) tends to reject more frequently than T1 (T 1 ) and
T3 (T 2 ) under the null hypothesis. This is because T2 (T 2 ) has more product terms in the
auxiliary regression model. Moreover, the rejection frequencies of the tests tend to increase
under the null hypothesis when the serial and contemporaneous correlations increase. As
the sample size grows, the empirical size of the tests improves.
The power of the tests improves as the sample size becomes larger.
The T3(T 3 ) test was designed in the hope that it has higher power when only the intercept
term is subject to smooth transitions. But even in this case, it does not show superior power.
When the regression coefficients are subject to non-linearity, it shows significantly lower
power than the rest.
T2(T 2 ) tends to reject more frequently than T1(T 1 ) under the alternative hypothesis.
However, considering the tendency of T2(T 2 ) to reject more often under the null hypothesis,
no one of the tests seems to dominate the other.
When only the intercept term is subject to non-linearity, T2(T 2 ) does not show any noticeably
higher power than T1(T 1 ) though it was designed to deal with the smooth transitions in the
intercept term.
The empirical size and power of the tests are not affected significantly by the choice of the
value of K.
The simulation results for T1µ(T 1µ ), T2µ(T 1µ ) and T3µ(T 3µ ) (tests for the case of drift)
reported in Tables 4–6 can essentially be summarized in the same manner as for the T1(T 1 ), T2(T 1 )
and T3(T 3 ) tests. The only notable differences are:
The tests tend to overreject relative to the tests without a linear time trend, but the size
properties improve as the sample size grows. Obviously, the overrejection of the tests stems
from the inclusion of a linear time trend term in the regression model and the detrending
of the transition variables.
The power of the tests are lower than that in Tables 1–3.
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Testing linearity in cointegrating smooth transition regressions 359
Table 4. Empirical size and power of linearity tests.
One transition function, one transition variable, two regimes and drift.
T1µ T2µ T3µ
ω T (ν, β) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0) 0.081 0.084 0.091 0.087 0.099 0.105 0.082 0.088 0.096
(0.0, 0.5) 0.662 0.666 0.671 0.617 0.625 0.627 0.658 0.665 0.667
(0.5, 0.0) 0.113 0.119 0.125 0.114 0.127 0.133 0.108 0.117 0.l21
(0.5, 0.5) 0.651 0.659 0.663 0.612 0.623 0.638 0.641 0.641 0.644
200 (0.0, 0.0) 0.068 0.073 0.073 0.062 0.067 0.067 0.063 0.062 0.065
(0.0, 0.5) 0.800 0.803 0.804 0.761 0.766 0.769 0.805 0.805 0.806
(0.5, 0.0) 0.115 0.114 0.118 0.114 0.116 0.118 0.115 0.114 0.118
(0.5, 0.5) 0.794 0.795 0.798 0.752 0.757 0.765 0.789 0.792 0.795
0.5 100 (0.0, 0.0) 0.089 0.096 0.107 0.098 0.102 0.116 0.082 0.087 0.095
(0.0, 0.5) 0.704 0.708 0.712 0.659 0.669 0.680 0.716 0.720 0.723
(0.5, 0.0) 0.103 0.109 0.117 0.107 0.117 0.129 0.106 0.112 0.121
(0.5, 0.5) 0.707 0.711 0.715 0.672 0.684 0.694 0.700 0.706 0.711
200 (0.0, 0.0) 0.061 0.065 0.067 0.067 0.069 0.075 0.070 0.074 0.076
(0.0, 0.5) 0.808 0.811 0.815 0.769 0.775 0.778 0.815 0.816 0.818
(0.5, 0.0) 0.121 0.122 0.126 0.114 0.120 0.120 0.116 0.116 0.121
(0.5, 0.5) 0.799 0.801 0.805 0.748 0.755 0.762 0.809 0.812 0.815
Note: Data were generated as for Table 1, and the same methods as for Table 1 were used except that a linear time trend
term is added to the regression model and that detrended regressors are used for the transition function.
In order to improve the size properties of the tests, one may use critical values from the
F-distribution 6 whose degrees of freedom are the number of product terms and the sample size
minus the number of regressors. For this approximation to work, the test statistics should be
multiplied by the number of product terms in the auxiliary regression models. According to some
simulation, this may or may not improve the size properties of the tests using the critical values
from the chi-square distribution. When there are many product terms, using the F-distribution
tends to slightly improve the size properties of the tests. Detailed results are not reported here to
save space.
6. APPLICATIONS TO THE U.K. MONEY DEMAND FUNCTION
Estimation of a money demand function has been an important research topic in macroeconomics
since the late 1950s. More recently, methods for nonstationary time series have been applied
to the money demand function and stable cointegrating relations of the relevant variables have
statistically been confirmed. A partial list of related studies includes Baba et al. (1992), Hendry
and Ericsson (1991a,b), Ericsson (1998), Ericsson et al. (1998), Hoffman and Rasche (1991),
6 One of the referees kindly suggested this idea.
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360 In Choi and Pentti Saikkonen
Table 5. Empirical size and power of linearity tests.
Two transition functions, one transition variable for each transition function, three regimes and drift.
T 1µ T 2µ T 3µ
ω T (ν 1, ν 2, β 1, β 2) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0, 0.0, 0.0) 0.090 0.095 0.094 0.096 0.105 0.113 0.092 0.103 0.109
(0.0, 0.0, 0.5, 1.0) 0.675 0.707 0.736 0.521 0.561 0.616 0.442 0.474 0.503
(0.5, 1.0, 0.0, 0.0) 0.129 0.134 0.139 0.133 0.146 0.155 0.111 0.118 0.122
(0.5, 1.0, 0.5, 1.0) 0.649 0.684 0.715 0.501 0.539 0.587 0.427 0.456 0.492
200 (0.0, 0.0, 0.0, 0.0) 0.576 0.060 0.063 0.066 0.064 0.068 0.063 0.065 0.066
(0.0, 0.0, 0.5, 1.0) 0.725 0.736 0.750 0.621 0.636 0.653 0.539 0.553 0.568
(0.5, 1.0, 0.0, 0.0) 0.198 0.203 0.199 0.215 0.220 0.222 0.179 0.185 0.183
(0.5, 1.0, 0.5, 1.0) 0.725 0.737 0.748 0.630 0.638 0.656 0.529 0.536 0.549
0.5 100 (0.0, 0.0, 0.0, 0.0) 0.089 0.093 0.097 0.091 0.103 0.112 0.089 0.094 0.098
(0.0, 0.0, 0.5, 1.0) 0.733 0.752 0.769 0.615 0.642 0.674 0.551 0.563 0.580
(0.5, 1.0, 0.0, 0.0) 0.139 0.144 0.150 0.136 0.165 0.168 0.133 0.143 0.151
(0.5, 1.0, 0.5, 1.0) 0.729 0.745 0.764 0.623 0.646 0.675 0.532 0.549 0.573
200 (0.0, 0.0, 0.0, 0.0) 0.061 0.066 0.068 0.071 0.067 0.074 0.059 0.061 0.065
(0.0, 0.0, 0.5, 1.0) 0.798 0.799 0.805 0.717 0.719 0.734 0.603 0.604 0.613
(0.5, 1.0, 0.0, 0.0) 0.207 0.208 0.213 0.234 0.258 0.251 0.191 0.200 0.201
(0.5, 1.0, 0.5, 1.0) 0.786 0.789 0.794 0.706 0.710 0.720 0.592 0.600 0.605
Note: Data were generated as for Table 2, and the same methods as for Table 2 were used except that a linear time trend
term is added to the regression model and that detrended regressors are used for the transition function.
Hoffman et al. (1995), Lütkepohl et al. (1999), Stock and Watson (1993) and Teräsvirta and
Eliasson (2001). Among these, Ericsson et al. (1998), Lütkepohl et al. (1999) and Teräsvirta and
Eliasson (2001) use non-linear models for their empirical results. Ericsson et al. (1998) employ a
non-linear specification for the error correction term in their error-correction equation for the U.K.
money demand, and the latter two articles use the STR model for the differences of the German
and U.K. M1, respectively. The error correction terms in Ericsson et al. (1998), Lütkepohl et
al. (1999) and Teräsvirta and Eliasson (2001) assume linear long-run relation between money
and other variables that is based on economic theories for money demand (e.g. Baba et al. 1992;
Baumol (1952) and Tobin (1956)). Thus, non-linearity in these studies is related only to short-run
dynamics.
Money is demanded for transactions motive. Holding money enables us to smooth out the
differences between income and expenditure stream. In a modern economy, money is also
demanded as an asset that avoids the volatility of stock and bond markets. The classical long-run
money demand function, building on these motives for holding money, can be written as
M d /P = f (Y , R,P), (29)
where M d denotes the nominal money demand, P the price level, Y the real income, R a vector of
rates of return on assets outside money and on money itself and P the inflation rate. The rate of
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Testing linearity in cointegrating smooth transition regressions 361
Table 6. Empirical size and power of linearity tests.
One transition function, two transition variables, two regimes and drift.
T 1µ
ω T (ν, β 1, β 2) K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
0.2 100 (0.0, 0.0, 0.0) 0.142 0.170 0.211 0.199 0.246 0.303
(0.0, 0.5, 1.0) 0.486 0.541 0.608 0.230 0.258 0.313
(0.5, 0.0, 0.0) 0.160 0.186 0.221 0.209 0.248 0.302
(0.5, 0.5, 1.0) 0.498 0.546 0.615 0.232 0.269 0.315
200 (0.0, 0.0, 0.0) 0.076 0.086 0.092 0.096 0.103 0.114
(0.0, 0.5, 1.0) 0.587 0.603 0.624 0.330 0.337 0.360
(0.5, 0.0, 0.0) 0.127 0.130 0.137 0.111 0.125 0.137
(0.5, 0.5, 1.0) 0.581 0.602 0.627 0.326 0.344 0.367
0.5 100 (0.0, 0.0, 0.0) 0.132 0.162 0.194 0.179 0.210 0.268
(0.0, 0.5, 1.0) 0.603 0.645 0.688 0.317 0.349 0.404
(0.5, 0.0, 0.0) 0.147 0.171 0.215 0.167 0.204 0.265
(0.5, 0.5, 1.0) 0.592 0.639 0.679 0.317 0.356 0.410
200 (0.0, 0.0, 0.0) 0.081 0.091 0.101 0.093 0.097 0.119
(0.0, 0.5, 1.0) 0.669 0.682 0.701 0.408 0.413 0.435
(0.5, 0.0, 0.0) 0.117 0.118 0.134 0.100 0.111 0.127
(0.5, 0.5,1.0) 0.681 0.691 0.715 0.418 0.420 0.442
Note: Data were generated as for Table 3, and the same methods as for Table 3 were used except that a linear time trend
term is added to the regression model and that detrended regressors are used for the transition function.
return on assets outside money signifies the opportunity cost of holding money rather than other
financial assets. The inflation rate is also the opportunity cost of holding money instead of goods.
It is normally expected that
∂ f
∂Y
> 0;
∂ f
< 0;
∂ Rout ∂ f
> 0;
∂ Rown ∂ f
∂P
where R out and R own denote the rates of return on assets outside money and on money itself,
respectively. In the empirical literature of money demand, M1, GDP deflator, real GDP (or total
final expenditure) and inflation rate (log difference of GDP deflator) have usually been used for
M d , P, Y and P. For the vector of interest rates R, rates of return on long term bonds and deposit
rate have been used. Baba et al. (1992) contain more detailed discussion on the choice of interest
rates for the money demand function.
Economic theories for money demand do not necessarily require the long-run money demand
function (29) be linear, though theories implying STR-like behaviour of the money demand
function do not exists either according to our knowledge. There are a couple of reasons that the
money demand function may show STR-like behaviour. First, it is well known that many cyclical
variables show an asymmetric behaviour over various phases of the business cycle (e.g. Neftci,
1984). Thus, the money demand function may display STR-like behaviour depending on real
income and inflation rate that show cyclical movements over the phases of the business cycle.
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T 2µ

362 In Choi and Pentti Saikkonen
Second, if the nominal deposit interest rate is so low (as in current Japan), the opportunity cost of
holding money is so close to zero that the public may change their money holdings without any
change in the interest rate. In other words, money demand may be less responsive to the deposit
rate when it is low than when it is high.
In empirical analyses, however, linear specifications have been used for the long-run money
demand equation, and possibilities of non-linear specifications of the long-run money demand
have not been explored. As a first step towards non-linear modelling of the long-run money
demand function, this section tests the null of linearity for the U.K. money demand function using
the tests developed in Sections 3 and 4.
In this paper, we focus on the U.K. quarterly data from 1982:Q3 to 1998:Q4. All the data were
taken from International Financial Statistics. For money, we used seasonally adjusted M4 which
comprises notes and coin in circulation outside the Bank of England, plus non-bank private sector
sterling deposits held with U.K. banking institutions. Seasonally adjusted GDP and GDP deflator
(1995 = 100) were used for nominal income and aggregate price, respectively. The log difference
of the GDP deflator was used as inflation rate. As a rate of return on money, the deposit rate
(per cent per annum) for instant access savings accounts was used. The 91-day Treasury bill rate
(per cent per annum) was used for the rate of return on assets outside of money. Five variables—
real money, real GDP, the Treasury bill rate, the deposit rate and the inflation rate—were used for
our linearity tests. Applications of the Dickey–Fuller test using Ng and Perron’s (1995) sequential
lag selection method indicate that the null of a unit root cannot be rejected at conventional levels
for any of these variables. We took natural logs of the real money and real GDP. Natural logs
are not taken of the interest rates when structural regressions are fit in most previous studies (e.g.
Stock and Watson, 1993). But cointegration tests are applied to the natural logs of the interest
rates in some other studies (e.g. Hoffman and Rasche, 1991). Thus, we considered both types of
the interest rates in this study.
The U.K. real GDP during the sample period shows upward trend which may indicate the
presence of the deterministic linear trend. Detrending the log of real GDP using the sample
mean of the difference of the log GDP eliminates the upward trend. These imply that it is more
appropriate to use the auxiliary model (14) for our linearity tests and that the detrended real GDP
needs to be used as a transition variable.
In Table 7, we report the test results. Transition variables in Table 7 imply the variables
appearing in the transition function. We assume that all the regression coefficients in the money
demand function are subjected to transitions under the alternative. To calculate the T1µ(T 1µ ) test,
the real money was regressed on the real GDP, Treasury bill rate, deposit rate, inflation rate, leads
and lags of these variables, and the products of the transition variables and the regressors. For
example, regressors used for the T1µ test in the first row of part (1) of Table 7 are the real GDP,
Treasury bill rate, deposit rate, inflation rate, leads and lags of these variables and the products
of the detrended real GDP (transition variable) and the four variables in level. For the T2µ test,
the detrended real GDP to the third power was additionally used as a regressor in the auxiliary
regression model.
Part (1) of Table 7 reports the results when no logs were taken of the interest rates. When
the detrended log real GDP, Treasury bill rate and deposit rate are used as transition variables
individually, the null of linearity is rejected at some lags. Because the detrended log real GDP and
deposit rate as transition variables show stronger evidence than the Treasury bill rate in terms of
frequency of rejections in the table, we tested the null using the two variables jointly as transition
variables. The result shows that the T 1µ test rejects the null at lag 3. The detrended real GDP and
Treasury bill rate as joint transition variables do not provide any rejection of the null. This again
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Testing linearity in cointegrating smooth transition regressions 363
Table 7. Linearity test results for the U.K. money demand function (1982:Q3–1998:Q4).
T1µ (T 1µ ) T2µ (T 2µ )
Transition variables K = 1 K = 2 K = 3 K = 1 K = 2 K = 3
(1) No log taken of the interest rates
Detrended rgdp 4.87 7.98 12.28∗ 6.18 10.55 14.71∗ TB 4.74 6.22 10.26∗ 5.79 8.53 13.21
DR 6.77 9.64∗ 13.13∗ 6.93 9.80 13.15∗ infl 1.82 2.48 5.44 2.64 3.96 7.80
Detrended rgdp, TB 6.16 9.36 14.39 7.48 12.37 19.01
Detrended rgdp, DR 8.37 12.62 16.30∗ 9.24 15.77 20.86
(2) Log taken of the interest rates
Detrended rgdp 10.53∗ 17.86∗∗ 30.19∗∗ 11.57 19.83∗∗ 34.34∗∗ tb 7.13 5.87 8.27 7.51 7.06 20.17
dr 10.99∗ 11.14∗ 13.57∗∗ 11.01 11.25 ∗ 17.21∗∗ infl 5.23 7.39 12.35∗ 6.30 9.25 16.36∗∗ Detrended rgdp, dr 15.93 ∗ 21.53 ∗∗ 37.16 ∗∗ 16.82 27.14 ∗∗ 60.00 ∗∗
Detrended rgdp, infl 10.80 19.58 34.20 ∗∗ 13.66 23.08 46.65 ∗∗
Notes:
(i) rgdp: natural log of real GDP, TB: 91-day Treasury bill rate, DR: deposit rate, infr: inflation rate (log difference of
GDP), tb: natural log of TB, dr: natural log of DR.
(ii) Transition variables imply the variables appearing in the transition function gj(·).
(iii) The long-run variance was estimated using Andrews’ (1991) methods with an AR(4) approximation for the prefilter.
(iv) K denote the number of backward and forward lags in the auxiliary regression model.
(v) ( ∗ ): significant at the 5% level; ( ∗∗ ): significant at the 1% level.
(vi) When there is one transition variable, T1µ and T2µ have the asymptotic chi-square distributions with degrees of
freedom four and five, respectively. When there are two transition variables, T1µ and T2µ have the asymptotic chi-square
distributions with degrees of freedom 8 and 12, respectively.
indicates that the evidence of non-linearity induced by the Treasury bill rate is not too strong. 7
When the inflation rate is used as a transition variable, there is no statistically significant evidence
of non-linearity. The results in part (1) indicate that the detrended real GDP and deposit rate are
important candidates for transition variables in modelling the STR-type money demand function.
Part (2) of Table 7 contains the results of using the interest rates taken log of, and shows
more evidence of non-linearity. When the detrended GDP and deposit rate are used as transition
variables either individually or jointly, we find strong evidence of non-linearity. The Treasury bill
rate as a transition variable does not provide any statistically significant evidence of non-linearity,
but the inflation rate does at some lags. Furthermore, the inflation rate and the detrended real GDP
as joint transition variables do provide some evidence for non-linearity. 8 According to the results
in part (2), the detrended GDP, deposit rate and inflation rate should be considered as transition
variables when modelling the long-run money demand function using the STR model.
7 The test results using the Treasury bill and deposit rates jointly as transition variables could not be produced because
the data matrix in the auxiliary regression model became almost singular.
8 Though it may be useful to obtain the test results using the deposit and inflation rates jointly as transition variables, we
could not produce them because the data matrix in the auxiliary regression model ran into the difficulty of singularity.
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364 In Choi and Pentti Saikkonen
The results in Table 7 indicate that the detrended GDP, deposit rate and inflation rate are
candidates for transition variables. Formal estimation of various smooth transition models needs
to be performed in order to find which variables, individually or jointly, produce the best smooth
transition modelling of the U.K. money demand function. The results in Table 7 provide rationale
for pursuing the non-linear modelling, but a separate paper seems to be required to study the issue
in depth.
7. CONCLUSION
We have developed tests that can be used to test linearity in a general cointegrating STR model
with I(1) regressors. The regressors and errors of the model are allowed to be serially and
contemporaneously correlated. In order to allow for this feature, an endogeneity correction based
on a leads-and-lags approach is employed. The model’s transition mechanism is assumed to
be more general than previously. The application of the proposed tests is simple because OLS
techniques and standard chi-square limiting distributions apply. Simulation experiments indicate
that the tests have reasonable finite sample properties. In empirical applications evidence of nonlinearity
in the U.K. money demand function is found especially when the detrended real GDP
and deposit rate are used as transition variables.
ACKNOWLEDGEMENT
Pentti Saikkonen thanks the Academy of Finland and the Yrjö Johnson Foundation for financial support.
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