COINTEGRATION AND NONLINEAR ERROR CORRECTION

In this section, we introduce some useful concepts of time-series that are integrated of order 0 and 1, I(0) and I(1), and we characterize the concept cointegration in the context of NEC models. The problem of introducing nonlinearities in nonstationary contexts is nontrivial since the concepts of integration and cointegration are based on linear time-series frameworks.

In general, the concept of I(1) is related to time-series variables that need to be differenced once to become covariance-stationary, I(0). Therefore, given a finite starting value, the partial sum of an I(0) process is I(1) since their variance and covariance grow over time. The basic idea now is to extend the usual concepts of I(0) processes to allow for certain types of heteroskedasticity and nonlinearities of the underlying sequences. To do that we use nonlinear measures of short-run dependence; see Davidson (1994).

For example, the concept of α-mixing introduced by Rosenblatt (1978), measures dependence between events that are separated by at least m periods (leads and lags). Another useful measure for the degree of temporal dependence in a time-series is based on the concept of near epoch dependence (NED); see Appendix A. Say that the coefficient [phi]_m is a measure of the worst mean square prediction error of ν_t based on the present, near past, and near future of η_t, E(ν_t/η_t−m, …, η_t+m). If the sequence is [phi]-NED, then the worst conditional mean square prediction error tends to zero as m→∞.

The concepts of α-mixing and NED are used now to define operative definitions of I(0) and I(1) in a nonlinear context, following Escribano and Mira (2002).

DEFINITION 1.

The sequence ε_rt is said to be I0) if it is an α-mixing sequence ([phi]-NED on the underlying α-mixing sequence ν_rt) and the partial sum x_rt = [sum ]^t_i=1 ε_ri is not α-mixing (is not [phi]-NED). We then say that x_rt = [sum ]_i=1^tε_ri is I(1).

Let ε_r0 = 0. Notice that from Definition 1 if x_rt is I(1), then Δ x_rt is I(0). Furthermore, if ε_rt is I(−1), then the partial sum x_rt = [sum ]_i=1^t ε_ri is an α-mixing sequence ([phi]-NED on the underlying α-mixing sequence ν_rt) and therefore not I(1). In Section 3.2, we use these concepts to discuss the asymptotic distribution of the two-step least-squares estimator of NEC models.

The concepts of I(0) based on α-mixing or [phi]-NED are related to the notions of “short memory in distribution” and “short memory in mean” discussed by Granger and Teräsvirta (1993) and Granger (1995). Our broad concept of I(0) has the clear advantage of having laws of large numbers (LLN) and functional central-limit theorems (FCLT) associated with them; see, for example, White (1984), Herrndorf (1984), Phillips (1987), Gallant and White (1988), Wooldridge and White (1988), Davidson (1994) and Dufrénot and Mignon (2002).

On the basis of those concepts, Escribano and Mira (1996) found the asymptotic distribution of the nonlinear least-squares (NLS) estimator of the nonlinear cointegrating relationship.

Let the N×1 vector X_t be equal to (y_t, q_t′)′, where y_t is an escalar and q_t is an (N−1)×1 vector.²

The following concept of cointegration is convenient for Theorem 1.

DEFINITION 2(COINTEGRATION).

If y_t and q_t are I(1) and there is a linear combination, z_t≡y_t−α ′q_t, that is α-mixing (or correspondingly [phi]-NED) for the parameters (1,α′) and is not α-mixing (or [phi]-NED) for any other vector (1,α*′) with α′≠ α*′, then y_t and q_t are cointegrated with cointegrating vector (1, −α′).

This definition can easily cover cases of nonlinear cointegration if we allow z_t ≡ g(y_t−α′q_t) in Definition 2. The concept of I(0) for [phi]-NED sequences is now used, instead of the asymptotic uncorrelation condition used by Escribano (1987), to properly characterize single-equation NEC models based on the representation theorem of Escribano and Mira (2002). The objective of Theorem 1 is to give sufficient conditions for NEC models to imply cointegration (Granger-type representation theorem).

THEOREM 1 (SINGLE-EQUATION NEC REPRESENTATION THEOREM).

Let the N×1 vector X_t = (y_t, q_t′)′, where y_t is a scalar and q_t is the (N−1)×1 vector be generated by

where

1. ε_yt is a martingale difference sequence relative, to the information generated by the explanatory variables (X_t−1) observed up to time t−1, with zero mean, constant variance, σ², and the (N−1)×1 vector ε_qt is α-mixing with constant variance–covariance matrix, Σ
2. the determinantal equation | 1 − a₁B − a₂B² + · + a_pB^p| = 0 has all its roots outside the unit circle;
3. the function f(z,γ) is continuously differentiable on z;
4. and −2<df(z_t−1,γ)/dz_t−1<0 (stability condition)
5. Δq_t, ε_yt and the cross products have finite second-order moments.

Then,

• (a) X_t is a vector of I(1) components and
• (b) z_t and ΔX_t are [phi]-NED and therefore y_t and q_t are cointegrated with cointegrating vector (1, −α′).

Proof. See Appendix A.

In Theorem 1, we show that when there is an NEC model like (1a), if the stability condition (iv) is satisfied, then under some regularity conditions the first difference of the variables, ΔX_t, must be I(0); y_t and the levels of some (or all) of the q_t variables are cointegrated; and z_t = y_t−α ′q_t (some but not all of the components of α could equal to 0).

Notice that when the maximum lags r and p are equal, and when certain common factor restrictions (COMFAC) are satisfied [see Hendry and Mizon (1978)], b₀′−α′ = 0 , b₁′− a₁α′ = 0 , …, b_p′− a_pα′ = 0, we can write the NEC—equation (1a)—in terms of z_t:

This nonlinear autoregressive process, AR(p+1), for z_t is a nonlinear extension of the augmented Dickey-Fuller (ADF) equation used by Engle and Granger (1987) to test the null hypothesis of unit roots (noncointegration) against roots lower than unity (cointegration). With the NEC models [equation (2)], the null hypothesis of noncointegration [(z_t, I(1))] implies that H₀: f(z_{t − 1},γ) = 0 whereas, under the alternative hypothesis of cointegration (z_t, [phi]-NED), H₁: f(z_{t − 1},γ) ≠ 0 with −2 < df(z,γ)/dz < 0.

Escribano (1985, 1986) suggested the use of cubic polynomials for f(z_{t − 1},γ)=[sum ]_{i = 1}³c_i zⁱ since polynomials can approximate general nonlinear functions based on Taylor-series expansions. However, a first-order autoregressive cubic polynomial representation for z_t cannot globally satisfy the useful [phi]-NED condition of Escribano and Mira (1997, 2002), −1< (dz_t/dz_{t − 1}) < 1, since polynomials are unbounded and explosive functions for large values of z_{t − 1}.

To solve the instability problem of polynomial approximations for large values of z_{t − 1}, we suggest the class of time-series models based on low-order rational polynomial functions that globally satisfy the stability condition (iv) of Theorem 1, while avoiding the three undesirable properties (i)–(iii) of linear EC models mentioned in the introduction. This property will become clear in the empirical application of Section 5.

Among the many nonlinear parametric functional forms to consider for f(z,γ), there is a also a sound argument in favor of rational polynomial functional forms based on Padé approximants. The argument goes as follows: Assume that the function f(z,γ) can be written as a power series; then, Padé's approximant is the rational fraction

which has a Maclaurin expansion that agrees with the power-series expansion. In total, we have L+M+1 unknown coefficients in (3). The rational fraction, {L/M}, should fit the power series so that

Power series have a circle of convergence |z|= R. If |z|< R, the series converges; if |z|> R, it diverges. When R = ∞, the function is analytic. The following version of Montessus's theorem [see Baker and Graves-Morris (1996)] gives a general approximation result.

THEOREM 2 (CONVERGENCE OF PADÉ APPROXIMANTS).

Let f(z,γ) be a function that is analytic at the origin and analytic in the entire z-plane except for a countable number of isolated poles and essential singularities. Suppose ε>0 and δ>0 are given. Then, M₀ exists such that any {L/M} Padé approximant of the ray sequence with L/M = λ(λ≠0λ≠∝) satisfies

on any compact set of the z-plane except for a set of measures less than δ.

Proof. See Appendix A.

However, we argue in this paper that cubic polynomial functional forms are very useful and flexible parametric approximations to unknown functional forms. The first justification is based on the simulation results obtained with Lagrange multiplier (LM) tests, when testing the null hypothesis of linearity against the alternative of nonlinearity; see, for example, Tsay (1986) and Escribano and Jorda (2001). Second, cubic polynomial approximations can be used as decision rules for selecting nonlinear parametric time-series models. Teräsvirta (1994) and Escribano and Jorda (1999, 2001) discussed the use of this type of LM tests when selecting between logistic and exponential smooth-transition autoregressive (STAR) models and between logistic and exponential smooth-transition regression (STR) models, respectively. Third, cubic polynomials can be used to detect asymmetries (linear or nonlinear) and threshold points (unique equilibrium or multiple equilibria); see Escribano and Pfann (1998). Fourth, they can be used to empirically check the stability condition ([phi]-NED condition) of the nonlinear adjustment, f(z,γ); see condition (iv) of Theorem 1. In the empirical applications discussed in Sections 4 and 5, we show how to use the flexibility of cubic polynomial functional forms in the context of NEC models.

SPECIFICATION, ESTIMATION, AND TESTING IN NEC MODELS

One of the main advantages of using the concepts of I(1), I(0), and cointegration defined in the previous section is that they allow us to check whether an estimated NEC model satisfies those conditions. This is so because we give explicitly sufficient conditions on the nonlinear function.

On the other hand, to make the broad concepts of I(1) and I(0) introduced in Definition 1 testable we could add extra conditions in the manner of Phillips (1987) and Lo (1991).

Assumption 1.

1. Let ν_t be I(0) and α-mixing so that the sequence of partial sums y_t = [sum ]_{i = 1}^tν_i is I(1);
2. E(ν_t) = 0, for all t;
3. sup E|ν_t|^β < ∞, for some β>2;
4. σ² = lim_{T → ∝} E(T⁻¹y_T²) < ∞, with 0 < σ ²< ∞;
5. [sum ]_{m = 1}^∞α_m^1−2/β < ∞, where the α_m–mixing coefficient is defined in Appendix A.

On the basis of Assumption 1, Herrndorf (1984) proved the following FCLT: T^−1/2([sum ]_{i = 1}^[Tr]ν_i)→σ W (˙) where W(·) is a standard Brownian motion.

Lo (1991) suggested the use of the test statistic T^−1/2 Q_T:

where μ_T and σ²_T(s) are the estimated mean and variance of ν_t, and the optimal value of s is obtained as the integer value of K_T, where K_T = (3T/2)^1/3(2ρ/(1−ρ²)^2/3.

Under the null hypothesis, H₀ : ν_t satisfies Assumption 1, Lo (1991) derived the nonstandard asymptotic distribution of the test statistic T^−1/2 Q_T.

Kwiatkowski et al. (1992) used a different test statistic to test the same null hypothesis (H₀),

where y_t=[sum ]_{i = 1}^t (ν_i−μ_T) and σ²_T(s) is given by (6b). The values of s that they consider are s=0, s=(4(T/100)^1/4), s=(12(T/100)^1/4), and the optimal value of s previously mentioned (integer value of K_T).

Using the statistics T^−1/2 Q_T and KPSS_T, we could test the null hypothesis, H₀: ν_T is I(0) and satisfies Assumption 1 against the alternative H₁: ν_T is I(1) and/or does not satisfy Assumption 1.

Similarly, we could use the same test statistics to test the hypothesis of I(0) versus I(1) based on [phi]-NED ν_t sequences, provided that they satisfy a FCLT. Wooldridge and White (1988) gave explicit conditions for a FCLT to hold under NED conditions; see also Davidson (1994).

Semiparametric Approach

Before we describe the semiparametric estimation procedure of NEC models based on somoothing splines, it is convenient to rewrite model (1a) as

where Δ Q_t′=(Δ q_t′, Δ q_t−1′,…,Δ q_{t − r}′) is a 1×(N−1)r vector and Δ Y_{t − 1}′ = (Δ y_{t − 1},…,Δ y_t−p) is a 1×p vector of lagged dependent variables, and b and a are the parameter vectors of orders r(N−1)×1 and p×1, respectively.

When the sample size is T+h, t=−h,…, T with h=max(r, p), model (8) in matrix notation becomes

where Δ y=(Δ y₁,…, Δ y_T)′, f(Z₋₁,γ)=(f(z₀,γ),…, f(z_{T − 1},γ))′ and ε_y=(ε_y1,·, ε_yT)′ are T×1 vectors, Δ Q=(Δ Q₁,…, Δ Q_{1 − r}) is a matrix of order T×(N−1)r and Δ Y=(Δ Y₋₁,…, Δ Y_−p) a matrix of order T×P.

Our objective is to estimate the unknown nonlinear equilibrium correction, f(Z₋₁, γ), by the semiparametric procedure of smoothing splines. Spline smoothing is a very flexible estimation technique. The first analysis of their statistical properties for nonperiodic nonlinear functions appeared in Wahba (1975) and Silverman (1985). Different empirical applications of this estimation procedure are in Granger et al. (1984), Engle et al. (1986), and Härdle and Linton (1994).

The smoothing-splines estimator that we use in the empirical section is obtained using the following procedure. Substitute equation (9) by a general parametric approximation,

where f(Z₋₁,γ)≈χ₋₁β, Δ H=(Δ Q, Δ Y) and [phi]=(b′ a′)′. In this approximation χ₋₁ is a T×K matrix of zeros and ones and k≤T is the number of intervals into which the domain of the function has been divided. The matrix χ₋₁ is formed by reading each of the T observations of z_t−1 and assigning the value 1 to the interval where the observation lies and 0 in the rest of the intervals. To estimate a smooth function, while avoiding overfitting the data, we can impose a smoothing penalty. To do that explicitly, we will first write equation (10) as

where W=(χ₋₁, ΔH) and Δ=(β′, [phi]′)′. The estimator of δ is obtained by minimizing the following objective function:

where the matrix V is chosen so that the smoothing constraint applies, in a direct way, only to the parameter vector β. Let V=,(B,0), where B is a K×K matrix and 0 is a K×L matrix where L=(N−1)xr+p; hence,

The first component is the usual error sum of squares and the second represents the penalty imposed for lack of smoothness. The content of the matrix B determines what type of smoothing constraints we want to impose. If, following the idea of smoothing splines, we impose a penalty based on the second derivative of the function f(z,γ), then, when the intervals are of equal size, the matrix B will be formed so that three adjacent β′s are close to a straight line, β_i+1+β_i−1−2β_i=0; see Engle et al. (1986) for the treatment of a more general case.

The solution to the optimization problem (12) is

which, in a more explicit form, can be written as

The estimator δ(λ) is, under some regularity conditions, consistent but biased. The size of the bias of δ(λ) depends positively on the value of the parameter λ. On the other hand, the variance decreases with λ getting the classical trade-off between bias and efficiency.

To ascertain what the optimal value of λ should be, Wahba (1975) developed the generalized cross-validation (GCV) procedure. This procedure chooses the value of λ that solves

where RSS(λ)=(Δ y − Δ ^y)′(Δ y − Δ ^y), Δ ^y is the fit and G(λ)=W(W′ W+λ V′ V)⁻¹ W′, see Härdle and Linton (1994) for a discussion on the properties of this procedure.

When working with macroecomic variables, the sample sizes are usually too small to obtain consistent and efficient nonparametric or semiparametric estimators. However, as we will see in the empirical application of the UK money demand, this semiparametric procedure can be very informative in finding appropriate types of nonlinear functional forms, cubic polynomials, rational polynomials, etc. The final empirical step should therefore consist of the estimation of NEC models from those suggested parametric specifications. This final step should provide more efficient estimates.

Parametric Approach

Assumption 2. The vector (Δ y_t, Δ q′_t)′ satisfies a FCLT based on NED sequences, and z_t satisfies Assumption 1.

THEOREM 3 (TWO-STEP ESTIMATOR OF NEC).

Under Assumption 2 and conditions (i)–(v) of Theorem 1,

Equations (17a) and (17b) can be estimated by OLS and NLS with the following properties:

• (a) The OLS estimator of the cointegrating vector α, in (17a), is superconsistent and its asymptotic distribution is nonstandard.
• (b) The OLS and NLS estimators of the parameters (b₀, b₁,·,b_r)′ and (a₁,·, a_p, γ) of equation (17b) have the same limiting distribution no matter what value of z_t−1 we use: the true value (z_t−1) or its OLS estimate, (U_t−1), from (17a).
• (c) The OLS or NLS estimators of the parameters (b₀, b₁,·,b_r)′ and (a₁,·, a_p, γ) of equation (17b), where z_t−1 was previously estimated (U_t−1) by OLS on (17a), are
  -consistent and asymptotically normal.

Proof. See Appendix A.

Those least-squares (LS) estimation results are based on Stock (1987) and Escribano and Mira (1996, 1997) and generalize Engle and Granger (1987) two-step estimator to single-equation NEC models.

Most of the previous results on estimation and inference in NEC models are based on parametric procedures (OLS, NLS) when the nonlinear functional form is known. However, the functional form is generally unknown and, if we start with the wrong class of nonlinear functions, it might be difficult to reach a good model. As will become clear in the next section, to overcome this problem we suggest starting the nonlinear specification with the semiparametric estimation procedure mentioned in Section 3.1 in order to find out plausible parametric functional forms for the nonlinear equilibrium correction. However, this makes sense if we have previously identified certain nonlinearities in the model. This procedure is explained in the next section.

Testing for the Existence of NEC Adjustments

The first step of the methodology is the estimation of a linear specification. Once we have evidence of cointegration and have specified a linear EC model, usual misspecification tests can be applied to test for nonlinearities in the equilibrium correction. For example, one can use an LM test based on the R² of auxiliary regressions (using the asymptotic χ² distribution) or its small-sample counterpart (based on the F distribution) to test the null hypothesis of linear error correction (H₀: EC) against the alternative hypothesis of nonlinear error correction (H₁: NEC). Alternatively, we could use the likelihood ratio (LR) or Wald (W) tests. Sometimes, LM tests have the advantage of being easier to compute in nonlinear cases. Furthermore LM tests can be used to identify certain types of parametric functional forms to be used. For example, when selecting between logistic and exponential specifications, Teräsvirta (1994) proposed an interesting decision rule, in the context of smooth-transition autoregression models (STAR). Escribano and Jorda (1999, 2001) suggested an alternative decision rule and extended the analysis to the class of smooth-transition regression models (STR). As we will see later, those decision rules based on LM tests could also be informative when deciding among alternative nonlinear functional forms in NEC models.

If the linear error-correction model have been rejected in favor of a nonlinear error correction, then we should follow a semiparametric estimation procedure (smoothing splines, kernels, etc.) to identify the class of parametric nonlinear functions. This strategy is most relevant in the absence of large sample sizes since parametric estimates will be more efficient and the corresponding test statistics more powerful.

Once we have estimated a plausible parametric nonlinear formulation of the error correction adjustment (NEC), the usual misspecification tests should be applied to test the null hypothesis of correct specification against the alternative misspecification hypotheses: autocorrelation, heteroskedasticity, wrong dynamic specification, parameter nonconstancy, etc.

In the following sections we apply this econometric methodology to different periods of the UK money demand and discuss competing estimates, some of which were mentioned critically by Friedman and Schwartz (1991).

EMPIRICAL APPLICATIONS OF NEC METHODOLOGY

We start this section with a brief historical review of some of the first unpublished competing UK money demand estimates based on annual observations from the year 1878 to 1970. The database was originally obtained from Friedman and Schwartz (1982) but rescaled as in Hendry and Ericsson (1983) to compensate for the Southern Ireland effect. Table 1 includes the three main unpublished competing specifications of the money demand that were independently proposed. One of them represents the first nonlinear and asymmetric error-correction model estimated. In what follows, we briefly highlight the basic differential elements of the three alternative money demand specifications.

Hendry and Ericsson (1983) estimated an EC money demand equation in one-step (see Table 1, col. 2) by OLS, as a critical reaction to the empirical approach followed by Friedman and Schwartz (1982). The implicit cointegration relationship was between the log-inverse velocity of circulation of money (m−p−i)_t−4 and short-run interest rate (RS)_t. To obtain parameter constancy, their specification needed a questionable dummy variable D2 (see Appendix B), which captures what Friedman and Schwartz (1982) called a “liquidity preference shift.” Based on this, Hendry and Ericsson (1983) realized that their specification was not satisfactory. This comment motivated other researchers to look for alternative money demand specifications.

Simultaneously, two different money demand equations were estimated, one at the London Business School in the United Kingdom, by Longbottom and Holly (1985), and the other at the University of California at San Diego (USA) by Escribano (1985, 1986).

Longbottom and Holly (1985) estimated a linear error correction in one-step by OLS (see Table 1, col. 3). The implicit cointegration relationship they considered was among real money stock (m−p)_t−1, real income (i)_t, and short-run interest rates in logs (rs)_t. Therefore, they did not impose a unit income elasticity of real money demand in the long-run. In particular, they suggested the following changes: (i) relax the restriction on the coefficient of the first lagged dependent variable, (ii) use interest rates in logs in both the long run (rl) and the short run (rs), (iii) relax the restriction on the coefficients of the rate of growth of prices, and (iv) impose the same coefficient on both world war dummies, D1 and D3. With those changes, they did not use the dummy variable D2 and reduced the estimated standard error of the regression from 1.7% to 1.4%.

Escribano (1985, 1986), followed an alternative strategy, estimating a NEC in two-steps (see of Table 1, cols. 3 and 4). The explicit nonlinear cointegration relationship was between log-inverse velocity of circulation of money (m−p−i)_t and short-run interest rates (RS)_t without logs. The dynamic formulation considered in the second step incorporated the following changes: (i) relax the restrictions on the coefficient of the first lagged dependent variable, (ii) relax the restriction on the coefficients of the growth rate of prices, (iii) include the first difference of both short-run and long-run interest rates (RS and RL), and (iv) incorporate the lagged residuals (U_t−1) from the cointegrating relationship in a nonlinear form (cubic polynomial error correction). The incorporation of the nonlinear equilibrium correction made the contribution of the dummy variable D2 insignificant and reduced the estimated standard regression error of the regression from 1.7% to 1.4%. Both alternative parameterizations—columns 3 and 5 of Table 1—represented significant improvements over previous money demand equations, as was later recognized by Friedman and Schwartz (1991) and Hendry and Ericsson (1991).

Empirical Application of NEC Methodology to the UK Money Demand from 1878 to 1970

This section is based on the seminal work of Escribano (1985, 1986, 1996) and therefore uses the original data set to describe the econometric methodology discussed in Sections 2 and 3 when searching for nonparametric and parametric functional NEC models. For the stability analysis, larger data sets will be considered. The usual first step in dynamic single-equation modeling is to analyze the univariate time-series properties of the variables. This is important to determine the class of models and the estimation and inference properties that we should consider. However, since the descriptive analysis of this data set can be found in Friedman and Schwartz (1982) and in Hendry and Ericsson (1991), we only discuss what is different in terms of the I(1) and I(0) properties of the variables.

Recall from Theorem 1 that the vector X_t=(y_t, q_t′)′, where y_t is a scalar and q_t is an (N−1)×1 vector. The dependent variable that we want to explain in the dynamic regression analysis is the log of the real money stock y_t ≡ (m−p)_t and its rate of growth Δ y_t ≡ Δ (m−p)_t. The set of explanatory variables form an 8×1 vector (q_t)′≡ (i, p, rs, rl, D1, D2, D3,1)_t′ where the first component (i) is the log of real income, (p) is the implicit price deflator of real income in logs, (rs) is the log of the short-run interest rate, (rl) is the long-run interest in logs, and the variables (D1), (D3), and (D2) are dummy variables to capture special events such as World War I, World War II, and a “liquidity preference shift,” produced by “economic depression and war,” following the work of Friedman and Schwartz (1982, p. 281). The precise definitions of the variables are given in Appendix B.

When the true model is a NEC model, we might first want to check that the two-step estimation procedure works correctly, giving consistent OLS estimates of the cointegration relationship in the presence of a nonlinear model of the residuals. Theorem 3 requires that the errors from the cointegrating relationship are α-mixing, satisfy Assumption 1, and the first difference of the variables of the model is I(0) and satisfies Assumption 1. All of these conditions are testable.

In Table 2, we report the numerical values of Lo (1991) and Kwiatkowski et al. (1992) test statistics obtained for the following components of the vector of first differences (Δ (m−p), Δ i, Δ p, Δ rs, Δ rl, Δ RL)_t′)′.

Given that all estimated values of the test statistics are inside the 95% confidence interval, we cannot reject the null hypotheses that the variables in first differences are I(0) and satisfy Assumption 1 against the alternative hypothesis of I(1). If we plot the first difference of the relevant series from 1878–2000, it is clear that none of the series in first differences look like I(1), confirming the results of the test statistics that the first difference of the variables is I(0) and therefore the partial sums are I(1).

Furthermore, from unit root tests and the evolution of the autocorrelations and partial autocorrelation of the univariate time series, we conclude that the components of the vector X_t=(y_t, q_t′)′, except for the dummy variables and the constant, can be characterized as I(1); see Escribano (1985, 1986), Hendry and Ericsson (1991) and Ericsson et al. (1998) for more details.

In the last two columns of Table 2, we report several cointegrating tests, under the assumption that RS_t (or rs_t) are strictly exogenous in their corresponding cointegrating regression. The results of testing the null hypothesis (H₀) of cointegration are based on the test statistic of KPSS_T extended by Shin (1994). From the results of column 8 of Table 2, we cannot reject that there is a cointegrating relationship (H₀) between inverse velocity of circulation of money in logs, (m−i−p)_t, and short-run interest rates in logs, (rn)_t. Similarly, column 9 of Table 2 shows that there is no evidence against a nonlinear cointegrating relationship (H₀) between inverse velocity of circulation of money in logs, (m−i−p)_t and short-run interest rates without logs, (RN)_t. In both cases the test statistics based on the residuals, u and U from the OLS-cointegrating regressions, are inside the 95% confidence interval. Therefore, in both cases, we cannot reject the null hypothesis of cointegration against the alternative hypothesis that the residuals are I(1) and/or do not satisfy Assumption 1. The power of the test and the small-sample critical value (C.V.) require a deeper analysis but the study is beyondthe scope of this paper.

Given that the variables are cointegrated, we can proceed by estimating a linear error correction model; see Table 3. Following the results of Escribano (1985, 1986), Table 3 confirms the previous evidence against the null hypothesis of no cointegration [U_t is I(1)] between log-inverse velocity of circulation of money (m−i−p)_t and short-run interest rates (RS)_t, with a t-ratio of U_t−1 equal to –3.2.

Once we have found a plausible linear error-correction model, we could test for the existence of an NEC model. We could do that by using standard testing principles based on Lagrange multipliers (LM), Wald (W), or likelihood ratio (LR). As we mentioned earlier the LM test is probably the simplest test against nonlinear alternatives since all we have to do is estimate an artificial regression using the residuals from the linear error-correction model as the dependent variable and the explanatory variables of the linear EC model as the regressors, plus polynomial terms of the first lag of the residuals from the OLS cointegrating relationship estimated in the first step, U_{t − 1}² and U_{t − 1}³. However, if the NEC model can also be estimated by OLS, this advantage of the LM test disappears. For example, in Table 3, we report the results of implementing an LR test for the null hypothesis of linearity against the alternative of NEC. The log likelihood, or its small-sample counterpart F-statistic, has p-values smaller than 0.05, rejecting the null of linearity in favor of NEC. As discussed by Teräsvirta (1994) and Escribano and Jorda (1999, 2001), the LM tests for linearity, based on the alternative hypothesis of cubic polynomials or fourth-order polynomials, have power against logistic or exponential nonlinear models. However, for reasons that will be mentioned later, we suggest considering a new class of nonlinear functions: rational polynomial error-correction models.

Once we have found evidence of a NEC model, we suggest estimating the unknown nonlinear relationship by the semiparametric procedure based on smoothing splines; see Section 2.

Given that we have rejected the null hypothesis of linear error correction in favor of NEC, we could maintain the parametric linear dynamic specification while allowing the nonlinearity to affect only the error correction term. Applying the smoothing-splines estimation procedure to the money demand of the United Kingdom (1878–1970), we find that the number of intervals K is 11 and that the estimated (optimal) value of the smoothing parameter λ is λ=0.00001216. The empirical results obtained are reported in Model A of Table 4.

The graphical representation of the nonlinear adjustment is in Figure 1. From the graph it is again clear that the two attractor points (equilibria) are 0 and 0.2. We could also interpret the values of U between 0 and 0.2 representing the multiple long-run equilibria of the money demand. It is interesting to realize that the nonlinear plot of this semiparametric estimation looks like the cubic polynomial plot previously obtained.

Therefore, the nonparametric estimation supports cubic polynomials as plausible parametric functional approximations of the nonlinear equilibrium correction. The next closest parametric alternative is the rational polynomial, which as we mentioned before, has an important advantage over cubic polynomials, since it is consistent with U_t being [phi]-NED.

One possible interpretation of the asymmetric adjustment is that it is piecewise nonlinear only with negative equilibrium errors but the adjustment is linear with positive errors greater than or equal to 0.2. The economic and statistical interpretation of this possibility is unclear but it could be justified by having certain asymmetric adjustment costs.

When imposing a strong smoothing constraint (large value of the smoothing parameter λ) on the splines estimator, we get a linear error-correction adjustment. This linear error-correction model was obtained by setting the value of λ at 10,000 while keeping 11 intervals; see Model 4B of Table 4. The graphical representation of the linear error-correction adjustment is in Figure 1. From the graph, it is clear that the whole adjustment is biased toward the value of U=0.2, thus explaining why the initial estimates of the UK money demand (1878–1970) required the use of the dummy variable D2 in their linear specifications.

The final money demand of Hendry and Ericsson (1991) is a NEC model estimated by the two-step procedure of Engle and Granger (1987); see columns 2 and 3 of Table 3. Following Escribano (1985, 1986), they incorporated the residuals from the OLS estimator (U_t), lagged once, as a cubic polynomial in the dynamic formulation. The NEC term of Hendry and Ericsson (1991) is in column 3 of Table 5, −2.55(U_{t − 1}−0.2)U²_{t − 1}=0.5 U²_{t − 1} − 2.55 U³_{t − 1}. Following Longbottom and Holly (1985), they also included the short run and the long-run interest rates in logs (rates of change) in the short run. In the last rows of column 3 of Table 5, we present several misspecification tests. We were unable to reject the null hypothesis of correct specification. That is, or AR(2) structure in the residuals, or ARCH(1), etc.

Following Granger and Lee (1989) in the last column of Table 5, we report the estimation of an asymmetric error-correction model with the threshold point (τ) at 0. However, this model is worse since the standard error of the regression increases from 1.42% to 1.46% and only the negative errors are significantly different from zero. In Section 5, we will give an explanation of this result using the extended data set going from 1878 to 2000.

Econometric Evaluation of Ericsson et al. (1998) UK Money Demand Estimates

Hendry and Ericsson (1991) also extended the model presented in column 3 of Table 5 to cover the period from 1970 to 1975. This is an important period due to the introduction of Competition and Credit Control regulations. To account for that, they define a dummy variable D4, see Appendix B, which also enters interactively with the rate of growth of the short-run (Δrs) interest rates, (D4Δrs). Additional data from 1975 to 1993 were compiled by Attfield et al. (1995) and examined by Ericsson et al. (1998). We replicate their two-step estimation in column 2 of Table 6. In step 1, they estimate the long-run parameters from the cointegrating relationship using the period of 1873–1970 and impose those estimated parameter values of the cointegrating vector for the period 1873–1993. In step 2, they estimate the error-correction model including cubic polynomial equilibrium correction terms, following Escribano (1985, 1986). However, as we can see in the misspecification test included in the last rows of Table 6, col. 2, there is clear evidence of some autocorrelation, AR(2), in the residuals of the model.

This problem is mitigated but not solved by estimating this model in one step by nonlinear least squares (NLS) (see column 3 of Table 6). Columns 4 and 5 of Table 6 show that their specification can be improved by getting rid of the constraint imposed on the lagged dependent variable at lags 2 and 3. To test for that, we included the variable Δ p_t − 2 as an extra regressor. It became significant with an absolute value of the t-ratio equal to 2.62, and after that the model passes all the misspecification tests (see last rows of columns 4 and 5 of Table 6). Therefore, from now on we will use the NEC specification obtained in columns 4 and 5 as a benchmark when searching for alternative nonlinear money-demand specifications during the extended period of 1878 to 2000.

NEC MODELS OF UK MONEY DEMAND FROM 1878 TO 2000

The first aspect we want to address in this section is the fact that we have a nonlinear cointegration relationship between the log of velocity of circulation of money, v=− (m−p−i), and short-run interest rates without logs, RN^a=RNA, Figure 2A. This implies that there is a long-run money demand relationship between real money in logs (m−p)_t and short-run interest (RN^a)_t with a constant unit income (i) elasticity. A linear cointegrating relationship is obtained if we include RN^a in logs, rn^a=LRNA, in the cointegrating relationship; see Figure 2B.

Nonlinear cointegration relationships raise difficult econometric questions:

1. What do we mean by nonlinear cointegration?
2. Can we have a representation theorem for NEC models, similar to Theorem 4 of Escribano and Mira (2002), when the variables are nonlinearly cointegrated?
3. Is it possible that two variables are simultaneously linearly and nonlinearly cointegrated?
4. If this is the case, which of the two cointegrating relationship is better?

The answer to question (i) was given after Definition 2 of Section 2, using the concepts of integration suggested by Escribano and Mira (1996, 2002).

For general nonlinear cointegrating functions, it is difficult to obtain a representation theorem. However, it is possible to give a simple answer to question (ii) for the particular linear and nonlinear cointegrating functions obtained with the UK money demand since 1878. All we have to do is to consider that in Theorem 2 of Section 2, the vector of explanatory variables q_t includes the short-run interest rates in logs(rn^a) as well as in levels (RN^a) and that they only enter in the cointegrating vector one at a time. It is an empirical issue which of two cointegrating specifications is the most appropriate. To answer question (iii), we first estimate both cointegrating vectors with very similar long-run results in terms of goodness of fit. The plots of both cointegrating series in Figures 2A and 2B support the same conclusion. Both pairs of series seem to move together in the long-run but the distance between them varies through time, giving support to the possibility of multiple equilibria and also to the possibility that the adjustment toward the multiple equilibria is not constant. To empirically support those hypotheses, we estimate different NEC models.

First we estimate a NEC model with all of the variables being entered as logs. Therefore, the log variables are linearly cointegrated. The estimation results are given in Model A, col. 2 of Table 7. The NEC model satisfies all the misspecification tests done (see the last rows of column 2 of Table 7). The NEC term is a cubic polynomial in u_{t − 1}, where u_t is the linear cointegrating error. The corresponding absolute value of the t-ratio of (u_{t − 1}− 0.2)u²_{t − 1} is 4.6 and the standard error of the regression model A is 1.71%.³

The equivalent analysis is done with the nonlinear cointegrating residuals U_t. The results are in column 3 of Table 7, where U_t represents the corresponding long-run equilibrium error. Model B of Table 7 shows that the absolute value of the t-ratio of the cubic polynomial error-correction term, (U_{t − 1}− 0.2)U²_{t − 1}, is 5.33 and the standard error of the equation is 1.61%. Therefore, in terms of goodness of fit, the answer is clear: The nonlinear cointegration relationship fits better than the linear one.

The cubic polynomial equilibrium correction of Model B, column 3, of Table 7, is plotted in Figure 3. On the horizontal axis, we have the values of the long-run equilibrium errors, U_t=(m−p−i)_t+ 0.35 + 6.16 RN^a_t, and on the vertical axis the cubic polynomial error-correction term, POL(U_t)=−2.02 (U_t−0.2)U²_t for t=1878–2000.

On the other hand, Model A of Table 7 has very similar nonlinear cubic polynomial adjustment (see Figure 4). The horizontal axis has the values of the other long-run equilibrium errors obtained from u_t=(m−p−i)_t+1.26+0.19 rn^a_t, with the short-run interest rates in logs (linear cointegration), and in the vertical axis the cubic polynomial equilibrium correction terms, POL(u_t)=−2.22 (u_t−0.2)u²_t.

Cubic polynomials are very flexible since they can be thought of as a parametric approximation to any function that is fourth-order continuously differentiable. In fact, for NEC modeling, they represent a very useful tool to identify threshold points. In particular, in the UK money demand since 1878, we saw that there is a continuum of equilibria between 0 and 0.2 and therefore we can identify two threshold points (attractors) τ₁=0 and τ₂=0.2 (see Figure 1, Model A and Figure 3). For more than two threshold points, we would need to consider a higher-order polynomial approximation or the nonparametric approach such as the smoothing splines of Section 3.1. Notice that if we specify an asymmetric error correction for the UK money demand with only one threshold point at τ=0 [see Granger and Lee (1989)], it will be misspecified. This is the reason why only adjustments to negative values of U_t−1 are significant in the Asymmetric NEC Model, column 5, of Table 5. On the contrary, cubic polynomials are robust to having one or two thresholds.

The main caveat of cubic polynomials is that they do not globally satisfy the equilibrium correction condition (stability condition) −2 < df(z,γ)/dz < 0 when z → ∞, although in sample it could satisfy it. Some other unstable reactions can be observed to the right of the zero equilibrium error; see Figures 3 and 4. Therefore, we would like to have a class of parametric models that satisfies the previous four desirable properties of cubic polynomials and the stability condition on df(z,γ)/dz, which is sufficient for z_t to be [phi]-NED; see Theorem 1. Simple nonlinear functions that satisfy those conditions are certain types of the rational polynomials introduced in Section 2. For example,

where if γ₂=− γ₁³ and (γ₃²+γ₄)≠ 0, then f(z=0, γ)=0 is the equilibrium. The adjustment can be asymmetric and/or nonlinear, generalizing Granger and Lee's (1989) asymmetric (piecewise linear) error correction.

If we want the rational polynomials to have multiple equilibria, we might prefer the following functional form:

where if γ₂=−γ₁³; then, f(z=0, γ )=0 and f(z=−γ₃, γ)=0 and f(z, γ) ≈ 0 for all z ∈ (0, γ₃). Therefore, we could have a continuum of equilibria between 0 and γ₃. Estimates of (18) and (19) are given by models C and D of Table 7. Figures 5 and 6 represent the nonlinear adjustments. It is clear that the adjustment is faster the farther we are from U=0, and it is clearly asymmetric. This type of asymmetry is not piecewise linear around a zero threshold point, but is piece wise nonlinear around a set of threshold points around zero.

The equilibrium value (threshold) U=0.2 of Figures 3 and 4 has an interpretation in terms of the role that the dummy variable D2 was playing in the linear money-demand equations of Friedman and Schwartz (1982) and Hendry and Ericsson (1983); see column 2 of Table 1. If we run a cointegrating regression with log-inverse velocity of circulation of money on a constant, the dummy D2 and short-run interest rates (RN^a), then the estimated coefficient of D2 is near 0.2, which corresponds to the positive extreme value (threshold) of the interval of the long-run equilibria obtained by estimating NEC models.

Two-Step OLS Estimation of NEC Money Demand (1878–2000)⁴

These estimation results and the misspecification test done with equations (20a) to (20b) were computed using PcGive 9.0; see Hendry and Doornik (1996).

The best two-step money demand equation estimated for the period 1878 to 2000 is Model B of Table 7, which is written below as equations (20a) and (20b). In what follows, we will evaluate this econometric specification in terms of the stability of the parameter estimates as well as other possible sources of misspecification (autocorrelation, heteroskedasticity, normality, functional form, etc.).

The actual values of (m−p)_t and the corresponding fitted values are ploted in Figure 7 with satisfactory results for such a long period of time.

Step 1: Nonlinear cointegration relationship:

Step 2:

Summary statistics of equation (20b). Absolute values of t-ratios are in parentheses under the coefficient estimates, T=123 [1878–2000], 14 variables, R²=0.87, 100σ=1.61%; AR1-2, F(2,107)=1.93 (p-value=0.15); ARCH(1), F(1,107)=0.25 (p-value=0.62); normality, χ² (2)=0.19 (p-value=0.91); functional forms X_i², F(24,84)=0.8 (p-value=0.72); and X_i*X_j, F(85,23)=0.75 (p-value=0.83); RESET, F(1,108)=0.60 (p-value=0.43); Jt=2.35; and Var=0.34. It is clear that equation (20b) passes all the misspecification tests, including the stability tests of Hansen (1992), (Joint statistic) Jt=2.35, and (variance statistic) var=0.3; see Hendry and Doornik (1996) for a detailed explanation of each test statistic.

The stability analysis of the parameter estimates of this NEC model is complemented with a graphical representation of the standardized innovations in the first graph of Figure 8. The second graph plots the one-step residuals with the 0±(S.E)_t. The third graph of the panel plots the one-step Chow statistics normalized by their one-off 1% critical values. Finally, the last graph represents the breakpoint Chow statistics similarly normalized. The stability of the parameter estimates of the model is clear.

To get stronger support on the empirical constancy of the model, we estimate the main parameters of the model using recursive least squares. The results are included in Figure 9. The eight graphs plot the recursive estimated coefficients plus-or-minus twice the recursively estimated standard errors, and all of the estimated coefficients show a remarkable stability over such a long period of time.

Even if the UK money demand estimated from 1878 to 2000 in equation (20b) behaves satisfactorily, it is possible to get more precise estimates by doing a joint estimation of the short-run and the long-run (cointegration) parameters. This could be especially important for certain types of nonlinearities. To explain this possible source of deviation between the OLS (two-step) and the NLS estimates, we can write equation (1a) as (21b),

Therefore, if the error z_t of the cointegrating relationship (21a) has important nonlinearities or breaks [see Campos et al. (1996) and Arranz and Escribano (2000)], then the OLS estimates in (21a) might not be reliable. In those circumstances, we should expect to have more precise estimates from the joint estimation of equations (1a) and (1c) or (20a) and (20b). The empirical NLS results are reported in equations (22a) and (22b) below.

One-Step NLS Estimation of NEC Money Demand (1878–2000)

The model (22b) fits marginally better than the corresponding two-step OLS contempart since the percentage standard error of the regression (22b) is reduced from σ=1.61% to σ =1.57%.

This money demand model passes all the misspecification tests done at the 5% significance level; see the summary statistics included below equation (22b). It is interesting to comment on the important stability of the long-run parameter estimates obtained during very different long time periods. For example, if we compare the coefficients of the cointegrating vectors from the three different periods (1878–1970), (1878–1993), and (1878–2000), the corresponding results are the following: from Table 5, col. 2, we have (1, α₁)=(1, 7.0); from Table 6, col. 5, we have (1, α₁)=(1, 7.0); and from equation (22a), we have (1, α₁)=(1, 7.3).

Joint estimation (NLS) of equations (17a) and (17b):

Nonlinear cointegration relationship:

Nonlinear error correction:

Summary statistics of equation (22b): Absolute values of t-ratios are in parentheses under the coefficient estimates; T=123 [1878–2000]; 17 variables; R²=0.87; 100σ= 1.57%; AR1-2, F(2,104)=2.43 (p-value=0.09); ARCH(1), F(1,104)=1.21 (p-value=0.27); normality, χ² (2)=0.07 (p-value=0.96); functional forms X_i², F(29,76)=1.03 (p-value=0.44).

Before concluding, we want to stress the three main issues that we identify as the key elements in order to obtain a parameter constant UK money-demand equation from 1878 to 2000. The first one is the inclusion of proxies for deregulation. The second one is the fact that we are using the updatted measure of the opportunity cost of holding money, RN^a, suggested by Ericsson et al. (1998). The difference of this measure with respect to the RN variable used until the year 1970 is clearly seen in Figures B.1 and B.2 of Appendix B. Finally, the third key element for getting parameter constancy is the fact that the equilibrium correction is nonlinear (NEC) around multiple equilibria identified by the interval with two clear threshold extreme values at 0 and 0.2. The cubic polynomial, (U_{t − 1}−,0.2)U²_t−1, error-correction model estimates those threshold values (0, 0.2) in a simple and flexible way since the term U²_{t − 1} measures deviations from 0 while the term (U_{t − 1}−0.2) measures deviations from 0.2.