Chen & Yen

Censored Models < C3 -Econometric Methods: Multiple/Simultaneous Equation Models < C -Mathematical and Quantitative Methods, C51 -Model Construction and Estimation < C5 -Econometric Modeling < C -Mathematical and Quantitative Methods

Keywords: JEL Classification Number: C15, C34, C51 multivariate sample-selection model, censored system of equations, Heckman-correction

 two-step estimator that account for cross-equation correlation and use the inverse Mills ratio as correction-term are consistent only if certain restrictions apply to the true error-covariance structure. An alternative class of generalizations to the classical Heckman two-step approach is derived that condition on the entire selection pattern rather than selection in particular equations and, therefore, use modified correction-terms. It is shown that this class of estimators is consistent. In addition, Monte-Carlo results illustrate that these estimators display a smaller mean square prediction error.

avoid inconsistent estimators, even if their mean square error is small. For this reason, addressing unbiasedness and consistency of Heckman-type two-step estimators for censored systems of equations is a relevant task.

The analysis presented in this article shows that some of the estimators proposed by Tauchmann (2005) are consistent only for restrictive error-covariance structures. It also shows that a modified two-step Heckman-type estimator is generally consistent and performs well in terms of the mean square prediction error. In order to yield these results, the remainder of this paper is organized as follows: Section 2 introduces the model to be analyzed in more detail and analyzes the properties of straightforward multivariate generalizations to the [START_REF] Heckman | The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for such Models[END_REF][START_REF] Heckman | Sample Selection Bias as a Specification Error[END_REF] two-step estimator.

In Section 3 an alternative class of generalized two-step Heckman-type estimators is derived. Section 4 presents results from Monte-Carlo simulations that illustrate the theoretical results and extends the analysis to the estimators' mean square error. Section 5 concludes.

2 An analysis of sample-selection models

y * it = x it β i + ε it (1) d * it = z it α i + υ it , (2)
characterize the latent model, that is y * it and d * it are unobserved. Their observed counterparts y it and d it are determined by

d it =        1 if d * it > 0 0 if d * it ≤ 0
(3)

y it = d it y * it . (4)
Here, i = 1, . . . , m indexes the m equations of the system, and t = 1, . . . , T indexes the individuals. (5)

The diagonal-elements of Σ υυ are subject to the normalization σ υυ ii = 1, i = 1 . . . m.

Inconsistency of Heckman-type estimators

For the model ( 1) through ( 4) Tauchmann ( 2005) suggests a class of system twostep estimators that -analogously to the original Heckman two-step approach -con- In order to analyze these estimators' properties, we consider α as known and focus

3 To simplify notation, E(y it |x it , d it = 1) is used as short term for E(y it |x it , υ it > -z it α i ) through- out this paper. Yet, it does not denote E z [E(y it |x it , υ it > -z it α i )]
, although z it is not explicitly mentioned in list of the conditioning variables. This analogously applies to any moment that is conditional on either

d it = 1, d it d jt = 1, d it , d it d jt , or d t .
4 Because of (4), which implies E(y it |x it , d it = 0) = 0, the original [START_REF] Heckman | The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for such Models[END_REF][START_REF] Heckman | Sample Selection Bias as a Specification Error[END_REF] estimator can well be interpreted as a procedure that conditions on d it in the full sample and, therefore, uses d it as a weighting variable rather than an estimation procedure that conditions on d it = 1 and uses the sub-sample of selected units; see Tauchmann (2005) for details. [START_REF] Heckman | The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for such Models[END_REF], the errors are heteroscedastic and SUR is not a proper GLS estimator.

5 Because of var(ε it |d it = 1) = σ εε ii 1 -σ ευ ii 2 σ εε ii -1 + σ ευ ii 2 σ εε ii -1 1 -z it α i λ(z it α i ) -λ(z it α i ) 2 , cf.
β = (X DΩDX) -1 X DΩY. (6)
Because of Y = D(Xβ + ε) equation ( 6) is equivalent to

β = β + (X DΩDX) -1 X Ξ, with Ξ ≡ DΩD ε. (7) F o r P e e r R e v i e w
Here, the condition E(Ξ|X) = 0 implies plim T -1 (X Ξ) = 0 and, therefore, implies consistency of β under standard regularity conditions. To check whether E(Ξ|X) = 0 holds, consider an arbitrary element from Ξ:

ξ it = ω iit d it ε it + m j=1 j =i ω ijt d it d jt ε jt (8) = ω iit d it [ε it -E(ε it |d it = 1)] + m j=1 j =i ω ijt d it d jt [ε jt -E(ε jt |d jt = 1)].
We apply the law of iterated expectations to (8). First, we take the expectation of ξ it conditional on x t as well as on the individual selection pattern

d t . E(ξ it |x t , d t ) = ω iit d it [E(ε it |d t ) -E(ε it |d it = 1)] + m j=1 j =i ω ijt d it d jt [E(ε jt |d t ) -E(ε jt |d jt = 1)] (9)
Second, we take the expectation with respect to d t , yielding (1964).

E(ξ it |x t ) = m j=1 j =i ω ijt Pr(d it d jt = 1)[E(ε jt |d it d jt = 1) -E(ε jt |d jt = 1)]. ( 10
Including these expressions and rearranging terms leads to the system of regression equations

y it = d it x it β i + d it m j=1 δ ij ψ jt φ(z jt α j ) Φ m-1 ( A jt , R jt ) Φ m (•) + d it ε it , i = 1, ..., m. (12)
As in the original Heckman-model, the coefficients δ ij attached to the correction-terms

ψ jt φ(z jt α j ) Φ m-1 ( e A jt , e R jt ) Φ m (•)
are subject to estimation. Here, φ denotes the probability density function of the univariate standard normal distribution, while Φ m denotes the cumulative density function of the m-variate standard normal distribution. ψ jt is defined as 2d jt -1 and distinguishes truncation from either below or above. A jt represents a vector which consists of m -1 elements In applied work α and Σ υυ are likely to be unknown. In order to calculate the auxiliary regressors ψ jt φ(z jt α j )

ψ lt (z lt α l -σ υυ lj z jt α j ) (1-(σ υυ lj ) 2 ) 1/2 ; l = 1 . . . m, l = j. Cor- respondingly, R jt is defined as Ψ jt R jt Ψ jt ,
Φ m-1 ( e A jt , e R jt ) Φ m (•)
, one has to replace the true parameters with estimates obtained from first-step multivariate probit estimation. In the special In order to be able to construct a proper GLS estimator, expressions for Var(ε t |d t ) are required from which one can calculate an appropriate weighting matrix Ω. Through the use of the normality assumption and the decomposition rule for variances in a joint distribution such an expression can easily be derived as

7 Since E(y it |x it , d t , d it = 0) = 0 holds
Var(ε t |d t ) = Σ εε -Σ ευ (Σ υυ ) -1 Σ ευ + Σ ευ (Σ υυ ) -1 Var(υ t |d t )(Σ υυ ) -1 Σ ευ . (13)

Monte-Carlo analysis

In addition to the theoretical analysis we carry out Monte-Carlo simulations. On the one hand, we want to illustrate the theoretical results derived in Section 2. Test results on the joint unbiasedness of the second-step coefficients are provided for this purpose. 10

On the other, we also want to address the estimators' performance beyond the issue of consistency. Therefore, we present estimates for the CP-conditional mean square error prediction criterion

CP( β) = E 1 T T t=1 m i=1 (β i -β i ) x it x it (β i -β i ) X , (14)
cf. Judge et al. (1980). CP( β) measures the mean squared deviation of the estimated conditional mean from its true counterpart E(y * it |x it ) and, therefore, translates an estimator's MSE-matrix to a scalar performance measure that takes into account its variance as well as a potential bias.

is effectively required for the construction of the GLS estimator in general is short-ranked and cannot ordinarily be inverted in order to obtain individual weighting-matrices Ω t . Yet, using a generalized Moore-Penrose inverse is appropriate for this purpose.

10 Both the tables of raw coefficients' estimates as well as the LIMDEP command file used for carrying out the MC-simulations are available from the author upon request. SUR allows for estimating the first step using univariate probit models. All other estimators require simultaneous estimation of all vectors α i along with Σ υυ .

The experimental setup

The design of the Monte-Carlo experiment is equivalent to the one used by Tauchmann (2005). 11 We consider the case m = 2. 12 The sample size is 4000. The size of the Monte Carlo experiment is 1000 iterations. The vectors of exogenous variables each consist of three elements:

z it = [1 z 1,it z 2,it ] , x it = [1 x 1,it x 2,it ] , i = 1, 2.
Here z 1,1t , z 2,1t , z 1,2t , and x 2,1t are independently drawn from the standard normal distribution, while z 2,2t = z 2,1t , x 1,1t = z 1,1t , x 1,2t = z 1,2t and x 2,2t = x 2,1t . These variables are drawn only once and then kept fixed. For the coefficient vectors

β i = [1 1 1] ,
11 In contrast to the analysis presented here, Tauchmann (2005) imposes restrictions on the coefficients' estimates β i . This does not allow for directly comparing estimated CP-measures.

12 For m ≥ 3, simulated ML were required for estimation the first-step multivariate probit models.

This would increase computing time for the Monte-Carlo experiments enormously. The value √ 0.5 is assigned to all coefficients α attached to z 1,it and z 2,it . In order to allow for different unconditional censoring probabilities Pr (d * it ≤ 0), the constants α 0,i are varied. We run two simulations with unconditional censoring probabilities that are uniform across equations, in particular 0.25 and 0.5, which corresponds to constants 0.9539 and 0, respectively. Another simulation is carried out for mixed unconditional censoring probabilities, i.e. 0.25 for equation one and 0.75 for equation two, which corresponds to constants 0.9539 and -0.9539, respectively. The errorcovariance structure is specified as

Σ εε =    1.5 -1 2    , Σ υυ =    1 -0.5 1    , Σ ευ =    0.75 -0.25 -0.25 0.75    .
As an alternative specification, the value zero is assigned to all off-diagonal elements of Σ υυ and Σ ευ everything else remaining unchanged, i.e.

Σ εε =    1.5 -1 2    , Σ υυ =    1 0 1    , Σ ευ =    0.75 0 0 0.75    .
This defines the four-variate N (0, Σ) distribution, from where the random components are drawn separately for each model. After drawing the error vector, the dependent variables are calculated as defined by model ( 1) through (4). Subsequently, the generated data serves as input to the estimators. 13 We do not vary these parameters, since -in contrast to the estimator proposed by Shonkwiler & Yen (1999) -the performance of generalized Heckman estimators does not depend on the true value of β, c.f. Tauchmann (2005).

  x it and z it are vectors of observed exogenous variables. The vector d t = [d 1t . . . d mt ] describes the entire individual selection pattern. Finally, ε t = [ε 1t . . . ε mt ] and υ t = [υ 1t . . . υ mt ] are normally distributed, zero-mean error vectors with the covariance matrix Var (ε t , υ t ) =

5

  Page d it equation-by-equation. That is, after first-step estimation of the vectors α i by univariate or multivariate probit, the second-step regressions yielding estimates for the vectors β i are based on the conditional expectations3 E(y it |x it , d it ) = d it x it β i + d it σ ευii λ(z it α i ). Each regression equation, therefore, includes the inverse Mills ratio λ(z it α i ) as an auxiliary regressor and the parameters σ ευ ii are estimated as regression coefficients. Note that d it serves as a weighting variable, i.e. censored observations are weighted by zero and are therefore effectively excluded from the regression. 4Tauchmann (2005) distinguishes three variants of this estimator: The first one uses ordinary least squares (OLS) and ignores cross-equation correlation of ε it , another variant accounts for it in a simplified SUR fashion, and a third accounts for crossequation correlation and heteroscedasticity using a proper generalized least squares (GLS) approach. 5

  -step regression. Let X denote the stacked, mT × mk regressor-matrix 6 consisting of rows 0 1×k(i-1) x it λ(z it α i ) 0 1×k(m-i) . Note that inverse Mills ratios are included to the list of regressors. Let D denote a mT × mT matrix with diagonalelements d it and zero off-diagonal elements. This matrix allocates zero weight to censored units. Ω denotes the mT × mT block-diagonal weighting-matrix with elements ω ijt . It coincides with the identity-matrix if the model is estimated using the classical Heckman approach equation-by-equation, i.e. OLS. In the case of SUR estimation, the individual m × m sub-matrices Ω t are uniform across all t. In the case of GLS estimation, these weighting matrices are individually derived through matrix-inversion from estimates for var(ε it |d it = 1) and cov(ε it , ε jt |d it d jt = 1). Finally, let Y denote the stacked mT × 1 vector of dependent variables y it and ε denote the corresponding mT × 1 error-vector. Because of the inclusion of λ(z it α i ) to the list of regressors and E(ε it |d it = 1) = σ ευ ii λ(z it α i ), the error vector ε consists of elements ε it -E(ε it |d it = 1) rather than ε it . Now the generalized Heckman-estimators for β proposed by Tauchmann (2005) can be written

  where R jt denotes the partial conditional correlation-matrix Cor(υ t |υ jt ) and Ψ jt denotes a diagonal-matrix with diagonal elements ψ lt , l = j. Finally, Φ m (•) denotes the joint probability of the observed pattern d t . Note that the regression equations are still weighted by d it . 7

  , the ith equation of the tth observation still receives zero weight if y it equals zero because of censoring. m = 2 the regression equations are equivalent to the one used by Poirier (1980), except for the fact that Poirier (1980) conditions on d 1t d 2t = 1 rather than d 1t and d 2t , i.e. ψ jt equals one for all j and t. 8 For m = 2, δ ij = σ ευ ij holds for the auxiliary regression coefficients. One may estimate the system (12) equation-by-equation using OLS. Yet, the simple equation-by-equation Heckman-estimator is consistent as well in this case. So, conditioning on d t makes sense only in the context of simultaneous estimation. As a simple variant, one can construct such a system-estimator in the standard SUR fashion. However, this ignores the heteroscedasticity of the individual conditional error-variances.

  Shah S.M. and N.T. Parikh (1964): Moments of Singly and Doubly Truncated Standard Bivariate Normal Distribution, Vidya 7, 51-91. Shonkwiler J.S. and S.T. Yen (1999): Two-Step Estimation of a Censored System of Equations, American Journal of Agricultural Economics 81, 972-982. Su S.-J. and S.T. Yen (2000): A Censored System of Cigarette and Alcohol Consumption, Applied Economics 32, 729-737. Tallis G.M. (1961): The Moment Generating Function of the Truncated Multi-Normal Distribution, Journal of the Royal Statistical Society (Series B) 23, 223-229. Tauchmann H. (2005): Efficiency of two-step Estimators for Censored Systems of Equations: Shonkwiler and Yen reconsidered, Applied Economics 37, 367(1997): Estimating Models with Sample Selection Bias: A Survey, Journal of Human Resources 33, 127-169. Vermeulen F. (2001): A Note on Heckman-type Corrections in Models for Zero Expenditures, Applied Economics 33, 1089-1092. Yen S.T. K. Kah and S.-J. Su (2002): Household Demand for Fat and Oils: Two-step Estimation of a Censored Demand System, Applied Economics 34, 1799-1806. Yen S.T. (2005): A Multivariate Sample-Selection Model: Estimating Cigarette and Alcohol demand with Zero Observations, American Journal of Agricultural Economics 87, 453-466. Zellner A. (1963): An Efficient Method of Estimating Seemingly Unrelated Regression Equations: Some Exact Finite Sample Results, Journal of the American Statistical Association 58, 977

  Estimated coefficients: censoring prob. 0.25 and 0.75, dense error variancedev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev.

  Estimated coefficients: censoring prob. 0.5, diagonal-matrices Σ dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. Estimated coefficients: censoring prob. 0.25 and 0.75, diagonal-matrices Σ dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev.

  jt |d it d jt = 1) equals E(ε jt |d jt = 1) for any pair i = j and any t; that is E(ε it |d t ) exclusively depends on d it , yet does not depend on any d jt , j = i.weights ω ijt is inconsistent, unless Σ υυ as well as Σ ευ are diagonal matrices. Clearly, the inconsistency of β originates from conditioning on d it equation-by-equation. it -E(ε it |d t ) rather than ε it -E(ε it |d it = 1), the condition E(ξ it |x t , d t ) = 0

	from the multivariate normal distribution have to be included to correct for sample-
	selection bias. Results for these truncated means are provided by Tallis (1961) as well
	as for the special case m = 2 -albeit in more detail -by Shah & Parikh
	3 A consistent generalized Heckman estimator
	The above discussion clearly suggests, how to construct a consistent system-estimator
	F o r as generalization to the original Heckman-approach. From (9) follows that if ε it were F o defined as ε and subsequently E(ξ it |x t ) = 0 would be satisfied for any weighting matrix Ω, rendering r
	P the entire class of estimators consistent. Uniformly conditioning on d t , i.e. condition-P
	e e r ing on the entire selection pattern, in all equations rather than conditioning on d it equation-by-equation and, correspondingly, including E(ε it |d t ) rather than the inverse e e r
	) From (10) it becomes obvious that the estimator β is biased and inconsistent unless either (i) E(ε This R e v i e w Mills ratio as correction-term would lead to errors defined as ε it -E(ε it |d t ). That is, the regression must be based on the conditional mean E(y it |x it , d t ) rather than E(y it |x it , d it ). In order to implement this estimator, an expression for E(ε it |d t ) is required. It is easily shown that R e v i e w
	E(ε t |d t ) = E(ε t ) + Σ ευ (Σ υυ ) -1 [E(υ t |d t ) -E(υ t )]	(11)
	requires that Σ υυ as well as Σ ευ are diagonal matrices. The estimator β does also not
	suffer from inconsistency if (ii) ω ijt = 0 holds for all i = j and t. Condition (ii) implies holds. Since the unconditional expectations of ε t and υ t equal zero, the expression
	that equation-by-equation Heckman is consistent, since cross-equation correlations are reduces to a linear-combination of truncated first moments E(υ t |d t ) from the multi-
	not taken into account. Yet, in contrast, any system estimator that involves non-zero variate normal distribution. Therefore, in each regression equation m truncated means
	8 9

  for the latter ones are provided byTallis (1961). Therefore, with estimates for α and Σ υυ in hand, one can calculate these auxiliary regressors.

	Submitted Manuscript
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Obviously, any element of Var(ε t |d t ) is a linear function of all elements of the truncated m-variate normal variance-covariance matrix Var(υ t |d t ). Therefore, estimates for the elements of Var(ε t |d t ) can be obtained as fitted values from regressing squared residuals and residual cross-products -which, in turn, have been obtained from initial OLS regressions -on a constant and on estimates for all elements of Var(υ t |d t ). 9 Results 8 See Vella (1997) for other related models. 9 Because of var(d it ε it |d it = 0) = 0, the variance-covariance matrix Var(d 1t ε 1t . . . d mt ε mt |d t ) that 11 Page 11 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

12 Page 12 of 56 Editorial

  Unknown values for α and Σ υυ rather than known ones appear to be the relevant case from the viewpoint of applied econometrics. In our Monte-Carlo simulations, therefore, these parameters are estimated by first-step probit models. We consider six different estimators. In particular, conditioning on either d it or d t is combined with OLS, SUR and, finally GLS estimation. Conditioning on d it combined with OLS or

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Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

14 Page 14 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Results for Wald-tests on the unbiasedness of the six estimators are displayed in

	Submitted Manuscript
	4.2 Simulation results
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Table 1 .

 1 These simulation results are consistent with the theoretical ones, obtained in Section 2. If Σ υυ and Σ ευ are dense matrices, unbiasedness is clearly rejected for those estimators that condition on d it equation-by-equation and use SUR or GLS.

	In contrast, the classical Heckman estimator employed equation-by-equation does not

display a significant bias. The estimators that condition on the entire selection pattern do not display a significant bias either. If, instead, Σ υυ and Σ ευ are diagonal-matrices, neither of the estimators display a bias that is significant at the 0.05-level. Therefore, the Monte-Carlo simulation confirms that system-estimators that condition on d t are consistent, while system-estimators that condition on d it equation-by-equation are biased, unless certain restrictions apply to the true error-covariance matrix.

< insert Table 1 about here >

In order to analyze the estimators' performance beyond the issue of unbiasedness, estimates for the CP-conditional mean square error prediction criterion are displayed in

Table 2 .

 2 Comparing the SUR estimator that conditions on d t with its counterpart that conditions on d it yields the following plausible result: If the true covariance-matrix is dense, the consistent estimator that conditions on d t yields smaller CP-measures than the inconsistent one that conditions on d it . If Σ υυ and Σ ευ are diagonal-matrices -in estimated CP-measures are statistically insignificant at the 0.05-level.The comparison of OLS estimators that either condition on d t or d it yields similar results. If the error-covariance matrix is dense, the first estimator seems to perform better, though both are consistent. If, instead, Σ υυ and Σ ευ are diagonal-matrices the latter displays smaller CP-measures. However, these differences never are statistically significant, except for one simulation. mean square error and computational simplicity are a researcher's main criteria, while consistency is of secondary relevance, one might even argue in favor of the inconsistent SUR estimator that conditions equation-by-equation on the outcome of the upstream choice problem. Finally, if both consistency and a small mean square error are desired, and the computational burden of full information maximum likelihood is to be avoided, then the GLS estimator that conditions on the entire selection pattern appears to be Judge G.G. et al. (1980): The Theory and Practice of Econometrics, 2nd ed., John Wiley, New York.

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i.e. both estimators are consistent -the more parsimoniously parameterized one that conditions on d it performs better except for one simulation. Yet, the latter differences 15 Page 15 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK < insert Table

2

about here > Finally, we examine the performance of GLS estimators. Here, we observe substantial deviations in estimated CP-measures. While GLS conditional on d t yields the smallest mean square prediction error among all considered estimators in any simulation, GLS conditional on d it , except for two simulations, displays the largest one.

Moreover, the deviations in CP-measures between both GLS estimators always are significant. In fact, if the error covariance-matrix is dense, GLS conditional on d t significantly outperforms any other estimator in any simulation. As the only exception to this result, in some cases SUR conditional on d t displays CP-measures which are not significantly lager.

Our key simulation result -that GLS conditional on d t displays the best performance in terms of the mean square prediction error -fits theory. Among the considered estimators, GLS conditional on d t is the only one that not only is consistent, but also properly accounts for cross-equation correlation and heteroscedasticity. 16 Page 16 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK step estimator that condition on the entire selection pattern rather than the selection of particular single equations -and, therefore, use generalized correction-terms -are shown to be generally consistent. Moreover, these estimators display a smaller mean square prediction error. These new estimators are computationally more demanding since they generally require simultaneous estimation of a multivariate probit model. Nowadays, however, hard-coded procedures for this estimation problem are provided by econometric software packages, rendering computational complexity a minor obstacle to the practical application of the suggested estimation procedure. concern, then equation-by-equation Heckman appears to be the best choice. If a small 17 Page 17 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 :

 1 Tests on joint unbiasedness of regression coefficients

		Submitted Manuscript
		OLS	SUR	GLS	OLS	SUR	GLS
		conditional on d it	conditional on d t
	dense error variance-covariance matrix
	censoring prob. 0.25	0.800 0.000 0.000 0.449 0.484 0.155
	censoring prob. 0.5	0.545 0.000 0.000 0.964 0.070 0.642
	censoring prob. 0.25 and 0.75 0.446 0.000 0.000 0.117 0.929 0.259
	Σ υυ and Σ ευ with zero off-diagonal elements
	censoring prob. 0.25	0.320 0.415 0.052 0.805 0.208 0.082
	censoring prob. 0.5	0.595 0.659 0.610 0.900 0.760 0.807
	censoring prob. 0.25 and 0.75 0.832 0.620 0.604 0.963 0.634 0.215

Note: P-values for Wald-tests reported. Page 22 of 56 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 2 :

 2 Estimated conditional mean square prediction errors

		OLS	SUR	GLS	OLS	SUR	GLS
		conditional on d it	conditional on d t
	dense error variance-covariance matrix
		6.499	6.038	6.455	5.826	5.468	5.357
	censoring prob. 0.25	(0.154) (0.149) (0.169) (0.130) (0.137) (0.140)
		12.810	13.444	15.816	12.275	11.776	11.178
	censoring prob. 0.5	(0.342) (0.377) (0.483) (0.309) (0.335) (0.330)
		23.747	23.129	35.492	23.095	21.877	19.387
	cens. prob. 0.25 & 0.75 (0.806) (0.755) (1.391) (0.784) (0.762) (0.786)
	Σ υυ and Σ ευ with zero off-diagonal elements
		6.420	5.520	5.872	6.304	5.452	5.269
	censoring prob. 0.25	(0.156) (0.135) (0.212) (0.147) (0.138) (0.140)
		13.451	11.776	15.880	13.533	11.932	11.702
	censoring prob. 0.5	(0.386) (0.346) (1.601) (0.364) (0.324) (0.352)
		23.219	20.627	28.690	24.156	21.091	17.748
	cens. prob. 0.25 & 0.75 (0.768) (0.744) (2.733) (0.853) (0.710) (0.655)
	Notes: Standard errors in parenthesis.
	Displayed CP-measures are scaled by the factor 1000.

Table 1 :

 1 Estimated coefficients: censoring prob. 0.25, dense error variance-covariance matrix . dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev.

		GLS
	t conditional on d	SUR
		OLS
		GLS
	it conditional on d	SUR
		OLS	mean st
		true value

Table 2 :

 2 Estimated coefficients: censoring prob. 0.5, dense error variance-covariance matrix . dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev.

	conditional on d it conditional on d t	true value OLS SUR GLS OLS SUR GLS	1 0.9963 0.0596 0.9886 0.0586 0.9670 0.0641 0.9993 0.0587 1.0014 0.0604 0.9994 0.0586 mean stβ 0,1	1 1.0016 0.0351 1.0036 0.0339 1.0079 0.0346 1.0006 0.0361 0.9992 0.0334 1.0008 0.0327 β 1,1	1 1.0006 0.0250 1.0001 0.0249 1.0000 0.0234 0.9995 0.0256 1.0013 0.0245 0.9999 0.0239 β 2,1	0.75 0.7543 0.0783 0.7579 0.0763 0.7793 0.0846 0.7507 0.0759 0.7455 0.0765 0.7507 0.0736 σ ευ 11	σ ευ 12

Table 4 :

 4 Estimated coefficients: censoring prob. 0.25, diagonal-matrices Σ . dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev. mean st. dev.

	ευ
	υυ and Σ

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

MU11,MU12/MU21,MU22] $ CALC ; cor= MUU(1,2) YD2,X1Y2,X2Y2,M2Y2 $ MATR ; OL= L1'L1 ; OR= Init(4,4,0) ; UL= Init(4,4,0) ; UR= L2'L2 $ MATR ; YO= L1'YS1 ; YU= L2'YS2 ; XW= [OL,OR/UL,UR] ; YW= [YO/YU]


Bert Scholtens
email: scholtens@rug.nl.

Wijtze Peenstra

Scoring on the Stock Exchange? The Effect of Football Matches on Stock Market Returns: An Event Study

Keywords: football, stock market, abnormal returns, performance Football, Results, Stock Market, Firm Performance

We analyse the effect of results of football matches on the stock market performance of football teams. We analyse 1274 matches of eight teams in the national and European competition during 2000-2004. We find that the stock market response is significant and positive for victories and negative for defeats. The response is significantly stronger in the case of defeat. The response is stronger for matches in the European competition than for those in the national competition. Unexpected results have a stronger impact for European matches than expected ones but this is not the case in the national competition.

Introduction

Money is key in football 1 . For example, AC Milan, the winner of the 2006/2007 UEFA Champions League received € 40 million and all participants together got € 580 million in prize money alone. 2 But the role of money in this game is nothing new. The English Football League, with professional players, dates back already to 1888 and since then the role of money seems not to have decreased [START_REF] Dobson | Performance and revenue in professional league football: Evidence from Granger causality tests[END_REF]. The main income of the football industry results from merchandising, sponsoring, media contracts and receipts from matches (for an overview of football teams' finances, see the yearly reviews by Deloitte and Touche). Competition is enormous and that is one of the reasons why clubs have turned to the stock exchange [START_REF] Mitchell | A competitive index for international sport[END_REF]. The emission of shares gives them money that can be used to improve their financial position [START_REF] Cooper | How to Invest in Sports Shares[END_REF]. Tottenham Hotspurs was in 1983 the first football club with a listing on the stock exchange. Since then, many clubs have followed. Good results during the matches may translate in financial rewards as success attracts media attention and the scope for sponsoring, etc. [START_REF] Dobson | The Economics of Football[END_REF].

In efficient markets, market participants respond to new information or news that in some way or another might regard the firms they invest in. The investors response can have an impact on the valuation of the firm. Given the enormous and growing amounts of money that are involved in football, it seems a legitimate question to ask whether losing or winning a football match impacts on the market valuation of the football club. The stock market participants can interpret the result as information and integrate it in their revaluation of the firm. A market reaction can be induced by the expected imminent cash flow associated with new information. So far, little research has been done into the 1 With football, we mean the game by which two teams of 11 players try to kick a round ball in the goal of the other team. In the USA, this game is called soccer. 2 Source: UEFA direct 7. 07 (www.uefa.com). that unexpected results on national and international games impact on the club's share returns. There is no difference between the results on national or European games. [START_REF] Renneboog | Share price reactions to sportly performances of soccer clubs listed on the London Stock Exchange and the AIM[END_REF] analyse 17 British teams during 1995-1998. They find that a win results in a positive reaction on the stock market, whereas a loss or a draw has a negative impact. The impact of losses is larger than that of wins. They do not detect different responses to national or European games. [START_REF] Palomino | Stock Price Reactions to Short-Lived Public Information: The Case of Betting Odds[END_REF] analyse 16 British teams for the period 1999-2002 and find statistically significant abnormal returns. Zuber et al. (2005) consider the game-related performance of ten listed football teams in the English Premier League and focus on fan behaviour. They find that there is no abnormal return from (unexpected) results.

We analyse the stock market reaction to the results in football matches of international football teams with a finance model. 3 We report results of 1247 matches of eight international football teams in the period 2000-2004. New is that we analyse both European and national matches for a group of international teams and conduct several robustness tests with respect to our results. Also, our period of analysis is much more recent than those of existing studies. Our research is complementary to Zuber et al.

(2005), as we investigate the actively traded football shares and, thus, concentrate on investor behaviour rather than on fan behaviour. Furthermore, we restrict ourselves to analysing the direct response of the football share and do not go into detail about the response to (changes in) betting quotes (see [START_REF] Forrest | Odd-setters as forecasters: The case of English football[END_REF][START_REF] Palomino | Stock Price Reactions to Short-Lived Public Information: The Case of Betting Odds[END_REF].

Method and Data

We use the event study methodology to analyse the effect of the results in the football matches on the stock market return of the listed football team. This methodology can be used to analyse the price reaction of a share from a specific event. Brown and (1980,1985) and [START_REF] Campbell | The Econometrics of Financial Markets[END_REF] give a thorough overview of this methodology (see also [START_REF] Mills | Misspecification testing and robust estimation of the market model and their implications for event studies[END_REF][START_REF] Mishra | Skewness preference and the measurement of abnormal returns[END_REF]. The event in our case is the result of the football match. The outcome is either win, loss or draw. The normal return of the share R jt is calculated as:

[ ] 1 / ) ( log + = jt jt jt jt P D P R (1)
Where t j P , is the price of firm j at the end of period t; t j D , is the dividend given to the shareholder during period t; 1 , t j P is the price of firm j at the end of period t-1.

To arrive at the abnormal return, we first must establish expected or normal returns. This is the return that is to be expected when the event would not have occurred.

The difference between the actual and the normal return is the abnormal return. The estimation period is the period that precedes the event period, i.e. the period at or during which the event does occur. The estimation period is used to estimate the model's parameters. It must not overlap with the event period as this would imply that it includes returns that are affected by the event. As the football matches are our events and as the events occur on a very frequent basis, it is rather difficult to choose an estimation period that does not include events that may impact on the returns. We follow the approach suggested by Brown andWarner (1980, 1985) and [START_REF] Campbell | The Econometrics of Financial Markets[END_REF] and use an estimation period of 250 trading days. This estimation period is used for each match.

We use the market model to arrive at normal returns (see Brown andWarner (1980, 1985), [START_REF] Beaver | Econometric properties of alternative security return methods[END_REF], [START_REF] Dyckman | A comparison of event study methodologies using daily stock returns: A simulation approach[END_REF]). This model is defined as:

mt j j jt R R ˆ = (2)
Where R mt is the return of the market index at day t. j

mt j j jt jt R R AR ˆ = (3)
The football matches are our events. Given that there are a large number of matches, it is not possible to arrive at an estimation period that does not include eventrelated returns of the football teams. Renneboog and Van Brabant (2000) nor the other studies mentioned above do account for this problem and follow the approach suggested by Brown andWarner (1980, 1985). [START_REF] Brown | Market reaction to public information: The atypical case of the Boston Celtics[END_REF] deviate from the standard approach and use all return data to arrive at their estimation of the normal return. We follow Brown and Warner and apply the [START_REF] Brown | Market reaction to public information: The atypical case of the Boston Celtics[END_REF] approach of using the whole sample period as the estimation period as a robustness check.

We use an event period of 1 single day, namely the first trading day after the match took place. This is defined as 'day 0'. We assess the impact of the event (the result of the football match) for this event period only. We take this extremely short event period in order to avoid that event periods overlap, which could result in misinterpretations of the outcome of the analysis. Short event periods -like ours of one single day -are not uncommon. [START_REF] Dyckman | A comparison of event study methodologies using daily stock returns: A simulation approach[END_REF] as well as [START_REF] Glascock | Sensitivity of the SCPE test statistic[END_REF] advise to look for such a short period in order to be able to focus on the direct and uncontaminated results of an event.

As to our data, we depart from the 42 teams in Appendix 1 and apply the following selection criteria:

-The team must have played in the European competition between August 1st, 2000 and December 31st, 2004.

-The matches must have a betting quotation.

-

The country where the teams play their matches has a reinvestment index that reflects the development of the stock market return of the domestic market. The football shares are not subject to so-called zero-return dates. This implies that the share is not traded. No-trades will affect the results of our analysis. 4 Zero returns reflect market illiquidity, which is an important market inefficiency [START_REF] Junker | Measurement of aggregate risk with copulas[END_REF]. Alternatively, under the semi-efficient market hypothesis, no price change means that no new information arrived at the market.

But we specifically investigate the impact of results from matches, i.e. news about the teams.

Given these criteria, we have eight teams from five countries: Ajax Amsterdam from the Netherlands, Borussia Dortmund from Germany, Lazio Roma, AS Roma and Juventus

Turin from Italy, Manchester United from England and Porto and Sporting Lissabon from Portugal. For these teams, we have results for 1274 matches. This regards 235

European and 1039 national matches. 5 The eight teams won 721 matches, lost 256 and 297 times there was a draw. Stock market returns are derived from DataStream. This database gives information about share prices from 1973 onwards. We use broad indices and not sector indices as the benchmark in line with the suggestion by [START_REF] Brown | Using daily stock returns: The case of event studies[END_REF]. We use the following indices as a benchmark for the market: for England the FTSE All-Share index, for Germany the Dow Jones Germany Index, for Italy the Dow Jones Italy Index, for the Netherlands the Amsterdam All-Share index and for

Portugal the Dow Jones Portugal Index. All indices are value weighted indices; they weight of the returns of a firm within the index is determined by the market value of the index (see [START_REF] Krueger | Parameter specifications that make little difference in anomaly studies[END_REF]. We require a stock market index for every country that has firms in our sample. This index is a reinvestment index, taking account of dividends and stock splits. [START_REF] Campbell | The Econometrics of Financial Markets[END_REF] shows that the event study methodology is robust to cross-sectional dependence and clustering. It is suitable for tackling the 4 Zuber et al. ( 2005) exactly go into this issue as they are particularly interested in the investor characteristics of the football fans. Therefore, they do include the zero-return dates. 5 The design of the European competition differs significantly from that of the national competition. Football teams play only two or a few matches before it is clear whether or not they may proceed to the next round, whereas in the national competition the number of matches is much larger before it is clear who ends as the champion and who is allowed to join the European competition. Furthermore, in the European competition, the teams may earn a bonus from each won match. In this paper we strictly focus on investor behaviour with respect to football stocks in relation to match results. [START_REF] Campbell | Measuring security price performance using daily NASDAQ returns[END_REF].

We use information from bookmakers to arrive at the expectation about the result (see [START_REF] Forrest | Odd-setters as forecasters: The case of English football[END_REF][START_REF] Palomino | Stock Price Reactions to Short-Lived Public Information: The Case of Betting Odds[END_REF]. As such, we do not account for the 'home ground advantage' (Vergin and Sosik, 1999), but assume that this effect is incorporated in betting quotes. [START_REF] Falter | Demand for football and intramatch winning probability: an essay on the glorious uncertainty of sports[END_REF] argue that the main 'football variables' have only a tenuous explanatory power concerning the final outcome of a given match. Match results are derived from http://www.soccerbase.com. Betting quotes for national games are derived from http://football-data.co.uk and for European ones from TotoSelect B.V., the Netherlands.

As to robustness, we will look into the impact of excluding outliers, we account for nonnormality and we undertake an alternative approach to determine normal returns.

Results

Table 1 gives the average abnormal returns on the first trading day after the football match. It reveals that a victory results in a mildly positive but very significant response on the stock market (+0.36%). This suggests that the market value of listed football teams increases by 0.36% after a victory. Defeat is punished (-1.41%), whereas a draw also results in a significantly negative response (-1.10%). When we compare the average abnormal return after a victory and that of a defeat, we find that the difference is statistically significant at the 1%-confidence level.

[ Insert Table 1 about here ]

Table 2 gives the abnormal returns on the first trading day after a match in the national leagues and relate these returns to whether or not the result was as expected. It reveals that unexpected wins result in slightly smaller positive abnormal returns than expected wins (namely +0.23 and +0.39% respectively). The difference is not et al. (2005). Surprisingly too is that unexpected losses result in a negative return that is smaller (in absolute terms) than the abnormal returns that accompany expected losses.

Again, we find that the stock market responds stronger to defeats in the national competition than to victories (-1.14 versus +0.38%). The asymmetry is statistically significant at the 1%-level. If there is a draw when victory was expected, we find a significantly negative abnormal return. If there is a draw when defeat was expected, the abnormal return is not significant.

[ Insert Table 2 about here ]

Table 3 gives the abnormal returns for matches in the European competition. It reveals that an expected victory has no significant impact whereas an unexpected victory has. Expected defeats in European football matches earn a significantly negative abnormal return, suggesting that the stock market does not account for all available information (see also [START_REF] Palomino | Stock Price Reactions to Short-Lived Public Information: The Case of Betting Odds[END_REF]. Unexpected defeats result in a significantly stronger negative reaction from the stock market. The difference between an unexpected victory in the national and an unexpected one in the European competition is not significant. However, we find that the difference between victories and defeats is significant at the 5%-level. We do not find a stronger response to expected than to unexpected results. As in the national leagues, we find that (un)expected defeats result in a stronger reaction than (un)expected victories. The stock market's reaction after a defeat in European matches is larger than that in national ones (-2.14% versus -1.14% for losses in national leagues). This may be related to the design of the European competition (see footnote 3). Table 3 also reveals that a draw where a victory was expected results in a significantly negative abnormal return. In contrast, when there is a draw when defeat was expected, this does not result in a significant abnormal return. Apart from a purely financial explanation we also may hold psychological factors responsible for our results:

There is both theoretical background and empirical evidence that people respond emotionally stronger after defeats than after victories [START_REF] Kahneman | Prospect theory: An analysis of decisions under risk[END_REF]; , 1989;Trovato, 1998;[START_REF] Reilly | Science and football: a review of applied research in the football codes[END_REF][START_REF] Pain | Knowledge and perceptions of sport psychology within English soccer[END_REF]. As this is not the particular subject of this study, we will not go further into this issue.

Robustness

As to the robustness of our findings, we look into the impact of excluding outliers, we account for nonnormality and we undertake an alternative approach to determine normal returns.

We find that the results are robust to outliers (robustness results are available upon request). Only in the case of games unexpectedly won at the national level and a draw in the national competition when defeat was expected do outliers impact on the conclusion.

The Student t-test and the Corrado rank test result in the same conclusions with respect to the statistical significance of our findings. Only for one type of matches they give different results: The average abnormal return after an unexpected victory is significant according to the Corrado test, not to the Student test. But when we exclude outliers they both point at significant abnormal returns. We are inclined to base our conclusions on the Corrado test as, with a limited number of observations, abnormal returns will depart more from the normal distribution than in the case of many observations [START_REF] Brown | Using daily stock returns: The case of event studies[END_REF].

We also use an alternative estimation procedure to arrive at the expected returns, namely the one suggested by [START_REF] Brown | Market reaction to public information: The atypical case of the Boston Celtics[END_REF]. They take the full observation period to arrive at the normal returns. The results for the Brown and Hartzell approach are not reported here for brevity sake but are available upon request. We find that the Brown-Hartzell approach gives results that are fully in line with those reported in the text above. Therefore, we conclude that our findings are robust to the estimation procedure.

Conclusion

Money is key in football. Losing or winning a match impacts on the expected cash flows of the team and may affect its market value. In order to find out how investors respond to the results of football matches, we investigated whether results from 1247 national and European football matches lead to abnormal returns of the shares of eight listed football teams. We conclude that the stock market responds positive to victories and negative to defeats. Second is that the stock market responds asymmetrically, that is the response to defeat is 'stronger' than that to victory. This may be related to the idea that the public is more sensitive to losses. However, it also might result from asymmetric results and returns. Furthermore, the stock market reacts stronger to the results in European matches than to those in the national leagues. Fourth, unexpected results in European matches do result in a stronger stock market response than expected results, whereas this is not the case in the national competition. We assume that the third and fourth conclusion can be related to the much larger importance of financial incentives in the European competition. Our results are robust to outliers, to nonnormality in the returns and to the estimation procedure.

When we relate our results to previous research, our observations are in line with [START_REF] Brown | Market reaction to public information: The atypical case of the Boston Celtics[END_REF] as we also found that matches indeed directly impact on the stock return and that there is an asymmetric reaction with respect to won or lost matches.

Brown and Hartzell, however, do not find a significant effect of matches in the national competition whereas we do find such an effect. This may result from the type of competition analysed, namely basketball in the case of Brown and Hartzell and football in our study. In line with [START_REF] Renneboog | Share price reactions to sportly performances of soccer clubs listed on the London Stock Exchange and the AIM[END_REF] and [START_REF] Palomino | Stock Price Reactions to Short-Lived Public Information: The Case of Betting Odds[END_REF], we find that victories are positively rewarded by the stock market, whereas defeat and draw are 'punished', i.e. earn negative returns. Also, we find that defeats result in higher (absolute) negative returns than victories. Furthermore, our results are in line with the analysis of [START_REF] Stadtmann | Frequent news and pure signals: Evidence of a publicly trade football club[END_REF], and generalize his findings to an international setting.

  results and financial performance. Stadtmann (2003) analyses 97 games of Borussia Dortmund, a German football club, during 2000-2002. He finds

  ˆand j ˆare estimated on the basis of the ordinary least squares of the returns during the estimation period. j ˆis an of the share in relation to the stock market. As such, the market model accounts for market and firm specific conditions in relation to the share. The market model defines the following abnormal return (AR jt ):

  result of one team on the abnormal returns of other teams (see Campbell

  . Apparently the stock market did not fully account for the information revealed by the betting quotes. This is in line with the findings of Palomino

  .C. and Sosik, J.J. (1999) No place like home: an examination of the home field advantage in gambling strategies in NFL football. Journal of Economics and Business, 51, 21-31. White, G. (1989) Media and violence: The case of professional football championship games. Aggressive Behavior, 15, 423-433. Zuber, R.A., Yiu, P. Lamb, R.P. and Gandar, J.M. (2005) Investor-fans? An examination of the performance of publicly traded English Premier League teams. Applied Financial Economics, 15, 305-313.

Page 10 of 16 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o
	r
	P
	e e r
	R e v i e w

Table 1 : Abnormal return for all matches (N = 1274)

 1 Our results contrast those ofZuber et al. (2005) who did not find abnormal returns from football teams' results. This difference can be explained by the fact that Zuber et al.

	include nonzero return days to focus on fan-behaviour, whereas we focus on investor-
	behaviour and exclude nonzeros. From this, we are inclined to conclude that football
	stock investors differ from football fans.

Table 2 : Abnormal return for matches in the national league (N = 1039) AR

 2

					Mann-
			Corrado	Student	Whitney
		%	t-value	t-value	t-value
	Win	0,38%	2,077 **	2,903 ***
	Win expected	0,39%	1,850 **	2,889 ***	0,343
	Win unexpected	0,23%	1,843 **	0,468
	Loss	-1,14%	-5,218 ***	-4,501 ***
	Loss expected	-1,87%	-4,468 ***	-3,963 ***	1,980
	Loss unexpected	-0,84%	-3,745 ***	-2,720 ***	**
	Draw	-1,14%	-5,660 ***	-5,160 ***
	Draw: win expected	-1,40%	-5,802 ***	-5,831 ***	1,536
	Draw: loss expected	-0,01%	-1,220	-0,014	*

Table 3 Abnormal return for matches in the European league (N = 235) AR

 3

					Mann-
			Corrado	Student	Whitney
		%	t-value	t-value	t-value
	Win	0,22%	0,724	1,337
	Win expected	-0,07%	0,023	0,307	1,950
	Win unexpected	1,19%	2,009 **	2,061 **	**
	Loss	-2,14%	-5,089 ***	4,447 ***
	Loss expected	-1,34%	-3,000 ***	-2,897 *** -2,015
	Loss unexpected	-3,07%	-4,244 ***	-3,376 ***	**
	Draw	-0,94%	-3,279 ***	-3,171 ***
	Draw: win expected	-1,32%	-3,724 ***	-3,405 ***	1.711
	Draw: loss expected	-0,27%	-0,499	-0,707	**
	.

Alternative approaches to assess performance are the econometric model (for example, see[START_REF] Koning | An econometric evaluation of the effect of firing a coach on team performance[END_REF] and data envelopment analysis (see[START_REF] Pestano Barros | Performance evaluation of the English Premier Football League with data envelopment analysis[END_REF].


IN UK SECTORAL OUTPUT

Keywords: business cycles, trend, cycle, growth, mean-reverting, decomposition, Kalman filter JEL No: E32, C50 business cycles, trend, cycle, growth, mean-reverting, decomposition, Kalman filter 1

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

This paper examines three business cycle elements of cycles, trends, and growth in UK GDP sectors. It decomposes sectoral output into cycles and trends and pays attention to growth, adopting a state space method executed through the Kalman filter. Most business cycle studies on trend-cycle decomposition of GDP follow the tradition of [START_REF] Clark | The cyclical component of the U.S. economic activity[END_REF] where the growth rate is specified as a pure random walk, implying that the time series in concern is I(2). Though, together with a stationary cycle component, the specification indicates the recognition of the fact that GDP is usually less persistent than a pure integral random walk. The decomposition procedure is the kind of the Kalman filter named after [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]. Recently the research has become more sophisticated and advanced technically, e.g., in [START_REF] Kim | Friedman's plucking model of business fluctuations: tests and estimations of permanent and transitory components[END_REF], but the construction elements have not been fundamentally amended. In this paper we propose alternative specifications for modeling output growth, which are featured by mean-reversion in 2 the growth rate for the stochastic trend. A pure random walk allows the growth rate to wander and reach any point, leading to a theoretically unjustifiable proposition. Although we are less critical to the nature of the growth rate in empirical research, alternative specifications of the growth rate must be tendered and the validity of the restrictions imposed on each of the alternatives be tested. With such modeling framework, the present paper possesses inclusiveness and simplicity in statistical formulation. The econometric models in this study are the simplest and most straightforward in the state space executed by the Kalman filter, clearly demonstrating the components and state transition mechanism in economic activity. Moreover, the specification of growth in the general model can easily reduce to two different growth formulations via imposing relevant restrictions. So the study accommodates diverse beliefs about growth paths. In theory and strictly speaking, the results of model estimation are valid only if the data set is infinite in time, which obviously cannot be met in any empirical investigation. Therefore, what we analyze are indeed in a short stretch within the infinite time; or put it another way, an instant in history. In this respect, the term "permanent" is not in the strict sense, allowing various results, some of which appearing to contradict conventional knowledge, to be made possible in empirical studies. From the viewpoint of usefulness and policy effectiveness, explanations and implications obtained from various models, even if they are controversial, can be acceptable.

The present paper is not only motivated by the above theoretical and methodological considerations in search for improved understanding of output growth and fluctuations.

Empirically, there are much less business cycle studies conducted at sectoral levels that are also almost confined to US data. It is an obvious fact that the US economy is to a great extent larger than most other economies in the world. Subtler are the implications in output data characteristics and, subsequently, business cycle features. Many typical economies of the UK size involve lesser aggregation in output data than the US economy, and may behave rather 3 differently due to this difference in the degree of aggregation. Previous research by [START_REF] Engle | A unified approach to the study of sums, products, time-aggregation and other functions of ARMA processes[END_REF] amongst others suggests that aggregation results in correlation even if the individual series are not correlated, implying that the larger the size of an economy, the higher is the degree of exaggeration of the cycle component in output data. This problem of aggregation is relevant to the studies using aggregate output data as against those using sectoral data too, as the cycle component may be exaggerated in aggregate output data. Moreover, [START_REF] Long | Sectoral vs. aggregate shocks in the business cycle[END_REF] suggest that the contribution of common shocks to the co-movement between sectors will appear to be greater than their true contribution; therefore the role of common or aggregate shocks may be over-estimated. All these point to the necessity of empirical business cycle research using non-US data and at sectoral levels, as carried out by the present paper.

Although the notion of business cycles started to attract attention from economists and governments alike in as early as the first half of the 20th century, in their search for an understanding of the patterns in economic activity and a possible therapy for mitigating the damage caused by severe economic downturns, a century's endeavor has not rendered great fruition. It seems that modern regulatory frameworks remain as fragile, futile and above all, primitive, as a century ago in tackling credit crisis; the fear of recession in business cycles remains as strong as ever; and the peril of credit crisis in severe business cycle downturns remains as real as in Marx's time and his analysis at the time. Hence go on the search and research. Recent empirical studies on business cycles with a sectors focus include Caporale (1997), [START_REF] Peel | The variance of economic activity over the business cycle: some further evidence[END_REF], Wang el al (1999), [START_REF] Wang | Shock persistence in property and related markets[END_REF] and [START_REF] Dibeh | Time delays and business cycles: Hilferding's model revisited[END_REF], among others. [START_REF] Caporale | Sectoral shocks and business cycles: a disaggregated analysis of output fluctuations in the UK[END_REF] sectoral deflators from the first quarter in 1963 to the fourth quarter in 1994, Wang el al (1999) find that variability between sectors cause uncertainty in economic aggregates, which may identify business cycle evolution. [START_REF] Peel | The variance of economic activity over the business cycle: some further evidence[END_REF] employ a joint model of bilinearity inconditional mean and generalized-autoregressive-conditional heteroscedasticity to test for the presence of non-linearities in UK and US industrial and sectoral production growth rates. They find bilinearity inconditional mean to be present in US industrial production and manufacturing, and significant conditional variance asymmetries in the majority of series considered such that conditional variance is higher during recessions and stronger in the more cyclically sensitive durable consumer goods sectors. More recently, [START_REF] Sensier | Inventories and asymmetric business cycle fluctuations in the UK: a structural approach[END_REF] investigates the movement of manufacturing inventories and production over the business cycle, [START_REF] Jenkins | Nominal inertia and shock persistence in UK business cycles[END_REF] attempt to identify and map out the effects of innovations in the money supply, employment, output, wages and prices, while [START_REF] Wu | Stylized facts on nominal term structure and business cycles: an empirical VAR study[END_REF] examines the importance of various macroeconomic The rest of the paper is organized as follows. Section 2 introduces the model and provides an analysis of business cycles features with regard to growth, trends and cycles, while the technical aspects of the model's state space representation and estimation are provided in the appendix. Section 3 presents the empirical results of this study and discusses the findings and their implications. Finally, Section 4 concludes.

Modeling of growth, trends, and cycles

The review of recent research on business cycles in the previous section has pointed out the importance of decomposition of business cycles data, though the approaches and the theoretical guidelines vary from one study to another. The essence is to capture the crucial features of the business cycle and its components and shed light on the issues such as output growth, fluctuations, and their patterns of persistence and durability. To this end, model specifications utilized in this study are introduced in 2.1, followed by analysis and groupings of business cycle features with regard to growth, trends and cycles in 2.2.

Model specifications

Unlike most previous studies reviewed earlier where the growth rate is a pure random walk, the model in this example has a stochastic growth rate that can be stationary or non-stationary depending on the value of γ in equation (3). Specifically, if γ is smaller than but close to one, the growth rate is persistent in its behavior. The model is as follows: assumptions. If λ is set to be zero, then the growth rate is constant over time when w t is zero as well 1 . So, in the empirical inquiries, there are three modes of growth, when two sets of restrictions are imposed against the general form of equation ( 3). According to [START_REF] Watson | Univariate detrending methods with stochastic trends[END_REF] and [START_REF] Clark | The cyclical component of the U.S. economic activity[END_REF], the two innovations in the trend and the cycle, u t and v t , are specified as independent processes, and innovation in growth,w t , is further assumed to be uncorrelated to u t and v t . [START_REF] Blanchard | The dynamic effects of aggregate demand and supply disturbances[END_REF] and [START_REF] King | Stochastic trends and economic fluctuations[END_REF] also follow this tradition in structural decompositions, while Beveridge and Nelson (1981) assume that the innovations from the trend component and the cycle component are perfectly correlated.

t t t C T Y + = (
Write equations ( 1) -( 4) in the state space form, the observation equation is:

[ ]             = - t t t t t g C C T Y 1 0 0 1 1 (5)
The state equation is:

1 When λ is set to zero, wt is not identifiable from ut.

F o r P e e r R e v i e w             +             +                         =             + + + - + + + 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 t t t t t t t t t t t w u v gc g C C T g C C T λ ϕ ϕ (6)
The construction elements of the model are:

t t Y y = , [ ] t t t t t g C C T 1 ' - = ξ , [ ] gc x t 0 0 0 ' = [ ] 0 0 1 1 = H ,             = λ ϕ ϕ 0 0 0 0 0 1 0 0 0 1 0 0 1 2 1 F , 0 = A , 1 = B 0 = t µ , [ ] t t t t w v u 0 ' = ν               = 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 w v u Q σ σ σ , 0 = R
The model will be estimated using the Kalman filter algorithms, and the empirical results will be reported and analyzed in the next section.

Analysis of business cycle features

Now let us discuss some general ideas about the behavior of output series in relation to the parameters in the model, and analyze business cycle features with regard to trends, cycles and growth. σ u and σ v , the standard deviation of the trend component and that of the cycle component, measure the contribution of trends and cycles and indicate their relative importance in the stochastic process. There is no stochastics in cycles when σ v is zero and there are no cycles when ϕ 1 and ϕ 2 are zero. If σ w , the standard deviation of the growth rate, and λ are zero, the time series collapses to a constant growth rate case. When λ<1 the time series is I(1) and when λ=1, i.e., a random walk growth rate is assumed, the time series is I(2). Therefore, the Durability of cycles is the sum of the two cycle equation coefficients ϕ 1 and ϕ 2 . When the sum of ϕ 1 and ϕ 2 , which is confined by (-1, 1), is large, the effect of a shock to the cycle, though will eventually disappear, would be long-lasting and the process of mean-reversion takes place slowly.

There are three indicators for growth features. Persistence of shocks to growth is measured by 1/(1-λ), the cumulative effect by a unit shock 2 . Impact of shocks to growth 3 has two 2 Notice the shock is to the growth rate so its effect on the level of output is cumulative. 3 Much of the empirical business cycle literature regards technology shocks as those of long-run effects that can be either positive or negative. Few argue, for technological progress, they can only be positive. We assume the former, considering technical advancement as well as technical obsoleteness. Evidence of obsolete techniques and their (persistent) effects on the sector and the economy can be easily found, e.g., the British Rail Track, and the British car manufacturing industry. The decline and eventual disappearance of the car manufacturing industry in Britain is in essence a technology problem, it is not due to a strong sterling, nor high costs of labour; if low efficiency is to be blamed, it has its roots in technology (political issues are beyond the consideration elements: σ w that gives a measure for the size of shocks and the immediate impact, and σ w /(1-λ), shock size adjusted cumulative effect on trend levels. Average growth rate, gc/(1-λ), is the average or stationary mean of the growth rate in the whole period.

{Table 1 about here}

In the general model, Model 0, the average growth rate is measured by gc/(1-λ). In Model 1 where restrictions gc=0 and λ=1 are imposed, the growth rate is a random walk and no stationary mean value exists for the growth rate. In Model 2, the mean value of the growth rate is simply gc. We regard the specifications of Model 0 and Model 2 as being more justifiable.

Because the growth rate is stationary in Model 0 and Model 2, and it is non-stationary in Model 1. In Model 0, the growth rate, either expected or realized, is time-varying. Although it will be subject to empirical verification, we favor the specification of Model 0, because changes in growth are usually the result of technological progress, the impact of which will be realized gradually over some time4 .

Empirical results and discussions

The data used in this study are UK sectoral and aggregate GDP, starting in the first quarter, 1955 and ending in the first quarter, 2002, seasonally adjusted at the 1995 constant price. In addition to the aggregate GDP, the seven main sectors used in the study are: Agriculture, Forestry and of this paper). One can immediately sense these when visiting the factories if s/he is not scienceilliterate. Technical obsoleteness comes in two forms, absolute and comparative. The case of the British railways is the former where some parts of the system are simply beyond normal usage and will continue to be so for a considerable period. The British car manufacturing industry is the latter; any technical progress elsewhere that is not to be realized, matched or surpassed in Britain leads to technical obsoleteness in the British car manufacturing industry. and Services (J-Q, including business services and finance, and government and other services).

The Mining and Quarrying sector (C) is excluded, as its weight in UK GDP is minimal and has being declining over decades; and more importantly, its change has been mainly influenced by unconventional economic forces and other factors.

The estimation results are reported in Table 2. Graphs for the aggregate GDP are plotted in Figure 1 and those for sectors are drawn in Figure 2 to being marginally significant), so the cycle is modestly durable and persistent. The effect of a shock to the cycle will eventually fade away but it will take time. The standard deviation of the cycle, σ v , is 0.8390e -2 ; the standard deviation of the trend, σ u , is not significant but the standard 11 deviation in growth, σ w , is 0.3341e -2 , the contribution of which to the standard deviation in the trend is σ w /(1-λ 2 ) 0.5 = 0.4571e -2 . Overall, the standard deviation of the cycle is about two times larger than that in the trend, indicating the cycle is the main source of the stochastic fluctuation in UK GDP 5 . The average quarterly growth rate over the whole period is gc/(1-λ) = 0.1932e - 2 /(1-0.6825)≈0.61%, or 2.4% annually. With λ being 0.6825, the growth rate is persistent (notice the different scales of the vertical axis for trend, cycle and growth), which appears to coincide with a diffusion process in technical changes. Tests on restrictions confirm that the general model, Model 0, is the best to fit UK GDP data. Model 1, where the growth rate is a random walk, performs worst and is rejected on the ground of a significant LR statistic at the 5% level;

while the LR test fails to reject the constant growth rate model, Model 2. The three graphs in Figure 1 exhibit the trend, cycle and growth reasonably well. Panel (b) clearly demonstrates UK business cycle features and panel (c) shows a stochastic growth rate that is persistent but meanreverting (appears to reject Model 2 also).

{Table 2 about here}

Having inspected UK aggregate GDP and gained a broad view of the British economy, we carry on to scrutinize sectoral output, with reference to Table 2 andFigures 2-8. While on average the standard deviation of the cycle is about two times larger than that in the trend (incorporating growth), as found in the aggregate GDP, the contribution of trends and cycles differ across borders. Industry E, Electricity, Gas and Water Supply, shows the largest cyclical fluctuations with its standard deviation (0.3743e -1 ) being more than 6.6 times larger than that in the trend [σ w /(1-λ 2 ) 0.5 = 0.5685e -2 as the contribution of growth uncertainty to the standard deviation in the trend equation, and a very small insignificant σ u ignored], whereas industry D, Manufacturing, has the largest relative contribution from the trend, and the standard deviation of the cycle (0.1083e -1 ) is smaller than that contributed by the trend overall [0.7176e -2 for σ u plus 5 The fluctuation in the fitted AR(2) equation is non-stachastic. [0.6257e -2 for σ u and σ w /(1-λ 2 ) 0.5 = 0.5496e -2 from growth]; and the contribution from the trend and that from the cycle (σ v is 0.1159e -1 ) are of comparable importance. On the other hand, industry F, Construction, is next to the energy sector (Electricity, Gas and Water Supply) to have considerably large contribution from the cycle, with the standard deviation in the cycle (0.2622e - 1 ) being 4.5 times larger than that in the trend (0.5931e -2 ). The behavior of the services sector J-Q is rather different. Some problems may arise from the data; in the first 15 years period it exhibits deterministic seasonal cycles. This may have affected parameter estimation, though the model is able to pick the deterministic cycle in the data. Given the results shown in the table, fluctuations in the services sector, both in the trend and the cycle, are much smaller and smoother compared with all other sectors in the economy. Furthermore, it is the only sector to have a much smaller contribution from the cycle (σ v = 0.1866e -2 ), either in absolute terms in comparison with all other sectors, or in relative terms in relation to its own trend (σ u = 0.4892e -

2

). These results and findings indicate that the energy sector (Electricity, Gas and Water Supply)

and the construction sector are mostly subject to cyclical fluctuations, though a considerable part of fluctuations in the energy sector could be seasonal rather than business cycles, and caused by a volatile world oil price. On the other hand, the services sector (including government services)

is the least vulnerable to suffer business cycles. We provide an indicative rank of the sectors, viewed purely from the trend-cycle standpoint of this section, which considers the relative contributions of the cycle and the trend, and takes into account the overall fluctuations in the time series. We place similar sectors in one category, as it is not helpful to be too trivial. Ranked from low to high with reference to fluctuations, they are: (1) Services; (2) Distribution, Hotels, We then inspect the durability of cycles across sectors. This, as discussed earlier, is measured by the sum of the two cycle coefficients ϕ 1 and ϕ 2 . We have reasonably found high durability or persistence of cycles in the Construction sector (0.9741) and the Transport, Storage and Communication sector (0.9731), and low durability or little persistence in the Services sector (ϕ 2 = -0.8590, ϕ 1 is a small insignificant number 0.09104). But cycles in the sector of Distribution, Hotels, Catering and Repairs also appear to be highly durable (0.9797). The lowest durable cycles are found in industry E, Electricity, Gas and Water Supply (0.5528); and cycles are modestly persistent in industry D, Manufacturing (0.8168) and industry A&B, Agriculture, Forestry and Fishing (0.8637). These three sectors A&B, D and E also have a positive ϕ 1 and negative ϕ 2 , indicating there would be more alternations in their cyclical fluctuations. These sectors constitute production sectors also, with A&B being agricultural production and D and E being industrial production while the mining sector is excluded.

In the above, the main consideration is the relative contribution of the trend and the cycle. Now we turn to the important issue of growth. The size of shocks to growth in Industry D, the manufacturing industry, is by far the largest (0.9885e -2 ). Although the shocks are not highly persistent in terms of the unit cumulative effect [1/(1-λ)=2.2026], its shock size adjusted cumulative effect [σ w /(1-λ) = 0.2177e -1 ] is also the largest. In other sectors where shocks to growth are of considerable consequence are Industry E, Electricity, Gas and Water Supply, with the size of shocks being 0.3396e -2 and shock size adjusted cumulative effect being 0.1715e -1 ; Industry I, Transport, Storage and Communication with 0.3434e -2 for the size of shocks and 0.1567e -1 for shock size adjusted cumulative effect; and Industry G&H, Distribution, Hotels, Catering and Repairs with 0.4161e -2 for the size of shocks and 0.1515e -1 for shock size adjusted cumulative effect. In industry F, Construction, the size of shocks is large but insignificant, so the Forestry and Fishing and sector J-Q, Services, the size of shocks to growth is much smaller and insignificant. There might be some data problems in the service sector of which we have been aware. With regard to average growth rates measured in gc/(1-λ), the highest growth sectors are found to be Electricity, Gas and Water Supply (3.3% annually) and Transport, Storage and Communication (3.2% annually); and Manufacturing (1.3% annually) is only second to Agriculture, Forestry and Fishing (1.2% annually) to be the lowest. In the middle, we find the broadly defined services, sectors G&H (2.1% annually) and J-Q (2.5% annually). We summarize these business cycle features by sector in Table 3. Sectors with similar features under each of the business cycle aspects are grouped into one category and painted with the same color.

{Table 3 about here}

Finally we examine model specifications. All the restrictions are tested against the general model for the sectors as well as the aggregate GDP. The likelihood ratio test, as reported in

Conclusions

In this paper we have examined the behavior of UK GDP sectors through decomposing the time series data into trend and cycle components using the Kalman filter. Unlike previous business cycle exercises, we determine the characteristics of trend growth empirically. That is, the trend growth rate in this paper is not pre-specified as either stationary as most researchers would insist, or a random walk as in [START_REF] Clark | The cyclical component of the U.S. economic activity[END_REF] and a few more recent studies, e.g., [START_REF] Kim | Friedman's plucking model of business fluctuations: tests and estimations of permanent and transitory components[END_REF]. The general model of this study encompasses a number of alternative specifications about trend growth, therefore accommodating diverse views on growth.

There is reasonable support in the results for the general model, and the rejection of the random walk growth rate model is overwhelming across UK GDP sectors. The growth rate is best specified as being mean-reverting and time-varying, though it is not materially different from a constant. This is sound, taking into account the economic behavior of output growth over an indefinite time horizon, a setting for the theory as well as for model estimation.

Overall, the results indicate that the Services sector is least to subject to business cycle fluctuations, and that the energy sector of Electricity, Gas and Water Supply and the Construction sector are most vulnerable to suffer cyclical fluctuations, though a considerable part of fluctuations in the energy sector could be seasonal rather than business cycles, and caused by a volatile world oil price. With regard to durability of cycles, the Services sector again exhibits low durability or little persistence in cycles; while the Construction sector, the Transport, Storage and Communication sector and the Distribution, Hotels, Catering and Repairs sector appear to be on the other end of the spectrum.

The largest impact of shocks to growth is reasonably found in the Manufacturing sector, a technology intensive and sensitive sector, which is followed by the Electricity, Gas and Water Supply sector, the Transport, Storage and Communication sector and the Distribution, Hotels, σ u 0.7958e -2 ‡ (0.4229e -2 ) 0.7176e -2* (0.5359e -3 ) 0.2357e -4 (0.2143e -1 ) 0.5931e -2 ‡ (0.3239e -2 ) 0.2787e -2 (0.1950e -2 ) 0.6257e -2* (0.3650e -3 ) 0.4892e -2* (0.6649e -3 ) 0.1237e -3 (0.4260e -2 ) σ v 0.1861e -1* (0.1855e -2 ) 0.1083e -1* (0.5221e -3 ) 0.3743e -1* (0.2496e -2 ) 0.2622e -1* (0.6503e -3 ) 0.1204e -1* (0.6224e -3 ) 0.1159e -1* (0.9448e -3 ) 0.1866e -2* (0.2373e -3 ) 0.8390e -2* (0.5070e -3 ) σ w 0.6466e -3 (0.4438e -3 ) 0.9885e -2* (0.1523e -2 ) 0.3396e -2* (0.1249e -2 ) 0.6216e -2 (0.8314e -1 ) 0.4161e -2* (0.6209e -3 ) 0.3434e -2* (0.7823e -3 ) 0.8512e -3 (0.9898e -3 ) 0.3341e -2* (0.3311e -3 ) Gc 0.6787e -4* (0.1571e -3 ) 0.1529e -2* (0.1396e -3 ) 0.1619e -2* (0.3435e -3 ) 0.2153e -2* (0.3740e -3 ) 0.1428e -2 † (0.6304e -3 ) 0.1761e -2* (0.7007e -4 ) 0.3441e -2* (0.2088e -3 ) 0.1932e -2* (0.3888e Impact of shocks to growth: Size of shocks to growth and shock size adjusted cumulative effect on trend levels. From top to bottom: increasing impact, small to large.

-Q) Q) Q) Q) (a) (b) (c) Log J-Q 3 3.5 4
J-Q J-Q J-Q J-Q E G&H E
Persistence of shocks to growth: Measure of persistence of the effect on the growth rate. From top to bottom: increasing persistence, low to high.

Average growth rate: Average or stationary mean of the growth rate in the whole period. From top to bottom: decreasing growth rate, high to low.

  modifies a linear real business cycle model to allow for disaggregate factors in the generation of macroeconomic fluctuations, and then makes attempts to determine the relative importance of aggregate and sectoral shocks by performing principal components analysis on the residuals from a VAR of output growth rates in 19 UK industrial sectors.Investigating shock persistence in property and related sectors, it has been revealed by Wang that shocks from the housing market have the largest effect on the persistence in commercial property, followed by the services sector, production sector and construction. Conversely, shocks in property company shares, i.e. the stock market investment in property, have relatively small effects on the persistence in commercial property. These findings indicate business cycle evolution patterns, especially in economic downturns, which the world economy experiences in the last few years of the first decade in the new millennium. They also point to the likely triggering of recessions. Based on Hilferding's theory of disproportionality in capital accumulation in a two-sector economy, Dibeh (2001) develops a Marxian model of the business cycle. The disproportionality arises from the existence of time delays in production generated by the differential capital intensity in the two sectors. The time delays produce an asymmetric price structure that causes overproduction and crisis. Numerical simulations show that the model produces an economy-wide business cycle phenomenon and various dynamics ranging from monotonic convergence to explosive oscillations. Analyzing UK quarterly GDP deflator and 29

  the movement of the term structure of nominal bond yields in the post-war USA and the channels through which such macro-shocks influence the yield curve.

.

  Overall fluctuations indicate the relative contributions of the cycle vis-à-vis the trend measured in standard deviations, and total fluctuations in the cycle and the trend. It consists of the sum of the standard deviations of the cycle and the trend, and the ratio of the standard deviation of the cycle to that of the trend. The first element measures the total fluctuations in the output series. The standard deviation in growth, σ w , also constitutes part of the standard deviation in the trend, which is σ w /(1-λ 2 ) 0.5 , so total volatility in the trend is The second element explains the relative contributions of the cycle and the trend and adjusts the total fluctuations according to the relative importance of the cycle. The larger the ratio of the standard deviation of the cycle to that of the trend, the larger the contribution of the cycle, and the larger is the indicator of overall fluctuations.

  ); Manufacturing (D); Electricity, Gas and Water Supply (E); Construction (F); Distribution, Hotels, Catering and Repairs (G&H); Transport, Storage and Communication (I);

  to Figure 8, with the top, middle and bottom panels for output and trend, cycle, and growth respectively. We concentrate the analysis on the more recent period, which people can recall vividly. The decline in the British economy appeared to have eased since the 1980s and, in the last twenty years since then, people have witnessed a rather large-scale cycle and hoped to iron out or alleviate the next. Starting in the early 1980s at the trough of the previous recession, the British economy, measured in aggregate GDP, climbed up steadily to last for almost one decade. The economic boom climaxed towards the end of 1989 shortly before the economy endured one of the most severe recessions in its modern history, undergone especially phenomenally in property market and stock market collapses. The recovery did not take place until two years later and the economy has being running smoothly since 1994, with reasonable GDP growth accompanied by (sometimes exceptionally) low inflation and an interest rate converging to the lower interest levels in the US and the rest of the EU. Now we analyze what the models tell us about trends, cycles and growth in UK aggregate GDP. The cycle coefficients ϕ 1 and ϕ 2 are 0.6222 and 0.2021 (the latter is only close

  1180e -1 from growth]. Sharp falls of output in sector E in 1985 reflect the fact of the world oil price collapse in that time. Industry I, Transport, Storage and Communication, is next to manufacturing to have the second largest contribution from the trend component

  ; Transport, Storage and Communication; Manufacturing; (3) Agriculture, Forestry and Fishing; (4) Construction; Electricity, Gas and Water Supply.

  to growth is not clear from viewing the results. With sector A&B, Agriculture,

  Figure 1. GDP Aggregate Figure 1. GDP Aggregate Figure 1. GDP Aggregate Figure 1. GDP Aggregate (a)

  fluctuations: Relative contributions of the cycle vis-à-vis the trend measured in standard deviations, and total fluctuations in the cycle and the trend. From top to bottom: increasing overall fluctuations. Durability of cycles: Sum of the two cycle equation coefficients. From top to bottom: increasing durability, short to long.

  size of σ u , σ v , and σ w , together with ϕ 1 and ϕ 2 , demonstrate the behavior of GDP series. To demonstrate and summarize the behavior of sectoral output with the above parameters in the model, we propose the following groupings of business cycle features, with Table1summarizing these groupings.

	relative importance and
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	8

  Table2, rejects Model 1, the random walk growth rate model, in all seven sectors and the aggregate GDP, and two of the rejections are at the high level of 1% and two at the modest level of 5%. On the other hand, restrictions imposed on Model 2, the constant growth rate model, are rejected only in three out of eight cases, none at a high level of significance. So there is reasonable support in the results for the general model, and the rejection of the random walk growth rate model is overwhelming. While the growth rate is best specified as being timevarying and mean-reverting, tests on the restrictions also indicate that the time-changing component may be fairly negligible. From human beings' perspective, this suggests that people believe the growth rate may change but are less confident in how it changes.

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	15

Table 2 .

 2 Decomposition of GDP sectors into trend and cycle with a stochastic growth rate using the Kalman filter Decomposition of GDP sectors into trend and cycle with a stochastic growth rate using the Kalman filter Decomposition of GDP sectors into trend and cycle with a stochastic growth rate using the Kalman filter Decomposition of GDP sectors into trend and cycle with a stochastic growth rate using the Kalman filter

	Page 25 of 28	Submitted Manuscript
	5
	4.5
	1956:01 1962:01 1968:01 1974:01 1980:01 1986:01 1992:01 1998:01
		J-Q	Trend
		Log J-Q
	0.01
	0.005
	0
	1956:01 1962:01 1968:01 1974:01 1980:01 1986:01 1992:01 1998:01
	-0.005
	-0.01
		Cycle
		Log J-Q
	0.007
	0.0065
	0.006
	0.0055
	0.005
	1956:01 1962:01 1968:01 1974:01 1980:01 1986:01 1992:01 1998:01
		Growth

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Page 27 of 28 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  significant at the 1% level; † significant at the 5% level; ‡ significant at the 10% level. Standard errors in brackets. LR is the likelihood ratio statistic

				Submitted Manuscript
	λ General model Likelihood Value	0.9772 * (0.3252e -1 ) 618.7923 F o r	0.5460 * (0.2792e -1 ) 658.1481	0.8020 * (0.3528e -1 ) 501.4905	0.4973 * (0.8767e -1 ) 575.2099	0.7253 * (0.4451e -1 ) 698.7701	0.7808 * (0.8810e -2 ) 686.6194	0.4576 * (0.7459e -1 ) 845.3897	-3 ) (0.5380e -1 ) 0.6825 * 759.3137
	Restrictions: gc=0, λ=1 Likelihood value LR Restrictions: λ=0, σ w =0 Likelihood value LR	616.2179 5.1488 ‡ P 652.1676 11.9610 * 616.9088 3.7670 656.9255 2.4452 e e r	494.3601 13.8384 * 496.8284 8.9018 †	572.7405 4.9388 ‡ 573.9711 2.4776	695.1590 7.2222 † 696.1267 5.2868 ‡	683.6324 5.9740 ‡ 683.7019 5.8350 ‡	842.2776 6.2242 † 844.8977 0.9840	756.2246 6.1782 † 757.4017 3.8240
			R
			e
				v i e
				w
						28

*

Table 3 .

 3 Table 3. Table 3. Table 3. Business cycle features by sector Business cycle features by sector Business cycle features by sector Business cycle features by sector

	Overall	Durability of	Impact of shocks	Persistence of	Average growth
	fluctuations	cycles	to growth	shocks to growth	rate

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

[START_REF] Lippi | Diffusion of technical changes and the decomposition of output into trend and cycle[END_REF] suggest a diffusion process of higher order moving averages for technological changes featured by growth in the trend. The AR(1) process, as we adopt in the study for trend growth, can approximate higher order MA and is parsimonious, and the experience tells us that the use of a parsimonious model is not only simpler, but also more effective.

Overall fluctuations Durability of cycles

Impact of shocks to growth

Persistence of shocks to growth Average growth rate