2. NONLINEAR STR ERROR CORRECTION MODELS

We begin with the following general nonlinear vector error correction (VEC) model for the n × 1 vector of I(1) stochastic processes, z_t:

where α(n × r), β (n × r), and Γ_i (n × n) are parameter matrices with α and β of full column rank and

is a nonlinear function. A conventional linear VEC model is obtained by choosing g(β′z_t−1) = 0 or g(β′z_t−1) = −αμ′ where μ is an n × 1 vector of level parameters.

Recently, analogues of Granger's representation theorem have been studied in the context of the nonlinear VEC model, (2.1), e.g., Escribano and Mira (2002) and Bec and Rahbek (2004). Here we follow Saikkonen (2005) and make the following assumption.

Assumption 1. (i) The errors ε_t in (2.1) are iid(0, Σ), with Σ being an n × n positive definite matrix and E|ε_t|^{[ell ]} < ∞ for some [ell ] > 6. (ii) The distribution of ε_t in (2.1) is absolutely continuous with respect to the Lebesgue measure and has a density that is bounded away from zero on compact subsets of

. (iii) The initial observations Z₀ ≡ (z_−p,…,z₀) are given. (iv) Let A(z) be given by

. If det A(z) = 0, then |z| > 1 or z = 1, where the number of unit roots is equal to n − r. (v) g([bull ]) is asymptotically no greater than a linear function of x_t.

Assumptions 1(i)–(iv) are standard in the cointegration literature, whereas Assumption 1(v) is imposed to deal with nonlinearity of the underlying model (2.1). Under Assumption 1, Saikkonen (2005, Thm. 2) proves that there exists a choice of initial values z_−p,…,z₀ such that Δz_t and β′z_t−1 are strictly stationary.

In this paper we aim to analyze at most one conditional long-run cointegrating relationship between y_t and x_t, and we focus on the conditional modeling of the scalar variable y_t given the k-vector x_t (k = n − 1) and the past values of z_t and Z₀, where we decompose z_t = (y_t,x_t′)′. For this we rewrite (2.1) as

where α is an n × 1 vector of adjustment parameters and

with β_x being a k × 1 vector of cointegrating parameters.

We now make the following assumption.

Assumption 2. (i) The nonlinear function g(·) in (2.2) follows the exponential smooth transition functional form

where we assume θ ≥ 0 for identification purposes and c is a transition parameter. (ii) Partition α = (ζ, α_x′)′ and φ = (γ, φ_x′)′ conformably with z_t = (y_t,x_t′)′. Then, α_x = φ_x = 0. (iii) There is no cointegration among the k-vector of I(1) variables, x_t. (iv) ζ < 0.

Assumptions 2(ii) and (iii) imply that the process x_t is weakly exogenous and therefore the parameters of interest in (2.6) are variation-free from the parameters in (2.7), e.g., Pesaran et al. (2001). Assumptions 2(i) and (iv) imply that Assumption 1(v) is satisfied but also ensure that the underlying (nonlinear) error correction mechanism in (2.2) is globally stationary. In practice various functional forms of g([bull ]) can also be considered to allow for the presence of nonlinear adjustment to the error correction mechanism, but we focus on ESTR here.

Next, partitioning ε_t conformably with z_t as ε_t = (ε_yt, ε_xt′)′ and its variance matrix as

, we may express ε_yt conditionally in terms of ε_xt as

where e_t ∼ iid(0,σ_e²), σ_e² ≡ σ_yy − σ_yx Σ_xx⁻¹σ_xy, and e_t is uncorrelated with ε_xt by construction. Substituting (2.5) and (2.4) into (2.2), partitioning Γ_i = (γ_yi′, Γ_xi′)′, i = 1,…,p, defining φ = ζ − γ, and under Assumption 2, we obtain the following conditional exponential smooth transition regression error correction model (STR ECM) for Δy_t and the marginal vector autoregression (VAR) model for Δx_t:¹

The exponential transition function in (2.6) is bounded between zero and 1, i.e.,

has the properties F(0) = 0; lim_x→±∞ F(x) = 1, and is symmetrically U-shaped around zero.

where ω ≡ Σ_xx⁻¹σ_xy and ψ_i′ ≡ γ_yi − ω′Γ_xi, i = 1,…,p. We notice that (2.6) is a reparameterization of (2.2) under Assumption 2, which will become more convenient for the construction of the cointegration tests we propose in the next section, and Assumption 2(iv) implies that φ + γ < 0. See also Kapetanios et al. (2003).

We call (2.6) the (conditional) nonlinear STR ECM. The representation (2.6) makes economic sense in that many economic models predict that the underlying system tends to display a dampened behavior toward an attractor when it is (sufficiently far) away from it but shows some instability within the locality of that attractor. We note in passing that Assumption 1 is trivially satisfied for our final model specifications (2.6) and (2.7). Therefore, applying Saikkonen's Theorem 2 directly to our model, we may conclude that there exists a choice of initial values z_−p,…,z₀ such that u_t−1 and Δz_t are strictly stationary.²

For a growing literature on the joint analysis of cointegration and STR models see also Gallagher and Taylor (2001), van Dijk, Teräsvirta, and Franses (2002), Choi and Saikkonen (2004), and Saikkonen and Choi (2004). In practice different lag orders for Δy_t and Δx_t in (2.6) can be selected using standard model selection criteria or significance testing procedures without loss of generality or change in the asymptotic analysis, e.g., Ng and Perron (1995).

3. TESTING FOR NONLINEAR COINTEGRATION UNDER STR ECM

To fix ideas for the motivation of the tests, we follow Kapetanios et al. (2003) and impose φ = 0 in (2.6), implying that u_t follows a unit root process in the middle regime; see also Balke and Fomby (1997) in the context of threshold ECMs. Note that for the operational versions of the tests we suggest later we consider both the case φ = 0 and φ ≠ 0. It is then straightforward to show that the test of the null of no cointegration against the alternative of globally stationary cointegration can be based on the single parameter θ. Hence, we set the null hypothesis of no cointegration as³

against the alternative of nonlinear ESTR cointegration of H₁ : θ > 0, where the positive value of θ determines the stationarity properties of u_t. We note in passing that the null hypothesis implies in terms of the preceding model that φ = 0 and θ = 0. Under the alternative hypothesis u_t follows a nonlinear but globally stationary process as described in Section 2.

We propose a number of operational versions of the cointegration test under the nonlinear STR ECM framework given by (2.6). To this end we follow Engle and Granger (1987) and take a pragmatic residual-based two-step approach. In the first stage, we obtain the residuals,

, with

being the ordinary least squares (OLS) estimate of β_x. In the second stage and to overcome the Davies (1987) problem that γ in (2.6) is not identified under the null, we follow Luukkonen, Saikkonen, and Teräsvirta (1988) and Kapetanios et al. (2003), approximate (2.6) by a first-order Taylor series approximation to (1 − e^{−θ(u_t−1−c)2}), while allowing φ ≠ 0 under the alternative hypothesis, and obtain the following auxiliary testing regression:

For this model, we consider an F-type test for δ₁ = δ₂ = δ₃ = 0 given by

where SSR₀ and SSR₁ are the sum of squared residuals obtained from the specification with and without imposing the restrictions δ₁ = δ₂ = δ₃ = 0 in (3.2), respectively. It is interesting to note that (3.2), which is called the cubic polynomial nonlinear error correction (NEC) model, has been analyzed by Escribano (1986, 2004) and his coauthors for the estimation of the UK money demand function although it is important to add that these authors did not provide the formal testing framework that we give in this paper.

There are prior theoretical justifications for restricting the switch point, c, to be zero in many economic and financial applications in the ESTR function, in which case we obtain the following (restricted) auxiliary testing regression:

and obtain the following F-type statistic:

where SSR₀ and SSR₁ are the sum of squared residuals obtained from the specification with and without imposing δ₁ = δ₂ = 0 in (3.4), respectively.

Finally, under the further assumption that φ = 0 (which is the maintained assumption made in Kapetanios et al., 2003) (3.4) is simplified to

For this model, we propose a t-type statistic for δ = 0 (no cointegration) against δ < 0 (ESTR cointegration), which is given by

where

,

is the T × T idempotent matrix, S = (ΔX,ΔZ₋₁,…,ΔZ_−p) is the T × (k + p(k + 1)) data matrix with ΔX = (Δx₁,…,Δx_T)′ and ΔZ_−i = (Δz_1−i,…,Δz_T−i)′, i = 1,…,p, and

, are the OLS estimates of δ, ω, ψ_i in (3.6).

Alternatively and in keeping with the tradition in linear cointegration, we propose a companion test statistic, one that is the analogue to the EG statistic for linear cointegration, which is obtained by estimating the following approximate regression:

where

is the error term associated with an infinite autoregressive (AR) representation of

, e.g., Phillips and Ouliaris (1990, (A.14)). This is so even if the underlying process z_t follows a finite-order VAR(p) model with p ≥ 2. Hence we now need to allow p(T) to tend to grow with T for consistent estimation of (3.8). We follow Phillips and Ouliaris (1990) and impose the condition p(T) = o(T^1/3),

where

,

,

are the OLS estimates of δ and φ_i in (3.8).

The main asymptotic distributional results for all these tests are described in Theorem 3.1 under the following additional assumption.

Assumption 3. Partition Γ_xi = (γ_xyi′, Γ_xxi′)′ in (2.7) conformably with z_t = (y_t,x_t′)′ and set γ_xyi = 0.

This assumption states that Δy_t does not Granger cause Δx_t. Under Assumptions 1(i) and (ii), 2(i) and (ii), and 3, the well-known multivariate invariance principle holds for e_t and ε_xt:

where [Ta] is the integer part of Ta, ⇒ denotes a weak convergence, and W(a) and W_x(a), defined on a ∈ [0,1], are independent scalar and k-vector standard Brownian motions. Without Assumption 3, W(a) and W_x(a) are no longer independent under the null of no cointegration because the weak exogeneity assumption guarantees only that

, but not that

, and therefore the long-run correlation between the partial sums of e_t and Δx_t is possibly nonzero. See also Banerjee et al. (1998) and Pesaran et al. (2001). At the end of this section we discuss a standard extension to the testing procedures that allows us to relax this assumption without affecting asymptotic results.

THEOREM 3.1. Consider the conditional nonlinear ESTR ECM, (2.6), and let Assumptions 1–3 hold. Under the null hypothesis of no cointegration, the F_NEC statistic, defined by (3.3), has the following asymptotic distribution:

where B and W are shorthand notations for

defined on a ∈ [0,1]. Next, imposing c = 0 and under the null of no cointegration, the F_NEC* statistic defined by (3.5) has the following asymptotic distribution:

Further, imposing both φ = 0 and c = 0, and under the null of no cointegration (3.1), the t_NEC and t_NEG statistics defined by (3.7) and (3.10) have the following asymptotic distributions, respectively:⁵

where

. Under the alternative hypothesis that the cointegrating relationship follows ESTR, all the tests are consistent.

To accommodate deterministic components in (2.3), we extend to consider regressions with an intercept and with an intercept and a linear time trend,

Alternatively, we take a simpler but equivalent approach and reexpress (3.15) and (3.16) as

where the superscripts * and + indicate the demeaned and the detrended data, respectively. The respective F_NEC, F_NEC*, t_NEC, and t_NEG statistics are then obtained as follows. First, the appropriate residuals are obtained from either (3.17) or (3.18), and then the corresponding regressions are constructed. For example, we have

where

. Then, the F_NEC statistics are obtained as the F-statistic for δ₁ = δ₂ = δ₃ = 0 in (3.19) or (3.20), respectively. The corresponding F_NEC*, t_NEC, and t_NEG statistics are obtained in an analogous way. For the regression with a nonzero intercept, (3.15), it is easily seen that the asymptotic distributions of the F_NEC, F_NEC*, t_NEC, and t_NEG statistics are the same as in Theorem 3.1, except that W(a) and W_x(a) are replaced by the demeaned Brownian motions, denoted

. Similarly, for the regression with nonzero intercept and nonzero linear trend coefficient, (3.16), the associated asymptotic distributions are such that W(a) and W_x(a) are replaced by the demeaned and detrended Brownian motions, denoted

. Asymptotic critical values of the F_NEC, F_NEC*, t_NEC, and t_NEG statistics for the preceding three cases have been tabulated for k = 1,…,5, via stochastic simulations with T = 1,000 and 50,000 replications in Table 1.

Asymptotic critical values for alternative statistics

We comment on one further modeling issue. An alternative popular nonlinear STR adjustment scheme to the one favored in this paper is given either by the first-order logistic function, 1 − b/(1 + exp(−θu_t−1)), or by the second-order logistic function, 1 − b/(1 + exp(−θu_t−1²)), where b < 1 is a scaling constant. We note that the resulting t-type tests against the alternative of the second-order logistic function have exactly the same form as in (3.7) and (3.10). It is also easily seen that adopting the first-order Taylor approximation to the first-order logistic terms as we did before gives the regression

and we could develop a t-type test for δ = 0 in (3.21). Though the asymptotics for this case are straightforwardly obtained via a simple extension of the analysis in this paper, the interpretation of the error correction coefficient in the first-order logistic STR error correction model is problematic. By contrast, in the exponential case the interpretation of this coefficient is both natural and intuitive (for a discussion of this point, see Kapetanios et al., 2003). Therefore, we focus on the exponential case.

Finally, we comment on how to relax Assumption 3 in practice. To purge the potentially nonzero long-run correlation between the partial sums of e_t and Δx_t, we should augment our conditional model (2.6) with leads of Δx_t, as is standard in the cointegration literature; see, e.g., Banerjee et al. (1998) and Saikkonen (1991). Then, following Theorem 4.1 of Saikkonen (1991), it is readily seen that the limit distribution of these ECM-based tests will be identical to those derived in Theorem 3.1, under the further condition that the number of leads grows at a rate less than T^1/3. We also notice that the asymptotic result for the t_NEG test in this situation would be intact too.

4. FINITE-SAMPLE PROPERTIES

In this section we undertake a small-scale Monte Carlo investigation of the finite-sample size and power performance of our proposed t_NEC, t_NEG, F_NEC, and F_NEC* tests in conjunction with the linear cointegration tests of Engle and Granger and Johansen denoted t_EG and JOH, respectively.

We consider experiments based on a bivariate ECM similar to that adopted by Arranz and Escribano (2000). We generate the data as follows:

Here we fix β_x = 1 and σ₁² = 1. In the tabulated experiments we fix φ = 0 but also discuss (rather than tabulate) results for nonzero φ. Under the null we set θ = 0, whereas we consider parameter values for θ = {0.01, 0.1, 1} and γ = {−1,−0.5,−0.3,−0.1} under the alternative. To investigate the impact of the common factor (COMFAC) restriction, λ = 1 and the impact of different signal-to-noise ratios we consider different parameter values for λ = {0.5,1} and σ₂² = {1,4}.

To save space we only report the results for the demeaned case (see (3.17)) with T = 100.⁶

Size of alternative tests for demeaned data (T = 100)

Turning to the power performance of the tests, which is summarized in Table 3, a general finding is that nonlinear-based tests dominate linear counterparts in terms of power for almost all cases considered. As expected, the power gain of our suggested NEC-based tests over the nonlinear analogue of the EG test becomes substantial where the COMFAC restriction (λ = 1) does not hold. The result that when the COMFAC restriction is violated the power superiority of t_NEC over t_NEG increases with the variance of the innovation in x is both intuitive (a higher variance increases the correlation with the regression error) and a reiteration of what is now well known to occur in the linear case.

Power of alternative tests for demeaned data (T = 100)

Next, we focus on the results for θ = 0.01, because this value is found to be close to empirically plausible estimates.⁷

5. EMPIRICAL APPLICATION TO ASSET PRICING IN THE PRESENCE OF TRANSACTIONS COSTS

In a seminal paper, Campbell and Shiller (1987) investigate the existence of linear cointegration between aggregate U.S. stock prices and U.S. dividends, as predicted by a simple equilibrium model of constant expected asset returns. Their results were ambiguous. A null hypothesis of no (linear) cointegration was marginally rejected in their data, but the implied estimate of long-run asset returns was implausible. Imposing a more credible long-run return caused nonrejection of the null of no cointegration. Subsequent literature has met with similar mixed results, e.g., Campbell and Shiller (1988), Froot and Obstfeld (1991), and Cuthbertson, Hayes, and Nitzsche (1997).

In this section we apply our proposed tests for cointegration to asset prices and dividends for 11 stock portfolios allowing for nonlinear adjustment to equilibrium of the STR variety. The theoretical motivation for STR nonlinearity has already been given in the introduction, and we reemphasize that theoretical reasoning suggests that it is STR adjustment rather than threshold adjustment that is likely to arise in our current application.⁸

We collected monthly data from January 1974 to November 2002 on end period real prices and within period real dividend yields for value weighted market portfolio indexes of stocks traded on the main exchanges of the following 11 countries: Germany, Belgium, Canada, Denmark, France, Ireland, Italy, Japan, Netherlands, United Kingdom, and United States. A dividend series was constructed as the product of dividend yield and prices. Although not presented here, augmented Dickey–Fuller (ADF) tests give overwhelming support to the hypothesis that all variates are I(1).

As alluded to before, we test for the existence of a linear cointegrating relationship between dividends and prices of the form

The existence of bid-ask spreads discussed previously motivates the following nonlinear dynamic STR ECM specification:

where u_t−1 = p_t−1 − βd_t−1 and ε_t is a (possibly autocorrelated) error term that captures other microstructure effects such as specific kinds of noise trading and dealer inventory control mechanisms whereby the adjustment of prices to elicit inventory-correcting trades generates autocorrelated price movements, e.g., Snell and Tonks (1998).

We computed three tests. The first two, t_EG and t_NEG, are the linear EG test and its nonlinear counterpart. The third, t_NEC, is the t-ratio on

in the STR ECM formulation (see (3.6)) where

is the residual from the first stage (spurious under the null) regression of p_t on d_t. The price and dividend series appeared to have an upward trend so that all series were demeaned and detrended before use.

The results for the tests are in columns 2–4 in Table 4. Looking at the results we see that viewed through the “eyes” of linear cointegration tests there is little support for the hypothesis that dividends and prices move together in the long run, with only two of the t_EG tests rejecting the null, albeit at the 1% level. Furthermore, none of the remaining nine t_EG statistics are significant even at the 10% level. By contrast the nonlinear t_NEG test rejects in seven out of eleven stock markets with four of these rejections occurring at the 1% level. A further three statistics are quite close to the 10% critical value. The success in rejecting the null of no cointegration is less marked for t_NEC, with only five rejections at standard significance levels, although three of these reject also at the 1%. A further two t_NEC statistics are quite close to the 10% critical value. If the alternative is really true then we could interpret this finding as being somewhat at odds with the Monte Carlo evidence presented before, which generally shows that t_NEC has more power than t_NEG. One possible explanation is that prior empirical investigation of the data suggested that the COMFAC restriction may hold here. If true, a bivariate ECM would lack parsimony compared with the univariate specification, and this may lead to a loss in power when moving from t_NEG to t_NEC.

Cointegration tests and estimates of the ESTR parameter for asset prices and dividends

Given the strength of evidence against the null and support for the alternative we could obtain estimates of adjustment parameters under the alternative. Focusing on the univariate model we obtained nonlinear least squares estimates of θ from the alternative ESTR model,

where the model has been specialized compared with the general ESTR considered previously by imposing a unit coefficient on γ. Early attempts to estimate γ jointly with θ foundered on severe identification problems, and our nonlinear algorithm failed to converge in most cases—hence the specialization. Under the alternative (and estimation of (5.3) only makes sense if the alternative is true), θ is scale dependent. To clarify its interpretation and to facilitate numerical convergence, we normalized the

series to have unit sample variance (a procedure that only makes sense under the alternative). The results for

and its t-statistic are given in Table 4. Although we cannot interpret the t-statistic as a significance from zero test (for obvious reasons) we refer to it as “significant” if an asymptotic 95% confidence interval around the estimate excludes zero. We see that

is “significant” in all cases and varies between 0.007 and 0.017.

To get a feel for what such values imply, Figure 1 plots IRFs for the error correction term for initial impulses of one, two, three, and four standard deviation shocks, respectively. To be explicit, we take the univariate specification in (5.3) and perturb its disturbance by one to four standard deviation shocks. We trace the time path implied by (5.3) following each of the four shocks and plot them on a single IRF diagram.¹¹

Impulse responses of error correction terms to k (= 1,2,3,4) standard deviation shock for various countries.

6. CONCLUDING REMARKS

The investigation of nonstationarity in conjunction with nonlinear autoregressive modeling has recently assumed a prominent role in econometric study. It is clear that misclassifying a stable nonlinear process as nonstationary can be misleading both in impulse response and forecasting analysis. Similarly, not allowing for the presence of cointegration when the speed of adjustment varies with the position of the system as in the case of nonlinear cointegration can be equally misleading. In this paper we have proposed a simple direct cointegration test procedure that is designed to have power against STR nonlinear error correction specifications. Our proposed tests are shown to have better power than the linear cointegration tests that ignore the nonlinear nature of the alternative. An empirical application clearly shows the potential of our approach. Unlike linear cointegration tests, the nonlinear tests find substantial evidence of cointegration in stock price and dividend systems. Further research to develop similar tests for alternative nonlinear models is currently under way.