The Labor Market Matching Function A Quantitative Assessment of Theoretical, Methodological, and Empirical Issues (preliminary and incomplete version) Raymond J.G.M. Florax, a Enrique López-Bazo, b Jordi López-Tamayo, b and Brigitte S. Waldorf c a Department of Spatial Economics, Master-point, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands, Phone +31 20 4446090, Fax +31 20 4446004, E-mail: [email protected] b Department of Econometrics, Statistics and the Spanish Economy, "Anàlisis Quantitativa Regional" Research Group, University of Barcelona, Avda. Diagonal 690, 08034 Barcelona, Spain, Phone +34 93 4024320 / 402 1984, Fax +34 93 4021821, E-mail: [email protected] / [email protected] c Department of Geography and Regional Development, University of Arizona, Harvill Building, Box #2, Tucson, Arizona 85721, USA Phone: +1 520 621 7486, Fax: +1 520 621 2889, E-mail: [email protected]
Working paper version, 11/5/ 日曜日 6:25 a.11./p.11. A BSTRACT The labor matching function has been the focus of a myriad of studies in labor economics. The matching function, usually of the Cobb- Douglas type, explains new hires as a function of the number of job seekers and the number of job vacancies. Typically, the efficiency of the labor market and the rate of return to matching attain considerable interest. Returns to scale play a pivotal role in explaining the existence of multiple underemployment equilibria, as it has been shown that increasing returns to matching are a necessary, though not sufficient condition for the existence of such inefficient equilibria. This paper employs a meta-analysis to investigate empirical estimates across studies of returns to scale in matching models of the labor market. The results of 18 studies, dealing with the US, Canada, Israel, and various European countries, over a time period extending from the late 1960s to the mid 1990s, are considered. The analysis uses an ordered probit model to assess the occurrence of (dis-)economies of scale. Subsequently, differences in magnitude of matching elasticities and scale economies across studies are analyzed in a continuous regression model with restrictions imposed in a simultaneous equation design. The analysis sheds light on theoretical, methodological and empirical issues raised in the literature. A flexible functional form, more comprehensive definitions of matching and the use of corrected vacancy figures increase the likelihood of finding increasing returns to scale. In contrast, studies conducted for more recent and longer time periods and for regions/countries with higher unemployment rate, tend to provide lower estimates for the returns to scale. JEL B49, C25, J41, J64 K EYWORDS labor market, matching function, meta-analysis 1. I NTRODUCTION In recent years the analysis of aggregate labor market performance has received considerable attention in the labor economics literature. A common model typically includes a matching function that specifies new hires as a function of the number of job seekers and the number of vacancies. The flow of new hires is related to the stocks of job searchers and job vacancies, in a dynamic process represented by a production function. On the basis of the production function framework traditional issues — labor market efficiency, elasticities of IVEEA - 11/5/ 日曜日 - 2
unemployment and vacancies to matching, the rate of return to the matching 'technology' — have been empirically investigated. 1 Empirically, the myriad of studies has produced rather mixed results. These empirical discrepancies make it difficult to unambiguously shed light on the theoretical and methodo- logical issues surfacing in the labor market matching literature. This paper, therefore, uses a meta-analytic framework to identify the sources of variation in empirical estimates of the labor market matching function. The analysis proceeds in two steps. First, it uses an ordered probit model to assess variations in estimated economies of scale (i.e., increasing or decreasing returns). The interest in such an analysis is inspired by the theoretical proposition that increasing returns to scale are a necessary, although not sufficient, prerequisite for the occurrence of multiple underemployment equilibria. Subsequently, differences in magnitude of matching elasticities and scale economies across studies are analyzed in a continuous simultaneous regression model. In both types of analysis the relevance of salient issues that are still unsettled in the literature (see Petrongolo and Pissarides 2000, for a recent review) will be assessed quantitatively: for instance, do the estimates depend on 'matches' be measured as filled hires or unemployment outflow, does a correction of official vacancy figures exert any influence in estimated returns, are matching elasticities lower in Western- Europe, are estimated elasticities and scale economies in any way dependent on the spatial aggregation level? The analysis is based on 190 estimates from 18 primary studies, dealing with the US, Canada, Israel, and various European countries, over a time period extending from the late 1960s to the mid 1990s. The remainder of this paper is organized as follows. Section 2 presents the theory of labor market matching, and discusses the state-of-the-art of empirical work reported in the 1 See, e.g., Devine and Kiefer (1991), Pissarides (2000), and Petrongolo and Pissarides (2000) for a review of the empirical literature. IVEEA - 11/5/ 日曜日 - 3
literature. In Section 3, the methodology of meta-analysis is introduced and its suitability in analyzing rates of return to matching is explained. Section 4 presents an exploratory account of the empirical results of 18 recent studies as well as the results of the meta-analysis. The final section highlights the conclusions of the analysis and makes recommendations for further primary research. 2. T HE THEORY AND EMPIRICS OF THE MATCHING FUNCTION Labor markets are characterized by the simultaneous existence of two components: workers searching for a job, and job vacancies that employers wish to fill. In the presence of incomplete information on either side of the market, heterogeneity and friction, the number of matches between job searchers and job vacancies will be lower than the minimum number of searchers or vacancies. The outcome of the matching process can be summarized in a matching function that relates the number of matches at any time to the number of job searchers and posted vacancies. Typically, the labor market matching function takes on the form: M=m( U, V, A ) (1) where M is the flow of matches formed in a time period, U is the stock of job searchers, V the stock of unfilled vacancies. A denotes the level of efficiency of the process that is supposed to be affected by a set of other exogenous variables. Therefore, it determines the number of hires for any given level of searchers and vacancies. The function m(·) is defined to be increasing and concave in U and in V , and m(0, V , A ) = m( U , 0, A ) = 0. IVEEA - 11/5/ 日曜日 - 4
Most empirical studies use a Cobb-Douglas functional form for the matching function in (1), often expanded to include variables that account for efficiency variability across observations: ln M = m + a ln U + b ln V + f(t) + dynamics + search decisions + mismatch (2) The parameters a and b are unemployment and vacancy elasticities, respectively, whereas returns to scale are the sum of both elasticities. The function f(t) captures changes in efficiency over time, dynamics refers to the inclusion of temporal (others than the ones captured by a time trend) and/or spatial dynamics. Search decisions and mismatch account for structural variables that may influence efficiency. Clearly, in its basic form the matching function summarizes a complex process that is conditioned by the micro-economic foundations of both individual (worker) and firm behavior. In a recent literature review on the labor market matching function Petrongolo and Pissarides (2000, p. 3) state: " the matching function is a modeling device that occupies the same place in the macroeconomist's tool kit as other aggregate functions, such as the production function and the demand for money function. Like the other aggregate functions its usefulness depends on its empirical viability and on how successful it is in capturing the key implications of the heterogeneities and frictions." The matching function has become increasingly popular for empirically analyzing the labor market. Its attractiveness is rooted in its simplicity, its ability to provide a better IVEEA - 11/5/ 日曜日 - 5
understanding of factors influencing the efficiency of the market, and its ability to estimate the returns to scale of the matching process. The inclusion of the exogenous factors, summarized in A in (1), enables the researcher to account explicitly for imperfections in the matching process —imperfections that can be caused by individuals' /employers' decision making, or by heterogeneities and barriers. Specifically, A often responds to assumptions regarding the search process of job searchers and employers, and thus includes variables measuring search intensity, job acceptance probability, length of unemployment spells, amount and duration of unemployment insurance, demographic variables, and costs of migration. Moreover, efficiency of the matching process can also be influenced by factors unrelated to individuals' search process. A worker searching a job and an unfilled job may not form a match because of heterogeneities in the labor market. For instance, in the case of technological change, new jobs often require skilled workers while the pool of searchers may be mainly comprised of unskilled workers. In addition to skill heterogeneities, mismatches may also occur due to sectoral barriers and spatial location. Whereas the interest in exogenous factors responsible for imperfections in the matching process is important for an understanding of labor markets, our analysis focuses on the evaluation of returns to scale. There are two main reasons for this focus. First, the literature indicates that increasing returns to scale are a necessary albeit not sufficient condition for multiple steady-state unemployment equilibria. The possible existence of more than one such equilibrium has been identified as a critical question in the literature (see for instance Anderson and Burgess, 2000). The second reason for focusing the meta-analysis presented in this paper on returns to scale is the variation of estimated returns across different economies in space and time. Whereas initial estimates of the matching function, using time series of an aggregate IVEEA - 11/5/ 日曜日 - 6
economy and restricted functional forms, support the idea of constant returns to scale (Pissarides, 1986 for UK; Blanchard and Diamond, 1989 for the US), there has been a growing number of studies that provide evidence for decreasing or increasing returns to scale in the matching function. Increasing or decreasing returns to scale are frequently encountered when analyzing cross-sectional or panel data, a specific sector of the economy, or when using a flexible form of the matching function (e.g. Blanchard and Diamond, 1989; Warren, 1996; Yashiv, 2000). Several theoretical and empirical issues can affect the estimate of the returns to scale. Considering the heterogeneity across studies, the comparison of the estimated returns should be made with caution. The key factors potentially influencing the estimates of returns to scale are: Measurement of matches, e.g., as outflow from unemployment only or as total new hires; Measurement of the pool of workers searching for a job, e.g., including only those unemployed or also considerating of the on-the-job search; Specification of set of right-hand side variables that may lead to shifts in the matching process; Scale of investigation, e.g., aggregate economy versus local labor markets, aggregate economy versus a specific sector of the economy; Functional specification of the matching function, e.g., restricted Cobb-Douglas specification versus flexible functional form; Dynamic elements, e.g., temporal and/or spatial dynamics; Type of data, e.g., time series for a national economy versus a pool of regions Estimation procedure, e.g., OLS estimators versus estimator controlling for endogeneity of some regressors. IVEEA - 11/5/ 日曜日 - 7
3. M ETA - ANALYSIS Meta-analysis, a standard methodological tool in medicine, marketing, psychology, and education, is a statistical analysis of a large collection of research results on a specific topic. Its goal is to synthesize and evaluate the empirical findings from previous individual studies (Glass 1976, p. 3). 2 By now, meta-analysis finds its way into economics, pioneered by the work of Button in industrial and transport economics (Button and Weyman-Jones 1992), and Smith in environmental economics (Smith 1989). Increasingly, meta-analysis is also used in labor economics. Jarrell and Stanley (1990) started off this trend with an assessment of studies dealing with the influence of union membership on wage levels. Card and Krueger (1995) investigate the impact of publication bias in minimum wage studies. Most recently, Ashenfelter et al. (1999) analyze the abundance of studies on the rate of return to schooling, giving special attention to publication bias as well. Many meta-analyses in economics make use of regression techniques. A meta- regression is typically based on least squares estimation (either ordinary or generalized least squares for a fixed or random effects specification) of a model in which a specific effect measure 3 observed in a series of studies is taken as the dependent variable. The explanatory variables include typical underlying causes of the phenomenon under consideration as well as moderator variables representing, for instance, research design, and time-period and geographical locations covered in the original studies (Stanley and Jarell 1989). Although a state-of-the-art review of the literature on rates of returns to labor market matching is useful and valid in its own right, there are two serious drawbacks attached to such literature reviews. First, the selection of topics and empirical results is rather arbitrary 2 Although Glass (1976) coined the term 'meta-analysis' in the second half of the 1970s, its history goes back to the beginning of the twentieth century (see Olkin 1990). 3 As meta-analysis was initially developed for experimental designs the statistical techniques are based on the analysis of effect sizes, typically defined as a standardized difference in means of an experimental and a control group. IVEEA - 11/5/ 日曜日 - 8
(Van den Bergh et al. 1997). Moreover, a literature review is potentially hampered by selectivity bias in the study sample selection process. Thus, assessing the external validity of inferences to a larger population is plagued with uncertainties. Whereas meta-analysis is similarly confronted with arbitrariness and selectivity bias, special techniques have been developed to assess and correct for selection bias (see below). Second, literature reviews frequently — more or less implicitly — apply 'vote- counting.' Vote-counting refers to counting significantly positive and negative, and insignificant results; the categories are simply tallied and the category with the plurality of studies is assumed to represent the true underlying relationship (Light and Smith 1971). Vote-counting obviously ignores the differing sample sizes of the underlying studies and does not account for differences in the magnitude of the observed effects. Furthermore, vote- counting has very low power for the sample sizes and effect sizes typically encountered in social science research (Bushman 1994). Finally, Hedges and Olkin (1980) point out that the vote-counting methodology, on average, leads to the wrong conclusion more often when the number of studies considered increases, as the Type-II errors of the underlying studies do not cancel one another (see also Hedges and Olkin 1985, pp. 48-52). The crudity of vote-counting techniques, which are essentially based on a discrete categorization of the direction of the effects (taking into account a pre-specified significance level), makes them insufficient in assessing the consistency of research results. Meta- analysis can rectify this shortcoming by specifically considering the magnitude of the estimated effects. Moreover, unlike in a traditional literature review employing vote- counting, meta-analysis accounts for non-sampling variation in the form of moderator variables. Meta-analysis thus emerges as a statistically rigorous approach to synthesizing research results. It can make a significant contribution to summarizing relationships and IVEEA - 11/5/ 日曜日 - 9
tracing factors responsible for differing results across studies. Consequently, meta-analysis is a promising tool to evaluate the empirical literature and shed light on theoretical and methodological issues. However, meta-analysis is also confronted with practical difficulties (Cooper and Hedges 1994), and with methodological pitfalls. An important practical difficulty is the lack of research registers in the social sciences, which makes it difficult to obtain an overview of the studies available for sampling. 4 It is even more cumbersome to obtain studies from the fugitive literature. Furthermore, management of the database and the systematic coding of highly heterogeneous information contained in the underlying studies is a daunting task. The heterogeneity in reporting research results is often so excessive and problematic that a recording of 'missing data' becomes inevitable. The most important methodological difficulties fall into three categories (Glass et al. 1981): sample selection bias, heterogeneity, and dependence. Sample selection bias may be due to a variety of factors such as language, date of publication, publication outlet, and research design. Publication bias has received the most attention in the literature and is identified as a main source of sample selection bias (Card and Krueger 1995; Ashenfelter et al. 1999). As sample selection bias prohibits the generalization of meta-analysis results to a larger population, it compromises the external validity of meta-analysis (Button 2001). Several techniques have been developed to assess the presence and impact of sample selection bias, ranging from graphical tools (the so-called 'funnel-plot'), to statistical tests and more sophisticated methods based on weighted distribution theory (see Begg 1994, for an overview). An obvious tool that, until now has only been applied by Smith and Huang (1995), is the two-stage sample selection estimator developed by Heckman (1979). It 4 The availability of literature databases and powerful search engines on the Internet nowadays mitigates this problem. IVEEA - 11/5/ 日曜日 - 10
accounts for the potential bias in the second stage (i.e., the meta-analysis) via the inverse Mills ratio resulting from an explicit modeling of the sample selection process in the first stage. There are various sources of heterogeneity. One is the inherent heteroscedasticity related to unequal sample sizes of the underlying studies. This is, however, easily remedied using a weighted least squares approach or a heteroscedasticity robust estimator for the covariance matrix (e.g., White-adjusted standard errors). Another source is the heterogeneity related to quality differences across studies, differences in research design, differences in types of data used (time series, cross section, panel data), and other study characteristics. These sources of heterogeneity can either be modeled through adequate specification of the set of explanatory variables, or through the use of a fixed or random effects model (or a combination of both). The use of a fixed or random effects model has, however, wider implications, as it also influences the external validity of the analysis. In a fixed effects model the sampling variation among studies is modeled through fixed effects, and the remaining sampling variation thus pertains to the 'within variation' among studies. The results can only be generalized to the population of studies. The meta-sample may actually consist of the whole population or, alternatively, constitute a random sample from a larger population of studies. In the random effects model the sampling variation is attributed to the sampling of studies from a universe of all possible studies, and the sampling related to the determination of the effect size on the basis of a sample of subjects. The results of the random effects model can thus be generalized to a universe of possible observations (Raudenbush 1994). It is remarkable that, with a few exceptions, 5 neither correction 5 Robust covariance or weighting techniques are sometimes used (see, e.g., Smith and Osborne 1996; Brouwer et al. 1997; De Blaeij et al. 2000), but the use of random and fixed effects models is still rare (Rosenberger et al. 1999; Jeppesen et al. 2000 are exceptions). IVEEA - 11/5/ 日曜日 - 11
mechanisms for heteroscedasticity nor fixed and random effects models are used in economic meta-analyses. The third type of methodological difficulties surrounding meta-analysis refers to dependency issues. Although most studies in economics use multiple estimates sampled from the same study, the resulting correlation between the estimated effect sizes is usually comfortably ignored. This approach can be justified since ignoring the dependence among estimates sampled from the same study does not affect the unbiasedness and consistency of ordinary least squares estimators. However, it does lead to inefficient parameter estimators. Discarding data is an obvious strategy of dealing with the dependency issue (Hedges and Olkin 1985). In most cases, however, this strategy is not particularly useful because of a rather limited number of studies and because of the resulting problem of determining an unambiguous criterion for selection. A second strategy uses pooled correlated estimates, and ensures that statistical inferences are rigorous by employing generalized least squares estimation techniques. It remains to be seen, however, how big the gains are from pooling correlated estimates. Hedges and Olkin (1985, pp. 220-222) point out that the efficiency gain is probably rather small, and the techniques for handling correlated estimates (such as generalized least squares) may be rather cumbersome to apply (see also Gleser and Olkin 1994). 4. A QUANTITATIVE ASSESSMENT OF THE LITERATURE 4.1 T HE META - SAMPLE : SELECTION AND DATA EXPLORATION A crucial step in a meta-analysis — as in a literature review — is to attain a good overview of the literature available. A common misperception is that the overview should be IVEEA - 11/5/ 日曜日 - 12
comprehensive. Comprehensive coverage is not necessary as long as the included studies are representative of the population. Common desiderata in literature retrieval are a high recall and a high precision. Recall is defined as the ratio of relevant documents retrieved to those in a collection that should be retrieved. Precision is defined as the ratio of documents retrieved and judged relevant to all those actually retrieved. Unfortunately, precision and recall tend to vary inversely (White 1994). Most researchers favor high precision, whereas researchers dealing with research synthesis typically desire a high recall. The collection procedure used in this study focuses on high recall. It is based on the traditional 'footnote chasing' (or rather 'reference chasing') and consultation of scholars. In addition, we extensively used more modern methods of literature retrieval, such as browsing Internet databases. In particular, we searched EconLit 6 and Sociological Abstracts 7 in order to obtain as many published articles, book chapters, books, and dissertations as possible up until 1999. In addition, we took special care to include unpublished studies, up until 1999, through a search in NetEc, RepEc, and websites of renowned universities and research institutes (for instance, CEPR, NBER, etc.). 8 Table 1 presents an overview of 38 empirical matching function studies found in these sources. From the studies in Table 1 we selectively include only those labor market matching studies that provide a test on whether the returns to scale estimate (usually defined as b a ˆ ˆ in a Cobb-Douglas production function) is significantly different from 1. If the 6 EconLit (see http://www.econlit.org/) is a comprehensive, indexed bibliography with selected abstracts of the world's economic literature, produced by the American Economic Association. It includes coverage of over 400 major journals as well as articles in collective volumes (essays, proceedings, etc.), books, book reviews, dissertations, and working papers licensed from the Cambridge University Press Abstracts of Working Papers in Economics. 7 Sociological Abstracts (see http://www.csa.com/) provides access to the world's literature in sociology and related disciplines, both theoretical and applied. The database includes abstracts of journal articles selected from over 2500 journals, abstracts of conference papers presented at various sociological association meetings, relevant dissertation listings from Dissertations Abstracts International, enhanced bibliographic citations of book reviews, and abstracts of selected sociology books. 8 NetEc includes BibEc, WoPEc, and WebEc (see http:/netec.wustl.edu). For RepEc see http://www.repec.org. IVEEA - 11/5/ 日曜日 - 13
Type-I error probability is greater than .05, then the estimate indicates constant returns to scale. In all other cases, the returns to scale are categorized as decreasing ( 1 ˆ ˆ b a ) or increasing ( 1 ˆ ˆ b a ). Based on the significance test criterion, the analysis includes 18 of the 38 studies, with a total of 190 estimates for returns to scale. 9 Of the 190 estimates, 81 (42.6 percent) show decreasing returns to scale, 59 (31.1 percent) indicate constant returns to scale and 50 estimates (26.3 percent) suggest increasing returns to scale in labor market matching. The average estimated returns to scale amount to 1.073 with a standard deviation of .725. Figure 1 shows the distribution of estimates for the three categories. For decreasing returns to scale, the estimates average .6 and vary between -.4 and .983 with a standard deviation of .307. The range of estimates is even wider for increasing returns to scale, varying between 1.1 and 3.734 with a standard deviation of .811 about the mean value of 2.035. The constant returns to scale estimates are nearly equally split between values smaller than one (30 estimates) and values greater than one (29 estimates). However, deviations from one are slightly more pronounced for estimates greater than one than for estimates smaller than one, yielding an average value of 1.085 and a standard deviation of .275. Figure 1 also shows the distribution of the two underlying components of returns to scale, i.e., unemployment elasticities and vacancy elasticities. As a general trend, unemployment elasticities exceed vacancy elasticities. In fact, on average, unemployment elasticities are more than twice as high than vacancy elasticities. Moreover, unemployment elasticities mirror the returns to scale distribution more closely than vacancies elasticities. Thus, it is not surprising that the correlations with returns to scale amount to r = .92 for 9 For six of the 190 estimates only categorical information on returns to scale (significantly decreasing, significantly increasing, constant) is available. For an additional eleven estimates, no information is available on the split between unemployment and vacancy elasticities. Where necessary, these estimates are excluded from descriptive statistics and the graphical display in this section. IVEEA - 11/5/ 日曜日 - 14
unemployment elasticities as opposed to only r = .47 for vacancies elasticities. This suggests that unemployment elasticities play a pivotal role in labor market matching. The specific contexts of the estimates reported in the literature and summarized in Figure 1 vary widely, based on characteristics such as sample sizes, geographic scales, geographic regions, and model specifications. Figures 2 to 7 suggest that some of these factors potentially play an important role in influencing the returns to scale estimates. The funnel plot displayed in Figure 2 relates the returns to scale estimates to sample sizes, disaggregated by publication status (published versus unpublished). The estimates from unpublished studies are further distinguished by whether or not they refer to the Czech Republic. 10 The funnel plot shows clearly that the very large returns to scale estimates refer to the Czech Republic. For the remaining estimates, those based on large sample sizes are closer to the mean, whereas smaller sample sizes produce a wider range of estimates. Such a funnel or triangular shape suggests that there is no correlation between the magnitudes of the estimates and sample sizes. This is indeed the case for the 66 estimates from published studies ( r = .113). In contrast, however, for estimates from unpublished studies (excluding those referring to the Czech Republic), there is a pronounced negative relationship ( r = -.395) suggesting that higher estimates are more likely to occur when using small sample sizes. A negative relationship ( r = -.349) is also found among estimates for the Czech Republic. In sum this indicates that a bias may be present in the publication of empirical results, as relatively small effect sizes are over-represented. As the population value is unknown — there may be even more than one population value — the occurrence of publication bias does not necessarily imply a deviation from the 'truth' in the published results. Figure 3 suggests an inverse relation between unemployment rates and estimated returns to scale in the labor matching function. The very high estimates for returns to scale 10 All estimates referring to the Czech Republic are from unpublished studies. IVEEA - 11/5/ 日曜日 - 15
are only found for low unemployment rates. However, the range of estimates for low unemployment rates is higher than for high unemployment rates. While these trends hold true for both regional and national estimates, Figure 3 also indicates that, on average, national estimates tend to be substantially lower than regional estimates. In fact, none of the national estimates exceeds 1.7, and all negative estimates refer to studies conducted at the national level. Figure 4 shows the distribution of returns to scale and elasticity estimates in different geographic regions. Compared to estimates for a North American context, those derived in a West European context tend to be smaller, with a maximum returns to scale estimate of only 1.39. The returns to scale estimates for East European 11 contexts show the widest range, from -.107 to +3.734. Interestingly, vacancy elasticities in Eastern Europe are very low, implying that the unemployment elasticity, rather than the vacancies elasticity, is the driving force behind the matching process. In addition to variations by geographic scale and region, the estimates also vary by temporal scale. Most importantly, as shown in Figure 5, none of the 25 estimates based on annual data exceeds one. Among the 138 estimates based on monthly data, 60 are greater than one. Forty-seven are even significantly greater than one and thus categorized as increasing returns to scale. Finally, Figures 6 and 7 suggest that model specification plays an influential role in explaining variations in the estimated returns to scale of the labor market matching function. Figure 6 shows that, if a flexible function is used in the model specification, then all returns to scale estimates can be categorized as increasing. In fact, all but one estimate derived from a flexible function are greater than those based on a non-flexible specification. 11 In this analysis, East Germany after unification with West Germany is categorized as Eastern Europe. IVEEA - 11/5/ 日曜日 - 16
Figure 7 differentiates returns to scale estimates by inclusion of spatial and/or temporal dynamic components. Models that include a temporal dynamic component yield, on average, higher returns to scale estimates than those not accounting for any dynamics. The average scale elasticity estimate in temporal dynamic models is 1.315 compared to the average estimate of 1.096 for non-dynamic models. Moreover, the temporal dynamic component accounts for about 20.1 percent of the scale elasticity estimate. In contrast, models including spatial dynamics for both unemployment and vacancies yield very low returns to scale estimates, averaging .787 only. Moreover, none of the estimates exceeds one. The contribution of the spatial dynamics component to the matching process is smaller than that of the temporal dynamic component. In fact, it is even negative in four of the twelve cases. Given the opposing impacts of spatial and temporal dynamics on the returns to scale estimate, it is not surprising that models including both components take on a middle position. The average returns to scale estimate is similar to that for models without dynamics (1.024 versus 1.096, respectively). Remarkably, in models including both space and time dynamics, the contribution of the dynamic element to the overall scale elasticity estimate is 50 percent, with most of it attributable to temporal dynamics. Although the discussion so far concentrates on bivariate descriptions, it clearly shows that a variety of factors influence the modeling results for labor market matching functions. Yet, it also makes it clear that bivariate descriptions are not sufficient to identify sources of variations in returns to scale estimates. The remainder of this paper thus turns to a multivariate analysis that simultaneously assesses the impact of salient covariates on estimated returns to scale categories (decreasing, constant, increasing) and estimated returns to scale parameters. Prior to the multivariate analysis, however, we pay attention to the possibility of combining information on effect size estimates and their degree of heterogeneity. IVEEA - 11/5/ 日曜日 - 17
4.2 C OMBINING EFFECT SIZES AND THE EXTENT OF HETEROGENEITY AMONG EFFECT SIZES The combination of effect sizes, in this case the rates of return to matching and the unemployment and vacancy elasticity, respectively, gives insight into the overall conclusion from the underlying studies. We can do this on the basis of the parameter estimates and their probability values. For the latter, many methods are available. tbc .............. First, combination of effect sizes by means of probabilities. Second, tests on homogeneity of effect sizes ( Q -test); note the possibility of weighting. 4.3 S PECIFICATION ISSUES IN DISCRETE CHOICE AND CONTINUOUS MODELS In Section 3 various methodological issues are raised that should be taken into account in the specification of a meta-regression model. The meta-regressions considered in this study are of two different types, focusing on different questions to be answered. The first type of meta- regression is concerned with a limited dependent variable model, primarily dealing with the salient theoretical issue identifying factors that influence the type of scale economies. In particular, the model contains a set of exogenous variables that affect the probabilities of finding increasing, constant, and decreasing returns to labor market matching. It centers on to the main theoretical issue of increasing returns to scale being a necessary, although not sufficient, condition for the existence of multiple underemployment equilibria. The second type of meta-regression investigates the structural factors that contribute to the magnitude of the effect sizes. As the literature reports estimated effects sizes for scale elasticities as well as for the constituent unemployment and vacancies elasticities, the IVEEA - 11/5/ 日曜日 - 18
analysis should focus on all three simultaneously. Many conceptual, methodological, and empirical issues raised in the literature can thus be addressed. Below we will discuss various ways of handling the methodological pitfalls of sample selection, heterogeneity, and dependence. Furthermore, we will present the set of explanatory variables used in the meta-regressions. S AMPLE SELECTION . The potential effects of sample selection bias are obviously detrimental to meta-analysis — as they are for a traditional literature review. As pointed out earlier, an advantage of meta-analysis is, however, that the occurrence of sample selection bias can be assessed empirically. In this study, two different avenues are explored. The first avenue is confined to the exploration of the impact of publication bias. The second is much broader, as it is deals with sample selection bias as such, and is thus not necessarily limited to publication bias alone. Both avenues are discussed below. In order to account for the possibility of publication bias we include a dummy variable that separates unpublished from published studies. The dummy variable reveals a shift in the probability of finding decreasing or increasing returns (in a discrete choice set- up), or alternatively the magnitude of the effect sizes (in the continuous case), based on the publication status of a paper. This is a straightforward albeit rather crude way of assessing the impact of publication bias. In particular, it is fairly restrictive to assume that the impact of publication bias is merely apparent in a shift in probability or effect size magnitude, leaving the influence of all other exogenous variables the same. There is also a degree of arbitrariness in the coding of the publication dummy variable, especially for more recent studies, as it is basically a variable that accounts for the publication status of a paper at a IVEEA - 11/5/ 日曜日 - 19
certain moment in time. Recent unpublished papers are 'right censored' as they may still attain the 'published' status in the future. 12 An alternative means of accounting for the potentially disturbing effects of publication bias, not suffering from the above, is based on the use of weighted distribution theory (Lane and Dunlap 1978; Hedges 1984, 1992). Unfortunately, it is only suitable in the (continuous) analysis of effect sizes. Each study i , with an estimated statistic X i , can be assigned to a weight function w ( X i ), which determines the probability of being observed, i.e., of being published. Typically, the weight function is determined by the p -value, rather than the magnitude of the effect size (Begg 1994). The publication selection process is modeled using the weight function in a flexible step function with a priori determined discontinuities. Hedges (1992) introduces the following weighting scheme, the weights i representing the relative probabilities of being published, as one of the weights is fixed to an arbitrary value: 1. if , if , 0 if 1 1 1 p a a p a a p p w i k k j i j j i i i (3) where a refers to the a priori determined endpoints. A logical choice is to set = 1 implying the values represent the chance that an estimate with a given p -value is observed relative to the chance that studies with p < a i are observed. Hedges (1992) derives the log-likelihood: 12 If sufficient degrees of freedom are available, the potential change in publication status can be captured by an interaction term of publication status and the year in which the study appears. IVEEA - 11/5/ 日曜日 - 20
(2) n i n i n i k j ij j i i i n i i i B X X w c L 1 1 1 1 2 1 2 1 , log log , log where B ij (  ) is the probability that a normally distributed random variable, with mean and variance 2 i is assigned a specific weight value. Hedges (1992) presents first and second derivatives for this log-likelihood, and Vevea and Hedges (1995) derive a test on grouped frequencies over the different intervals of the weight function, and a Likelihood Ratio test comparing a specification with and without publication bias. In addition to the publication status of a paper various other characteristics of the underlying studies may be relevant in the sample selection process. The extensive sample of empirical studies on labor market matching summarized in Table 1 is reduced following a strict selection rule: a studies is only included in the meta-analysis if it based on unrestricted estimation of the matching function comprises an explicit test on constant returns to matching. In order to establish the influence of potential biases in the sample selection process, Heckman's two-stage sample selection estimator can be applied. The two-stage Heckman procedure is a sequential estimation procedure. In a first step, estimates from all 38 studies are categorized as to whether or not they are included in the meta-analysis and the inclusion status of an estimate is related to a set of potential sampling criteria. The results of this step are summarized in the inverse Mills ratio that, in a second step, is entered as an additional exogenous variable in the meta-analysis. H ETEROGENEITY . The second methodological pitfall mentioned in the preceding section is heterogeneity. If heterogeneity is restricted to unequal variances it can be easily remedied through the use of White-adjusted variances or a weighted least squares estimator. A less IVEEA - 11/5/ 日曜日 - 21
restrictive perspective on heterogeneity calls for the use of a fixed or random effects estimator. The sample consisting of (typically multiple) estimates from a series of studies is treated as an unbalanced panel. Fixed or random effects thus account for unobserved heterogeneity among studies. As the fixed effects estimator is a 'within' estimator it is not very adequate for the meta-regression, given the limited number of observations per study as well as the lack of within-variation for some of the explanatory variables. D EPENDENCE . As mentioned above, Hedges and Olkin (1985) maintain that the gain of using multiple, highly correlated, estimates from the same study, is likely to be rather limited. However, given the limited number of available primary studies the use of multiple estimates cannot be avoided. The potential dependence problem in discrete dependent variable models is ignored, as this is an issue that has not been solved to date. The effect on the estimated parameters and variances is likely to be relatively small, as the estimator remains unbiased and consistent. Hence, the lack of efficiency does not preclude drawing statistically warranted conclusions. However, care should be taken given the fact that the Type-II errors are relatively large. E XPLANATORY DESIGN . Table 2 provides a comprehensive overview of the characteristics of the studies included in the meta-sample. Based on this overview, seven categories of explanatory variables are distinguished. These categories are defined as: spatio-temporal domain, econometric aspects, data characteristics, definitions, sectoral detail, publication status, and labor market situation. Table 3 summarizes the list of variables for each category and the respective definitions. The influence of spatio-temporal characteristics of the primary study is assessed through six variables. BEGINT and ENDT depict the beginning and end of the time period IVEEA - 11/5/ 日曜日 - 22
considered, whereas TIMESPAN measures the length of the period covered in the study (in months). Three dummy variables ( NAM , WEUR , EEUR ) distinguish the studies according to the larger geographic region, i.e., North America, Western Europe, and Eastern Europe, and hence measure spatial variation. 13 Econometric aspects are accounted for by defining variables related to: the number of observations in the underlying studies ( NOBS ), the use of ordinary least squares as compared to more complex estimators ( OLS ), the use of a flexible function specification as compared to all other functional forms (usually of the Cobb-Douglas type; FLEX ), and a series of variables related to model dynamics. The differences in model dynamics are captured by dummy variables indicating whether the specification in the primary study includes a spatially or temporally lagged variable ( DYNLAG ), fixed space or time effects (eventually including varying slopes), random effects or a dynamic error correction model ( DYNFRE ), an AR(MA) error specification ( DYNAR ), or a time trend ( DYNTREND ). Data characteristics are distinguished on the basis of dummy variables revealing the use of monthly, quarterly or yearly data ( MONTH , QUARTER , YEAR ), regional or national data ( REGION ), and time series, cross section or panel data ( TSERIES , CROSS , PANEL ). The impact of varying definitions and measurements of the dependent and exogenous variables is operationalized as follows. A 'narrow' definition of the dependent variable measures hires in terms of unemployment outflow, whereas a 'broad' definition centers on hires or filled vacancies. This difference is captured by the dummy variable DEFYOUTF , which is unity in case the first definition is used. In addition, some studies focus on hires from the stock of unemployed, or alternatively use a 'broader' definition, including hiring from the stock of employed and/or those not in the labor force. This is measured by the 13 In addition we used a dummy variable for the Czech republic that did, however, not turn up as a significant variable. IVEEA - 11/5/ 日曜日 - 23
dummy variable DEFYSTOC , which is unity in case mere hires from the stock of unemployed are considered. Furthermore, differences in the functional specification of the dependent variable are captured in a dummy variables indicating whether the dependent variable is measured as the logarithm of the level of hires ( DEFYLOGL ), the logarithm of the rate of hires ( DEFYLOGR ), or the change in the logarithm of the level of hires ( DEFYCHLL ). Finally, the dummy variable CORRVAC signals the use of corrected vacancies as opposed to 'official' vacancies. The last three categories of variables are less elaborate, comprising zero-one variables indicating the sectoral coverage in terms of the manufacturing sector, other sectors or all sectors ( MANSECT , OTHSECT , and ALLSECT , respectively), and a dummy variable that reveals the publication status of the study considered ( UNPUBL ). Continuous variables indicate the year of publication ( PUBYEAR ), and the mean unemployment rate and its standard deviation for the study considered ( UEMEAN and UESTDEV , respectively). 14 4.4 A N ASSESSMENT OF THE LITERATURE USING META - REGRESSIONS The large number of explanatory variables available and the inevitability that many of them are operationalized as dichotomous variables seriously complicates the estimation. The occurrence of multicollinearity precludes using all variables, and a subset for which the results are robust is therefore used. Table 4 presents an overview of an ordered probit specification, that allows for unobservable random effects for each primary study. The first column presents the parameter estimates and the p-value for the significance test, whereas the last three columns show the marginal effects derived from the estimates, for the three possible outcomes: decreasing, constant and increasing returns. It can 14 Unemployment rates have been obtained from the OECD dataset. We have computed the average rate and the standard deviation corresponding to the sample of observations used in each primary study to obtain the estimated returns. This is because such information is not always available in the primary studies while, additionally, we ensure certain homogeneity in the measurement of such variable. IVEEA - 11/5/ 日曜日 - 24
be observed how the spatio-temporal domain of the analysis has a rather limited impact on the type of estimated returns. Only TIMESPAN is significant at 10% and the probability of observing the different types of effects are almost unaffected by the period or the geographical location, when conditioned to the other characteristics. More remarkable is the contribution of the specification of the functional form ( FLEX ). When the estimates come from a flexible specification there is a reduction in the probability of observing decreasing returns, while increases the one for constant and especially increasing returns. Regarding the characteristics of the dataset, our results reveal that the probability of obtaining decreasing, constant and increasing return is significantly influenced by the use of a panel of data versus a cross-section or a time series. Studies using panel data have more probability of achieving constant and increasing. It was previously commented that the way in which the main variables of the empirical matching function are measured is crucial for understanding differences in the estimated elasticities and returns to scale (Anderson and Burgess, 2000). This statement is confirmed by our results: the probability of concluding decreasing, constant or increasing returns is highly determined by variables used in the empirical analysis to proxy for the matches and the pool of searchers and vacancies. The probability of observing increasing returns decreases when outflows from unemployment are used to proxy for matches. This seems to be also the case when the dependent variable in the matching function is limited to hires from the stock of unemployed (excluding those from the pool of already employed and from out of the market). Furthermore, a remarkable effect is detected for differences in the functional specification of the endogenous variable. When the logarithm of the level of hires is used, the probability of decreasing returns increases largely, in respect to the definition of the endogenous as the log of the hires rate or the growth rate of hires. Finally, a large effect is IVEEA - 11/5/ 日曜日 - 25
observed for the definition of vacancies as well. The use of alternatives instead of the official figures seems to attenuate the probability of decreasing returns. From these estimates, it can also be deduced that the absence of controls for changes in the level of efficiency in the process of matching might bias the estimates towards constant and increasing returns. That is, higher probability of decreasing returns is observed when the estimates come from specifications that include regressors aiming at capturing such changes. Finally, high unemployment rates seems to be related to decreasing returns while there are more chances of obtaining constant and increasing returns in samples with moderate levels of unemployment. Results for the continuous analysis are summarized in Table 5. The table collects the estimated coefficients and the corresponding p-value for the null of non-significance associated to the same set of variable than in the discrete analysis. As described in the previous section, the model includes the restrictions in the parameters across the 3 equations, given that returns to scale is obtained as the sum of the elasticities to unemployment and vacancies. In brief, results are in accordance to the ones obtained for the discrete analysis. Interestingly, a flexible functional form increases the estimated returns due to the increase in the estimate of both elasticities, although to a greater extend the one of unemployment. The use of panel data also increases the returns, with a similar influence in both elasticities. But we can also observed how there are some factors that have a different effect in the elasticity to unemployment and the one to vacancies. For instance, the important decrease in the unemployment elasticity when using the log of the level of hires ( DEFYLOGL ) is somewhat counterbalanced by the increase in the vacancy elasticity. Tbc ...... 5. C ONCLUSIONS : THEORY , METHODOLOGY AND EMPIRICS IVEEA - 11/5/ 日曜日 - 26
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