The Impact of Model Choice on Estimates of Regional TFP

Abstract

A large range of different approaches to determine a region’s total factor productivity (TFP) is currently used in the literature. As there is no generally recognized method, it is an important issue whether different approaches produce comparable results or whether results depend heavily on the chosen method. By employing a data set of 220 European regions over the period from 1990 to 2007, the present work compares a set of TFP estimation approaches and examines whether regional TFP levels and TFP growth rates obtained from the various approaches differ considerably from each other. More specifically, the neoclassical accounting approach, the cross-section approach, the pooled panel approach, and the fixed-effects approach are investigated. Furthermore, this article proposes an extension of the fixed-effects approach that allows estimation of both regional TFP levels and region-specific, long-term TFP growth rates in a single estimation procedure. The results clearly demonstrate that the choice between the examined approaches has an essential impact on estimation results for both TFP levels and TFP growth rates. From the apparent differences between the obtained results, it is reasonable to expect that follow-up analyses based on TFP estimates, for example, studies investigating the determinants of regional TFP, are affected by the model choice to a considerable extent. Therefore, it is important to examine whether one model is superior to others or whether it is only up to the scholar to choose a model.

Keywords

total factor productivity European regions model comparison production function regional disparities

Introduction

Disparities of productivity performances across regions are central to many questions addressed by researchers in regional science as well as by policy institutions concerned with regional economic development such as the European Union’s cohesion policy: are less developed regions catching up to productivity levels of more prosperous regions (Di Liberto and Usai 2013; Vogel 2015)? What drives regional disparities in productivity levels and growth and which determinants cause these differences (Dettori, Marrocu, and Paci 2012; Marrocu and Paci 2013; Mitze 2014)? Before starting with the investigation of these issues, researchers have to choose a productivity measure which then forms the basis for the actual analysis. Many empirical works on the regional level prefer total factor productivity (henceforth TFP) over alternative measures, as TFP takes into account cross-regional differences in factor endowments of capital and labor (to name a few: Antonelli, Patrucco, and Quatraro 2011; Brixy 2014; Capello and Lenzi 2015; Marrocu and Paci 2013; Mitze 2014). However, as TFP is not a directly observable quantity and there is no recognized standard method to estimate TFP at the current state, researchers have to choose from a set of available TFP estimation approaches. Obviously, in the case that different estimation approaches produce equivalent or at least similar TFP estimates, results will not be affected by the model choice to a great extent, and consequently follow-up analyses, for example, regarding the determination of productivity drivers, are independent of the chosen estimation approach. If, however, different approaches produce TFP estimates that differ considerably from each other, it is reasonable to expect that also results of follow-up analyses are influenced by the TFP model choice. In this context, the major aim of the present work is to compare various TFP estimation approaches with regard to the obtained results and highlight the advantages and shortcomings of the examined approaches.

TFP literature dates back to the mid-late 1950s, focusing mainly on productivity growth within individual countries (Abramovitz 1956; Solow 1957). In the decades that followed, the focus of TFP studies shifted toward the analysis of cross-country differences in TFP growth (Aghion and Howitt 2007; Baier, Dwyer, and Tamura 2006; Moro and Nuño 2012) and TFP levels (Caselli 2005; Gundlach, Rudman, and Wößmann 2002; Hall and Jones 1999). Starting mainly with studies on the country level, TFP has recently also become a common concept in studies analyzing productivity differences between regions (Antonelli, Patrucco, and Quatraro 2011; Bernardini Papalia and Bertarelli 2010; Byrne, Fazio, and Piacentino 2009; Dettori, Marrocu, and Paci 2012; Melachroinos and Spence 2013). The increasing interest in TFP differences across both countries and regions has given rise to the development of a broad range of TFP estimation approaches and led to an extensive set of currently available models. Del Gatto, Di Liberto, and Petraglia (2011) provide a comprehensive overview of available approaches. In this survey, TFP approaches are classified according to three different criteria, that is, methods used at the aggregate level (countries/regions/industries) versus individual level (firms/plants), frontier versus nonfrontier approaches, and deterministic versus econometric methods.

In the context of these distinctions, the present study restricts its analysis to a relatively homogeneous field in the broad area of TFP approaches: first, the analysis examines TFP at the regional (aggregate) level. Second, regarding frontier versus nonfrontier approaches, the focus is on the latter, and frontier models such as the data envelopment analysis developed by Farrell (1957) are not considered. Finally, this work considers both deterministic and econometric approaches, however, only those employing a conventional Cobb–Douglas production function including the traditional inputs labor and physical capital. Although the examined approaches appertain to an apparently homogeneous set of models, they differ in their modeling setup to a considerable extent. Nevertheless, most works in the literature employing one of these models do not discuss model choice in an extensive manner and assume their employed model as a kind of state of the art. A critical assessment and evaluation of the chosen TFP approach is generally missing. Obviously, in the case that different approaches produce equivalent, or at least similar, TFP estimates, the application of different approaches as well as the lack of a critical assessment would not be much of a problem. Furthermore, the disregard for model choice implicitly suggests that the choice between TFP approaches plays only a minor role for the actual results. However, is this correct?

In this context, the current work attempts to answer the following two questions: first, are TFP levels and TFP growth rates obtained from various estimation approaches de facto equivalent or do they substantially differ? Second, is it possible to identify a “most suitable” model from the set of TFP approaches based on considerations relating to theoretical and econometric characteristics of the different models?

The reason to restrict the analysis to a rather homogeneous set of estimation methods is that if within such a homogeneous set of methods the results show substantial differences, this can be expected largely for more heterogeneous models. For this purpose, we investigate approaches relying on the Cobb–Douglas production function including two inputs, which, to the best of our knowledge, is the most frequently used method of regional TFP estimation (just to name a few applications, see Dettori, Marrocu, and Paci 2012; LeSage and Fischer 2012; Marrocu and Paci 2013; Mitze 2014; Vogel 2015). More specifically, five different TFP models are examined, and regional TFP levels and TFP growth rates obtained from these models are compared. The first model is the classic model in TFP estimation, that is, the neoclassical accounting approach (applied on a regional level, among others, by Byrne, Fazio, and Piacentino 2009; Derbyshire, Gardiner, and Waights 2013; LeSage and Fischer 2012; Scoppa 2007). Furthermore, the cross-section approach (used by Berlemann and Wesselhöft 2012; Capello and Lenzi 2015) and the pooled panel approach (Marrocu, Paci, and Usai 2013; Mitze 2014) are investigated. As a fourth model, the fixed-effects model (Dettori, Marrocu, and Paci 2012; Ladu 2012; Marrocu and Paci 2012, 2013) is analyzed. Finally, an extension of the fixed-effects model is introduced which allows estimating both regional TFP levels and TFP growth rates as model parameters of the production function rather than as a residual. In order to compare the five approaches, a data set of 220 European regions over the period from 1990 to 2007 is employed.

The remainder of the article is organized as follows: the second section presents the five examined TFP estimation approaches. The third section describes the employed data set and discusses the econometric methodology. The fourth section compares the results obtained with the various models. The fifth section evaluates the alternative TFP models and discusses their shortcomings. Finally, the sixth section provides concluding remarks.

TFP Estimation Approaches

The present work examines and compares five models for estimating regional TFP: the neoclassical accounting model (1), the cross-section approach (2), the pooled panel approach (3), the fixed-effects approach (4), and the fixed-effects approach with region-specific time trend (5). Models (1) to (4) are taken from the literature and model (5) is an extension of the fixed-effects approach. All of the presented approaches start with the conventional Cobb–Douglas production function with the inputs capital and labor. However, they show clear differences in their modeling setup, that is, in their assumptions regarding the heterogeneity of factor elasticities and regional efficiency levels. Furthermore, the five approaches allow for two different interpretations of TFP: on the one hand, the first three approaches obtain TFP as the residual of output that cannot be explained by inputs. As such, TFP comprises many components, some wanted (e.g., the effects of technological innovation) and others unwanted (such as measurement errors or missing variables). On the other hand, TFP of both fixed-effects models is estimated as a model parameter of the production function excluding the error term.

Neoclassical Accounting Approach

The neoclassical accounting model developed by Solow (1957) employs the conventional Cobb–Douglas production function of equation (1).

Y_{r t} = A_{r t} K_{r t}^{α_{r t}} L_{r t}^{β_{r t}},

where r = 1, ..., N and t = 0, ..., T are the regional and time subscripts, Y represents the output level, K and L denote the stock of physical capital and the labor input, A is the TFP level, and α and β are the output elasticities of capital and labor. By assuming perfect competition, output elasticities can be set equal to the factor shares in total income. Thus, the elasticity of labor is computed according to equation (2) as the ratio of total employment compensation, denoted by W, to total output. With the further assumption of constant returns to scale (CRS), the regional capital share is computed as given in equation (3).

β_{r t} = \frac{W_{r t}}{Y_{r t}},

α_{r t} = 1 - β_{r t} .

By computing equations (2) and (3) for each region and every point in time, the model allows output elasticities to vary both across regions and over time (see, e.g., Antonelli, Patrucco, and Quatraro 2011; Byrne, Fazio, and Piacentino 2009; Quatraro 2010). With values for α and β and rearranging equation (1) for A_rt, it is an easy task to calculate regional TFP levels. Note that by obtaining TFP as the residual of output that cannot be explained by inputs, any type of error is incorporated into the TFP measure. The accounting approach still is one of the most popular approaches to examine productivity differences between regions in the form of TFP levels (Byrne, Fazio, and Piacentino 2009; Derbyshire, Gardiner, and Waights 2013; LeSage and Fischer 2012; Scoppa 2007) and TFP growth rates (Antonelli, Patrucco, and Quatraro 2011; Brixy 2014; Melachroinos and Spence 2013; Quatraro 2010; Vogel 2015).

Cross-section Approach

While the Solowian approach is a purely deterministic approach, the cross-section approach applies a so-called quasi-growth accounting procedure where, in a first step, elasticities of factor inputs are estimated through econometric techniques. In a second step, estimated elasticities are employed within the accounting framework to compute regional TFP levels. The cross-section approach (used by Berlemann and Wesselhöft 2012; Capello and Lenzi 2015) estimates factor elasticities within the Cobb–Douglas production function of equation (4) for each cross section in the data set separately.¹

Y_{r t} = A_{t} K_{r t}^{α_{t}} L_{r t}^{β_{t}} u_{r t},

where u represents the remainder noise. Equation (4) is estimated separately for each point in time within the entire observation period, allowing factor elasticities α_t and β_t to vary across time but not across regions. In the second step of this approach, estimated (time-varying) factor elasticities are employed in the traditional accounting framework according to equation (5) in order to obtain regional TFP levels.

A_{r t} = \frac{Y_{r t}}{K_{r t}^{\hat{α_{t}}} L_{r t}^{\hat{β_{t}}}} = \hat{A_{t}} u_{r t} .

Note that the obtained TFP level A_rt of equation (5) equals the estimate of A_t times the error term u_rt of equation (4). Consequently, the regional variation of TFP of this model is determined only by the independent and identically distributed (i.i.d.) error term.

Pooled Panel Approach

In contrast to the previous model, the pooled panel approach (Marrocu, Paci, and Usai 2013; Mitze 2014) exploits the panel structure of the data set rather than estimating elasticities for each cross section separately. More precisely, equation (6) is used to estimate factor elasticities.

Y_{r t} = A K_{r t}^{α} L_{r t}^{β} θ_{t} u_{r t},

where θ_t represent the time fixed effects that account for possible shocks affecting all regions simultaneously. In this model, factor elasticities are assumed to be constant across both regions and over time. Again, estimated elasticities of factor inputs are employed in the accounting framework in order to obtain regional and time-specific TFP levels according to equation (7).

A_{r t} = \frac{Y_{r t}}{K_{r t}^{\hat{α}} L_{r t}^{\hat{β}}} = \hat{A} \hat{θ_{t}} u_{r t} .

Note that similar to the previous model, the TFP level A_rt of equation (7) is defined by the estimates of A and of the time dummy θ_t, times the error term u_rt of equation (6). Thus, the regional variation of TFP is determined only by the random error term.

Fixed-effects Approach

As with the previous model, the fixed-effects approach assumes factor elasticities to be constant across both regions and over time. However, in contrast to the preceding approach, the fixed-effects approach (Dettori, Marrocu, and Paci 2012; Ladu 2012; Marrocu and Paci 2011) obtains regional TFP levels directly from estimated time and regional fixed effects, denoted by A_r, which allows for an alternative interpretation of TFP. While in the former two approaches the error term is implicitly included in the TFP measure, this approach considers TFP as a model parameter of the production function excluding the error term. Thus, the fixed-effects approach estimates equation (8) and obtains regional TFP levels according to equation (9).

Y_{r t} = A_{r} K_{r t}^{α} L_{r t}^{β} θ_{t} u_{r t},

A_{r t} = \hat{A_{r}} {\hat{θ}}_{t} .

Obviously, as the temporal change in TFP levels is defined only by the time effect common to all regions, the TFP growth rate is assumed to be equal across regions. However, as the focus of the present work is also on regional differences of TFP growth rates, the same approach as in Ladu (2010) and Marrocu and Paci (2012, 2013) is applied. More specifically, estimated factor elasticities from the fixed-effects approach are employed in the accounting framework in order to obtain both heterogeneous TFP levels and TFP growth rates, which corresponds to the inclusion of the error term into the right-hand side of equation (9). However, TFP levels obtained with this procedure are used only to compute regional TFP growth rates. For comparison of TFP levels between the various models, TFP levels obtained from estimated regional and time fixed effects are used. Thus, while the regional variation of TFP levels is defined by the estimated parameter $\hat{A_{r}}$ , regional variation of TFP growth rates is determined by the error term similar to previous approaches.

Fixed-effects Approach with Region-specific Time Trend

As an alternative approach to obtaining heterogeneous TFP growth rates across regions, the fixed-effects approach with region-specific time trend is proposed. In contrast to the three preceding approaches, this model estimates both regional TFP levels and TFP growth rates as individual model parameters of the production function. For this purpose, the TFP term A_rt of the fixed-effects approach is extended by the region-specific time trend tγ_r which represents the regional long-term TFP growth rate. Thus, the regional TFP level is defined according to equation (11) by estimated regional and time fixed effects as well as by the region-specific time trend. Again, factor elasticities are assumed constant over time and across regions.

Y_{r t} = A_{r} K_{r t}^{α} L_{r t}^{β} θ_{t} e^{t γ_{r}} u_{r t},

A_{r t} = \hat{A_{r}} \hat{θ_{t}} e^{t \hat{γ_{r}}} .

Estimation of (Mean) Regional TFP Growth Rates

After having obtained regional TFP levels for each of the various models, (mean) regional growth rates of TFP, denoted by $\bar{g_{r}}$ , are computed according to equation (12) for models (1) to (4) as the average logarithmic growth rate of the region’s TFP levels between the first and the last observation period.

\bar{g_{r}} = \frac{1}{T} ln (\frac{A_{r T}}{A_{r 0}}) .

In model (5), regional TFP growth is defined according to equation (13) by the time fixed effects common to all regions as well as by the region-specific time trend.

\bar{g_{r}} = \hat{γ_{r}} + \frac{ln (\hat{θ_{t}})}{T} .

Data and Method

This article uses a panel data set of 220 European regions belonging to the fifteen member states of the EU-15 plus Norway and Switzerland observed from 1990 to 2007. Employing a data set that excludes the period of the financial crisis and the subsequent recession allows for better comparability between our results and those of quoted studies within this work, which generally do not include the postcrisis era. At the suggestion of two anonymous reviewers, a separate analysis was conducted that includes the postcrisis era up to 2014. Results for this extended observation period are generally in line with those presented in this article and are available by the authors upon request. Regional data are employed at the Nomenclature of Territorial Units for Statistics (NUTS) level 2. Spanish exclaves and the Canary Islands, French overseas departments and territories, and the Portuguese regions Azores and Madeira are not included in the sample.

Data from the output variable value added, the input variable labor units, and employment compensation, which are required for the accounting approach, stem from Cambridge Econometrics. The stock of physical capital is calculated according to the perpetual inventory method:

K_{t} = (1 - δ) K_{t - 1} + I_{t - 1},

where δ is the depreciation rate, which, similar to Dettori, Marrocu, and Paci (2012); LeSage and Fischer (2012); Marrocu and Paci (2012); and Capello and Lenzi (2015), is assumed to be 10 percent. A robustness check is conducted by rerunning all estimations and computations of the present work with alternative regional capital stocks obtained by applying depreciation rates of 6 and 8 percent, respectively. Results obtained with these alternative capital stocks are nearly identical with those obtained by applying a depreciation rate of 10 percent. I_t−1 is the flow of gross investments from the previous period obtained from Cambridge Econometrics. The initial capital stock is computed similar to Dettori, Marrocu, and Paci (2012); Marrocu and Paci (2012); and Capello and Lenzi (2015), as the cumulative sum of investment flows over the preceding ten-year period from 1980 to 1989. For the models that require econometric estimations, models (2) to (5), the respective Cobb–Douglas function is estimated in its log-linearized form. Furthermore, in order to control for the size of regions, input and output variables are normalized to population.

It is worth noting that for the empirical analysis, all five TFP approaches are using the same full data set, and hence, the availability of information is the same for all approaches. However, the use of the available information differs between the models. While the accounting approach (1) as a purely deterministic model is a class on its own, the use of available information increases with the number of estimated parameters for the econometric approaches. As the cross-section approach (2) is estimated for each time period separately, less information is used than for models (3) to (5), which exploit the panel structure of the data set. For the remaining approaches (3), (4), and (5), which present nested models, the use of information increases from model (3) to (5). While model (4) adds regional fixed effects to the pooled panel approach (3), the fixed-effects approach with region-specific time trend (5), on the other hand, additionally includes regional time trends. Thus, the number of estimated parameters and consequently the use of information are the highest for model (5).

A well-known issue occurring when estimating production functions is the possible endogeneity of the production inputs capital and labor, which causes ordinary least squares (OLS) estimates to be inconsistent. A frequently used method for tackling this issue is to apply an instrumental variable approach with two-stage least squares (2SLS) (see, e.g., Dettori, Marrocu, and Paci 2012; Marrocu and Paci 2012; Marrocu, Paci, and Usai 2013). Consequently, the 2SLS approach is adopted, and, similar to Marrocu and Paci (2012) and Marrocu, Paci, and Usai (2013), one-year lagged independent variables are employed as instruments for the production factors. In order to check the appropriateness of the employed instruments due to the high time persistence in the input variables, as an alternative, the three-group approach proposed by Kennedy (2008), and applied by Marrocu and Paci (2011) and Dettori, Marrocu, and Paci (2012) to estimate regional TFP levels, is adopted. Regarding the three-group method, instruments of each factor input take the value of −1, 0, and +1 depending on whether the observations are in the bottom, middle, or top third of the distribution. Results from this method are quantitatively similar to those obtained by employing lagged variables as instruments.

Initially, a simple 2SLS model was considered. However, Moran’s I test on 2SLS residuals as well as robust Lagrange multiplier (LM) tests for both spatial error and spatial lag dependence support the choice of a spatial regression model. Regarding the choice between the spatial error model (SEM) and the spatial lag model (SAR) proposed by Anselin (1988), the former is preferred for two reasons. First, after evaluating robust LM error and robust LM lag tests, the SEM (estimated with 2SLS) appears to be more appropriate than the SAR (estimated with 2SLS). The second reason is based on more content-related considerations: including a spatial error term has the advantage that the concept of the production function remains relatively unaffected. The inclusion of a spatial lag term, on the other hand, requires modification of the production function by adding the spatial lag term, which makes interpretation of the obtained TFP level difficult. Thus, all estimates of models (2) to (5) reported in the remainder of this article are obtained with an SEM estimated with 2SLS. In the estimation procedure, similar to Marrocu and Paci (2011) and Dettori, Marrocu, and Paci (2012), an inverse square distance matrix normalized by its largest eigenvalue is employed. Furthermore, the weighting matrix is truncated at 300 kilometer. This distance is chosen, as Bottazzi and Peri (2003) show that innovation spillovers exist for regions within the distance of 300 kilometer.

As it is not reasonable to expect that the five TFP estimation approaches produce TFP levels and TFP growth rates that are equal in absolute values, the focus is on comparisons in ordinal, rather than cardinal, terms. Thus, ordinal rankings of TFP levels and TFP growth rates obtained from the various approaches are compared. For this purpose, Spearman’s rank correlation coefficients and the distribution of absolute rank differences between TFP rankings of each pair of approaches are analyzed. Such an operationalization is a rather mild claim for the comparability between models.

Results

Model Specification

Estimated output elasticities of capital and labor as well as the estimates of the spatial error coefficient and R² values of models (2) to (5) are reported in Table 1. As the cross-section model (2) is estimated for each time period separately, all values are reported as averages over time. Furthermore, the table shows elasticities of the accounting approach (1) as averages across regions and over time.

Table 1.

Estimated Coefficients of the Five TFP Approaches.

Model	Accounting Approach (1)	Cross- section Approach (2)	Pooled Panel Approach (3)	Fixed- effects Approach (4)	Fixed-effects Approach with Region-specific Time Trend (5)
α (capital coefficient)	.44	.50	.49	.19	.17
β (labor coefficient)	.56	1.00	1.02	.37	.36
Spatial error coefficient	—	.88	.87	.89	.86
(Pseudo) R ²	—	.71	.72	.98	.99
Adjusted R ²	—	.70	.72	.98	.99

Note: Factor elasticities of model (1) are “calculated” values, which are reported as average values across regions and over time. All estimates of the models (2) to (5) are obtained with an SEM estimated with 2SLS. The spatial weight matrix is the inverse square distance between two regions normalized by its largest eigenvalue. All estimates of the models (2) to (5) are statistically significant at the 1 percent level. Coefficients and (pseudo) R² of model (2) are reported as averages over time. (Pseudo) R² is computed as the square correlation between fitted and actual values. Input and output variables are normalized to population in order to control for the size of the regions. SEM = spatial error model. 2SLS = two-stage least squares. TFP = total factor productivity.

The large spatial error coefficients of all four models indicate the need to use a spatial regression model. However, as error terms are relatively small in all models, even large spatial error coefficients are of little relevance for interpreting the final results. The high adjusted R² values of the fixed-effects models (4) and (5) compared to those of models without fixed effects, models (2) and (3), indicate that the introduction of regional fixed effects leads to a considerable increase in the models’ statistical fit, suggesting that TFP is driven largely by regional characteristics. R² values of models (4) and (5), however, differ only marginally, and hence, the models’ statistical fit is of little help in differentiating between these two models. Between models (3), (4), and (5), which present nested models, likelihood-ratio tests are conducted to determine the suitability of considering regional fixed effects, that is the test of model (3) versus (4), and regional time trends, that is the test of model (4) versus (5). Highly statistically significant test results indicate that model (5) is preferable over model (4) and obviously over (3) and hence provide support for approaches that consider more specific features of TFP, that is, using more information through a higher number of estimated parameters.

Regarding the estimated elasticities of factor inputs, both the cross-section (2) and the pooled panel model (3) produce factor elasticities that add up roughly to 1.5, indicating increasing returns to scale and exceeding average elasticities of the neoclassical accounting model (1), which relies on the assumption of CRS and hence, by definition, produces elasticities that add up to unity. On the other hand, the sum of factor elasticities of both fixed-effects models (4) and (5) is evidently smaller (cf. Table 1). The inclusion of regional fixed effects produces smaller coefficients for both factor inputs, which is in line with previous findings in the literature (see, e.g., Dettori, Marrocu, and Paci 2012). By conducting a small simulation study, it can be shown that models without regional fixed effects produce output elasticities that are upwardly biased. On the other hand, the simulation confirms that estimated output elasticities of models including regional fixed effects are, unsurprisingly, close to true values. Based on these results, estimates of models (2) and (3) presented in Table 1 are considered to be severely biased due to model misspecification. A detailed description of the simulation study and the corresponding results will be made available by the authors upon request.

The indication for misspecification of models (2) and (3) is supported by an analysis of the residuals of the different models. Under the assumption of i.i.d. residuals, 50 percent of a region’s residuals are expected to be of positive sign and 50 percent of negative sign. Therefore, the probability of a region having k residuals of the same sign follows a binomial distribution (p = .5, n = 17). For each number of equal residual signs, the expected number of regions is computed, and this (theoretical) frequency distribution is compared to the observed frequency distributions of the different models. Comparison of the observed frequency distributions and the theoretical frequency distribution gives evidence of the severity of the misspecification. The cross-section model (2) and the pooled panel model (3) clearly show signs of being severely misspecified, as residuals of more than half of all regions show the same sign over the entire observation period. The fixed-effects model (4) shows a distribution very similar to the theoretical distribution, while the distribution of model (5) exceeds the expected distribution. This discrepancy between the two fixed-effects models may be due to a higher time autocorrelation of residuals of model (4) compared to model (5).

Finally, we wanted to assess the generalizability of the results or whether severe heterogeneity between regions is present. Therefore, the whole estimation procedure is performed multiple times for subsamples of the regions. More specifically, subsamples are drawn randomly containing either 50 percent of the regions (i.e., 110 regions) or 70 percent of the regions (i.e., 154 regions), and all TFP approaches are applied. For both subsample sizes, 500 subsamples are drawn. The obtained results for both 50 percent and 70 percent subsample sizes are in line with the results obtained using the entire data set. When comparing, for example, the ranking of the fixed-effects approach with region-specific time trend (5) of the full sample with its corresponding subsample rankings, the largest median rank difference of the 500 subsamples is 3.5 (TFP level) and 7.5 (TFP growth) for the 50 percent subsamples and 4.5 (TFP level) and 12 (TFP growth) for the 70 percent subsamples. Comparisons of the other model rankings with their corresponding subsamples lead to similar results.

Comparison of TFP Levels and TFP Growth Rates

Regional rankings of TFP levels and TFP growth rates are compared between the five estimation approaches by means of Spearman’s rank correlation coefficients and absolute rank differences. Table 2 presents the comparison of regional TFP level rankings for each pair of estimation approaches. TFP levels from the last year of the observation period, that is, 2007, are compared. Alternatively, the rankings of mean regional TFP levels are compared with basically unchanged results.

Table 2.

Comparison of Regional TFP Level (2007) Rankings for Each Pair of Estimation Approaches.

Model	Model of Comparison		Distribution of Absolute Rank Differences
Model	Model of Comparison	Spearman’s Rank Correlation Coefficient	25 Percent Quartile	Median	75 Percent Quartile
Accounting model (1)	Cross-section model (2)	.68	13	29	49
Accounting model (1)	Pooled panel model (3)	.67	13	30	53
	Fixed-effects model (4)	.60	12.5	29.5	61
	Fixed-effects with time trend (5)	.65	14	28	55
Cross-section model (2)	Pooled panel model (3)	.98	2	6	11
Cross-section model (2)	Fixed-effects model (4)	.57	12	33	65.5
	Fixed-effects with time trend (5)	.62	10	29	64
Pooled panel model (3)	Fixed-effects model (4)	.64	9	29	63
Pooled panel model (3)	Fixed-effects with time trend (5)	.68	10	26	57
Fixed-effects model (4)	Fixed-effects with time trend	.95	3.5	9	19

Note: TFP = total factor productivity.

While some models produce very similar rankings, other rankings differ substantially from each other, as indicated by Spearman’s rank correlations ranging between .57 and .98 and median absolute rank differences ranging from 6 to 33. By looking at the pairwise ranking comparisons individually, three types of models can be identified: first, the accounting model (1); second, the two models without fixed effects, that is, models (2) and (3); and third, the two approaches including fixed effects (4) and (5). Both models without fixed effects produce nearly identical rankings. Similarly, approaches including fixed effects show rankings that do not differ substantially from each other. However, when comparing one of the fixed-effects models (4) and (5) with one of the models without fixed effects (2) and (3), the differences in the rankings are striking. Finally, the accounting model (1) is in a class of its own, as the rankings differ from all other rankings to a large extent.

A similar pattern is shown by rankings of regional TFP growth, as displayed in Table 3. Again, the same three types of models can be identified. Both fixed-effects models produce very similar rankings. The same holds for the two models without fixed effects. Comparison of the rankings of one of the fixed-effects models with one of the models without fixed effects gives correlation coefficients ranging from .72 to .79. Hence, differences between TFP growth rankings of these models are slightly less evident than between their relative TFP level rankings. Finally, the TFP growth ranking of the accounting model (1) differs substantially from other rankings as shown by very low rank correlation coefficients and high absolute rank differences.

Table 3.

Comparison of the Rankings of Regional TFP Growth Rates for Each Pair of Estimation Approaches.

Model	Model of Comparison		Distribution of Absolute Rank Differences
Model	Model of Comparison	Spearman’s Rank Correlation Coefficient	25 Percent Quartile	Median	75 Percent Quartile
Accounting model (1)	Cross-section model (2)	.20	28	53	95.5
Accounting model (1)	Pooled panel model (3)	.09	31	59.5	109
	Fixed-effects model (4)	.29	24	52	86
	Fixed-effects with time trend (5)	.27	24	54.5	90
Cross-section model (2)	Pooled panel model (3)	.91	5	14	24.5
Cross-section model (2)	Fixed-effects model (4)	.79	7	22.5	45.5
	Fixed-effects with time trend (5)	.72	10	26	53
Pooled panel model (3)	Fixed-effects model (4)	.77	10	26	49
Pooled panel model (3)	Fixed-effects with time trend (5)	.73	11	26.5	50
Fixed-effects model (4)	Fixed-effects with time trend	.97	3	7.5	15

Note: TFP = total factor productivity.

Discussion

The results clearly demonstrate that the model choice between the examined TFP approaches has an essential impact on the obtained results, even though the approaches pertain to a set of similar models. This applies, first, to estimated factor elasticities and, as a consequence, to the attribution of regional (and temporal) disparities to differences either in factor endowments or in TFP.

The same holds for obtained TFP levels and TFP growth rates. Two types of models, firstly the models without fixed effects and secondly the models including fixed effects, produce rankings that differ only marginally from each other. However, comparison of one of the two fixed-effects models and one of the models without fixed effects reveals striking differences between these two model types. In each pairwise comparison, half of the regions change their TFP level ranking by at least 26 ranks and their TFP growth ranking by at least 22.5 ranks. In addition, the rankings of the accounting model (1) differ considerably from the rankings of the remaining models.

The results show that the model choice has a striking impact on both TFP levels and TFP growth rates. This is true not only for TFP estimates but also for regional rankings based on these estimates. This implies that estimates obtained by the different approaches cannot be considered as monotone transformations of each other. Thus, the first research question regarding the comparability of TFP levels and growth rates obtained from various approaches can be answered clearly. The fact that the different models are contemporaneously employed in the literature shows that currently there is no evident state-of-the-art TFP approach. From the apparent differences between the model rankings, it is reasonable to expect that follow-up analyses, for example, regarding the determination of TFP drivers, are affected by the model choice to a considerable extent. This implies that studies relying on TFP estimations have to discuss their model choice in a more considerable manner than they currently do. Furthermore, it cannot be ruled out that some results hold only in connection with a particular model and hence should be interpreted in this context.

The second research question, that is, the choice of a “most suitable” model based on considerations related to theoretical and econometric characteristics of the different models, requires a more differentiated view. The conventional accounting model (1) is in a class of its own, as the rankings of TFP levels and TFP growth rates differ from rankings of the remaining models to a large extent. The accounting model has three major shortcomings: first, as a purely deterministic approach, the model, by definition, has no stochastic component and hence incorporates any type of error into the results. Furthermore, the lack of an error term prevents the quality of the model from being evaluated. Second, the model relies on the restrictive assumptions of perfect competition and CRS, which are doubted to adequately represent real-world economies (Hall 1988; Roeger 1995). Third, as the accounting model (1) allows output elasticities to vary over time and across regions, its traditional measure of TFP (in terms of A) is incomplete as productivity varies with the TFP term A and with the factor exponents (Bernard and Jones 1996). As a consequence, comparing TFP levels (in terms of A) and TFP growth rates across observation units can produce misleading implications. Results of model (1) as well as the comparison of results with those of the four remaining approaches should therefore be treated with caution.

The latter argument holds also for the cross-section model (2), where factor elasticities are allowed to vary over time and hence TFP growth rates should be interpreted cautiously. In addition, the regional variation of TFP of the cross-section model (2), along with the pooled panel model (3), is determined only by the i.i.d. error term. By including the error term in the TFP measure, unwanted components such as measurement errors are incorporated in TFP obtained by these models. The analysis of residuals of both approaches provides evidence that both models are heavily misspecified. The argument of misspecification is supported by the results of the simulation study, which show that estimated output elasticities are biased upward in these approaches. The evidence for misspecification disappears when regional fixed effects are introduced. Due to the arguments just mentioned, models (1) to (3) seem to be much less suited to analyze regional differences in TFP levels and growth rates than are the fixed-effects models (4) and (5). This is supported by empirical results of the present work, as TFP estimates of former models using less information differ considerably from estimates of the latter, more complex approaches, indicating that by ignoring parts of the available information estimates are substantially biased.

Choosing between the remaining two models (4) and (5) is quantitatively less relevant, as both approaches produce similar results regarding TFP levels and TFP growth rates. Both approaches obtain TFP levels as model parameters of the production function, which allows for an alternative interpretation of TFP than that of a residual. By estimating TFP levels directly without considering the i.i.d. error term, TFP of these approaches should be free from unwanted components, which will be included in the error term. However, the fixed-effects approach (4) estimates only TFP levels as model parameters, while heterogeneous TFP growth rates are obtained by including the i.i.d. error terms into the TFP measure. In contrast, the fixed-effects approach with region-specific time trend (5) estimates TFP growth rates, in addition to the TFP levels, as individual components of the production function. Thus, confidence intervals and information on statistical significances of both TFP estimates are obtained. Furthermore, model (5) has the advantage that it endogenously determines two parts of a region’s TFP change, that is, a time component affecting all regions simultaneously and a region-specific, long-term growth rate. In addition, model (5) reduces the time autocorrelation of the residuals. These characteristics provide an interesting approach for further regional economic research.

Conclusion

TFP levels and their temporal changes are highly attractive measures for determining the efficiency of regional production processes. As the literature provides a large number of approaches to estimate TFP levels and growth rates, researchers have to make a choice within the set of available approaches. However, as the present work has shown, the model choice is by no means independent with respect to the obtained results. Thus, it is reasonable to expect that follow-up analyses, for example, regarding the determination of TFP drivers, are affected by the model choice to a considerable extent. In this context, a detailed justification of a particular model choice is absolutely essential, and results obtained with a specific approach have to be tested for sensitivity in a more profound manner than is currently the case.

A further implication for future research refers to the two different theoretical concepts of TFP discussed in the previous sections. Approaches that obtain TFP as the residual of output that cannot be explained by inputs such as models (1), (2), and (3) produce TFP estimates that differ considerably from those of more sophisticated models that estimate TFP as a model parameter of the production function such as the fixed-effects models (4) and (5). Follow-up analyses based on TFP estimates of former approaches risk obtaining blurred results when analyzing the relationship between regional productivity and potential drivers. In the worst case, nonexisting relationships may be accepted as “true,” while other substantial relationships may be overlooked. Thus, future works in regional science should avoid TFP approaches based on the residual interpretation of TFP and instead use more sophisticated approaches.

After analyzing the shortcomings of the examined models, the fixed-effects approach with region-specific time trend (5) is proposed. This approach allows estimating endogenously regional TFP levels, time changes in TFP affecting all regions simultaneously, and region-specific TFP growth rates. In addition, the model avoids the identified shortcomings of the remaining approaches. As both regional TFP levels and TFP growth rates are estimated as individual model parameters, confidence intervals and information on statistical significances of both TFP estimates are obtained. This provides some new opportunities for analyzing regional production processes and, consequently, for future work in regional science. For example, follow-up works investigating drivers of TFP growth may focus on regions with significant positive TFP growth rates, which should lead to more robust results. Similarly, convergence analyses based on regional TFP estimates may benefit from information on the statistical significance of TFP levels and growth.

A prerequisite of the proposed approach is the availability of a data set that has to be “sufficiently long” in the time series dimension in order to adequately estimate region-specific, long-term TFP growth rates. This condition, however, holds for alternative approaches as well, as the estimation of TFP growth rates over a short time horizon is not very instructive. If such a time series is not available, growth rates cannot be estimated in a robust manner. In this case, an estimation of TFP levels based on the fixed-effect approach (4) is the first choice. Alternatively, if only one observation period is available, the application of an approach that estimates TFP levels as a residual such as the cross-section model (2) is inevitable, even though there is a considerable risk of obtaining regional TFP levels blurred by too much noise. Finally, it has to be noted that the proposed fixed-effects approach with region-specific time trend (5) must be tested and validated externally similar to the alternative models presented in this work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Note

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