Computational Economics 7: 23-35, 1994 (~ 1994 Kluwer Academic Publishers. Printedin the Netherlands.

Cointegration Tests on MARS P E T E R S. S E P H T O N Department of Economics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 (Received: June 1992) Abstract. Multivariate adaptive regression spline (MARS) models due to Friedman (1991) are

employed to examine non-linear cointegration. Critical values of the Dickey-Fuller cointegration test statistics, appropriate to the MARS model, are presented. Several empirical examples demonstrate the gains to the non-linear modelling of economic time series.

I. In~oducfion Engle and Granger (1987) provided a set of residual based tests of whether two or more series that appear to be random walks were in fact tied together by a long-run linear relationship. Since this seminal work there has been an explosion of empirical research on both testing for cointegration and in modelling cointegrated series, a Relatively rudimentary residual based tests have been extended and replaced, in great measure, by those based on systems methods of estimation. The purpose of this paper is to provide yet another extension to the Engle and Granger test to examine the extent to which non-linear cointegrating relationships can be identified and examined. Attempts to model non-linear cointegrating relationships have already begun to appear in the literature. Meese and Rose (1991) used the relatively simple ACE (alternating conditional expectations) algorithm of Breiman and Friedman (1985) to compare models of exchange rate determination, while Chinn (1991) used A C E to estimate a monetary model of the exchange rate. The ACE algorithm searches for data transformations that maximize the R -~ between the transformed series. Granger and Hallman (1991) and Hallman (1991) employed ACE in an attempt to identify non-linear cointegration, and suggested that ACE may be capable of capturing time-varying cointegration. Granger and Hallman (1991) provided some experimental evidence on the behaviour of the Dickey-Fuller and Augmented Dickey-Fuller cointegration tests based on ACE residuals, but their Monte Carlo simulations used only 100 replications on 100 data points, after which they applied ACE to a dataset of over 400 observations. The primary aim of this paper is to use the more sophisticated MARS algorithm to examine non-linear cointegration, since there may be stable long-run equilibrium relationships that escape additive analysis. MARS allows for both additive

24

P.S. SEPHTON

and interactive effects: that is, it is capable of identifying the (possibly non-linear) individual and interactive effects of a number of series on a dependent variable. ACE, on the other hand, only investigates the additive relationships between variables. Since the appropriate functional form is rarely identified by economic theory, it is reasonable to suggest that MARS is a sensible estimation strategy. Separability and additivity need not necessarily be the proper assumptions under which to examine the data. The classic example is of a liquidity preference schedule, which is almost always specified in natural logarithms, with variations on income as the scale variable 2 and interest rates as the opportunity cost variable. The transactions and precautionary motives are assumed to be captured through the scale effect, while the interest rate demonstrates the speculative motive, yet we know the money demand function cannot be innocuously separated in this manner. 3 MARS would allow the estimation of the appropriate functional form and variable interactions, demonstrating that it may be preferred to traditional linear models, and purely additive non-parametric methods. The MARS model is presently capable of analysis on a single dependent variable, and as such, univariate residual based tests of cointegration will be investigated rather than their multivariate counterparts. In addition, while it is well known that the Engle-Granger 'two-step' testing procedure is outdated and neglects short-run dynamics that may be of interest in finite samples, the methods here represent a first step in the application of the MARS model to cointegration analysis, and for this reason, the two step approach will be the vehicle used to test for cointegration. Subsequent extensions of the process to multivariate dynamic models incorporating error correction mechanisms will be the subject of future work in this area. One might question whether it is appropriate to assess the order of integration of untransformed series, and then attempt to find a non-linear attractor to which they are drawn. However, Hallman (1991) has shown that non-linear transformations of integrated series can be cointegrated. Thus it of some interest to determine whether tests of non-linear cointegration identify long-run behaviour that escapes linear analysis. A second object of the paper is to provide a set of critical values to the Dickey-Fuller cointegration tests for use in subsqeuent applications of the MARS model to economic data. The outline of the paper is as follows. The next section provides a brief summary of the MARS algorithm, its ANOVA representation, and model selection. This is followed by a review of the Dickey-Fuller cointegration test. Critical values of the test are derived for a number of MARS models. Applications to an aggregate production function, short-term and long-term interest rates, and spot exchange rates demonstrate the benefits to MARS modelling. Final remarks follow.

COINTEGRATION TESTS ON MARS

25

II. The M A R S Model The issues of model specification versus estimation were argued separately by Fisher and Pearson in the early 20th century, with Fisher the clear winner of the debate. Pearson was inclined towards the specification question, whereas Fisher believed that estimation should be the researcher's paramount concern. Econometrics has progressed primarily in the estimation direction, with the specification issue taking a relatively cursory role in data analysis. Non-parametric regression takes the alternate view that the appropriate specification of a model is of vital import, and to that end, the data is allowed to dictate functional form. Used in conjunction with economic theories that imply certain types of behaviour, for example over concavity and homogeneity, nonparametric regression can advance our understanding of non-linear economic behaviour.4 There are a number of attractive non-parametric regression models, the most recent introduced by Friedman (1991a, 1991b). For an exhaustive description of the algorithm the reader is referred to those papers. What follows is an attempt to provide an intuitive review of the modelling strategy. The multivariate adaptive regression splines (MARS) model uses an extension of recursive partitioning regression to construct a set of truncated power basis functions using cubic spline approximations to the data.5 For example, when modelling the relationship between a predictor x t and the dependent variable Y t , a general model might take the form y~ = a I + a 2 x t + a 3 x 2 + . . . + aM X M - 1 + e t

(1)

Here, Yt is a polynomial in xt. This can be written in a more general form, as M

Y t = ~'~ a k B k ( X t ) + et

(2)

k=l

where B k ( x t ) is the k-th basis function of x r These basis functions can be highly non-linear transformations of x , but note that yt is a linear (in the parameters) function of the basis functions. The parameters a k can be chosen on the basis of a sum of squared residuals metric. The advantages to MARS are in its ability to estimate the basis functions so that both the additive and the interactive affects of the predictor space are allowed to determine the response variable. With piecewise polynomial approximation, the predictor space is broken into sub-regions with knots separating each region. A cubic polynomial spline function is fit to the data within that region. Summing over the predictor space yields a set of spline functions that form the fitted model. Johnston (1984, p. 392) provides a pedagogic example of the application of spline functions in economics. MARS recognizes that spline fitting routines are sensitive to knot location, and

26

P.S. S E P H T O N

employs a truncated power basis function to fit a cubic polynomial to the series. Knots are chosen on the basis of a multiple regression that is analogous to retaining knots which reduce the sum of squared residuals. For example, with a single predictor the sum of squared residuals would be

Y, - Z bjx j - Z ak(Xt -- tk) q i=1

j=O

(3)

k=l

where bj and a k are multiple regression coefficients on cubic (q = 3) splines of x,, and x, relative to knot location t k. Note that the notation ( x ~ - tk) q includes the cubic spline of x t relative to knot location t k if the difference is positive, and otherwise zero. From (3) it is clear that the addition of a knot can be viewed as adding the corresponding variable ( x t - tk)q+. A forward and backward stepwise search is incorporated in the algorithm, with the forward step purposely overfitting the data. Insignificant terms are deleted on the backward step of the routine. Another benefit to the MARS algorithm is that the parent region is retained after it has been split into its sibling regions. This has the advantage of producing overlapping regions, without gaps, should a sibling be excluded after it has been determined to be insignificant. Many other non-parametric routines are constrained so as to disallow overlapping regions. Model selection can be based on N-fold cross-validation, or alternatively, a modification of the generalized cross-validation (GCV) criterion of Craven and Wahba (1979). The GCV can be expressed as N

GCV = ( l / N ) ~ ([y,-fM(xt)]2/[1 - C ( M ) / N ] 2}

(4)

t=l

where there are N observations, and the numerator measures the lack of fit on the M basis function model fM(X,), and denominator contains a penalty for model complexity, C(M). F o r present purposes, the GCV will be used for model selection: the model with the lowest GCV score is automatically selected for subsequent analysis.

INTERPRETATION

MARS estimates can most readily be interpreted from the ANOVA representation of the model (5), where the fitted function is a linear combination of additive basis functions in single variables denoted by f(xi) and interactions between variables, denoted by fij(xi,, xjt), fijk(X,, X#, Xk~), and so on.

Yt = f i G ) + e, where

(5)

27

COINTEGRATION TESTS ON MARS

f(xt) = ao + •

KM=I

fi(xit) + E

KM=2

fij(xit, xjt) +

jk(x,, xj,, xk,) + " ' . KM=3

Each f~(xi,) is a spline representation of a univariate function. A plot of f~(x,) against x, would demonstrate the optimal transformation of the series x,. This graph illustrates the degree to which to optimal transformation is non-linear, a major benefit to non-parametric regression. The summations over the K M denote additive (KM = 1), bivariate interactive (KM --2), and higher order interactions, respectively. For interactions involving two variables a surface can be derived to illustrate the joint effects of the series on the endogenous variable. Beyond bivariate designs the models can be 'sliced' to provide a set of surfaces that show the contribution of the two series to a smooth of the dependent variable on all variables in the system. The choice of interaction terms can be made through a comparison of the low order (K~ = 1) and high (KM 1>2) order models. Friedman suggests a comparison of the generalized cross validation scores, with a model involving interaction terms chosen over an additive model only if its GCV score is lower. As part of the M A R S output, the relative contribution of each variable is determined, as are estimates of the model's GCV given that a particular ANOVA function (variable) has been omitted from the model. This assists in interpreting the significance of each ANOVA function. MARS has been extended to incorporate categorical variables, logit regression, and missing data, and has been applied to the estimation of autoregressive threshold models (Lewis and Stevens (1991)), and exchange rate prediction (Sephton (1992)). While these extensions might have important applications in other fields of economics (transportation, consumer theory, industrial organization, natural resources), our purpose is to demonstrate how MARS can be used to investigate non-linear cointegration. The next section reviews the DF and A D F cointegration tests. This is followed by examples demonstrating that MARS models adequately capture non-linear behaviour in a number of cases.

III. Engle-Granger Cointegration Tests Engle and Granger (1987) proposed the use of Dickey-Fuller (DF) and Augmented Dickey-Fuller (ADF) tests. If two or more series are integrated of an identical order, the DF and ADF tests will determine whether the residuals contain a unit root: if they do, the original data cannot be linearly cointegrated, since the error between them is not stationary. The DF and A D F tests are constructed from three regressions. In a bivariate model the first, the cointegrating regression (CR), examines the integrated series Yt and xt, where a constant (and time trend) may be included:

28

P.S. SEPHTON Yt = a +/3xt + e,

(6)

The second regression constructs the DF test as a regression of the first difference of the fitted residuals from (6) on the lagged residual and a constant: A E t = 6 -]- ")/Et- 1 q- r e s i d u a l

(7)

To account for serial correlation in the residuals, the ADF test adds p-lagged changes in the residuals to equation (7) P

A~t = • + 7~t_ 1 + ~ A~t_1 + residual

(8)

i=1

The t-statistics on the fitted 3' in (7) and (8) provide the DF and ADF tests of a unit root in the CR residuals: if the null of a unit root is not rejected, the series are not linearly cointegrated. Critical values of these tests were provided by Engle and Granger (1987) for the bivariate case and 100 observations: Engle and Yoo (1987) for the multivariate case and 50, 100, and 2000 observations, and MacKinnon (1991) provides response surface estimates that can be used to obtain critical values for the multivariate model (over any number of observations).

CRITICALVALUES ON MARS Critical values of the DF and ADF tests using the MARS algorithm depend on a number of factors. The first is related to whether the model is additive or whether it allows for both additive and interaction effects. The test statistics will most certainly be different for additive and interaction models, so critical values must be constructed for both cases. The second factor affecting the test statistics is the maximum number of basis functions allowed in the estimation process. Clearly the higher the setting, the greater the chance of a very good fit to the data, and hence the more stationary the residual will appear. In many ways this is akin to controlling for degrees of freedom in a linear regression: the lower the degrees of freedom, the better the fit. The critical values of the DF and ADF tests based on MARS residuals should be constructed for a wide range of settings of the maximum number of basis functions. Other factors affecting the test statistics include the number of variables in the system, and sample size. For present purposes the dimensions of the models are set from two variables to five variables. Tests are constructed for a sample size of 2000 observations so that critical values approximate their asymptotic limits. Since Gregory (1991) has shown that DF and ADF small sample test values are sensitive to a number of nuisance parameters and departures from the classical regression model, only the 2000 observation test values will be reported. Computational considerations relegate a response surface approach to simulating critical values to subsequent analysis.

29

COINTEGRATION TESTS ON MARS

To simulate the critical values of the DF and ADF tests, IMSL routines CHFAC, RNSET and RNMVN were employed to create from two to five variables with a covariance matrix taken to be the identity matrix. As Engle and Yoo (1987) note, the test statistics will be the same for any value of the covariance matrix. The data are generated by X t = X t _ 1 "~- 12t, X 0 =

with

vt ~

IN(0,

IN)

0,

,(9)

X t = ( X l t , X2t , X3t , X4t , X5t )

and the CR takes the form of equation (10)

X l t = ~ 0 "~- ~ I X 2 t -~- ~ 2 X 3 t ql- ~ 3 X 4 t -~- ~ 4 X 5 t -~

residual

(10)

Since the asymptotic distribution of the ADF test coincides with the DF test, and we are not interested in examining the powers of the tests, this data generating mechanism will be used in the simulations. Tables 1 and 2 contain the one, five, and ten percent critical values of the DF and ADF tests for the additive and (two-variable) interaction models based on one thousand replictions, omitting the negative signs for convenience. As Davidson and MacKinnon (1993, Ch 21) note, with one thousand replications, the length of a 95% confidence interval on the estimated size of the test statistics can be determined to be 0.12 at one percent, 0.027 at five percent, and 0.037 at the ten percent level.6 As is the case in the linear DF and ADF critical values, the larger the system, the larger (in absolute value) are the critical values. There is also a clear Table 1.

Critical test values for additive model

Number of Variables

Maximum Number of Basis Functions

1%

5%

10%

2 2 2 2

5 10 15 20

4.909 5.094 5.334 5.296

4.043 4.462 4.588 4.618

3.703 4.115 4.157 4.323

3 3 3 3

5 10 15 20

5.145 5.886 6.008 6.629

4.491 5.281 5.421 5.654

4.147 4.924 5.119 5.311

4 4 4 4

5 10 15 20

5.236 5.784 6.582 6.561

4.759 5.298 5.702 6.034

4.467 4.993 5.373 5.625

5 5 5 5

5 10 15 20

5.525 6.311 6.949 7.219

4.866 5.687 6.263 6.499

4.635 5.397 5.905 6.148

30

P.S. SEPHTON Table 2.

Critical test values for interactive model

Number of Variables

Maximum Number of Basis Functions

1%

5%

10%

2 2 2 2

5 10 15 20

4.909 5.094 5.334 5.296

4.043 4.462 4.588 4.618

3.703 4.115 4.157 4.323

3 3 3 3

5 10 15 20

5.145 5.886 6.008 6.629

4.491 5.281 5.421 5.654

4.147 4.924 5.119 5.311

4 4 4 4

5 10 15 20

5.236 5.784 6.582 6.561

4.759 5.298 5.702 6.034

4.467 4.993 5.373 5.613

5 5 5 5

5 10 15 20

5.577 6.519 7.139 7.333

4.964 5.906 6.535 6.775

4.721 5.626 6.193 6.458

difference between the critical values associated with the interactive versus the additive models once the system extends beyond four variables For low order systems there does not appear to be a significant difference in the distribution of the test statistics between the additive and interaction models. This is most likely a function of the data generation process.7 As expected, the choice of maximum number of basis functions affects the critical values. This result is intuitively pleasing, since the larger the number of basis functions the better the fit, and the greater is the potential for their erroneous inclusion in the simulation experiment. IV. Applications Three examples will demonstrate the potential benefits to MARS modelling of non-linear cointegrating relationships. As will be shown, in some cases the inferences drawn from non-linear modelling are identical to those based on the traditional linear regression approach to cointegration, yet the cointegrating relationships are significantly different. This might suggest that in some applications, cointegration tests are insensitive to functional form. In two cases, inferences drawn from linear analysis are at odds with those based on a non-linear analysis of the data, suggesting there are merits to MARS modelling.

AGGREGATE PRODUCTION FUNCTION T h e first a p p l i c a t i o n o f M A R S is to th e C h r i s t e n s e n an d J o r g e n s e n (1970) d a t a o n a n n u a l U S o u t p u t , capital and l a b o u r inputs, s p a n n i n g 1929 to 1967. M a d d a l a

31

COINTEGRATION TESTS ON MARS

(1992) provides a paedological analysis of the data, using a double log transformation. Assuming natural logarithms to be the appropriate transformation of the data, it is possible to show that all series are I(1) at or about the five percent level of signficance. The OLS based DF cointegration test is -3.657 relative to an asymptotic five percent critical value of -3.743. This suggests the series are not cointegrated, since the CR residuals contain a unit root. When MARS is applied to the untransformed data, the best MARS model is interactive and allows a maximum of ten basis functions, of which seven are retained. The MARS DF test is -5.774, relative to a five percent critical value of -5.2809. This indicates the series are cointegrated, since the MARS residuals do not contain a unit root. Figure 1 illustrates the optimal transformations of the capital and labour inputs, with the surface contour clearly demonstrating diminishing returns to labour, given a value of the capital stock. It also appears that the transformation is not logarithmic. For this data set, the question of whether the sereis are cointegrated yields different inferences across the 'linear' and the non-linear methods, and it is clear that the optimal transformations are not those typically employed in the area.

SHORT AND LONG INTEREST RATES

Engle and Granger (1987) found 20 year and one month interest rates were not cointegrated, using monthly data spanning 1952 to 1982. Granger and Hallman (1991) used ACE to demonstrate that there was evidence of a non-linear cointegrating relationship between short term and long term interest rates that escaped linear tests of cointegration. Here the US long-term government bond yield (over 10 years) and the 30 day commercial paper rate are examined from January 1969 to November 1992. Taking the long rate as the dependent variable, the linear DF cointegration test leads to a calculated value of -2.375 relative to a five percent critical value of -3.34. The best MARS model allows a maximum of

Capital Labour

\

Fig. i.

Aggregate production function surface.

32

P.S. S E P H T O N

20 basis functions of which 7 are retained in estimation. The MARS cointegration test yields a DF value of -3.23 relative to the five percent value of -4.6179. The results indicate the absence of both linear and non-linear cointegrating relationships. However, Figure 2 demonstrates the optimal transformation of the short rate on the long rate, and it is clearly non-linear. This suggests that functional form may not be of vital importance in the construction of cointegration tests, since both linear and non-linear analysis leads to the same inference.

SPOT E X C H A N G E R A T E S

There have been a number of recent studies into whether the natural logarithms of spot exchange rates are linearly cointegrated. Baillie and Bollerslev (1989) find evidence of one cointegrating vector in a system of seven daily exchange rates, while Hakkio and Rush (1989) have shown that monthly spot exchange rates are not cointegrated. Sephton and Larsen (1991) demonstrated the sensitivity of these findings to sample period, with a 'rolling' test approach indicating periods during which cointegrating might be suggested by the data. The final example of non-linear cointegration will employ the Baillie and Bollerslev (1989) daily 6.4

5.6

4.13

4,0

3.2

2.4

1.6

0.8

0.0 1

25

49

73

97

121

14S

lfi9

193

217

0bservQtion Fig. 2.

Transformed short interest rate.

241

265

COINTEGRATION TESTS ON MARS

33

Franc~ 0

Fig. 3. Spot exchange rate surface: Deutschemark and Franc. exchange rate data on four currencies: the British pound, the Deutschemark, the Yen, and the French Franc. Taking the logarithms of the data, it can be shown that all series are I(1) at reasonable levels of significance. The cointegrating regression (normalized on the British Pound) yields residuals that have a D F test value of -1.4963 relative to a five percent critical value of - 4 . 1 0 . Hence the traditional linear analysis suggests that the four currencies are not cointegrated, since the CR residual contains a unit root. The best M A R S model allows a maximum of 20 basis functions of which 18 are retained in an interactive model. Two surfaces presented in Figures 3 and 4 demonstrate the optimal transformations of combinations of the Deutschemark

Franc

~ ~I Fig. 4. Spot exchange rate surface: Yen and Franc.

o

34

P.S. S E P H T O N

and the Franc, and the Yen and the Franc, respectively. The MARS DF test is -6.995 relative to a five percent critical value of -5.7018. This indicates that the four currencies are cointegrated, since the MARS residuals do not contain a unit root. Once more, the results of non-linear cointegration tests are at odds with the traditional linear approach to testing whether series hang together over time. The results suggest that non-linear cointegration should be the subject of further analysis.

V. Conclusions The purpose of this paper was to extend the DF and ADF cointegration tests to residuals based on MARS. Critical values of the test statistics were derived for a variety of MARS models. Additivity, basis function selection, and system design played an important role in the simulation experiments. Several empirical examples demonstrated that non-linear cointegration may be found in systems that do not display evidence of linear cointegration. While this latter result did not hold in every case, it is clear that eointegration tests are sensitive to the methods used to obtain the residuals used in the tests. The present analysis can be extended in a number of directions. Response surfaces can be estimated to obtain a clear picture of how sample size affects the critical values of the test statistics. Other residual based cointegration tests, such as those advocated by Kwiatkowsi et al. (1992) that examine stationarity under the null should be examined to determine their properties on MARS. Finally, it may be interesting to replicate previous studies purporting to provide conclusive evidence in favour or against cointegration, to determine whether their conclusions survive on MARS.

Acknowledgements I thank an anonymous referee for useful comments, and the Social Sciences and Humanities Research Council of Canada for financial assistance.

Notes I See E n g l e - G r a n g e r (1991) 2 See Mankiw and Summers (1986) on the specification of the scale factor. 3 See Laidler (1985). 4 See Hardle (1990) for a review of non-parametric regression. 5 Spline functions were so named because draftsmen used them to fit a smooth line through diverse data points. Cubic splines can give smooth approximations to very noisy data. Schumaker (1981) provides an excellent discussion of splines. 6 This uses the normal approximation to the binomial. Covering two standard errors (2"1.96), the length is constructed as 3.92 [p(1 - p)/1000] s, where p is the size of the test statistic. 7 Future work examining the power properties of M A R S cointegration tests will examine how the data generation process affects the distribution of the test statistics for higher interaction models.

COINTEGRATION TESTS ON MARS

35

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