A Spatial Autoregressive Stochastic Frontier Model for Panel Data with Asymmetric E ciency Spillovers

Transcription

1 A Spatial Autoregressive Stochastic Frontier Model for Panel Data wh Asymmetric E ciency Spillovers Anthony J. Glass, Karligash Kenjegalieva y and Robin C. Sickles zx December 204 Abstract By blending seminal lerature on non-spatial stochastic frontier models wh key contributions to spatial econometrics we develop a spatial autoregressive stochastic frontier model for panel data. The speci cation of the spatial autoregressive frontier allows e ciency to vary over time and across the cross-sections. E ciency is calculated from a composed error structure by assuming a half-normal distribution for ine ciency. The spatial frontier is estimated using maximum likelihood taking into account the endogenous spatial autoregressive variable. To maximize the concentrated log-likelihood function we propose a new multivariate erative search procedure which is a combination of algorhms used in spatial econometrics and stochastic frontier analysis. We apply our spatial estimator to an aggregate production frontier for 4 European countries over the period The tted models are used to discuss, among other things, the asymmetry between e ciency spillovers to and from a country. Key words: Spatial Stochastic Frontiers; Global and Local Spatial Dependence; Supply Chain Management; Asymmetric ectional E ciency Spillovers; Aggregate Production; European Countries. JEL Classi cation: C23; C5; D24. School of Business and Economics, Loughborough Universy, Leics, UK, LE 3TU. A.J.Glass@lboro.ac.uk. y School of Business and Economics, Loughborough Universy, Leics, UK, LE 3TU. K.A.Kenjegalieva@lboro.ac.uk. z Corresponding author x Department of Economics, Rice Universy, Houston, US, and School of Business and Economics, Loughborough Universy, Leics, UK, LE 3TU. rsickles@rice.edu

2 Introduction The presence of omted variable bias from ignoring spatial autoregressive (SAR) dependence in cross-sections has long been recognized. Among other reasons, this motivated the development of the SAR model in key contributions by Cli and Ord (973; 98). Since then other spatial models have been proposed such as a model that includes spatial lags of the dependent variable and the error term (e.g. Kelejian and Prucha, 200; Su, 202; Baltagi and Breeson, 20), and also the spatial Durbin model (Anselin, 988; Anselin and Bera, 998). The latter nests the SAR model as augments the SAR model wh exogenous spatial lags of the independent variables. For stochastic frontier models, biased parameter estimates due to the omission of the SAR variable also have implications for the e ciency scores. We therefore merge techniques used in spatial econometrics wh those from the stochastic frontier lerature to develop spatial stochastic panel data frontiers wh SAR dependence. The composed error structure of stochastic frontiers consists of ine ciency and an idiosyncratic error, where we follow much of the lerature on non-spatial stochastic frontiers and make two distributional assumptions to distinguish between the components of the composed error. Our motivation for the spatial Durbin speci cation of the stochastic frontier is the compelling econometric case for this speci cation over, for example, the SAR and spatial error speci cations (see LeSage and Pace, 2009). There are various arguments for the inclusion of the SAR term, or more speci cally, the spatial Durbin speci cation. Firstly, the spatial Durbin speci cation yields unbiased parameter estimates even if the true data generating process is the spatial error model or the SAR speci cation. Secondly, the global spillovers in the spatial error model relate to the latent nuisance term, whereas global spillovers from a model which contains the SAR term have a structural economic interpretation as these spillovers can be related to the independent variables. Thirdly, the ratio of indirect and direct marginal e ects from the SAR model are the same for all the independent variables, which is unrealistic and not the case for the spatial Durbin model. 2 The case which LeSage and Pace (2009) make in favor of the spatial Durbin model is at odds wh the view that spatial model selection is a matter for LM test procedures. They make the case for the spatial Durbin model because of concerns about the robustness of these tests to misspeci cation of the spatial dependence. Early LM tests for spatial error dependence (Burridge, 980) and SAR dependence (Anselin, 988) ignore the possibily of SAR and spatial error dependence, respectively. Anselin et al. (996) develop cross-sectional LM tests for spatial error dependence and SAR dependence, where both tests are robust to misspeci cation of local spatial dependence (i.e. st order neighbor e ects as modelled by spatial lags of the exogenous variables) but not global spatial dependence (i.e. st order neighbor e ects through to (N )th order neighbor e ects as modelled by the SAR term or the spatial error term). Furthermore, there is a debate about whether to base spatial model selection on a speci c-togeneral testing procedure (Florax, et al., 2003), a general-to-speci c approach (Mur and Angulo, 2009) or a mix of the two (Elhorst, 204). 2 A direct elasticy is interpreted in the same way as an elasticy from a non-spatial model, although a direct elasticy takes into account feedback e ects (i.e. e ects which pass through st order and higher order neighbors and back to the un which iniated the change). An indirect elasticy can be 2

3 As a comparator to the spatial Durbin stochastic frontier model wh an assumption about the ine ciency distribution, in the application section of the paper we also estimate the corresponding SAR frontier model rather than the spatial error frontier speci cation. This is because, rstly, the SAR model is a closer comparator to the spatial Durbin speci cation because, as we noted above, in both cases the indirect marginal e ects relate to the independent variables. Secondly, as is the case in the spatial non-frontier lerature the single estimator which we develop encompasses both the SAR and spatial Durbin stochastic frontiers. We develop a maximum likelihood (ML) estimator for these spatial stochastic frontiers which combines the ML estimation procedures for the SAR nonfrontier model (Anselin and Hudak, 992; Elhorst, 2009) and the non-spatial stochastic frontier (Greene, 982). The lerature on spatial stochastic frontier modeling is rather sparse. A small number of studies estimate spatial stochastic frontiers and calculate e ciency using the crosssectional speci c e ects. The rst of these studies is due to Druska and Horrace (2004). By extending the spatial error model for cross-sectional data as set out by Kelejian and Prucha (999), they develop a GMM spatial error stochastic frontier model wh xed e ects. They then calculate time-invariant e ciency from the cross-sectional speci c effects using the Schmidt and Sickles (984) (SS from hereon) estimator. The SS e ciency estimator assumes a composed error which consists of the usual idiosyncratic disturbance and time-invariant ine ciency. The un wh the largest (smallest) xed/random e ect is placed on the concave (convex) frontier and the e ciency estimates are the exponential of the di erence between the best performing un s xed/random e ect and the corresponding e ect for each of the other uns in the sample. Glass et al. (203) adopt a similar approach by following Cornwell et al. (990) (CSS from hereon) and using the crosssectional speci c e ects from a SAR stochastic frontier model to estimate time-variant e ciency. We are not aware of a stochastic frontier model that accounts for global spatial dependence via the endogenous SAR variable, or via the endogenous spatial ine ciency term and the endogenous spatial idiosyncratic error; whilst also calculating e ciency from a composed error structure by making an assumption about the ine ciency distribution. Such models which account for SAR dependence are therefore developed in this paper. To the best of our knowledge, Adetutu et al. (204) is the only study that introduces a spatial relationship into a stochastic frontier model where an assumption is made about the distribution of the ine ciency component of the error structure. Their model, however, unlike the estimator we develop, overlooks global spatial dependence by omting calculated in two ways yielding the same numerical value. This leads to two interpretations of an indirect elasticy: (i) the average change in the dependent variable of all the other uns following a change in an independent variable for one particular un; or (ii) the average change in the dependent variable for a particular un following a change in an independent variable for all the other uns. The total elasticy is the sum of the direct and indirect elasticies. 3

4 the endogenous SAR variable and/or the endogenous spatial ine ciency term and the endogenous spatial disturbance. That said, although their model speci cation represents a simple way of accounting for spatial interaction they lim the analysis to local spatial dependence by including only spatial lags of the exogenous variables. Such local spatial stochastic frontier models can therefore be estimated using the standard procedures for the non-spatial stochastic frontier model. The spatial stochastic frontier estimator which we present represents an alternative to using the cross-sectional speci c e ects from a spatial frontier to estimate e ciency. In addion, there is plenty of scope to extend the spatial stochastic frontier estimator which we develop. For example, rather than assume that the ine ciency distribution is halfnormal, as we do here, we could use the time-varying decay e ciency estimator (Battese and Coelli, 992) or assume that ine ciency follows a truncated normal distribution (e.g. Stevenson, 980) or, alternatively, is Gamma distributed (e.g. Greene, 990). In addion, the SAR and spatial Durbin stochastic frontiers which we propose capture supply chain management issues across space such as outsourcing to a rm in another location or, at the aggregate level, importing from another country. This is because the speci cation of, for example, SAR and spatial Durbin stochastic production frontiers is such that via the indirect input elasticies, a un s output depends on the inputs of the other uns in the sample. The issue then is how e ciently does a un use the inputs of other uns in di erent locations (i.e. e ciency across space). Spatial e ciency is central to the SAR and spatial Durbin stochastic frontiers which we develop but such rigorous econometric techniques and economic concepts are rarely applied to analyze supply chain management issues. 3 The SAR and spatial Durbin frontiers which we propose would therefore appear to be examples of spatial econometric models which could be used more widely in quantative analysis of supply chain management issues. We apply our spatial estimator to an aggregate production frontier using balanced panel data for 4 European countries over the period E ciency varies over time and across the cross-sections, and we account for the endogeney of the SAR variable in the ML estimation. In addion to the above econometric case for the spatial Durbin speci cation, we can also make an economic case for this speci cation when analyzing country production. A number of studies have extended the standard Solow (956; 957) neoclassical growth set-up by endogenizing technical change and modeling this change for the representative rm as a function of aggregate capal per worker in the economy (López-Bazo et al., 2004; Egger and Pfa ermayr, 2006; Ertur and Koch, 2007; Koch, 2008; Pfa ermayr, 2009). The resulting reduced form equation for output per worker is approximated empirically by the spatial Durbin model, where a spatial lag of capal per worker captures capal per worker spillovers from rst order neighbors and the SAR variable accounts for, among other things, the e ect of capal per worker 3 An exception is Williamson s (2008) application of transaction cost economics (Williamson, 979). 4

5 spillovers from higher order neighbors. Of the aforementioned studies only Ertur and Koch (2007) and Koch (2008) explicly model the structural form of the output per worker equation by estimating a spatial Durbin model. López-Bazo et al. (2004), Egger and Pfa ermayr (2006) and Pfa ermayr (2009), on the other hand, estimate a spatial error/autoregressive model, which involves assuming that the spatial error/autoregressive term captures the e ect of global (st order through to the (N )th order) capal per worker spillovers. Rather than estimate the reduced form equation for output per worker, from above follows that if we estimate the assumed underlying spatial Cobb-Douglas production technology, as we do here, this is approximated empirically by the spatial Durbin model. The e ciency estimates from our SAR and spatial Durbin stochastic frontiers are directly comparable to the e ciency estimates from the corresponding non-spatial stochastic frontier. For such spatial and non-spatial stochastic frontiers where an assumption is made about the ine ciency distribution, the e ciencies are calculated relative to an absolute best practice frontier. In contrast, when the cross-sectional speci c e ects are used to calculate time-invariant e ciency, e ciency is estimated relative to the best performing un in the sample. There are two novel features of our approach. Firstly, we calculate e ciency by making an assumption about the ine ciency distribution, and, secondly, we adapt the SS method and apply to the e ciencies to calculate time-varying relative direct, relative indirect and relative total e ciencies in order to examine spatial e ciency. The SS method assumes e ciency is time-invariant but we adapt this approach to obtain time-varying estimates of relative direct, relative indirect and relative total ef- ciencies by placing the best performing un in the sample in each time period on the frontier, where the best performing un and thus the benchmark may change from one period to the next. The relative direct, relative indirect and relative total e ciencies are calculated and interpreted along the same lines as the direct, indirect and total marginal e ects. There are two ways of estimating relative indirect e ciency which yield estimates of di erent magnude (i.e. asymmetric directional e ciency spillovers). In turn this leads to two estimates of total e ciency which di er in magnude. 4 Intuively, relative indirect e ciency benchmarks how successful a un is at exporting/importing productive performance vis-à-vis s peers in the sample. For example, rms in di erent countries may export and import e ciency via knowledge spillovers. 5 To provide an insight into 4 Glass et al. (204) show how e ciencies which are estimated using the cross-sectional speci c e ects from a SAR stochastic frontier can be used to compute relative direct, relative indirect and relative total e ciencies. In this paper we show that this approach to calculate relative direct, relative indirect and relative total e ciencies can also be applied to a SAR stochastic frontier when an assumption is made about the ine ciency distribution. Unlike Glass et al. (204), however, we do not con ne our analysis to just one way of estimating the relative indirect and relative total e ciencies. 5 The lerature on knowledge spillovers is vast but key references include Coe and Helpman (995), Keller (2002) and Keller and Yeaple (203). For other studies on knowledge spillovers see Branstetter 5

6 the type of conclusions that can be made from our spatial stochastic frontiers; we nd that the average technical e ciencies from the non-spatial and local spatial frontiers non-negligibly underestimate the average technical e ciencies from the SAR and spatial Durbin frontiers. The remainder of this paper is organized as follows. In Section 2 we present the SAR stochastic frontier, which encompasses the spatial Durbin frontier, and we set out the key features of our ML estimation procedure. In Section 3, we describe how the e ciencies from the SAR frontier can be used to estimate relative direct, asymmetric relative indirect and asymmetric relative total e ciencies. We apply our SAR stochastic frontier estimator to an aggregate production frontier for European countries in Section 4 and in Section 5 we conclude. 2 Model, Estimation and E ciency 2. SAR Stochastic Frontier Consider the following SAR stochastic frontier model for panel data: P y = x + N w ij y + v u ; () j= i = ; :::; N; t = ; :::; T: N is a cross-section of uns operating over a xed time dimension T, y is an observation for the objective variable (e.g. output, pro t, revenue, etc.) for the h un at time t, where lower case letters denote logged variables, x is a ( K) vector of exogenous independent variables and is a vector of parameters. For the SAR stochastic frontier model x will include variables which together wh y represent the frontier technology, and also any variables which shift the frontier. For example, in the case of a production frontier x will include observations for the inputs. P N j= w ijy is the endogenous spatial lag which also shifts the frontier technology, where w ij is a non-negative element of the spatial weights matrix, W, and is the SAR parameter. Eq. also encompasses the spatial Durbin stochastic frontier model if x also includes exogenous spatial lags of the non-spatial independent variables as addional variables which shift the frontier. In light of this, as is standard practice in the spatial econometrics lerature, we have presented the SAR speci cation in Eq. Robinson, 200; Robinson and Thawornkaiwong, 202). rather than the spatial Durbin speci cation (see e.g. The error structure in Eq. is " = v u and is for production, revenue, standard and alternative pro t, and input distance frontiers, where v is an idiosyncratic disturbance and u represents ine ciency. The error structure for cost and output distance (200), Bottazzi and Peri (2003), Blazek and Sickles (200) and Bahar et al. (204). 6

7 frontiers would be " = v + u. We assume that v and u are both i:i:d, and that v N (0; 2 v) and u jn (0; 2 u)j. This asymmetric error structure is adopted directly from the classic stochastic frontier framework in Aigner et al. (977) (ALS from hereon). Furthermore, we follow ALS and obtain the log-likelihood function for Eq. by using 2 = 2 v + 2 u and = u v to reparameterize Eq.. As a result, 2 u = and 2 v = W captures the spatial arrangement of the cross-sectional uns and also the strength of the spatial interaction between the uns. W is speci ed prior to estimation and is usually based on some measure of geographical or economic proximy. As is standard in the spatial lerature, all the diagonal elements of W are set to zero to rule out self-in uence. In addion, W is normalized to have row sums of uny so that the endogenous spatial lag of the dependent variable is a weighted average of observations for the dependent variable for neighboring uns which facilates interpretation. We make the following assumptions wh regards to Eq.. (i) (I N W ) is non-singular and the parameter space of is r min ;, where I N is the (N N) identy matrix and r min is the most negative real characteristic root of W. The inverse of the largest real characteristic root of W is the upper lim of the parameter space of. Since we use a row-normalized W in the application, is the largest real characteristic root of W which simpli es the computation of log ji N W j in the estimation (see subsection 4:). (ii) The row and column sums of W and (I N W ) are bounded uniformly in absolute value before W is row-normalized. As a result of this assumption the spatial process for the dependent variable has a fading memory (Kelejian and Prucha, 998; 999). 2.2 Estimation of the Spatial Frontier and E ciency Assuming that the panel is balanced, the log-likelihood function associated wh Eq. is: LL yj; ; ; 2 = NT 2 log 22 NP TP + T log ji N W j " i=t= NP TP " log ; (2) i=t= where is the standard normal cumulative distribution function. The scaled logged determinant of the Jacobian of the transformation from " t to y t (i.e. T log ji N W j) is included in the log-likelihood function (see Anselin, 988, and Elhorst, 2009) to: (i) ensure that lies whin s parameter space; (ii) adjust for the endogeney of the SAR variable; and (iii) account for the fact that " is not observed. The partial derivatives of the log-likelihood function are given in Eqs. 3 6: 7

8 = NP TP 2 i= t=x " + NP TP x i=t= = 2 = NT NP TP " 2 + NP TP " 2 3 = 0; (4) = NP TP " 2 i=t= = = NP NP TP " 2 w ij y T i=j=t= I N W W + NP NP TP w ij y = 0; (6) i=j=t= where = ", = ", is as de ned previously and is the associated probabily densy function. Following ALS we use Eqs. 4 and 5 to obtain the following ML solution for 2 : 2 = NT NP TP " 2 : (7) i=t= The ML expression for is obtained by pre-multiplying Eq. 7 by 2 (X 0 X) : = (X 0 X) X 0 y (I T W ) y + ; (8) where X is an (N K) matrix of observations for the independent variables stacked in the cross-sectional and time dimensions. Rewring Eq. 3 in = 2 [X0 y X 0 (I T W )y X 0 X] + X0 = 0; (9) where I T is the (T T ) identy matrix and is the Kronecker product. As shown by Greene (982), by rewring the term in square brackets in Eq. 9 as X 0 " and then pre-multiplying Eq. 9 by 0, we can solve for the implic ML solution for : = 0 X 0 " 0 X 0 : (0) There are a number of possible ways of implementing ML estimation of Eq.. The approach which we employ involves developing a new multivariate search strategy using the concentrated log-likelihood function. To derive the concentrated log-likelihood function we follow the approach for the panel data SAR non-frontier model (Elhorst, 2009). By substuting the above expressions for and 2 into Eq. 2 we obtain the following 8

9 concentrated log-likelihood function: LL C = NT 2 log [(e 0 e ) 0 (e 0 e )] + T log ji N W j + log (e 0 e ) : () is a constant that does not depend on, and e 0 and e are the OLS residuals from successively regressing y and (I T W )y on X. For the SAR non-frontier model, the concentrated log-likelihood function contains just one unknown parameter, which has to be computed numerically as a closed form solution does not exist. However, the concentrated log-likelihood function for the SAR stochastic frontier model contains three unknown parameters,, and, so we must compute these parameters numerically, which involves developing a new multivariate search procedure. The six-step numerical search procedure which we employ is set out brie y in the Appendix. A feature of our search procedure involves drawing on Pace and Barry s (997) ML estimation procedure for spatial non-frontier models, where before estimation we calculate log ji N W j for a vector of values for over the interval r min ;. As suggested by Pace and Barry (997) we calculate log ji N W j for values of based on 0:00 increments over the above feasible range for. 6 For the inial values of and there will be various possibilies. In the application, the inial values of and for each value of in the grid are obtained using the ML estimates of u and v for the non-spatial stochastic frontier model. The estimator appears to perform well in the application as the SAR and spatial Durbin speci cations of Eq. both converge, and for both these model speci cations the estimates of the direct marginal e ects for the key variables are in line wh estimates in the relevant lerature. A Monte Carlo analysis is thus an area for further work which would provide a comprehensive assessment of the statistical performance of the estimator. 7 The asymptotic variance-covariance matrix for Eq. is given in Eq Following 6 As previously noted, we evaluate log ji N W j using a vector of values for, which LeSage and Pace (2009) refer to as the vectorized approach. Another approach to compute log ji N W j which Ord (975) proposed is ji N W j = Q N i ( r i ) ; where r i is a root of W. Wh the vectorized approach, knowing only the largest and smallest real characteristic roots of W, log ji N W j is evaluated for the vector of values of prior to the erative procedure. The values of log ji N W j are stored and then used in the erative procedure. Wh the Ord approach log ji N W j must be computed for each eration. In addion, all the roots of W are needed to calculate log ji N W j, and for some asymmertric speci cations of W there is the added di culty of calculating the complex roots of W. LeSage and Pace (2009) are therefore of the opinion that is only feasible to compute all the roots of W and thus use the Ord approach to evaluate log ji N W j when N is small or moderate. Even though N is que small in the application (4) we still use the vectorized approach because this approach can also be used when N is very large (e.g. N > 3; 00 for counties and county equivalents in continental Uned States). 7 Accounting for incorrect skewness in the SAR stochastic frontier model is another area for further work. See Almanidis and Sickles (202) and Almanidis, Qian and Sickles (204) on how to address incorrect skewness in the non-spatial stochastic frontier model in cross-sectional and panel data settings. 8 An alternative to Eq. 2 is a sandwich expression for the variance-covariance matrix. The advan- 9

10 LeSage and Pace (2009) we obtain the standard errors using a mixed analytical-numerical Hessian, where all the second order derivatives are computed analytically, wh the exception 2 LL=@ 2 which is evaluated numerically. Evaluating the second order derivatives of the log-likelihood function analytically rather than numerically is less sensive to badly scaled data, and numerical rather than analytical evaluation 2 LL=@ 2 when N is very large avoids any di culties (or lengthy computation time) associated wh evaluating the dense (I N W ) matrix in tr( W f W f + W f0 W f ) in Eq. 2. Asy:V ar(; ; ; 2 ) = X 0 X X 0 X 2 2 X(I T f W )X+ 2 2 X(I T f W )X X 0 (I T f W )X+ T tr( f W f W + f W 0 f W ) 2 2 ( ) % 0 X 0 (I T f W )X X (IT f W )X NT 2 3 X (I T W f )X (2" + ) 2 4 % NT ( ) 2 NT NT " 4 5 (NT 2 3) ; where f W = W (I N W ), tr denotes the trace of the relevant matrix, = (2) ( ) and % = + ( ) ".9 Having estimated,, and 2 we compute the composed prediction error, " = y P N j= w ijy x. To estimate e ciency,, we use the following Jondrow et al. (982) method to predict u condional on " : where = exp( E (u j" ) = u v " ; (3) u ). Finally, for SAR stochastic cost and output distance frontiers, the log-likelihood function and the estimation procedure are the same as above but wh tage of a sandwich estimator is that yields a consistent estimate of the variance-covariance matrix under misspeci ed variance-covariances and heteroskedastic errors. It is well-known, however, that, in general, a sandwich variance-covariance matrix yields a larger variance than classical estimators of the variance-covariance matrix (Kauermann and Carroll, 200). Various methods have been used in the stochastic frontier lerature to estimate the variance-covariance matrix. Our approach can be viewed as an adaptation to the spatial case of the information matrix used by ALS. 9 Even though N is que small in the application section, we 2 LL=@ 2 numerically rather than analytically to demonstrate how to compute Eq. 2 when N is very large. We 2 LL=@ 2 numerically as 2 2 LL where & = (0 ; ; 2 ). 0

11 " replaced by " in and, replaced by and = exp(u ). 3 ect, irect and Total Relative E ciencies LeSage and Pace (2009) demonstrate that for models such as Eq. the coe cients on the independent variables cannot be interpreted as elasticies. This is because the marginal e ect of an independent variable is a function of the SAR variable. To address this issue they propose disentangling the e ect of an independent variable from the e ect of the SAR variable by using the tted parameters to calculate direct, indirect and total marginal e ects. This approach involves using the reduced form of Eq. : y t = (I N W ) X t + (I N W ) v t (I N W ) u t ; (4) where the i subscripts are dropped to denote successive stacking of cross-sections, X t is an (N K) matrix of stacked observations for x and all the other variables and parameters are as above. For details on what is now the standard approach to calculate the direct, indirect and total marginal e ects by di erentiating Eq. 4 see LeSage and Pace (2009) and Elhorst (204). We note, however, rather than follow their approach to calculate the t statistics for the direct, indirect and total marginal e ects via Bayesian MCMC simulation of the distributions of the marginal e ects we calculate the t marginal e ects using the delta method. statistics for the By adapting the approach to estimate the marginal e ects we compute the timevaryng estimates of the relative direct, relative indirect and relative total e ciencies. This involves recognizing that (I N W ) u t = u T t ot, where u T t ot is the (N ) vector of total ine ciencies. It follows that (I N W ) exp ( u t ) = T ot t, where T ot t is the (N ) vector of total e ciencies. Thus we can wre: 0 (I N W ) 2 : : N 0 = C B : : + N : : + 2N : + : + : : + : : + : + : : + : N + N2 + : : + NN 0 = C B T ot T ot 2 : : T ot N ; (5) C A where ij (i = j) and ij (i 6= j) are direct and indirect e ciencies. By making an assumption about the distribution of u t, as we have done here, the benchmarking exercise yields t, which is relative to an absolute best practice frontier., and T ot, however, are not relative to absolute best practice frontiers. To aid the interpretation of, and T ot we adapt the SS method and apply to, and T ot to obtain time-varying estimates of the relative direct, relative indirect and relative total

12 e ciencies. The relative direct, relative indirect and relative total e ciencies are relative to the best performing un in the sample in each time period and are calculated as follows. RE RE = max i RE = RE = RE T ot = RE T ot = max i max j = RE = max i max j NP j= NP NP j= i= N P i= + N P j= + N P + N P max j ; (6)!; (7) ; (8) j= i= + N P i=!; (9) : (20) refers to e ciency spillovers to the h un from all the h uns. It is also valid to interpret relative indirect e ciency as RE the h uns from a particular h un. Since asymmetric directional e ciency spillovers. Hence RE T ot and RE as shares of RE T ot, which we denote SRE RE SRE and RE and SRE as shares of RE T ot we calculate RE, which refers to e ciency spillovers to all, which we denote SRE, RE 6= j, RE 6= RE 6= RE T ot and SRE and RE T ot where this un is the best performing un in the calculation of RE T ot lines, to estimate SRE and SRE we calculate RE, RE so there are. We calculate RE. We also calculate and SRE. To estimate relative to the same un, and RE T ot. Recognizing that RE T ot. Along the same relative to the best performing un in the calculation of RE T ot and RE T ot can be disaggregated into relative direct and relative indirect e ciency components as follows: RE T ot = max RE T ot i + NP max RE T ot i i= NP j=!; (2) 2

13 RE T ot = max RE T ot j + NP max RE T ot j j= N P i= : (22) It is possible for eher term in Eq. 2 or Eq. 22 to be greater than if the un wh the highest total e ciency is not also the un wh the highest direct e ciency and/or the highest indirect e ciencies. If this were the case would indicate that there are uns that operate beyond the direct and/or indirect reference levels in Eq. 2 and/or Eq SRE SRE is the rst term on the right-hand side of Eq. 2 as a share of RE T ot. SRE, and SRE are calculated in the same manner. 4 Application to Aggregate Production in Europe As is standard in applied spatial econometrics we use panel data which is balanced. Using aggregate data for 4 European countries for the period , which we describe in subsection 4:, we estimate SAR and spatial Durbin stochastic production frontiers (denoted SARF and SDF, respectively, from hereon). We also estimate the corresponding local spatial and non-spatial stochastic production frontiers (denoted LSF and NSF, respectively, from hereon), where the LSF is the SDF wh the SAR term omted. The LSF therefore only takes into account st order spatial interaction (e.g. the rst order Germany-France spatial e ect is the e ect of Germany s capal on France s output) via the exogenous spatial lags. interaction (st order through to (N The LSF does not account for global spatial )th order spatial interaction where a second order Germany-France spatial e ect would be, for example, the e ect of Germany s output on the Netherlands output which in turn via the spatial multiplier matrix a ects France s output). The SARF and the SDF, on the other hand, account for global spatial interaction via the SAR variable. The LSF can therefore be estimated using the standard procedure which we use to estimate the NSF. We use the production technology which underlies the Solow (956; 957) neoclassical growth model and employ a Cobb-Douglas speci cation. The speci cation of our SDF is very similar to the speci cation of the spatial Durbin production function in Koch (2008). That said, in Koch s speci cation spatial lags of the exogenous inputs shift the technology, 0 In the application there are no cases from the tted spatial autoregressive and spatial Durbin stochastic frontier speci cations of Eq. where a term from Eq. 2 or Eq. 22 is greater than. The asymptotic properties of global spatial estimators (e.g. spatial error, SAR and spatial Durbin non-frontier models, as well as the spatial stochastic frontier estimator which we propose here) can become problematic for unbalanced panels if the reason why data are missing is not known (Elhorst, 2009). Extending global spatial estimators to an unbalanced panel data setting therefore involves making a strong assumption about why observations are missing. For example, Pfa ermayr (203) assumes that data are missing at random for an unbalanced spatial panel. 3

14 whereas in our SDF a number of addional variables also shift the technology. 2 Since the SDF nests the NSF, SARF and LSF we can present the models which we estimate in the application using only the following SDF, where lower case letters denote logged variables unless otherwise stated in the discussion of the data in subsection 4: y = + t + 2 t 2 + g $ + z + NX w ij g + j= NX w ij h + NX w ij y + v u : (23) j= y is output of the h country at time t and g is a vector of inputs. The remaining variables in Eq. 23 shift the frontier technology and are as follows: t is a time trend, where t and t 2 are included to account for Hicks-neutral technological change, P N j= w ijg is a vector of spatial lags of the inputs, z is a vector of non-spatial variables, P N j= w ijh is a vector of spatial lags of a subset of elements of z and P N j= w ijy is as de ned for Eq.. is the intercept and, 2 $,,, and are parameters or vectors of parameters. Turning now to a more detailed discussion of the variables. j= 4. Variables, Data and the Spatial Weights Matrix All the data was extracted from version 8:0 of the Penn World Table, P W T 8:0 (Feenstra et al., 203a), which is rst version of the Penn World Table to include estimated data for capal stock. Output is output-side real GDP, y (in 2005 million U.S. dollars at 2005 PPPs, rgdpo), where P W T 8:0 notation for the variables are given in parentheses. As recommended in the documentation which accompanies P W T 8:0 we use rgdpo to analyze productivy across countries rather expendure-side real GDP (rgdpe) or GDP at 2005 national prices (rgdpna) (see page 3 in Feenstra et al., 203b). g is a ( 2) vector of input levels. The rst input is the labor input and is the number of people engaged, g (emp). Real capal stock at current PPPs is the second input, g 2 (in 2005 million U.S. dollars, ck). 3 The spatial lags of the inputs in Eq. 23 are denoted as (I T W ) g and (I T W ) g 2 in the presentation of the tted models. z is a ( 3) vector of variables, where z measures net trade openness and is net exports of merchandise as a share of GDP (z = csh_x+csh_m because all the observations for csh_m in P W T 8:0 are negative to signify that imports are a leakage). z 2 is government spending as a share of GDP (csh_g) and 2 See the unrestricted regression results in Table 2 of Koch (2008). 3 Following the documentation which accompanies P W T 8:0 (see page 3 in Inklaar and Timmer, 203) we use ck as our measure of real capal stock rather than real capal stock at 2005 national prices (in 2005 million U.S. dollars, rkna). 4

15 z 3 is a dummy variable for EU membership. (I T W ) h is a ( 2) vector of spatial lags of z and z 2 denoted (I T W ) z and (I T W ) z 2, respectively. 4 All the continuous variables which are not shares are logged. The descriptive statistics for the continuous variables are presented in Table and are for the raw data. [Insert Table ] In summary, the di erence between the SARF, SDF, LSF and NSF models relates to the variables which shift the frontier technology. For the NSF only t, t 2 and z shift the frontier technology compared to t, t 2, z, (I T W ) g and (I T W ) h for the LSF, t, t 2, z and (I T W ) y for the SARF, and t, t 2, z, (I T W ) g, (I T W ) h and (I T W ) y for the SDF. As is evident from Eq. 23, we do not capture unobserved time-invariant heterogeney via xed or random e ects. See Chen et al. (204) for a non-spatial stochastic frontier model wh consistent estimates of the xed e ects, and Greene (2005) for a non-spatial stochastic frontier model wh random e ects. 5 We do, however, account for spatial heterogeney via the SAR variable in the SARF and the SDF. This is because measures the degree of spatial dependence of y in the cross-sections. Assuming is posive, which is usually the case, a high degree of spatial heterogeney in the sample will not only have implications for the signi cance of but will also have a downward e ect on the magnude of. 6 Along similar lines, Zimmerman and Harville (99) argue that the spatial error term soaks up spatial heterogeney, but for the reasons given above we have a strong preference for the SDF over a spatial error stochastic frontier speci cation. W. 7 In the application we use a single row-normalized inverse distance speci cation of The speci cation of W is symmetric and dense as the spatial weights are the row-normalized inverse distances between each pair of capal cies. 8 Such a dense speci cation of W is appealing as captures the spatial interaction between every pair of countries and does not therefore involve using an arbrary cut-o for the number of neighbors (e.g.. the three, four or ve nearest neighbors). Using a row-normalized W also 4 There are no spatial lags of t, t 2 and z 3 in the speci cations of the LSF and the SDF. In all three cases this is for reasons of perfect collineary, e.g. (I T W ) t = t. 5 A logical extension of Eq. is to control for unobserved heterogeney using xed or random e ects. Another approach may entail the development of alternative ways to control for unobserved heterogeney and common shocks in spatial stochastic frontier models, possibly building on the factor model approach of Kniep, Sickles and Song (202). 6 Kao and Bera (203) consider what they describe as the curious case of negative spatial dependence, whilst acknowledging that posive spatial dependence dominates in spatial data and therefore also dominates in the spatial lerature on estimation, testing and forecasting. 7 As we noted in subsection 2:, the speci cation of W is row-normalized so that the SAR variable is a weighted average of observations for the dependent variable for neighboring uns which facilates interpretation. Speci cally, spillovers are therefore inversely related to the relative (and not the absolute) great circle distance between uns. 8 Since the speci cation of W which we use in the application is symmetric all s roots are real. In general, aysmmetric speci cations of W have complex roots but row-normalizing such matrices yields speci cations of W which have only real roots. 5

16 simpli es the calculation of the lims of the values of used to evaluate log ji N W j. This is because we know that the largest real characteristic root of a row-normalized W is, so we only need to nd the most negative real characteristic root of W. In addion, since the speci cation of W which we use in the application is an inverse distance matrix the spatial weights are exogenous. Finally, we note that even though the speci cation of W which we use is dense inverting (I N W ) in the estimation posed no problems. 4.2 Fted Models In this subsection we present and analyze the tted NSF, LSF, SARF and SDF models. Moreover, we use the tted models to make a case for the SDF speci cation. In Table 2 we present the tted SDF which is our preferred speci cation for the reason which we will discuss shortly. To enable comparisons wh our preferred SDF in Table 2 we also present the NSF and the corresponding LSF and SARF models. Like the SARF and SDF models, the NSF and LSF models have a normal half-normal disturbance structure so there is a direct correspondence between the four model speci cations. In Table 3 we present the direct, indirect and total marginal e ects for the SARF and SDF models. [Insert Tables 2 and 3] The estimates of in Table 2 are not spillover elasticies. The spillover elasticies from the SARF and SDF models are the indirect marginal e ects which are a function of, among other things, (see LeSage and Pace, 2009, and Elhorst, 204). Estimates of, however, indicate how the spatial dependence of y is a ected by the model speci cation and the speci cation of W. It is evident from Table 2 that for the SARF and SDF models, the estimates of are signi cant at the 0:% level. The estimates of from the SARF and SDF models are also of a similar order of magnude, which suggests that the degree of global spatial dependence is robust to model speci cation. To choose between the tted NSF, LSF, SARF and SDF models, we use the approach which Pfa ermayr (2009) uses to choose between di erent speci cations of the spatial weights matrix for a SAR non-frontier model and base our model selection on the Akaike Information Creria (AIC). To check the robustness of the model selection using the AIC we also use the Schwarz/Bayesian Information Creria (BIC). It is apparent from Table 2 that we have a strong preference for the SDF over the NSF, LSF and SARF models as the SDF yields the lowest AIC and BIC values. The NSF is the least preferred of the four models wh the largest AIC and BIC values. We can see from Table 2 that all the local spatial parameters from the tted LSF (, 2, and 2 ) are signi cant at the 0:% level. Interestingly, the BIC (AIC) favors the SARF (LSF) over the LSF (SARF). These BIC results therefore favor including the SAR variable over spatial lags of the exogenous variables, whereas the AIC values suggest the oppose. 6

17 The remainder of the discussion in this subsection tends to focus on the marginal e ects from the preferred SDF model. From Table 2 we can see that the own labor (capal) elasticies from the NSF and the LSF are of the order 0:289 (0:668) and 0:320 (0:633), respectively. Table 3 reveals that the direct labor (capal) elasticies from the SARF and the SDF models are 0:277 (0:684) and 0:30 (0:653), respectively. All the own/direct labor and capal parameters from the NSF, LSF, SARF and SDF models have the expected posive signs and are signi cant at the 5% level or lower. To be in line wh convention, however, the own/direct labor elasticy should be about 2 and the 3 own/direct capal elasticy should be around. Our results, on the other hand, point 3 more towards own/direct labor and capal elasticies which are of the oppose order of magnude. In particular, we report own/direct labor elasticies which are much closer to than 2, and own/direct capal elasticies which are approximately 2. Using data for 9 non-oil countries for 995 from version 6: of the Penn World Table, Koch (2008) estimates non-spatial and spatial speci cations of the Cobb-Douglas production function and reports own labor elasticies of the order 0:268 ranging from 0:679 input elasticies. 9 0:30 and own capal elasticies 0:78, thus providing support for our estimates of the own/direct Romer (987) argues that conventional growth accounting where the own labor elasticy is around 2 and the own capal elasticy is about substantially underestimates 3 3 the role of capal accumulation in growth. He is instead in agreement wh our estimates of the own/direct input elasticies (and the above estimates reported by Koch (2008)), arguing that innovation associated wh capal accumulation is sensive to wages so when wages are relatively high, as is the case in many European countries, innovative investment economizes on labor pushing the own/direct labor elasticy substantially down and the own/direct capal elasticy substantially up. We nd that the direct net trade openness (z ) and EU membership (z 3 ) parameters from the SDF model have the expected posive signs and are both signi cant at the 0:% level. Also, the direct government size (z 2 ) parameter from the SDF model has the expected negative sign and is signi cant at the 0:% level, which is consistent wh the signi cant negative relationship which F½olster and Henrekson (200) observe between government size and economic growth from a non-spatial non-frontier model. Wh regard to the indirect input elasticies from the SDF model, the picture is similar to what we observed for the corresponding direct input elasticies. This is apparent because the indirect labor and indirect capal elasticies are both posive and signi cant at the 5% level or lower, and the indirect labor elasticy (0:076) is smaller in magnude than the indirect capal elasticy (0:66). It is therefore evident that the indirect labor and 9 Koch (2008) estimates a spatial Durbin Cobb-Douglas production function but he does not proceed to calculate the direct marginal e ects, among other things. The ranges for the own labor and own capal elasticies from Koch (2008) which we report are therefore from the tted non-spatial and spatial error Cobb-Douglas production functions. 7

18 indirect capal elasticies from the SDF model are smaller than the associated direct elasticy. The SDF model yields indirect z z 3 parameters which have the same sign as the corresponding direct e ect but only the indirect z and z 2 parameters are signi cant. Moreover, the indirect z and z 2 parameters from the SDF model are much smaller in magnude than the corresponding direct elasticy. Finally, we note that the direct time parameter from the SDF model is negative and signi cant at the 0:% level. This is consistent wh the time parameter from Kumbhakar and Wang s (2005) non-spatial stochastic production frontier analysis of 87 countries, and also the time parameters for various country groupings in recent studies of world productivy growth (Sickles et al., 203; Duygun et al., 204). We do not interpret the negative direct time e ect from the SDF model as evidence of technological regress. Instead the interpretation is in the context of the application and we pos that the negative direct time e ect is due to our sample consisting of a large number of Eastern European countries that underwent major reform during the study period. 4.3 Technical E ciency and ect, irect and Total Relative Technical E ciencies Throughout this discussion of the technical e ciencies and the direct, indirect and total relative technical e ciencies, the emphasis is twofold. Firstly, we show for our application that technical e ciency very much depends on whether SAR dependence is accounted for in the model speci cation. Secondly, we demonstrate that the SARF and SDF models are a rich source of addional information about e ciency as they yield estimates of direct, indirect and total relative technical e ciencies. We nd a high correlation between the average technical e ciencies of the countries and the average technical e ciency rankings for the countries from the NSF, LSF, SARF and SDF. This is evident because the lowest correlation between two sets of average technical e ciencies for the countries from the NSF, LSF, SARF and SDF is 0:886. Moreover, the lowest Spearman rank correlation between two sets of average e ciency rankings for the countries from the NSF, LSF, SARF and SDF is 0:863. The average technical e ciencies from the NSF, LSF, SARF and SDF models are 0:773, 0:804, 0:888 and 0:920, respectively. This suggests: (i) Accounting for local spatial dependence (LSF) yields a higher average technical e ciency than when we overlook global and local spatial dependence (NSF). (ii) Accounting for global spatial dependence (SARF) yields a higher average technical e ciency than when we account for local spatial dependence (LSF). (iii) Finding that the average technical e ciency when we account for global and local spatial dependence (SDF) is higher than the average technical e ciencies from the SARF and the LSF is consistent wh ndings (i) and (ii) above. 8

19 Comparing the technical e ciency kernel densy plots in Figure reveals that the e ciency distribution from the NSF is the most negatively skewed followed by the ef- ciency distribution from the LSF, then the distribution from the SARF and nally, the distribution from the SDF. 20 This suggests that when we control for global and local spillovers (SDF), and to a lesser extent when we account for just global spillovers (SARF), countries wh low e ciency rankings have higher technical e ciencies. Accordingly, as we can see from Figure the technical e ciencies from the SDF are the least dispersed followed by those from the SARF, then those from the LSF and nally, those from the NSF. [Insert Figure ] In Table 4 we present for selected countries the average technical e ciencies and the corresponding average e ciency rankings from the NSF, LSF, SARF and SDF. It is evident from the average e ciency rankings in Table 4 that the rankings for several countries are robust across the NSF, LSF, SARF and SDF models. To illustrate, in Table 4 there are countries wh consistently low average e ciency rankings (Albania, Armenia, Belarus, Moldova, Portugal and Ukraine), consistently high average e ciency rankings (Norway, UK and possibly Sweden) and consistently mid-ranging e ciency rankings (Czech Republic and possibly Ireland and Slovenia). In addion, is evident from Table 4 that although some countries have consistently low average e ciency rankings, when we control for SAR dependence (SARF and SDF) absolute average technical e ciency for these countries is much higher than we observe from the NSF and the LSF. This we suggest is because the SDF and the SARF account for the geographical correlation of y and thus yield technical e ciencies which take into account the disadvantageous location of countries such as Albania, Armenia, Belarus, Moldova and Ukraine. [Insert Table 4] Finally, for the SARF and the SDF in Table 5 for selected countries we present the average relative e ciencies, and the average direct and average indirect relative e ciency shares. Speci cally, the rst row of results refers to estimates of RE, RE and RE T ot (Eqs. 6, 7 and 9) and the second row consists of estimates of RE, RE (Eqs. 6, 8 and 20). Using Eq. 2 we calculate the estimates of RE the third row, which are then used to calculate SRE Similarly, using Eq. 22 we compute the estimates of RE which are then used to compute SRE and SRE [Insert Table 5] and SRE and RE in the sixth row. and RE T ot in and RE in the fourth row. in the fth row, 20 To plot the kernel densies we use the Gaussian densy and obtain the bandwidth using the Sheather and Jones (99) solve-the-equation plug-in method. 9