Carsten Pohl
email: pohl@ifo.de

Gerhard Kempkes
email: gerhard.kempkes@tu-dresden.de

Some Evidence from Non-Parametric and Parametric Methods

Keywords: Universities, Efficiency, Data Envelopment Analysis, Stochastic Frontier Analysis Universities, E¢ ciency, Data Envelopment Analysis, Stochastic Frontier Analysis I21, M11

The Efficiency of German Universities -Some Evidence from Non-Parametric and Parametric Methods

Introduction

Human capital is an important factor for economic growth and consequently the e¢ ciency of the educational system plays a major role in the political and scienti…c discussion. The number of students as well as their educational achievements are crucial in the formation of human capital and thus for the future competitiveness of the German economy.

The vast majority of German universities are publicly …nanced. Due to tight public budgets, university …nance has come under severe pressure in recent years. 1 The share of university spending to total government spending has decreased from about 6.6% in 1975 to about 5% in 2003 while the number of students has more than doubled. In the [1975][1976][1977][1978][1979][1980][1981][1982][1983][1984][1985][1986][1987][1988][1989][1990] period aggregate university spending has increased by about 2.2% on average in real terms. Spending growth from 1998 to 2004 has decreased to a yearly average of about 1.3% while student numbers have increased by 2% in the same period. 2 Moreover, international comparisons show that in Germany public spending on tertiary education as a share of GDP is below the OECD mean. Consequently, there is a public and academic discussion about more private funding in the German university landscape. In this context, many federal states are currently introducing tuition fees in public universities.

While additional fund raising represents an option to improve the …nancial situation of the universities, we focus on the e¢ ciency of public universities in Germany. Regardless of privatization or tuition fees, information about university e¢ ciency performances is essential in times of scarce public resources. For instance, knowledge about university e¢ ciency enables university management to recognize shortcomings and improve their performance.

In addition, state governments, which are responsible for …nancing universities in Germany, can use e¢ ciency indicators as a guideline for the distribution of funds among the universities and thus create an atmosphere of competition within the public university system.

1 University …nance is a major responsibility of the German states, which account for about 90% of spending on higher education. The federal level contributes about 10% to university …nance while the local governments are not involved at all.

2 German reuni…cation complicates pre-1990 and post-1990 comparisons. In addition, in 1997 a major change in the university statistics was introduced that makes it virtually impossible to compare data from the period before 1997 to data after 1997. The vast majority of e¢ ciency analyses for higher education institutions have employed data envelopment techniques. 3 Due to the fact that universities are producers of at least two outputs -teaching and research -DEA seems to be an appropriate method. Early studies have predominantly examined the performance of single departments across universities since these are supposed to have similar structures. For instance, admitted research grants are subject to a faculty bias because some departments are more inclined to receive earnings from research grants (e.g. medicine or engineering) than others (e.g. languages).

The …rst studies were conducted for universities in Anglo-Saxon countries. [START_REF] Johnes | Research Funding and Performance in U[END_REF] performed a cross-sectional investigation on the e¢ ciency of economics departments in 36 British universities in the year 1989. [START_REF] Beasley | Determining Teaching and Research E¢ ciencies[END_REF] also considered a single year (1992) to study the relative e¢ ciency of chemistry and physics departments of 32 British universities. [START_REF] Madden | Measuring Public Sector E¢ ciency: A Study of Economics Departments at Australian Universities[END_REF] used panel data in order to evaluate the e¢ ciency of economics departments at Australian universities.

More recent studies have focused on evaluating entire universities since it is often associated with the availability of panel data in the …rst place, which in turn permits a study of e¢ ciency change over time. [START_REF] Athanassapoulos | Assessing the Comparative E¢ ciency of Higher Education Institutions in the UK by Means of Data Envelopment Analysis[END_REF] analyzed the relative e¢ciency of 45 British universities for the years 1992/1993, indicating that there are signi…cant di¤erences across higher education institutions. [START_REF] Abbott | The E¢ ciency of Australian Universities: A Data Envelopment Analysis[END_REF] considered 36 Australian public universities for their investigation. [START_REF] Flegg | Measuring the E¢ ciency of British Universities: A Multi-period Data Envelopment Analysis[END_REF] used data envelopment analysis to examine changes in technical e¢ ciency of 45 British universities in the period 1981-1993.

However, whereas DEA is able to capture multiple outputs and multiple inputs at the same time, this method has its drawbacks. First, DEA does not account for stochastic noise in the data. For instance, the results may be severely biased when measurement errors are present. Second, in the DEA approach the heterogeneous structure of the university landscape -i.e. the department composition may di¤er considerably from one university to 3 Worthington (2001) provides an extensive review on the e¢ ciency analysis of universities. (2005), these studies only partially accounted for faculty composition by distinguishing between arts students and science students. One drawback in the SFA is that it requires an assumption regarding the functional form of the cost or production function. The selection of a function is not a clear-cut task in higher eductaion as [START_REF] Kraus | Schätzung von Kostenfunktionen für die bundesdeutsche Hochschulausbildung: Ein konzeptioneller Ansatz im empirischen Test[END_REF] points out.

The e¢ ciency of German higher education institutions has only recently been investigated

by [START_REF] Warning | Performance Di¤erences in German Higher Education: Empirical Analysis of Strategic Group[END_REF] and Warning (2005) applying DEA techniques. In these studies, the number of graduates as well as admitted research grants served as outputs. In addition, di¤erentiated information on SSCI (Social Science Citation Index) and SCI (Science Citation Index) publication were taken as a research output whereas current expenditure and the number of professors were considered as inputs. However, possible changes in e¢ ciency and/or technical change cannot be captured by these investigations due to the use of crosssectional data. In particular, the use of cross-sectional data can be considered a problem in analyzing university e¢ ciency due to the volatile output of publications or PhD completions.

Research output often varies considerably over the years. Thus, e¢ ciency scores based on cross-sectional data of adjacent periods may yield quite di¤erent results. The use of panel data should mitigate this problem.

Since our data set encompasses the period 1998-2003, we are able to extend the scope of e¢ ciency analysis for the German university landscape. We will not only indicate the relative e¢ ciency of universities for a speci…c year but also show to what extent total factor productivity has changed over time. In addition, we control for faculty composition of universities. Moreover, the e¢ ciency scores from DEA and SFA will be compared in our study. Thus, we can o¤er a comprehensive picture on the e¢ ciency of public higher education institutions in Germany. In this section we will …rst describe our data set as well as specify our input and output variables. In a second step we will provide descriptive statistics on the universities in Germany.

We use data on public universities for the years 1998-2003 provided by the Federal Statistical O¢ ce of Germany. We con…ne our analysis to public universities since we are especially interested in the e¢ ciency of public universities as set out in the introduction. In addition, private universities in Germany are highly specialized, i.e. oriented towards business management and/or medical studies, so that their inclusion represents a possible source of bias. For the same reason we also drop other specialized universities such as universities of …ne arts and music. Universities of applied science are also excluded from our investigation since these are more oriented towards teaching instead of research. In particular, universities of applied science are in general not enabled to train doctoral students so that their consideration would have created a more heterogeneous sample.

The selection of input and output variables shows only limited variation in the studies on e¢ ciency analysis of universities, since a university is in general assumed to accomplish two major duties. On the one hand, it is responsible for the production of human capital of the enrolled students. For this reason, we use the number of graduates as an approximation of the teaching output (see Table 1). On the other hand, the university serves as an organization in the …eld of research and development. Higher education institutions are involved in applied research, e.g. cooperating with private enterprises, as well as fundamental research in order to create knowledge. Research grants may be considered as a market price that gives information on the quality as well as on the quantity of research output, e.g. see [START_REF] Johnes | Costs and Industrial Structure in Contemporary British Higher Education[END_REF] and [START_REF] Koshal | Economics of Scale and Scope in Higher Education: A Case of Comprehensive Universities[END_REF].5 (2005).

For an overview on the …nancial as well as personnel structure of German universities we report descriptive statistics in Due to di¤ering faculty compositions, the universities in our sample are quite heterogeneous. Table 3 reveals these dissimilarities by grouping universities into sub-samples according to their faculty structure. Speci…cally, universities with engineering and medical faculties seem to have a di¤erent …nancial and personnel structure than universities without these two faculties. In the second line of Table 3 we included all 72 universities and calculated three different ratios with regard to the number of graduates. On average, a single university spends about 66,700 euros per year and per graduate in the considered time period 1998-2003. When restricting the sample to universities that have an engineering and a medical department, we …nd expenditures of 92,200 euros per graduate. Accordingly, higher education institutions without such cost-intensive faculties clearly spend the least money per graduate (17,800 euros). In addition, not only the …nancial but also the personnel endowment is higher among universities that o¤er studies in engineering and/or medicine. In microeconomic theory it is usually assumed that production units operate e¢ ciently. In particular, the decision making unit (DMU), e.g. a company or in our case a university, is supposed to allocate available capital and labor in such a way that no increase in the output level is possible without adding more inputs. Accordingly, given a speci…c output level, the DMU is supposed to use all inputs e¢ ciently, i.e. no reduction in inputs is possible without diminishing the output level. However, in reality companies are subject to ine¢ ciencies in the production of outputs as [START_REF] Coelli | An Introduction to E¢ ciency and Productivity Analysis[END_REF] indicate.

In order to detect ine¢ ciencies within companies in an industry, data envelopment analysis provides a suitable method. This non-parametric approach, which is based on linear programming assumes that the e¢ ciency of a production unit can be measured by calculating the ratio of (weighted) outputs over (weighted) inputs. In other words, the main idea is to construct a non-parametric frontier over the available input and output data of the considered production units. The e¢ ciency of a particular university can then be calculated relative to this frontier. Due to this general approach no functional form has to be assumed, which represents a considerable advantage compared to parametric methods. However, it is important to notice that only relative and not absolute e¢ ciency is measured, i.e. the e¢ ciency of a particular university is only calculated relative to the e¢ ciency of the other higher education institutions in the sample. E¢ cient production units lie on the frontier whereas ine¢ cient DMUs are envelopped by the frontier.

The choice of the input-or output-oriented approach in DEA is basically subject to the (possible) control of the administrator of the university. In the case of publicly funded higher education institutions in Germany, current expenditure as well as research and technical sta¤ can be considered as given. Hence, the output oriented approach seems to be appropriate for our investigation, i.e. we assume that given the amount of inputs, the university pursues the maximization of its outputs. the e¢ cient frontier. Accordingly, an e¢ ciency score below 1 implies that the university is ine¢ cient compared to the other higher education institutions in the sample. 6 The vector y rj stands for the output r of university j whereas the vector x ij represents the input i of university j (see equation 1). The parameter j indicates the university-speci…c weight for the input and output factors, which are determined endogenously. Hence, the linear optimisation problem is solved, resulting in an e¢ ciency score e k for each university.

Restriction one and two in our DEA model (1) state that the reference universities produce at least as much of all outputs as the observed university. At the same time e¢ cient universities do not consume as much input as the observed university. In addition, we introduce the restriction that the sum of the parameter j equals 1, ensuring that variable returns to scale are possible. Otherwise constant returns to scale would be the benchmark case, assuming that every university already operates on the optimal scale level.The fourth and …fth restriction indicate that the weights as well as the inputs and outputs have to be positive, respectively. Since we are assuming, an output-oriented approach the ine¢ cient university would have to increase its output by the factor 1 e k in order to attain the e¢ cient frontier. x ij j x kj (i = 1; 2; :::; m)

n X j=1 j = 1 (j = 1; :::; n) j 0 y rj ; x ij 0 (i = 1; 2; :::; m); (r = 1; 2; :::; s); (j = 1; :::; n)

Due to the availability of a panel data set, we are not only interested in the relative performance of a single university in a particular year but also in how the e¢ ciency of 6 See [START_REF] Banker | Returns to Scale in Di¤erent DEA Models[END_REF] for a detailed presentation of the model. universities have developed over time. Hence, we apply the Malmquist index, which is able to capture total factor productivity change from one year to another. 7

The Malmquist index is constructed in such a way that the radial distance of observed output and input vectors in periods t and t + 1, relative to a reference technology, is measured. The output-oriented approach of the Malmquist index considers the maximum level of outputs with a given input vector and a given production technology relative to the observed outputs. To measure the distance between realized and hypothetical output, we use distance functions (d t and d t+1 ).

Applying the Malmquist index, the ratio of the distances of each data point relative to a common technology is calculated. In other words, we measure total factor productivity change over time. In order to avoid the necessity of choosing one of the time points as reference period, the geometric mean is calculated. A value of m greater than one indicates positive TFP growth from period t to period t + 1, while a value less than one stands for a TFP decline.

m(y t+1; x t+1 ; y t ; x t ) = d t (y t+1 ; x t+1 ) d t (y t ; x t ) d t+1 (y t+1 ; x t+1 ) d t+1 (y t ; x t ) 1=2 (2)
The use of the Malmquist index not only permits to measure total factor productivity change but we are also able to decompose this change into technical e¢ ciency change and technical change, rearranging equation ( 2) as follows:

m(y t+1; x t+1 ; y t ; x t ) = d t+1 (y t+1 ; x t+1 ) d t (y t ; x t ) d t+1 (y t+1 ; x t+1 ) d t (y t ; x t ) d t (y t+1 ; x t+1 ) d t+1 (y; x) 1=2 (3)
The …rst term on the right-hand side in this equation represents the e¢ ciency change while the term in brackets measures technical change.

Results

The results of the data envelopment analysis of German universities in the year 2003 are presented in Appendix A1 with an ordering of universities by their size, i.e. by the number of students. From the distribution of the e¢ ciency scores, we conclude that large universities (e.g. University of Cologne, University of Munich) as well as small universities (e.g. TU Clausthal, University of Vechta) are operating on the e¢ cient frontier. Hence, the size of a university is not necessarily associated with its e¢ ciency.

With regard to the regional distribution, we …nd that some universities in East Germany are operating on the e¢ cient frontier, e.g. FU Berlin, TU Berlin while others are ine¢ cient, e.g. University of Magdeburg, University of Greifswald.8 On average, the East German universities are less e¢ cient than their West German counterparts.

Since data envelopment analysis does not consider the faculty composition of each university, e¢ ciency scores are likely to su¤er from a faculty bias. In descriptive statistics we found that universities with a medical or engineering department display higher costs than the average university. Hence, in order to detect the impact of the faculty composition, we extend our investigation by taking university speci…c as well as environmental characteristics into account.

In a …rst step of our cross-sectional analysis, we calculated e¢ ciency scores for every university. In a second step we regress the obtained e¢ ciency scores on regional gross domestic product per capita. 9 We test the in ‡uence of regional GDP per capita on university e¢ ciency since we consider GDP per capita an overall proxy for the characteristics of university location. The idea is that there might be bene…cial or adverse e¤ects from university location on e¢ ciency due to spillovers. In particular, cooperations with research intensive companies in the region as well as the existence of laboratories, research institutions and big libraries or think tanks might result in positive spillover e¤ects in regions with high GDP per capita.

Alternatively, GDP per capita could prove to be cost enhancing due to wage/price elevating e¤ects in agglomeration areas. In addition, we include dummy variables in the regression since only about 75% of them can be characterized as districts ("Kreise"), comprising rural areas as well as villages and smaller cities. In contrast, the remaining quarter can be referred to as larger cities ("Kreisfreie

Städte") This institutional peculiarity makes local GDP per capita a poor proxy for the location e¤ect that we want to test because local GDP per capita is systematically biased downwards in cities that include surrounding areas. Thus, we choose regional GDP per capita at the level of "Raumordnungsregionen", which is supplied by the [START_REF] Bbr | Inkar pro CD ROM, Bundesamt für Bauwesen und Raumordnung[END_REF] 11 in order to control for the existence of an engineering and/or medical department for the reasons explained above. On the one hand, these two faculties are more expensive than other faculties; on the other hand, they earn more research grants on average than other faculties.

Moreover, especially medical faculties (e.g. university hospitals) have a di¤erent structure when compared to other departments.

Since the e¢ ciency scores are right-censored, we run Tobit regressions and for sensitivity checks also conduct ordinary least squares (OLS) regressions. 10 The estimation results reveal that the coe¢ cient for gross domestic product per capita is signi…cant but only explains a small amount of the variation (see Table 4). Thereby, gross domestic product per capita has a positive in ‡uence on the e¢ ciency of the universities. Hence, institutions of higher education that are located in economically prosperous regions are likely to bene…t from the environment through spillover e¤ects. This …nding holds true for the OLS as well as for the Tobit regression. Including the dummy variables for engineering and medical faculties, we …nd both coe¢ cients to be signi…cant and negative. The existence of one of these faculties has a signi…cant impact on the e¢ ciency of a university. Hence, when analyzing relative e¢ ciency among German universities the faculty composition should be taken into account.

Table 4 about here

In Appendix A2 we report results from the Malmquist index. On average, universities in East Germany display higher scores in total factor productivity change than their West 5 Stochastic Frontier Analysis

Model Speci…cation

Since parametric techniques allow us to control for faculty composition by including dummy variables, we conduct a stochastic frontier analysis. As aforementioned, e¢ ciency analysis in higher education has to take into account at least 2 outputs -research and teaching. This renders impossible estimating a production function, although the output-oriented approach is an appropriate assumption for German universities (see section 4.1). Thus, to include multiple outputs, we select the cost function approach. 11 However, choosing a functional form for the cost function is not straightforward in higher education as pointed out by [START_REF] Kraus | Schätzung von Kostenfunktionen für die bundesdeutsche Hochschulausbildung: Ein konzeptioneller Ansatz im empirischen Test[END_REF]. A functional form that o¤ers a ‡exible functional relationship and especially allows factor substitution to be unrestricted should be applied. We use the translog cost function, which was also considered appropriate by [START_REF] Stevens | A Stochastic Frontier Analysis of English and Welsh Universities[END_REF], for our investigation: 11 We choose to estimate a cost function compared to a distance function as we prefer to interpret the coe¢ cients -especially the coe¢ cients from the faculty controls -in the intuitive context of a cost function.

13

= + t + 2 X j=1 j ln Q jit + 1 2 2 X j=1 2 X k=1 jk (ln Q jit ln Q kit ) + 1 ln w it + 2 1 2 (ln w it ) 2 + 2 X j=1 3j (ln w it ln Q jit ) + ! 1 M ED it + ! 2 M ED it ln w it + 2 X j=1 ! 3j (M ED it ln Q jit ) (4) +' 1 EN G it + ' 2 EN G it ln w it + 2 X j=1 ' 3j (EN G it ln Q jit ) + ! 4 M ED it EN G it + u it + v it
In equation ( 4) i denotes universities. The sample period is from 1998 to 2003 and is referred to as t. C it represents the costs in university i and time period t. As discussed above we choose total costs less research grants as our cost variable. Q it denotes the same research and teaching outputs as described in the DEA section (graduates and research grants, j).

Costs as well as research grants and the number of graduates are normalised by the number of students. As to wage or price information (w it ) we only have a limited access to data.

Thus, we follow [START_REF] Stevens | A Stochastic Frontier Analysis of English and Welsh Universities[END_REF] dividing total wage spending by the number of employees to get a proxy of wages. Wages for university employees are regulated at the federal level in Germany. However, our approach is able to capture di¤erences in the structure of sta¤ across universities: While some universities might prefer to employ a larger quantity of assistant professors and fewer full professors, other universities might have a higher density of full professors with a smaller number of total research/teaching sta¤. M ED and EN G are dummy variables that control for the faculty composition of universities, speci…cally for medical and engineering faculties. As a sensitivity check, we also estimate the model, excluding the controls for faculty structure. Additionally, we include a constant ( ) and a linear time trend ( t ) to account for technological change. 12 v it in equation (4) denotes an error term which is i.i.d. N (0; 2 v ) and independent of u it , where u it represents a non-negative random variable that is assumed to display cost ine¢ ciency in the production of teaching/research in university education. Speci…cally, u it 12 Unfortunately, we are not able to include a proxy for incoming students'educational background. This information is impossible to obtain in Germany as there is no general admission test procedure. displays total economic ine¢ ciency, i.e. technical ine¢ ciency plus allocative ine¢ ciency. We follow the [START_REF] Battese | A Model for Technical Ine¢ ciency E¤ects in a Stochastic Frontier Production Function for Panel Data[END_REF] methodology assuming u it to be independently distributed and following a truncated normal distribution: 13 N ( it ; 2 u ). The [START_REF] Battese | A Model for Technical Ine¢ ciency E¤ects in a Stochastic Frontier Production Function for Panel Data[END_REF] model permits to test hypotheses with respect to the determinants of ine¢ ciency, i.e.

it is assumed to be determined by Z it variables:

it = Z it (5)
Z it denote structural/environmental variables that might play an important role in the production of teaching/research but that typically cannot be in ‡uenced directly by the university. We select regional GDP per capita as a proxy for bene…cial or adverse e¤ects from university location (see section 4.2). Moreover, we include a linear time trend as an environmental variable to account for e¢ ciency change over time. Information about the share of deviations from the cost function that are due to ine¢ ciencies is reported by:

= 2 u 2 u + 2 v (6)
Consequently, if is zero, all deviations are caused by noise rather than ine¢ ciency; if is one, all deviations are due to ine¢ ciencies.

Results

Table 5 reports the results from the estimation of the cost function. The estimation results indicate that the controls for faculty composition, i.e. the dummies for engineering and medical faculties, are signi…cant at the 5% and 1% level, respectively. The speci…cation with faculty dummies is also strongly supported by a likelihood ratio test, which rejects the null hypothesis of the dummy variables and interaction terms being restricted to zero at every conventional signi…cance level (see Table 6). Thus, we consider the model with faculty 13 Truncated at zero to display ine¢ cient performances "above"the estimated cost function, see also Coelli controls as our baseline model. As suggested by the descriptive statistics, costs in universities with these faculties are higher than in universities without engineering or medical faculties.

In addition, several of the interaction terms of faculty dummies with output/wage variables are signi…cant, indicating that universities with medical and/or engineering faculties not only have di¤erent cost levels but also di¤erent marginal cost structures. For instance, research grants only substitute for state money in universities with engineering faculties whereas this e¤ect is not signi…cant in universities without medical and engineering faculty.

In universities with medical faculties, research grants even crowd in additional state funds. 14 Not controlling for the faculty composition of universities signi…cantly biases the estimation results and thus the predictions for university level e¢ ciency scores. This can be seen by comparing the two models in Table 5. For instance, in the model without controls for faculty composition, research grants have a highly signi…cant and highly positive e¤ect on total costs less research grants (4:41). However, after controlling for faculty structure, this e¤ect decreases signi…cantly in size and turns insigni…cant. Moreover, a negative e¤ect of the number of graduates on total costs disappears after controlling for faculty structure.

Both models suggest that there is a strong negative e¤ect of wages per employee on costs.

The model without dummies indicates that a one percent increase in wages per employee results in a 12:7% decrease in costs per student. Of course, this e¤ect could be due to cost savings that arise from high quality sta¤. However, after controlling for faculty composition, this e¤ect declines to about 7:3%. Thus, faculty composition accounted for nearly half of this e¤ect. Medical and engineering faculties employ an increased number of lower cost sta¤ when compared to other faculties, e.g. nurses or technical sta¤. At the same time, these faculties have higher costs as shown in the descriptive statistics. Accounting for faculty composition therefore reduces this cross-section variation. In part, the remaining e¤ect could be due to di¤erences between social sciences and sciences. The model with faculty dummies as well as the model without dummy variables both indicate that there are economies of scope between teaching and research. This e¤ect increases in size and signi…cance after controlling for faculty structure ( 0:46). In both models the gamma coe¢ cient is high. After controlling for faculty structure the share of deviations that is due to ine¢ ciencies increases from 95% to 99%, indicating almost all deviations from the cost function are caused by ine¢ ciency.

Table 7 presents the results for the in ‡uence of environmental variables on university e¢ ciency. Regional GDP per capita has a positive e¤ect on e¢ ciency suggesting that universities might indeed bene…t from positive spillover e¤ects as discussed in section 4.2. This prediction is in accordance with the results from the Tobit regression (see Table 4). Recall that e¢ ciency scores that are based on a cost function are bound between 1 and 1. An e¢ ciency score of 1 denotes an e¢ cient performance. The e¢ ciency scores obtained from the DEA are bound between 0 and 1 where 1 indicates an e¢ cient performance. Thus, in fact, the results from the Tobit regression and the determinants of ine¢ ciency in the [START_REF] Battese | A Model for Technical Ine¢ ciency E¤ects in a Stochastic Frontier Production Function for Panel Data[END_REF] model both indicate that regional GDP per capita has a bene…cial e¤ect on university e¢ ciency. However, this e¤ect is not statistically signi…cant at reasonable levels of signi…cance in the SFA speci…cation. The coe¢ cient of the time trend suggests that universities improved their e¢ ciency in the sample period. But again, this e¤ect is only signi…cant at the 15% level in the baseline model and at the 10% level in the model without faculty controls Table 7 about here Appendix B reports the predictions for the university level e¢ ciency scores based on our baseline model. We do not discuss the university level e¢ ciency scores in detail. However, the e¢ ciency predictions suggest that East German universities improved their e¢ ciency performance considerably from 1998 to 2003 and are catching up with the West German universities (see Table 8). However, West German universities were still more e¢ cient in 2003. Although data envelopment and stochastic frontier analysis are quite di¤erent approaches to e¢ ciency analysis and although our models di¤er considerably in assumptions and even in input and output de…nitions, we found many similarities concerning the e¢ ciency performance of German universities. This is also con…rmed by the correlation coe¢ cients for the e¢ ciency scores obtained from DEA and SFA, which are signi…cantly and positively correlated by up to 60%.

One robust result of our investigation is that East German universities only appear in medium ranking positions, i.e. in the year 2003 they are still less e¢ cient than the higher education institutions in West Germany. This …nding is also consistent with the crosssectional evidence presented by Warning (2005) for the year 1998. However, focusing on total factor productivity in the years 1998 to 2003 we …nd that East German universities have outperformed their West German counterparts. Thereby, both the Malmquist index as well as the SFA models suggest that the main determinant has been e¢ ciency change rather than technological change.

With regard to university location, Tobit regression results as well as our SFA model indicate that regional GDP per capita has a small but positive impact on the e¢ ciency of higher education institutions. Thus, universities that are located in a relatively rich region bene…t from their environment probably due to spillover e¤ects.

Accounting for faculty composition of German universities revealed that those institutions with medical and/or engineering faculty not only have higher cost levels but also di¤erent marginal cost structures. In our stochastic frontier analysis, not accounting for university faculty composition biases the cost function estimates in several aspects and thus gives biased e¢ ciency scores. Besides the evidence presented for German universities, our discussion of the results from Tobit regression of DEA e¢ ciency scores as well as the introduction of faculty controls in the stochastic frontier analysis might be an important contribution to the e¢ ciency analysis of higher education institutions. At least for German universities, Tobit regression and SFA faculty controls suggest that DEA is not an appropriate technique to analyse e¢ ciency at the level of entire universities. While it may be appropriate for the e¢ ciency analysis of single university departments, we suggest that entire universities are too heterogeneous to be compared using non-parametric methods. The present empirical investigation is the …rst approach to apply data envelopment as well as stochastic frontier techniques to a panel data set of 72 German universities over the years 1998-2003. We …nd that accounting for faculty composition is fundamental for obtaining unbiased e¢ ciency scores. Thus, our model and the evidence presented for Germany suggest that results from earlier studies, which analysed the e¢ ciency of entire universities without accounting for faculty structure might be biased.

With respect to German universities, our estimation results indicate that total factor productivity has been increasing more rapidly in East German universities compared to their West German counterparts. Due to the upcoming demographic changes -the number of high-school graduates in East Germany is expected to decrease by about 40% within the next 15 years -the universities in East Germany must continue their good dynamic e¢ ciency performance. On a university level West German universities appear at the top end of our e¢ ciency rankings.

E¢ ciency rankings of German universities have several channels by which university performance could be improved. On the one hand, university management is informed about its own performance in the …rst place and can thus implement measures to improve e¢ cient spending of funds. On the other hand, state governments could distribute at least a share of public funds according to university e¢ ciency performance, rewarding e¢ cient allocation and creating an atmosphere of competition in the German public university system. Moreover, the current introduction of tuition fees will bring about a third channel: Students will demand that their fees be spent e¢ ciently. In short, the very existence of e¢ ciency rankings of German public universities and a recurrent update could contribute to improve the allocation of public resources in higher education.

H 0 :! 1 =! 2 =! 31 =! 32 =' 1 =' 2 =' 31 =' 32 =! 4 = 0 21.67 437.5
The LR test statistic is given by -2{ln[Likelihood(H 0 )-ln[Likelihood(H 1 )]}, where H 0 denotes the model without dummy variables and H 1 denotes the model including faculty dummies. This statistic has asymptotically a chi-square distribution with degrees of freedom equal to 9 (parameters assumed to be zero), see [START_REF] Berndt | The Practice of Econometrics. Classic and Contemporary[END_REF]. The ln(Likelihood) can be obtained from

y

  rj j e k y rk (r = 1; 2; :::; s) n X j=1

9

  Page

7

  See Coelli et al. (2005) for a detailed presentation of the Malmquist index.

  German counterparts. For instance, the University of Potsdam or the TU Cottbus display values in TFP change above one. Note that no university in East Germany shows a TFP change below one, whereas some universities in West Germany, e.g. University of Köln or University of Passau, show a relatively strong decline. However, there are also West German universities that improved their total factor productivity during the considered time period, e.g. TU Darmstadt or TU Braunschweig.10 Simar and Wilson (2006) indicate that results from the second stage, i.e. the regression of productive e¢ ciency on environmental variables, might be subject to serial correlation within the e¢ ciency estimates.They propose a double bootstrapping procedure that improves statistical e¢ ciency in the second stage regression. However, in our study the second stage regression is accompanied by the estimates of the[START_REF] Battese | A Model for Technical Ine¢ ciency E¤ects in a Stochastic Frontier Production Function for Panel Data[END_REF] model, which can be considered as a sensitivity check to the results from the Tobit regression (see section 5). Disentangling total factor productivity change into e¢ ciency change and technical change, we …nd that the change in e¢ ciency represents a main determinant of TFP change. Obviously, universities in Germany improved the allocation of …nancial means and/or adapted their spending, which resulted in a positive TFP change. In contrast, technical change was considerably low in the considered periodTo summarize, the DEA results show that universities in East Germany have improved their e¢ ciency considerably in the sample period. In this respect they also outperform the West German universities. With regard to the e¢ ciency level, we …nd higher education institutions in the West to appear in top positions. Since Tobit regression results indicate that there are important di¤erences in e¢ ciency between universities that are due to university structure, in the following sectio we employ parametric techniques, which allow us to control for faculty composition by including dummy variables.

  Acknowledgements: We thank Martin Beck, Federal Statistical O¢ ce of Germany, for providing the data as well as for his helpful remarks. Moreover, we acknowledge support from Christian von Hirschhausen, Helmut Seitz, Marcel Thum and especially Astrid Cullmann and Borge Hess for valuable comments and intensive discussions. We thank the participants of the 1 st Halle E¢ ciency and Productivity Analysis Workshop (HEPAW), June20-21 (2006), especially Tim Coelli and Matthias Staat for very helpful remarks.

  t-statistics in parentheses, p-value: * = 0.10, ** = 0.05, *** = 0.01 Sources: All data has been supplied by the Federal Statistical O¢ ce of Germany with the exception of GDP data, which has been taken from Bundesamt für Bauwesen und Raumordnung(2006). C, RG and W have been de ‡ated using the government consumption de ‡ator supplied by the German Council of Economic Experts (2005); GDP has been de ‡ated using the GDP de ‡ator from the same source.

  another -cannot be adequately taken into account.4 Stochastic frontier analysis permits accounting for faculty composition by including dummy variables for faculties. Although recent investigations applied SFA, e.g. Izadi,Johnes, Oskrochi, and Crouchley (2002) and Stevens

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Table 1

 1 corporated as a third input variable. Alternatively to these three inputs, total costs less research grants will be used in the SFA approach. All monetary variables are de ‡ated using the government consumption de ‡ator as provided in the 2004/05 annual report of Council

	about here
	The number of research personnel and the number of technical personnel serve as input
	variables. Additionally, other expenditures (total expenditures less wage spending) are in-

Table 2 .

 2 Since the German states are responsible for education policy, we present the statistics at the state level. The ratios refer to single universities within the relevant state and represent the averages for the years 1998-2003. E and W denote East

	and West Germany, respectively. Hence, considering the ratio expenditure over graduates,
	we …nd that universities in East Germany display higher expenditures per graduate than
	their West German counterparts. With regard to the personnel endowment, we also observe
	that universities in East Germany have more research and total sta¤ per graduate.

Table 2

 2

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Table 5

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	14 A one percent increase in research grants per student decreases total costs less research grants (divided
	by the number of students) by 0.21% in universities with engineering faculties. In universities with medical
	faculties, a one percent increase in research grants per student increases total costs less research grants (per
	student) by 0.11%.
	16

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Table 1 :

 1 Input and output variables

	Variable	Description of Variable
	Outputs
	Graduates	Number of graduates
	Grants	Amount of research grants
	Inputs
	Technical sta¤	Number of technical personnel (DEA)
	Research sta¤	Number of research personnel (DEA)
	Current expenditure	Financial means (DEA)
	Total costs -research grants Total costs -research grants (SFA)
	Source: Own represenation.

Table 2 :

 2 Ratios of outputs and inputs in federal states

	State (1998-2003)	Expenditures Graduates	a)	Research staf f Graduates	T otal staf f Graduates
	Baden-Württemberg (W)	76.3	2.0	4.8
	Bayern (W)	69.0	1.5	3.7
	Berlin (E)	84.8	1.9	4.4
	Brandenburg (W)	38.0	2.5	4.0
	Bremen (W)	37.9	1.8	2.7
	Hamburg (W)	66.7	1.7	4.0
	Hessen (W)	59.9	1.6	3.9
	Mecklenburg-Vorpommern (E)	127.4	2.6	7.7
	Niedersachsen (W)	36.9	1.3	2.8
	Nordrhein-Westfalen (W)	60.4	1.3	3.0
	Rheinland-Pfalz (W)	44.8	1.4	3.1
	Saarland (W	88.2	2.3	6.3
	Sachsen (E)	75.3	2.2	5.0
	Sachsen-Anhalt (E)	183.4	3.3	9.6
	Schleswig-Holstein (W)	82.1	1.3	4.2
	Thüringen (E)	82.5	2.3	5.7
	a) measured in 1,000 e with base year 1995.

Source: Federal Statistical O¢ ce of Germany; own calculations. 22 Page 22 of 35 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 :

 3 Ratios of outputs / inputs for di¤erent university samples

	1998-2003	Expenditure Graduates	a)	Re search Sta¤ Graduates	T otal Staf f Graduates
	All universities	66.7	1.6	3.9
	Universities
	engineering and medical faculty	92.2	2.1	5.3
	no engineering faculty	71.2	1.5	3.9
	no medical faculty	29.5	1.4	2.5
	no engineering and no medical faculty	17.8	1.2	1.9
	a) measured in 1,000 e with base year 1995.
	Source: Federal Statistical O¢ ce of Germany; own calculations.

Table 4 :

 4 Results from OLS and Tobit regressions

	OLS	Tobit
	(1)

Table 5 :

 5 Maximum Likelihood Estimates of the Cost Function Coe¢ cients

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	Coe¢ icient	Baseline model	Model without dummies
	Constant	13.02**	20.11**
					(2.38)	(2.00)
	Time		-0.004	0.02
	Linear time trend	(0.41)	(1.22)
	RG			0.56	4.41***
	Research grants divided by the number of students	(0.80)	(7.07)
	GRA		-0.33	-3.36*
	Final degrees completed divided by the number of students	(0.25)	(1.65)
	RGSQ	0.29***	0.08
	RG	2		(4.66)	(1.31)
	GRASQ	0.58***	0.08
	GRA	2	(3.27)	(0.29)
	RGGRA	-0.46***	-0.17*
	RG * GRA	(6.78)	(1.75)
	W				-7.33***	-12.69***
	Wage spending per university employee	(2.95)	(2.64)
	WNWSQ	2.67***	4.30***
	W	2		(4.50)	(3.71)
	WNWRG	-0.34*	-1.20***
	WNW * RG	(1.90)	(7.20)
	WNWGRA	0.57**	1.06**
	WNW * GRA	(1.97)	(2.33)
	ENG		2.17**	-
	Engineering faculty: yes = 1, no = 0	(1.98)
	MED	3.90***	-
	Medical faculty: yes = 1, no = 0	(3.30)
	MEDENG	-0.16**	-
	MED * ENG	(2.11)
				24

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Table 6 :

 6 Likelihood ratio -test of faculty restrictions

	Null hypothesis	2 0:99	Test statistic

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Table 7 :

 7 Impact of environmental variables

	Coe¢ cient	Baseline model	Model without dummies
	Constant	23.71 (1.17)	15.56 (0.95)
	Time trend	-0.76 (1.54)	-0.06 (1.90)*
	Regional GDP p.c.	-3.01 (1.21)	-0.08 (0.49)
	LR of the one-sided error	31.56	29.29
	t-statistics in parentheses, p-values: * = 0.10, ** = 0.05, *** = 0.01
	Source: Own calculations.

Table 8 :

 8 E¢ ciency scores: Mean e¢ ciency of East and West German universitiesNote that e¢ ciency scores that are based on cost function estimates range from 1 to 1; a score of 1 indicates an e¢ cient performance.

		1998	1999	2000	2001	2002	2003
	East German universities	1.66	1.62	1.50	1.35	1.35	1.30
	West German universities	1.35	1.34	1.34	1.31	1.28	1.25
	Note: E¢ ciency predictions based on baseline cost function
	model as set out in Table 5.
	Source: Own calculations.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

35 Page 35 of 35 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Previous investigations also speci…ed the number of publications, in some cases weighted by journal ranks or the number of pages. However, due to data availability we did not include publications in our investigation. The number of students has been used ambiguously as an output of teaching activities or as an input into teaching production.

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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Note that all three universities of Berlin are considered here as universities located in East Germany.

On the local level, Germany is divided into 434 regional authorities. Yet, a distinction has to be made

< M1 -Business Administration < M -Business Administration and Business Econ;