Measuring local employment multipliers and informal employment: a stochastic frontier approach

ABSTRACT This paper offers policymakers a novel tool for calculating employment multipliers. A theoretical model incorporating a non-tradeable employment function is combined with a stochastic frontier methodology to estimate an accurate multiplier. The advantage of this model is that it allows a consideration of unobserved informal employment when estimating the multiplier. We find an employment multiplier effect of 1.2 jobs in the non-tradeable sector for one job in the tradeable sector. Also, the greater the number of skilled consumers, the higher the multiplier indices and the lower the level of informal employment. Moreover, specialised sectors requiring skilled workers also present less informal employment. We use provincial data for Spain over the period 1995–2013.


INTRODUCTION
The EU funds various programmes such as cohesion policies to foment the economy of specific areas, industries or sectors, expecting positive employment outcomes.Likewise, states and local governments dedicate substantial taxpayer monies to fiscal incentives channelled towards business and job creation.Invariably the latter incentives are directed towards tradeable goods producing sectors that supposedly 'provide greater economic benefits' (Bartik, 2003(Bartik, , 2004)).However, according to Moretti (2010), the effects of these policies and their actual effects in terms of employment are not always fully understood, because there is little systematic evidence on the effects of successfully attracting a new firm on other parts of the local economy. 1Following this seminal paper, some progress has been made on the econometric estimation of multipliers, opening new research fields with respect to other variables, impacts or empirical techniques.Multipliers can be harnessed to obtain insights into public expenditure transmission mechanisms and how these transform into additional gains for permanent employment, a real concern for citizens, public authorities and governments (e.g., see Faggio and Overman, 2014;and Faggio, 2019, for the UK).The availability of more disaggregated spatial databases together with more sophisticated econometric models has to some extent redeemed past criticisms dismissing multipliers as a viable option in the long-established regional economics literature.
However, in the study of employment multipliers, one must bear in mind that when reporting the number of workers, it is not possible to account for informal employment, understanding this as being those jobs outside the regulatory framework because they are not subject to labour legislation, social protection, taxes or employment benefits (Jütting et al., 2007).Officially, according to the International Labour Organization (ILO) (2003), informal employment can be defined as the total number of informal jobs, whether carried out in formal sector enterprises, informal sector enterprises, or households. 2 This type of employment is not only important in developing countries but also is a relevant issue in developed countries (Julià et al., 2014).Moreover, Jütting et al. (2007) also highlight some economic sectors as being especially prone to informal employment relations (e.g., domestic servants, construction workers and non-tradeable jobs in general).In this vein, we postulate the potential existence of informal jobs in the Spanish non-tradeable sector.
Thus, given that the objective of this study is to explain the creation of non-tradeable jobs via the employment multiplier, it is necessary to allow for the possibility that the more traditional or conventional method for estimating employment multipliers may, in fact, be biased due to the existence of non-tradeable informal employment, since this is not observable.That is, non-tradeable jobs may be systematically subject to underreporting, and this must be considered in the model in order to obtain an accurate measure of the 'true' employment multiplier.
To tackle this issue, we propose a stochastic frontier analysis (SFA) approach, traditionally applied in the production economic literature (e.g., Kumbhakar & Lovell, 2000).The SFA approach can be used to control for the existence of non-tradeable informal employment because as it is not observed by econometricians, reported non-tradeable employment will always be lower than the total real number of workers.Therefore, the informal or unobserved workers can be proxied using a one-sided random term in the same fashion as firm inefficiency in production economics.In this way, our estimate of the non-tradeable employment function is one that includes both observed and unobserved jobs, which represents a step forward in the literature on employment multipliers.To the best of our knowledge, the empirical approach used here is the first to use an SFA-based application for employment multipliers.As such, we consider this to be our main contribution to the literature on the subject.
For this purpose, within a suitable theoretical framework, we present an employment multiplier function to analyse how an increase in the number of tradeable jobs via a demand shock implies additional income generated by these workers and, consequently, an increase in demand for non-tradeable goods, leading to a multiplier effect.To do this, we extend the model of Guccione and Gillen (1980) and assume that skilled consumers may possess different preferences to unskilled ones.Thus, for example, they may make more use of other cultural (museums, exhibitions) or health services (Raghupathi & Raghupathi, 2020) that can create additional local jobs.Therefore, we assume that consumers may display different consumption habits coinciding with their academic and cultural background.The proposed theoretical model concludes that these differences can have repercussions on the demand of the non-tradeable sector and, therefore, on employment multipliers.Therefore, our hypothesis is that places with more skilled consumers will present a greater demand for local services and will therefore generate higher employment multipliers.

LITERATURE REVIEW
A major part of studies on employment multipliers use as their starting point the work of Moretti (2010), which estimates employment multipliers for US cities.Consequently, research, whether methodological or empirical in the form of replicas or critiques of Moretti, has focused for the most part on the United States (Gerolimetto & Magrini, 2016;van Dijk, 2017van Dijk, , 2018;;Bartik & Sotherland, 2019;Osman & Kemeny, 2022).The latter two studies are of particular interest.Bartik and Sotherland (2019) calculate US employment multipliers based on different demand shocks and offer in tandem an excellent review/critique of the related literature.They estimate new values for US employment multipliers, one quarter lower than those typically estimated in other studies and tending to fall in a range closer to 1.5 than to 2.0.Moreover, their study confirms higher local employment multipliers for high-tech industries, albeit lower than those estimated in other papers as well as similar employment multipliers across geographical areas (the latter differ in other studies depending on whether the area considered is at local labour market, county or state level).The recent work of Osman and Kemeny (2022) corroborates in part the findings of Bartik and Sotherland (2019) in terms of smaller employment multiplier estimates for the United States.Likewise, they suggest that results can differ from previous studies due to a reliance on arbitrary periods of observation, the limiting of samples to more populous regions, the common use of relatively aggregated industrial categories and some methodological issues.The authors also offer a useful synopsis of multiplier values for US and non-US employment multipliers.
A handful of studies analyse the topic for Europe: Frocrain and Giraud (2019) and Malgouyres (2017) analyse employment multipliers in France, De Blasio and Menon (2011), Auricchio (2015) and Cerqua and Pellegrini (2020) explore employment multipliers in Italy.Faggio and Overman (2014) and Faggio (2019) examine public employment multipliers for the UK; Senftleben-König (2014) and Becker et al. (2021) for Germany; and Auricchio et al. (2020) for Italy.Moretti and Thulin (2013) undertake a comparative study of local multipliers and capital for the United States and Sweden.
However, all the previous literature does not consider the existence of informal employment, a factor that could, in part, explain the different results found in the literature.As already mentioned, this challenge as the main objective of the present paper is addressed below.To do this, we use data for Spain over the period 1995-2013.In this way, a broad period is considered spanning not only the pre-financial crisis period but also the immediate aftermath.Moreover, Spain proves to be an interesting case study given that, in line with other European member states such as France (Frocrain & Giraud, 2019), it has evolved into an increasingly service-orientated economy.Hence, the present study's primary focus on the creation of non-tradeable employment.

THE THEORETICAL MODEL
and Gillen (1980), facilitated the development of our own theoretical framework.
Our model uses a general equilibrium framework for a regional economy, based on the Rosen-Roback spatial general equilibrium model (Moretti, 2011;Roback, 1982;Rosen, 1979)  With the entire economy divided into tradeable and non-tradeable activities, employment can accordingly be assigned to either sector.In the remaining discussion, for notational purposes, the aforementioned relationship will be denominated as follows: where N is total employment; with N T and N NT tradeable (T) being employment and non-tradeable (NT) employment, respectively.The model assumes that each area is a competitive economy using the factor labour and capital to produce both a nationally tradeable exogenously priced good and a non-tradeable locally priced good.Each area uses the factors labour and capital to produce both nationally tradeable goods, X T with a price P T , and non-tradeable goods, X NT with a price P NT .Labour is considered mobile across sectors within an area signifying that marginal products and wages are equalised locally across sectors in the long run.Using the foregoing assumptions, we further develop the theoretical framework supporting our employment multiplier model.Profit maximisation in both sectors, under competitive conditions and with labour and capital inputs yields the following equations: where π is profit, X NT (N NT , K NT ), X T (N T , K T ), are production functions for tradeable and non-tradeable output, respectively, with K T and K NT representing each sector's capital stock; W are wages; and r is the capital price.
From the first order conditions we have: From (3) we obtain: Rearranging N NT in equation ( 5) we get the labour demand factor: Substituting into the non-tradeable production function we obtain a supply function for the non-tradeable sector dependent on the real wage and capital: Furthermore, we assume that demand in the non-tradeable sector (D NT ) is a function of relative prices P NT /P T , the real local income for which is approximated by the real income (WN /P NT ); and consumer preferences.
We extend the model of Guccione and Gillen (1980) to additionally assume that skilled consumers may possess different preferences to unskilled ones.Thus, for example, they may make more use of other educational services (museums, exhibitions) that can create additional local jobs.Literature also exists that finds that the higher the level of education, the higher the demand for health services (e.g., Raghupathi & Raghupathi, 2020).Hence, we also take into account the percentage of skilled individuals (Sk) in the demand function: In this way, by ( 7) and ( 8) general equilibrium yields: From equations ( 1), ( 3), ( 4) and ( 5) we obtain N NT , a relationship between the non-tradeable workers and tradeable employment, capital and the percentage of skilled individuals as follows: 4 where: Finally, from (10) we can define the employment multiplier for the non-tradeable sector as follows: Expressed in words, M is the employment multiplier for the non-tradeable sector which indicates how employment in the non-tradeable sector varies with changes in tradeable employment, ceteris paribus the level of capital and the degree of education of the population.As already explained, here the mechanism of the multiplier is based upon the idea that an increase in the tradeable employment sector via a demand shock will positively affect employment in the non-tradeable sector.However, as already pointed out, when empirically approaching the theoretical model of the proposed employment multiplier, it is necessary to consider a potential error in its measurement due to the existence of informal employment.The informal sector does not appear in this theoretical framework because the set of traditional assumptions (e.g., perfect competitive markets, completely informed agents, etc.) used to develop the general equilibrium model does not contemplate or allow for the existence of such an informal sector.Therefore, the proposed SFA specification of the model proposed below controls not only for measuring errors, but also for the lack of fulfilment of some of the theoretical assumptions.In the next section we address this problem and explain the methodology that underpins our novel proposal and contribution to the literature on employment multipliers.

EMPIRICAL MODEL
Under the assumptions of the theoretical model presented above, non-tradeable employment is modelled as a function of tradeable employment, capital and the share of the skilled population (equation 10).However, due to the potential existence of non-tradeable informal employment, the dependent variable in equation ( 10) will be measured with error, due to the existence of non-tradeable workers without any written contracts and, therefore, unaccounted for in the official statistics.That is, the dependent variable may be systematically under-reporting, and this must be taken into account in the estimates to obtain an accurate measure of the true employment multiplier.That is, the researcher only observes N NT o (observed non-tradeable employment) that will be equal to N NT * (real non-tradeable employment) if there is no informal employment, with N NT o proving lower than N NT * if informal employment exists in the non-tradeable sector.That is, for equation (10): That is, in a standard model of the employment multiplier, defined according to equation (10), the dependent variable will be measured with error in the presence of non-tradeable informal employment, since the observed number of non-tradeable jobs will be lower than the real one (which considers both, legal and illegal employment).
The problem appears because although the classical measurement error (random) reduces precision but does not bias estimates, when the outcome is systematically over-or underreported, it implies an error term that is skewed with a non-zero mean (i.e., non-random), that bias the estimates and leads to erroneous conclusions (Millimet & Parmeter, 2022).As these authors point out, a solution to this issue is to model the error term assuming it stems from a particular parametric distribution as is traditionally done in the SFA approach (Aigner et al., 1977).The main characteristic of this approach is that the random error is divided into two components: v is the random error term that represents random events, and the term u which is non-symmetric.This methodology has been widely applied in the production and efficiency economics literature, but to our knowledge it has not yet been developed for the analysis of employment multipliers.In this paper we propose the use of this model.Thus, to convert inequality (12) to equality and taking logs we obtain: where i indicates province; and t is time.The expression is the stochastic function frontier that represents the real non-tradeable employment (N NT * it ) level that province i may achieve with a given level of tradeable employment, ceteris paribus.We expect the data generating processes behind both random terms u and v to be quite different: v it ≈ iid N + (0, s 2 vit ) is a standard random disturbance term; and u it ≈ iid N + (0, s 2 uit ) is a non-negative random term that allows the possibility of informal employment.Thus, the model allows us to differentiate between random differences between observed (N NT o it ) and real non-tradeable (N NT * it ) employment (captured by v it ) and one-sided differences between both concepts due to the existence of informal employment (captured by u it ).
Linearising and adding time and regional dummy variables 5 in equation ( 13) we have: where β is a vector of parameters to be estimated; D AC are autonomous community dummies; and D T are time dummies.Finally, once equation ( 14) has been estimated, for equation (11) we can define an unbiased employment multiplier (M*) from the SFA model proposed as follows: Note that by equation (13): (16) Finally, the gap between N NT o it (observed non-tradeable workers) and N NT * it (real non-tradeable workers) can be measured by rearranging equation ( 13), and for (16) we Measuring local employment multipliers and informal employment: a stochastic frontier approach 81 REGIONAL STUDIES have: where EGI is defined as the employment gap index for province i in year t.This index takes values between 0 and 1 (given that u it is non-negative) and indicates the difference between real employment (which includes formal and informal employment) and observed employment (without accounting for informal employment).The greater the number of illegal workers, the lower the value of the EGI, and vice versa.

Modelling error terms
Applying an SFA approach makes it also possible to include potential heteroskedasticity in the skewed error of the estimation.This is important if we consider that although informal employment is present amidst all the social strata involving persons with different personal characteristics, those who are forced into these extreme situations are usually low-income workers, with little preparation or non-existent qualifications; that is, circumstances in which the worker feels more vulnerable when it comes to gaining access to a work contract.This will imply heteroscedasticity in u it .Caudill and Ford (1993), Caudill et al. (1995) and Hadri (1999) proposed to parameterise the variance of the pre-truncated u distribution in the following way: where we model the variance of u it as a linear function of a set of covariates h that can explain the differences between real and observed employment, with ϕ being the set of parameters to be estimated.Vector h includes variables relating to skilled individuals.We expect areas with a high percentage of skilled workers to reflect smaller differences between real and observed employment, that is, a higher value of the EGI, and vice versa.Similarly, we assume that the random error component (v it ) is heteroskedastic with its variance depending on the same variables used in equation ( 18): where 4 is a set of parameters to be estimated.

Endogeneity issues
The SFA model proposed above will be estimated using maximum likelihood (MLE) methodology.The consistency of the estimator depends on the exogeneity of the explanatory variables.However, prevalent in employment multiplier literature is the presence of endogeneity problems due to the potential simultaneity between tradeable and non-tradeable employment (here N T and N NT ).Consequently, we have to tackle the potential endogeneity of the N T variable.
In the context of endogeneity problems related to stochastic frontier models, the literature is relatively recent.Managing endogeneity in the aforementioned models is more complex than for standard regression models, given the special definition granted to the error term (Karakaplan & Levent, 2017). 6Amsler et al. (2016) suggest an approach using a two-step procedure.This procedure departs from equation ( 14) where we assume that cov (N T it , v it ) = 0. Thus, as a first step, we regress N T using some instruments z it considered exogenous in the sense that E(v it |z it ) ¼ 0. We can now think in terms of a reduced form for the endogenous variable, which we write in matrix form as: where Z is a vector that includes the other exogeneous variables used in the model, and where h is uncorrelated with Z. Then endogeneity of N T corresponds to cov(hv) = 0. Estimating (20) by ordinary least squares (OLS), yields: The second step is to estimate the equation ( 14) by inserting the standardised 7 residual ĥ * it in the model as follows:

DATABASE AND VARIABLES
To estimate the employment multiplier model, the study uses NUTS-3 data covering 50 Spanish provinces provided by the Spanish Regional Accounts, which is elaborated by the Instituto Nacional de Estadística (INE -National Institute of Statistics).Use has also been made (Capital Stock and Education variables) of the database of the Instituto Valenciano de Investigaciones Económicas (IVIE).
The database is broken down into seven major specialisation sectors: agriculture, manufacturing, industry, other industry, construction, services and non-market services.In order to benefit from a longer time series, the two Spanish regional accounting databases were linked in order to express provincial employment in persons as appearing in the Spanish National Accounts, base 2010 (Contabilidad Nacional de España). 9 We use a balanced homogeneous panel dataset comprising 950 observations spanning the period 1995-2013.The database (in persons) for 50 Spanish provinces was divided into non-tradeable (N NT ) and tradeable employment (N T ), our principal variables for estimating the employment multiplier.For this purpose, use is made of the location quotient (LQ), 10 one of the principal approaches advocated by economic base theory (North, 1955(North, , 1956;;Tiebout, 1956aTiebout, , 1956b)).The LQ has various applications in regional economics, multiplier analysis 11  and the construction of national input-output tables (e.g., Flegg & Tohmo, 2016;Kowalewksi, 2015;Lehtonen & Tykkyläinen, 2014).
Another principal variable is K representing the net capital stock per province (in thousands of constant euros, base 2010) from the IVIE database. 12Finally, the variable Sk is the percentage of the total population with university studies. 13 The availability of NUTS-3 Spanish labour market data (number of jobs in persons) justifies the choice of an LQ-based measurement.The coherence of our classification of non-tradeable and tradeable employment was analysed for each province with the authors basing their criteria on (1) an understanding of the underlying Spanish provincial production structure, and (2) the idea inherent in the methodology of Jensen and Kletzer (2005) and Faggio and Overman (2014) (see Appendix A in the supplemental data online for the breakdown of tradeable and non-tradeable employment by sectors and regions).
As already explained, equation ( 22) is estimated taking into account the specifications of the random terms provided by equations ( 18) and ( 19), which analyse those determinants that can affect informal non-tradeable employment.Our hypothesis is based upon the idea that provinces with a highly educated population (Sk) could have less non-tradeable informal employment.Similarly, a high degree of specialisation in skilled jobs may explain lower levels of illegal employment.Because of this, equations ( 18) and ( 19) include Sk and the cross between the coefficient of specialisation (COS) 14 and the percentage of workers with university education (N_Sk), that is, (COS × N_Sk).Finally, we take time into account (via the use of dummy variables).Descriptive statistics are shown in Table 1.
A graphical representation of tradeable and non-tradeable employment for Spain is shown in Figure 1.The latter identifies Spain's production structure and the abundance of non-tradeable employment compared with tradeable employment.Both series show considerable growth until 2008, more so for non-tradeable employment, but fall dramatically thereafter spurred by the crisis.

RESULTS
The SFA model based on the estimation of equations ( 18), ( 19) and ( 22) was estimated using the Stata 16 programme.First the Wald test, which contrasts the validity of the frontier model against an average function model, is highly significant, indicating that omitting the asymmetric error term might bias the results. 15As already explained in section 4.2, we have applied the model based on a two-step procedure approach suggested by Amsler et al. (2016) in order to tackle the potential endogeneity of the model.In this way, the residual calculated according to equation ( 21) has been included at a second stage in equation ( 22).As instruments (vector Z ), the exogenous variables in equation ( 22) have been used, also including efficiency determinants and time and regional dummies.Table 2 shows the results obtained.The coefficient of this residual proves significant, which confirms the endogeneity of the variable T and the need to carry out the adjustment made.
Specifically, Table 2 indicates the frontier results, with the tradeable employment coefficient, indicating the expected positive relationship with non-tradeable employment.The positive coefficient of the capital variable also yields the expected sign.Greater capital stock is associated with higher levels of non-tradeable employment.Likewise, the percentage of the population possessing university education (Sk) which we can proxy as the skilled portion of the population, has a positive impact on nontradeable employment.This indicates that, according to our initial hypothesis, a more educated population implies a greater demand for non-tradeable services and, therefore, increases potential non-tradeable employment, thereby displacing the frontier.This result is in line with other studies that find evidence that technical industries and skilled groups produce larger employment multipliers (e.g., Goos et al., 2018;Moretti, 2010;Moretti & Thulin, 2013;van Dijk, 2018;Lee & Clarke, 2019).
where s is the sectorial specialisation by province, N si is provincial employment in sector s, N i is total provincial employment, N s is national employment in sector s, and N is total national employment.It takes values between 0 (minimum specialisation) and 1 (maximum specialisation).
Measuring local employment multipliers and informal employment: a stochastic frontier approach 83 REGIONAL STUDIES

The local employment multiplier
Once equation ( 22) has been estimated and applying equation ( 15), the value of the employment multiplier at the sample mean is as follows: That is, at the sample mean this result suggests an employment multiplier effect of 1.2 jobs in the non-tradeable sector as a result of the creation of 1.0 job in the tradeable sector.
In addition, it is possible to analyse the time evolution of the multiplier as shown in Figure 2, where a clear positive progression is observed throughout the period.This positive evolution could potentially be explained by any technological advances occurring during the period analysed.Said advances usually introduce new tasks that create new jobs (e.g., application programmers, digital marketing managers, cybersecurity experts, data scientists or digital privacy lawyers).Thus, for example, Acemoglu and Restrepo (2018) analyse the importance of new tasks in employment growth using data from the US labour market, finding that approximately half of the total employment growth in the United States between 1980 and 2007 can be explained by the creation of new types of tasks related to new technologies.These new jobs created by technology industries increase the demand for services in the local economy and therefore create indirect jobs, that is, increase the multiplier effect.Likewise, Goos et al. (2018) estimate that for Europe every job in the high-tech industry creates five additional low-tech jobs in the region where the industry is located.In a comparative study of the United States and Sweden, Moretti and Thulin (2013) show that the multiplier effect is significantly larger for tradeable jobs with high levels of human capital and for high-tech industries.Moretti (2010) finds similar results for the US economy.All this suggests the importance of new technologies (such as the digital economy) for determining the size of the local multiplier.

Average regional multipliers
Although our database is at the provincial level, in order to simplify the analysis and for comparison purposes, the present study centres on average regional multipliers (by autonomous community). 16The average regional multiplier values and their evolution over time are shown in Table 3.For a clearer presentation of the results, we have divided the sample into three time periods of six years each.
In general terms, an improvement in the multiplier is observed throughout the period considered.Asturias and Cantabria show the best values for the employment multiplier, while Extremadura reports the worst values.Thus, for example, for each tradeable job, Extremadura employs 0.68 non-tradeable workers while Asturias hires 3.5.The multiplier is highest for several regions, mainly those situated in northern Spain (such as Asturias, Cantabria and Galicia) or the regions of Murcia and Valencia, object of industrial restructuring in the 1980s.During that decade, numerous reconversion policies targeted primarily to the secondary sector led to significant job losses.To mitigate these negative effects, a series of fiscal and labour policy measures was adopted.The latter meant that, despite a destruction of tradeable employment, this did not result in a proportional destruction of non-tradeable employment.This may explain the high values recorded for the multiplier in the aforementioned regions.
In terms of a European comparison, we review those studies that consider the impact of demand shocks for the tradeable and non-tradeable sectors.Malgouyres (2017), assuming only manufacturing jobs as tradeable, calculates a local multiplier of 1.46 for France over the period 1995-2007.For a similar period to our own paper (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015), Frocrain and Giraud (2019) estimate an employment multiplier of 0.80, that is, for every 100 tradeable jobs created, 80 additional non-tradeable jobs were created in the same French employment area.For the Italian case, De Blasio and Menon (2011) research employment multipliers for both northern and southern Italy in both the tradeable and non-tradeable sectors based on an exogenous shift in the tradeable sector finding no evidence of positive spillovers from the tradeable sector to the non-tradeable sector, that is, no multiplier effect.Likewise, Auricchio (2015) quantifies the effect of a local labour demand shock in the tradeable sector on the employment in the non-tradeable sector for Italy for 1991, 2001 and 2011, finding similarly that on average the effect of said exogenous shift in local tradeable employment upon non-tradeable employment is zero.However, in line with Moretti (2010) and Moretti and Thulin (2013), using Eurostat data for technology in the manufacturing sector, the study reveals that high-tech jobs have a positive and significant local employment multiplier of 2.2 additional jobs.Over the period 1996-2006, Cerqua and Pellegrini (2020) reveal positive multipliers for Italy in the ranges of 0.26-0.33 and 0.88-1.13for the tradeable sector (manufacturing) and the non-tradeable sector (construction and services), respectively, values that are lower than the United States but higher than some Asian and European countries.
Studies for the Spanish case are somewhat scarce.Along the lines of Moretti (2010), Gerolimetto and Magrini (2014) compute the jobs in the non-tradeable sector created by an additional job in the tradeable sector.Controlling for endogeneity and spatial dependence and after drawing a comparison with Moretti (2010) and De Blasio and Menon (2011), they find no multiplier effect for Spanish local labour market areas, attributable in their opinion to less flexible labour markets than the United States and lower job and geographical mobility.Jofre-Monseny et al. (2014, 2020), one of the few authors studying multipliers for Spain, research the quantification of the long-run impact of public sector employment on local labour markets in Spanish cities.As a preliminary study, Jofre-Monseny et al. ( 2014) complement the British study by Faggio and Overman (2014) who estimate the multiplier effects of public sector job relocations outside London.For the Spanish replicate, Jofre-Monseny et al. ( 2014) consider the period between 1980 and 2001 following General Franco's death in 1975, focusing on very longrun changes in employment.Their OLS results indicate that Spanish public administration employment had a positive multiplier effect for the non-tradeable sector (1.7 additional jobs) and a negative effect for the tradeable  (1980-90 and 1990-2001) in the private employment and population of Spanish cities.They find that one additional public sector job increases non-tradeable employment by 0.9 jobs but with hardly any effect on tradeable employment.Overall, the large expansion in Spanish public sector employment (133%) exerted only a moderate impact on local unemployment rates due to a 'crowding-in' effect on private jobs in cities.For the precrisis period 1995-2008, Bashford-Fernández (2014), using a shorter period, obtains a multiplier effect of 1.13 non-tradeable jobs for one tradeable job.However, it should be noted that all the above references do not contemplate informal employment.Thus, in order to compare our results with the case in which informal employment is not taken into account, we reestimate the model using a non-frontier approximation  (the results are presented in Appendix D in the supplemental data online).In order for both models to be as homogeneous as possible, we use the same instruments as those applied in the frontier model.From the results, a difference is observed between the value of the multiplier obtained by the frontier and the non-frontier approach (specifically 1.2 in the frontier approach versus 1.04 in the non-frontier approach).That is, the frontier model yields a value of the multiplier that is 15.4% higher than the value obtained with the non-frontier approximation.This result confirms the robustness of our proposal.Thus, the discrepancy between both multipliers is due to the existence of informal employment, since, with the observed data, it is not possible to capture all non-tradeable employment (which includes both observed and informal non-tradeable employment).The non-frontier model ignores, by construction, the existence of such informal employment, which implies a value of the multiplier lower than the real one.Instead, the frontier model allows us to obtain, by means of a one-sided error, a multiplier that does take informal employment into account.

The employment gap index
Having reviewed the frontier results, we analyse the EGI defined by equation ( 17) that analyses informal employment (i.e., the difference between real and observed nontradeable employment).At the mean, the EGI is 0.90.This indicates that, on average, 10% of informal nontradeable employment exists, which is not being taken into account in the official statistics.This result is similar to that obtained in the literature that has attempted to make an approximate estimate of this type of employment in Spain (e.g., Julià et al., 2014). 17In order to understand the determinants of this unobservable type of employment, we analyse the results obtained from the estimation of equation (18). 18 Table 4 presents the results relating to the determinants of the variance for the error term u.A negative sign is interpreted as reducing the distance to the frontier, and vice versa.On the one hand, the coefficient Sk (representing the percentage of skilled persons in the total population) is negative.This indicates that, according to our initial hypothesis, a more educated population implies greater opportunities for workers and less informal employment.On the other hand, the specialisation coefficient denoted by COS is positive and statistically significant suggesting that a degree of sectoral specialisation would seem to limit work opportunities thereby favouring the emergence of more informal employment.Nevertheless, the coefficient COS × N_Sk, representing the skilled portion of the workforce involved in sectorial specialisation, is also negative, meaning that the existence of specialised sectors with a high percentage of skilled workers reduces the difference between observed and real non-tradeable employment.That is, those sectors that recruit more qualified personnel are those that have less non-tradeable informal employment.This result is in line with other studies that analyse the relationship between education level and informal employment, finding a negative relationship between both (Funkhouser, 1996).
Finally, Table 5 shows the values of the EGI by region.Considering the extreme values, Extremadura has, on Measuring local employment multipliers and informal employment: a stochastic frontier approach 87 average, 15% of informal non-tradeable employment compared with Asturias with only 3%.

CONCLUSIONS
In spite of the ubiquity of the local multiplier effect in arguments favouring industry-oriented place-based policies, it was not until the work of Moretti (2010) that this topic saw a resurgence in the literature on employment multipliers.
The objective of this study was to analyse the creation of non-tradeable jobs via the employment multiplier.With this aim, our paper proposes a novel model that estimates employment multipliers motivated by the idea that, due to the potential existence of non-tradeable informal employment, it is not possible to observe all the non-tradeable jobs that are being generated in the economy.This fact, if not taken into consideration, can lead to a biased estimate of the multiplier.To address the problem, this paper proposes the use of a new methodology in the employment multiplier literature based on an SFA approach.The latter enables us to obtain the value of the multiplier while, at the same time, allowing for the one-sided error that inevitably occurs when estimating the employment multiplier in situations where unobserved non-tradeable informal employment exists.That is, with this approximation, it is possible to obtain an unbiased employment multiplier which takes into account both observable and unobservable (informal) employment.
We validate our approach using data for Spanish provincial employment over the period 1995-2013 and obtain, at the sample mean, a multiplier of 1.2 non-tradeable jobs per one tradeable job.In addition, the results confirm the hypothesis raised in the theoretical model that the creation of non-tradeable employment is positively and significantly related to a higher proportion of university educated population.These results highlight the importance of education in explaining the multiplier and corroborate the findings of previous studies which suggest larger employment multipliers in the presence of skilled workers.
Over time, the average non-tradeable multiplier sees a gradual upward trend reflecting the improved capacity of the Spanish provinces in the creation of non-tradeable employment.This increase, simultaneous with the country's technological advances, seems to indicate that the latter will, in net terms, boost the creation of non-tradeable employment.
Regarding the gap between the real and observed multiplier, results indicate that the presence of university educated individuals in the population decreases informal employment.This positive result is also obtained when industrial specialisation is combined with skilled individuals.That is, specialisation in sectors that require highly qualified personnel tends to reduce informal employment.
In sum, our results confirm the validity of the SFA alternative-based research approach which could tackle some issues regarding the more standardised methodologies used in the recent empirical literature on employment multipliers.Our proposed methodology allows obtaining an unbiased estimate of the multiplier in the presence of informal employment.This in turn demonstrates that in order to increase non-tradeable employment levels via employment multipliers, sectors employing higher levels of human capital should be targeted.Furthermore, the methodology presented here permits an approximation to the presence of illegal employment providing some insight as to its determinants.As such, it could prove a useful tool in the development of statistics on the informal economy, thereby serving to improve labour statistics and national accounts.This information may prove valuable to policy makers when tackling this issue.

NOTES
1. Kline and Moretti (2014) discuss the issues of efficiency, social welfare and employment in relation to place-based policies.2. On the basis of this definition, several types of workers are identified: own-account workers and employers of informal firms, contributing family workers, informal employees (of formal and informal firms), and members of informal producers' cooperatives (Hussmanns, 2004).3. See Thulin (2015) for a revision of this literature.4. K = K NT + K T is assumed in the empirical model because, due to data restrictions, it is not possible to obtain more disaggregated data.Note also that relative prices do not appear in equation ( 10) due to the model assumptions.5. Spain's 50 provinces make up Spain's 17 autonomous communities.Given the complexity of the empirical model, the estimation gives rise to convergence issues if the frontier estimated includes 50 provincial dummies.6.The Bartik instrument, although widely used in the multiplier literature, has not yet been implemented in the case of SFA models.7.As Amsler et al. (2016) point out, in this two-step procedure the standard errors from step 2 need to be adjusted.Following Wooldridge (2010), we construct the standardised residual as: ĥ * it = ĥit / ŝh .8. The capital variable has been lagged one year (predetermined variable).Hence, N NT in a given year should not affect capital in a preceding year.We thank a referee for suggesting this lag for the capital variable.

Figure 2 .
Figure 2. Evolution of the multiplier (M) over time.Source: Authors' own elaboration.
sector (0.6 job losses).In subsequent research, Jofre-Monseny et al. (2020) use a spatial equilibrium calibration (search and matching model) to simulate the impact of public sector expansions via decennial adjustments

Table 3 .
Multiplier (M)values by regions and periods.

Table 5 .
Employment gap index (EGI) values by regions.
Note: All (non-continuous) variables are in logs.***Significant at the 1% level; and **at the 5% level.