Yannis Georgellis
email: yannis.georgellis@brunel.ac.uk

Andros Gregoriou

Nikolaos Tsitsianis

Adaptation towards Reference Values: A Non-linear Perspective

Keywords: JEL Classification: D60, I31, J60, C20 Reference Points, Relative Utility, Adaptation, ESTAR Models, Job Satisfaction

Using large-scale panel data, we examine the dynamics of adjustment towards reference points for key workplace attributes. We discover that an Exponential Smooth Transition Autoregressive (ESTAR) model is superior to a linear model in characterizing such a process. The speed of adjustment increases nonlinearly with the distance from reference points and adjustment is faster for job satisfaction shocks compared to shocks in earnings or work hours. Our findings lend further credence to the relative utility hypothesis and highlight the inherently nonlinear nature of the dynamic path of adjustment towards reference points, a previously neglected issue in the adaptation literature.

Introduction

The notion that utility is relative in nature continues to gain a wider acceptance among economists and to be supported by a growing number of empirical studies. 1 Clark and Oswald (1996), for example, show that utility depends on income relative to some reference or comparison income, while [START_REF] Easterlin | Income and happiness: Toward a unified theory[END_REF] advocates that earnings expectations should be explicitly introduced in a utility function. In the same vein, [START_REF] Stutzer | The role of income aspirations in individual happiness[END_REF] finds that higher income aspirations, determined both by individuals' past income and the average income in their community, reduce utility, a conclusion also supported by the findings of [START_REF] Ferrer-I-Carbonell | Income and well-being: An empirical analysis of the comparison income effect[END_REF] who shows that income of the reference group is as important as one's own income for individuals' happiness. 2 Recent work on adaptation further highlights the importance of reference points in determining actual behavior in that a divergence between individuals' current status and their aspiration or reference level causes disutility that could spur them into action to change their current status. For example, [START_REF] Rizzo | Reference incomes, loss aversion, and physician behavior[END_REF] find that physicians take strong, often unappealing actions to boost their earnings when they experience shortfalls from their reference income. Strikingly, for physicians who are at or above their reference income, reference income does not seem to influence future income growth. In a different context, [START_REF] Fiegenbaum | Strategic reference point theory[END_REF] also reaffirm the prediction that the risk behavior of individuals depends on both their reference point and their current situation. Individuals who believe that their current state is below their reference point tend to exhibit risk-seeking behavior, while individuals who believe that their current state is above their reference point exhibit risk-averse behavior. 1 This trend follows the publication of the [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF] article on prospect theory, which according to the 2002 Nobel Prize committee, is the most cited paper ever published in Econometrica. 2 [START_REF] Van De Stadt | The relativity of utility: Evidence from panel data[END_REF] offer some of the earlier evidence consistent with the relative utility hypothesis, although they acknowledge the possibility that utility might be partly absolute. Other notable studies supporting the relative utility hypothesis include [START_REF] Clark | Is utility absolute or relative?[END_REF] and [START_REF] Solberg | Wanting, having, and satisfaction: Examining the role of desire discrepancies in satisfaction with income[END_REF].

A c c e p t e d M a n u s c r i p t 2 Although psychological theories of adaptation, often encapsulated in the set point or baseline model, examine how departures from reference points affect individuals' utility, there is little empirical evidence on the dynamics of the adaptation process. The set point model postulates that people react to or are affected psychologically by events, but they eventually adapt back to their baseline level of well-being. This baseline level is essentially a reference point determined by individuals' past, pre-event experience, or by individuals' standing relative to others, "people like you". [START_REF] Brickman | Lottery winners and accident victims: is happiness relative?[END_REF], [START_REF] Headey | Personality, life events, and subjective well-being: Toward a dynamic equilibrium model[END_REF], and [START_REF] Lykken | Happiness is a stochastic phenomenon[END_REF], among others, provide strong supporting evidence for the set point theory, concluding that adaptation is usually quick and complete. 3 The study by [START_REF] Lucas | Reexamining adaptation and the set point model of happiness: Reactions to changes in marital status[END_REF] is one of the first to support the set point theory using large-scale panel data, thus overcoming some of the limitations of earlier empirical work. However, more rigorous testing of adaptation theories and the dynamics of adjustment towards reference points is still in its infancy.

In this paper, we use data from the German Socio-Economic Panel (GSOEP), a large-scale panel survey, to examine the dynamics of the adjustment process towards reference values for key workplace characteristics. 4 Focusing on earnings, hours of work, and overall job satisfaction, three factors often quoted as important determinants of labor market behavior, we model the dynamics of adjustment as a non-linear process. More specifically, we fit an Exponential Smooth Transition Autoregressive (ESTAR) model, which we find to be superior to a linear model in characterizing the path of adjustment towards reference points. The estimated ESTAR coefficients imply that the speed of adjustment increases in a non-linear fashion with the distance from reference points.

Further comparisons across the three main job attributes under consideration reveal that a one 3 Van Praag (1971) and Van Praag and Kapteyn (1973) were the first, among economists to explore the 'preference drift' phenomenon, which refers to the same mechanism that [START_REF] Brickman | Hedonic relativism and planning the good society[END_REF] termed 'hedonic treadmill'. In recent years, economists have embraced adaptation as an important issue that could further our understanding of economic behavior. Recent evidence by [START_REF] Easterlin | A puzzle for adaptive theory[END_REF] suggests that there is a clear distinction to be made between adaptation in the economic domain and adaptation in the life domain. Easterlin finds that aspirations about economic wealth and other pecuniary aspects of one's well-being tend to change with the level of actual circumstances, suggesting almost complete adaptation, but this is not the case with marriage, number of children and other non-pecuniary aspects of one's life. In another study, [START_REF] Dunn | Loss aversion and adaptation in the labor market: Empirical indifference functions and labor supply[END_REF] uses data on income and leisure tradeoffs and finds evidence of adaptation in the labor market. [START_REF] Van Praag | The anatomy of subjective well-being[END_REF] explore the link between general well-being and well-being in various domains, including health, financial situation, job, leisure, housing, and environment, thus providing a broader context for the study of adaptation.

3 standard deviation change in job satisfaction results in a much higher speed of adjustment towards reference values and a higher probability of job switching than a one standard deviation change in wages or work hours. Gender differences in the adjustment process towards reference values are also apparent. Although shocks in wages and working hours seem to be equally important in terms of the speed of adjustment towards reference points for both men and women, we find that, compared to men, the speed of adjustment towards reference wages and hours is lower for women.

Most interestingly, our results show that an asymmetry in the adjustment process towards reference values exists depending on whether actual values are above or below their reference values. While our results support the hypothesis of a non-linear adjustment when actual values are below the reference values, there is no adjustment at all, linear or non-linear, when actual values are above their reference values.

The remainder of the paper is structured as follows. Section 2 describes the GSOEP sample and outlines the empirical methodology. Section 3 presents the main results, and section 4 concludes.

2

Data and Methodology

Data

We use data from the first 19 waves of the German Socio-Economic Panel (GSOEP) for the period 1984-2002. The GSOEP is a nationally representative panel data set of individuals and households residing in the old and new federal German states. The original sample in 1984 was selected following a stratified random sampling procedure, and the panel is periodically refreshed with new samples so that successive waves are representative of the relevant population in Germany. In 4 2002, there were approximately 12,000 households and 22,000 persons in the panel. 5 The dataset provides detailed information on personal and labor market characteristics, including information on the three main variables of interest: labor earnings, work hours and job satisfaction. 6To provide a context for our empirical analysis, it is worth noting that during the period 1984-2002 there has been a real increase in average earnings, accompanied by a remarkable increase in the gap between the mean and median values and the dispersion of the earnings distribution. This is indicative of an increase in the proportion of people in the lowest quartile, explained by the slowdown of the German economy and rising unemployment rates. In terms of reported job satisfaction, there is a downward trend in Germany in the 1990s, for both men and women. This downward trend in job satisfaction is also confirmed independently in the ISSP series: the proportion of workers completely or very satisfied with their job was 39 percent in 1997, down from 44 percent in 1989. Finally, there is an increase in the number of workers who are not satisfied with the work hours, with an increasing preponderance of both male and female workers who wish to work fewer hours, and from a smaller base an increase in the proportion wishing to work more hours.

Methodology

For our empirical analysis, we limit the sample to include only West German workers who are between 18 and 65 years of age and working full-time in either the public or private sectors.

Excluding the self-employed and those in training results in a sample of 49,177 and 20,903 personyear observations for males and females. We track workers in this sample up to the point when job switching occurs, that is, when workers' full-time spell with their current employer is terminated followed by an employment spell with a new employer. Using the above proxies for workers' reference values, we then explore the dynamics of adjustment towards these reference values, adopting the methodology by [START_REF] Granger | Modelling non-linear economic relationships[END_REF] and [START_REF] Terasvirta | Specification, estimation, and evaluation of smooth transition autoregressive models[END_REF] on the Smooth Transition Autoregressive (STAR) model. Although alternative non-linear models could be considered, we adopt the STAR methodology because it allows for the speed of adjustment to vary in a non-linear fashion with the distance between actual and reference values. Moreover, STAR models imply a smooth non-linear adjustment process, as opposed to Markov switching type of models that assume that adjustment occurs instantly from one state to another. 8 Before adopting the STAR model as the preferred model for capturing the dynamics of adjustment of actual to reference values, we perform a three-stage testing procedure for the presence of non-linearities. First, we specify the appropriate lag length for the linear 7 Other studies on reference groups include [START_REF] Easterlin | Will raising the income of all increase the happiness of all[END_REF], who implicitly assumes that individuals compare themselves with all the other citizens of the same country, and [START_REF] Persky | Local status and national social welfare[END_REF], who assume that the reference group consists of all individuals living in the same region. Formally, consider two possible regimes comprising a pure 'small' and pure 'large' adjustment of individuals' current status with respect to changes in their reference points.9

Following Granger and Terasvirta, we write a STAR model of order , k for t e as

    0 1 0 1 ' ' t t t t d t e x x F e w           , (2)
where

        2 1 2 1 1 2 1 1 2
, ,... , , ,... ', , ,... ', ~0, ,

t t t t k k k t x e e e w iid                  . F is the continuous transition function, t d
e  is the switching variable, and d is the delay parameter.

 

.

F is a monotonically increasing function with   0 F   and   1 F  , which yields a non-linear asymmetric adjustment.

Consider the following LSTAR function

      1 1 exp t d t d F e a e c            , (3)
where a measures the smoothness of transition from one regime to another and c is a threshold value for e that indicates the halfway point between the two regimes.

The LSTAR model assumes that different regimes may have different dynamics and that adjustment takes place in every period, but the smoothness of adjustment varies with the extent of

 . 0.5 F  if . t d e c  
The smaller is a , the smoother the transition. In the extreme, 0 a  means that   . F becomes a constant, so (2) becomes a linear model. On the other hand, as a  there is an even sharper transition at t d e c

  where  

. F jumps from 0 to 1. [START_REF] Terasvirta | Characterizing nonlinearities in business cycles using smooth transition autoregressive models[END_REF] define the Exponential STAR (ESTAR) function as:

    2 1 exp t d t d F e a e c           , (4)
where, as previously, a measures the speed of transition from one regime to another and c is some threshold value for e that indicates the halfway point between the two regimes. The ESTAR function in (4) defines a transition function about c where   .

F

is still bounded between 0 and 1.

As in the case of the LSTAR model above, the main property of the ESTAR model, described in equation ( 4), that makes it an attractive model in the present context is the fact that it captures the non-linear, smooth adjustment process towards reference values.

The initial testing for the presence of non-linearities in t e involves three stages. First, a linear autoregressive model for e is specified in order to determine the lag length . : ' 0 H   is important for an LSTAR specification, it is insignificant for an ESTAR model (see page 209, equation 3.10 in Terasvirta). However, [START_REF] Granger | Modelling non-linear economic relationships[END_REF] and [START_REF] Terasvirta | Specification, estimation, and evaluation of smooth transition autoregressive models[END_REF] show that application of this sequence of tests may lead to incorrect conclusions because the higher order terms of the Taylor expansion used in deriving these tests are disregarded. Thus, they recommend that the choice of STAR model should be made on the basis of the lowest p-value among the values computed for all the F tests of ( 6)-( 8).

To account for potential asymmetries in the adjustment process, we implement the STAR methodology separately when actual values are below reference values and when actual values are above reference values. When actual values are above reference values, the model does not reject the null hypothesis of no non-linear adjustment. In this case, a further test of the null hypothesis H o :  1 = 0 for the presence of linear adjustment shows no adjustment at all. Therefore, all subsequent results reported in the empirical section focus on non-linear adaptation of earnings, job satisfaction and work hours when actual values are below reference values. 10 3 Empirical results

Reference values

In the first stage of our empirical analysis, we construct reference values for earnings and job satisfaction based on the estimated coefficients of standard earnings and job satisfaction equations respectively. 11 These results are shown in Tables A1 andA2 in the appendix, and they are broadly consistent with the findings of previous studies. Briefly, as the results in Table A1 suggest, both males and females earnings are higher for those with longer job tenure, those in higher level occupations and for skilled workers, and those working in larger firms. A high secondary degree followed by apprenticeship seems to yield, ceteris paribus, a positive and significant return for men. Age seems to have a positive, non-monotonic effect on earnings, while the effect of marriage is stronger for men than that for women. As expected, health limitations have a negative effect on earnings.

The results in Table A2 reveal that earnings have a positive effect on job satisfaction, while reference earnings have a negative effect. 12 Preferences for work hours are important determinants of job satisfaction, consistent with the findings of previous studies. 13 Although marriage has a 10 The results of linear and non-linear tests for the case of actual values above the reference values are available upon request.

11 Estimated using a panel fixed effects estimator. A Hausman test rejected the random effects specification. 12 The additional list of variables included in the earnings equation and not included in the job satisfaction equations help to identify the model. A number of alternative specifications of the earnings and job satisfaction equations produce very similar results, but the specifications in Tables A1 and A2 dominate, based on standard log-likelihood comparisons among alternative specifications. 13 For example, [START_REF] Green | An investigation of national trends in job satisfaction in Britain and Germany[END_REF] find that working more hours than the desired level lowers job satisfaction for men by an average of 0.34. Interestingly, [START_REF] Clark | Your money or your life: Changing job quality in OECD countries[END_REF] summarizes evidence in OECD countries, showing that overall job satisfaction is either stable or declining despite rising wages and falling hours of work.

positive effect on men's job satisfaction, it has a negative effect for women. Consistent with the findings of Clark et al. (1996), job satisfaction increases with age, but in a non-linear fashion. (falling), conditional on the assumption that their norms are changing a little or not at all. The same assumption can be made using the interpretation of job satisfaction adopted by [START_REF] Levy-Garboua | Reported job satisfaction: What does it mean?[END_REF]. Whether this assumption is valid depends on the circumstances, which partly depend on the time horizon. Taken over the very long-term, this assumption may be dubious, but over the medium term, within a decade or so, it may be reasonable to assume comparatively stable norms.

[Figures 1, 2 and 3 about here]

Figure 3 summarizes the trend in actual and reference values for hours. It is evident that, for both males and females, actual hours are consistently above the reference values for the whole period under consideration. An obvious explanation might be that workers are obliged to work more than they wish because of structural and institutional constraints. Numerous studies in the economics literature focus on labor demand as a main factor restricting workers' income-leisure choices [START_REF] Altonji | Labor supply, hours constraints and job mobility[END_REF][START_REF] Kahn | The causes of hours constraint: Evidence from Canada[END_REF][START_REF] Hamermesh | Adjustment costs in factor demand[END_REF][START_REF] Euwals | Desired and actual labour supply of unmarried men and women in the Netherlands[END_REF]). However, one would expect that this explanation is less relevant in contemporary labor markets where part-time work, flexibility and family friendly work schedules become increasingly more important aspects of the workplace. Alternative explanations why workers' actual hours exceed their desired hours are based on social rationality arguments and goal-framing effects [START_REF] Lindenberg | Social rationality versus rational egoism[END_REF]. Such explanations are based on the premise that workers are unlikely to view work simply as a choice between income and leisure, resulting in a deviation between actual and preferred hours. A similar explanation emphasizes the dynamics of the post-Fordist workplace environment that promotes over-employment as workers increasingly focus more on finishing tasks and less on income-leisure tradeoffs.

Tests For Linearity and STAR Model Selection

The application and estimation of the STAR models require stationary time series. Indeed, univariate ADF panel unit root tests on t e reject the null of non-stationarity at all levels of significance. 14 For this reason, we proceed to apply the linearity tests and the STAR models to . The Ljung-Box statistic suggests white noise residuals for all autoregressive models. Using 0.05 as 14 The unit root tests indicate that t e is clearly stationary at all significance levels. For each test, the lag length is chosen using the Schwartz Bayesian Information Criterion. The results (not reported) are available upon request. 15 The Ljung-Box statistic reports the correlogram of the residuals of the model. The null hypothesis is that the residuals are white noise. If the p-value is greater then 0.05, then there is no serial correlation present in the residuals.

12 a threshold p-value, the test rejects linearity, classifying the series as non-linear. We can therefore proceed to build non-linear models for .

t e [TABLE 1 about here]

The tests for the choice between LSTAR and ESTAR models are shown in Table 2. Using the hypothesis tests outlined in equations ( 6)-( 8), the results indicate that the ESTAR model is the most appropriate non-linear model in all cases. The ESTAR model implies that individuals' adjust their status of employment from high to low levels towards the middle ground in a similar fashion.

[TABLE 2 about here]

Estimates of the non-linear models

The ESTAR models are estimated by non-linear least squares, using the Marquandt algorithm. Granger and Terasvirta (1993, pp. 123-124) and [START_REF] Terasvirta | Specification, estimation, and evaluation of smooth transition autoregressive models[END_REF] point out that estimation of the parameter a may cause particular problems such as "overestimation". As Terasvirta explains, if F(.) is not rescaled, then estimation results in very large values for F(.), given that it is obtained from the solution of an exponential function. The problem of very large values for F(.) is threefold.

First, the non-linear adjustment of the model is overestimated. Second, the non-linear model does not converge as easily. Third, the standard errors of the non-linear model are very large resulting in poor estimation. Therefore, in order to achieve convergence and obtain reasonable and reliable estimates, we scale the argument of the transition function F(.) by dividing it by 2 ( ). e  Hence, the scaled transition function used in the estimation of the ESTAR model is

         2 2 1 exp 1/ t d t d F e a e e c              . ( 9
)
Because of this scaling, we have used 1 a  as an initial value and the sample mean as a starting value for the parameter c. 16 The estimates of the autoregressive model are used as initial values for the  and  parameters.

[TABLE 3 about here]

Table 3 represents the non-linear estimation of equation ( 2) by the Gauss Newton method.

The standard errors of the non-linear models are smaller then the standard errors of the linear models for all the series, suggesting that the non-linear models provide a better fit of the data. The Jacque Bera normality test indicates that the residuals are normally distributed for all the series examined. Hence, the non-linearities in individuals' job satisfaction, wages and work hours are not the outcome of any outliers in the data. We further test the residuals for first order serial correlation and ARCH effects. The p-values (using 0.05 as the threshold) reject the presence of serial correlation and the presence of ARCH non-linearity in the residuals for all series. In all cases a is correctly signed and significantly different from zero at the 5 per cent level. This implies that individuals' have a faster speed of adjustment the further they are away from their reference points, thus justifying the use of the non-linear model.

It is also apparent from the examination of the ESTAR model estimates that the estimated a values vary. Small values of a are indicative of a very slow and smooth transition from one state to another. On the other hand, larger values of a imply sharper and more abrupt changing of states.

The results suggest that first, males are more inclined to change status faster than females given their larger values of a with reference to working hours, wages and job satisfaction. This is consistent with the findings in previous studies pointing to differences in how such job attributes affect men's and women's labor turnover decisions (see, for example, Clark 1997). Second, both males and females have a quicker speed of adjustment with changes to job satisfaction than changes 16 The scaling of  

t d e c   in the transition function makes it possible to judge the size of a (Granger and Terasvirta,pp 123, in wages and working hours. Table 3 also reports estimates of the halfway points, c, or thresholds between the two pure regimes. In all cases c was insignificantly different from the mean adjustment rate, e , implying that observations are distributed roughly equally between the left-hand and righthand tails of their respective exponential functions.

To gain further insight on what the estimated values of a actually mean, we summarize individuals' adaptation propensities as a result of a one standard deviation shock to job satisfaction, working hours and earnings. Let   0 t d F e   and   1 t F e   be the probabilities of individuals changing their current status. Suppose that we are initially in equilibrium with equal weights attached to the two pure regimes, implying that individuals have a 50% chance of changing status.

Based on the estimated values for c and a, in Table 3, in the case of job satisfaction for males, a one standard deviation shock to t d e  yields F(e t-d ) = 0.71. The corresponding values for male earnings and hours of work are 0.68 and 0.67, suggesting that, for men, the speed of adaptation to job satisfaction shocks dominates the speed of adaptation to wages and hours of work shocks. Such a result is consistent and reinforces the view that job satisfaction has emerged in recent years as one of the main determinants of labor market behavior and quits in particular. Inspired by earlier work by [START_REF] Freeman | Job satisfaction as an economic variable[END_REF] and [START_REF] Akerlof | Job switching and job satisfaction in the U.S. labor market[END_REF] who used US data, [START_REF] Clark | Job satisfaction, wage changes and quits: evidence from Germany[END_REF] and [START_REF] Clark | What really matters in a job? Hedonic measurement using quit data[END_REF] use German and British data respectively to explore the link between job satisfaction and workers' quit behavior. Both studies find that even after controlling for wages and other job and personal characteristics, job satisfaction remains a robust predictor of quitting behavior. As Clark (2001) argues, job satisfaction seems to be a suitable index of a poor job match: people with low job satisfaction are significantly more likely to quit their jobs in the future than otherwise identical persons with high job satisfaction.

A similar story emerges for women, with estimated values for F(e t-d ) of 0.58, 0.50 and 0.50 for a one standard deviation shock in job satisfaction, earnings and hours respectively. The estimated probabilities suggest that the speed of adjustment towards reference values for women is 153) and to interpret the estimated parameters of the ESTAR model as elasticities.

generally lower than that for men for all three attributes under consideration. There is ample evidence in the existing literature that women tend to report higher levels of job satisfaction despite experiencing apparently worse objective job conditions. A proposed explanation relies on expectations. If job outcomes are evaluated relative to expectations, women's higher job satisfaction may reflect their traditionally poorer position in the labor market that may lower their expectations. Furthermore, although the job characteristics of men and women may be the same, their expectations of what their job should be like, or the reference points against which they compare, are different. Because men, on average, tend to earn more than women do, it is likely that men and women may aspire to different levels of reference income. [START_REF] Mccue | Married women's hours and participation revisited[END_REF] document significant reference point heterogeneities as well as different reservation wages between men and women. [START_REF] Long | The effects of tastes and motivation on individual income[END_REF] reports that a strong longing for financial well-being substantially increases males ' earnings, whereas McGoldrick (1996) finds that women are unlikely to give up more with a view to gaining higher earnings. Regarding hours and effort requirements of jobs, two recent developments have been significant: changes in the extent to which workers' preferences are matched to employers' demands, and the intensification of work effort. First, while there have been no major increases in average weekly hours of workers, there has been an increase in the concentration of hours within households. In both Britain and the US, the weekly hours worked by dual-earner households have substantially risen since the early 1980s [START_REF] Jacobs | Overworked individuals or overworked families?[END_REF].

Because employment opportunities within firms are normally constrained, and given that employers' and employees' preferences can fluctuate, some individuals may need to change jobs in order to attain their desired level of labor supply [START_REF] Altonji | Labor supply, hours constraints and job mobility[END_REF][START_REF] Stewart | Constraints on the desired hours of work of British men[END_REF][START_REF] Euwals | Testing the predictive value of subjective labour supply data[END_REF]. It is well established that these labor market rigidities have a more negative impact on female workers because of the different set of constraints and occupational strategies they face, including childcare responsibilities and a desire for flexible work schedules [START_REF] Bianchi | Maternal employment and time with children: dramatic change or surprising continuity?[END_REF].

Concluding comments

A closer and more rigorous examination of the dynamics of adjustment towards reference points in the context of labor markets has been long overdue. While previous studies have acknowledged the importance of aspirations and reference points in influencing workers' labor market behavior, there is very little empirical evidence on the dynamics of such an adjustment process. To the best of our knowledge, this is one of the first studies to model the adjustment process as an ESTAR process, a non-linear process that allows for the speed of adjustment to vary with the distance of workers' actual values from their reference points. The non-linear model, proposed here, has the additional advantage of capturing the potential asymmetry in individuals' responses depending on whether current values are below or above their reference values. Our results show that non-linear adjustment takes place only when actual values are below reference values. In contrast, when actual values are above reference values, there is no adjustment at all, linear or non-linear. Perhaps not surprisingly, workers with current wages or job satisfaction below their reference values will seek ways to change their current status, while workers whose current values are above their reference points will adopt a more risk-averse attitude towards changing their current status. In many respects, this is a similar prediction to that of efficiency wages models whereby efficiency wages deter risk-seeking behavior on the part of workers and deter labor mobility. Although the asymmetric adjustment in hours of work, implied by our findings, is perhaps less intuitive, it is still broadly consistent with a substantial volume of research in both the economics and sociological literature explaining hours constraints and over-employment. Thus, our analysis offers not only a novel and rigorous test of the dynamics of adjustment towards reference points, but also the importance of job satisfaction and other job attributes as determinants of labor market behavior.

However, while we have established that the ESTAR model is superior to a linear model in characterizing the path of adjustment towards reference points, the novelty of our approach highlights the need for a detailed examination, in future research, of the performance of alternative non-linear models in characterizing the adaptation process. Given the sharp transitions from one regime to another, as our results for males suggest, a particularly fruitful avenue for future research would be also to investigate in more detail non-linearities within a Markov switching framework.

Finally, empirical testing for the presence of non-linearities in the process of formation of reference points remains a largely unexplored area in the adaptation literature that we aim to explore in future work.

A c c e p t e d M a n u s c r i p t

                    1
                    1
                    1

Notes

With the exception of hours, earnings and reference earnings, all explanatory variables are dummy variables with value 1 for the stated characteristic and 0 otherwise. Reference categories: 61≤Age≤65, actual hours equal to desired hours; other reference categories as in Table A1.

A

  c c e p t e d M a n u s c r i p t 7 the deviation from equilibrium. The transition function of LSTAR increases monotonically in t d e  and yields asymmetric adjustment towards equilibrium in the model. Moreover,

Figures 1 -

 1 Figures 1-3 illustrate how actual and reference values for earnings, job satisfaction and work hours evolve over time. As Figure1illustrates, reference earnings for both males and females tend to follow actual earnings, in broad agreement with[START_REF] Easterlin | A puzzle for adaptive theory[END_REF] claim that reference or aspiration income tends to increase proportionately with individuals' actual income. In Figure2, reference values for job satisfaction fluctuate between 7 and 8, consistent with the findings of Ferrel-i-Carbonell who predicts job satisfaction in the same range for West Germans. Interestingly, Figure2reveals a slight downward trend in job satisfaction for both males and females, with reference values to follow this trend. It is worth noting that the particular value of job satisfaction data lies in its trends. If it can be assumed that the norms against which job satisfaction judgments are made are stable in the medium term, the trend data then convey information about changes in well being. If job satisfaction is rising (falling) we could conclude that workers' well being is rising

A

  c c e p t e d M a n u s c r i p t 25

Figure 3 :

 3 Figure 3: Work hours -actual and reference values

k

  The lag length selection is based on the Schwarz information criteria and the Ljung-Box statistic for serial correlation. The residuals are saved from the chosen autoregressive model and denoted as . v

	where the linearity test is on the null hypothesis 0 2 : ' H 			3	'			4	' 0.  Equation (5) is estimated
	across a range of values for d where the smallest p-value attached to the linearity test determines
	d in the estimation of (2). The final stage of the non-linearity test is to determine which smooth
	transition model, LSTAR or ESTAR, is appropriate for the data. This is done by running the
	following sequence of nested tests:
	04 H   4 : ' 0																			(6)
	03 H  : ' 0 / ' 0 3 4   												(7)
	02 H  : ' 0 / ' 2 4  			3	' 0  .									(8)
	Second, having determined k , the next stage is to test for the presence of non-linearities. This is
	done through the estimation of
	t v	0    	1	'	t x			2	'	t t d x e 			3	'	2 t t d x e 			4	'	3 t t d x e 	t w  ,	(5)

The ESTAR model does not contain a cubic term (see equation

3.16 in Terasvirta)

. Therefore, if we reject (6) there is a cubic term in the non-linear model suggesting that we have an LSTAR specification. If we accept (6) and reject (7), then the non-linear model possess a squared term but not a cubic term. Since a squared term is required for an ESTAR specification and we have no cubic term, then we can conclude that accepting (6) and rejecting (7) implies acceptance of the ESTAR model (see equation 3.16 in Terasvirta). Accepting (

6

) and (

7

) and rejecting (8) leads to an LSTAR model because although 02 2

  Notes‡ denotes the lowest p-value associated with the variable-deletion tests and therefore the determination of the relevant STAR model. The values of k (lag length) and d (delay parameters) are reported in Table1. Type of model in all cases: ESTAR.

	TABLE 2 -SPECIFICATION OF NON-LINEAR MODELS
			MALES			FEMALES
	Job Satisfaction	Job Satisfaction
	H	04	0.136	H	04	0.157
	H	03	0.005 ‡	H	03	0.004 ‡
	H	02	0.80	H	02	0.168
	Earnings	Earnings
	H	04	0.154	H	04	0.150
	H	03	0.004 ‡	H	03	0.004 ‡
	H	02	0.127	H	02	0.160
	Work Hours	Work Hours
	H	04	0.173	H	04	0.164
	H	03	0.003 ‡	H	03	0.002 ‡
	H	02	0.143	H	02	0.139
				22

TABLE 3 -

 3 ESTIMATES OF THE ESTAR MODELS

					MALES	FEMALES
	Job Satisfaction								Job Satisfaction
	t e	0.50 0.672	t e	1 		0.84 0.792	t e	1 	
		(0.16) (0.30) (0.32) (0.33)
		 x 1-exp -4.193 1/ 		2	  e e  t	1 	0.459		2	
		(1.02)						(0.25)

  is the p-value for the first order autoregressive conditional heteroscedasticity (Engle F-test). NORM(2) is the p-value for the Jacque-Bera normality test. S/SL is the ratio of the standard errors for the non-linear and linear models.

		t e		0.55 0.731 e t 			0.79 0.753 e t 	1 
				(0.10) (0.08) (0.32) (0.24)
			x 1-exp -2.27 1/    	2	t e e	1 		0.4502	2	  
				(1.02)				(0.22)
	S = 0.422, Q(1) = 0.532, ARCH(1) = 0.172, NORM(2) =	S = 0.401, Q(1) = 0.478, ARCH(1) = 0.173, NORM(2) =
	0.191, S/SL = 0.762	0.163, S/SL = 0.798

Notes

Values under regression coefficients are standard errors. Q(1) is the p-value for first order serial correlation (Ljung-Box Q statistic).

ARCH(1)

TABLE A1

 A1 NotesWith the exception of age and tenure, all explanatory variables are dummy variables with value 1 for the stated characteristic and 0 otherwise. Reference categories: 61≤Age≤65, single, untrained, no qualifications, manufacturing, between 20 and 200 employees.

	-EARNINGS EQUATIONS
		MALES	FEMALES
		Coefficient |t-ratio| Coefficient |t-ratio|
	Constant	10.215	28.59	9.664	13.05
	Personal Characteristics
	18 ≤ Age ≤ 24	0.093	3.43	0.276	4.88
	25 ≤ Age ≤ 34	0.259	11.12	0.371	7.16
	35 ≤ Age ≤ 40	0.237	11.70	0.326	7.00
	41 ≤ Age ≤ 45	0.184	10.40	0.255	6.02
	46 ≤ Age ≤ 50	0.137	9.00	0.187	4.80
	51 ≤ Age ≤ 60	0.068	5.45	0.082	2.35
	Married	0.109	13.13	0.039	2.77
	Married, but separated	0.115	7.61	0.016	0.62
	Divorced	0.073	5.57	0.048	2.25
	Widowed	0.146	4.74	-0.015	0.41
	Spouse Absent	0.111	4.62	0.007	0.14
	No of Children	0.010	4.10	-0.060	9.51
	Limited by Health	-0.037	4.00	-0.042	2.28
	Labor Market Characteristics
	Semi-trained Employee	0.065	8.60	0.083	7.71
	Trained Employee	0.098	11.59	0.153	10.46
	Foreman	0.136	12.66	0.230	6.05
	Qualified Professional	0.135	12.95	0.203	14.19
	Highly Qualified Professional	0.237	19.82	0.248	12.56
	Manager	0.342	19.31	0.327	8.18
	Civil Worker (Low/Medium Scale)	0.069	3.16	0.217	4.53
	Civil Worker (High Scale)	0.134	5.65	0.244	5.52
	Tenure/10 Tenure/1,000 2	0.019 -0.001	25.31 23.37	0.032 -0.001	20.33 19.99
	Education
	No Qualifications +Apprenticeship	-0.039	1.13	0.003	0.05
	Lower Secondary	-0.077	4.38	-0.171	5.00
	Lower Secondary +Apprenticeship	-0.020	1.23	0.032	0.97
	High Secondary	-0.217	6.01	-0.442	6.90
	High Secondary + Apprenticeship	0.077	2.63	-0.074	1.55
	High Secondary + University	0.016	0.74	-0.007	0.16
	Other Degree	-0.067	2.95	-0.098	2.57
	Other Degree + Apprent./Univ.	-0.041	1.92	0.053	1.17
	Industry
	Energy-Mining	0.053	2.83	-0.055	0.81
	Construction	0.012	1.32	-0.041	1.15
	Trade	-0.026	3.01	-0.048	3.11
	Transportation	-0.030	2.29	0.041	1.10
	Banking-Finance	0.004	0.19	0.000	0.00
	Services	-0.031	3.54	-0.026	1.73
	Firm Size
	200 ≤ Size < 2000	0.058	8.57	0.050	4.11
	Size ≥ 2000	0.089	11.95	0.095	7.43
	Region Dummies	Yes		Yes
	Year Dummies	Yes		Yes
	Number of Observations:	49,177	20,903
	Number of Groups:	8,796		4,905
	F test that all the fixed effects are equal to zero:	F-stat=5.56 (0.000)	F-stat=4.16 (0,000)

TABLE A2 -

 A2 JOB SATISFACTION EQUATIONS

		MALES	FEMALES
		Coefficient	|t-ratio|	Coefficient	|t-ratio|
	Constant	21.468	10.09	18.472	8.59
	Log (Wages)	0.112	3.91	0.343	1.06
	Log (Reference Income)	-0.936	7.39	-0.692	5.30
	Log (Hours)	-0.109	1.98	-0.230	0.30
	Preferences for Hours of Work
	Actual > Desired (Over-employed)	-0.341	6.99	-0.440	7.20
	Actual < Desired (Under-employed)	-0.211	3.48	-0.314	2.13
	Personal Characteristics
	18 ≤ Age ≤ 24	0.364	2.13	0.338	0.61
	25 ≤ Age ≤ 34	0.270	4.70	0.295	1.25
	35 ≤ Age ≤ 40	0.216	5.72	0.272	1.56
	41 ≤ Age ≤ 45	0.209	5.45	0.295	2.00
	46 ≤ Age ≤ 50	0.442	4.60	0.344	1.57
	51 ≤ Age ≤ 60	0.517	3.18	0.421	0.28
	Married	0.135	3.54	-0.115	2.11
	Limited by health	-0.361	6.79	-0.411	4.37
	Labor Market Characteristics
	Semi-trained Employee	0.218	4.76	0.154	2.66
	Trained Employee	0.412	7.73	0.323	3.88
	Foreman	0.569	8.27	0.879	4.34
	Qualified Professional	0.674	10.08	0.573	6.56
	Highly Qualified Professional	0.902	11.89	0.799	6.86
	Manager	0.949	11.54	0.799	3.68
	Civil Worker (Low/Medium Scale)	0.152	1.20	0.844	3.36
	Civil Worker (High Scale)	0.668	4.80	0.936	4.43
	Education
	No Qualifications +Apprenticeship	-0.299	1.51	-0.489	1.59
	Lower Secondary	-0.240	2.36	-0.051	0.29
	Lower Secondary +Apprenticeship	-0.146	1.58	0.310	1.83
	High Secondary	-0.275	1.29	-0.701	2.06
	High Secondary + Apprenticeship	-0.006	0.04	0.142	0.58
	High Secondary + University	0.045	0.36	-0.245	1.13
	Other Degree	-0.137	1.04	-0.419	2.13
	Other Degree + Apprent./Univ.	-0.179	1.47	0.043	0.19
	Industry
	Energy-Mining	0.109	1.01	0.017	0.05
	Construction	0.042	0.83	0.248	1.34
	Trade	-0.042	0.84	0.070	0.87
	Transportation	-0.007	0.09	0.791	4.10
	Banking-Finance	0.092	0.74	0.412	2.50
	Services	-0.044	0.85	0.271	3.49
	Firm Size
	200 ≤ Size < 2000	0.124	3.03	0.192	3.02
	Size ≥ 2000	0.288	6.10	0.376	5.40
	Region Dummies	Yes		Yes
	Year Dummies	Yes		Yes
	Number of Observations:	49,177		20,903
	Number of Groups:	8,796		4,905
	F test that all the fixed effects are equal to zero:	F-stat=4.47 (0.000)	F-stat=3.38 (0.000)

As a referee pointed out, there is a subtle distinction to be made between the use of the terms adaptation and adjustment. Although adaptation usually refers to the formation of preferences / reference points, our emphasis in this paper is on the adjustment process towards reference points.

For a detailed description of the GSOEP data see[START_REF] Burkhauser | Soziale sicherung in einer dynamsichen gesellschaft (Social insurance in a dynamic society[END_REF].

Job satisfaction is an ordinal variable, based on respondents' answers to the following question: "How satisfied are you today with the following areas of your life? Please answer by using the following scale, in which 0 means totally unhappy and 10 means totally happy. If you are partly happy and partly not, select a number in between". Desired hours of work are based on responses to the question: "If you could choose the extent of your hours at work, taking into account that your earnings would change corresponding to the time: How many hours per week would you like to work."

The main alternative non-linear formulation to the STAR models is the Markov Switching Regime Model, first established by[START_REF] Hamilton | A new approach to the economic analysis of non-stationary time series and the business cycle[END_REF]. Markov switching models are popular in the finance literature where high frequency stock price data is used to explore second-by-second adaptation in financial markets.

The ESTAR model is essentially a dynamic gap model, allowing for both the actual and reference values of the variables of interest to change overtime.


A c c e p t e d M a n u s c r i p t 1 Introduction

Mainstream economics adopts the classical mechanical approach of 19th century physics, based upon the reductionist principle, according to which one can understand the aggregate simply by analyzing its single elements. The microfoundation of macroeconomics in the (New) Classical tradition is based on the hope that the aggregate behavior is the magnification of the single agent's behavior on a larger scale. The application of the reductionist framework implies that the so-called overlapping principle holds true, that the dynamics of a (linear) model can be decomposed into its constituent parts through the representative agent (RA) framework.

The microeconomic foundations of general equilibrium models must be based, according to mainstream economics, on an optimizing RA, fully rational and omniscient. Unfortunately, "there are no assumptions on [...] isolated individuals which will give us the properties of aggregate behavior which we need to obtain uniqueness and stability. Thus we are reduced to making assumptions at the aggregate level which cannot be justified by the usual individualistic assumptions. This problem is usually avoided in the macroeconomic literature by assuming that the economy behaves like an individual. Such an assumption cannot be justified in the context of the standard economic model and the way to solve the problem may involve rethinking the very basis on which this model is founded" (Hildenbrand and Kirman 1988, p. 239).

The quantum revolution of the last century radically changed the perspective in contemporary physics. According to the holistic approach, the aggregate is different from the sum of its components because of the interaction of particles. In the social sciences, a step in this direction is taken by the agent-based modeling (ABM) strategy.

Agent-based models, which are increasingly applied in economics (Tesfatsion 2007, Axelrod 1997), have been developed to study the interaction of many heterogeneous agents. In a sense they are based on new microfoundations, according to a bottom-up approach. They follow a holistic methodology as opposed to the reductionist approach of the mainstream economics. One builds a model starting from simple behavioral rules at the single agent level. Through interactions some aggregate statistical regularities emerge so that they can not be inferred from the individual level. This emergent behavior often feeds back to individual agents, making their rules change (they may evolve in an adaptive way). According to this approach, macroeconomics is not a set of equations that occurs by summation and averaging of the individual decisions, but it is a SOC (Self-Organized Critical) phenomenon that rises from the micro-level.

As already mentioned, ABM and simulations have been extensively used in many scientific fields, including economics, in the last decade (Axelrod 1997, Axtell 2000). However, in recent years only, researchers have started considering the issue of validation: that is whether a model and its results may be considered correct. As Sargent (1998, p. 53) puts it, "This concern is addressed through model verification and validation. Model validation is usually defined to mean substantiation that a computerized model within its domain of appli-1

Blinded Manuscript (NO Author Details)

A c c e p t e d M a n u s c r i p t cability possesses a satisfactory range of accuracy consistent with the intended application of the model". This is not at all a secondary problem; in fact, only a correct model can be considered a suitable model.

In this paper we deal with some validation experiments of a simplified version of the model proposed in Gallegati et al. (2003a, 2005), the so called CATS (Complex Adaptive Trivial System) model.

The CATS model has been extensively used (see, for example, Gallegati et al. 2003b, 2004, 2005) to replicate a large number of scaling type stylized facts with a remarkable degree of precision, and for these purposes, the simulation of the model has been performed entering ad hoc parameters' values and using the same initial set up for all the agents involved in the experiments. It must be recalled that the above mentioned analyses have been performed following Kaldor's suggestion: "construct a hypothesis that could account for these stylized facts, without necessarily committing himself on the historical accuracy" (1965, p. 178).

In this paper our intentions are more ambitious: using an initial set up of actual data (a sample of Italian firms in 1996) we aim to verify whether the CATS model, simulated over a period for which actual data are fully available (the interval 1996-2001), is an acceptable representation of the real system. In other words we intend to perform an ex-post validation of the model.

Alternative distributional and goodness-of-fit tests, discussed in Prabhakar et al. (2003) and Kleiber and Kotz (2003), are performed, and some graphical tools (Embrechts et al. 1997) are proposed in order to give the reader a quick comprehension of actual and simulated data.

In the validation exercise, over the simulation period 1996-2001, we use a sample of 6422 Italian firms included in the AIDA database. The model parameters have been estimated using actual data, and the initial set up consists of the sample data of the year 1996. The CATS model is then simulated over the period 1996-2001, and the simulations' results are ex-post validated with respect to actual data.

The model reproduces, in a short (medium) term horizon, a good percentage (81% in 2001) of the output actual data. The two samples (simulated and observed data) belong to the same distribution with a confidence interval of 95%. Moreover the model also reproduces the firms' growth dynamics at a micro level, while less satisfying is the simulation for the behavior of the very small and very large firms.

The papers is organized as follows. Section 2 presents the state of the art for the validation of agent-based models, Section 3 introduces the model we have studied and validated, Section 4 describes the database we used and the empirical evidence we aim to investigate, Section 5 shows the proceeding of the validation procedure, and Section 6 concludes.

2

A c c e p t e d M a n u s c r i p t 2 Empirical Validation of Agent-based Models

As Leigh Tesfatsion points out in her important website on Agent-based Computational Economics1 , the validation of ACE models is becoming one of the major points in the agenda of those researchers who work according to the agent-based approach.

In the literature, looking at the main methodological aspects, there are three different ways of validating computational models:

1. Descriptive output validation, matching computationally generated output against already available actual data. This kind of validation procedure is probably the most intuitive one, and it represents a fundamental step towards a good model's calibration.

2. Predictive output validation, matching computationally generated data against yet-to-be-acquired system data. Obviously, the main problem concerning this procedure is essentially due to the delay between the simulation results and the final comparison with actual data. This may cause some difficulties when trying to study long time phenomena. Anyway, since prediction should be the real aim of every model2 , predictive output validation must be considered an essential tool for an exhaustive analysis of a model meant to reproduce reality.

3. Input validation, ensuring that the fundamental structural, behavioral and institutional conditions incorporated in the model reproduce the main aspects of the actual system. This is what we can call ex ante validation; the researcher, in fact, tries to introduce the correct parameters in the model before running it. The information about parameters can be obtained analyzing actual data, thanks to the common empirical analysis. Input validation is obviously a necessary step one has to take before calibrating the model.

In this paper we analyze points 1 and 3. As far as predictive validation is concerned, we hope to develop it in the near future.

Since the empirical validation of agent-based models is still a new topic, at the moment there are only a limited number of contributions in the literature dealing with it, as summarized below.

In their paper, Axtell et al. (1996) develop the basic concepts and methods of an alignment process for agent-based models. In particular they show how alignment can be used to test whether two different computational models can be considered similar in terms of behavior and output.

In Carley (1996), there's a first stress on model validation issues, even if the attention of the author is still focusing on computational modeling in general.

A very interesting experiment can be found in the paper by Gilli and Winker (2003), in which the authors present an agent-based exchange market model and introduce a global optimization algorithm for calibrating the model's parameters via simulation.

In Troitzsch (2004), there is a comprehensive list of all the issues concerning the validation of simulation models to describe and predict real world phenomena.

In Fagiolo et al. (2007), one can finally find a very interesting discussion about the ways agent-based modelers have tried to face the empirical validation of their models. The authors briefly review the standard approaches to model validation employed by mainstream economists and then point out the main differences dealing with ABM validation. The paper concludes with some suggestions regarding the methodological aspects of validation.

Finally, without any presumption of being complete and exhaustive, we cannot forget the mainly theoretical and methodological contributions by Sargent (1998), Klevmarken (1998), Epstein (1999) and Judd (2006).

The CATS model

The model we present was first introduced in Gallegati et al. (2003a) to study financial fragility and power laws. Here, it's modified to better reproduce actual data, according to the input validation principle we have mentioned above.

Following the ACE philosophy, it is a simple model, since it makes use of straightforward and clear-cut assumptions. Simplicity is one of the main qualities of agent-based models, which are considered good models only if able to reproduce and explain empirical evidences without being too complicated or making too many assumptions. In other words, the simpler the model is, the easier are reading and interpreting the results.

Consider a sequential economy3 , with time running t = 1, 2, ..., populated by firms and banks. Two markets are opened in each period: the market for a homogeneous produced good and the market for credit. As in the levered aggregate supply class of models first developed in Greenwald and Stiglitz (1990, 1993), our model is fully supply-determined in the sense that firms can sell all the output they optimally decide to produce. Due to informational imperfections in the equity market, firms can raise funds only from the credit market, apart from retained profits from previous periods. This assumption seems to reflect the examined reality in Italy since new equity issues were rarely a financial option for Italian firms in the observed period. Moreover, the full distribution of dividends to shareholders was expensive due to the fiscal system. In a perfect environment, without taxes, corporations would not have preferences among these financial options, as shown by the Modigliani-Miller theorem.

Hence, in our setting, the demand for credit is related to investment expenditure, and it is fully satisfied at the fixed banks' interest rates (i.e. total credit A c c e p t e d M a n u s c r i p t supply always equals the demand for it). This hypothesis helps us to identify a suitable proxy of the individual interest rates (namely, the average interest rate) since we have no reliable information related to them.

Let us then briefly describe the main features of the model in the remaining part of the section.

At any time t, the economy consists of N t firms, each belonging to two different sets (small firms and large ones) [START_REF] Axtell | Aligning simulation models: a case study and results[END_REF] , depending on their size, and facing different levels of risk (price shocks). This assumption is different from the original one (Gallegati et al. 2003a) with a single risk level.

Every firm i ∈ N t produces the output Y according to a linear production function, in which capital (K it ) is the only input 5 :

Y it = φ it K it . (1)
For each firm i the productivity φ it in t = 1 corresponds to its actual productivity (estimated on the AIDA data in 1996), and it evolves according to the following formula:

φ it = φ it-1 + ̺ it φ 2 it-1 , where ̺ it = M 2 , (2)
with M ∼ U (0, 1), if the firm is small, and to

φ it = φ i1 , (3)
if large. All this reproduces the evidence from our database, reported in Figure 1, where one can see the evolution of the average productivity for small and big firms for actual and simulated data. It clearly emerges that the growth rates of productivity are decreasing in firms' size, contradicting Gibrat's law for means [START_REF] Bottazzi | Explaining the distribution of firm growth rates[END_REF] .

Figure 1: Evolution of average productivity for small and big firms from 1996 to 2001. Actual (straight lines with pluses) and simulated (dotted lines with diamonds) data.

Each firm's demand for goods is affected by an iid idiosyncratic real shock. Since arbitrage opportunities are imperfect, the individual selling price is the random outcome of a market process around the average market price P t of the output, according to the law P it = u it P t , where E(u it ) = µ and σ 2 uit < +∞. Actual data suggest splitting the price generator process into two different processes, depending on firms' size. For the sake of simplicity we assume that u it follows two different uniform distributions; small firms get a high average price and a stronger volatility, while big firms face more concentrated prices with a lower mean. This assumption has a justification in the analysis of actual data: small firms, in fact, show a stronger volatility in their revenues and profits.

Summarizing, if U 1 is the distribution of u it if i is small and U 2 if i is large, we have that µ U1 > µ U2 and σ 2 U1 > σ 2 U2 . Since, by assumption, credit is the only external source of finance for firms, the firm can finance its capital expenditure by recurring to net worth (A it ) or bank loans (L it ), that is

K it = A it + L it .
At the exogenous real interest rate r, at each time t debt commitments for every firm are equal to rL it . Since for the sake of simplicity there are no dividends distributed to shareholders, financing costs equal debt commitments. Therefore, profit/loss (π it ) in real terms is

π it = u it Y it -rL it . (4)
In our model a firm goes bankrupt if its net worth becomes negative, that is to say A it < 0. The law of motion of A it is, for hypothesis,

A it = A it-1 + π it . (5)
As in Greenwald and Stiglitz (1993), we assume that the probability of bankruptcy (Pr b ) is directly incorporated into the firm's profit/loss function: bankruptcy is costly and increasing with the firm's size. In particular we have chosen a quadratic cost function:

C b = cY 2 it c > 0. (6)
Finally, each firm, by maximizing its objective function, determines its optimal capital stock

K * it , max Kit Γ it = E(π it ) -E(C b ), (7)
and the demand for credit. Thanks to several queries on the database, we have collected a sample of 6422 Italian non-financial firms, all satisfying the following: (i) no missing data in each year, and (ii) reliable data for capital, employees and costs. For each firm A c c e p t e d M a n u s c r i p t and year, we have data on equities, long term debts and loans, short term debts, total capital, gearing ratio, solvency ratio, debt ratio, number of employees, cost of employees and revenues.

The Database and the Empirical Evidence

Recent explorations (Gallegati et al. 2007) in industrial dynamics have detected three empirical regularities that are so widespread across countries and persistent over time as to be characterized as universal laws:

1. The distribution of firms' size is right skewed and can be described by a Zipf or power law probability density function (Gallegati et 2005) have shown analytically that 1-3 determine several industrial, financial and business cycle facts (see that paper for a review of the empirical literature.) A model should therefore be able to replicate the empirical evidence 1-3, and our validation exercise is focused on it.

The following section will present the validation exercise (i.e. if the above presented CATS model successfully deals with the evidence 1-3).

Simulation and Results

Our validation exercise is run with a sample of 6422 firms over the period 1996-2001.

The validation procedure we have used is new for agent-based models [START_REF] Cirillo | Some considerations about Gibrat's law in Italy[END_REF] , but it is based on some well-known results of extreme value theory, mainly as far as the analytical tests are concerned (for a reference see Embrechts et al. 1997).

In t = 1996 (t = 1 in simulations), each firm is initialized with its actual data from 1996: net worth, loans, productivity and so on. The market interest rate is exogenous and equal for all the firms. [START_REF] Embrechts | Modelling Extremal E vents[END_REF] In each period actual data from the AIDA database are compared with the simulated data produced by the model. In particular our analysis can be divided into two different approaches: a pointwise analysis, meant to evaluate the evolution of the single firm in order to study the predictive power of the model, and a distributional analysis, whose aim is to look for regularities.

Our experiments can be considered a first ex-post validation of the CATS model, that is to say a first step, necessary to develop all the subsequent analysis.

A c c e p t e d M a n u s c r i p t

Capital Distributions

As far as the aggregate output is concerned, the model underestimates it slightly (average aggregate actual output over six years on log scale is equal to 10.3486; average aggregate simulated output over six years on log scale is equal to 10.1132), while the output volatility is almost identical (≃ 1.205 vs. ≃ 1.207). This is a first interesting result that underlines the ability of the model in replicating actual data 10 .

Total Capital

Let us now consider the total capital dynamics. Accepting a maximum deviation of ±20% between observed and simulated data in 2001 (that is a composite yearly deviation rate of 3.5%), we succeed in reproducing 5201 firms over 6422 (81%). As Figure 4 shows, the tails of the firms' size distribution is not adequately fitted. Similar results can be found in the previous years (in 1997, for example, the percentage of fitted firms is 74%, while in 1999 it's 77%) and by analyzing the pooled distributions (78%). 11 In order to verify the real goodness of these results (verifying if they are due to the goodness of the model rather than to the characteristics of data such as few years, few firms with respect to the universe and so on), we have performed an empirical analysis of actual data. In Figure 3 one can observe the comparison, by the means of a Zipf's plot, between actual total capital in 1996 and 2001. The evidence is quite clear: there is a substantial difference between the two amounts of data (see also the boxplot in Figure 4). Accepting the usual 20% deviation, only 25% of the firms (essentially the smallest ones) can be considered as fitting the data. In fact, even if both distributions belong to the Paretian case [START_REF] Fujiwara | Zipf law in firms bankruptcy[END_REF] , their parameters are different. Several analytical tools, such as the Kolmogorov-Smirnov's statistics and the Kruskall-Wallis' test, confirm all this.

Even if we calculate the average growth rate of firms from 1996 to 2001 and then we multiply the initial data in 1996 for this coefficient, in 2001 we succeed in fitting "only" 62% of the firms. Since our model can fit 81% of them, it must be considered as performing better. [START_REF] Epstein | Agent-based computational models and generative social sciences[END_REF] Obviously, even if our sample is representative of the universe, we cannot completely exclude some slight dependence between output volatility and the number of firms in the system, which is surely influenced by the conditions we have imposed on the database (no lack of data, plausible capital values and so on). However, our experience of simulations with different datasets induce us to think that this detail should not be sufficient to invalidate our results.

11 As in Ijiri and Simon (1997), the use of pooled distribution is possible since the single distributions show similar slopes.

In this paper, almost all the figures refer to year 2001.

12 This is quite obvious; in only six years one cannot expect a distribution to change its shape and family. Finally this is what we can find in most of all the empirical studies about firms. We have performed many graphical and analytical tests to check if our two samples (observed and simulated data) may be considered belonging to the same distribution.

A first Quantile-Quantile plot (Figure 5) supports the idea of a unique distribution for both samples, since there is a clear linear relationship between the observed and simulated data. Another graphical test, the Box Plot (Figure 6), shows that the two distributions have many similarities. For example, the median (thick line [START_REF] Gaffeo | Power law scaling in the world income distribution[END_REF] ) is almost the same and it is not centered in the box, indicating two skewed distributions. Moreover, both distributions present a great number of outliers (red plus) in the right tails, underling the possible presence of fat tails. The same results are supported by the Generalized Kolmogorov-Smirnov Test [START_REF] Gallegati | Financial fragility, patterns of firms' entry and exit and aggregate dynamics[END_REF] with a confidence interval of 95%. Therefore, it is possible to say that our two samples belong to the same distribution.

F N = n(i) N , (8)
where n(i) represents the number of points less than Y i . As one can see, this step function increases by 1 N for each data point. The Kolmogorov-Smirnov test is based on the maximum distance between the ECDF and the theoretical cumulative distribution one wants to test (F T ):

D = max 1≤i≤N F T (Y i ) - i N . ( 9
)
On the contrary, the two sample K-S test, instead of comparing an empirical distribution

9

A c c e p t e d M a n u s c r i p t

In particular, excluding the right Paretian tails [START_REF] Gallegati | Complex dynamics and financial fragility in an agent-based model[END_REF] , we found out that our data follow a Generalized Pareto Distribution (GPD) [START_REF] Gallegati | Business cycles fluctuations and firms' size distribution dynamics[END_REF] , in particular a Pareto II type (ξ ≤ 0). Figure 7 shows a linear Pareto II Plot 18 of the observed capital distribution (again, after excluding the largest firms). The presence of Paretian behavior in the right tails of the two distributions is also supported by the Mean Excess Function versus Threshold Plot (MEPLOT). An upward sloping mean excess function, as in Figure 8, indicates a heavy tail in the sample distribution. That is why, thanks to the Hill's method [START_REF] Gallegati | Complex dynamics and empirical evidence[END_REF] , we have decided to estimate the shape parameters of the two samples in order to see whether data have a similar behavior in the right tails. function to the theoretical one, compares two different ECDF, that is

D = |F 1 (i) -F 2 (i)| , ( 10
)
where F i is the empirical distribution for sample i.

The generalised K-S statistic can be defined as:

H 0 : F 1 = F 2 →
the two samples come from the same distribution;

H 1 : F 1 = F 2 → the two samples come from different distributions.
To decide the results of the test, the values of D are compared to the critical values obtained from Kolmogorov and Smirnov's table. [START_REF] Gallegati | Complex dynamics and financial fragility in an agent-based model[END_REF] We have trimmed them after a threshold study based on graphical (for example looking for linearity in meplots) and analytical (studying the behavior of Hill's and Pickands' estimates) analysis. A very detailed description of the methodology is presented in Embrechts et al. (1997). [START_REF] Gallegati | Business cycles fluctuations and firms' size distribution dynamics[END_REF] Starting from the well-known Fisher-Tippett Theorem, which deals with the convergence of maxima, the GPD distribution represents one of the most important limiting cases.

Its functional form is

H(x) = 1 -(1 + ξ x β ) -1 ξ if ξ = 0 1 -e -x β if ξ = 0 , (11)
where β > 0 and x is such that 1 + ξx > 0 and ξ is the shape parameter (tail index α = 1 ξ ). There are three different situations:

1. ξ > 0 → GPD distribution becomes the classical Pareto distribution and shows fat tails;

2. ξ = 0 → GPD distribution converges to the exponential distribution;

3. ξ < 0 → GPD distribution; it is then known as Pareto II. [START_REF] Gallegati | A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility[END_REF] That's a quantile-quantile plot with Pareto II coefficients. [START_REF] Gallegati | Complex dynamics and empirical evidence[END_REF] The well-known Hill's Estimator ξ, together with the Pickands' one, is the most used way to determine the shape parameter α = 1 ξ of a distribution belonging to the GEV family. In particular

ξ = 1 k -1 k-1 i=1 ln x i,N -ln x k,N f or k ≥ 2, ( 12
)
where k is the upper order statistics and N the sample size.

10 Figure 9 reports the Hill's estimates of the shape parameter for the simulated capital, while Figure 10 refers to observed data. In the first case α = 1.61, while in the second one α = 1, 68. Hence, the two parameters are very similar (Figure 11) and belong to the Paretian field (0.8 < α < 2) [START_REF] Gilli | A global optimization heuristic for estimating agent-based models[END_REF] , but we cannot claim that the two tails behave in the same way. Simulated capital, in fact, shows a slightly heavier tail (since its alpha is lower [START_REF] Gourieroux | Simulation-based Econometric Methods[END_REF] ), demonstrating that we slightly overestimated the observed values.

Net Worth and Loans

As far as net worth is concerned, accepting a maximum deviation of ±20% between actual and simulated data in 2001, we succeed in reproducing 4944 firms over 6422 (77%) [START_REF] Greenwald | Macroeconomic models with equity and credit rationing[END_REF] . This number is lower than that of total capital, indicating some more problems of fitting [START_REF] Greenwald | Financial market imperfections and business cycles[END_REF] .

Other positive results, see Figure 12, are the skewness of the two distributions and the presence of a clear Paretian behavior in both actual and simulated net worth. Hill's estimates of the shape parameters both show heavy right tails: actual data present α = 1.52, while the simulation produces α = 1.48. As far as the possibility of a unique distribution for the two samples, the twosided generalized Kolmogorov-Smirnov test rejects such a null hypothesis. On the contrary the one-sided right version of the test [START_REF] Hahn | Money and Inflation[END_REF] is not rejected, indicating that we get a better fitting of medium and big firms, but we fail in forecasting the smallest one.

The results we get about loans are very similar to those of the total capital: we succeed in fitting 5137 firms out of 6422 (80%). [START_REF] Gilli | A global optimization heuristic for estimating agent-based models[END_REF] Once again the results concerning the pooled distributions are very similar. The reason can be found in the words of Ijiri and Simon (p. 19): "We conclude that when two or more Pareto distributions are pooled together, the resulting distribution is Pareto if and only if all the distributions have similar slopes [...]. This result is important in dealing with the aggregation of empirical firm size distributions." [START_REF] Gourieroux | Simulation-based Econometric Methods[END_REF] As clearly showed in Kleiber and Kotz, the Pareto density has a polynomial right tail that varies at infinity with index (-α -1), implying that the right tail is heavier as α is smaller.

22 66% in 1997, 73% in 1999 [START_REF] Greenwald | Financial market imperfections and business cycles[END_REF] In particular this may depend on the hypothesis of the model that 1) firms cannot raise funds on the equity market, 2) profits are entirely retained in the firm and 3) as suggested by the referee, that all firms face the very same interest rate. However, these simplifying hypothesis, typical of CATS model, do not seems to affect too much the robustness of our validation results. [START_REF] Hahn | Money and Inflation[END_REF] H 0 :

F + 1 (x) = F + 2 (x). H 1 : F + 1 (x) > F + 2 (x).
11 Moreover, similarly for total capital, both graphical and analytical tests support the idea of a unique distribution for both actual and simulated debt data (Figure 13). As in Fujiwara (2004), the distribution of loans is also power law. The Hill's estimates of the shape parameters of the Paretian right tails are α = 1.71 for the actual data and α = 1.58 for the simulated ones, demonstrating an overestimate of biggest firms.

Finally, analyzing the ratio between net worth and debt we find out that, apart from some exceptions [START_REF] Hall | The relationship between firm size and growth[END_REF] , it is almost constant for each firm over time. In other words, if firm i has a ratio of x% in 1996, it shows a very similar ratio in 2001.

Growth Rates

As far as firms' growth rates are concerned, several studies (Bottazzi and Secchi 2005, Axtell 2001, Hall 1987) find a tent-shape behavior. In particular, the Laplace and Lévy distributions seem to provide the best fit (Gabaix et al. 2003).

We have investigated whether the empirical distributions of growth rates (in terms of capital) belong to the well-known Subbotin's family (Subbotin 1923), which represents a generalization of several particular cases such as Laplace and Gaussian distributions. The functional form of Subbotin's family is

f (x, a, b) = 1 2ab 1 b Γ 1 + 1 b e -1 b | x-µ a |b , ( 13
)
where µ is the mean, b and a two different shape parameters and Γ is the standard Gamma. If b → 1 the Subbotin distribution becomes a Laplace, a Gaussian for b → 2.

Using the maximum likelihood method [START_REF] Hildebrand | Equilibrium Analysis[END_REF] , we have estimated the three Subbotin's parameters on our data. Table 1 contains the results. At a first glace, observed and simulated growth rates show several similarities:

1. The two means are very close to zero; [START_REF] Hall | The relationship between firm size and growth[END_REF] While validating our model, we have conducted several experiments on interest rates, finding out that those firms showing a decreasing net worth/debt ratio are the same that obviously go bankrupt if the interest rates rise. This is interesting since the decreasing ratio is almost completely due to a monotonically deteriorating equity ratio (Beaver 1966, Gallegati et al. 2005). Moreover, surprisingly, all the firms that went bankrupt in our simulations were the same as those that really went bankrupt in 2002, showing a decreasing equity ratio.

Unfortunately, as already said, our data are not complete for 2002, so we prefer not to state this as a result. [START_REF] Hildebrand | Equilibrium Analysis[END_REF] The results are very similar, using the method of moments.

A c c e p t e d M a n u s c r i p t 2. Since b is very near to 1, both distributions are in the field of attraction of the Laplacian case [START_REF] Ijiri | Skew Distributions and the Size of Business Firms[END_REF] . Figure 14 supports this evidence since it is tentshaped;

3. The values of a, the Laplacian shape parameter, are not very different in both cases, even if simulated data show slightly fatter tails (0.061>0.059); see Figure 14.

Overall, the CATS model is able to mimic firms' growth dynamic, once again with some discrepancies as far as the tails are concerned.

Growth Rates and Firms' Size

In order to analyze the relationship between firms' size and firms' growth rates, we followed the methodology suggested by Gabaix et al. (2003). Stanley et al. (1996) find that large firms show a lower volatility of their growth rates; moreover, they show that this volatility (σ rates ) linearly decreases with size (S), or ln σ rates = -α ln S + β,

with α ≃ 0.15.

In order to investigate whether this relationship holds true for our actual and simulated data, we divided firms' size in four bins. Then we computed the standard deviation of their growth rates. Finally, we plotted a log-log graph of the average standard deviation of growth rates versus the average size in each bin.

For both observed and simulated data, our results are very similar to those presented in Gabaix et al. Figure 15 shows how the relationship we have found decreases with size. Our estimates of α are 0.1643 for the actual data and 0.1621 for the simulated ones (not far from Gabaix's 0.15). Once again, the CATS model successfully reproduces the empirical data.

In sum, we may say that the CATS model successfully passes the ex-post validation exercise of this section, with the only exception for very small firms. More reliable results could be obtained improving the specification of the model, better calibrating some key parameters using simulation based methods discussed, for example, in Gouriéroux and Monfort (1996) and in Klevmarken (1998), and carefully adjusting the dimensions of the sample used in the initial set up.

In future validation experiments, we intend to modify the model specification, endogenizing the banking sector (see Vagliasindi et al. 2006) and the price generator process and including a labor market module. Moreover, we hope to be able to work on a richer and more reliable dataset in order to generalize our results and correctly calibrate the model, taking also into account the specificity of the Italian situation.

A Validation Procedure: some notes

The aim of this appendix is to describe briefly the procedure we have used to validate the CATS model.

All the codes and the programs have been written in Fortran90 c , while all the graphics have been developed with Matlab7 c .

As far as the simulation of the CATS model is concerned, it can be useful to stress the following aspects:

1. In t = 1 (1996), when the simulation starts, every firm is initialized with its actual data from the database. These data are net worth, loans and productivity. The current version of the model has a recursive structure so that parameters φ it have been consistently estimated using, firm by firm, ordinary least squares. Then productivity evolves according to the laws of motions presented in 2 and 3;

2. The parameter M in 2 follows an uniform distribution, whose support (0, 2) has been ad hoc calibrated, thanks to several replications;

3. The interest rate is equal to 11,5% in 1996 (t = 1) and decreases every year, arriving at 10% in 2001. This reproduces the average behavior of the interests paid every year by firms in our database;

4. The two different uniform distributions we have used to model the idiosyncratic shocks on prices show support (0.8, 2.8) for small firms and support (0.5, 1.5) for the large ones. These supports have been inductively calibrated with a grid method, considering the results of several alternative replications, in order to get the best fitting values; 14 5. Every year the following data are stored in order to be compared with actual data: net worth, loans, total capital, productivity, growth rates, paid interests, total output, aggregate output.

Our analysis of data can be divided into two different approaches: a pointwise analysis, meant to evaluate the evolution of the single firm, in order to study the predictive power of the model; and a distributional analysis, whose aim is to look for general regularities.

In Embrechts et al., one can find a quite complete list of all the tests a researcher should perform in analyzing data, while Kleijen (1998) deals with the theoretical implications of validation.

  All our validation experiments, together with the subsequent empirical analysis, are based on firm-level observations from the AIDA database for the period 1996-2001, AIDA, formerly developed by the Italian Chambers of Commerce, is now a subset of AMADEUS, a comprehensive pan-European database elaborated by Bureau Van Dijk 7 .

Figure 2 :

 2 Figure 2: Zipf's Plot of the total capital distributions: observed (plus) and simulated (diamonds).Figure 3: Comparison between actual total capital in 1996 (plus) and 2001 (diamonds) Figure 4: Boxplot of actual total capital in 1996 and 2001.

Figure 3 :Figure 4 :

 34 Figure 2: Zipf's Plot of the total capital distributions: observed (plus) and simulated (diamonds).Figure 3: Comparison between actual total capital in 1996 (plus) and 2001 (diamonds) Figure 4: Boxplot of actual total capital in 1996 and 2001.

Figure 3

 3 Figure 3 also shows that both observed and simulated capital distributions are particularly skewed, with fat right tails (decreasing linear relationship in the plot). This reproduces a widely accepted result (Zipf 1932), according to which firms' size is power law distributed 13 (Axtell 2001, Gaffeo et al. 2003, Gabaix et al. 2003).We have performed many graphical and analytical tests to check if our two samples (observed and simulated data) may be considered belonging to the same distribution.A first Quantile-Quantile plot (Figure5) supports the idea of a unique distribution for both samples, since there is a clear linear relationship between the observed and simulated data.

Figure 5 :

 5 Figure 5: Q-Q Plot of the two Capital distributions

Figure 6 :

 6 Figure 6: Box Plot of simulated (left) and actual (right) Capital.

Figure 7 :

 7 Figure 7: Pareto II Plot of the actual capital (biggest 10% trimmed)

Figure 8 :

 8 Figure 8: Meplot of observed (above) and simulated (under) Capital.

Figure 9 :

 9 Figure 9: Hill Plot of the simulated capital Figure 10: Hill Plot of the actual capital Figure 11: Comparison of the two Hill Plots.

Figure 12 :

 12 Figure 12: Zipf's Plots of the net worth distributions: observed (plus) and simulated (diamonds) data

Figure 13 :

 13 Figure 13: Zipf's plot of loans: observed (plus) and simulated (diamonds).

Figure 14 :

 14 Figure 14: Empirical distributions of actual and simulated growth rates.

Figure 15 :

 15 Figure 15: Growth Rates Std vs Size (observed data) 27 Some authors prefer a truncated Lévy distribution. The querelle is open. See Kleiber and Kotz.

  al. 2003b, Gaffeo et al. 2003, Axtell 2001, Ramsden and Kiss-Haypal 2000, Okuyama et al. 1999; Quandt 1966a-b, Mandelbrot 1960, Simon 1955); 2. Firms' growth rates are Laplace distributed, belonging to the Subbotin's Family (Stanley et al. 1996, Bottazzi and Secchi 2005);3. There is a power law relation between the variance of the size growth rates and the size itself(Stanley et al. 1996, Gabaix et al. 2003).

Gallegati et al. (

Table 1 :

 1 Estimated Subbotin's Parameters (standard errors in brackets)

		observed data Simulated data
	µ	-0.0030 (0.0013)	0.0048 (0.0021)
	a	0.0587 (0.0244)	0.0614 (0.0238)
	b	1.0184 (0.3495)	1.0626 (0.3664)
	-loglik	1.1528	1.1549

http://www.econ.iastate.edu/tesfatsi/empvalid.htm

Validation is not the end of the study process. Indeed, it must be considered an intermediate step, necessary to ameliorate the model in order to make predictions.

In a sequential economy[START_REF] Hahn | Money and Inflation[END_REF] spot markets open at given dates, while future markets do not operate.

According to the Italian Fiscal Law, to which we referred in writing this paper, a firm is considered "small" if it has less than

employees, "medium" if it has between

and 250 employees and "large" if it has more than 250 employees. In our sample, the percentage of firms is: ≈ 56% small, ≈ 31% medium, ≈ 13% large. In 1996 the smallest firm shows 2 employees, while the largest one 7308.[START_REF] Beaver | Financial ratios as predictors of failure. Empirical Research in Accounting: Selected Studies[END_REF] Capital stock never depreciates.[START_REF] Bottazzi | Explaining the distribution of firm growth rates[END_REF] This is not the only evidence in the AIDA database contradicting Gibrat's law. As we will see in what follows (section 5), power law behaviors in firms' size distributions and Laplacian growth rates are an unambiguous negation of Gibrat's prediction, at least in its strongest version. For further details see[START_REF] Cirillo | Some considerations about Gibrat's law in Italy[END_REF], section 2.5

For more information: http://amadeus.bvdep.com 6

On JEBO website an appendix describing the procedure in detail is available.

It decreases every year starting from 11.5% (1996) and arriving at

10% (2001).7

A power law behaviour in firms' size is essentially due to the random iid micromultiplicative shocks[START_REF] Solomon | The microscopic representation of complex systems[END_REF] and the presence of the (bankruptcy) lower bound we have modelled. It is possible to show that a system with power laws tails distributions have divergent first and second moments, so the law of large numbers does not hold and the system is not ergodic.

[START_REF] Gaffeo | Power law scaling in the world income distribution[END_REF] Line is red in the website

version.[START_REF] Gallegati | Financial fragility, patterns of firms' entry and exit and aggregate dynamics[END_REF] The Generalised or Two Sample Kolmogorov-Smirnov test is a variation of the classical Kolmogorov-Smirnov test.Given N data points Y 1 , Y 2 , ..., Yn the empirical distribution function (ECDF) is defined as