Government expenditures and revenues: evidence of fractional cointegration in an asymmetric modeling.(ORIGINAL PAPER)(Report)

Abstract In this paper, we examine the relationship between the US government expenditures and revenues using a fractional cointegration framework. In doing so, we permit a much richer degree of flexibility in the dynamic adjustment process toward equilibrium than in the classical case of cointegration. Moreover, we relax the assumption of a symmetric adjustment process throughout the use of threshold autoregressive (TAR) and momentum threshold autoregressive (M-TAR) models in the error correction representation of the process. The results show that both individual series are non-stationary I(1) and we do not find evidence of cointegration of any degree. However, if we take into account a structural break at 1973(2), fractional cointegration is found if the underlying process is autocorrelated, especially in the asymmetric modeling.

Keywords Public revenues * Public expenditures * Fractional integration * Asymmetric modeling * Structural breaks

JEL C32 * E62 * H60

Introduction

This paper re-examines the relationship between government expenditures and revenues by using new econometric techniques based on fractional integration and cointegration. According to Payne (2003), four alternative approaches have been used to describe the relationship between these two variables. The tax-spend approach suggests that changes in government revenues lead to changes in government expenditures. In line with this approach, Friedman (1978) argues that increasing revenues have a positive effect on expenditures, while Buchanan and Wanger (1978) suggest the contrary, which is, increases in government revenues lead to a decrease in expenditures throughout fiscal illusion. The second approach (spend-to-tax hypothesis) proposes that causality runs in the opposite direction, from expenditures to revenues. (See, Barro, 1979; Peacock and Wiseman, 1979). The third approach is usually called the fiscal synchronization hypothesis (Meltzer and Richard, 1981) and says that government simultaneously chooses the desired package of expenditures along with the revenues necessary to finance the spending programs. Finally, the institutional separation hypothesis (Hoover and Sheffrin, 1992) argues that government decisions to spend are independent from decisions to tax; therefore, the two variables are not linked to each other.

From an empirical point of view, and focusing on the US case, the evidence of the relationship between government revenues and expenditures is as controversial as it is in theory. A survey of the literature can be found in Payne (2003). Traditionally, authors used unrestricted vector autoregressive (UVAR) models to describe the relationship between the two variables and the conclusions were mixed. For example, Von Furstenberg, Green and Jeong (1986), Anderson, Wallace and Warner (1986) and others found evidence of causality running from expenditures to revenues, while Blackley (1986), Ram (1988a,b) and others found causality running in the opposite direction. Later, and especially after the seminal paper on cointegration by Engle and Granger (1987) numerous authors have found that expenditures and revenues are cointegrated. However, the issue of causality still remains controversial within this approach. Thus, authors such as Bohn (1991), Garcia and Henin (1999) and Chang et al. (2002) find support to the tax-and-spend hypothesis while Jones and Joulfaian (1991), Ross and Payne (1998) support the spend-to-tax interpretation. The fiscal synchronization hypothesis is supported by Miller and Russek (1989) and Hasan and Sukar (1995). Finally, Hoover and Sheffrin (1992) argue in favor of the institutional separation model.

Payne (2003) argues that these differing results can be due to the different methodologies employed as well as the time periods examined. However, these papers are all based on the same concept of cointegration that assumes that the individuals series are nonstationary I(1) while the cointegration relation is stationary I(0). In this paper, we argue that these differences may be due to the fact of imposing a strict distinction between the I(0) and the I(1) specifications, and since the possibility of fractional cointegration is ruled out, the models can be misspecified, leading to inconsistent estimates in the cointegrating relationship. In fact, we argue that the cointegrating relationship is fractionally integrated, with an order of integration located in the interval (0, 1), producing then a hyperbolic adjustment process, unlike the exponential rate obtained in the classical cointegration framework through the autoregressive (AR) structure. Moreover, we relax the assumption of symmetry in the adjustment process by using a threshold autoregressive (TAR) and the momentum threshold autoregressive (M-TAR) models developed by Enders and Granger (1998) and Enders and Siklos (2001). In the context of standard I(1)/I0) cointegration, Ewing, Payne, Thompson and Al-Zoubi (2006) also examined the tax-spend debate using TAR and M-TAR models, with the corresponding asymmetric error correction models. These authors, however, do not consider the possibility of fractional differencing.

According to Ewing et al. (2006), there are various arguments that can be employed to justify an asymmetric modeling in the budget and in the responses of its components to equilibrium. Policymakers can respond differently to a deviation of the deficit or surplus from its long run trend; asymmetries in the business cycle (e.g. Neftci, 1984; Potter, 1995; Hansen and Prescott, 2005; etc.) can also be associated with asymmetric changes in the budget. Sichel (1993) documented asymmetric adjustment paths in macroeconomic variables leading to changes in the budget. Finally, tax revenues can also be affected by changes in the effective tax rates. Bertola and Drazen (1993), Giavazzi et al. (2000) argue that abrupt changes in fiscal policy, though observed infrequently, may generate possible nonlinear effects in fiscal policy actions.

The outline of the paper is as follows: In Section 2, we briefly describe the concepts of fractional integration and cointegration. Section 3 deals with the error correction model in the context of fractional cointegration and asymmetric modeling. Section 4 contains the empirical application based on the relationship between the US government expenditures and revenues, while Section 5 contains some concluding comments and extensions.

Fractional Integration and Cointegration

For the purpose of the present paper, we define an 1(0) process {[u.sub.t], t=0, 1, ...} as a covariance stationary process with a spectral density function that is positive and finite at the zero frequency. In this context, we say that {[X.sub.t], t=0, 1, ...} is I(d) if:

[(1 - L).sup.d][x.sub.t] = [u.sub.t] t = 1, 2, ... (1)

with [x.sub.t] = 0, t[less than or equal to]0, where L is the lag operator ([Lx.sub.t] = [x.sub.[t-1]]) and where d can be any real number. Expressing the polynomial in the left-hand-side in (1) in terms of its Binomial expansion, we have that, for all real d,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that,

[(1 - L).sup.d][x.sub.t] = [x.sub.t] - d[x.sub.[t-1]] + [d(d - 1)/2] [x.sub.[t-2]] - ....

If d=0 in (1), [x.sub.t] = [u.sub.t], the series is said to be short memory (or I(0)) and a weakly autocorrelated (e.g. ARMA) [u.sub.t] is allowed. If d>0, [x.sub.t] is said to be long memory, so-named because of the strong association between observations widely separated in time. The higher the d is, the higher will be the level of association between the observations. (1)

To correctly determine d is crucial from a statistical viewpoint. Thus, if d [member of] (0, 0.5) in (1), [x.sub.t] is covariance stationary and mean-reverting, with the effect of the shocks disappearing in the long run; if d [member of] [0.5, 1), the series is no longer stationary but it is still mean-reverting, while d[greater than or equal to]1 means nonstationarity and non-mean-reverting.

Robinson (1994a) proposes a Lagrange Multiplier (LM) method of testing for unit roots and other forms of nonstationary or even stationary (fractional) hypotheses. A simple version of his test consists in testing the null hypothesis:

[H.sub.o]: d = [d.sub.o] (2)

in a model given by (1) for any real value [d.sub.o]. The specific form of the test statistic (denoted here by [^.R] = [[^.r].sup.2]), is given in the Appendix. He shows that under very mild regularity conditions:

[^.R] [[right arrow].sub.d] [[xi].sub.1.sup.2], as T [right arrow] [infinity]. (3)

Consequently, a one-sided 100[alpha]% level test of [H.sub.o] (2) against the alternative: [H.sub.a]: d>[d.sub.o] (d<[d.sub.o]) will be given by the rule: Reject [H.sub.o] if [^.r]> [z.sub.[alpha]] ([^.r] < -[z.sub.[alpha]]), where the probability that a standard normal variate exceeds [z.sub.[alpha]] is [alpha]. Robinson (1994a) also shows that the above test is efficient in the Pitman sense against local alternatives from the null.

Having defined fractional integration and described a way of testing I(d) statistical models, next we introduce the concept of fractional cointegration. By adopting the simplest possible definition, it can be said that a given time series vector [X.sub.t] is fractionally cointegrated if:

a) all its components ([x.sub.it]) are integrated of the same order (say d), i.e.,

[(1-L).sup.d][x.sub.it] = [u.sub.it], t = 1, 2,..., for all i

with I(0) [u.sub.i]'s, and

b) there is at least one linear combination of these components, which is fractionally integrated of order b, with b < d.

We use a two-step procedure proposed in Gil-Alana (2003), for testing the null hypothesis of no cointegration against the alternative of fractional cointegration, which is based on the univariate tests of Robinson (1994a). In the first step, Robinson's (1994a) tests are used to test the order of integration of the individual series and, if all of them are integrated of the same order (say d), in a second step, the degree of integration in the residuals from the cointegrating regression is tested. In standard cointegration analysis (where cointegration of order 1,0 is considered), Stock (1987) and Phillips (1991) showed that the least squares estimate of the cointegrating parameter was consistent and converged in probability at the rate [T.sup.[1-[delta]]] for any [delta]>0, rather than the usual [T.sup.[1/2]]. In the context of fractional degrees of integration, the estimation of the parameter of the cointegrating relationship is more delicate, and crucially depends on the values of d and b. Thus, if d<0.5, the least squares estimate (LSE) produces an inconsistent estimate. To overcome this problem, Robinson (1994b) showed that a narrow-band frequency-domain least squares estimate (NBLSE) is consistent, and the same method was subsequently studied by Robinson and Marinucci (2001) in the case of d>0.5. Here, the LSE is consistent, with the convergence rate depending on the specific locations of d and b, though the NBLSE still sometimes converges faster. In this paper, we focus on cases where the individuals series are I(1), and, given the consistency of the LS estimators, we can use Robinson (1994a) for testing the integration order of the equilibrium errors. A difficulty in this context is that the residuals are not actually observed but obtained from the cointegrating regression, and there might be a bias in favor of stationary residuals; this problem is similar to the one encountered by Engle and Granger (1987) when testing cointegration. In order to solve this problem, finite-sample critical values of the tests were computed in Gil-Alana (2003). We consider the model,

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