Econometric and Statistical Methods: Special Topics < C -Mathematical and Quantitative Methods, I12 -Health Production

Keywords: Data Envelopment Analysis, Hospitals Efficiency, Bootstrapping

During the 1990s health related expenditures in Germany amounted to more than 10% of GDP; 1 the largest single item among all outlays was in-patient care, which added up to around 3.5% of the GDP. In 1995, this equalled € 45bn (OECD, 2001). 2 Therefore, cost containment in the hospital sector is a key issue in stabilising health related expenditures at a sustainable level.

Several studies comparing health care systems internationally concluded that the German system is not efficiently managed. 3 Especially the overcapacity of hospitals beds in Germany is considered to be a source of inefficiency. There are numerous reasons behind the excess -the OECD (1997) estimates 14% but much higher estimates exist -of beds in Germany implied by these figures.

A number of studies on the efficiency of hospitals in Germany attempted to quantify the savings potential. Neither of these studies gives a result representative for all German hospitals or for a specific segment of hospitals. For instance Henke, Wettke and Paffrath (1995) assess the cost efficiency of German hospitals by comparing the average case cost of hospitals in different cities. They give examples of two cities where cost for treating one particular ICD exceed the national average by 20% and 53%, and conclude that there are "dramatic differences in efficiency" between German hospitals. Swart et al. (1996) derive a ranking for 50 hospitals which treated patients insured with the Magdeburg (Saxony-Anhalt) regional subdivision of the Allgemeine Ortskrankenkasse (AOK), the leading German sickness fund. They rank hospitals by their length of stay (LOS) for the most common ICDs. The ICDs receive a number of points equal to #observations + 1 -rank; these points are then summed and divided by the maximum number of points which results in a score between 0 and 1. The scores for a subsample of nine homogeneous hospitals with a focus on internal medicine range between .77 and .27. According to the authors, these differences cannot be explained by Recently, a few studies on the relative efficiency of German hospitals using Data Envelopment Analysis (DEA) have been published (for surveys on health care studies using DEA, see Hollingsworth, Dawson and Maniadakis, 1999 and Chilingerian, 2004). Steinmann et al. (2004) Both DEA studies on Germany hospitals are based on a regional selection comprising very different hospitals in terms of departments and specialization. Therefore, one may expect their results to be downward biased because samples with similar hospitals would generate results that would have a higher average efficiency. On the other hand, DEA results are biased upward (see section III for details) and it is not possible to tell which of the two effects dominates. In the sequel, a DEA study on German hospitals will be carried out employing a precursor data set to the one collected for the hospitals benchmark, which was introduced into Germany law in 1998. 5 Unlike other DEA studies, the results derived are representative for hospitals in the old federal states in Germany. This paper is organised as follows: in the next section the data are described in detail. This is followed by an introduction to the bootstrapping method employed for the empirical analysis. 6 Next, the results of the analysis are presented and their implications are discussed. A brief summary of the findings concludes.

II DATA

The data used in the present study are a precursor data set to the one used for the mandatory hospital benchmark in Germany. This benchmark is carried out at the Wissentschaftliches Institut der Ortskrankenkassen (WIdO), the research institute of the AOK. The data are from 1994. 7 These are the latest data available that allow for a consistent benchmarking of hospital performance. Hospitals were remunerated with a base per diem that applied for all departments of the hospital. This negotiated per diem is available in the data contained in Arnold and Paffrath (1996).

The information 8 for some 1700 hospitals includes type of ownership (public, private or owned by a non-profit organisation), size (number of beds) and structure of the hospital (number and type of departments). For the following five departments, the average LOS and the case mix cluster (see the explanation below) are known: internal medicine, surgery, gynaecology, orthopaedics and ENT.

It is obvious that the entire services a hospital provides are reflected by its per diem. In order to ensure general comparability of hospitals, the analysis is restricted to type I, local, and type II, regional hospitals. Also, the information on structural classes (Strukturgruppen -hospitals with a similar departmental structure) is used. The concept of structural classes was developed for the purpose of identifying "comparable" hospitals; the information on structural classes is contained in the data supplied with Arnold and Paffrath (1995). 9 There are 77 different classes and it is suggested that "only hospitals belonging to the same structural class should be compared …" (Arnold and Paffrat, 1995, p. 273, translation by the author).

Only groups 11 to 13 of the 77 structural groups are included in the analysis. All these hospitals provide basic care. They have two main departments: one for medicine and one for surgery. The only difference between the three structural classes is the fraction of beds which are made available for patients treated by external specialists. 10 This leaves 160 hospitals, 108 of type I and 52 of type II. 11 Summary statistics on the hospitals in our data are given in Table 1. One input indicator used to compare these hospitals is their per diem rate. This differs from the WIdO-Benchmark, which is based on departmental case cost. A specification that treats the per diem rate and LOS separately is preferred as this makes it possible to analyse the efficiency of the hospitals in more detail. 12 The second input is the number of beds. Since there may be economies of scale and/or differences between the learning rates of institutions of different size it is necessary to ensure that the individual hospital and the benchmark technology are comparable in this respect. As the number of beds cannot be changed in the very short run it is treated as a non-discretionary variable (see Banker and Morey, 1986, and Staat 1999). The number of cases per year and the reciprocal of LOS, R-LOS, in the two fields of specialisation, model output. 13 R-LOS is used to ensure that hospitals with a low per diem but high LOS do not appear efficient or conversely hospitals with a higher per diem but low LOS do not appear inefficient. Also, departments with an adverse case mix should only be compared to other departments with a like case mix. Departments with an adverse case mix cannot be expected to have the same output-to-cost ratio as departments with an ordinary case mix. Departments treating ordinary case mix can be expected to have no higher resource use than other departments, however. Therefore, an indicator for adverse case mix is used as an additional output descriptor. Gerste (1996) divides surgery departments into six case mix clusters (for a description of the case mix clusters, see Gerste, 1996, Table 2). Only one of the clusters differs substantially from the others w. r. t. either LOS, intensive care days per 100 cases, minutes of care per day, the fraction of patients older than 75 years or the per diem rate.

The cluster with the most heterogeneous case mix has about twice as many intensive care days (59.1) as the remaining 5 clusters (between 22.6 and 33.1 days). The fact that this is the only cluster with cases that may require significantly more resources than the patients in any of the other clusters is reflected in the comments made by Gerste (1996, p. 123; translation by the author): "Although hospitals treat different case mixes there is no difference in the services they provide. ... Only the heterogeneous class with its high values for intensive care differs from the others. "

Of the four case mix clusters into which the internal medicine departments are divided again only the most heterogeneous group with an average LOS of 16.1 days differs from the other three groups for which LOS ranges from 12.9 to 13.8 days. 14 An output dummy for adverse case mix is set to one for the internal medicine or surgery departments with a heterogeneous case mix. Because the hospitals in the data belong to the segment of the hospital system that provides the most basic form of clinical care and involved cases are likely to be referred to more specialised hospitals only very few hospitals face an adverse case mix (see Table 1). A significant difference for the proportion of departments with an adverse case mix indicates that surgery departments in type II-hospitals treat a more heterogeneous case mix than departments in type I-hospitals as was to be expected. Hospitals may admit patients who are treated by external specialists of various disciplines. Hospital management will require the more resources the more disciplines; in order to take this potentially efficiency relevant factor into account, an indicator for the number of all fields of specialisation including those represented by external specialists is used.

The two input (discretionary: per diem; nondiscretionary: beds) and six output indicators (cases, R-LOS and case mix for the medicine and the surgery department as well as fields of specialisation) described above are used to assess the efficiency of the hospitals.

Table 1 contains summary statistics of the sample. On average, hospitals of type II have a slightly lager number of beds than hospitals of type I but the smallest hospital in the sample is of type II. The latter hospitals on average have slightly more fields of specialisation than type I hospitals but the hospital with the most fields of specialisation is of type I (see Table 1).

Except for LOS in the surgery department, which takes a maximum of three weeks in the local and of only two weeks in the regional hospitals, all indicators have very much the same range for the two types of hospitals. Treatment duration is longer for the inter- The parameters used in the analysis ensure that the hospitals to be compared will be very similar w. r. t. structure, size, case mix and tasks performed. The effects of any remaining differences on the results will be further reduced by the way in which DEA constructs the reference technologies. Since the observations will be compared only to reference technologies with an almost identical input-output-structure (see formula (4) below) the results should only reflect differences in the efficiency of service provision.

III DEA & BOOTSTRAP

This brief section on the DEA-estimation of the efficiency of production 15 starts with some basic definitions. A production set (1)

{ }

( , ) can produce , p q

x y x y

+ + Ψ = ∈ R
describes which amounts of some p inputs x can produce some q outputs y. An input requirement set ( ) X y is defined as:

(2)

{ } ( ) ( , ) p X y x x y + = ∈ ∈Ψ
The standard assumptions w. r. t. these sets maintained here are a) the convexity of ( )

X y for all y, b) that nonzero production of y requires some nonzero inputs x, and c) strong disposability of x and y. The efficient boundary of the input requirement set, ( ),

X y ∂ is defined as: (3) { } ( ) ( ), ( ) 0 1 X y x x X y x X y θ θ ∂ = ∈ ∉ ∀ < < and { } min ( ) k k k x X y θ θθ = ∈
is the input-oriented efficiency measure for a given combination of inputs and outputs, ( )

, . k k x y
The sets Ψ and ( ) X y as well as the efficient boundary ( ) As can be seen from formula (4), the efficiency measure is calculated as the maximum proportional reduction of inputs for observation k, given that the benchmark units (the terms containing the i λ ) produce at least as much output with no more input than ˆ.

X
n n n k k i i k i i i i i i i y y x x i n θ θ λ θ λ θ λ γ = = = ⎧ ⎫ = ≤ ≥ > = ≥ = ⎨ ⎬ ⎩ ⎭ ∑ ∑ ∑ Non-
( ) ( ) ( ) ( ) ( ) constant. where , , 2 ˆ of bias asymp. 1 1 3 1 2 0 0 0 3 2 0 = ′ ′ - - = - c c x g x f x g n x g
This bias 18 depends on sample size n as well as on "the curvature of the frontier and the magnitude of the density at the frontier" (Simar and Wilson, 2000, p. 59). It should be intuitively clear that that the bias decreases in sample size and density and increases in curvature. That is, in large samples with a high density of observations around a frontier with a mild curvature, one should expect a relatively small bias; when the sample is small, the frontier exhibits kinks (changes in curvature) and the density of observations around the frontier is low, a relatively large bias is to be expected.

For the case of one input and output, it is possible to derive a bias corrected estimator on the basis of (5). The effect of the number of observations on bias will be even stronger for the case of multiple inputs and outputs. For this case, however, no expressions equivalent to (5) can be derived and in order to obtain bias corrected estimates for the multiple-input-multiple-output case, the bootstrap method must be applied. The estimates ˆk θ and the bootstrap estimates * ˆk θ are related in the following way:

. *

ˆˆ( ) ~(

) . Simar and Wilson, 2000).

IV RESULTS

The results presented are representative for two important subsections of German hospitals. No representative results were obtained in previous studies (see the introductory section). Although some hospitals can potentially reduce input, i.e. lower per diem by almost 50%, the efficiency deficits on average range between 10% and 25% according to the bias corrected estimates.

The average score calculated with the bootstrap method (the standard DEA results in parentheses) for type I institutions is 0.75 (0.87) and 0.89 (0.94) for type II hospitals.

The mean for all hospitals is 0.79 (0.89). Both differences of the two means were significant. 19 In all cases, the MSE test (see section III) was passed, which indicates the homogeneity of the samples.

The scores of the ordinary model are between the ones found by Steinmann et al. (2004) and Kuntz and Scholtes (2004). This is remarkable, as the hospitals in the dataset used for the present study are much more homogeneous than the ones used for the two German DEA studies just mentioned. This indicates that the efficiency of German hospitals was lower than previously thought because much of the inefficiency found in the two other studies could be attributed to the fact that the datasets comprised very different hospitals. The bias corrected figures cannot be compared to the results of the two other studies but they again indicate that the degree of inefficiency is much higher than previously thought.

None of the facilities in this study had a score below 50%, which would have been surprising given the homogeneity of the data. However, about half of the type I low. Since differences in case mix and hospital structure are controlled for it can be ruled out that the low efficiency of these units is due to an atypical case mix. The theoretical evidence on the relevance of the type of ownership for the efficiency of hospitals is mixed (see Burgess and Wilson, 1996; Mobley and Magnussen, 1998).

German for-profit hospitals are thought to be significantly more efficient than other hospitals but there are too few of them in the data to allow any firm conclusions. Table 3 shows that efficiency does not differ significantly with the type of ownership for the hospitals in this sample.

Table 4 compares the characteristics of efficient and inefficient units. The average parameter values for the upper (efficient) quintile, the lower (inefficient) quintile and the three remaining quintiles are tabulated. Efficient hospitals in the upper quintile have lower per diem rates but LOS is not shorter in efficient hospitals compared to inefficient ones. Both the per diem rate and the mean cost per case (the latter variable is used merely for verification) differ significantly between the upper and lower quintile, mean LOS, however, is about the same.

Table 4: Distribution of characteristics by efficiency score

Efficient regional-care hospitals, especially their surgery departments, have slightly longer LOS compared to their inefficient counterparts. The difference is, however, not significant. Nevertheless, some efficient hospitals have a higher average LOS in some 12 departments than inefficient ones highlighting the fact that isolated benchmarks are inadequate when inefficient institutions are to be identified. In this case, the criterion "duration of stay (in the surgery department)" would lead to efficient institutions being placed at the end of a duration-based ranking (one such study is Swart et al., 1997, see the introduction). Therefore, only a simultaneous benchmarking of all relevant variables (for instance by DEA) can lead to a useful assessment of efficiency.

Type I hospitals have a lower (standard) average score than type II hospitals. Given that the efficiency in both samples is assessed on the basis of the same parameters and since the expected upward bias is stronger the fewer observations are available it was to be expected that the dataset with more observations has a lower average efficiency.

However, the difference between the standard and the bootstrapped DEA results is much larger for type I hospitals, i.e. (relative) bias was actually stronger in the larger sample. For this to be the case, the observations in the larger sample must be more heterogeneous than the ones in the smaller sample. This may be interpreted in the following way: given the same case mix, type I hospitals have a less homogeneous structure of outputs and inputs, i.e. cases, LOS and cost than type II-hospitals. Whatever the effects that lead to this relatively higher degree of heterogeneity -these may be a lower qualification of the staff on which there is no information in the data, a lower learning rate due to the lower number of cases treated, etc.-it must be stronger than effects that may favour smaller institutions like the possibility of cream skimming (if they cream-skim at all) by referring more problematic cases to more advanced facilities.

V CONCLUSION

The conclusions derived in the present study are still largely valid today because the German hospital system has largely resisted attempts by policy makers to induce more efficiency. The main obstacle was that hospitals were entitled by law to full reimbursement of their cost; this was abolished only recently. The main finding of the 13 study is that significant productivity differences between nearly identical hospitals exist. These differences are less dramatic than some findings in other studies on German hospitals; on the other hand, the bias-corrected results imply a much larger inefficiency than the results obtained in other DEA studies with German data. This is remarkable because the results of the present study were derived with rather homogenous data; consequently one may have expected smaller differences in relative efficiency compared to results obtained in other studies based on more heterogeneous samples.

The comparison of the average efficiency of the two samples showed that type I hospitals were on average less efficient than type II facilities. Therefore, the largest effects for the improvement of efficiency in the German hospital system could be expected if efforts followed a "bottom up" approach, concentrating on type I facilities NOTES 1 The average of this figure was 5.2 during the 1960s, 7.9 during the 1970s and 9.1 during the 1980s.

Taking some additional health care related expenditures into account, one seventh of the German GDP was spent on health care (OECD, 1997, p. 71). Sickness funds paid for about half of that. The latest figure is for 10.8% for 2002 (WHO, 2004).

2 Occasionally, figures from 1994 are given instead of presenting up-to-date information because the data for the subsequent analysis are from this year. Most up-to-date figures are available in OECD (2001).

3 Puig-Junoy (1998, p. 257) observes: "…, our study suggests that the most inefficient producers of health are Sweden, Denmark, Iceland, Germany and Finland". 4 The details of their methodology are given in Kuntz and Scholtes (2000). 5 There are many advantages of using DEA rather than parametric alternatives for the evaluation of health care providers (see Steinmann and Zweifel, 2003) F o r P e e r R e v i e w 14 6 One study on hospitals efficiency using an alternative bootstrap procedure is Löthgren and Tambour (1999). For a discussion of some problems with their bootstrap approach, see Simar and Wilson (2000). 7 Beginning in 1996, departmental per diem rates were introduced but no data are available. 8 For an extensive description of the data source, see Arnold and Paffrath (1995) p. 273ff. and Arnold and Paffrath (1996), p. 279 ff. 9 More than 2000 hospitals are contained in the 1995 raw data set and some 1800 in the 1996 data. There were 1700 hospitals that could be matched from both data sets. Due to the fact that reporting data was not mandatory at the time, the quality of the data was insufficient for some hospitals to be included in the 1996 report. The data in both reports are based on a 1994 sample of German hospitals. hospitals with any of the five departments for which the information on LOS and case mix is available but the data sets would be smaller than the ones used here.

12 See Chilingerian (2004) for some general comments on specification issues of DEA models used for evaluations in health care and Hollingsworth and Smith (2003) on ratio indicators. 13 If capacity use was at 100% one of the two indicators would be redundant. But average capacity use in hospitals is well below 100%, varies considerably and exceeds 100% in some cases.

Unfortunately, cases per year are rounded to the accuracy of 1000 cases in the data which creates some imprecision. Therefore an alternative measure calculated as number of beds times days per year times average capacity use in the respective federal state (see Krankenhaus Report, 1995, table 17-11) is used to check the robustness of the results. No significant differences were found.

14 Data kindly provided by Bettina Gerste, WIdO. 15 Introductions into DEA can be found in a number of recent textbooks, e.g., Thanassoulis (2004) This section reiterates the DEA bootstrap set-up developed in Simar and Wilson (1998, 2000). 16 Alternatively, one could solve the primal version of the linear programme. This so called multiplier form is often presented when weight restrictions are used. For a health application, see Womer et al. (2003).

  differing age of patients; consequently, they consider this variation as an indication of a substantial savings potential.

  report results for the federal state of Saxony for the years 2000 to 2002. Their sample comprises 105 hospitals with at least 20 beds which do not provide tertiary care. They assume constant returns to scale (CRS) and use academic, administrative and nursing staff as well as other expenses, patient days and beds as inputs. The number of cases treated in the internal medicine, paediatric, gynaecological, surgical or intensive care department function as (separate) outputs. The average efficiency for these hospitals ranges between 79 % (2002) and 83 % (2000). Kuntz and Scholtes (2004) report results on 92 hospitals located in Rhineland-Palatinate based on data from 2001. They assume a CRS technology and employ two different model specifications. 4 Their input parameters are the number of beds and total cost. One model is based on the total number of cases in 20 different ICD-clusters (model 1) as outputs; the other contains patients who did not stay over night and were treated in one of 20 different departments (model 2). Model 1 results in an average efficiency score of 95 % with 72.83 % of the observations considered to be fully efficient. Less than 8 % of the hospitals have an efficiency score below 90 %.

  Hence, only hospitals with a similar range of services can be compared. Hospitals are grouped into four categories by their function in German health care: hospitals of only local importance providing basic care without any large scale technical facilities (type departments which are of central importance for the region (type III) and tertiary care hospitals (type IV).

  and shorter for the surgery departments of the type I hospitals when compared to type II hospitals. These differences are not significant, however.

  zero weights i λ are assigned to efficient producers on the frontier. These producers jointly constitute the reference technology. evaluation based on a technology with variable returns to scale, i.e. the hospital that is evaluated and the reference technology operate on the same scale. Nondiscretionary inputs can be included in the model by changing the condition , the benchmark based on observations will in all likelihood be weaker than ( ); y X ∂ hence, the upward bias of the efficiency scores . θ

θ

  To gain some intuition on how bias arises one may picture a situation where, in addition to the observations in this specific sample, other hospitals exist, which are not contained in the sample and at the same time are efficient when compared to the DMUs (decision making units: observations in a DEA) in the sample. Thus, the observed efficiency of inefficient producers calculated on the basis of what is observed is an upward biased estimate of their true efficiency.

  the sampling properties of the efficiency estimates are available only for the one-input-one-output case. Assuming a monotone, concave production function with a frontier function ( )gthat is twice continuously differentiable at x 0 Simar and Wilson (2000, see section 3 and the results obtained byGijbels et al., 1999, cited therein) derive the following expression 17 for the asymptotic bias(5)

  an improvement, the bias corrected estimator should not have a mean square error (MSE) larger than the ordinary estimates. For this to be the case, the condition

  75 %. Only three of the type II facilities (or 6%) have a score this

F

Table 1 : Summary statistics

 1

Table 2 : Characteristics of case mix clusters

 2

Table 3 : Efficiency by group and ownership

 3

Table 2 :

 2 This followsSimar and Wilson (2000) in giving expressions for the output oriented case. Characteristics of case mix clusters

	17 18	0 b	=	( ) 0 f x g x ( 0 ,	)	,	2 b	= -	( ) 0 g x ''	/ 2,	1 c	=	( ) constant and . : densitiy funtion f
				o
							r
								P e
									e
										r
											R e
													v i
													e w
													15

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

Table 3 :

 3 Efficiency by group and ownership

			Submitted Manuscript
	Structure group	Type I θ (σ) obs.	Type II θ (σ) DMUs	total θ(σ) DMUs	Owner-	ship	Type I θ(σ) DMUs	Type II θ(σ) DMUs	total θ(σ) DMUs
		.74 (.10)	.87	.78	Non-	.73	.88	.77
	11	29	(.07) 13	(.11) 42	profit	(.09) 53	(.07) 18	(.11) 71
	12 .75 (.07)	.89	.80	private	.74	.93	.78
		48	(.06) 23	(.10) 71			(.10) 04	(.00) 01	(.12) 05
	13 .76 (.09)	.89	.80	public	.77	.89	.82
		31	(.06) 16	(.10) 47			(.08) 51	(.06) 33	(.09) 84
	total .75 (.09)	.88	.79	total	.75	.88	.79
		108	(.06) 52	(.10) 160			(.09) 108	(.06) 52	(.10) 160

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

Table 4 : Distribution of characteristics by efficiency-score

 4

							Page 22 of 22
		type I (basic)		type II (standard)
	µ (σ)	0 -20% 21 -80%	81 -	0 -20% 21 -80%	81 -
				100%			100%
	per diem rate	438.31	385.22	373.33	435.20	384.13	362.38
		(52.76)	(40.48)	(56.26)	(39.53)	(50.45)	(30.47)
	avg. cost per	5772.73	4585.94	4136.36	5800	4421.88	4000
	case	(1031.96)	(492.44)	(515.98)	(1619.33)	(623.59)	(577.35)
	number of	157.77	165.47	158	218.9	186.47	203.9
	beds	(35.36)	(44.51)	(53.47)	(56.34)	(66.29)	(105.08)
	average LOS	12.77	11.14	10.76	12.74	10.89	10.98
		(2.24)	(1.62)	(1.51)	(2.91)	(1.21)	(1.30)
	"Inner	13.27	12.27	11.86	12.89	11.73	11.42
	Medicine"	(2.09)	(1.78)	(2.07)	(1.90)	(2.00)	(2.04)
	"Surgery"	11.18	10.77	10.65	11.79	10.78	11.58
		(2.95)	(1.90)	(2.32)	(1.83)	(1.63)	(1.43)
	Adverse mix
	of cases
	"Inner	0	.02	.05	0	.06	.10
	Medicine"	(-)	(.13)	(.21)	(-)	(.25)	(.32)
	"Surgery"	0	.02	.18	.30	.25 (.44)	0
		(-)	(.13)	(.39)	(.48)		(-)
	Number of	2.95	3.78	4.27	3.90	4.19	4.5
	departments	(.90)	(.97)	(1.49)	(1.20)	(1.18)	(1.27)
	Observations	22	64	22	10	32	10

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

Submitted Manuscript

In many hospitals, a certain number of beds is reserved for patients of specialists who are not employed by the hospital but perform usually minor surgical procedures on in the hospital's facilities. The fraction of this type of beds is below 10% for group

11, between 10% and 20% for group 12 and exceeds 20% for group 13.11 The analysis therefore covers less than 10% of all German hospitals. It could be extended to some other

For the standard DEA results, this is based on simple t-tests. For an evaluation of various tests regarding the significance of differences of efficiency scores, see[START_REF] Kittelsen | Monte Carlo simulations of DEA efficiency measures and hypothesis tests[END_REF].


Alan E H Speight
email: a.speight@swan.ac.uk

Piers Thompson

Prof Alan Speight

Is Investment Time Irreversible? Some Empirical Evidence for Disaggregated UK Manufacturing Data

Keywords: Time Irreversibility, Nonlinearity, Investment , Disaggregated Data Time Irreversibility, Nonlinearity, Investment, Disaggregated Data. JEL: C10, E22, E32, L60

It has long been suggested that investment may be time irreversible, and consideration of the option value of waiting to invest has aroused renewed interest in this issue. This paper tests for time irreversibility in UK investment according to disaggregation by type of investment expenditure and across manufacturing sector groupings. The test results reported indicate that the irreversibility of investment patterns varies not only from industry to industry but also according to the type of capital being purchased, with significant time irreversibility detected in gross fixed capital formation and aggregate vehicles expenditure, and industrial sector groupings comprising fuels & oil refining, engineering & vehicles, and textiles & leather. However, only in the first and last of these series is time irreversibility attributable to non-linearities in the underlying data generating process, and consistent with threshold effects which may be associated with (S,s) type models of investment dynamics.

Introduction

Interest in the possibility of economic time series displaying asymmetric behaviour over the course of the business cycle is long-standing [START_REF] Mitchell | Business Cycles: The Problem and its Setting[END_REF]Keynes 1936;Burns and Mitchell 1946), and this interest has been formalized statistically in more recent years with the development of a variety of procedures for the statistical testing of business cycle asymmetry (Neftci, 1984;DeLong and Summers, 1996;Sichel, 1993;McQueen and Thorley, 1993;[START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF]Rothman, 1996, Verbrugge, 1997;[START_REF] Hinich | A frequency domain test of time reversibility[END_REF][START_REF] Psaradakis | P-value adjustments for multiple tests for nonlinearity[END_REF]. Applications of these test procedures have been widespread and varied, and have included, for example, applications to international industrial production and other coincident business cycle indicator data [START_REF] Speight | Testing for business cycle asymmetry: some international evidence[END_REF]Adreano and Savio, 2002;Pieró, 2004), national macroeconomic data (Mills, 1995;[START_REF] Stanca | Asymmetries and nonlinearities in Italian macroeconomic fluctuations[END_REF]Cook, 2000;[START_REF] Bodman | Steepness and deepness in the Australian macroeconomy[END_REF][START_REF] Olekalns | Cyclical asymmetries in Australian macroeconomic data[END_REF], inflation data [START_REF] Verbrugge | Longtitudinal inflation asymmetry[END_REF] and consumers' expenditure data in particular (Holly and Stannett, 1995;[START_REF] Speight | Are there asymmetries in consumption?: A closer look[END_REF]Cook, 2000aCook, . 2000b;;2000c). In distinction from more general business cycle modelling, the latter applications are in part motivated by the well-recognized distinction in theoretical models between consumers' expenditure and consumption as a service flow from a stock of goods and services, such that the purchase of durables may be more closely related to investment theory than consumption theory. Indeed, theoretical models of investment asymmetry are well developed, based on the 'option value of waiting to invest' and the resulting 'irreversible' nature of investment under uncertainty [START_REF] Pindyck | Irreversible investment, capacity choice, and the value of the firm[END_REF]Dixit, 1992;[START_REF] Dixit | Investment Under Uncertainty[END_REF][START_REF] Abel | Optimal investment with costly reversibility[END_REF][START_REF] Vandenbroucke | General trigger values of optimal investment[END_REF]. As potentially the most volatile component of aggregate expenditure, and in view of the attendant consequences for simulation exercises and the policy implications of such asymmetries being incorporated into the sectoral relationships embodied in large scale macroeconometric models, it is therefore perhaps of some surprise that the In view of the foregoing considerations, this paper provides an empirical examination of time irreversible asymmetry in UK investment data at the aggregate and disaggregated industry level through application of the time reversibility test introduced by Ramsey and [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF], which possesses appealing properties in the current context. In addition to characterizing the nature of asymmetric adjustment in investment, the test has the particular advantage of enabling discrimination between time reversibility which is driven by a nonlinear data generating process, in which case a non-linear model, possibly of threshold autoregressive form may be appropriate, and time reversibility which is instead driven by innovations drawn from a non-Gaussian probability distribution in the presence of a linear data generating process. Further, the test has power in discriminating between certain classes of nonlinear model, in particular the bilinear extension of the linear ARIMA model and the threshold autoregressive model.

The remainder of this paper is organised as follows. Section II provides an overview of the concept of the option value of waiting to invest and its implications for the lumpy and time irreversible nature of investment, and the associated potential for nonlinear dynamics of threshold form. Section III describes the statistical concept of time reversibility and the associated test procedure for time irreversibility and its underlying form. The data to be examined are discussed in Section IV, while Section V reports the time irreversibility test 1 Notable exceptions to this remark are provided by the work of Price (1995[START_REF] Price | Aggregate uncertainty, investment and asymmetric adjustment in the UK manufacturing sector[END_REF], who examines the consequences of aggregate uncertainty for capacity utilisation and investment, and demonstrates in particular the existence of asymmetric adjustment dynamics governing UK manufacturing sector investment in the context of a non-linear time series model exhibiting threshold effects. Relatedly, [START_REF] Sensier | Inventories and asymmetric business cycle fluctuations in the UK: a structural approach[END_REF] investigates the asymmetric properties and time-series behaviour of UK manufacturing inventories and production over the business cycle in the context of a model of asymmetric adjustment, as well as providing test statistics for the asymmetric properties of that data. Section VI summarizes the findings and conclusions of the paper.

II. Irreversible Investment

Capital adjustment costs were first introduced into neoclassical investment theory by [START_REF] Eisner | Determinants of business fixed investment[END_REF] in order to preclude the possibility of infinite rates of investment and, for simplicity, these adjustment costs were assumed to take a symmetric convex form. 2 The incompatibility of such assumed adjustment costs with the patterns observed in investment data subsequently encouraged authors such as [START_REF] Rothschild | On the cost of adjustment[END_REF] to develop investment functions with linear piecewise or fixed adjustment cost components. The assumption of this type of adjustment cost results in firms only undertaking investments (disinvestments) when the desired capital stock is a certain level above (below) the actual capital stock. In extension of this approach, '(S,s)' models such as those initially developed by [START_REF] Dixit | A simplified treatment of the theory of optimal regulation of Brownian motion[END_REF] and Abel andEberly (1996, 1997) attempt to model the way firms operate in three regimes: invest, don't invest, and divest.

Why should capital adjustment costs include a fixed component? One suggestion is that investment is at least partially irreversible. Thus, once made, a firm cannot immediately retrieve the whole cost of an investment. The explanations typically offered for this irreversibility of investment are that it is due either to investments being firm or industry specific, or firms facing a 'lemon's' style problem when reselling capital [START_REF] Akerlof | The market for lemons: quality uncertainty and the market mechanism[END_REF].

Whilst the potential irreversibility of investment is not a new concept, with a number of authors having previously drawn attention to the difficulty firms face in reselling capital 2 [START_REF] Lucas | Adjustment costs and the theory of supply[END_REF], [START_REF] Gould | Adjustment costs in the theory of investment of the firm[END_REF] and [START_REF] Mussa | External and internal adjustment costs and the theory of aggregate and firm investment[END_REF] also provide important contributions to the subject of capital adjustment costs, particularly the separation of costs associated with capital good supply (external) and disruption costs to the firm (internal). It should also be noted that Eisner and Strotz recognized the potential inadequacy of symmetric costs, and appreciated that adjustment costs may indeed be asymmetric in practice. [START_REF] Bernanke | Irreversibility, uncertainty and cyclical investment[END_REF][START_REF] Mcdonald | The value of waiting to invest[END_REF][START_REF] Dixit | Investment Under Uncertainty[END_REF][START_REF] Pindyck | Irreversible investment, capacity choice, and the value of the firm[END_REF]. In particular, if investment is irreversible, there is a return to waiting. Hence, a firm will only invest if the value of an investment is at least equal to the value of waiting for further information or better investment opportunities to become available (the option value of waiting to invest at a later date). 3

The empirical consequence of investment being irreversible should be that, rather than smoothing capital accumulation through time to move towards a new optimum level of capital (as would be the case under standard neo-classical theory), investment will instead be undertaken discretely in time, leading to spikes or 'lumpiness' in investment. More generally, the aggregate effect of firms making 'lumpy' investments under (S,s) rules at the microeconomic level will depend upon whether these investments are clustered in time or across industrial sectors. If the factors that induce firms to invest are wholly idiosyncratic there is likely to be little affect upon the patterns of aggregate or industry investment [START_REF] Bertola | Irreversibility and aggregate investment[END_REF][START_REF] Caballero | Plant-level adjustment and aggregate investment dynamics[END_REF]. 4 However, if investment inducing shocks occur predominantly at the industry or aggregate level then the bunching of firms' investment 3 See, for example, [START_REF] Dixit | Investment Under Uncertainty[END_REF] for a review of this earlier literature, and [START_REF] Caballero | Uncertainty, investment and industry evolution[END_REF] for an examination of the effects of irreversible investment on total investment and firm entry in a competitive industry subject to both industry-wide and idiosyncratic uncertainty. Relatedly, [START_REF] Gale | Delay and Cycles[END_REF] has demonstrated that where the profitability of investment is dependent on the level of economic activity, the resulting incentive to delay investment in a recession may deepen the recession and lengthen the period of economic recovery. [START_REF] Abel | An exact solution for the investment and value of a firm facing uncertainty, adjustment costs, and irreversibility[END_REF] have shown that irreversible investment in the presence of convex costs leads to regions of behaviour in which investment in a homogeneous good is not responsive to Tobin's q, and other regions where it is. For extensions and empirical studies of the nonlinear relationship of investment to q, including regions of insensitivity, see for example [START_REF] Barnett | Non-linear response of firm investment to q: testing a model of convex and non-convex adjustment[END_REF] and [START_REF] Corrado | Nonlinearities and inactivity in aggregate investment: some theoretical analysis and time-series evidence[END_REF].

On the consequences of irreversible investment and idiosyncratic uncertainty for differences in firms' capital stocks and the aggregate capital stock in a dynamic general equilibrium setting model, see [START_REF] Jamet | Irreversibility, uncertainty and growth[END_REF]. For a recent application of the real options methodology in the context of the valuation of agricultural investment decisions, see [START_REF] Tzouramani | Employing real options methodology in agricultural investments: the case of greenhouse construction[END_REF]. 4 In particular, [START_REF] Bertola | Irreversibility and aggregate investment[END_REF] and [START_REF] Caballero | Plant-level adjustment and aggregate investment dynamics[END_REF] show that under certain distributional assumptions, and in the presence of a large amount of idiosyncratic microeconomic uncertainty in particular, there is likely to be little synchronisation at the aggregate level but the history of accumulated past shocks can nevertheless greatly affect the elasticity of investment response to future shocks. spikes is likely to produce much more pronounced peaks in the industry or aggregate investment pattern [START_REF] Cooper | Machine replacement and the business cycle: lumps and bumps[END_REF]. 5 Alternatively, where firms face stochastic fixed adjustment costs, the typically sharp (S,s) rules are replaced a probability of adjustment (adjustment hazard) that responds smoothly to the firms' capacity gap, but yields nonlinear aggregate investment dynamics [START_REF] Caballero | Explaining investment dynamics in US manufacturing: a generalized (S,s) approach[END_REF]. 6 These considerations suggest that one way to empirically assess microeconomic investment irreversibility and its aggregate consequences is to test for the presence of time irreversibility in investment patterns at various levels of industrial sector aggregation, and in the aggregation of particular types of investment good expenditures.7

Moreover, the lumpiness of firms' investment patterns under the assumption of irreversible investment, as implied by (S,s) models, implies that some form of threshold model may be most appropriate for modelling investment. The most obvious class of time series model that would readily accommodate such behaviour in the modelling and forecasting of aggregate and industry level investment are of threshold autoregressive form, including threshold autoregressive (TAR), self-exciting threshold autoregressive (SETAR)

and smooth transition autoregressive (STAR) models. However, the adoption of one or more of these non-linear models instead of some linear alternative may first depend on whether the

III. Time Irreversibility

A dynamic stationary system is said to be time reversible if it is possible to substitute -t for t into the equation describing motion and leave the solution invariant, while the converse is true for a time irreversible system. 9, 10 More formally, [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] . In the presence of stationarity, the vectors ( )

n t t t X X X , , , 2 1 K and ( ) 1 1 , , , t t t X X X n n K
will also have the same joint probability distributions, which are assumed to be uniquely characterized by the respective sequence of moments and cross moments, such that time reversibility holds

only if [ ] [ ] i k t j t j k t i t X X E X X E = for all N k j i , ,
. The existence of a lag k for which these two moments are not equal provides a sufficient though not necessary condition for time 8 For further discussion of threshold models see [START_REF] Tong | Non-Linear Time Series: A Dynamical Systems Approach[END_REF]. 9 [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] offer the time path of a round projectile in (windless) flight as an intuitive example of a time reversible process, and the dispersal of ink in water as an intuitive example of a time irreversible process. Investment aside, the diffusion of technology provides an obvious further example of a time irreversible economic process. 10 In relation to the extensive empirical literature concerned with testing for asymmetries in economic time series, time irreversibility therefore captures those measures of asymmetry that are 'longitudinal', whilst purely 'transversal' asymmetries are time reversible; in particular, longitudinal 'steepness' asymmetry is time irreversible, while transversal 'deepness' asymmetry is time reversible. These alternative definitions refer to differing speeds of adjustment in expansions and contractions; for example, the business cycle has long been thought to be characterised by steeper recessions and longer more gentle expansions. A particular advantage following from the representation of asymmetries in terms of time irreversibility, as noted in Section I, is that the formulation in terms of time irreversibility lends itself to a discriminating test between circumstances where the process innovations are asymmetric but the impulse transmission mechanism is linear, and circumstances where innovations are symmetric and the impulse transmission mechanism is nonlinear. In contrast, the representation of asymmetries in terms of 'steepness' associated with the properties of the third moment of a series distribution does not permit such discrimination. See [START_REF] Mittnik | Asymmetries in business cycles: econometric techniques and empirical evidence[END_REF] and [START_REF] Psaradakis | P-value adjustments for multiple tests for nonlinearity[END_REF] for further discussion of a number of commonly employed tests of asymmetry. (1)

( ) [ ] [ ] i k t j t j k t i t j i X X E X X E k = , such that } { t X is time reversible if ( ) 0 , = k j i for all N k j i , , .
Given that the comparison of all bicovariances for a time series is impractical, a process is said to be time reversible to order m and degree K if

( ) 0 , = k j i
under the limits ( )

m j i + and K k
. 11 The test statistics introduced by [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] to test for time irreversibility are based on a method of moments sample estimator of the symmetric-bicovariance function in (1):

(2)

( ) ( ) ( ) k B k B k 2 , 1 1 , 2 1 , 2 ˆ = where ( ) ( ) + = = T K t k t t X X k T k B 1 2 1 1 , 2 ˆ, ( ) ( ) + = = T K t k t t X X k T k B 1 2 1 2 , 1
ˆand T denotes sample size. Standardizing by the small sample variance of ( )

k 1 , 2 ˆ then yields the ( ) k TR
statistic for a given lag order, k, as:

(3) [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] suggest that the appropriate values for m and K that should be selected in practice are 3 = m and 5 = K

( ) ( ) [ ] ) ( / ˆ2 , 1 2 , 1 k Var k k TR = 11
. The choice of order 3, and similarly degree 5, arguably provides the best compromise value for identifying time reversibility given the degrees of freedom typically available for economic time series. random variables then the exact small sample variance of ( )

k 1 , 2 ˆ is given by ( ) [ ] ( ) ( ) ( ) ( ) 2 3 2 2 3 2 4 2,1 / 2 2 / 2 Var k T k T k T k = µ µ µ µ where [ ] 2 2 t X E = µ , [ ] 3 3 t X E = µ and [ ] 4 4 t X E = µ
are replaced by their sample counterparts. 12 However, this result does not hold where } { t X is serially correlated, as is the case for the data under consideration here, and we therefore follow the recommendations of [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] and [START_REF] Rothman | FORTRAN programs for running the TR Test: a guide and examples[END_REF] , which we denote by TR(k*), 12 For further details, see Theorem 2 of [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF], and for a frequency domain variant of the TR test statistic based upon the bispectrum, see [START_REF] Hinich | A frequency domain test of time reversibility[END_REF]. 13 Under this latter approach an estimate of the variance of the variance of ( )

k 1 , 2 ˆ
is calculated by fitting a linear autoregressive (AR) model to the data, obtaining an estimate of the innovations variance, and then simulating a series using the estimated AR coefficients and generating a Gaussian error process with zero mean and variance equal to that estimated in the preceding stage. Values of ( )

k 1 , 2 ˆ
are calculated for each such replication for N replications, where N=100, permitting straightforward computation of the estimated variance using the replicated values for ( )

k 1 , 2 ˆ
. If the process is truly linear Gaussian, and time reversible, this is an exact simulation procedure. If the series is truly nonlinear (Type I time irreversible), the linear model constitutes a local approximation to the unknown nonlinear model, but the procedure should nonetheless provide asymptotically unbiased estimates of the variance of ( )

k 1 , 2 ˆ
in the presence of uncorrelated innovations. 14 Note that it is a requirement of the TR test statistic that the data possess a finite sixth moment. [START_REF] Chen | Testing time reversibility without moment restrictions[END_REF] have proposed an alternative to the TR test statistic which does not have any moment restrictions. However, as Chen et al. note, their test is not directly applicable to model residuals because it is a test of unconditional symmetry, and cannot therefore be used in order to discriminate between Type I and Type II time irreversibility. Where statistically significant time irreversibility is identified by the portmanteau version of the TR test it may be due to one of two causes. The underlying data generating process may be non-linear even though it has symmetric innovations or, alternatively, a linear model may possess innovations which are drawn from a non-Gaussian probability distribution. The existence of time irreversibility due to the underlying conditional mean model being non-linear is referred to as Type I irreversibility, whilst time irreversibility due to the presence of non-Gaussian innovations in the context of a linear model is described as Type II irreversibility. Both these forms of time irreversibility will lead to rejections of the null of time reversibility for the TR-test applied to the raw stationary data.

The test procedure advocated by [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] permits discrimination between the Type I and II alternatives based on a further application of the test to the residuals of a linear model fitted to the raw data. If the rejection of the null in the raw data is due to Type I irreversibility (a non-linear model with symmetrically distributed innovations) then the approximation using the linear model should generate residuals which, when the TRtest is applied to those residuals, will reject the null of time reversibility symmetry with probability greater than the power of the test. 15 If, on the other hand, the rejection of the null is due to Type II time irreversibility (a linear model with non-Gaussian innovations) the underlying innovations are i.i.d. and the approximation using the linear model will produce residuals that should be (approximately) time reversible. Thus, under Type I time irreversibility the TR-test applied to the linear model residuals should lead to rejections of the 15 Note that whilst Type I time irreversibility implies nonlinearity, the converse is not necessarily true, since there exist stationary nonlinear processes that are time reversible (e.g. [START_REF] Lewis | Gamma processes[END_REF]. The TR test cannot therefore be considered as equivalent to a test for nonlinearity of unknown form. , for low values of k are positive, steepness asymmetry in the form of long slow expansions and short steep declines is evident, whilst where such initial values are negative the asymmetry takes the form of fast sharp expansions and longer gradual declines. Further, [START_REF] Rothman | Characterization of the time irreversibility of economic time series[END_REF][START_REF] Hinich | A frequency domain test of time reversibility[END_REF] demonstrates by Monte Carlo simulation that where Type I irreversibility is identified, the appropriate class of non-linear model to represent the time series can be determined from the pattern of significant TR statistics generated by the TR2 test. Specifically, if the TR2 statistics are found to decline exponentially with k this is indicative of non-linear models of the bilinear (BL) class. However, where the underlying nonlinear model is of threshold form, as is our a priori expectation as discussed in Section I, and of the self exciting threshold autoregressive (SETAR) form in particular, TR2 tests will produce a large significant statistic for k=1 but further statistics will fall immediately near to zero for values of k>1. 17

IV. Data

In an effort to ascertain whether time irreversibility is more closely associated with certain disaggregated categories of investment expenditure or with certain industrial groupings, two 16 For more detailed discussion of this test procedure and its rationale, see [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF]. On the uses and possible limitations of the TR test statistic as a guide to model specification tool in application to nonlinear conditional mean and conditional variance model residuals, see [START_REF] Rothman | Higher-order residual analysis for simple bilinear and threshold autoregressive models with the TR test[END_REF] and [START_REF] Belaire-Franch | Higher-order residual analysis for AR-ARCH models with the TR test[END_REF], 2003, 2004). 17 Note that whilst Type I time irreversibility implies nonlinearity, the converse is not necessarily true, since there exist stationary nonlinear processes that are time reversible (e.g. [START_REF] Lewis | Gamma processes[END_REF]. The TR test cannot therefore be considered as equivalent to a test for nonlinearity of unknown form. and the last of these three categories includes much investment that is often referred to as plant and equipment investment. Two additional categories are also included: total business investment (the combined total of these three investment good categories), and gross fixed capital expenditure. 18 The National Accounts data for these investment series aggregated by expenditure type is quarterly, seasonally adjusted, expressed in constant 2001 £million prices, and runs from 1966:1-2004:2. 19 The second group of investment series analyzed are ONS First Release investment series for industry groups. The seven industry groups considered are: fuels; metals; chemicals; engineering; food; textiles; and other manufacturing. Table 1 shows industrial compositions of these industry groups by two-digit standard industrial classification (SIC)

codes. This data is also quarterly, seasonally adjusted and expressed in constant 2001 £million prices, but covers the shorter available sample period 1979: 1-2004:2. 20 Whilst non-stationarity due to drift is a pervasive feature of economic time series, and would lead them to necessarily exhibit time irreversibility, the essential question is whether the drift-free stationary components of those process are time reversible. Thus, as emphasized in Section II, the series being tested must be stationary in order for the TR test not to give misleading results, and we therefore employ logarithmic first differences of all series in the 18 Whilst the first four of these investment series are net of disposals of capital, gross fixed capital expenditure refers only to acquisitions.

V. Empirical Results

Tables 2 and3 present the results obtained from the application of the TR1 and TR2 tests discussed in Section II to the stationary transformations of the various investment data series described in Section III. 22 More specifically, Table 2 3.

It is immediately apparent from the TR1 test results reported in Panel A of Table 2 that highly significant time irreversibility is present in vehicles expenditure whilst new building work and other capital expenditure are time reversible. However, the aggregate of these three expenditure types, total business investment, is time reversible. In contrast, aggregated gross fixed capital formation is time irreversible. What do these results mean for the patterns observed in the series through the business cycle? The sign of the initial TR1 test 21 Whilst not reported here in full, the appropriateness of this transformation is confirmed by the results of Phillips-Perron test statistics which are unable to reject the presence of a unit root in the logarithmically transformed data but are able reject a unit root in the first difference of logarithms, for all the investment series considered. These inferences are also robust to variation in the test equation specification concerning inclusion or exclusion of a trend, constant or both, for the logarithmic data and the differenced logarithmic data. Full details of these unit root test results are available on request from the authors. 22 Recall from the discussion in Section II that an appropriate linear specification is necessary in order for the significance of the standardized test statistic results to be calculated and the alternative hypothesis of time irreversibility to be tested against the null of time reversibility, as well as in discriminating between Type I and Type II time-irreversibility. In particular, the results reported employ an estimate of the variance of ( )

k 1 , 2
ˆ calculated by Monte Carlo simulation using a a linear autoregressive AR(p) model fitted to the data of order p determined by reference to the Akaike Information Criterion (AIC). Alternative results based on application of the Schwarz (Bayesian) Information Criterion (BIC) which provide qualitatively equivalent results are omitted here in the interest of conserving space, but are available from the authors on request. 13 statistic is found to be negative for vehicles expenditure which (assuming the investment series are pro-cyclical) would suggest that vehicles expenditure follows a fast up and slow down pattern, whilst the positive sign associated with initial test values for gross fixed capital formation implies a slow up but fast down pattern over the business cycle. The pattern exhibited by vehicles expenditure is therefore most similar to that expected due to irreversible investment whereby firms make 'lumpy' investments, whilst that displayed by fixed capital expenditure is the more akin to the widely perceived asymmetry of the business cycle in general. On the basis of the TR2 test results in Panel B of Table 2 the time irreversibility detected in vehicle expenditure is of Type II form (implying an underlying linear model with non-Gaussian distributed innovations) whilst for gross fixed capital formation the time irreversibility is of Type I form (implying an underlying non-linear model with symmetrically distributed innovations). From the discussion in Section I, a possible causes of Type I time irreversible investment is the lemons' problem, which is likely to be a particularly acute issue in relation to vehicles. In contrast, it is of particular note that the pattern of TR2 test statistics for gross fixed capital formation, which imply an underlying nonlinear time series structure, are broadly indicative of some form of threshold process, and consistent with (S,s) type models of investment behaviour [START_REF] Rothman | Higher-order residual analysis for simple bilinear and threshold autoregressive models with the TR test[END_REF].

The test results for investment broken down by industry groupings in Table 3 As with the earlier results for capital formation, the TR2 test results for textiles & leather show that the largest TR statistic is found for k=1 with mostly much reduced statistics for k>1 implying that a threshold (S,s) process may provide an appropriate time series model to describe the investment dynamics of this industry grouping, but the sizable TR2 test statistic for k=4 suggests that the most appropriate non-linear representation for this series may not be of specifically SETAR form.

VI. Conclusion

It has long been suggested that investment may be time irreversible in the presence of a fixed capital adjustment cost component, due to capital being specific to a particular industry or firms' facing a lemons'-type problem in attempting to resell capital, and these rationales have been supplemented more recently by the introduction of the 'option value' of waiting to 23 Indeed, along with other industries based in the primary sector such as mining [START_REF] Brennan | Evaluating natural resource investments[END_REF] and forestry [START_REF] Morck | The valuation of forestry resources under stochastic prices and inventories[END_REF], where natural resources lend themselves easily to the explanation of the option value of an investment, the oil industry provides one of the 'benchmark' examples of irreversible investment. invest. However, depending on whether the dynamics governing firm investment decisions and shocks to the firms operating environment are common to an industry (or the macroeconomy) or wholly idiosyncratic, such irreversibilities may or may not be manifest in aggregate investment data. This paper has therefore sought to assess the empirical evidence for time irreversibility in UK investment not only at the aggregate level but according to disaggregation by type of investment expenditure as well as by disaggregation across manufacturing sector groupings, using the time reversibility test methodology of Ramsey and Rothman (1986) and [START_REF] Rothman | FORTRAN programs for running the TR Test: a guide and examples[END_REF][START_REF] Rothman | Higher-order residual analysis for simple bilinear and threshold autoregressive models with the TR test[END_REF] [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] applied to the stationarity adjusted logarithmic data for k=1,…,5 under the null hypothesis of time reversibility. TR2 refers to the portmanteau version of the standardized time reversibility test statistic applied to the residuals of an AR model of order determined by the AIC criterion as a discriminating test between Type I and Type II time irreversibility under the null of time reversibility in those residuals. In both cases p(TR(k*)) refers to the p-value for rejection of the null associated with the portmanteau version of the test based on the largest absolute value of the standardized test statistics (indicated in italics). An asterisk denotes portmanteau test significance at the 5% level.

Where only significant TR1 portmanteau test significance holds Type II time irreversibility is indicated (implying an underlying linear model with non-Gaussian innovations), where both TR1 and TR2 portmanteau test significance holds Type I time irreversibility is indicated (implying an underlying non-linear model with symmetrically distributed innovations), and where neither statistic is significant the process is fully time reversible (and linear). For further details, see Section II and [START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF].

  and specifically time-irreversible, properties of investment have not received closer scrutiny in the empirical literature. 1

  to the aggregate and disaggregate UK investment series considered.

  in question are empirically time reversible or not, which remains an open empirical question. 8

  Ramsey and Rothman to propose the a test procedure based on consideration of the difference between the symmetric bicovariances for a zero-mean stationary process:

  is a stationary sequence of zero mean independently and identically distributed(i.i.d.)

  associated probability of rejecting the null of time reversibility for that value of k, denoted by p(TR(k*)).

  reversibility, whilst Type II time irreversibility should not.16 To clarify these differing uses of the TR-test, and to facilitate tabulation of our results, we denote the TR test resulting from application to the raw data as TR1, and we denote the TR test resulting from application to the approximated linear model residuals by TR2.In relation to tests of asymmetric behaviour, it is of particular interest that simulation results noted by Ramsey and Rothman indicate that where the values obtained for

  data are considered here. The first comprises Office for National Statistics (ONS) National Accounts data for aggregate investment across three investment good categories: new building work; expenditure upon vehicles; and other investment. These series cover investment made by all sectors of the UK economy not just the manufacturing sector,

  indicate significant time irreversibility on the basis of TR1 test results for three groupings, fuels & oil refining, engineering & vehicles, and textiles & leather, whilst the remaining industry groups exhibit time reversible investment patterns (metals & metal goods; chemicals & man made fibres; food, drink & tobacco; and other manufacturing). The oil industry in particular hasbeen used previously as an example of an industry strongly affected by irreversible investment[START_REF] Paddock | Option valuation of claims on real assets: the case of offshore petroleum leases[END_REF]), so the finding of time irreversibility for the fuels industry sector

  unexpected.23 The patterns of investment suggested by the results displayed in Table3Panel A for the three industry groups with significant time irreversibility results suggest that for both engineering & vehicles and textiles & leather investment follows the slow up fast down dynamic widely accepted as characterising the business cycle. The implications are less clear for fuels & oil refining where the alternating TR1 test statistic signs for k=1, 2 suggests that this may be the only industry grouping with at least the potential for a fast up slow down pattern, as might be expected with irreversible investment, though the presence or absence of this property in this series cannot be definitively asserted by the test values reported. On the basis of the TR2 test results in Panel B of Table 3, Type I time irreversibility is found only for the textiles & leather industry grouping, with Type II time irreversibility present in the fuels & oil refining, and engineering & vehicles industry groups.

  in performing computation of the test statistic's empirical variance by means of Monte Carlo simulation of the hypothetical linear model which generated the data.13 Under the null hypothesis of time reversibility the expected value of

				ˆ	2	1 ,	( ) k	is zero
	and, under certain mixing conditions	TR	( ) k	is asymptotically distributed as	N	(	) 1 , 0	,
	permitting straightforward calculation of probability values associated with rejection of the
	null. K k ,..., 2 , 1 =

14

However, in order to account for possible interdependence among the test statistics for different values of k, we follow

[START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF]

and

[START_REF] Rothman | FORTRAN programs for running the TR Test: a guide and examples[END_REF]

in reporting a portmanteau version of the TR test statistic based on the largest absolute value among individual TR test statistics calculated for

  presents the results of TR1 and TR2 tests for time irreversibility to order m=3 and degree K =5 applied to the investment series disaggregated by expenditure type, including the standardised test statistics TR(k) for

	=	, 1	5 ,..., 2

k , and the portmanteau version of the TR test statistic based on the probability of rejecting the null of time reversibility for the largest absolute value among individual TR test statistics, denoted by p(TR(k*)). Corresponding results for investment disaggregated by broad industrial sector grouping are similarly presented in Table

Table 1 Industrial Sector Composition by Standard Industrial Classification (SIC) 2-Digit Code Sector Name (and ONS Code) 2-digit SIC(92) Industries contained within sector

 1 . The test statistics reported provide evidence of statistically significant time irreversibility in aggregate gross fixed capital formation, aggregate vehicle expenditure, and the industrial sector groupings covering fuels & oil refining, engineering & vehicles, and textiles & leather. These findings suggest that the irreversibility of investment patterns varies not only from industry to industry but also according to the type of capital being purchased. However, with the exception of gross fixed

	capital formation (and possibly the textiles & leather sector grouping), such time
	irreversibility is mostly attributable to non-linearities in the shocks to the processes governing
	such investments rather than non-linearities in the underlying data generating process
	associated with threshold effects consistent with (S,s) type models of investment, with
	attendant implications for the time series modelling of such processes. In particular, such
	findings are consistent with the generalized (S,s) theoretical framework developed by
	Caballero and Engel (1999). However, it remains possible that the levels of disaggregation
	entertained here are insufficient to fully reveal the consequences of the irreversible and
	potentially non-linear nature of investment at firm or plant level, and further analyses might
	usefully extend our results by considering data at finer levels of disaggregation by sector and
	expenditure type, and in their combination, data availability permitting.

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

Table 2 Time Irreversibility Test Results for Investment Data Disaggregated by Expenditure Type

 2 TR1 refers to the time reversibility standardized TR(k) test statistic of[START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF] applied to the stationarity adjusted logarithmic data for k=1,…,5 under the null hypothesis of time reversibility. TR2 refers to the portmanteau version of the standardized time reversibility test statistic applied to the residuals of an AR model of order determined by the AIC criterion as a discriminating test between Type I and Type II time irreversibility under the null of time reversibility in those residuals. In both cases p(TR(k*)) refers to the p-value for rejection of the null associated with the portmanteau version of the test based on the largest absolute value of the standardized test statistics (indicated in italics). An asterisk denotes portmanteau test significance at the 5% level. Where only significant TR1 portmanteau test significance holds Type II time irreversibility is indicated (implying an underlying linear model with non-Gaussian innovations), where both TR1 and TR2 portmanteau test significance holds Type I time irreversibility is indicated (implying an underlying non-linear model with symmetrically distributed innovations), and where neither statistic is significant the process is fully time reversible (and linear). For further details, see Section II and[START_REF] Ramsey | Time irreversiblity and business cycle asymmetry[END_REF].

	TR(k)

Notes:

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

Table 3 Time Irreversibility Test Results for Investment Data Disaggregated by Industry Group

 3

	TR (k)

Notes: TR1 refers to the time reversibility standardized TR(k) test statistic of

For an interesting recent applied analysis of the within-industry coordination problem and strategic behaviour implications that may arise as the result of such lumpy investment patterns in the context of the British brick industry, see[START_REF] Wood | Investment interdependence and the coordination of lumpy investments: evidence from the British brick industry[END_REF].

More specifically, and in some parallel with option value of waiting approach,[START_REF] Caballero | Explaining investment dynamics in US manufacturing: a generalized (S,s) approach[END_REF] explain investment delays as the strategic response of firms' to heterogeneous adjustment hazards which have the capacity to either magnify or dampen the response of investment to an aggregate shock, depending on the shock size. The presence of such adjustment hazards implies asymmetry due to nonlinearity in aggregate investment, consistent with threshold effects, in that there are occasional sharp responses of investment to current or cumulative shocks.

In the pursuit of evidence for such behaviour,[START_REF] Doms | Capital adjustment patterns in manufacturing plants[END_REF], for example, examine plant level investments in US manufacturing over the period 1973-1988 and find the distribution of investment rates to be symptomatic of irreversible investment, in that very few firms exhibit negative gross investment rates, whilst many firms group the majority of their investment in just three years of the sample period covered. The same study also found there to be a correlation between the number of investment spikes and the aggregate investment rate. In an interesting cliometric application of the Doms and Dunne methodology,[START_REF] Süssmuth | Are procyclical lumpiness and asymmetry in capital adjustment dateless phenomena? The case of firms in German industrialization: 1880-1913[END_REF] similarly reports evidence of lumpiness and asymmetry as defining characteristics of German firm level capital adjustment patterns for the period1880-1913.

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

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

Submitted Manuscript


The disappearance of the border barrier in some European Union countries' bilateral trade

Keywords: EU countries, Trade, Gravity Equation, Border effects

This paper estimates the tariff equivalent of the border barrier in each bilateral trade among European Union countries. The results show that there are large differences in the border barrier across European Union countries' bilateral trade. In some bilateral trade flows the border barrier has almost disappeared, whereas in other cases it is still equivalent to a 75 per cent tariff. The results also show that some countries have low border barriers in most of their bilateral trade flows with other EU members, whereas other countries persistently present large border barriers.

Introduction

In a famous paper published in 1995, John McCallum found that Canadian provinces trade 22 times more among themselves than with US States, once differences in income and distance where taken into account. This finding was striking considering that the Canadian -US border seemed to be one of the "thinnest" in the world, due to very low trade barriers, the use of a common language and the existence of highly integrated transport infrastructures. In fact, [START_REF] Obstfeld | The Six Major Puzzles in International Macroeconomics. Is there a Common Cause?[END_REF] considered this case as one of the six major puzzles of open economy macroeconomics.

Motivated by McCallum's finding, different studies have aimed to estimate the extent of the border effect among European Union (EU) countries [START_REF] Wei | Intra-National Versus International Trade: How Stubborn are Nations in Global Integration?[END_REF][START_REF] Head | Non-Europe: the causes and magnitudes of market fragmentation in the EU[END_REF][START_REF] Nitsch | National borders and international trade: evidence from the European Union[END_REF][START_REF] Evans | The Economic Significance of National Border Effects[END_REF][START_REF] Chen | Intra-National Versus International Trade in the European Union: Why Do National Borders Matter?[END_REF]. The EU has been considered a relevant case to test the pervasiveness of home-bias in trade, as this economic union has undertaken a large range of measures in order to facilitate trade among country members. These studies have estimated the average border effect among EU countries; that is, they have assumed that the barrier that borders impose is equal in all trade flows among EU countries. The aim of this paper is to test the validity of this assumption. We use a structural gravity model to estimate the tariff equivalent of the border barrier in each bilateral trade among EU countries. Our results show that there are large differences in the border barrier across EU countries' bilateral trade: in some cases the border effect has almost disappeared whereas in other cases large barriers remain. The remainder of the paper is organised as follows. The next section explains the model that is used for the analysis. Section 3 analyses the data-base. Section 4 presents the empirical results and the last section summarises the main findings.

The model

In order to estimate the border effect in each bilateral trade among EU countries we use the gravity model developed in [START_REF] Anderson | Gravity with Gravitas: A Solution to the Border Puzzle[END_REF]. In this model each good is differentiated by the place of origin and each country is completely specialised in the production of one good, whose supply is fixed. Demand is modelled through a constant elasticity of substitution (CES) utility function.

With these building blocks the model leads to the following exports' equation:

(1) ) 1 ( j j ij i i ij y P t p x σ β - =
where xij is exports from country i to country j, βi a distribution parameter, pi the supply price of the good produced in country i, tij the trade cost factor between i and j, σ the elasticity of substitution, Pj the consumer price index in country j and yj the gross domestic production in country j. According to equation (1) the value of exports from country i to country j is positively related with income in country j and negatively related with the price of country i's product relative to the price of the other competitors' products in country j. The consumer price index is calculated as:

(2) ) ( 1 1 1 σ σ β - - = i ij i i j t p P
Anderson and van Wincoop use market clearance conditions (country i's income should equal the value of its exports plus the value of the production sold in the domestic market), and assume that trade barriers are symmetric, that is tji=tij. These steps allow, firstly, to define each country's consumer price index as a function of partner countries' price indexes and trade barriers:

(3) j 1 , 1 1 ∀ = - - - i j i i i j t P P σ σ σ θ
where θi is country i's share in world income. 4), the value of exports from country i to country j, after controlling for size, is negatively correlated with the trade barrier between both countries and positively correlated with each country's multilateral resistance. It is clear that, given a trade barrier between country i and country j, a rise in country's j multilateral resistance makes country i's products relatively cheaper and, hence, the value of exports is increased. However, it is less obvious why a higher multilateral resistance in the exporter country i also rises the value of exports to j. According to the model, if a country i exporter faces higher trade barriers the demand for its products in partner countries will lower and, hence, the supply price will decline.

Therefore, for a given trade barrier between country i and country j, an increase in i's multilateral resistance will increase the value of exports.

Finally, the unobservable trade cost factor is modelled as a function of observables.

Among the elements that may explain the trade cost factor distance and the existence of a border are considered to exert a very important role. Taking a log-linear function, tij can be modelled as:

(5)

ρ ij ij ij d b t =
where d is distance, ρ is the distance parameter and bij is a parameter which takes the value 1 if the trade partners are the same country and l plus the tariff equivalent of the border barrier if i and j are not the same country. Equation ( 5) can be expanded to include other factors, such as linguistic similarity or adjacency, that may affect the bilateral trade cost. 4) and taking logs we derive the gravity equation that will be used to estimate the distance coefficient and the border barrier coefficient in each bilateral trade among EU countries; if n is the number of countries that are included in our sample, we estimate [n*(n-1)]/2 bilateral border barriers 1 .

The gravity equation to estimate becomes:

(6) ln ln ... ln ln 1 1 ) 1 ( 12 ij j i ij n n ij ij j i ij P P a a d y y x ε δ δ µ α σ σ + + + + + + + = - - -
where α is the constant term 2 , µ the distance coefficient: (1-σ)ρ and ε the error term.

Parameters a12 to a(n-1)n denote the border coefficient of each bilateral trade ((1σ)lnbij); δij takes the value of 1 if country i and country j are those that correspond to a coefficient's subscripts (independent of the order) and zero otherwise. The difficulty in estimating equation ( 6) lies in the quantification of multilateral resistances. Some authors use price indices to proxy them [START_REF] Head | Non-Europe: the causes and magnitudes of market fragmentation in the EU[END_REF][START_REF] Baier | The growth of world trade: tariffs, transport costs, and income similarity[END_REF]. However, as explained in [START_REF] Feenstra | Advanced International Trade: Theory and Evidence[END_REF] 2004), this procedure is problematic. Firstly, price indexes, such as the consumer price index, include both tradable and non-tradable products; moreover, prices may differ across countries due to local taxes; in addition to that, short term fluctuations in the exchange rate may introduce big differences in the price index across countries. Secondly, [START_REF] Anderson | Gravity with Gravitas: A Solution to the Border Puzzle[END_REF] argue that differences in multilateral resistances may also reflect non-pecuniary costs, as consumer preferences for domestically-produced goods.

Another approach is to substitute multilateral resistances by country specific dummies [START_REF] Hummels | Towards a Geography of Trade Costs[END_REF][START_REF] Rose | National Money as a Barrier to International Trade: The real Case for Currency Union[END_REF][START_REF] Feenstra | Border Effects and the Gravity Equation: Consistent Methods for Estimation[END_REF][START_REF] Chen | Intra-National Versus International Trade in the European Union: Why Do National Borders Matter?[END_REF]). More specifically, let e i τ denote an indicator variable whose value is 1 if country i is the exporter and zero otherwise, and m j τ an indicator variable whose value is 1 if country j is the importer and zero otherwise. Then the gravity equation ( 6) can be re-written as:

(7) .... ln ln 2 1 ) 1 ( 12 ij m j e i ij n n ij ij j i ij a a d y y x ε τ λ τ λ δ δ µ α + + + + + + + = -
+ + + - -= i a a d i i j ij n n ij ij e P P δ δ µ σ σ θ
Multilateral resistances implicit solutions are substituted in equation ( 6). Now, the right-hand side of equation ( 6) is written as a function of observables and parameters to estimate. Parameters are estimated with non-linear least squares, which minimizes the sum of squared errors 3 .

Although the second and third procedures yield consistent estimates, as Anderson and van Wincoop point out, the latter procedure is more efficient because it uses information of the full structure of the model. Hence, we use this approach in our estimation.

Finally, it is important to highlight that the [START_REF] Anderson | Gravity with Gravitas: A Solution to the Border Puzzle[END_REF] model is not free of limitations. Some authors have pointed out that some assumptions of the model, such as the one-sector economy and the absence of a non-tradable sector, are not very realistic; on the other hand, other authors have highlighted that some of the predictions of the model, such as too high trade barriers or large differences in countries' multilateral resistances, cast some doubts on the validity of the model [START_REF] Balistreri | Trade frictions and welfare in the gravity model: How much of the iceberg melts?[END_REF][START_REF] Helliwell | Border Effects: Assessing Their Implication for Canadian Policy in a North American Context[END_REF]. Although [START_REF] Anderson | Trade Costs[END_REF] defend the validity of their theoretical framework against these criticism, they also recognise that some extensions of the model that overcome some of its limitations would improve the estimation of border barriers.

Data

As we do not have real data on intra-national trade in EU countries, following previous studies [START_REF] Wei | Intra-National Versus International Trade: How Stubborn are Nations in Global Integration?[END_REF]Head y Mayer, 2000;[START_REF] Nitsch | National borders and international trade: evidence from the European Union[END_REF][START_REF] Chen | Intra-National Versus International Trade in the European Union: Why Do National Borders Matter?[END_REF] When available, gross production data are transformed to constant values using STAN data-base's gross production volume index; when this index is missing we use the value-added volume index provided by the same source. In order to compute gross merchandise production we subtract gross production of services, gross production of electricity, gas and water supply, and gross production in construction from the total gross production figure 5 . Trade figures are also converted to constant values using each EU country's trade deflator from the OECD National Accounts 6 . Following [START_REF] Helliwell | Measuring internal trade distances: a new method applied to estimate provincial border effects in Canada[END_REF] and [START_REF] Chen | Intra-National Versus International Trade in the European Union: Why Do National Borders Matter?[END_REF], we calculate intra-national distance as a regional GDP weighted average of the great circle distance among a country's regions and the great circle distance within each region. The distance among countries is calculated as a regional GDP weighted average of the great-circle distance between one country's regions and the partner country's regions 7 .

Estimation

Table 1 presents the result of the estimation. In the first estimation we assume that the border barrier is equal in all EU countries' bilateral trade (a12 = a13 =....=an(n-1)).

The border coefficient is -1.60. If we assume, as in [START_REF] Anderson | Gravity with Gravitas: A Solution to the Border Puzzle[END_REF], that the elasticity of substitution between domestic and foreign goods is five, the existence of borders impose a barrier which is, as average, equivalent to a 49% tariff.

In a second analysis, we drop the common border barrier assumption, and estimate the model with a border coefficient for each bilateral trade among EU countries 8 . As can be seen in the table, there are large differences across EU countries' bilateral border coefficients. In some cases, such as Netherlands-Finland and Netherlands-Denmark, the border coefficients are almost zero. According to this result, the costs for a Dutch firm when it trades with a Danish or Finnish firm are the same as those This is a remarkable result, as it means that in some bilateral trade flows the additional costs that borders may entail, such as differences in language, legal systems and the existence other information costs, seem to have already disappeared.

On the other hand, in other bilateral trade flows, such as Austria-France, Austria-Italy and Austria-Spain, the border coefficient is higher than two; if we assume, as before, an elasticity of substitution equal to five, in those bilateral trade flows borders still impose a barrier equivalent to more than a 70 per cent tariff. It is important to note, as well, that the distance coefficient obtained in the second estimation (-1.49) is larger than the one obtained in the first estimation (-1.16).

Table 2 offers a descriptive analysis of the tariff equivalent of the bilateral border barriers (henceforth border tariffs) estimated by the model; once again, the elasticity of substitution is set at five. As can be seen in the table, the border tariffs are evenly distributed across the 10 percentage point percentiles, except for the first one. The highest number of border tariffs are found in the 10%-20% and in the >60% percentile (eight) and the lowest number in the 0-10% percentile (three). The average border tariff in EU countries' bilateral flows is 38%.

Table 2 also presents how each country's border tariffs are distributed across the percentiles and its average. The country with the lowest border tariffs is Netherlands: six of its border tariffs are in the 0%-20% range and the average is 15%. The second country is Denmark where five of its border tariffs are in the 0%-20% range and the average is 22%; these countries are followed by Germany (28% average), United Kingdom (33%) and Finland (34%). The countries with the highest border tariffs are Austria, where seven of its border tariffs are above 40% (with a 54% average), Spain (52% average), France (50%), Italy (48%) and Sweden (44%).

Conclusions

The studies that have analysed the extent of home-bias in the EU have assumed that the barriers that borders impose on trade were equal in all trade flows among EU countries. In this paper we test the validity of this assumption and calculate the tariff equivalent of the trade barrier in each bilateral trade among EU countries. We found that there are large differences in the barrier that borders impose on bilateral trade. In some cases, such as in Netherlands -Denmark and Netherlands -Finland bilateral trade the border barrier is nil; this result is outstanding, as it means that in some bilateral trade flows the additional costs that borders entail seem to have already disappeared. In other cases, such as Austria-France, Austria-Italy and Austria-Spain, borders still impose a barrier equivalent to more than a 70 per cent tariff. The results also show that some countries have low border barriers in most of their bilateral trade flows with other EU countries, whereas other countries persistently show large border barriers.

F

  Anderson and van Wincoop refer to consumer price indexes as multilateral resistances, as they depend on all bilateral resistances. Secondly, a gravity equation is derived:

  The main difference between the equation estimated byAnderson and van Wincoop and previous studies, such as McCallum (1995), is the introduction of multilateral resistance terms in the equation. The model developed by Anderson and van Wincoop shows that multilateral resistances are correlated with distance and the border effect and, hence, if they are not included in the equation an omitted variable bias can arise. On the other hand, as opposed to McCallum's type equations, the Anderson and van Wincoop's model imposes unitary income elasticities.

  , followed by[START_REF] Anderson | Gravity with Gravitas: A Solution to the Border Puzzle[END_REF], is to solve multilateral resistances as an implicit function of observables and the parameters of the model. More specifically, if the trade cost function is substituted in equation (3) multilateral resistances are expressed as:

  we have to input them. The assumption is that intra-national trade is the difference between production and exports. Production data is obtained from the OECD STAN Industrial Structural Analysis data-base version 2004 release 4. The STAN data-base is primarily based on OECD member countries' Annual National Accounts by activity tables and uses data from other sources, such as national business surveys/censuses, to estimate any missing detail. This data-base provides complete gross production data for ten EU countries in the 1991-2001 period: Austria, Denmark, Finland, France, Germany, Italy, Netherlands, Spain, Sweden and United Kingdom 4 . Total exports and bilateral exports data among EU countries are obtained from the OECD ITCS data-base and the STAN bilateral trade data-base version 2004 release 1. The sample covers a total of 1100 observations [((10 countries x 9 partners) trade flows +10 intra-national trade flows) x 11 years].

  countries' real GDP is obtained as well from the OECD National Accounts.

  with a Dutch firm, once we control for differences in transport costs.

Table 1 . Results of the estimation (non-linear least squares)

 1 robust standard errors are given in parenthesis. The regression includes time dummies. They are not reported in the table.

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

Table 2 . Distribution of the bilateral border barriers' tariff equivalents

 2

	Average	tariff equivalent	(%)
	Total		45
	>60%		8
	0%-10% 10%-20% 20%-30% 30%-40% 40%-50% 50%-60%		3 8 7 5 7 7
			All

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

Note that, by assumption, tij=tji and hence bij=bji.

The natural logarithm of w

It should be pointed out that Austria, Finland and Sweden were not members of the European Union until 1995.

These figures are also transformed to constant values using, when available, the volume indices provided by the STAN data-base.

From the total merchandise trade figure we subtract trade in Scrap metal, trade in Electricity, gas and water supply and trade in Others. When these sectoral data were not available we inputted it using the average share of each sectoral trade in total merchandise trade from the years where the data were available.

The regional dissagregation matches Eurostat's NUTS 2 classification. We use the average regional GDP in the 1995-1999 period as weight. Regional GDP was obtained from Eurostat's Regio data-base. As Denmark only provides GDP data on a country-level we defined as regions the four most populated cities and population was used as weight.

We estimate 45 bilateral border coefficients. However, we present them twice in the table to make it easier to read.