Laura Romeu
email: laura.romeu-gordo@dza.de

Laura Romeu Gordo

Compression of Morbidity and the Labor Supply of Older People

Keywords: Health Production: Nutrition, Mortality, Morbidity, Disability, etc. < I1 - Compression of morbidity, Disability, Older workers JEL Classification Number: J14 J21 I12 Compression of morbidity, disability, older workers

This paper tests whether there is evidence of compression of morbidity using HRS data and analyzes the effects of this on the labor supply of older people. We find younger cohorts to suffer less from functional problems than older cohorts at given ages.

Furthermore, we observe that instrumentalized disability has a negative effect on labor force participation. According to the cohort analysis and the multivariate analysis, it can be concluded that individuals will be able to work longer because of the delay in the onset of disability problems.

Introduction

In recent decades there have been major changes in the age composition of the population in most of the OECD countries. Increases in life expectancy have compensated for the decrease in birth rates, causing population growth and increases in the average age of the population.

On average across the OECD countries, life expectancy at birth has increased by over 8 years since 1960. These increases in life expectancy are the result of the reduction in mortality rates due to improvements in medical care, better lifestyles and improved living conditions. At the same time, there has been a significant decrease in fertility rates throughout the OECD, in some countries falling below replacement levels. The average fertility rate [START_REF] Bell | Life tables for the United States social security area 1900-2100[END_REF] across OECD countries in 1960 was 3.2 children per woman, while in the year 2000 it was 1.6 children per woman. As a result of the increasing life expectancies and the decreasing fertility rates, the proportion of older individuals has increased considerably in most of the OECD countries. This tendency has triggered considerable debate about the necessity of reforms in the pension systems with contribution schemes. Furthermore, these trends in longevity lead to other important questions: what are the health consequences of the increase in longevity? Are increases in life expectancy translated into increases in morbidity?

Knowing the health consequences of increases in life expectancy has important policy implications. First, if morbidity in older ages is changing, it is necessary to adapt the social and health care services in order to deal with the changing demand [START_REF] Benjamin | Ceaseless Toil? Health and labor supply of the elderly in rural China[END_REF] . Second, if the F o r P e e r R e v i e w 3 onset of severe health problems is delayed, this may have an impact on the labor supply of older people. Recently, the necessity of increasing the retirement age in order to compensate for the increase of the ratio between individuals receiving public pensions and contributors has been under discussion. Improvement in the health status of older people may also help to increase the labor supply of older people before and after the legal retirement age.

The impact of increases in life expectancy on health has traditionally been analysed in medical literature (Crimmins and Saito, 2001; Freedman et al. 2002; Fries, 1980; Fries, 2003; Manton et al., 1997; Manton and Gu, 2001). However, there is an increasing interest on this topic in the economic literature since it has very important economic implications (Lakdawalla and Philipson, 2001; Bhattacharya et al., 2004).While recent studies analyse the consequences of increasing longevity on health care costs, until now there has not been evidence of how compression of morbidity affects the labor supply of older people.

Health consequences of increases in life expectancy

As pointed out in the introduction, people are living longer. The question is whether the delay in death implies longer periods of morbidity for the individuals.

There are three major hypothesis dealing with this question. The first one is the expansion of morbidity hypothesis (Gruenberg, 1977; Kramer, 1980), which argues that the decline in mortality is translated into an increase in morbidity. It has been argued

that health policy has focused on preventing death without preventing morbidity. Most efforts have been focused on reducing fatal diseases like cardiovascular diseases, smoking-related respiratory illnesses and cancers. The consequences of these achievements is that the death rates of the younger olds (younger than 80) are decreasing. However, since the proportion of individuals older than 80 is increasing there is a higher proportion of the population affected by diseases which usually appear in older ages and which are usually related to disability problems [START_REF] Berndt | The practice of econometrics: Classic and contemporany[END_REF] .

The compression of morbidity hypothesis argues that the decline in mortality is not necessarily associated with an increase in morbidity. Fries (1980) argued that if the onset of chronic illness (typical for later life) is postponed and if this postponement is greater than increases in life expectancy, then the lifetime burden of illness could be reduced.

The consequence of this trend would be a reduction in the cumulative lifetime morbidity.

The third hypothesis (Manton, 1982) suggests a sort of status quo. There is a decline in mortality in younger ages, and at the same time, there is a decrease in the incidence and progression of chronic diseases.

These three hypotheses are represented in figure 1. The reference point in this figure is the hypothetical present situation of a representative individual in which the onset of morbidity occurs at age 55 and death occurs at age 76. In a context of expansion of morbidity, the period of morbidity is longer, given that the onset of morbidity does not change (55 years) but the death is delayed (80 years). In a situation of status quo, the onset of both morbidity and death are delayed (60 and 81 years respectively). In this case the total period of morbidity remains unchanged with respect to the reference situation. Finally, in a context of compression of morbidity, the onset of both morbidity and death are delayed (65 and 78 years). However, the delay in the onset of morbidity is larger than the delay of death, causing a reduction of the total period of morbidity. [START_REF] Berndt | The practice of econometrics: Classic and contemporany[END_REF] Like osteoarthritis, depression, isolation, Alzheimer. Increasing awareness of these issues has led to the development of population specific indicators. For example, the concept of health expectancy, which combines mortality and morbidity aspects. Health expectancy is an extension of the concept of life expectancy to measure the expectation of years of life lived in various health states (Mathers, 1996; Robine et al., 1996). The most widely used of the health expectancy indicators is disability-free life expectancy.

The concept of disability (or limited functional capacities) is very important when considering older ages. Disability increases with age (especially after 80) and implies that these individuals are dependent on others in order to carry out ordinary activities.

Furthermore, when comparing different cohorts, disability is a more reliable indicator than chronic diseases, since the debilitating effects of chronic conditions may change from cohort to cohort (Costa, 2002). At the same time, according to the epidemiological literature, disability summarizes the differential cohort accumulation of health effects of lifetime exposures (Manton et al., 1997).

In the disability literature, a distinction is made between severe and moderate disability (Jacobzone et al., 2000). 'Severe disability' includes individuals with problems carrying out Activities of Daily Living (ADL) such as eating, dressing and getting in and out of bed. Severe disability is usually associated with the need for personal help, either at home or in institutions. 'Moderate disability' includes individuals experiencing problems in carrying out Instrumental Activities of Daily Living (IADL) such as shopping, daily financial accounting or preparing meals. The trends in the prevalence of disability have been the instruments most used in order to test whether or not there has been compression of morbidity in recent decades (e.g. Jacobzone et al., 2000; Freedman et al., 2002; Crimmins et al., 1997; Vita and Campion, 1998 and Manton and Gu, 2001). In the following section, we test the hypothesis of compression if morbidity using differences between cohorts. We want to analyze whether younger cohorts suffer less from functional problems than older cohorts at given ages.

3 Is compression of morbidity occurring? In this section, we intend to analyze, using data from the HRS [START_REF] Blau | The effect of health on employment transitions of older men[END_REF] (Health and Retirement Study), whether there are differences between cohorts, and more concretely, whether younger cohorts suffer less from functional problems than older cohorts at a given age.

We argue that this would be prima facie evidence of compression of morbidity. [START_REF] Campolieti | Disability and the labor force participation of older men in Canada[END_REF] The HRS is a panel survey carried out in the United States of individuals aged 50 and above and their spouses. The panel was started in 1992 and is repeated every two years.

The objective of the panel is to support research on retirement, health insurance, saving and economic well-being.

The RAND Center for the Studying of Ageing has developed what is called the RAND HRS Data files, which is a processed collection of variables derived from the Health and Retirement Study (HRS) (St. Clair et al., 2003). In the RAND version of the HRS, all [START_REF] Blau | The effect of health on employment transitions of older men[END_REF] Concretelly, the first 5 waves of the RAND version of the HRS. [START_REF] Campolieti | Disability and the labor force participation of older men in Canada[END_REF] The delay in the onset of functional problems might be also evidence of what we have defined in the last section as Status Quo, where the delay in the onset of health problems is equal to the delay in death. However, since we are comparing birth cohorts which are close we don't expect large delays in longevity and therefore, we consider significant delays in the onset of health problems as prima facie evidence of compression of morbidity. The HRS offers very rich information on health status and functional problems (Fallace and Herzog, 1995). It also includes very complete information about difficulties in Activities of Daily Living (ADLs) and Instrumental Activities of Daily Living (IADLs).

Our objective is to compare ADLs and IADLs measures for different cohorts at given ages. Unfortunately, it is not possible to compare the functional status of every cohort at every age since the panel is not long enough. However, we can compare two different cohorts for every age group. In figure 2 For the construction of the cumulative ADL index, we first selected the following ADLs items: difficulties in bathing, dressing, eating, getting in and out of bed and walking across the room. We added up these (0/1) variables and divided the sum by the number of items (5). Next, we added up this result across waves for every individual, correcting for the number of waves in which the individual was present in the panel [START_REF] Case | Economic Status and Health in Childhood: The Origins of the Gradient[END_REF] .

In order to homogenize the sample of our analysis, it would be interesting to analyze all individuals who reach a certain age. Functional problems rise significantly before death, and having individuals in the sample who die before a certain age may lead to an overrepresentation of persons with functional problems. However, since the panel is not long enough, such a selection is not possible. In figure 4 we carry out the same analysis for the cumulative IADL index. This index was constructed in a similar way to the cumulative ADL index. The 5 IADLs items selected were: difficulties in phoning, managing money, taking medication, shopping for groceries and preparing hot meals. Again, we observe that older cohorts have more functional problems than younger cohorts, with the exception of the War Babies (born between 1942 and 1947), who have higher disability means than the preceding cohort.

Based on the analysis of both disability indicators, we can conclude that older cohorts seem to be worse off. Therefore, there is prima facie evidence of compression of morbidity. However, we also observe that War Babies are not always better off than the preceding cohort. This shows that although there is a general trend of reduction of morbidity, this reduction is not equal for every cohort due to the different health technologies with which individuals are faced over the course of their life. One plausible explanation for the results obtained for the War Babies is the argument offered by Case et al. (2002) [START_REF] Costa | Health and labor force participation of older men 1900-1991[END_REF] . According to the authors, health in childhood is determined by household income. Furthermore, this relationship becomes more pronounced as children grow older.

As a result, children in poor households are more likely to have worse health in adulthood than children from households with higher income. Individuals belonging to the War Babies cohort were probably exposed to adverse economic situations which may have affected their health not only in childhood but also at older ages.

Finally, we complete the analysis of compression of morbidity with a further exercise.

Up to now we have used the mean of the cumulative ADL and IADL indexes to test whether there are differences between cohorts. Now, we analyze whether the proportion individuals with no functional problems (ADL) we find no evidence of compression of morbidity since there are no significant differences between cohorts in the percentage of individuals with no functional problems. However, analyzing the percentage of individuals with high functional problems (cumulative ADL index higher than 0.2), we do find evidence of compression of morbidity, since younger cohorts have lower percentages of individuals with high functional problems.

These results, together with the conclusions obtained from the analysis of the mean differences between cohorts, lead us to the following overall conclusion: there is evidence of compression of morbidity in the sense that younger cohorts have on average fewer functional problems than the older ones at given ages 10 .

Furthermore, the percentage of individuals with high functional problems in carrying out

ADLs is lower for younger cohorts. However, after this descriptive analysis we do not find evidence of a reduction of the percentage of individuals with no functional problems. Although we will not analyze this specific result in detail we note its importance in terms of policy recommendations.

Compression of morbidity and labor supply

Theoretical framework

In the last section, we concluded that there is evidence of compression of morbidity in the sense that younger cohorts have on average fewer functional problems than the older 10 These results coincide with the results obtained in the medicine literature ( Furthermore, there is another important implication, which is how labor supply responds to these changes in health status. Given the increase in the proportion of individuals 65 and older the need of an increase in the retirement age has recently been the topic of discussion. The improvement of the health status of older people is a factor that may help to increase employment rates of individuals before and after the legal retirement age.

As known from the literature (e.g. Costa, 1996 This relationship has been shown to be endogenous and this aspect has conditioned the empirical works dealing with this topic.

The endogeneous relationship between the two variables derives from two sources. First, there is a double causal relationship between health and labor force participation. Individuals' decision to work at older ages may be affected by health but at the same time, work may have an effect on health. Second, individual unobserved factors may be associated with both the likelihood of working and with having a particular health status.

Although most of the empirical works conclude that good health has a positive effect on labor force participation at older ages, from a theoretical point of view, the relationship between health and labor supply is not trivial. In order to better understand the relationship between the two variables, we describe next the model developed by Currie and Madrian (1999). In this model, individuals derive utility from health, leisure and The intertemporal utility function of the consumers is:

) 1 ( 1
1 1 ∑ =       + T t t t U λ
where, λ is the discount rate. U t is defined by:

) 2 ( ) , , ; , , ( 1t t t t t t t u X L C H U U ε =
where H is the stock of health, C is consumption of other goods and L is leisure. X is a vector of exogenous factors affecting preferences, u 1 is a vector of permanent factors affecting individual preferences, and ε 1 denotes shocks to preferences. Each individual maximizes his or her utility subject to the following constraints:

) 3 ( ) , , ; , ( 2 2 1 t t t t t t u V TH M H H H ε - =
This restriction is the health production function where M are material health inputs and TH are time health inputs. V is a vector of exogenous factors affecting productivity, u 2 is a vector of permanent individual factors affecting productivity and ε 2t denotes a productivity shock. The time constraint is:

) 6 ( 1 = + + + t t t t S TW TH L
where TW is working time and S is sick time which at the same time depends on the health stock: health status determines the time an individual spends sick and therefore, determines the total time available for market (TW, TH) (and non-market (L)) activities. Poor health may also affect the marginal rate of substitution between leisure and health increasing the 'marginal disutility for work', in this way reducing the labor supply. A negative health shock may also have a negative effect on productivity, which may be translated into lower wages. A reduction in wages has income and substitution effects on labor supply, so that the net effect is not clear. Illhealth may increase the necessity of increasing material health inputs (M) which could increase labor supply due to an adverse income effect. Furthermore, ill-health may have an effect on non-labor income (I), depending on how non-labor income is obtained. At the same time, labor supply affects health in different ways. First, labor supply determines labor income, which at the same time determines the income available to purchase material health inputs. Furthermore, labor supply also determines the time available to produce time health inputs. Labor supply could also be considered as a direct input in the health production function, especially when considering jobs which are physically demanding and when analyzing the effects of labor supply on mental health.

According to this theoretical framework, there is no clear net effect of health on labor supply given the endogenous relationship between the two variables, although in the empirical literature it has been shown that ill-health negatively affects labor participation.

In the next subsection we carry out an empirical analysis of the effects of functional (ADL and IADL) problems on labor participation, correcting for endogeneity. If functional problems have a negative effect on labor participation after controling for other covariates and for endogeneity, we will expect compression of morbidity to have a positive effect on the employment rates of older people.

Empirical analysis

In this section, we analyse how disability affects labor participation, and where individuals are participating in the labour force, what the effect on the number of hours worked is. When including disability as a explanatory variable it must be taken into account that there is a possible endogenous relationship between disability and labor participation and between disability and number of worked hours. The most commonly used method to overcome the problem of endogeneity is IV (Instrumental Variables). The endogenous disability model for labor participation consists of a two-equation system:

) 8 ( ) ( 2 1 1 i i i i D X F LP ε α α + + = ) 9 ( 2 1 1 i i i i X Z D µ β β + + =
where LP is labor participation, X 1i is a vector of exogenous variables, D i is disability and Z i are the instruments. The endogenous disability model for the number of hours worked can be defined as following:

) 10 ( 2 1 2 i i i i D X WH η γ γ + + = ) 11 ( 2 2 1 i i i i X Z D µ β β + + =
where WH i are the number of hours worked per week and X 2i is a vector of exogenous variables. The estimation of the equation ( 8) by probit and the estimation of the equation (10) by OLS will yield to biased estimates of α 2 and γ 2 if D i is not exogenous. Therefore, IV is used in order to handle this endogenity problem. The main question when using this method is to define the right instruments. The instruments chosen for the present analysis are health measures that have been already used in the literature (Campolieti, 2002). Concretely, information is used about whether individuals have been diagnosed in the past one of the following health problems: high blood pressure, diabetes, cancer, lung disease, heart problems, stroke, psychical problems and arthritis and the individuals' body mass index. The main properties that these instruments must fulfil in order to be valid as instruments are the following: first, they must be highly correlated with disability and second, they must be uncorrelated with the error terms ε i and η i .

These instruments present a strong correlation with disability, as the F-tests on excluded [START_REF] Crimmins | Trends in disability-free life expectancy in the United States, 1970-90[END_REF] . Furthermore, it is not likely that these instruments are correlated with the error term of the equations ( 8) and (10), as has been already discussed by Campolieti (2002).

In table 1, the results of two different models of labour participation are presented. First, the effect of disability on labour participation is estimated without instrumentalizing disability. [START_REF] Crimmins | Trends in Disability-Free Life Expectancy in the United States, 1970-1990: Gender, Racial, and Educational Differences[END_REF] Next, in the second model, an IV probit is estimated.

In the first model we observe that non-instrumentalized disability has a significant and negative effect on labour participation. In the second model disability is intrumentalized using the health measures specified above. Instrumented disability has a significant and negative effect on labor participation. The higher the disability index, the lower the probability of participating in the labor market. By comparing the marginal effects obtained from the estimation of the two models, we observe that instrumentalized disability has a higher effect on labour force participation than non instrumentalized disability. This result indicates that in the second model, endogeneity leads to a downsize bias in the effect of disability on labour participation. The magnitude of the impact of disability on labor participation derived from our results is similar to the magnitude obtained by Campolieti (2002).

Other controls have been introduced in the estimation of the models. In the second model these controls present the following results: year of birth has a significant and positive effect on labor participation. The younger an individual is, the higher the probability of their working. We control for year of birth and not for age because in this [START_REF] Crimmins | Trends in disability-free life expectancy in the United States, 1970-90[END_REF] F-tests are listed on the regression tables. [START_REF] Crimmins | Trends in Disability-Free Life Expectancy in the United States, 1970-1990: Gender, Racial, and Educational Differences[END_REF] Disability is an index which can take values between 0 (non disability) and 2 (when the ADL index takes value 1 and the IADL index takes value 1). 17 way, we also control for the cohort effects. Being a woman and being partnered or married have a significant and negative effect on labor participation. Higher education and longer work experience have a significant and positive effect on labor participation. At the same time, the higher the number of individuals in the household, the higher the probability of working. Region dummies have been introduced in order to control for labour market disparities across the regions. However, these dummies are not significant.

Next, we estimate how disability affects the self-reported number of hours worked. In order to correct for selection bias, we calculate the inverse Mills ratio (λ) using the probit parameter estimates (Berndt, 1991). Then, λ is included in the estimation models of the number of hours worked per week [START_REF] Currie | Health, health insurance and the labor market[END_REF] . Again, two different models of number of worked hours are estimated and the results are presented in table 2. In the first model, the effect of disability on the number of hours worked is estimated. In the second model disability is instrumentalized using different health measures (Z). Disability and instrumentalized disability have a significant and negative effect on the number of hours worked per week. Again, the effect of instrumentalized disability is greater, which indicates that endogeneity leads to a downsize bias in the estimation of the effect of disability on the number of hours worked.

From these results we can conclude that disability has a negative effect on labour participation, and furthermore, has also a negative effect on the number of hours worked per week for those who participate in the labour market, which confirms the results obtained in the descriptive analysis. [START_REF] Currie | Health, health insurance and the labor market[END_REF] For persons working only. However, there is another aspect that has to be taken into account in order to predict which the effect of the delay in the onset of health problems on labor participation will be. The effect of disability on labor participation may change with time. This is due to different reasons: first, there are changes in leisure preferences between cohorts and second, there are also changes in labor conditions so that individuals with certain disability problems in the present may be have a different likelihood to participate in the labour market than in the past. In order to corroborate this idea, a model of labour participation is estimated separately for different cohorts (the results are presented in table 3). Not all the cohorts can be compared, given the age selection of the sample.

Only War Babies (born between 1942 and 1947) and HRS (born between 1931 and 1941) are compared. [START_REF] Fallace | Overview of the health measures in the health and retirement study[END_REF] By comparing the marginal effect of disability on labor participation from both cohorts, we observe that the negative effect of disability on labor participation is greater for the younger cohort. This result confirms the idea that the effect of disability on labor status changes with time.

In order to better illustrate how different levels of disability affect labour supply, we calculate changes in predicted probabilities of labor participation for different disability levels [START_REF] Freedman | Recent trends in disability and functioning among older adults in the United States: A systematic review[END_REF] . We first calculate how labor participation would change for the War Babies cohort if disability were to increase to the same level as the HRS cohort. Next, we calculate how labor participation would change for the HRS cohort if disability is reduced to the War Babies disability level.

We find that labor participation for the War Babies cohort would decrease by 1.8% if the level of disability had not decreased between cohorts (if there had been no compression [START_REF] Fallace | Overview of the health measures in the health and retirement study[END_REF] There are not enough observations in order to also include CODA (born between 1924 and 1930). [START_REF] Freedman | Recent trends in disability and functioning among older adults in the United States: A systematic review[END_REF] Calculations have been carried out following Long and Freese (2001). [START_REF] Fries | Aging, natural death and the compression of morbidity[END_REF] . On the other hand, labor participation for the HRS cohort would have been 1.4% higher had they the same disability levels as the War Babies cohort.

Summary and Conclusions

The increase in longevity has led to important debates. One of these debates is about the health consequences of increases in life expectancy. If the proportion of individuals with problems in carrying out daily activities increases as result of increases in life expectancy, we should be prepared to meet need for their social and health care. However, it is not clear that the recent trends in longevity cause increases in morbidity.

Fries argues that increases in life expectancy are not necessarily translated into increases in morbidity. Due, among other reasons, to improvements in medical care and in lifestyles, the onset of morbidity in older ages is delayed. If this delay is greater than the increase in life expectancy, there is no increase in morbidity.

In this paper we analyze, using the HRS, whether younger cohorts suffer less from functional problems at given ages. From this analysis we conclude that there is evidence of compression of morbidity, since younger cohorts suffer less from functional problems than older cohorts.

The delay in the onset of disability in older ages may have important consequences for the labor market. Currently, the increase in the retirement age is being discussed in most developed countries in order to overcome the financial problems of the public pension systems. At the same time, early retirement is discouraged so that the individuals must [START_REF] Fries | Aging, natural death and the compression of morbidity[END_REF] The change in disability is the difference in disability level between two cohorts for ages between 51 and 60 (ages at which both cohorts are comparable). work until the legal retirement age if they do not wish to be financially penalized. The idea is that older workers are necessary in order to maintain the public pension systems in economies where the old age dependency ratio is increasing dramatically.

In the second part of the paper, we analyze the effect of disability on labor force participation and on the number of hours worked per week. We do observe after controlling for endogeneity that disability has a negative effect on labor force participation and on the number of worked hours per week for those who work. Furthemore, we have observed that the negative effect of disability on labor participation changes between cohorts, having a greater effect for younger cohorts.

According to the cohort analysis and the multivariate analysis carried out in this paper, individuals will be able to work longer because of the delay in the onset of disability problems which cause lower participation in the labor force. However, this effect will be partly counteracted by changes in the effect disability will have on labor participation through time.

To summarize, given the improvements in the health of older workers in recent decades, there is an increasing number of potential labor force participants who would formerly have left the labor market due to health reasons. The labor market should be able to absorb this labor force in order to alleviate the financial problems of the public pension systems. (1) Probit of labor force participation with disability as explanatory variable.

F

(2) IV Probit of labor force participation (instrumentalised disability). (1) Estimation of the number of hours worked with disability as explanatory variable.

(2) IV Estimation of the number of hours worked (instrumentalised disability).

3. 1

 1 Health differences between cohorts Individuals of different cohorts are faced with different health technologies over the course of their life. For example, we expect an individual aged 70 today to have a different health status if compared to an individual who was 70 years old 20 years ago, due to the different circumstances in which these individuals live.

  in 5 waves, after merging the second wave of the HRS cohort (started in 1994) and the first wave of the AHEAD cohort, which was started in 1993. In the HRS, 4 different cohorts are interviewed: the initial HRS cohort, with individuals born between 1931 and 1941; the AHEAD cohort with individuals born before 1924; the children of depression cohort (CODA) born between 1924 and 1930, and finally the war baby (WB) cohort with individuals born between 1942 and 1947.

  we can see the age groups in which different cohorts are comparable. RAND has constructed yearly ADL and IADL indexes based on the functional information contained in the HRS, which are the usual indexes used in the empirical literature. However, we propose to use cumulative ADL and IADL indexes. The reason for using such cumulative indicators and not yearly indicators is that in this way, we capture in only one indicator the average functional difficulties suffered by the individual during the period in which he or she is observable for us. The idea of cumulative disability has been already used and validated in the medical literature (Vita and Campion, 1998).

Figure 3

 3 Figure3shows the mean of the cumulative ADL index by age for the different cohorts.

  no functional problems and the percentage of individuals with high functional problems at given ages change between cohorts. Analizing the percentage of

  Crimmins and Saito, 2001; Freedman et al., 2002; Fries, 2003; Manton and Gu, 2001).Page 10 of 33Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, ages. This result has important policy implications since the social and health care needs of the elderly change with time, and adequate responses must be given to these changing needs.

  ; Sickles and Yazbeck, 1998; Blau et al., 2001; Wilson, 2001; Mete and Schultz, 2002; Miah and Wilcox-Gök, 2007), there is an important link between labor force participation and health, especially in the elderly.

  By investing time and other health inputs in order to produce health, individuals reduce the total time being sick, which increases the total available time for leisure and market activities. At the same time, hours of work are necessary in order to increase the available income which allows the acquisition of material health inputs and other commodities from which individuals also derive utility.

u 3

 3 is a vector of individual factors determining illness, and ε 3t are shocks that cause illness.There are different possible effects of health on labor supply(Benjamin et al., 2003):

12 of 33 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  is the price vector of the health material inputs, A denotes assets and Y is total income. The different income sources are unearned income (I), labor income (WTW) and interest derived from assets (rA):

	Submitted Manuscript
	where P ) 5 ( t t t t t rA TW W I Y + + =
	F o
	r
	P
	e e r
							R e v i e w
	The budget constraint is:
	C	t	+	t P	M	t	+	( 1 t A	-	t A	)	=	t Y	(	4	)
																13

+

Page

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	F o
	r
	P
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21 Page 21 of 33 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Theories about the evolution of morbidity. Source:Fries (2003).

			Submitted Manuscript Submitted Manuscript
	Figure 3. Cumulative ADL per age group and cohort. Source: own calculations. Figure 4. Cumulative ADL per age group and cohort. Source: own calculations.
		0,045
		0,046
		0,040
	Cumulative ADL Cumulative IADL	o r P e e r F o r P e e r F o r P e e r F o r P e e r F o r P e e r Figure 1. Morbidity 0,041 55 y Present Morbidity Expansion of Morbidity F o r P e e r F o r P e e r 0,010 0,015 0,020 0,025 0,030 0,035 0,036 F 0,031 o r P r 0,011 e e 0,016 0,021 0,026	Death 76 y
		0,005 0,006
		0,000 0,001 -0,004	R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w 55 y 60 y R e v i e w R e v i e w 60-64 65-69 70-74 Age Groups AHEAD (before 1924) CODA (1924-1930) HRS (1931-1941) War Babies (1942-1947) 65 y Status Quo Compression of Morbidity 50-54 55-59 75-79 R 50-54 55-59 60-64 65-69 70-74 75-79 e Age Groups v w e i AHEAD (before 1924) CODA (1924-1930) HRS (1931-1941) War Babies (1942-1947)	80 y 81 y 78 y >80 >80
				26 28 29

27

Figure 2. HRS cohorts and ages at which they are comparable. Source: own calculations. Page 27 of 33 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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Table 1 .

 1 Estimates of labor force participation. Dependent variable: labor participation (=1) if working for pay, (=0) otherwise.

	(1)	(2)

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Table 2 .

 2 Estimates of the number of hours worked weekly.

	(1)	(2)

*p<0.1, **p<0.05, ***p<0.001. Standard Errors in parenthesis.

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Table 3 .

 3 IV Probit of labor force participation separated by cohorts. Dependent variable: labor participation (=1) if working for pay, (=0) otherwise.

	(1)	(2)

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For women aged 15-49. Source: OECD.

[START_REF] Zweifel | Aging of populations and health care expenditure: a red herring?[END_REF] argue that what is determining in terms of health care expenditure is the proximity of death and not the age at which death occurs, so aging of the population will not have major effects on aggregate health care expenditures.

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

We repeat this calculation for every wave, so that this cumulative measure changes for every age at which the individual is observable.

See also[START_REF] Almond | The costs of low birth weight, Working Paper 10552[END_REF]. Page 9 of 33 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Acknowledgements

The author acknowledges the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2002-00235, [AGE]. This paper was written during my stay at the University Ca'Foscari of Venice. I thank the Economics Department and specially Agar Brugiavini for her support and her contributions. I also thank David M. Blau, Mette Christensen, Christina Wübbeke and Johann Fuchs for their comments.

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


Kristian Jönsson
email: kristian.jonsson@konj.se

Time-Specific Disturbances in a Panel Data Stationarity Test

Keywords: Panel Data, Stationarity, Unit Root, Cross-Sectional Dependence, Monte Carlo Simulation JEL Classification: C12, C15, C22 Panel Data, Stationarity, Cross-Sectional Dependency, Monte Carlo Simulation

published or not. The documents may come

Introduction

When testing for stationarity, unit roots and cointegration in the time-series setting, the power of the tests is to a large extent influenced by the number of observations available for the considered time series. When the time-series sample is short, the tests can have very low power, often failing to reject the null hypothesis even when it is false. However, by using time-series data for several cross sections, i.e. by applying panel data when testing for unit roots, stationarity and cointegration, more powerful inference is enabled. This was recognized e.g. by [START_REF] Quah | Exploiting Cross-Section Variation for Unit Roots Inference in Dynamic Data[END_REF]; Levin andLin (1992, 1993); [START_REF] Levin | Unit Root Tests in Panel Data: Asymptotic and Finite-Sample Properties[END_REF]; [START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF][START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF], who suggest various panel data unit root tests; by [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF], [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF], [START_REF] Harris | Panel Stationarity Tests for Purchasing Power Parity with Cross-Sectional Dependence[END_REF] and [START_REF] Shin | Mean Group Tests for Stationarity in Heterogeneous Panels[END_REF], who suggest panel data stationarity tests; and by [START_REF] Pedroni | Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors[END_REF], [START_REF] Pedroni | Panel Cointegration: Asymptotic and Finite Sample Properties of Pooled Time Series Tests with an Application to the PPP Hypothesis[END_REF], [START_REF] Mccoskey | A Residual Based Test of the Null of No Cointegration in Panel Data[END_REF], [START_REF] Westerlund | A Panel CUSUM Test of the Null of Cointegration[END_REF], [START_REF] Larsson | Likelihood-based Cointegration Tests in Heterogenous Panels[END_REF] and [START_REF] Groen | Likelihood-Based Cointegration Analysis in Panels of Vector Error-Correction Models[END_REF], who suggest panel data cointegration tests. In the early panel data unit root, stationarity and cointegration literature, the assumption of cross-sectional independence was common. However, a growing body of research has documented the problems that can arise when dependency exits between the different cross-sectional units in the sample (see e.g. O 'Connell, 1998;[START_REF] Jönsson | Testing for Stationarity in Panel Data Models when Disturbances are Cross-Sectionally Correlated[END_REF]. As a consequence, several different methods, that address the issue of cross-sectional dependency in the panel data unit root and stationarity framework, have been developed (see e.g. Bai and Ng, 2004a,b;[START_REF] Chang | Nonlinear IV Unit Root Tests in Panels with Cross-Sectional Dependency[END_REF][START_REF] Chang | Bootstrap Unit Root Tests in Panels with Cross-Sectional Dependency[END_REF][START_REF] Jönsson | Cross-Sectional Dependency and Size Distortion in a Small-Sample Homogeneous Panel Data Unit Root Test[END_REF][START_REF] Jönsson | Time-Specific Disturbances and Cross-Sectional Dependency in a Small-Sample Heterogeneous Panel Data Unit Root Test[END_REF]; [START_REF] Moon | Testing for a Unit Root in Panels with Dynamic Factors[END_REF][START_REF] Pesaran | A simple panel unit root test in the presence of cross-section dependence[END_REF][START_REF] Phillips | Dynamic Panel Estimation and Homogeneity Testing under Cross Section Dependence[END_REF]. The methods that are used to deal with cross-sectional dependency (CSD) involve orthogonalization of the series under consideration, extraction of common factors, augmentation of test regressions and bootstrap procedures that take crosssectional dependency into account.

Within the panel data stationarity framework, [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF] suggest that the stationarity test of [START_REF] Kwiatkowski | Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root[END_REF] can be applied to the cross-sectional units of a panel and that a panel data stationarity test can be constructed as the standardized mean over the cross-section specific test statistics. The test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF] differ from e.g. the panel unit root test of [START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF] in that the former takes stationarity as the null hypothesis while the latter takes a unit root as the null. Moments for the asymptotic [START_REF] Kwiatkowski | Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root[END_REF] statistic are supplied by e.g. [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF] and, by utilizing the sequential limit arguments T → ∞ followed by N → ∞, the author shows that the panel data test statistic has a standard normal limiting distribution. 3;4 In order to account for a finite time-series dimension, [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] provide finite-T moments for the KPSS statistic under the assumption that the time series under consideration are serially independent, while approximate moments for the KPSS statistic with serially dependent time series are supplied by Jönsson (2006a). The moments of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] can be used to obtain a panel data test statistic that is standard normally distributed as N → ∞ for fixed T , while the moments of Jönsson (2006a) can be used to obtain an approximate standard normal distribution for the test statistic when T and N are fixed. By utilizing results from Monte Carlo simulations, [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] and Jönsson (2006a) show that inference can be improved by using the fixed-T moments, instead of the asymptotic moments, of the KPSS statistic.

When cross-sectional correlation, in the form of a time-specific disturbance, is allowed for in the panel data unit root and stationarity framework, [START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF] and [START_REF] Shin | Mean Group Tests for Stationarity in Heterogeneous Panels[END_REF] suggest that demeaning over cross sections should be employed in order to eliminate the dependence between cross sections. Even though some results indicate that demeaning over cross sections affects the small-N distribution of the panel data test under consideration (see e.g. Jönsson, 2006a,b), there is little knowledge about the small-N behavior of the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] test when time-specific disturbances are present and accounted for.

In this paper, we study the performance of the panel data stationarity test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] when cross-section dependence is modelled by a time-F o r P e e r R e v i e w specific disturbance that is common to all cross sections. We find that the commonly used cure for this type of dependency, namely demeaning across the cross-sectional dimension, induces a size distortion when the cross-sectional dimension is small. This applies even if no time-specific disturbance is present. That is, the mere fact that a time-specific factor is allowed for affects the performance of the panel data stationarity test. To mitigate this size distortion, critical values, that can be used when cross-sectional correlation is modelled by a time-specific factor, are supplied. The new set of critical values work well regardless of whether a time-specific factor is present of not. Furthermore, the size distortions, that arises when a time-specific factor is allowed but not accounted for, are eliminated without any major sacrifices in the power of the panel data stationarity test.

The rest of this paper is organized as follows. In Section 2, the econometric model and the panel data stationarity test are introduced. Furthermore, in this section the effects of allowing for a time-specific disturbance is analyzed. Critical values that can be used when a time-specific disturbance is present are supplied in Section 3. The performance of the panel data stationarity test, using the supplied critical values, is studied in the same section. Finally, Section 4 offers some concluding remark.

2 Panel stationarity with a time-specific factor

The panel data model

The current paper is centered around the following panel-data unobserved components model:

y i;t = α i + δ i t + ξ i;t + ε i;t (1) ξ i;t = ξ i;t-1 + ν i;t (2) ε i;t = θ t + η i;t (3)
The cross-section time-series, y i;t , consists of three main components. 5 First, there are cross-section specific deterministic components, α i and δ i t, which take the form of an intercept and a time trend. Second, y i;t includes of a random walk term, ξ i;t . The development of this term is described in (2), where ν it is i.i.d. N (0, σ 2 º ;i ). For all cases where σ 2 º ;i > 0, the cross-section time series y i;t will contain a unit root term, while it follows that y i;t is stationary, or trend stationary, if σ 2 º ;i = 0. In the latter case ξ i;t will reduce to ξ 0;i for all t, which amount to a constant term that will be captured in the intercept when y i;t is detrended. Finally, y i;t includes a stochastic disturbance term, ε i;t , which is described in (3). From (3) it can be seen that the composite disturbance term, ε i;t , can be broken down into two components. The first of these is a time-specific disturbance, θ t , which is assumed to be distributed i.i.d. N (0, σ 2 µ ). The second component, η i;t , is assumed 5 Here, and in the remainder of the paper, we let i denote the cross-sectional dimension and t denote the time-series dimension. Throughout we will also assume that i ∈ {1; : : : ; N} and t ∈ {1; : : : ; T }. to be an idiosyncratic disturbance that is distributed i.i.d. N (0, σ 2

´). Hence, from (3) it can be seen that cross-sectional dependence between disturbances ε i;t and ε j;t , i = j, is introduced by the time-specific factor θ t . It is the presence of θ t that makes this model differ from the model in [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF]. If we let θ t = 0 for all t ∈ {1, . . . , T }, the model in ( 1)-( 3) would reduce to the model of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF], i.e. to the model with cross-sectionally independent disturbances, and hence it would be possible to apply the panel data stationarity test suggested by these authors. However, within the extended model where θ t = 0, the issue of cross-sectional correlation has to be addressed. In the following sections, we elaborate on the test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF], discuss how it can be extended to allow for a time-specific factor and study how such an extension affects the small-sample behavior of the test.

The panel data stationarity test

The panel data stationarity tests of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF] and [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] are developed within the unobserved components framework in ( 1)-( 3) under the assumption that θ t = 0.6 The panel data stationarity test, here denoted LM, is constructed by applying the stationarity test of [START_REF] Kwiatkowski | Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root[END_REF] to each of the N time-series in the sample. By utilizing a standardized mean over the individual test statistics, a panel data stationarity test with a standard normal limiting distribution can be obtained. For the test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF], the asymptotic distribution is obtained by letting T → ∞ followed by N → ∞, while [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] obtains the limiting distribution by letting N → ∞ for fixed T . In the current paper, we investigate the finite-sample behavior of the panel data stationarity test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] when time-specific disturbances are present, and hence stick to the fixed-T setting. The general expression for the panel data test statistic is given in (4), while the cross-section specific KPSS test statistics are calculated as in ( 5)-( 7).

LM = 1 N N i=1 LM i -E(LM i ) V ar(LM i ) N (4) LM i = T -2 T t=1 S 2 i;t σ2 ";i
(5) In ( 6) and ( 7), e i;t denotes the OLS residual obtained after detrending y i;t with either an intercept or an intercept and a time trend. In (4), E(LM i ) and V ar(LM i ) are the moments of the cross-section specific KPSS statistics. These moments depend, in the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] setting, on the time-series dimension of the panel and on whether an intercept or an intercept and a time trend are used when detrending the series.

S i;t = t j=1 e i;j (6) σ 2 ";i = T -1 T t=1 e 2 i;t (7)
As noted above, the limiting distribution of the test statistic in (4) will not be standard normal as N → ∞ if a time-specific factor, θ t , is present. More specifically, the presence of cross-sectional correlation has been shown to cause size distortions in the panel data stationarity test (see e.g. [START_REF] Jönsson | Testing for Stationarity in Panel Data Models when Disturbances are Cross-Sectionally Correlated[END_REF]. Hence, it is desirable to account for the cross-sectional dependence in some way. Several methods for doing this are available. If a common factor structure is assumed, one could apply the panel data stationarity test Bai and Ng (2004a), which amounts to finding common factors by the method of principal components and then testing both the common factors and the de-factored series for stationarity. This method rests on asymptotics that require that both the cross-sectional and the time-series dimensions tend to infinity. Hence, it falls outside the framework of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF]. Another test that allows for cross-sectional correlation is that of [START_REF] Harris | Panel Stationarity Tests for Purchasing Power Parity with Cross-Sectional Dependence[END_REF]. These authors propose a test based on autocovariances that has an asymptotic standard normal limit when T → ∞ for fixed N . The same asymptotics applies for the test of [START_REF] Jönsson | Testing for Stationarity in Panel Data Models when Disturbances are Cross-Sectionally Correlated[END_REF], which is based on orthogonalization of ε i;t and hence of y i;t . Evidently, these tests also fall outside the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] framework. However, when the cross-sectional correlation in the panel data unit root and stationarity framework is caused by a time-specific factor, θ t , [START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF], [START_REF] Shin | Mean Group Tests for Stationarity in Heterogeneous Panels[END_REF] and [START_REF] Jönsson | Time-Specific Disturbances and Cross-Sectional Dependency in a Small-Sample Heterogeneous Panel Data Unit Root Test[END_REF] suggest that demeaning over the cross-sections provides a viable way of dealing with the size distortions that arise. Demeaning over cross sections amounts to constructing ỹi;t = y i;t -N -1 N j=1 y j;t , and performing the test in (4)-( 7) using the residuals obtained after detrending these series. In the next section, we go on by investigating how well the demeaning solution works in the context of the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] panel data stationarity test.

Effects of a time-specific factor

When the cross-sectional dependence is modelled by a time-specific chock, i.e. when ε i;t = θ t + η it , where θ t ∼ i.i.d. N (0, σ 2 µ ) and η i;t ∼ i.i.d. N (0, σ 2 ;i ), demeaning over the cross-sectional dimension resolves the cross-sectional correlation problem. 7 The cross-sectionally corrected test is performed as above, with the only difference being that ỹi;t = y i;t -1 N N j=1 y j;t is used instead of y i;t when performing the test. Recent results (see e.g. [START_REF] Jönsson | Time-Specific Disturbances and Cross-Sectional Dependency in a Small-Sample Heterogeneous Panel Data Unit Root Test[END_REF] have indicated that the demeaning procedure work unsatisfactory in small-N situations unless moments or critical values, especially obtained for the cross-sectionally corrected test, are used. More specifically, the demeaning procedure has been shown to introduce cross-sectional correlation, and hence a size distortion, in the panel data unit root framework even when disturbances are cross-sectionally independent. If this is the case also with the panel data stationarity test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF], a new set of critical values is required for situations where the demeaning procedure is employed and the cross-sectional dimension is small.

In order to study the size of the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] panel data stationarity test applied to demeaned series, data is generated according to the data generating process (DGP) in (8) below.

y i;t = α i + δ i t + i;t (8)
In ( 8), it denotes a random disturbance term, which is assumed to be i.i.d. N (0, 1). Without loss of generality, we let α i = 0 and δ i = 0 in the DGP. Using the moments supplied by [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF], the panel data test for the null hypothesis of stationarity is calculated as described above. In addition to calculating a test statistic for the case where the demeaning procedure is employed, we also obtain a panel data test statistic for the case where no cross-sectional correction is made. This is done in order to be able to single out the effects that are attributable to demeaning the data from the effects that are caused by making the normal approximation in a situation where N is fixed. These two tests are performed on 50,000 test statistics for panel dimensions where T ∈ {10, 20, 50, 100}, N ∈ {2, 3, . . . , 10, 25, 50} and various deterministic components are used in the detrending procedure. The resulting size of the tests, on the 10% significance level, is presented in Table 1.

As seen in Table 1, the panel data stationarity test is always over-sized whenever N is small, regardless of how large T is. For example, when T=10 and N=2 it can be seen that the size of the corrected test, i.e. the test applied to demeaned series, is 16.1% and 16.7% for the cases where an intercept and an intercept and a time trend are included in the detrending procedure. The same pattern can be seen for every parameter setup, and the conclusion that the test is upward size-distorted is undisputable. However, when we study the size results for the test based on series that have not been demeaned, it can be seen that a small upward size distortion exists also in this case, at least when N is small. Hence, the size distortion that was documented for the test employing a demeaning procedure obviously have two sources. The first source of the size distortion is the fact that we introduce some degree of cross-sectional correlation by demeaning the cross-section time series. The second source of the size distortion is the fact that N is small, maybe too small for the normal distribution to be accurately applied for inference. The relative importance of these two sources can be seen by comparing the size of the test that employs demeaning to the size of the test that does not.

As seen from Table 1, the size is somewhat higher than the significance level also for the uncorrected test, i.e. for the test denoted 'No corr.' in Table 1. Since the data is generated without a time-specific factor, the relative importance of the two sources for the size distortion can be easily identified. Comparing the size for the corrected and the uncorrected tests, it can be seen that the major part of the size distortion that occurs when N < 10, arises from the introduction of crosssectional dependency through demeaning. However, the size distortion arising from 6 F o r P e e r R e v i e w inaccurate normal approximation is non-negligible. This fact suggests that the use of critical values for inference about the null hypothesis that all cross-section time series are stationary is preferable to the use of standardizing moments that take the demeaning procedure into account. While the use of especially obtained standardizing moments, as in [START_REF] Jönsson | Time-Specific Disturbances and Cross-Sectional Dependency in a Small-Sample Heterogeneous Panel Data Unit Root Test[END_REF], will mitigate the effects of cross-sectional demeaning, the size distortions caused by inaccurate normal approximation will not be addressed. By using critical values, on the other hand, both sources for the size distortion can be mitigated.

In the next section, we supply response surface parameters that can be used to obtain critical values for the cross-sectionally corrected panel data stationarity test.

Small-sample distribution

As seen from the results in Table 1 and as concluded by [START_REF] Jönsson | The Accuracy of Normal Approximation in a Heterogeneous Panel Data Unit Root Test[END_REF], it can be suspected that the inappropriateness of normal approximation give rise to some size distortions in the [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] test, especially when N is small. This fact implies that it is desirable to work with critical values instead of relying on the normal approximation when conducting inference. Hence, we suggest holding on to the moments supplied by [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] when calculating the panel data stationarity test and, instead of critical values from the standard normal distribution, use finite-sample critical values obtained especially for the cross-sectionally corrected test. This approach will eliminate the size distortions arising from the poor normal approximation as well as the size distortions arising when introducing the demeaning procedure. Below, we will provide finite-sample critical values that can be used for inference about the stationarity hypothesis when a time-specific factor is modelled.

Critical values for the corrected test

In order to obtain critical values for the panel data test statistic, the cross-sectionally corrected panel data stationarity test statistic in ( 4) is calculated based on a set of artificially generated data series that are obtained as described in Section 2.3. In order to avoid vast tabulations with critical values, response surface regression can be fit and the parameters of the response surface regressions supplied. Hence, to be able to fit response surface regressions to the critical values, we obtain 50 critical values on the 10%, 5%, 2.5% and 1% significance levels based on 5,000 test statistics in each replication. The panel data dimensions considered are such that T ∈ {10, 20, . . . , 100, 250, 500, 1000}, while N ∈ {2, 3, . . . , 15, 20, 25, 50}. Critical values are obtained both for the case where an intercept is the only deterministic component and for the case where an intercept and a time trend are accounted for. This setup implies that each response surface regression is fitted to a sample consisting of 11,050 observations.

Based on the critical values obtained in the simulation we fit the response surface 7

F
= β 0 + 2 i=1 β i;T S T -i + 2 i=1 β i;CS N -i=2 + β CS;T S N -0:5 T -2 + ® j;N;T (9)
When estimated, the response surface parameters that are presented in Table 2 are obtained. The critical values calculated by using the response surface parameters of Table 2 can now be applied to obtain a panel data stationarity test that allows and accounts for cross-sectional correlation in the form of a time-specific disturbance. In order for the critical values based on the parameters in Table 2 to be useful, they should apply regardless of the degree of cross-sectional correlation that is present in the disturbances. Furthermore, in order for the the panel data stationarity test to work under heterogeneous alternatives, the cross-section demeaning procedure should be robust even if some of the cross sections are nonstationary, which means that the demeaning procedure should not annihilate the power of the test under such circumstances. To assure this, we set up and run a Monte Carlo simulation of the finite-sample size and power performance of the panel data stationarity test when a time-specific factor is allowed for and when the critical values that are supplied in the current paper are used for inference.

Finite-sample performance

In order to assess the performance of the demeaned panel data stationarity test when cross-sectional correlation is present, artificial data sets are once again generated and it is studied how well the panel data stationarity test performs when the critical values of the previous section are used.

In order to investigate how well the response surface parameters can reproduce accurate critical values, we run a Monte Carlo simulation where the data is demeaned prior to running the panel data stationarity test. In the simulations, 50,000 data sets are generated as in (8) for N ∈ {2, 3, 4, 5, 10, 25, 50} and T ∈ {25, 50, 100, 250, 500}. To be able to assess the influence of cross-sectional dependence, as opposed to the mere influence of demeaning, we generate data both with and without cross-sectional dependence. That is, we consider the case where θ t is added to the series and let θ t = 0 and θ t ∼ i.i.d. N (0, 1), respectively. Furthermore, as it is desirable to consider the power of the panel data stationarity test, we also investigate the case where a number, N H 1 , of the cross sections contain a random walk component generated according to ξ i;t = ξ i;t-1 + υ i;t , where i ∈ {1, . . . , N H 1 } and υ i;t ∼ i.i.d. N (0, 1).10 The size and power properties of the test, when no cross-sectional dependence is present, are given in Table 3 andTable 4, while the corresponding properties, when cross-sectional dependence is present, are seen in Table 5 andTable 6. 11 When comparing Table 3 andTable 5, it can be seen that the actual presence of cross-sectional correlation plays no role for the size properties of the test. Once a time-specific factor is accounted for, the test performs just as well when crosssectional dependence is present as when it is not. The same results emerge for the power properties of the panel data stationarity test. As seen in Table 4 andTable 6, the power of the test is unaffected by the actual presence of cross-sectional correlation, as long as we allow for it by accounting for a time-specific disturbance term. Moreover, it can be seen from Table 4 and Table 6 that, in both cases, the test fares well when a subset of the cross sections display a unit root behavior.

Conclusions

In this paper, we show that the panel data stationarity test of [START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] has a size distortion when the cross-sectional dimension is small and a time-specific factor is accounted for by demeaning over cross sections. We supply response surface parameters that can be used to calculate critical values that take this correction for cross-sectional correlation into account. The supplied critical values make the size distortions disappear and it is also proved that the test maintains good power properties even after allowing for cross-sectional correlation through a time-specific factor. 1.000 1.000 1.000 1.000 1.000 50 25

1.000 1.000 1.000 1.000 1.000 Notes: a We use N H 1 to denote the number of cross sections that are non-stationary under the alternative hypothesis. b Data is generated as y i;t = i;t + ξ i;t , with i;t ∼ N (0, 1) and ξ i;t = 0 for N -N H 1 cross-sectional units and ξ i;t = ξ i;t-1 + υ i;t , with υ i;t ∼ N (0, 1), for N H 1 cross-sectional units. 1.000 1.000 1.000 1.000 1.000 50 25

1.000 1.000 1.000 1.000 1.000

Intercept and trend N N H 1 T=25 T=50 T=100 T=250 T=500 2 1 0.508 0.852 0.985 1.000 1.000 3 1 0.513 0.858 0.985 1.000 1.000 4 2 0.756 0.981 1.000 1.000 1.000 5 2 0.737 0.980 1.000 1.000 1.000 10 5 0.969 1.000 1.000 1.000 1.000 25 12

1.000 1.000 1.000 1.000 1.000 50 25

1.000 1.000 1.000 1.000 1.000 Notes: a We use N H 1 to denote the number of cross sections that are non-stationary under the alternative hypothesis. b Data is generated as y i;t = i;t + θ t + ξ i;t , with i;t ∼ N (0, 1), θ t ∼ N (0, 1) and ξ i;t = 0 for N -N H 1 cross-sectional units and ξ i;t = ξ i;t-1 + υ i;t , with υ i;t ∼ N (0, 1), for N H 1 cross-sectional units.

  12

Table 1 :

 1 Size of panel data stationarity tests. a;b 'No corr.' refers to the case where no correction is made in order to account for cross-sectional correlation. 'Corr.' refers to the case where the cross-sectional mean is subtracted from each time-series observation. b Data is generated as y i;t = i;t , with i;t ∼ N (0, 1).

		Intercept	Trend			Intercept	Trend
	T N No corr. Corr.	No corr. Corr.	T	N No corr. Corr.	No corr. Corr.
	10 2	0.116	0.161	0.114	0.167	20 2	0.111	0.144	0.108	0.157
	3	0.116	0.148	0.111	0.150		3	0.109	0.138	0.108	0.142
	4	0.113	0.136	0.110	0.137		4	0.111	0.133	0.109	0.136
	5	0.110	0.129	0.112	0.128		5	0.111	0.132	0.108	0.129
	6	0.110	0.125	0.111	0.125		6	0.113	0.127	0.108	0.125
	7	0.108	0.121	0.106	0.121		7	0.109	0.120	0.107	0.121
	8	0.107	0.116	0.107	0.119		8	0.109	0.118	0.108	0.119
	9	0.109	0.118	0.111	0.119		9	0.107	0.119	0.107	0.118
	10	0.109	0.117	0.106	0.116		10	0.109	0.119	0.107	0.116
	25	0.105	0.107	0.104	0.107		25	0.105	0.109	0.106	0.110
	50	0.103	0.104	0.103	0.105		50	0.106	0.109	0.105	0.107
		Intercept	Trend			Intercept	Trend
	T N No corr. Corr.	No corr. Corr.	T	N No corr. Corr.	No corr. Corr.
	50 2	0.106	0.137	0.104	0.149	100 2	0.104	0.136	0.106	0.148
	3	0.107	0.135	0.106	0.135		3	0.106	0.133	0.105	0.135
	4	0.107	0.129	0.105	0.129		4	0.107	0.127	0.107	0.133
	5	0.108	0.127	0.108	0.128		5	0.108	0.126	0.107	0.125
	6	0.108	0.125	0.106	0.122		6	0.106	0.122	0.107	0.121
	7	0.108	0.120	0.109	0.124		7	0.107	0.120	0.105	0.119
	8	0.107	0.120	0.105	0.117		8	0.106	0.117	0.106	0.118
	9	0.108	0.118	0.106	0.116		9	0.108	0.118	0.105	0.118
	10	0.107	0.116	0.106	0.114		10	0.108	0.116	0.106	0.116
	25	0.105	0.109	0.107	0.111		25	0.108	0.112	0.105	0.110
	50	0.102	0.104	0.105	0.107		50	0.105	0.107	0.106	0.107
	Notes: a 13

Table 2 :

 2 Response surface parameters for critical values.

		Intercept
	Sig. Level 10%	β 0 1.317 0.387 -15.564 -0.023 0.900 38.309 0.889 β 1;T S β 2;T S β 1;CS β 2;CS R 2 β CS;T S
	5%	1.793 -0.304 -9.666 -0.383 2.596 19.337 0.957
	2.5%	2.241 -1.501 7.247 -0.828 4.530 -24.375 0.967
	1%	2.784 -3.711 41.326 -1.386 7.174 -107.421 0.967
		Intercept and trend
	Sig. Level 10%	β 0 1.347 0.233 -11.488 -0.278 1.302 26.713 0.914 β 1;T S β 2;T S β 1;CS β 2;CS R 2 β CS;T S
	5%	1.805 -0.490 -4.170 -0.593 2.757	6.036	0.958
	2.5%	2.223 -1.416 9.018 -0.945 4.361 -32.051 0.966
	1%	2.739 -3.181 39.163 -1.454 6.639 -111.125 0.965

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

Table 3 :

 3 Size of demeaned test. a Data is generated as y i;t = i;t , with i;t ∼ N (0, 1).

	Intercept
	N T=25 T=50 T=100 T=250 T=500
	2 0.052 0.053 0.050	0.050	0.052
	3 0.049 0.049 0.050	0.049	0.050
	4 0.048 0.047 0.050	0.048	0.049
	5 0.049 0.049 0.050	0.049	0.049
	10 0.052 0.050 0.050	0.050	0.052
	25 0.050 0.049 0.049	0.049	0.050
	50 0.047 0.049 0.048	0.048	0.049
	Intercept and trend
	N T=25 T=50 T=100 T=250 T=500
	2 0.052 0.052 0.050	0.049	0.048
	3 0.048 0.048 0.048	0.048	0.048
	4 0.047 0.048 0.050	0.048	0.050
	5 0.048 0.050 0.050	0.050	0.050
	10 0.050 0.051 0.050	0.052	0.051
	25 0.050 0.050 0.050	0.047	0.049
	50 0.048 0.046 0.046	0.048	0.048
	Notes: a 15

Table 4 :

 4 Power of demeaned test.

				a;b
			Intercept
	N N H 1 T=25 T=50 T=100 T=250 T=500
	2	1	0.693 0.896 0.982	0.999	1.000
	3	1	0.705 0.903 0.984	1.000	1.000
	4	2	0.905 0.990 1.000	1.000	1.000
	5	2	0.900 0.990 1.000	1.000	1.000
	10	5	0.997 1.000 1.000	1.000	1.000
	25 12	1.000 1.000 1.000	1.000	1.000
	50 25	1.000 1.000 1.000	1.000	1.000
			Intercept and trend
	N N H 1 T=25 T=50 T=100 T=250 T=500
	2	1	0.510 0.850 0.983	1.000	1.000
	3	1	0.518 0.854 0.986	1.000	1.000
	4	2	0.758 0.982 1.000	1.000	1.000
	5	2	0.737 0.978 1.000	1.000	1.000
	10	5	0.969 1.000 1.000	1.000	1.000
	25 12

Table 5 :

 5 Size of demeaned test when CSD is present. a Data is generated as y i;t = i;t + θ t , with i;t ∼ N (0, 1) and θ t ∼ N (0, 1).

	Intercept
	N T=25 T=50 T=100 T=250 T=500
	2 0.052 0.052 0.051	0.051	0.050
	3 0.047 0.049 0.049	0.048	0.047
	4 0.049 0.050 0.049	0.050	0.051
	5 0.050 0.049 0.052	0.049	0.050
	10 0.050 0.050 0.049	0.052	0.049
	25 0.050 0.048 0.050	0.050	0.049
	50 0.048 0.046 0.049	0.049	0.047
	Intercept and trend
	N T=25 T=50 T=100 T=250 T=500
	2 0.052 0.051 0.049	0.050	0.048
	3 0.049 0.048 0.046	0.049	0.050
	4 0.048 0.051 0.048	0.048	0.049
	5 0.048 0.049 0.049	0.049	0.050
	10 0.050 0.050 0.050	0.050	0.051
	25 0.049 0.050 0.049	0.051	0.050
	50 0.046 0.047 0.046	0.049	0.047
	Notes: a

17 Page 17 of 18 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
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Table 6 :

 6 Power of demeaned test when CSD is present. a;b

			Intercept
	N N H 1 T=25 T=50 T=100 T=250 T=500
	2	1	0.689 0.896 0.982	1.000	1.000
	3	1	0.704 0.904 0.985	1.000	1.000
	4	2	0.905 0.991 1.000	1.000	1.000
	5	2	0.901 0.990 1.000	1.000	1.000
	10	5	0.997 1.000 1.000	1.000	1.000
	25 12

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

It is important to note that[START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF] allows for the covariance stationary disturbance to be serially correlated, while[START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data where the Time Dimension is Finite[END_REF] consider the case where the disturbance is serially independent.

It should be noted, however, that other forms of cross-sectional dependency might not be addressed by demeaning the data (see e.g.[START_REF] Strauss | Shortfalls of Panel Unit Root Testing[END_REF].5

The final specification in[START_REF] Breitung | Unit Roots and Cointegration in Panels[END_REF] was obtained after trying out various specifications for the response surface regressions. The final specification was chosen on the basis of in-sample fit.

In[START_REF] Breitung | Unit Roots and Cointegration in Panels[END_REF], TS and CS refer to 'time series' and 'cross section', respectively.

To reduce the influence of the initial condition, » i;0 = 0, where i ∈ {1; : : : ; N H 1 }, for the random walk component, T + 100 time series observations are generated, while only the last T are used to construct the time series.

Acknowledgements

The author is grateful for the help obtained from discussions with Tommy Andersson. Financial support, from The Crafoord Foundation, The Royal Swedish Academy of Sciences and The Jan Wallander and Tom Hedelius Foundation, research grant number P2005-0117:1, is gratefully acknowledged. All the simulations in the current paper are performed in GAUSS and run on the LUNARC cluster.


Carmelo Reverte

Isidoro Guzman

THE PREDICTIVE ABILITY OF RELATIVE EFFICIENCY FOR FUTURE EARNINGS:AN APPLICATION USING DATA ENVELOPMENT ANALYSIS TO SPANISH SMEs

Keywords: relative efficiency, earnings prediction, DEA, SMEs

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

ability of a firm -as compared to the other firms under analysis-to generate the maximum level of outputs given a certain level of inputs. In this paper we use a non-parametric technique based on linear programming referred to as Data Envelopment Analysis (DEA) in order to provide a numerical measure of the relative efficiency of business units in using available inputs to generate the maximum level of outputs. This technique has the advantage over other methodologies (such as the stochastic frontier analysis -SFA-) of avoiding the use of any a priori assumptions about the functional form of the production function.

Our findings highlight that our efficiency measure has an incremental predictive ability over and above current earnings and book value of equity for predicting future earnings. Moreover, we have further validated the models in a holdout sample and our results evidence the highest forecast accuracy of the model that includes our DEA-based efficiency measure as an additional predictor to current earnings and book value of equity.

The rest of the paper is structured as follows. In the next section we discuss the DEA methodology used in our empirical analysis to measure relative efficiency. In Section 3 we present the research design, including the methodology, variables, hypotheses and a description of the In section 4 we present the results from the empirical analysis. Finally, we summarise the main conclusions of our study.

EFFICIENCY MEASURE: DATA ENVELOPMENT ANALYSIS

Data Envelopment Analysis (DEA) is a non-parametric technique developed by [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF] that using linear programming determines the relative efficiency of decision making units (DMUs) in the use of single or multiple inputs for the production of single or multiple outputs1 . The advantage of this technique is that there is no need to make any a priori assumptions about the functional form of the production function. Instead, an empirical production function, in the form of a piece-wise linear frontier, is fitted to the observed data based on the amounts of outputs and inputs of the business units under analysis so that the relative efficiency is assessed against the benchmark of the so-called "best-practice frontier". Those DMUs placed on the frontier are deemed to be efficient while those DMUs situated below the frontier are regarded as inefficient2 .

Assuming that there are n DMUs that use a vector of m inputs to produce a vector of s outputs, DEA allows to determine the efficiency (h z ) of each DMU (DMU z ) by solving the following linear programming problem:

Max = i iz i s rz z x v y u h r [1] subject to: 1 = i if i s rf z x v y u h r f = 1, 2, …..., n [2]
u r 0, r = 1,...,s v i 0, i = 1,...,m where y r and x i represent the amount of output r and input i, respectively while u r and v i represent their corresponding weights.

The previous formulation corresponds to the original model developed by [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF], known as the CCR model, which assumes that all DMUs operate under constant returns to scale. This assumption was relaxed in the model proposed by [START_REF] Banker | Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis[END_REF] -known as the 3 BCC model-that allows for variable returns to scale. [START_REF] Banker | Maximum likelihood, consistency and data envelopment analysis: A statistical foundation[END_REF] proved that in large samples the BCC estimator is consistent because the distance between the DEA estimate and the true efficiency score goes towards zero.

The output-oriented BCC model [START_REF] Banker | Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis[END_REF] to estimate the efficiency of each DMU is given by solving the following linear programming:

Max z [3]
s.a.:

iz n f i if f x S x = + =1 i = 1, ...,m [4] 0 1 = + = o n f rf f rz z S y y r = 1, ...,s [5] 1 1 = = f n f [6] 0 ; 0 f z f = 1, ...,n [7]
where z is the efficiency score for the DMU z being evaluated, x if is the amount of input i consumed by DMU f, y rf is the amount of output r produced by DMU f, x iz is the amount of input i consumed by the DMU z being evaluated, y rz is the amount of output r produced by the DMU z being evaluated, f is the weight on the inputs/outputs of DMU f and S o and S i are the slacks of the linear programming model.

The optimal value ( z ) of the linear program ranges between 1 and , such that it is equal to 1 for efficient units and higher than 1 for inefficient units. The estimated relative efficiency measure ( z = 1/ z ) varies between 0 and 1. Therefore, a DMU with a relative efficiency measure ( z ) equal to 1 is regarded as efficient whereas if it takes a value of, for example, 0.85 this would imply that the DMU being evaluated will have to increase its output an 15% of the estimated value in the efficient frontier.

RESEARCH DESIGN

Methodology, variables and hypothesis

As efficiency is sector-sensitive, we have splitted our sample of SMEs in several sectors and computed the efficiency of each firm in comparison with the rest of firms in the same sector. After analysing the data, firms were grouped in the following three sectors: a) Manufacturing; b) Service sector and c) Construction. The methodology used in our empirical study is based on three stages. In the first one, we obtain for each year and firm the DEA-based efficiency scores. As in [START_REF] Abad | Eficiencia relativa como indicador del rendimiento futuro[END_REF], the proposed DEA model takes the inputs and outputs that correspond with those of a typical production function (see Table 1). Therefore, our model will be oriented towards achieving the maximum level of revenues for a given level of inputs.

In the second stage, we estimate the following regression models (see Table 2) in order to test the incremental predictive ability of our relative efficiency measure over and above current earnings and book value of equity. The first model includes current earnings and book value of equity as regressors. Regarding the former, there is a huge number of studies that have shown the predictive ability of current earnings for predicting future earnings (e.g., [START_REF] Ball | Some time series properties of accounting income[END_REF][START_REF] Albretch | The time series properties of annual earnings[END_REF][START_REF] Watts | The time series of annual accounting earnings[END_REF]. Concerning book value of equity, the Ohlson valuation model (1995) shows its usefulness for predictive purposes, being that higher the lower the persistence of current earnings.

In this respect, Shroff (1999) -for the US-and Reverte ( 2003 DEA efficiency measure in year t is positively associated with next year's (t+1) earnings, after controlling for the effect of current earnings and book value of equity. To the extent our efficiency measure captures the permanent component of earnings, it should be able to predict future earnings over and above current earnings and book value. Therefore, we expect 3 to be positive and statistically significant.

Finally, in the last stage of our study, we validate our models in a holdout sample different to that used in their estimation by computing the differences in earnings forecasted by models ( 1) and ( 2) and actual earnings corresponding to our sample of firms for year 2004. In order to assess the forecast accuracy of both models, we compute the mean absolute percentage error, whose formulation is shown on Table 3. This measure has two advantages. First, as forecast errors are taken in absolute value, it avoids negative and positive errors to compensate each other. Second, it is not sensitive to scale as errors are divided by the actual value of the forecasted variable. Notes: e i : forecast error (difference between the actual and forecasted value). y i : actual value of forecasted variable. N: number of observations in the forecast period.

Once we have computed the mean absolute percentage error, we examine whether the differences in the forecast errors derived from models (1) and ( 2) are statistically significant by applying the non-parametric Wilcoxon signed-rank test, which is appropiate when the distribution of the errors departs from normality.

Sample

Data used in our study are obtained from the database SABI. We have chosen small and medium-sized firms (SMEs) with the legal form of public or limited liability companies 3 . The sample period ranges from 1999 to 2003, including additionally the year 2004 in order to validate our models. To ensure reliability of our data, we include only SMEs with a favourable audit report. 4 reports the descriptive statistics of our regressors (i.e., earnings, book value of equity and efficiency) for each of the three sectors and also for the aggregate sample. As it can be observed, the mean efficiency scores are quite similar across sectors, ranging from 81.7% for the manufacturing sector to 85.4% for the service sector. The mean efficiency value for the pooled sample is 83.6%. Table 5 shows the correlation matrix among the regressors. It can be observed that our efficiency measure is not highly correlated with both current earnings and book value of equity in the three sectors. Specifically, for the pooled sample, that correlations are 0.39 and 0.17, respectively. Hence, we can assert that the information captured by our efficiency measure is not already contained in both current earnings and book value of equity, avoiding any collinearity problem.

4.RESULTS

Cluster analysis

In order to get a first approximation of descriptive nature on the relationship between our efficiency measure and future earnings for our sample of Spanish SMEs, we have performed a kmeans cluster analysis. Specifically, for each of the three sectors and for the pooled sample, we have identified three clusters based on the current value of the efficiency measure (EFFIC t ). Then, we compute for each cluster the mean value of next-year's earnings (NI t+1 ) and, finally, we perform the Kruskal-Wallis test in order to examine whether the differences in the mean value of next-year's earnings across the three clusters are statistically significant. Results from this cluster analysis are reported on Table 6. 8 reports, for the three sectors and the pooled sample, the validation results as well as the Wilcoxon signed-rank test in order to examine whether the differences in the forecast errors derived from models (1) and ( 2) are statistically significant. Our results document that the model including the efficiency measure (model 2) has the highest forecast accuracy for the three sectors since the forecast errors are lower than model 1. Moreover, the Z-statistic of the Wilcoxon test indicates that the differences between the forecast errors of both models are statistically significant (p-value= 0.0001 for the pooled sample).

CONCLUDING REMARKS

Research on earnings prediction has documented that the transitory component of current earnings makes that figure less useful for predicting future earnings (e.g., [START_REF] Brooks | Further Evidence of the Time Series Properties of Accounting Income[END_REF][START_REF] Beaver | What determines price-earnings ratios?[END_REF]. The implication of this finding is that a measure that better captures the underlying persistent component of earnings might prove very useful in predicting future earnings when used along with current earnings. Our study, based on a large sample of 1939 Spanish SMEs, investigates whether an alternative mesure of performance ignored in previous research on earnings forecasting -i.e, relative efficiency-has predictive ability over and above current earnings and book value for predicting future earnings. Relative efficiency captures the inherent ability of a firm to make the most productive use of available resoureces as compared to the rest of firms in the sample. In order to measure the relative efficiency of our sample firms we use the Data Envelopment Analysis (DEA) non-parametric technique.

Our findings highlight that our efficiency measure has an incremental predictive ability over and above current earnings and book value of equity for predicting future earnings in the three sectors considered (manufacturing, services and construction). Moreover, we have further validated the models in a holdout sample and our results evidence again in the three sectors the highest forecast accuracy of the model that includes our efficiency measure as an additional predictor to current earnings and book value of equity. Therefore, our research suggest the usefulness of relative efficiency to capture the persistent component of earnings and, as a result, support its inclusion as an additional predictor in earnings forecasting models.

  ) -for Spain-have corroborated the predictive ability of book value of equity for the case of listed firms. The second model includes our relative efficiency measure as an additional regressor in order to test the following hypothesis: The

  As a result of the previous filters, our sample comprises 1939 SMEs and a total of 7756 firm/year observations for our four-year estimation period. The composition of the sample by sector is as follows: a) Manufacturing sector (802 firms/3208 observations), b) service sector (957 firms /38283 In order to define a firm as SME, we have followed the Recommendation 2003/361/EC of the European Commission on SME definition issued on 6 th May 2003, selecting the maximum levels proposed in this Recommendation, namely: total assets, up to 43 million euros; sales revenues, up to 50 million euros, and average number of employees, up to 250.

  c) construction sector (180 firms/720 observations). Table

Table 1 . DEA model (output orientation)

 1

	Output:
	-Revenues
	Inputs:
	-Cost of Materials consumed	Materials
	-Personnel expenses	Labor
	-Depreciation expense	Capital
	-Other operating expense	Overhead

Regarding the DEA methodology, we have computed the efficiency scores based on

[START_REF] Banker | Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis[END_REF]

that consider variable returns to scale (BCC model). Moreover, we have chosen an output orientation as we are interested in relating our efficiency measure to future earnings.

Table 2 . Earnings prediction models Model 1 : NI i,t+1 = 0 + 1 NI i,t + 2 BV i,t + e i,t Model 2 : NI i,t+1 = 0 + 1 NI i,t + 2 BV i,t + 3 EFFIC i,t + e i,t

 2

'

Variables: NI i,t+1 : Net income of firm i in year t+1; NI i,t : Net income of firm i in year t; BV: Book value of equity for firm i at the end of year t; EFFIC t : Efficiency score of firm i in year t.

Table 3 . Forecast accuracy measure

 3

	Mean	absolute	percentage	error	=	N 1	y e	i i

Table 4 . Descriptive Statistics by Sectors and for the Aggregate Sample

 4

	Panel A: Manufacturing sector (N=3208 obs.)
		NI t	BV t	EFFIC t
	Mean	611.22	5018.69	0.817
	Median	374.00	3680.00	0.810
	Standard deviation	838.69	4475.07	0.099
	Minimum	-4650.00	-2654.00	0.082
	Maximum	10440.00	33853.00	1.000
		Panel B: Service Sector (N=3828 obs.)
		NI t	BV t	EFFIC t
	Mean	453.69	3493.08	0.854
	Median	277.00	2360.05	0.856
	Standard deviation	680.56	3603.72	0.088
	Minimum	-4201.00	-1459.00	0.171
	Maximum	7379.00	31869.00	1.000
		Panel C: Construction sector (N=720 obs.)
		NI t	BV t	EFFIC t
	Mean	502.80	3019.87	0.828
	Median	328.00	2246.00	0.826
	Standard deviation	696.75	2835.88	0.121
	Minimum	-4048.00	-733.00	0.088
	Maximum	5857.00	19916.00	1.000
		Panel D: Aggregate sample (N=7756 obs.)
		NI t	BV t	EFFIC t
	Mean	523.42	4080.29	0.836
	Median	316.00	2785.00	0.835
	Standard deviation	755.05	4009.44	0.097
	Minimum	-4650.00	-2654.00	0.082
	Maximum	10440.00	33853.00	1.000

Table 5 . Correlations among explanatory variables Panel A: Manufacturing sector (N=3208 obs.) NI

 5 NI t : Net income of firm i in year t (thousands of euros); BV: Book value of equity for firm i at the end of year t (thousands of euros); EFFIC t : Efficiency score of firm i in year t.

	t	BV t	EFFIC t
	NI t	0.624	0.425
	BV t		0.209
	Panel B: Service Sector (N=3828 obs.)
	NI t	BV t	EFFIC t
	NI t	0.567	0.413
	BV t		0.216
	Panel C: Construction Sector (N=720 obs.)
	NI t	BV t	EFFIC t
	NI t	0.638	0.383
	BV t		0.207
	Panel D: Aggregate sample (N=7756 obs.)
	NI t	BV t	EFFIC t
	NI t	0.604	0.386
	BV t		0.170
	Notes:

Table 6 . Cluster Analysis Panel A: Manufacturing Sector (N=3208 obs.)

 6

	Cluster	N	EFFIC t	NI t+1
	1	1045	0.711	266.12
	2	1448	0.824	549.21
	3	715	0.955	1218.17
	Kruskal-Wallis test for NI t+1 : 2 = 460.03 (p-value: 0.0001)
	Panel B: Service Sector (N=3828 obs.)
	Cluster	N	EFFIC t	NI t+1
	1	845	0.737	208.19
	2	1819	0.845	370.76
	3	1164	0.952	810.86
	Kruskal-Wallis test for NI t+1 : 2 = 452.56 (p-value: 0.0001)
	Panel C: Construction Sector (N=720 obs.)
	Cluster	N	EFFIC t	NI t+1
	1	229	0.694	333.26
	2	278	0.826	550.34
	3	213	0.975	938.12
	Kruskal-Wallis test for NI t+1 : 2 = 49.12 (p-value: 0.0001)
	Panel D: Aggregate sample (N=7756 obs.)
	Cluster	N	EFFIC t	NI t+1
	1	2096	0.718	261.17
	2	3559	0.835	454.68
	3	2101	0.956	949.51
	Kruskal-Wallis test for NI t+1 : 2 = 798.55 (p-value: 0.0001)
	Notes: N: Number of observations in each cluster; NI t+1 = Net income in t+1
	(thousands of euros); EFFIC

t : Efficiency score in year t.

The DEA model can have two different orientations: a) an output orientation, in which the maximum amount of outputs for a given level of inputs is pursued; b) an input orientation, in which it is pursued to consume the minimum amount of inputs to achieve a given level of outputs.

For a review of the DEA approach and its methodological problems, see for instance[START_REF] Färe | Input aggregation and technical efficiency[END_REF],[START_REF] Sengupta | Economics of efficiency measurement by the DEA approach[END_REF],[START_REF] Färe | Aggregation bias and its bounds in measuring technical efficiency[END_REF] andBarnum and Gleason (2005, 2006).

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

As can be shown on Table 6, in the three sectors the higher the value of efficiency in the current period the higher the value of next-year's earnings. For cluster 1, comprised by those observations with the lowest values of the efficiency measure (0.718 for the aggregate sample), the mean value of next-year's earnings is 261.17 millions euros. On the contrary, in cluster 3, comprised by those observations with the highest values of the efficiency measure (0.956 for the aggregate sample), the mean value of next-year's earnings is 949.51 millions euros. The Kruskal-Wallis test evidences that the differences in the mean value of next-year's earnings across the three clusters are statistically significant at the 1% level for the three sectors and also for the aggregate sample.

Estimation results

Table 7 reports the results for the pooled sample period (1999)(2000)(2001)(2002)(2003) regarding the estimation of the two earnings prediction models (1) and (2) for the three sectors and for the aggregate sample. It can be observed that, as hypothesised, our efficiency measure is positively and significantly associated with next-year's earnings. The F-test indicates that efficiency has an incremental explanatory power over and above current earnings and book value of equity for predicting next-year's earnings in the three sectors (manufacturing, services and construction). For the pooled sample, the F-test is statistically significant at the 1% level (p-value=0.00). This finding corroborates our main hypothesis, i.e., the more efficient firms in the use of available resources -as compared to the rest of firms in the sample-have higher earnings in the following period than less efficient firms.

Validation results

In the predictive ability studies the relevant test is the validation of the models in a holdout sample different to that used in their estimation. In this respect, our hypothesis is as follows: The model including our efficiency measure (model 2) predicts more accurately next-year's earnings than the model that only includes current earnings and book value of equity (model 1).

We first compute the differences in earnings forecasted by models ( 1) and ( 2) and actual earnings corresponding to our sample of firms for year 2004. In order to assess the forecast accuracy of both models, we compute the mean absolute percentage error.