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One of Denis Sargan's lasting contributions to macroeconometric modelling is the idea of separating the long run from the short run when formulating dynamic relationships between economic variables; see Sargan (1964). The error-correction model has since gained widespread popularity which has been further enhanced by the emergence of the theory of cointegrated variables. In the meantime, the basic error-correction model has been extended by introducing the possibility of the error-correction being non-linear (see Escribano, 1985). The current paper consider non-linear error-correction, parameter constancy and long macroeconomic time series. Hendry and Ericsson (1991) developed a non-linear error-correction model with constant parameters for the UK money demand in 1878–1975. Ericsson, Hendry and Prestwich (1998), henceforth EHP, recently successfully extended the model to cover the years 1878–1993. Their model is a single-equation error-correction model in which short-term dynamics are built around a theory-based long-run equilibrium relationship. The model contains a non-linear error-correction mechanism which is specified following Escribano (1985). This type of model involves higher powers of the disequilibrium term; see EHP. It turns out that non-linear error-correction is a prerequisite for parameter constancy. If it is replaced by an ordinary linear error-correction mechanism, other things equal, the hypothesis of parameter constancy, when tested by an appropriate stability test, is rejected. Non-linearity thus plays a key role in the modelling effort of EHP.
Our aim is to reconsider non-linear error-correction in EHP. Teräsvirta (1998a) recently suggested that the Escribano-type error-correction in EHP may be seen as an approximation to an error-correction characterized by smooth transition regression (STR); see, for example, Granger and Teräsvirta (1993) and Teräsvirta (1998b). We shall examine this possibility by generalizing the error-correction mechanism in EHP directly using the STR framework. Sarno (1998) applied a similar idea when modelling Italian money demand using long annual (1861–1991) time series. Alternatively, one may start with a step back and model the non-linearity in the data with an STR model, applying the modelling cycle described in Teräsvirta (1998b). This implies that we begin with a model with linear error-correction and test linearity against STR. If it is rejected, we find the appropriate transition variable by a specification search. It will be seen that this procedure leads to an error-correction mechanism different from that in EHP. The results of the two approaches are compared with each other and with the Escribano-type model in EHP. Encompassing tests, among other things, are applied for this purpose. It appears from the results for the two STR models that the STR-based error-correction is an improvement over the specification in EHP.
The outline of the paper is as follows. Section 2 reviews the economic theory EHP applied and reminds the reader of their dummy variables. Section 3 introduces the statistical model and discusses the actual modelling, and Section 4 presents conclusions.
2 ECONOMIC THEORY, THE DATA AND EHP'S MODEL
EHP pointed out that in a modern economy, money may be demanded as an inventory to smooth differences between income and expenditure streams but it also forms an asset in a multi-asset portfolio. These demands yield a classical long-run money demand specification Md = g(P, I, ΔP, R) where Md is the nominal money demanded, P is the price level, I is the income variable, ΔP is inflation and R is a vector of rates of returns of a set of assets. EHP specified a log-linear version (interest rates in levels) of Md as
where Rown and Rout are the own and the outside rates of interest, and lower-case letters denote logarithms. Equation (1), when completed with an error term and estimated without inflation, defines the long-run relationship, whose linearity is not called into question. Non-linearity in this article concerns the strength of the attraction, not the form of the attractor itself.
In this paper we use the same data set as EHP. Their article contains a detailed discussion on the definitions and quality of the time series. The model we focus on is the non-linear error-correction equation (20) in EHP. It has the form
where the long-run disequilibrium relationship, ũt, is defined as
For details concerning the estimation of the long-run, see EHP. Furthermore, is the logarithm of the opportunity cost of holding money, whereas rlt is the logarithm of the long-term nominal interest rate RLt (for definitions see the Appendix). The definitions of the dummy variables also appear in the Appendix.
3 RECONSIDERING THE ERROR-CORRECTION MECHANISM
3.1 Smooth Transition Regression Model
As mentioned in the Introduction, one possibility in reconsidering the money demand equation of EHP is to follow the modelling strategy outlined in Teräsvirta (1998b). We begin by briefly introducing the STR model. It is defined as
where et ∼ nid(0, σ2), xt = (1, yt − 1, …., yt − k;z1t, …. zmt)′ = (1, x̃′t)′ with p = k + m is the vector of stationary explanatory variables, some of which may be linear combinations of nonstationary variables. Furthermore, φ = (φ0, …, φp)′ and θ = (θ0, …, θp)′ are parameter vectors. G(st; γ, c) is the transition function which is continuous in st and bounded between zero and unity. The transition variable st is either stationary or a deterministic time trend (t). The transition function of a kth-order logistic smooth transition regression, LSTR(k), model is
where k = 1 yields the LSTR(1) and k = 2 the LSTR(2) model. These are the two parameterizations of transition functions that are considered in this paper. For an exhaustive description of STR models we refer to Granger and Teräsvirta (1993) and Teräsvirta (1994, 1998b).
Testing linearity against the alternative of an LSTR(k) model amounts to testing if γ = 0 in equation (3). The model is not identified under the null hypothesis due to the nuisance parameters θ and c. A Taylor series approximation about γ = 0 is used as a substitute to circumvent this problem, and the tests are based on this transformed equation:
where , but under H0. The null hypothesis H0: γ = 0 in equation (2) implies H′0: β1 = β2 = β3 = 0 within equation (4) because where is a function of the parameters in the original STR specification. In order to decide between k = 1 and k = 2, one continues by testing a sequence of subhypotheses within equation (4). This sequence and the decision rule are given in Table III. Tests of parameter constancy against smoothly changing parameters are based on the same idea while assuming st = t; see Lin and Teräsvirta (1994) or Teräsvirta (1998b).
3.2 Testing linearity and specifying an STR model
As mentioned in the Introduction, we begin the modelling with a linear equation containing, with a single exception, the same variables as EHP included. This is in line with our stated goal, which is to see whether we can improve their specification of the EHP non-linear error-correction mechanism. We test linearity of that equation against STR and, if rejected, specify and estimate an error-correcting STR model for the money demand. In order to form the linear baseline model, equation (ehp) is modified in two ways. First, the term is replaced by the straightforward lag Δ1(m − p)t − 2 because that slightly improves the fit. Second, the non-linear error-correction term in (ehp) is replaced by ũt − 1. Note that ũt − 1 is estimated directly from the stochastic equivalent of equation (1) after excluding the inflation (the price level is assumed to be I(1)). The ensuing statistical inference will thus be conditional on the parameter estimates of equation (1). Estimation of the linear model yields
The errors of model (5) are normal (LJB is the Lomnicki–Jarque–Bera test of normality; the p-value in parentheses). While Table I shows that the errors of equation (5) are not serially correlated, the model does not seem to have constant parameters. Rejection of parameter constancy against smoothly changing parameters using the tests discussed in the previous section (Teräsvirta, 1998b) is very strong, as the results in Table II indicate. We also performed the Hansen (1992) coefficient stability test against the alternative that the parameters of model (5) are stochastic random walk processes.1 The test rejects the constancy of all parameters at the 20% and the constancy of the error variance at the 10% level of significance; see Table I. If these test results are compared with those obtained for (ehp), Tables I and II, it seems that introducing a non-linear error-correction mechanism is a step towards a model with constant parameters.
Table I. Diagnostic statistics
Maximum lag q
Hansen's test statistic
Notes: (a) p-values of the LM test of no error autocorrelation against an AR(q) and MA(q) error process, and (b) values of the Hansen (1992) statistic of structural stability against the alternative that all parameters are stochastic random walk processes (H92A) and the alternative that the error variance is a random walk (H92V), for equations (ehp), (5), (6) and (7) for the UK money demand, 1878–1993. Hansen's test: * significant at the 20% level; ** significant at the 10% level; *** significant at the 2.5% level of significance.
Table II. p-values of parameter constancy tests of the equations (ehp) and (5) for the UK money demand, 1878–1993, against STR type non-constancy
Parameter constancy test
Notes: The null hypotheses are:
(1)All parameters except the coefficients of the dummy variables are constant
(2)Coefficient of the error-correction component is constant
(3)Coefficients of the error-correction terms are constant
(4)The intercept is constant
The remaining parameters not under test are assumed constant in each case
LM3 is the test of hypothesis β1 = β2 = β3 = 0 in (4)
LM2 is the test of hypothesis β1 = β2 = 0|β3 = 0 in (4)
LM1 is the test of hypothesis β1 = 0|β2 = β3 = 0 in (4)
when st = t
Linearity tests in Table III show that linearity is rejected against STR for four potential transition variables. One of them is ũt − 1, which given the results of EHP is not surprising, and and , cause a rejection as well. Furthermore, the specification test sequence with ũt − 1 as the transition variable leads to an LSTR(2) model as H03 is rejected more strongly than H02 and H04. In accord with this outcome the same sequence with as the transition variable suggests an LSTR(1) model. Nevertheless, the rejection of linearity is strongest for Δ1it, so that we first construct an LSTR model for Δ1(m − p)t with Δ1it as the transition variable. The specification test sequence in this case does not unambiguously point at either one of LSTR(1) and LSTR(2), and we tentatively select the higher-order one of the two.
Table III. p-values of linearity tests in model (5) for the UK money demand, 1878–1993, for a set of transition variables
Linearity test Null hypothesis
Δ1pt − 1
ũt − 1
Δ1(m − p)t − 1
Δ1(m − p)t − 2
Notes Specification sequence : H04:β3 = 0, H03:β2 = 0|β3 = 0, H02:β1 = 0|β2 = β3 = 0. Choose k = 2 in if the p-value of the test of H02 is the smallest of the three, otherwise choose k = 1.
As for equation (ehp), the results in Table I reveal some evidence of autocorrelation in the errors. On the other hand, it is seen from Table II that the parameters in the conditional mean appear constant, whereas constancy of the error variance is rejected at the 2.5% level of significance; see Table I. We also tested linearity of (ehp) for comparison and obtained some strong rejections, but for space reasons those results are not reported.
Next we estimate an LSTR(2) model with Δ1it as the transition variable. The result, after removing some insignificant variables, is
where is the sample variance of the transition variable. The residual standard deviation of (6) is 82% of that of (5) while the corresponding ratio for (ehp) is 93%. The errors of (6) appear normal and the LM tests of no conditional heteroscedasticity do not cause any alarm. Results of misspecification tests can be found in Table I. The errors of (6) are free from autocorrelation. As for the Hansen (1992) test, the cumulative scores are obtained from the estimated individual scores of the log-likelihood of the STR model. The model also passes the test of no additional non-linearity and parameter constancy (Eitrheim and Teräsvirta, 1996; Teräsvirta, 1998b), but for space reasons the results are not reported here in detail.
The transition function of (6) as a function of the transition variable Δ1it is depicted in Figure 1. The transition between the extreme regimes is rather smooth. Broadly speaking, the error-correction mechanism is only operating when the growth rate of the economy (real income) is sufficiently high. There are, however, some very low (< −0.04) values of Δ1it, for which the local dynamics of the system are the same as for high growth rates. These correspond to 1931 and the rather exceptional years following the end of the First World War. Figure 1 suggests that downweighting those observations while specifying the STR model is likely to lead to an LSTR(1) model. Against this background it is not surprising that the model-selection test sequence does not offer a clear choice between the LSTR(1) and LSTR(2) alternatives. According to (6) the short-term dynamic behaviour of real money is also a non-linear function of the growth rate of the economy. The impact of inflation on real money increases with the growth in nominal income, as does the negative impact of the long interest rate.
3.3 Direct Generalization of the Error-correction Component in EHP
In this section we consider the possibility of modelling the error-correction mechanism by a straightforward generalization of (ehp). As mentioned above, EHP formulated their non-linear error-correction mechanism following Escribano (1985). This formulation may be seen as a first-order approximation to an STR-type parameterization in which the error-correcting variable ũt − 1 itself is the transition variable; see Teräsvirta (1998a). Indeed, the results of linearity tests in Table III show that linearity is also rejected against STR when ũt − 1 is the transition variable.
We specify and estimate a corresponding STR model for Δ1(m − p)t. It is seen from Table III that the specification sequence points at an LSTR(2) model, as one might expect from EHP. However, the estimation yields a model in which the estimate of c1 is so small and far outside the observed range of ũt − 1 that the observed transition function has value zero for the lowest value of the transition variable (for LSTR(2), the value of the transition function approaches unity as the value of the transition variable becomes sufficiently small). Thus we estimate an LSTR(1) model which has the form
where is the standard deviation of the transition variable. The residual standard deviation of (7) is 84% of that of equation (5), so that (7) fits better than (ehp). Normality of the errors is close to being rejected at the 5% significance level due to observations 1936 and 1981. (If these are dummied out, we have LJB = 2.2 (0.34).) The tests of no ARCH do not indicate any model misspecification. The errors are not autocorrelated (Table I), which is an improvement upon (ehp). On the other hand, the test of no additive non-linearity (not reported) rejects the null hypothesis when Δ1it is transition variable. This suggests that equation (6) characterizes the non-linearity of the error-correction mechanism better than (7). The short-term dynamic structure of the model closely resembles that of (ehp). The only major difference is the negative impact of the long interest rate on the change in money demand: it is non-linear even according to (7), almost non-existent for low values of the transition function and increasing in strength with ũt − 1. The graph of the transition function of (7) in Figure 2 shows that the transition is smooth.
As model (7) offers yet another way of characterizing the error-correction mechanism in the UK demand for money, it is interesting to look at differences between the equations. Interpreting the workings of the error-correction term in (6) is not straightforward, as the error-correcting combination contains the level of income while its annual change is the transition variable. To obtain an idea of the contribution of the error-correction term to the dependent variable, we have graphed the values of the error-correction component over time for equations (ehp), (6) and (7) in Figure 3. Note that in both (6) and (7), the intercept in the non-linear part of the equation is included in the error-correction mechanism as it acts as a counterbalance to the change in the coefficient of ũt − 1 as a function of Δ1it (equation (6)) or ũt − 1 itself (equation (7)). It is seen that in the beginning of the observation period (6) error-corrects more than the other two models. Between 1920 and 1960 the error-correction components in (ehp) and (7) remain practically constant. The largest differences between the models occur after 1973 when the estimation period in Hendry and Ericsson (1991) comes to an end. The wide amplitude of the error-correction component in (ehp) is the most conspicuous feature of Figure 3. The amplitude of the error correction component in (6) during that period is just about one-third of that of (ehp). The same component of (7) has a large negative value in 1975, otherwise it follows that of (6). Should the error-correction be considerably stronger from 1973 onwards than elsewhere in the sample remains a question without a definite answer, as the true model is not known.
3.4 Results of the Encompassing Tests
The LSTR(2) model (6) has the best fit of the three non-linear error-correction models considered here. It would therefore be interesting to see if (6) wholly explains the results obtained by the other two models. In order to consider that possibility, we could test each of the other two models against (6) and vice versa, using non-nested tests. Several tests exist; for a recent review, see Pesaran and Weeks (2000). In connection with money demand equations, the tradition has been to construct the so-called Minimal Nesting Model (MNM) nesting the two alternative models (see Mizon and Richard, 1986), and carry out the testing within that model. In this case, the MNM is an additive STR model (Eitrheim and Teräsvirta, 1996; Teräsvirta 1998b) with two transition functions:
where the linear part is split into two components. Vector x1t contains the variables that appear in the linear part of all of the three equations, variable x2t = Δ1(m − p)t − 2 is only present in the linear part of equations (6) and (7), whereas vector contains the variables that only appear in model (ehp). Furthermore, G1(Δit; γ1, c1) is the transition function of equation (6), and G2(ũt − 1; γ2, c2) that of (7). Vectors x4t and x5t contain the variables present in the linear combination of variables included in the non-linear part of the two STR models. They are combinations of ordinary exclusion restriction and linearity tests, in which the identification problem is solved by the standard Taylor expansion approximation of the transition function(s).
The test results can be found in Table IV. We do not report results on testing this equation against a combination of (ehp) and (7), because the latter models have a rather similar error-correction mechanism, that of equation (ehp) being in a way an approximation to the one in the LSTR1 model (7). The main outcome is that having the growth rate determining the strength of the error-correction is essential. Neither of the other two equations comes close to encompassing model (6). On the other hand, even if this model fits better than the other two, it does not encompass (ehp). Thus the latter equation does capture some features in the data that are not fully explained by model (6). One might even argue in favour of a combination of equations (6) and (7), which would imply a multiple LSTR model. But then, in that case overfitting could already emerge as a counterargument, as the total number of observations in the time series is just 116.
Table IV. Results of the Simplification Encompassing Tests (F-values, p-values and degrees of freedom) of models (ehp), (6) and (7) for the UK money demand, 1878–1993, based on equation (8)
H0[(7) against (ehp) + (6)]: α3 = 0, γ1 = 0 in (8)
[2 × 10−4]
[1 × 10−3]
[1 × 10−4]
In this article we show how viewing the Escribano-type error-correction as an approximation to a specific STR-type parameterization as Teräsvirta (1998a) suggested leads to the STR model (7) which encompasses the EHP equation. On the other hand, modelling the UK money demand with the same variables as those EHP used, but adopting a more general non-linear approach leads to another STR model (6). This model variance dominates the others and the results in the previous section do suggest that the growth rate of the UK economy has been an important factor in explaining the fluctuations in the UK money demand during the long time-span considered in EHP.
Finally, we should like to point out that the present choice of non-linearity is just one of many potential alternatives. It does not follow that all the other non-linear specifications would necessarily be inferior to the STR model. It may be concluded, however, that in this application, STR-based specifications provide a useful way of refining previous models of the demand for broad money in the UK.
The first author acknowledges support from the Swedish Council for Research in the Humanities and Social Sciences. The research of the second author has been supported by Jan Wallander's and Tom Hedelius's Foundation for Social Research. An earlier version of this paper was presented at the EEA98 conference, Berlin, September 1998. We are grateful to David Hendry and Neil Ericsson who generously allowed us to use their data set. We also wish to thank both for stimulating discussions and the three anonymous references and the editor (Hashem Pesaran) for insightful comments that have improved the presentation. The responsibility of any errors or shortcomings in this article remains ours.
APPENDIX: LIST OF VARIABLES (all variables concern the United Kingdom)
Broad money stock, £million, series obtained by splicing
Broad money stock, £million, actual series
Real net national income, £million in 1929 prices
National income price deflation, P = 1.00 in 1929
Short-term interest rate (fraction)
Long-term interest rate (fraction)
‘High-powered money’ (money earning a higher interest than the short-term interest rate RS), series obtained by splicing
‘High-powered money’ (money earning a higher interest than the short-term interest rate RS), actual series (no splicing)
Opportunity cost of holding money: RNa = (Ha/Ma)RS
Zero-one dummy variable for First World War
Zero-one dummy variable for Second World War
Zero-one deregulation dummy variable, non-zero for 1971–1975
Zero-one deregulation dummy variable, non-zero for 1971–1975 and 1986–1989
Lower-case letters denote logarithms. For more detailed definitions of the variables, see EHP.