Long-run matching relationship in the Japanese labor market

Transcription

1 Empirical Economics (2004) 29: DOI /s Long-run matching relationship in the Japanese labor market A panel cointegration approach Shigeki Kano 1, Makoto Ohta 2 1 Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka , Japan ( Kano@eco.usakafu-u.ac.jp) 2 Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki , Japan First version received: November 2002/Final version received: October 2003 Abstract. This paper investigates the long-run relationship among new hiring, unemployment (job seekers), and unfilled vacancies in Japan, using an annual panel data on 47 prefectures for We find that these three variables are Ið1Þ processes, and are cointegrated in our panel data. Further, we estimate the panel cointegration equation derived from a Cobb-Douglas matching function by the heterogeneous fully modified OLS and heterogeneous dynamic OLS. The estimation results reveal that conventional within estimates could have non-negligible biases. Key words: Matching function, Japanese labor market, panel cointegration analysis JEL classification: C23, E24, J41, J60 1. Introduction In the theory of equilibrium unemployment, the matching function relates unemployed job seekers and unfilled vacancies positively with new hiring under the presence of search frictions of a labor market. Since a prominent study by (Blanchard and Diamond 1989), dozens of authors have estimated the matching functions of various countries, and they provide statistical All correspondence to Shigeki Kano. The authors are grateful to the associate editor and two anonymous referees of this journal for helpful comments. Doctor Stephen J. Turnbull (University of Tsukuba) is also acknowledged for correcting English errors in this paper. Remaining errors are due to the authors. The data set and GAUSS programming code used in this paper are available upon request.

2 922 S. Kano, M. Ohta evidence on the existence of stable matching relationship among new hiring, unemployment, and vacancies. Recently, Petrongolo and Pissarides (2001) provide a detailed survey which lists a wide range of previous studies concerning the matching function. However, these studies are concentrated on the cases of the United States and European countries. No such studies seem to have been made on Asian countries such as Japan. This paper investigates the empirical relationship among new hirings, unemployment, and vacancies in Japan by estimating the matching function. Our paper is distinguished from previous empirical studies in that we use recently developed econometric methods for analyzing nonstationary panel data to estimate the matching function, as explained below. This is the methodologically original point of our paper. Most of the previous studies estimate the matching function by using time series data. However, they did not thoroughly investigate the stationarities of new hiring, unemployment, and vacancies. Stationarity is important, since it is well known that if the time series data in question is nonstationary, then the conventional techniques of statistical inference are no longer applicable. Unemployment, one of the explanatory variables in the matching function, is often characterized as a unit root process in existing studies on macroeconomic time series analysis. 1 The high persistency of unemployment series has been expressed frequently as hysteresis. Therefore estimating the matching function involves the possibility of problems due to non-stationarity. Gross (1997) might be the first and the only accessible study which sheds light on the non-stationarity problem in this empirical field. He applied the so-called Johansen procedure Johansen (1988) to German time series, and found that new hiring, unemployment, and vacancies are Ið1Þ processes and cointegrated. 2 But he did not use panel data. On the other hand, although recently estimations of the matching function have come to use panel data more and more, they have not considered the non-stationary problem (see Burgess and Profit, 2002, and Boeri and Burda, 1996, among others). Since the mid 1990s, econometric methods for panel unit root tests and panel cointegration analysis have grown rapidly so that they have become one of the common empirical tools used by applied researchers. Baltagi (2001, Chapt. 12] and Phillips and Moon (2000) provide concise surveys on recent developments in this field. An instructive application of panel cointegration analysis is found in Kao et al. (1999). Panel non-stationarity analysis enjoys the various advantages of conventional panel data analysis over pure time series analysis. Especially, authors in this field (e.g., Im et al. 2002) argue that adding a cross sectional dimension to the data can resolve the well-known low power problems of conventional, pure time series unit root tests. 1 There are vital debates as to whether the time series of unemployment (as well as other macroeconomic variables) is stationary or non-stationary. See, for example, Papell et al., (2000) for the recent argument. 2 The values of estimated cointegration vectors in his study are far from what is considered to be plausible by the equilibrium unemployment theory. He found two cointegration vectors, and interpreted them as the upper and lower bounds of elasiticites of new hiring with respect to unemployment and vacancies during the sample period.

3 Long-run matching relationship in the Japanese labor market 923 Exploiting these fruitful developments in empirical tools for nonstationary panels, this paper investigates the long-run relationship among new hiring, unemployment, and vacancies with Japanese annual, prefectural panel data. We find that new hiring, unemployment (actually job seekers), and vacancies are Ið1Þ processes, and are cointegrated. This implies the existence of a long-run matching relationship among these variables, as assumed by the equilibrium unemployment theory. Further, the panel cointegration equation derived from the matching function is estimated by heterogeneous fully-modified OLS and heterogeneous dynamic OLS, which we will explain briefly in section 2. It is shown that conventional within estimates are seriously biased. The remainder of this paper is organized as follows. Section 2 introduces our estimation model and provides a brief review of estimation methods used in the analysis. Section 3 assesses the data used in our analysis. Section 4 shows our results of panel unit root tests, panel cointegration tests, and panel cointegration estimations. Section 5 compares our analysis with previous panel studies on estimating the matching function. Section 6 concludes the paper. 2. Estimation model and method 2.1. Matching function: Introduction and specification In the equilibrium unemployment theory, the matching function plays a key role. The equilibrium unemployment theory assumes that current hiring levels are determined by a stochastic rationing of vacancies to job seekers (or equivalently rationing of job seekers to vacancies) under the presence of search frictions in a labor market. This assumption casts a sharp contrast to the conventional determination process of equilibrium labor input, i.e., the Walrasian wage mechanism which clears the market so that workers marginal disutility of labor is equated to firms marginal productivity of labor. Tentative evidence of labor market frictions and searching behaviours of firms and workers are found in the fact that vacancies have never vanished even in the presence of a vast pool of unemployed job seekers (e.g., Japan in 1990s), and job seekers have never vanished even in the presence of excess vacancies (e.g., Japan in 1980s). Major sources of labor market search frictions might be as follows: (1) Poor labor mobility among different industries, which occurs because the skills acquired in one industry can not utilized easily in other industries; (2) Poor labor mobility among different regions, which occurs because information about labor market conditions in one region may not be available easily in other regions, and moreover there are costs to going to workplaces in other regions. The former is called a skill mismatch and the latter a regional mismatch. The matching function is generally given by H ¼ MðU; V Þ; ð1þ where H, U, and V denote new hiring, unemployment, and vacancies, respectively. Variables may be structured as time series, cross section, or both

4 924 S. Kano, M. Ohta dimensions. We can view U as an inventory of labor supply, with V as a queue of labor demand. The following is a set of natural and testable V > V > 0; Mð0; V Þ¼MðU; 0Þ ¼0: It is an implicit but important assumption that at every point in time, a fraction of already matched worker-job pairs are separated at some (given) rate so that they drop into the pool of unemployment or vacancies. Otherwise, unemployment and vacancies will disappear in due course, if the size of labor force and total jobs are fixed. One can see how the matching function works in the equilibrium unemployment model such as Blanchard and Diamond (1989) and Pissarides (2000). For the review of empirical literature on the matching function and related fields, see Petrongolo and Pissarides (2001). Following previous empirical studies (e.g., regional panel data analysis of the matching function by Burgess and Profit 2001, among others), we specify the matching function as a Cobb-Douglas form: For individual unit i ¼ 1; 2;...; N and time period t ¼ 1; 2;...; T, H i;t ¼ A i;t U b u i;t V b v i;t expð i;t Þ; ð2þ where A i;t denotes the matching technology, to be treated as a fixed effect. i;t is a stochastic shock hitting the labor market of region i at period t. Of course, there might exist omitted variables affecting a region s hiring performance seriously, such as educational levels, on-going labor market policies, population structures, and so on. However, we do not include such variables in the analysis due to data unavailability. Instead, we control their effects on matching efficiencies by including individual effects in regression. The matching function is often said to be an analogous concept of production function (e.g., Petrongolo and Pissarides 2001). Unlike production function, however, the matching function expresses not an engineering but a social, institutional relationship among the set of input (unemployment and vacancies) and output (new hiring). So the relationship among these variables might be more fragile than expected by the proponents of equilibrium unemployment theory. Therefore, it is worthwhile investigating whether or not there exists a stable relationship among these variables to which a combination of them converges. Generally, a stable relationship among variables in the long-run can be dealt with as a cointegration relationship in econometrics. Thus we mean a cointegration relationship among variables when we say a long-run relationship in this paper. Further, from the statistical view point, variables in the matching function are often considered as Ið1Þ i.e., non-stationary. If this is the case, as mentioned earlier, conventional econometric methods cannot be applicable. Theses two problems (the long-run relationship and the non-stationarity) can be merged by the representation theorem Engle and Granger (1987) that showing the existence of cointegration relationship is equivalent to showing the existence of an error correction mechanism. So the cointegration analysis reviewed below is a suitable econometric tool to investigate our empirical concerns.

5 Long-run matching relationship in the Japanese labor market Review of panel cointegration estimation Let h i;t denote the log of new hiring during period t in the i-th prefecture, and u i;t and v i;t denote the log of unemployment and vacancies at the beginning of t in the i-th prefecture, respectively. Assume that these variables are Ið1Þ. Then, the following panel cointegration system is our interest; for i ¼ 1; 2;...; N and t ¼ 1; 2;...; T, y i;t ¼ x 0 i;t b þ l i þ e i;t x i;t ¼ x i;t 1 þ x i;t ; ð3þ ð4þ where y i;t ¼ h i;t and x 0 i;t ¼ðu i;t; v i;t Þ,andl i denotes individual effects. e i;t and x i;t in each equation denote stationary error terms with zero means. Notice that the stationarity of e i;t is equivalent to y t being cointegrated with x t by the definition of cointegration. Further e i;t and x i;t are assumed to have a longrun covariance matrix, X ¼ P 1 s¼ 1 Eðe i;se 0 i;0 Þ, where e i;t ¼ðe i;t ; x i;t Þ 0. Here X is assumed to be homogeneous among individual units, but it will be relaxed later. Equation (3) comes from the log-linearized Cobb-Douglas matching function, given by Eq. (2). So b denotes the vector of hiring elasticities with respect to unemployment, b u, and to vacancies, b v,andb > 0 is assumed. On the other hand, Eq. (4) states that the explanatory variables have unit roots. It is likely that other types of non-stationarity, e.g., unit root process with drift or with trend, are more relevant. Indeed, the panel unit root tests in the later section allow more general forms of non-stationary process. However we specify the process as (4) so that non-stationarity of the explanatory variables is neatly merged to regression Eq. (3). This approach is often called the single equation method of cointegration analysis (e.g., Engle and Granger 1987, Stock and Watson 1993, among others), distinguished from Johansen s (1988) VAR method. It is well known that the OLS estimation of the cointegration regression equation is biased and has a non-normal limiting distribution due to underlying endogeneity and serial correlation, so the use of conventional statistics, such as t-values, is misleading. Recently Kao (1999) and Phillips and Moon (1999) show that this statement holds also for panel data analysis. In order to obtain an asymptotically efficient estimator whose limiting distribution is normal, Phillips and Moon (1999) and Kao and Chiang (2000) develop alternative estimators, i.e., fully-modified OLS (FM) and dynamic OLS (DOLS), respectively. Particularly, Kao and Chiang (2000) provide detailed and comprehensive descriptions of these two estimators, including their limiting distributions, estimation procedures, small sample performances, and so on. 3 Basically, FM and DOLS are OLS with some kind of bias correction. In the FM estimation, based on the estimated long-run covariance matrix, a data transformation is done on the dependent variable y t before running OLS. On the other hand, in the DOLS estimation, extra terms are added to original 3 Notice that in the literature of non-stationary panel data analysis, the within estimator is commonly referred to as OLS, since actually it uses the OLS principle. This paper follows this tradition.

6 926 S. Kano, M. Ohta cointegration Eq. (3) so that bias is corrected. They consist of lags and leads of the first order differences of the explanatory variables: y i;t ¼ x 0 i;t b þ Xp 2 j¼ p 1 c j Dx i;j þ l i þ e i;t ; where c j is the coefficient of a lead or lag of first differenced explanatory variables, and e i;t is an appropriately re-defined error term. The assumption of homogeneous long-run covariance matrices among individual units might seem to be implausible, since regressions with a cross sectional dimension occasionally give rise to heterogeneity in the error term. Kao and Chiang (2000) develop generalized versions of FM and DOLS which allow the heterogeneity of the long-run covariance matrix: They are called heterogeneous FM and heterogeneous DOLS, respectively. We employ these techniques. So, for the panel cointegration estimation, the long-run covariance matrix X depends on the individual prefecture i. ð5þ 3. Data description The data used in our analysis is drawn from Referentials and Placements by Prefecture, Employment Referral Statistics in Year Book of Labour Statistics, issued by The Ministry of Labour (Currently, Ministry of Health, Labour, and Welfare), Japan. Our variables correspond to the published data as follows; h t ¼ log(placements), u t ¼ log(active Applicants), and v t ¼ log(active Openings). The data is collected from 1972 to 1999, and so the number of time periods is T ¼ 28 for panel unit root tests. But one period lag is used in explanatory variables for the analysis of panel cointegration and so T ¼ 27 in that analysis. The number of individual units, namely of prefectures in Japan, is N ¼ 47. For the purpose of summing up the dynamics of these three variables in Japan, we plot their annual aggregates over time in Fig. 1. A casual inspection reveals that unemployment and new hirings are counter-cyclical while vacancy is pro-cyclical, which is a common observation for other countries. For example, at the beginning of the 1990s, i.e., of the Japanese lost decade, both unemployment and new hirings exhibited a sharp rise while vacancies apparently have been decreasing. During the sample period, the correlations of the logs of new hirings, unemployment, and vacancies with the log of GDP growth are 0:20, 0:77, and 0:32, respectively. The figure gives us another important insight: During 1990s the growth rate of unemployment is greater than that of new hiring. It suggests an increase in average unemployment duration over time, i.e., unemployment divided by new hirings. This tendency toward increasing long-term unemployment is also confirmed by other data sources. For example, according to Report on the Special Survey of the Labour Force Survey, August 2001, recent rates of those who are unemployed for more than a year to the total unemployment are as follows: 18.1% in 1995; 19.6% in 1996; 20.9% in 1997; 20.7% in 1998; 22.4% in 1999; 25.1% in This problem is now one of major concern for both Japanese economists (e.g., Genda and Rebick 2000) and policy makers (e.g., Ministry of Labour 2000), although we do not discuss this problem in this paper.

7 Long-run matching relationship in the Japanese labor market Ht * 10 Ut Vt Fig. 1. New hiring (H t ), unemployment (U t ), and vacancies (V t ), in thousands. Note: 10 (H t )is shown in the figure Apparent drifts of unemployment and hiring growth in Fig. 1 during 1990s raises the possibility of structural changes in the cointegration relationship among variables. Indeed, during this period, the Japanese labor market experienced unprecedented many changes in its environment and employers and employees attitude toward hiring, e.g., decline in participation rates of labor unions, collapse of lifetime employment, and so on. Therefore we perhaps should do the empirical analysis in the following sections by splitting the sample period into the first half (before 1992) and the second half (after 1992). Gregory et al. (1996), for example, propose a Chow type procedure for testing structural breaks of cointegration for pure time series analysis. However, we did not do so because the sample period of this analysis is too small. Instead, we proceed the analysis by assuming that a recent rise in unemployment is not a permanent tendency. Of course, we cannot be sure whether this assumption is correct or not until further sample observations are accumulated. Finally, we must assess the quality of our data, e.g., how well it represents the total behaviour of the Japanese labor market. Table 1 shows the main instruments of job search for the unemployed. Though 34.5% of the unemployed regard the public referral office as a major source of their job search, a similar fraction of them resort to job advertisement leaflets or magazines. On the other hand, Table 2 shows the composition of new job openings for regular employees by occupation at public referral offices. Specified fields and Management, which are considered high skilled jobs, occupy less than 13% of the total. In addition, public referral offices are used mainly by employers as a means of recruiting mid-career job seekers such as those who were fired. In 2001, 59% of mid-career labor are recruited through the public referral office, while only 18% of university graduates were (source: Report on the Special Survey of the Labour Force Survey 2001, by Statistics Bureau).

8 928 S. Kano, M. Ohta Table 1. Main instruments of job search for the unemployed Methods % Public unemployment referral office 34.5 Private unemployment referral office 1.8 Registered to temporary stuff offices 2.1 Job advertisement or magazine 35.7 Referral by school or acquaintance 8.9 Direct application to job opening firms 4.5 Preparation for opening new business 0.9 Others 8.6 Source: Report on the Special Survey of the Labour Force Survey 2001, by Statistics Bureau, Ministry of Public Management, Home Affairs, Posts and Telecommunications, Japan. In short, our data represents the tendencies of total Japanese labor market to some extent, but may be biased toward low-skilled, low-wage jobs. Indeed, it is a common sense among Japanese that job seekers go to the public referral office as a last resource. Nevertheless we use such data, because there are no alternatives that are superior to the data by the public referral office. 4. Empirical results 4.1. Panel unit root tests As a pre-test for the cointegration analysis, we first investigate panel nonstationarity of h t, u t, and v t. Here two types of panel unit root tests are employed: The t-bar test proposed by Im et al. (2002, IPS) and Hadri (2000). The former, a panel analogous of Said and Dickey (1984), tests the null hypothesis of non-stationarity, while the latter, a panel analogue of Kwiatkowski et al. (1992, KPSS), tests the null hypothesis of stationarity. Therefore, for a panel data in question, if the former accepts the null and the latter rejects it, that implies a strong evidence of panel non-stationarity. On the other hand, if the former rejects and the latter accepts the null, that gives rise to a mixed result. The IPS t-bar is designed to test H 0 : all individual units have unit roots against H 1 : some individual unit does not have a unit root. Formally, H 0 : a i ¼ 0 8i; H 1 : 9i such that a i < 0; where a i is the coefficient of the Augmented Dickey-Fuller (ADF) regression for each individual unit, i.e., for a time series y t, Dy i;t ¼ l i þ a i y i;t 1 þ Xp k¼1 / i Dy i;t k þ c i t þ e i;t ; t ¼ 1; 2;...T ; ð6þ where c i could be zero or not. In our case h t, u t,orv t are assigned to y t. Im et al. (2002) show that the IPS t-bar consists of the average of ADF t-values for each individual, and after an appropriate normalization, IPS t- bar test statistic is asymptotically distributed as N ð0; 1Þ under the null hypothesis.

9 Long-run matching relationship in the Japanese labor market 929 Table 2. New job openings for regular employees by occupation, at public referral office Occupation % Specified fields 11.6 Management 0.5 Administration 26.2 Sales 12.4 Service 6.3 Maintenance of safety 1.2 Agriculture, forestry, and fisheries 0.4 Transport and communication 6.1 Skilled work, mining, manufacturing, construction and labor 29.8 Source: Year Book of Labour Statistics 2000, by Ministry of Health, Labour and Welfare, Japan. At present, there are no clear-cut procedures to determine the lag length of the IPS t-bar test, p, though the test result is often sensitive to it. It is also difficult to make a statistical decision on whether the ADF regression should include time trend or not in the panel data framework. Therefore we try to test the non-stationarity with several lag lengths and with or without trend. In order to utilize the table of first and second moments of IPS t-bar statistic by Im et al. (2002), we use a common lag length for all individual units. The Hadri (2000) test is based on quite different assumptions on the data generating process of the series from the IPS test, i.e., y i;t ¼ r i;t þ c i t þ t ; r i;t ¼ r i;t 1 þ n t ; where a time trend c i t may or may not be included. For descriptive simplicity, error terms t and n t are assumed to follow iidð0; r 2 Þ and iidð0; r2 nþ, respectively. However, in practice, the iid assumption is relaxed so that heteroskedasticity and serial correlation are permitted by familiar nonparametric procedures, e.g., Andrews (1991). It is obvious that if r n ¼ 0 in the above state space representation of Eq. (7) and (8), then r i;t ¼ r i;t 1 ¼ r i (i.e., r i is a simple time invariant individual effect) and y i;t is reduced to be stationary series with drift or with trend. So, Hadri s procedure tests the following hypothesis: H 0 : k ¼ r n ¼ 0; H 1 : k > 0: r The test is conducted by using the cross sectional average of individual KPSS Lagrange multiplier test statistics. Hadri (2000) shows that with additional assumptions, the normalized test statistic is distributed asymptotically as Nð0; 1Þ under the null. The results of IPS tests are listed in Table 3. We vary the lag length form zero (i.e., no lags) to four. As expected the results are sensitive to the selected lag length and inclusion/exclusion of the deterministic trend. In some cases the null is statistically rejected at conventional significance levels, 5% or 1%. However, overall, they seem to support the null hypothesis of panel nonstationarity. In order to check the possibility of Ið2Þ, we implement the IPS test on the first differenced series. These results are shown in the right-mostthree columns of Table 3. Except for only one case, the null hypothesis of non stationarity is strongly rejected. ð7þ ð8þ

10 930 S. Kano, M. Ohta Table 3. IPS t-bar test results (1) Without trend lag h t u t v t Dh t Du t Dv t 0 0:991 3:169 1:195 17:001 19:049 23:931 ð0:161þ ð0:001þ ð0:116þ ð0:000þ ð0:000þ ð0:000þ 1 1:767 3:115 5:847 18:765 13:093 18:419 ð0:039þ ð0:001þ ð0:000þ ð0:000þ ð0:000þ ð0:000þ 2 3:225 1:448 1:090 8:159 7:727 11:903 ð0:001þ ð0:074þ ð0:138þ ð0:000þ ð0:000þ ð0:000þ 3 1:348 2:632 1:867 7:012 7:885 11:772 ð0:089þ ð0:004þ ð0:031þ ð0:000þ ð0:000þ ð0:000þ 4 1:121 4:682 0:315 7:815 0:295 9:570 ð0:131þ ð0:000þ ð0:376þ ð0:239þ ð0:000þ ð0:273þ (2) With trend lag h t u t v t Dh t Du t Dv t 0 1:508 5:699 0:392 15:951 14:634 20:813 ð0:066þ ð0:000þ ð0:347þ ð0:000þ ð0:000þ ð0:000þ 1 5:743 0:990 16:201 16:778 8:471 12:641 ð0:000þ ð0:161þ ð0:000þ ð0:000þ ð0:000þ ð0:000þ 2 0:941 0:935 4:052 5:395 3:374 5:569 ð0:173þ ð0:175þ ð0:000þ ð0:000þ ð0:000þ ð0:000þ 3 0:894 2:268 1:949 3:859 3:764 6:463 ð0:186þ ð0:012þ ð0:026þ ð0:000þ ð0:000þ ð0:000þ 4 0:711 6:245 0:603 6:757 3:841 4:479 ð0:239þ ð0:000þ ð0:273þ ð0:000þ ð0:000þ ð0:000þ Note: H 0 : non-stationary. The normalized IPS t-bar statistic distributes as Nð0; 1Þ. p-values are in parentheses. Asterisk ( ) and double asterisk ( ) denotes that the hypothesis of non-stationarity is not rejected at 5% and 10% significance levels, respectively. Table 4. Hadri test results (1) Without trend h t u t v t Dh t Du t Dv t 14:471 22:355 16:850 9:831 2:016 3:188 ð0:000þ ð0:000þ ð0:000þ ð0:000þ ð0:022þ ð0:001þ (2) With trend h t u t v t Dh t Du t Dv t 19:770 8:296 12:518 0:854 6:404 1:753 ð0:000þ ð0:000þ ð0:000þ ð0:208þ ð0:000þ ð0:040þ Note: H 0 : stationary. The normalized Hadri test statistic distributes as Nð0; 1Þ. p-values are in parentheses. Asterisk ( ) and double asterisk ( ) denotes that the hypothesis of stationarity is rejected at 5% and 10% significance levels, respectively. The long-run variance is estimated by Andrews s (1991) automatic bandwidth selection procedure. The results of Hadri tests are shown in Table 4. Here we estimated the long-run covariance of n t in (7) and t in (8) by Andrews s (1991) non-parametric procedure to allow for heteroskedasticity and serial correlation. For all the variables and specifications, the null of stationarity is rejected strongly.

11 Long-run matching relationship in the Japanese labor market 931 Table 5. Allowance of heterogeneity of b and X and null hypothesis of cointegration tests/ estimators b X Null hypothesis Panel coint. tests Kao (1999) No No No cointegration Pedroni (1999) Yes Yes No cointegration McCoskey and Kao (1998) Yes Yes cointegration Panel coint. estimator Kao and Chiang (2000) No Yes - Note: Yes means that the test (or estimation) procedure in question assumes that a cointegration vector b (or long-run covariance matrix X) is heterogeneous among individual units. While No means that b (or X) is homogeneous. Kao and Chiang (2000) include heterogeneous FM and heterogeneous DOLS estimators. We also implement the Hadri test on the first differenced series. Contrary to IPS results on the first differenced data, these results are sensitive to specifications. Our results of the panel unit root tests seem sensitive to specifications of the tests, and it is difficult to judge which specification could be closer to the true data generating process of each series. However, at least, we cannot deny the possibility that h t, u t, and v t are characterized as Ið1Þ process. Therefore we conclude that it is worthwhile investigating a panel cointegration relationship among these variables of the matching function Panel cointegration tests Several authors have recently proposed alternative procedures for panel cointegration tests. In order to ensure robustness of results, we employ different types of residual-based tests developed by Kao (1999), Pedroni (1999), and McCoskey and Kao (1998). The former two procedures test the null hypothesis of no cointegration, while the latter one tests the null of cointegration. Therefore, if the former two reject their null hypothesis and the latter one accepts its null hypothesis, it implies a strong statistical evidence of longrun matching relationship. An appealing property shared by all of these tests is that after appropriate first and second moments adjustments, they distribute asymptotically as N ð0; 1Þ under the null hypothesis; in other words, these distributions are free from nuisance parameters. Since assumptions on which these tests are based are quite different from each other, we summarise these differences in Table 5. The panel cointegration tests proposed by Pedroni (1999, Panel m, Panel q, Panel t, Group q, and Group t) and by McCoskey and Kao (1998, LM) permit not only heterogeneity of long-run covariance matrices but also of slope coefficients among individual units. So b in Eq. (3) should be replaced by b i, for i ¼ 1; 2;...; N, in the cases of these tests. Indeed, like IPS and Hadri tests of panel unit roots, Pedroni and McCoskey and Kao test statistics consist of individual units residual-based cointegration tests. On the other hand, the tests by Kao (1999, DF q,df t,df q, and DF t ) are conducted by direct use of the pooled residual

12 932 S. Kano, M. Ohta Table 6. Panel cointegration test results (1) DF q DF t DF q DF t 4:15 3:728 11:516 7:115 ð0:000þ ð0:000þ ð0:000þ ð0:000þ (2) without trend Panel m Panel q Panel t Group q Group t 147:826 17:059 16:294 16:161 22:156 ð0:000þ ð0:000þ ð0:000þ ð0:000þ (0.000) with trend Panel m Panel q Panel t Group q Group t 171:728 13:715 19:627 11:160 24:145 ð0:000þ ð0:000þ ð0:000þ ð0:000þ (0.000) (3) LM 0:385 ð0:350þ Note: p-values are in parentheses. Statistics in block (1), (2), and (3) are given by Kao (1999), Pedroni (1999), and McCoskey and Kao (1998), respectively. DOLS residuals of individual units are used for (2) and (3). All the long-run covariance matrices are estimated by Andrews s (1991) procedure except for DF q and DF t (DF q and DF t do not use the long-run covariance matrices). As for (1) and (2), H 0 : no cointegration. As for (3), H 0 : cointegration. of Eq. (3) based on the assumption of the homogeneous coefficients and longrun covariance matrix. The results are reported in Table 6. 4 All of the Kao and Hadri tests reject the null hypothesis of no cointegration, and the McCoskey and Kao test accepts the null of cointegration. It should be stressed that the cointegration relationship is supported statistically by the tests with various types of assumptions, i.e., degree of heterogeneity of coefficients and of long-run covariance matrices. These results support the existence of a long-run relationship among new hiring, unemployment, and vacancies, which is required by the matching function in the equilibrium unemployment theory Panel cointegration estimation Table 7 presents the estimation results of the cointegration equation (3) or its dynamic OLS representation (5). FM and DOLS in the table denote heterogeneous fully-modified OLS and heterogeneous Dynamic OLS, respectively. We set the lag ¼ 3 and lead ¼ 2 for the DOLS. (We tried other lags and leads, but our estimates are not sensitive to the selected lag and lead length.) Their t-values are in parentheses. For comparison, conventional OLS estimates without bias correction, which are equivalent to the within estimates reported by Kano and Ohta (2002), are also listed in the table. 5 Notice, however, that OLS regression in the non-stationary case is just spurious in that conventional t-values are meaningless. 4 The long-run variance is estimated by Andrews s (1991) procedure, i.e., nonparametric estimation with a quadratic spectral kernel and automatic bandwidth selection. 5 Actual regressions include time effects.

13 Long-run matching relationship in the Japanese labor market 933 Table 7. Panel cointegration estimation OLS FM DOLS Unemployment 0:560 0:563 0:658 ð20:313þ ð16:735þ ð15:800þ Bias (%) 0:543 14:910 Vacancies 0:302 0:221 0:270 ð12:398þ ð6:568þ ð6:483þ Bias (%) - 36:581 11:770 RTS 0:862 0:784 0:929 Wald test: 10: :024 8:797 constant RTS (p-value) ð0:001þ ð0:000þ ð0:003þ R 2 0:968 0:942 0:929 Note: t-values are in parentheses. The long-run covariance matrix is estimated by Andrews s (1991) method.for DOLS, we set lag ¼ 3 and lead ¼ 2.The reported bias is calculated by ð1 b ols =b j Þ100 ð%þ, where b j is replaced by each estimator. Estimated coefficients of unemployment and vacancies, ^b u and ^b v, exhibit correct signs and are statistically significant for all the cases. However, the values of estimates by each estimation method differ from each other. Compared with the conventional OLS estimation, the heterogeneous FM remarkably reduces the value of ^b v, approximately 37%. In the heterogeneous DOLS, ^b u rises by approximately 16%, while ^b v falls by approximately 12%. The Monte Carlo experiment implemented by Kao and Chiang, 2000) shows that the finite sample performance of DOLS greatly dominates those of alternatives. 6 Specifically, even in the case with shorter time period than in ours such as T ¼ 20, the deviation of DOLS estimate from the true parameter value is surprisingly small. Therefore, following the estimation result of DOLS, the OLS estimate of the hiring elasticity with respect to unemployment seems under-estimated, while that with respect to vacancies seems overestimated. The returns to scale of the matching function, RTS ¼ b u þ b v, is important in the equilibrium unemployment theory, since increasing RTS is consistent with multiple, Pareto-rankable natural rates of unemployment. For example, in the model of pure exchange economy with search frictions, Diamond (1982) shows that if the matching function has regions of increasing returns, there exists optimistic and pessimistic equilibrium trading levels depending upon agents expectations. In addition, constant RTS is often assumed for convenience. So testing constant RTS has been of particular interest. We test the assumption H 0 : RTS ¼ 1 against H 1 : RTS < 1byKao and Chiang s (2000) Wald test for linear restrictions on the heterogeneous FM and heterogeneous DOLS estimator. See the test results in Table 7. It is found that constant RTS is rejected for all cases, supporting decreasing RTS. 6 Note that both FM and DOLS have the same asymptotic normal distribution. Kao and Chiang (2000) discuss that resulting poorer finite sample performance of the FM could be attributed to the failure of its non-parametric data transformation based on the estimated long-run covariance matrix.

14 934 S. Kano, M. Ohta This empirical finding of the decreasing RTS allows us not to consider the case of Diamond (1982). For comparison, we estimate the cointegration equation using aggregated data by pure time series DOLS of Saikkonen (1991) and Stock and Watson (1993). The result is as follows: h A t ¼ 35:822 ð 1:531Þ R 2 ¼ 0:915; þ 0:693 ð6:062þ u A t þ 0:230 v A t ; ð5:698þ where h A t ¼ P N i¼1 h i;t, u A t ¼ P N i¼1 u i;t, v A t ¼ P N i¼1 v i;t, and three lag and two lead terms are suppressed for simplicity. The result of Wald test for the constant RTS is with p-value = 0.379, so the constant RTS hypothesis is not rejected although estimated returns to scale is less than unity (0.923). Interestingly, this result is inconsistent with that of the above panel cointegration analysis. This may imply that some information contained in the cross sectional dimension of the panel data is lost in the pure time series cointegration analysis. This tells us the importance of employing panel data. 5. Comparison with estimates of other studies Recently, the estimation of the matching function have shifted its emphasis toward utilizing more disaggregated data. We list the selected results of previous studies using regional panels, namely, Burgess and Profit (2001), van Ours (1995), and Boeri and Burda (1996), all of which are about European countries. 7 The main purpose of estimating the matching function is to test the empirical foundation of the equilibrium unemployment theory; whether statistically significant relationship among new hiring, unemployment, and vacancies exists or not, and theoretically imposed assumptions are met or not. Especially returns to scale of the matching function is of particular interest, since, as mentioned earlier, increasing returns to scale suggests the possibility of sun spot equilibria (see, for example Diamond 1982). In addition, it is often the case that constant returns to scale (CRS) is assumed so that unemployment or vacancy hazard rate depends only on the vacancy-unemployment ratio, not on their sizes (see, for example Pissarides 2000). To study the above problems, we apply panel cointegration approach to the matching function, which has not been used previously in this field. We find that these variables are cointegrated and estimates exhibit theoretically expected signs, although the CRS assumption is statistically rejected. In Burgess and Profit (2001) for Britain, ^b v seems to be extremely small though all the estimates exhibit correct signs and are significant. The CRS assumption is also rejected. In Boeri and Burda (1996) for Czech Republic, ^b v is not significant, which means that new hiring is inelastic to the variation of 7 Of course, each author tries to estimate with several different specifications. We do not list all these results in the text except for the selected ones. See their papers for further details. Anderson and Burgess, (2000) also estimate using US data, but we do not list it because they involve industrial panel.

15 Long-run matching relationship in the Japanese labor market 935 vacancies. Only van Ours s (1995) estimates for Netherlands seem to support the CRS assumption, though he does not test it formally. We find the long-run relationship among new hirings, unemployment, and vacancies for the case of Japan. However, an international comparison made in this section suggests that the relationship among these variables, on average, may not be so stable in the following sense: (1) The estimates vary among countries to a large extent, and there is a case being unable to reject the null of no effect of vacancies on new hirings; (2) The CRS does not seem to be a fully empirically supported assumption. 6. Concluding remarks This paper investigates non-stationarity problems in estimating and testing the Japanese job matching function. For this purpose, we use the panel unit root tests and panel cointegration analysis. Our methodologically original point is the application of recently developed econometric techniques for analyzing non-stationary panel data to the empirical study of the matching function. Our main findings are as follows. First, new hiring, unemployment, and vacancies are Ið1Þ processes. Second, they are cointegrated, in other words, there exists a long-run relationship corresponding to the matching function in the theory of equilibrium unemployment. Third, conventional within estimates of hiring elasticity with respect to unemployment and vacancies have non-negligible biases. Finally, decreasing returns to scale of the matching function is statistically supported. Among the implications of this paper are the following two: empirical one and theoretical one. Empirically, the time average of each prefectural residuals of the estimated matching function measures the difference in the prefectural matching efficiencies. Now these residuals are found to be stationary in our study. This implies that these residuals do not involve non-stationary Table 8. Previous studies results (1) (2) (3) Country Britain Netherlands Czech Republic ^b u / 0:33 a ^b v (84.0) (9.6) / ð2:5þ a (11.633) / 0:72 a (13.5) (9.5) / ð9:0þ a (1.794) H 0 : CRS rejected n.a. n.a. other variables Spillover effect n.a. ALMP b method LSDV Non-liner LS IV Coverage 85:Sept. to 95:Dec. 81,83,85 92:Q1 to 93:Q4 303 travel-to-work areas 8 regions 72 districts Note: Estimates in column (1), (2), and (3) are given by Burgess and Profit (2001), van Ours (1995), and Boeri and Burda (1996), respectively. t-values are in parentheses. For studies not reporting t-values, they are computed by reported standard errors. a results for employed job seekers. b ALMP means active labor market policy measures.

16 936 S. Kano, M. Ohta omitted variables. This, in turn, means that our residuals will serve as reliable measures of the differences of the matching efficiencies across prefectures. So, as a possible extension of this paper, if data of labor market policies (e.g., prefectural job-training expenditures) are available, we will be able to estimate the effect of these policies on the matching efficiency by regressing these residuals on the data of these policies (Boeri and Burda 1996, 1996, by extending to the non-stationary panel analysis). It is important for the theoretical analysis of the policy effect whether there is a unique equilibrium or not. If there are multiple equilibria, it will be difficult to assess the policy effect on the equilibrium. We found that the Japanese matching function exhibits decreasing returns to scale. This implies a unique equilibrium Diamond (1982). So we have only to consider one equilibrium when we analyze the policy effect theoretically based on the equilibrium unemployment models. A technical extension of this paper will be to apply a likelihood-based, vector autoregressive approach for panel cointegration analysis by Larsson et al. (2001), which is a panel version of Johansen s (1988) procedure. Our paper has assumed implicitly that there exists a unique cointegration relationship, but does not explicitly confirm this assumption. In the likelihood based approach, on the other hand, the number of cointegration relationships is itself of interest, and dealt with as one of the central topics to be inferred. Finally, this paper leaves an important problem unsolved, i.e., the treatment of structural breaks in a panel cointegration relationship. We suggest only that pure time series version of cointegration-structural break tests, e.g., Gregory et al. (1996), might be applicable to the panel data with appropriate arrangements. References Anderson PM, Burgess SM (2000) Empirical matching functions: Estimation and interpretation using state-level data. Review of Economics and Statistics 82: Andrews DWK (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59: Baltagi BH (2001) Econometric Analysis of Panel Data, 2nd ed. Wiley, Chichester, England Blanchard OJ, Diamond P (1989) The beveridge curve. Brookings Papers on Economic Activity 1:1 76 Boeri T, Burda M (1996) Active labor market policies, job matching and the czech miracle. Europian Economic Review 40: Burgess S, Profit S (2001) Externalities in the matching of workers and firms in britain. Labour Economics 8: Diamond P (1982) Aggregate demand management in search equilibrium. Journal of Political Economy 90: Engle RF, Granger CWJ (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55: Genda Y, Rebick M (2000) Japanese labour market in the 1990s: Stability and stagnation. Oxford Review of Economic Policy 16: Gregory AW, Nason JM, Watt DG (1996) Testing for structural breaks in cointegrated relationships. Journal of Econometrics 71: Gross DM (1997) Aggregate job matching and returns to scale in germany. Economics Letters 56: Hadri K (2000) Testing for stationarity in heterogeneous panel data. Econometrics Journal 3:

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