María Manuela González
email: mmgonzalez@daea.ulpgc.es

Sergio Perelman
email: sergio.perelman@ulg.ac.be

Lourdes Trujillo
email: ltrujillo@daea.ulpgc.es

Maria Manuela

González Daea

Tracking the

Keywords: C89 -Other < C8 -Data Collection Regulatory changes, Stochastic Frontier, Efficiency

come

Introduction

The 1990s were characterized by widespread efforts to liberalize public services in the hope of increasing their efficiency. In most instances, reformers were aware of the need for a residual regulatory role for the public sector to ensure a fair distribution of the efficiency gains between the users and the natural monopolies (transport and distribution service in water and energy or the supply of infrastructure in port services). Sector differences and differences in the complexity of the political context have however resulted in different strategies to get to the regulatory role. In many instances, the reformers have had to adopt a gradual and slow distancing from self regulated monopolies rather than a shock approach to reform.

The different stages of the gradual approaches all contribute in different ways in determining the total efficiency gains achieved by reform packages. Getting a "broad" sense of where the biggest bang for the buck can be achieved can come in handy in designing new reform packages. Data limitations are however typically so strong that it is difficult to get much more than anecdotal evidence on the relative impact of the various reform stages. This paper proposes a simple "back door" method to address the problem. It allows an unbundling of the total gains achieved by a reform package and to assign credit for the relative contribution to each stage of the reforms. Several software programs allow to apply the method proposed. The approach is illustrated with an assessment of the impact of the gradual regulatory reforms adopted in the Spanish port sector during the 1990s. This paper is structured as follows: section 2 offers a survey of the literature analyzing the evolution of the efficiency over time. Section 3 describes our proposal.

Section 4 discusses the distance function we rely on to get to the total efficiency gains.

Section 5 shows how we can unbundle the sources of efficiency gains with an application to the Spanish port reform experience. The discussion includes a description of the data, the model and the most important results. Section 6 concludes.

A brief survey of the literature

The first models designed to assess the temporal variation of efficiency emerged in the early 1990s. They were based on the traditional models of panel data. These models estimate the frontier by means of corrections of the least square method, and therefore the definition of the temporal structure of efficiency is made using the independent term (through the individual effects).

The pioneering work in this field was that of Cornwell, Schmidt and Sickles (1990).

They suggested that the evolution of efficiency should follow a quadratic function of time, making it possible for efficiency to also vary between firms. This made the model highly flexible, although the price to be paid was that it required a lot of data. This is a major inconvenience in regulated sectors, where lack of data is a common problem, both in the cross-section and longitudinal dimensions. Lee and Schmidt (1993) propose another type of panel data model in which efficiency varies in the same way between firms. Although this model is more flexible than the previous one, it imposes that the efficiency varies in the same way for all the firms. These models do not allow for any distinction between the effects of a change in efficiency and the effects of neutral technical progress. This is an important issue in processes of regulatory changes; as such changes can be the result of variations in productivity due to an apparent technological change rather than the result of improvements in efficiency. A second type of model often used in this context is stochastic frontier models in which the pattern of variance in efficiency is modeled using the error term, assuming a distributional assumption. In stochastic frontier models, also a distinction can be made between those that specify a common pattern of variance of efficiency between firms and those that allow the efficiency to vary in a differentiated way for each firm. The formers include the proposal suggested by Kumbhakar (1990) in which the inefficiency term is specified as an exponential function of time and the Battese and Coelli (1992) model, who made an attempt to come up with a simpler proposal that implies an unique time parameter to catch efficiency variability over time for all firms.

The main difficulty for the practical implementation of these proposals is the restriction that efficiency follows similar pattern overtime -for all firms, which is not a very plausible supposition in most cases, as less inefficient firms have a greater margin for improvement than more efficient ones.

To overcome the inconveniences of the previous models, new proposals have been developed that make it possible to vary the differentiated efficiency for each firm (Kumbhakar, 1991; Heshmati and Kumbhakar, 1994; Cuesta, 2000 1 ). In addition to major theoretical econometric problems, from the applied work point of view, the fundamental issue with these models is that they require a large quantity of data, as the number of parameters to be estimated increases with the number of firms.

A more general solution to the problems of the above models could be offered however by the Battese and Coelli (1988) proposal. This is a model with time invariant efficiency. Therefore, although it is applied to panel data, it evaluates the efficiency of each firm at a single moment in time and does not resolve the question of measuring regulatory changes as a temporal evolution of the efficiency of each firm. However, this 1 For a more detailed survey about temporal variation in inefficiency see Cuesta (2001). model can be made to be even more general than their authors suggested through a simple data transformation, as discussed in the next section.

The model transformation

None of the models surveyed in Section 2 allows a diagnostic of the relative reaction of firm specific performance to the various components of a reform package. There is thus a need to transform time invariant models into models that are time "variant" over chunks of periods. This is a major limitation for a wide range of policy applications.

It turns out that a simple transformation of the data samples is enough to make Battese and Coelli's model (1988) capable of measuring the efficiency of firms in different time periods. This is achieved simply by creating groups of firms by periods.

These groups, despite actually containing the same firms, are tabulated in a different way for each period, so that what are simulated are different firms. In this way, the model, on recognizing "new firms", evaluates the corresponding efficiency of each group of firms which simply correspond to the same set of firms at different observation periods (i.e. the impact of a sequence of reforms can then be tracked over time).

For example, assume that there is a panel of data made up of 3 firms (called F 1 , F 2 and F 3) , for which we have information for four years (T 1 , T 2 , T 3 and T 4 ). These firms produce product Y ij by means of input X ij (i refers to the firm and j to the time period).

The usual way of incorporating the information into the software program is shown in la

Table 1. Original sample presentation

It is also supposed that in year 3, a change of regulation occurs, whose effects carry over into the next year, so we have to consider that the firms in years 3 and 4 are 2 At least three software programs allow the estimation of the Battese and Coelli's model (1988): FRONTIER (Coelli, 1996), LIMDEP and STATA. independent of the firms in the first two years. This is done by naming the firms F 4 , F 5 and F 6 , respectively. Thus, the information should be incorporated as shown in Table 2.

Table 2. Restructured sample

It should be observed that, as the firms are really the same, despite being assigned different codes, the data referring to the variables is entered in the same order. This means that the result is an incomplete panel of data that is expressed as in Table 3, where T 1 to T 4 refer to the time period and F 1 to F 6 to the firms. This way, the end result is the time invariant efficiency of firms F 1 , F 2 , F 3 , F 4 , F 5 and F 6 . As firms E 4 , E 5 and E 6 , are in fact E 1 , E 2 and E 3 , in another period, the change in efficiency obtained is really the time variation of the efficiency of firms F 1 , F 2 and F 3 .

The main assumption justifying the new sample organization is that regulatory reforms have had a major impact on firms' operations, such as that the same firm must be considered as a different and independent entity (decision making unit) before and after the reform.

Table 3. Structure of the panel of data

Empirical application

In order to demonstrate the suitability of the proposed transformation, we model Spain's two stage approach to port reform during the 1990s. The specific of these reforms are not of interest here. 3 The fact that it involves two main stages is. The first took place in 1992 and it was characterized by the development of new management procedures and organization structures with the objective of decentralize the port system and reinforce the autonomy of the port authorities. The second was in 1998 and insisted on the autonomy of the port authorities, regulated the participation of the regional 3 For more details see Gonzalez (2004). Government in the port organization and encouraged the participation of the private sector in port activities. Both stages were designed to have an impact on efficiency. To estimate the total efficiency gains and to track the differentiated effect of each stage, we estimate a distance function using the data of a representative sample of port authorities in the Spanish port system.

The distance function

The distance function, introduced by Shephard (1953Shephard ( , 1970)), allows the estimation of the relative efficiency of firms in relation to the technological frontier. This function is selected because of its advantages over the more standard alternatives approaches to assessing frontiers. Of particular interest here is that it makes it possible to capture multioutput processes and that it does not require the use of optimizing assumptions.

Moreover, it only uses physical data and, therefore, information is not necessary for output or factor prices.

The analysis of the conditions under which port authorities perform their activities led us to the estimation of an output-oriented distance function. An output-oriented distance function is defined as the smallest scalar by which all outputs can be proportionally divided, using the same level of productive factors. This is because in the provision of infrastructure services, port authorities have some power to decide on the production level through the use of two mechanisms: commercial policies and concessions. The port authorities also perform a significant amount of marketing for their services and facilities to attract new traffic. The commercial policies complement these efforts with tariff discounts offered within limits allowed. Furthermore, as long as port authorities decide on the type of firm that can operate at the different ports, they are also deciding on the ships and goods that will be handled. For instance, a port intended to attract fish to be processed needs that freezing companies be established there, relying the final decision on that is subject on the port authority's board of directors. Considering this capacity to influence output, port authorities encounter certain difficulties in adjusting the productive factors used in the provision of infrastructure services, basically: berths, area and labor. The first two are quasi-fixed factors that, due to their indivisibility, find it difficult to adapt to any changes in production, especially if the changes are downward. Furthermore, although investment decisions are made by the board of directors of each port authority, in reality these decisions are coordinated by the State Ports (Puertos del Estado), which has the margin to decide whether to allow or limit the financing of the construction of infrastructure work. As for the labor factor, this is generally made up of officers, which makes it difficult to make adjustments, particularly when the number needs to be reduced.

The functional form

The empirical application of a distance function calls for the definition of an appropriate functional form. It is desirable for the functional form to present the following advantages: it must be flexible, it must be easy to calculate and, lastly, it must make it possible to impose the homogeneity condition. The translogarithmic functional form (hereinafter translog) meets these conditions, which is why, at present, most authors use it in all fields of research. It consists of a flexible functional form that provides a local second-order approximation to an unknown functional form. In other words, no a priori restrictions about production technology are assumed and, thus, the criticisms associated with certain restrictive properties of the Cobb-Douglas function are overcome.

For all these reasons, we estimate a translog distance function that, when outputoriented, can be expressed as follows:

F o r P e e r
f = == = == == = = = + + + + + + + + (1)
where y is a vector of M outputs, x is a vector of K factors, i relates to the i-th firm, t refers to the t-th year, d relates to the H environment dummy variables and f to the time period dummies; , , , and are coefficients to be estimated and it is an error term which is discussed later. Continuous variables are expressed in relation to their deviation from the geometric mean; therefore, the first order term coefficients correspond to distance function elasticities at the sample mean points.

In order to determine the frontier, D O needs to be equal to the unit and, in that case, the term on the left of the equation, according to the neperian logarithm, will equal zero.

By definition, output distance functions assume radial expansion of outputs, therefore the homogeneity condition of degree 1 must be imposed. Following Lovell et al. (1994), 4 this condition has been imposed by standardizing the distance function with one of the outputs. This works on the assumption that homogeneity implies that:

( ) ( ) y , x wD wy , x D O O = , (2)
for any w>0.5

If in a translog distance function (1), any output is chosen, say y M , so that w = 1/y M , the following expression results: (4)

( ) ( ) ln , , , , , , , , O M it it Mit D y TL x y y d f = , (3)
In equation ( 4), the -ln(D O ) term can be interpreted as an error term which captures the technical inefficiency.

The distance function estimated is stochastic. For the purpose of estimating the equation ( 4), the random disturbance term must be determined. We applied the methodology developed by Battese and Coelli (1988) for panel data and applied an additive term, as suggested by Cuesta and Orea (2002), to account for the fact that we are estimating an output oriented distance function. The error term thus has the following form:

i it it u v + = (5)
where, v it is a symmetrical error term, iid with a zero mean (which represents the random variables that the operator cannot control) and u i is a one-sided negative error term (which measures the technical inefficiency of each operator that is constant over time) and is distributed independently of v it .

Applied to the distance function, this yields ,,,,,, ,

( ) ( ) ln ,
Mit it it Mit it i y TL x y y d f v u = ++ (6)
This equation can be estimated by the maximum likelihood method, which requires distributional assumptions of the random shock. This assumes that v it follows an N(0, # v 2 ) distribution and u i follows an NN(0, # u 2 )Ndistribution (Ritter and Simar, 1997).

This model thus assumes that the inefficiency effects are constant over time. To be able to assess the effects of policy changes on inefficiency levels, we structured the time horizon into 3 periods and considered the port authorities to be independent across periods. This way, any change resulting from reform can be assessed within the period.

The three time periods are:

(i) before the reform (1990)(1991)(1992) (ii) after the first reform (1993)(1994)(1995)(1996)(1997) (iii) after the second reform (1998-2002)

The data 6

The heterogeneity of activities performed in ports and the diversity of commodities handled suggests the idea of limiting the study to a certain number of ports and a specific type of cargo. Given the aforementioned recommendation, this study centers its analysis on the Port Authorities of Spanish ports, which are particularly relevant in terms of container traffic.

The ports included in the sample are the country's major commercial ports and cover a broad typology of ports. The time period under analysis is from 1990 to 2002, which makes it possible to analyze the effects that the modifications to the port system carried out in the nineties had on the efficiency of each of the ports in particular and the port system in general.

The unit for analysis is the port authority. More than 70% of the ships going through Spanish ports come under the control of the nine port authorities in the sample (Algeciras, Alicante, Balearic Islands, Barcelona, Bilbao, Las Palmas, Santa Cruz de Tenerife, Valencia and Vigo). These sample authorities also handle 96% of container traffic, all of which is proof of the high concentration of this kind of traffic. We also need to model a number of relevant changes that occurred during the period of analysis. These include, economic booms, a liberalization of maritime cabotage within the European Union, changes to ship building technology, technological changes to handling equipment to address the large expansion of container traffic. These effects are accounted for by a time dummy for each year covered by the sample. This allows us to capture the effect of factors that influence all ports equally at different points in time, other than the deregulation process.

The results

Table 4 shows the parameters estimates obtained with the output oriented distance function, which was estimated by maximum likelihood. Globally, it shows that the output distance function is well behaved.

The first-order parameters present the expected signs and are also significant. In other words, the parameters of output variables are positive and, thus, indicate that distance from the frontier increases when production grows (the output-oriented distance function takes a value between zero and one). On the contrary, first order input parameters are negative, suggesting that if inputs increase, for a given output level, the distance will be They both have a negative and significant coefficient.

The coefficients for the time dummies show the effects of factors that evolve over time and influence all the ports simultaneously. These coefficients are significant as of 1995, with the strongest effects taking place over the last 4 years of the total sample period.

Table 4. Parameters estimated

Table 5 reports a summary of the information that can be generated from the method proposed here to generate a picture of the relative impact of the various reforms and the results presented in Table 4. It aggregates the information generated at the level of each port.

The average story that emerges is that technical efficiency has dropped somewhat also not significantly so but that very different levels of technological change have been associated with each reform. The increase from 0.4% between 1990 and 1992 to 9.9% right after the first package of reform and the slow down (2.7%) in the gains associated with the second wave of reforms illustrates the interest of this methodology in obtained differentiating impact assessments.

Table 5: Impact of reform waves in the Spanish Port system

Concluding comments

The evaluation of the impact of reforms on efficiency levels in regulated industries has developed into a key issue. The underlying notion is to verify whether the proposed regulatory targets are being achieved. These objectives include the simulation of markets that are competitive for those services that are characteristically provided in terms of a monopoly. Regulators therefore need to guarantee that similar levels of efficiency to Methods for analyzing the development of efficiency are varied, and each has its pros and cons as detailed here. This study has presented a proposal that involves transforming the data in such a way that Battese and Coelli's time invariant model ( 1988) is shown to be variant, at least by periods of time. Our approach allows overcoming the restriction imposed by alternative modeling approaches which imply that the rate of changes in efficiency is constant over time. Our approach also has the advantage of being simple and much less data demanding that the alternatives. These characteristics are particularly useful for regulators who typically enjoy limited access to data and need to rely on relatively simple approaches.

To illustrate the proposal, a distance function was estimated because it captures the multi-output nature of the port sector without assuming somewhat implausible hypotheses regarding the economic behavior of port authorities, but instead using physical data that is much more reliable than economic data. The aim of this application was to show the impact that reforms to the Spanish port system have had on the technical efficiency of the analyzed ports.

Empirically, the paper has shown that the restructuring and the substantial reforms introduced not only changed the conditions for the development of port activities subject to regulation but also led to significant improvements in technological change. However, technical efficiency has not improved in a similar way and has in fact changed little on average. The regulatory change can be credited with a statistically significant incentive to

T 1 Y 11 X 11 F 2 T 1 Y 21 X 21 F 3 T 1 Y 31 X 31 F 1 T 2 Y 12 X 12 F 2 T 2 Y 22 X 22 F 3 T 2 Y 32 X 32 F 1 T 3 Y 13 X 13 F 2 T 3 Y 23 X 23 F 3 T 3 Y 33 X 33 F 1 T 4 Y 14 X 14 F 2 T 4 Y 24 X 24 F 3 T 4 Y 34 X 34 Table 2. Restructured sample Firm Year Output Input F 1 T 1 Y 11 X 11 F 2 T 1 Y 21 X 21 F 3 T 1 Y 31 X 31 F 1 T 2 Y 12 X 12 F 2 T 2 Y 22 X 22 F 3 T 2 Y 32 X 32 F 4 T 3 Y 13 X 13 F 5 T 3 Y 23 X 23 F 6 T 3 Y 33 X 33 F 4 T 4 Y 14 X 14 F 5 T 4 Y 24 X 24 F 6 T 4 Y 34 X 34
Table 3. Structure of the panel of data

T 1 T 2 T 3 T 4 F 1 F 1 0 0 F 2 F 2 0 0 F 3 F 3 0 0 0 0 F 4 F 4 0 0 F 5 F 5 0 0 F 6 F 6
Page Tracking the stepwise effects of regulatory reforms over time: a "back-door" approach

Introduction

The 1990s were characterized by widespread efforts to liberalize public services in the hope of increasing their efficiency. 1 In most instances, reformers were aware of the need for a residual regulatory role for the public sector to ensure a fair distribution of the efficiency gains between the users and the natural monopolies (e.g the supply of infrastructure in port or railways services, water or energy distribution services). Sector differences and differences in the complexity of the political context have however resulted in different strategies to get to the regulatory role. In many instances, the reformers have had to adopt a gradual and slow distancing from self regulated monopolies rather than a shock approach to reform.

The different stages of the gradual approaches all contribute in different ways in determining the total efficiency gains achieved by reform packages. Getting a "broad" sense of where the biggest bang for the buck can be achieved can come in handy in designing new reform packages. Data limitations are however typically so strong that it is difficult to get much more than anecdotal evidence on the relative impact of the various reform stages.

This paper proposes a simple "back door" method to address the problem. It allows an unbundling of the total gains achieved by a reform package and to assign credit for the relative contribution to efficiency changes to each stage of the reforms. The approach is illustrated with an assessment of the impact of the gradual regulatory reforms adopted in the Spanish port sector during the 1990s.

1 There are some papers that illustrate the effects in regulatory changes [START_REF] Estache | Efficiency Gains from Port Reform and the Potential for Yardstick Competition: lessons from México[END_REF][START_REF] Anstine | The impact of the regulation of the cable television industry: the effect on quality-adjusted cable television prices[END_REF] or the efficiency or cost structure in regulated industries [START_REF] Grosskopf | Estimating the efficiency of the system of healthcare financing in achieving better health[END_REF]Jara-Díaz et al., forthcoming). The paper is structured as follows: section 2 offers a survey of the literature analyzing the evolution of the efficiency over time. Section 3 describes our methodological proposal. Section 4 discusses the distance function we rely on to get to the total efficiency gains. Section 5 shows how we can unbundle the sources of efficiency gains with an application to the Spanish port reform experience. The discussion includes a description of the data, the model and the most important results. Section 6 concludes.

A brief survey of the literature

The literature offers roughly two groups of models to track the temporal variation of efficiency that emerged in the early 1990s. The first group of models was based on the traditional models of panel data and estimated the frontier by means of corrections of the least square method. In these models, the definition of the temporal structure of efficiency is derived from the independent term (through the individual effects). There are however various ways of getting to the evolution of efficiency within this first group of models.

Cornwell, Schmidt and Sickles (1990) were the pioneers. They suggested that the evolution of efficiency should follow a quadratic function of time, making it possible for efficiency to also vary between firms. This made the model highly flexible, although the price to be paid was that it required a lot of data. This is a major inconvenience in regulated sectors, where lack of data is a common problem, both in the cross-section and longitudinal dimensions. Lee and Schmidt (1993) proposed another type of panel data model in which efficiency varies in the same way between firms. Although this model requires less data than the previous one, it imposes that the efficiency varies in the same way for all the firms. From the perspective of the analysis of the impact of regulatory reforms, the main drawback of these models is their failure to allow for any distinction between the effects of a change in efficiency and the effects of neutral technical progress.

This is an important limitation since reforms can influence both dimensions and do so differently and at different points in time. A second group of models often used in this context consists of stochastic frontier models. In these models, the pattern of variance in efficiency is modeled using the error term, assuming a distributional assumption. They can also be separated into those that specify a common pattern of variance of efficiency between firms and those that allow the efficiency to vary in a differentiated way for each firm. The formers include the proposal suggested by Kumbhakar (1990) in which the inefficiency term is specified as an exponential function of time and the Battese and Coelli (1992) model, who propose a unique time parameter to catch efficiency variability over time for all firms. The main difficulty for the practical implementation of these approaches is the restriction that efficiency follows similar patterns overtime for all firms. Assuming similar patterns for all firms is not a very plausible assumption in most cases, as less inefficient firms have a greater margin for improvement than more efficient ones.

To overcome the inconveniences of the previous models, new proposals have been developed that make it possible to vary the differentiated efficiency for each firm (Kumbhakar, 1991; Heshmati and Kumbhakar, 1994; Cuesta, 2000 2 ). While they address the basic issues conceptually, they do so at some costs. First, they suffer from major theoretical econometric problems which have been discussed in [START_REF] Coelli | An Introduction to Efficiency and Productivity Analysis[END_REF] for instance. Moreover, for applied work, the fundamental issue with these models is that they require a large quantity of data, as the number of parameters to be estimated increases with the number of firms.

A most general solution to the econometric problems of the above models could be offered however by the Battese and Coelli (1988) proposal. This is a model with time invariant efficiency. Therefore, although it is applied to panel data, it evaluates the efficiency of each firm at a single moment in time and does not resolve the question of measuring regulatory changes as a temporal evolution of the efficiency of each firm. For gradual reforms impact assessments, however, the model cannot be used as is because it 2 For a more detailed survey about temporal variation in inefficiency see Cuesta (2001). yields time invariant efficiency estimates. Indeed, although it applies to panel data, it evaluates the efficiency of each firm at a single point in time. Their method is however much more flexible than what these authors recognized and could be used, thanks to a simple data transformation, to track the impact of reforms on the evolution of efficiency at the firm level as discussed in the next section.

The model transformation

None of the models surveyed in Section 2 allows a diagnostic of the relative reaction of firm specific performance to the various components of a reform package. There is thus a need to transform time invariant models into models that are time "variant" over chunks of periods. This is a major limitation for a wide range of policy applications.

It turns out that a simple transformation of the data samples is enough to make Battese and Coelli's model (1988) capable of measuring the efficiency of firms in different time periods. This is achieved simply by creating groups of firms by periods.

These groups, despite actually containing the same firms, are tabulated in a different way for each period, so that what are simulated are different firms. In this way, the model, on

recognizing "new firms", evaluates the corresponding efficiency of each group of firms which simply correspond to the same set of firms at different observation periods (i.e. the impact of a sequence of reforms can then be tracked over time).

For example, assume that there is a panel of data made up of 3 firms (called F 1 , F 2 and F 3) , for which we have information for four years (T 1 , T 2 , T 3 and T 4 ). These firms produce product Y ij by means of input X ij (i refers to the firm and j to the time period).

The usual way of incorporating the information into the software program is shown in la Table 1. 3

3 At least three software programs allow the estimation of the Battese and Coelli's model (1988): FRONTIER (Coelli, 1996), LIMDEP and STATA.

Table 1. Original sample presentation

For the sake of the argument, we also assume that a change of regulation occurs in year 3 and that its effects carry over into the next year. This implies that we have to consider that the firms in years 3 and 4 are independent of the firms in the first two years. This is done by naming the firms F 4 , F 5 and F 6 , respectively. Thus, the information should be incorporated as shown in Table 2.

Table 2. Restructured sample

It should be observed that, as the firms are really the same, despite being assigned different codes, the data referring to the variables is entered in the same order. This means that the result is an incomplete panel of data that is expressed as in Table 3, where T 1 to T 4 refer to the time period and F 1 to F 6 to the firms.

This way, the end result is the time invariant efficiency of firms F 1 , F 2 , F 3 , F 4 , F 5 and F 6 . As firms F 4 , F 5 and F 6 , are in fact F 1 , F 2 and F 3 , in another period, the change in efficiency obtained from the comparison between the two sets is really the time variation of the efficiency of firms F 1 , F 2 and F 3 . The main assumption justifying the new sample organization is that regulatory reforms have had a major impact on firms' operations and the same firm can be considered as a different and independent entity (decision making unit) before and after the reform.

Table 3. Structure of the panel of data

Empirical application

In order to demonstrate the suitability of the proposed transformation, we model Spain's gradual approach to port reform during the 1990s. The specifics of these reforms are not of interest here but the fact that it involves two main stages is since it allows a simple illustration of our methodological proposal. 4 The reforms, their sequence and of their timing can be summarized as follows for our purpose. The first stage of the reform took place in 1992. It was characterized by the development of new management procedures and organization structures. Its main purpose was to decentralize the port system and to reinforce the autonomy of the port authorities. The second stage was in 1998. It further increased the autonomy of the port authorities, regulated the participation of the regional Government in the port organization and encouraged the participation of the private sector in port activities. Both stages were designed to have an impact on efficiency. To estimate the total efficiency gains and to track the differentiated effect of each stage, we estimate a distance function using the data of a representative sample of port authorities in the Spanish port system.

The distance function

The distance function, introduced by Shephard (1953Shephard ( , 1970)), allows the estimation of the relative efficiency of firms in relation to the technological frontier. This function is selected because of its advantages over the more standard alternatives approaches to assessing frontiers. Of particular interest here is that it makes it possible to capture multioutput processes and that it does not require the use of optimizing assumptions.

Moreover, it only uses physical data and, therefore, information is not necessary for output or factor prices. 5

The analysis of the conditions under which port authorities perform their activities demands the estimation of an output-oriented distance function. An output-oriented distance function is defined as the smallest scalar by which all outputs can be 4 For more details see Gonzalez (2004). 5 This methodology has increasingly common in empirical papers on infrastructure industries, e.g. (Coelli andPerelman, 1999, 2000; or on regulated industries, e.g. Morrison et al. 2000, among others). proportionally divided, using the same level of productive factors. This is because in the provision of infrastructure services, port authorities have some power to decide on the production level through the use of two mechanisms: commercial policies and concessions. In addition, port authorities encounter certain difficulties in adjusting the productive factors used in the provision of infrastructure services, basically: berths, area and labor.

The functional form

The empirical application of a distance function calls for the definition of an appropriate functional form. It is desirable for the functional form to present the following advantages: it must be flexible, it must be easy to calculate and, lastly, it must make it possible to impose the homogeneity condition. The translogarithmic functional form (hereinafter translog) meets these conditions. It consists of a flexible functional form that provides a local second-order approximation to an unknown functional form.

In other words, no a priori restrictions about production technology are assumed.

The output oriented translog distance function can be expressed as follows:

0 0 1 l 1 1 1 1 1 1 1 1 ln ln 1 2 ln ln ln 1 2 ln ln ln ln M M M m mit mn mit mit m m n K K K K M k kit kl kit lit km kit mit k k l k m H T h h t t it h t D y y y x x x x y d f = == = == == = = = + + + + + + + + (1)
where y is a vector of M outputs, x is a vector of K factors, i relates to the i-th firm, t refers to the t-th year, d relates to the H environment dummy variables and f to the time period dummies; , , , and are coefficients to be estimated and it is an error term which is discussed later. Continuous variables are expressed in relation to their deviation The time period under analysis is relatively long since it spans from 1990 to 2002. This is long enough to track down the effects of a gradual regulatory reform. For that period, we have data on 9 port authorities. This dataset makes it possible to analyze the effects that the modifications to the port system carried out in the nineties had on the efficiency of each of the ports in particular and the port system in general. The authorities covered by our sample handle 96% of container traffic. Hence, the policy implications of our diagnostic should be quite representative.

The data available for each port authority for this period is standard in the literature on ports economics. 8 We have four variables representing port output (container cargo, liquid bulk, other cargo and passengers) and three productive factors (work, berths and area). Occasionally, certain specific factors may influence production activities without any possible interference from the port authorities. This study has included the existence of oil refineries and geographical location. A dummy was thus introduced to explicitly account for the oil refineries. Another dummy variable is included for the island ports.

To ensure the good quality of our econometric work, we also modeled a number of relevant changes that occurred during the period of analysis. These include, economic booms, a liberalization of maritime cabotage within the European Union, changes to ship building technology, technological changes to handling equipment to address the large expansion of container traffic. These effects are accounted for by a time dummy for each year covered by the sample. This allows us to capture the effect of factors that influence all ports equally at different points in time, other than the regulation process. In other words, it improves our estimates of the assessment of the impact of regulatory reforms. 7 For more details about the data and variables see Gonzalez (2004)

The results

Table 4 shows the parameters estimates obtained with the output oriented distance function, which was estimated by maximum likelihood. Globally, it shows that the output distance function is well behaved. The first-order parameters present the expected signs and are also significant. In other words, the parameters of output variables are positive and, thus, indicate that distance from the frontier increases when production grows (the output-oriented distance function takes a value between zero and one). On the contrary, first order input parameters are negative, suggesting that if inputs increase, for a given output level, the distance will be reduced. The regression results also show that the refinery and location variables matter. They both have a negative and significant coefficient.

The coefficients for the time dummies show the effects of factors that evolve over time and influence all the ports simultaneously. These coefficients are significant as of 1995, with the strongest effects taking place over the last 4 years of the total sample period.

Table 4. Parameters estimated

To better focus on the main concern of this paper, Table 5 reports a summary of the information that can be generated from the method proposed here. It gives a picture of the relative impact of the various reforms, aggregating the information generated at the level of each port. The average story that emerges is that technical efficiency has dropped somewhat also not significantly so but that very different levels of technological change have been associated with each reform. In other words, the explicit modelling of the gradual approach of reform is proving to be relevant since it reveals that each stage has a different impact on various sources of efficiency changes. For the record, our results suggest that the first stage of the reforms delivered the biggest impact and that this was quite significant. The increase from 0.4% between 1990 and 1992 to 9.9% right after the first package of reform is indeed strong. The slow down

(2.7%) in the gains associated with the second wave of reforms is notable but it shows that efficiency gains from the second stage of reforms have nonetheless been worth the effort.

Table 5: Impact of reform waves in the Spanish Port system

Concluding comments

The evaluation of the impact of reforms on efficiency levels in regulated industries is important for at least two reasons. The first is that the debates around the desirability of reforms in regulated industries often lack the necessary quantitative support necessary to assess where efficiency gains have actually been achieved or not. Measuring the efficiency effects of reforms, even gradual, allows more technical and less ideological assessments of reforms. Second, the measurement of efficiency has recently developed into a core business issue for regulators since they are expected to assess the scope for redistribution to the users of possible rents achieved by monopolistic providers. Gains can come from better efforts to improve performance, from the adoption of better technologies or from both. Any method of relevance to regulators needs to distinguish between these two sources of efficiency changes. The upshot is that the measurement of the evolution of efficiency levels is now a basic objective for many regulators and policy analysts.

In practice, analysts and regulators face major data constraints to conduct robust efficiency evaluations which allow fair assessments of the regulated operators. The data constraints are even stronger when the effects of various stages of reforms need to be isolated. The main contribution of this paper is the development of a simple method The relevance of the proposal has been illustrated with a case study of the gradual reform of the Spanish port system. The assessment has revealed very significant differences in the impact achieved by the two stages of reforms both in terms of the effort levels achieved by the operators (catching up effects) and in terms of the adoption of new technologies (frontier shift effects). From an analytical perspective, it is thus interesting to be able to monitor the effects of the various stages of reforms. From a regulatory view point, the method is particularly useful in that it allows the regulator to distinguish the evolution of the behavior of individual operators over time and to use this information as part of its regulatory decisions.

  technology, we have used four variables representing port output (container cargo, liquid bulk, other cargo and passengers) and three productive factors (work, berths and area). Occasionally, certain specific factors may influence production activities without any possible interference from the port authorities. This study has included the existence of oil refineries and geographical location. A dummy was thus introduced to explicitly account for the oil refineries in Algeciras, Bilbao and Santa Cruz de Tenerife. Another dummy variable is included for the island ports (Balearic Islands, Las Palmas and Santa Cruz de Tenerife).

  regression results also show that the refinery and location variables matter.

  achieved and that the gains in efficiency achieved are shared with users. Thus, the measurement of the evolution of efficiency levels becomes a basic objective for regulators. Simultaneously, regulatory changes may also lead to the adoption of better technologies by the operators (frontier shifts). This diversity of possible effects is what creates a demand for a measure which allows the unbundling of these effects into the pure efficiency and the technological effects.

  by the port authorities. This has resulted in significant improvement in their productivity.These results are particularly relevant in practice because a third wave of reforms has just been implemented and many more changes are expected to come from future EU guidelines for the liberalization of port activities, with a potentially strong influence on container traffic.

  data requirements for a wide range of situation in which it is important to assess the efficiency of gradual reforms at each stage of these reforms. Our proposal involves the transformation of the available data in such a way thatBattese and Coelli's time invariant model (1988) becomes variant, at least by blocks of time which can each be matched with a specific reform stage.

Table 1 . 2

 12

Table 1 . Original sample presentation

 1

	Firm	Year	Output	Input
	F 1

17 of 36 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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Table 4 . Parameters estimated

 4

	Variables and parameters		t-test	Variables and parameters		t-test
	Intercept	W0	0.2283	3.69 ln (liquid bulk). ln (other goods)	W34	0.2622	3.73
	Outputs*				ln (liquid bulk). ln (berth)	X31	0.4988	4.96
	ln(passenger)	W1	0.1636	(a) ln (liquid bulk). ln (surface)	X32	-0.1723	-2.34
	ln (container)	W2	0.2454	5.15 ln (liquid bulk). ln (labor)	X33	-0.1941	-1.83
	ln (liquid bulk)	W3	0.1051	2.96 ln (other goods). ln (berth)	X41	-0.9739	-5.18
	ln (other goods)	W4	0.4860	8.32 ln (other goods). ln (surface)	X42	-0.3557	-2.06
	Inputs				ln (other goods). ln (labor)	X43	0.3064	1.53
	ln (berth)	Y1	-0.3658	-2.69 ln (berth). ln (surface)	Y12	0..9816	3.14
	ln (surface)	Y2	-0.2564	-4.02 ln (berth). ln (labor)	Y13	0.4278	1.12
	ln (labor)	Y2	-0.7728	-6.48 ln (surface). ln (labor)	Y23	0.3913	1.03
	Cross terms				Temporal effects
	ln (passenger). ln (passenger)	W11	0.0399	(a) D 1991	Z1	0.0149	0.43
	ln (container). ln (container)	W22	-0.3432	-2.07 D 1992	Z2	-0.0081	-0.20
	ln (liquid bulk). ln (liquid bulk)	W33	-0.0573	-2.02 D 1993	Z3	0.0783	1.66
	ln (other goods). ln (other goods)	W44	-0.8545	-6.60 D 1994	Z4	-0.0592	-1.15
	ln (berth). ln (berth)	Y11	-2.0697	-4.15 D 1995	Z5	-0.2107	-3.65
	ln (surface). ln (surface)	Y22	-1.2459	-4.15 D 1996	Z6	-0.2862	-4.62
	ln (labor). ln (labor)	Y33	-0.7509	-0.99 D 1997	Z7	-0.3158	-4.95
	ln (passenger). ln (container)	W12	-0.0420	(a) D 1998	Z8	-0,3845	-5.74
	ln (passenger). ln (liquid bulk)	W13	-0.1766	(a) D 1999	Z9	-0.4828	-7.33
	ln (passenger). ln (other goods)	W14	0.1787	(a) D 2000	Z10	-0.5065	-7.42
	ln (passenger). ln (berth)	X11	0.3250	(a) D 2001	Z11	-0.5089	-7.34
	ln (passenger). ln (surface)	X12	0.0160	(a) D 2002	Z12	-0.5034	-7.15
	ln (passenger). ln (labor)	X13	-0.0633	(a) Environmental variables
	ln (container). ln (liquid bulk )	W23	-0.0283	-0.50 Location	[1	-0.2523	-3.33
	ln (container). ln (other goods)	W24	0.4135	3.09 Refinery	[2	-0.4868	-7.74
	ln (container). ln (berth)	X21	0.1501	0.75 Other ML parameters
	ln (container). ln (surface)	X22	0.5119	2.61	standard deviation		0.0164	3,21
	ln (container). ln (labor)	X23	-0.0489	-0.22	2 u / 2	\	0.7415	7.21

(a) indicates parameters calculated by application of the homogeneity condition.

Table 5 : Impact of reform waves in the Spanish Port system Periods Average technical efficiency (%)

 5

	Technological change
	(%)

Table 1 . Original sample presentation

 1

	Firm	Year	Output	Input
	F 1	T 1	Y 11	X 11
	F 2	T 1	Y 21	X 21
	F 3	T 1	Y 31	X 31
	F 1	T 2	Y 12	X 12
	F 2	T 2	Y 22	X 22
	F 3	T 2	Y 32	X 32
	F 1	T 3	Y 13	X 13
	F 2	T 3	Y 23	X 23
	F 3	T 3	Y 33	X 33
	F 1	T 4	Y 14	X 14
	F 2	T 4	Y 24	X 24
	F 3	T 4	Y 34	X 34

Table 2 . Restructured sample

 2

	Firm	Year	Output	Input
	F 1	T 1	Y 11	X 11
	F 2	T 1	Y 21	X 21
	F 3	T 1	Y 31	X 31
	F 1	T 2	Y 12	X 12
	F 2	T 2	Y 22	X 22
	F 3	T 2	Y 32	X 32
	F 4	T 3	Y 13	X 13
	F 5	T 3	Y 23	X 23
	F 6	T 3	Y 33	X 33
	F 4	T 4	Y 14	X 14
	F 5	T 4	Y 24	X 24
	F 6	T 4	Y 34	X 34

Table 3 . Structure of the panel of data

 3

	Page 35 of 36	Submitted Manuscript
	T 1	T 2	T 3	T 4
	F 1	F 1	0	0
	F 2	F 2	0	0
	F 3	F 3	0	0
	0	0	F 4	F 4
	0	0	F 5	F 5
	0	0	F 6	F 6

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

Table 4 .

 4 Parameters estimated

	Variables and parameters		t-test	Variables and parameters		t-test
	Intercept	0	0.2283	3.69 ln (liquid bulk). ln (other goods)	34	0.2622	3.73
	Outputs*				ln (liquid bulk). ln (berth)	31	0.4988	4.96
	ln(passenger)	1	0.1636	(a) ln (liquid bulk). ln (surface)	32	-0.1723	-2.34
	ln (container)	2	0.2454	5.15 ln (liquid bulk). ln (labor)	33	-0.1941	-1.83
	ln (liquid bulk)	3	0.1051	2.96 ln (other goods). ln (berth)	41	-0.9739	-5.18
	ln (other goods)	4	0.4860	8.32 ln (other goods). ln (surface)	42	-0.3557	-2.06
	Inputs				ln (other goods). ln (labor)	43	0.3064	1.53
	ln (berth)	1	-0.3658	-2.69 ln (berth). ln (surface)	12	0..9816	3.14
	ln (surface)	2	-0.2564	-4.02 ln (berth). ln (labor)	13	0.4278	1.12
	ln (labor)	2	-0.7728	-6.48 ln (surface). ln (labor)	23	0.3913	1.03
	Cross terms				Temporal effects
	ln (passenger). ln (passenger)	11	0.0399	(a) D 1991	1	0.0149	0.43
	ln (container). ln (container)	22	-0.3432	-2.07 D 1992	2	-0.0081	-0.20
	ln (liquid bulk). ln (liquid bulk)	33	-0.0573	-2.02 D 1993	3	0.0783	1.66
	ln (other goods). ln (other goods)	44	-0.8545	-6.60 D 1994	4	-0.0592	-1.15
	ln (berth). ln (berth)	11	-2.0697	-4.15 D 1995	5	-0.2107	-3.65
	ln (surface). ln (surface)	22	-1.2459	-4.15 D 1996	6	-0.2862	-4.62
	ln (labor). ln (labor)	33	-0.7509	-0.99 D 1997	7	-0.3158	-4.95
	ln (passenger). ln (container)	12	-0.0420	(a) D 1998	8	-0,3845	-5.74
	ln (passenger). ln (liquid bulk)	13	-0.1766	(a) D 1999	9	-0.4828	-7.33
	ln (passenger). ln (other goods)	14	0.1787	(a) D 2000	10	-0.5065	-7.42
	ln (passenger). ln (berth)	11	0.3250	(a) D 2001	11	-0.5089	-7.34
	ln (passenger). ln (surface)	12	0.0160	(a) D 2002	12	-0.5034	-7.15
	ln (passenger). ln (labor)	13	-0.0633	(a) Environmental variables
	ln (container). ln (liquid bulk )	23	-0.0283	-0.50 Location	1	-0.2523	-3.33
	ln (container). ln (other goods)	24	0.4135	3.09 Refinery	2	-0.4868	-7.74
	ln (container). ln (berth)	21	0.1501	0.75 Other ML parameters
	ln (container). ln (surface)	22	0.5119	2.61	standard deviation		0.0164	3,21
	ln (container). ln (labor)	23	-0.0489	-0.22	2 u / 2		0.7415	7.21

(a) indicates parameters calculated by application of the homogeneity condition.

Table 5 : Impact of reform waves in the Spanish Port system Periods Average technical efficiency (%)

 5

	Technological change
	(%)

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

This methodology has been applied in some empirical papers(Coelli and Perelman, 1999, 2000; Morrison et al. 2000; Orea, 2002, among others).

As stated byCuesta and Orea (2002) the chosen output does not influence the results.

For more details about the data and variables seeGonzalez (2004) andGonzalez and Trujillo (2005).

from the geometric mean; therefore, the first order term coefficients correspond to distance function elasticities at the sample mean points.

In order to determine the frontier, D O needs to be equal to the unit and, in that case, the term on the left of the equation, according to the neperian logarithm, will equal zero.

By definition, output distance functions assume radial expansion of outputs, therefore the homogeneity condition of degree 1 must be imposed. Following Lovell et al. (1994), this condition has been imposed by standardizing the distance function with one of the outputs. This works on the assumption that homogeneity implies that:

for any w>0. 6 If in a translog distance function (1), any output is chosen, say y M , so that w = 1/y M , the following expression results:

yielding the final expression: (4)

In equation ( 4), the -ln(D O ) term can be interpreted as an error term which captures the technical inefficiency.

The distance function estimated is stochastic. To estimate equation ( 4), the random disturbance term must be determined. We applied the methodology developed by Battese and Coelli (1988) for panel data and applied an additive term, as suggested by Cuesta and 6 As stated by Cuesta and Orea (2002) the chosen output does not influence the results. [START_REF] Orea | Parametric Decomposition of a Generalized Malmquist Productivity Index[END_REF], to account for the fact that we are estimating an output oriented distance function. The error term thus has the following form:

where, v it is a symmetrical error term, iid with a zero mean (which represents the random variables that the operator cannot control) and u i is a one-sided negative error term (which measures the technical inefficiency of each operator that is constant over time) and is distributed independently of v it .

Applied to the distance function, this yields ,,,,,, ,

This equation can be estimated by the maximum likelihood method, which requires distributional assumptions of the random shock. This assumes that v it follows an N(0, v 2 ) distribution and u i follows an N(0, u

2

) distribution (Ritter and Simar, 1997).

This model thus assumes that the inefficiency effects are constant over time. To be able to assess the effects of policy changes on inefficiency levels, we structured the time horizon into 3 periods and considered the port authorities to be independent across periods. This way, any change resulting from reform can be assessed within the period.

The three time periods are:

(i) before the reform (1990)(1991)(1992) (ii) after the first reform (1993)(1994)(1995)(1996)(1997) (iii) after the second reform (1998)(1999)(2000)(2001)(2002)


Forecasting with quantitative methods the impact of special events in time series

Keywords: C5 -Econometric Modeling < C -Mathematical and Quantitative Special Events, Forecasting, Quantitative methods, Artificial intelligence, Expert methods

published or not. The documents may come

Forecasting with quantitative methods the impact of special events in time series

Introduction

There is a fundamental difference between a shock and an expected/forthcoming irregular event: a non-periodic/ event that could be announced and therefore expected at a given time (e.g. sales promotion activity). The term shock is used, predominantly in the economics literature, for events that are not expected at all (e.g. an earthquake, a stock-market crash)thus the timing of a shock is not known a-priori. When it comes to shocks there are two issues to address: when the shock is going to happen and what is going to be the impact of it?

As far as forthcoming irregular events are concerned, the only interest is on the potential impact of such an event on the baseline forecast. This latter category of events is usually mentioned in the forecasting literature as special events [START_REF] Armstrong | Principles of Forecasting: A handbook for researchers and practitioners[END_REF].

The current study examines the relative performance of Artificial Neural Networks, Multiple Linear Regression and Nearest Neighbor approaches for estimating the impact on baseline series of expected irregular future events. Furthermore, building on the results of this comparison, an expert method is proposed which combines the strengths of regression and neural networks.

Literature review

In the forecasting literature, quantitative models have generally been very successful in extrapolating the basic trend-cycle component of business and economics time series [START_REF] Assimakopoulos | The theta model: a decomposition approach to forecasting[END_REF][START_REF] Allen | Forecasting profitability and earnings: a study of the UK market (1982-2000)[END_REF][START_REF] Halkos | Forecasting the stationary AR(1) with an almost unit root[END_REF][START_REF] Fildes | Generalising about univariate forecasting methods: further empirical evidence[END_REF], Jaesun and Kim 2006, Makridakis et al. 1982[START_REF] Makridakis | The M3-Competition: Results, conclusions and implications[END_REF], Maris et. al. 2004&2007, Petropoulos et. al. 2005&2007, Sadorsky 2005). However, time series models have been found to be deficient when it comes to handling special events, which are non-periodic such as promotions, announcements, changes in regulations and strikes [START_REF] Lee | Providing support for the use of analogies in demand forecasting tasks[END_REF], Nikolopoulos et. al. 2007a) Forecasting with quantitative methods the impact of special events in time series 3 Special events introduce an additive or multiplicative impact in the trend-cycle component [START_REF] Webby | Judgemental and statistical time series forecasting: a review of the literature[END_REF]. Both the literature and common practice suggest that forecasters tend to use their own judgment to adjust time series for irregular events [START_REF] Armstrong | Principles of Forecasting: A handbook for researchers and practitioners[END_REF]. However, the accuracy of these adjustments is often low [START_REF] Goodwin | Improving the voluntary integration of statistical forecasts and judgment[END_REF][START_REF] Goodwin | Integrating management judgment and statistical methods to improve short-term forecasts[END_REF].

Forecasting with neural networks has been very popular in academic circles in the last decade [START_REF] Aiken | Using a neural network to forecast inflation[END_REF][START_REF] Heravi | Linear versus neural network forecasts for European industrial production series[END_REF][START_REF] Curry | Simple' neural networks for forecasting[END_REF][START_REF] Kim | Financial time series forecasting using support vector machines[END_REF][START_REF] Sanders | The efficacy of using judgmental versus quantitative forecasting methods in practice[END_REF][START_REF] Wang | Prediction of maximum daily ozone level using combined neural network and statistical characteristics[END_REF][START_REF] Yao | Option price forecasting using neural networks[END_REF][START_REF] Zhang | Forecasting with artificial neural networks: The state of the art[END_REF][START_REF] Zhang | An investigation of neural networks for linear time-series forecasting[END_REF]. While batch forecasting was not one of the success areas for neural networks (Balkin and Ord 2000, Makridakis and[START_REF] Makridakis | The M3-Competition: Results, conclusions and implications[END_REF], the estimation of the impact of irregular events proved to be a very fertile area of research. In particular, [START_REF] Lee | Judgmental adjustment in time series forecasting using neural networks[END_REF] have proposed a formal framework for judgmental adjustments in time series using Artificial Neural Networks (ANN) in order to estimate the additive impact of irregular events. This framework has been enhanced by [START_REF] Nikolopoulos | Theta Intelligent Forecasting Information System[END_REF], to incorporate a variety of supportive models, such as Multiple Linear Regression (MLR - [START_REF] Makridakis | Forecasting, Methods and Applications[END_REF] or Nearest Neighbors approaches [START_REF] Härdle | Applied Nonparametric Regression[END_REF], as well as to be able to adjust to take into account any kind of business-related irregular events. However this feature was not properly evaluated in due to the lack of real special events data.

More recently Nikolopoulos et al. (2006b) have examined a case study with TV ratings data where the use of ANNs proved quite competitive to unaided judgment, MLR and Nearest Neighbor approaches. Furthermore in a follow up study simulation results have been submitted indicating that neither linear nor non-linear methods can effectively used solely, as both has advantages and disadvantages when applied to certain types of special events (Nikolopoulos et. al. 2007a, b) F o r P e e r R e v i e w

A framework for adjusting for past and future special events

A successful forecasting function within an organization is usually conditional on the participation of a variety of experts. A forecasting expert should be able to spot the appearance of an irregular event in the past of a series; this person will be the forecasting manager. In addition, marketing experts and production managers should be able to identify the event as well as the parameters that affected it and potentially quantify them. For example a forecasting expert should spot that the sales in a specific period in the past are more than expected for that period of the year, while the marketing expert should be able to identify the promotional activity that took place in this period and be able to recall the dimensions of that event (e.g. budget spend, media used, duration).

Ideally, the impact of such an event in the past would be quantified and removed from the series resulting in a filtered series that was more appropriate for extrapolation [START_REF] Armstrong | Principles of Forecasting: A handbook for researchers and practitioners[END_REF]. On the other hand, the future appearance of irregular events could be known in advance, and if an expert could forecast or even know the values of the event's parameters, an appropriate model could estimate the potential impact on the baseline series. A baseline series is defined as a filtered series embedding only trend and cycle components [START_REF] Makridakis | Forecasting, Methods and Applications[END_REF]. [START_REF] Lee | Judgmental adjustment in time series forecasting using neural networks[END_REF] have proposed all the necessary entities so as to describe, adjust and forecast the impact of an irregular event. An expert in the specific market (someone possessing all the necessary domain knowledge for the time series under consideration) should:

• Define Judgmental Factors; the expert should be able to identify promotions as a judgmental factor -that is an irregular factor that can influence the time series of historic sales for a specific product. • Identify Historical Judgmental Events; the expert should be able to identify when exactly an irregular event has happened as well as to define which factors characterize the event (and the values of the parameters for each factor). For example, a promotion could have been applied from May 2001 for two months spending in total 10.000€ in TV advertisements.

• Break events' impact into periods; an expert should be able to split the events into instances per period (week, month, etc) and define the impact on the baseline series for each instance either additively or multiplicatively. For example, a promotion might have been applied from May 2001 for two months with an impact of a 12% increase in the sales of May and an increase of 3% in June, respectively (relatively to the average sales of those months in the previous years).

• Remove the Impact of Judgmental Instances from Time-series; an expert or an automated FSS should be able to remove the impact of the judgmental instances of the irregular events from the historic data. As a result a filtered series should be created, a series that is far more appropriate for extrapolation via time series models.

On the expectation of a future irregular event, an expert should adjust his statistical forecasts [START_REF] Armstrong | Principles of Forecasting: A handbook for researchers and practitioners[END_REF]. This is a very tricky procedure that potentially introduces major errors into the forecasting process due to biases and inefficiencies of the adjusters [START_REF] Goodwin | Improving the voluntary integration of statistical forecasts and judgment[END_REF][START_REF] Goodwin | Integrating management judgment and statistical methods to improve short-term forecasts[END_REF]. Thus, there is need for automated support from FSS so as to enhance this adjustment process [START_REF] Nikolopoulos | Theta Intelligent Forecasting Information System[END_REF]. The forecaster/expert should be able to:

• Record expected irregular events; for example, the expert should know when the next promotion is due to take place, as well as all the relevant information that relates to the promotion. Forecasting with quantitative methods the impact of special events in time series 6

• Build a model for judgmental adjustment; a model should be built so as to estimate the impact of the future events, based on the historic impacts of similar events in the past. There are very few studies on this topic. Those that have been carried out mainly refer to case studies where a very specific time-series is examined (e.g.. oil sales in country X). Methods used from time to time include complex methods such as ANN [START_REF] Lee | Judgmental adjustment in time series forecasting using neural networks[END_REF] as well as approaches such as the Delphi method [START_REF] Rowe | The Delphi technique as a forecasting tool: issues and analysis[END_REF] and AHP [START_REF] Flores | Judgmental adjustment of forecasts: A comparison of methods[END_REF]).

• Apply adjustments to the baseline extrapolation; forecasts produced by the model should be superimposed with the baseline forecasts so as to constitute the final forecasts.

Forecasting the Impact of future irregular events

In this section we discuss four different approaches that seem appropriate for forecasting the impact of irregular events. The events are characterized by a single variable standing for the impact of the event accompanied by a set of cues that an act as independent variables.

Nearest Neighbors

Nearest Neighbors (NN) approaches [START_REF] Härdle | Applied Nonparametric Regression[END_REF], are a very simple way of estimating the impact of events. They involve looking at the past in order to assess the impact of similar events [START_REF] Green | Forecasting decisions in conflict situations: a comparison of game theory, role-playing, and unaided judgement[END_REF].

Finding the neighbors

This task requires ranking historic events in terms of similarity. Similarity requires a metric of distance. Various metrics have been used in order to measure distance in the nD-space as Euclidian norms [START_REF] Härdle | Applied Nonparametric Regression[END_REF]. nD-space metrics are required since our events involve in general more than one attribute. In this study a very straightforward metric, based on the Forecasting with quantitative methods the impact of special events in time series 7 Absolute Percentage Error (APE - [START_REF] Makridakis | Forecasting, Methods and Applications[END_REF], was used to measure the distance between similar events,

= = n i Ev i Ev i Ev i P P P Ev Ev 1 2 , 2 , 1 , 2 1
, where Ev: event, P: attribute value.

For example, in the case of a promotion characterized only by budget and duration, the above formula becomes: The lower the value of this measure, the more similar the events.

2 2 1 2 2 1 2 1 2 , 2 ,

Estimating Impact

Two different methods will be used in order to estimate the impact of future events:

• BN This is the simplest approach, where the impact of the nearest neighbor provides the impact for the future event.

• B3N

Here, the three nearest neighbors are used for the estimation of the impact of the future event. The final impact is constructed as 50% of the impact of the nearest neighbor and 25% from the second and third nearest neighbors, respectively. In mathematically terms, a triangular kernel function is used [START_REF] Härdle | Applied Nonparametric Regression[END_REF], assigning weights equal to (1/2, 1/4, 1/4) to the three nearest neighbors. Forecasting with quantitative methods the impact of special events in time series 8

Regression

Classical Multiple Linear Regression (MLR - [START_REF] Makridakis | Forecasting, Methods and Applications[END_REF]) can be used in order to estimate the impact of the holdout events:

= + = n i Ev i Ev i Ev i i i P P P P b c impact 1 2 , 2 , 1 ,
For example, in the case of a promotion characterized only by budget and duration, the above formula becomes:

impact = constant + b 1 * Budget + b 2 * Duration
The model parameters are estimated from the values of the cues for past events using the least squares criterion.

Artificial Neural Networks

One of the simplest topologies of ANN, a fully connected multilayer perceptron with two hidden layers and an output layer can be used for the estimation of the impact of irregular events [START_REF] Haykin | Neural Networks: A Comprehensive Foundation[END_REF]; the input signals would be the event's attribute values while the output would be the impact of the event under consideration (Figure 1).

[Insert Figure 1 about here]

In this study the size of the input signal is equal to the maximum number of parameters explaining the historic irregular events. In order to forecast the impact of a possible future event, the network is trained first with all historic data as the training set, using the Back Propagation (BP) algorithm [START_REF] Haykin | Neural Networks: A Comprehensive Foundation[END_REF]. Each historic event is considered to be one training example while the impact is the response of the network. As soon as the training of the network is completed, and the weights of the nodes have been determined, the future

Training

All different fully connected networks up to two hidden layers and no more neurons per hidden layer than the input size are trained ten times each. The number of alternative networks is limited by the constraint that the second hidden layer is not used if the first hidden layer does not contain neurons equal in number with the input size. Each time, the Mean Square Error (MSE) over the training sample is calculated [START_REF] Makridakis | Forecasting, Methods and Applications[END_REF]).

The network with the smallest MSE error is used to produce the forecast.

ANNs and MLR

There are similarities between ANNs and MLR. Neural networks can be seen as a form of nonlinear regression [START_REF] Ripley | Neural Networks and Related Methods for Classification[END_REF]). It has also been observed that MLR can be expressed as a simple ANN with only one layer [START_REF] Warner | Understanding Neural Networks as Statistical Tools[END_REF]. However, even if MLR and an ANN model have exactly the same topology, the weights that would be given would only be identical by chance as there are thousands of training methods suggested in the ANN literature.

Expert method

Our proposal for an expert approach is based on the assumption that linear as well as nonlinear relationships exist between an event's impact and the values of its attributes. MLR is expected to be better at capturing the linear relationships, while ANN should be superior at capturing the non-linearities, as their topology permits the adaptation to more complex We propose two different approaches:

• EXPRT1
Here the MLR model is fitted to the past events and R 2 is calculated. If R 2 > 95%, the relationship between the impact and its determinants is considered linear and MLR is used to estimate the impact of the future event. Otherwise ANN is used for that estimation.

• EXPRT2

Since the noise of the series is not known a priori, it might be inadvisable to set up a threshold as high as 95% for R 2 In this approach the MLR model is fitted to the past events and R 2 is calculated. The ANN is trained with the past events and R 2 is also calculated (Since estimations for all past events can be calculated with the optimized synaptic weights, R 2 can be calculated). If R 2 MLR > R 2 ANN the relationship between the impact and its determinants is considered linear and MLR is used to estimate the impact of the future event. Otherwise ANN is used for that estimation.

Evaluation

This section covers the results of the simulation experiment used to evaluate the alternative forecasting methods as well as evidence from an application to real life data via a case study in forecasting TV ratings.

Simulation setup

The Data Sets Each data set is constructed from 50 special events -the 35 first are used as a learning set and the remaining 15 used for testing out of sample accuracy -measured in terms of the Mean Absolute Percentage Error (MAPE) and the Median Absolute Percentage Error (MdAPE) (Makridakis et. al. 1998). So the first 35 special event can be seen as past events while the remaining 15 as forthcoming/announced special events withi known parameters however unknown impact.

The 50 special event correspond to non-consecutive points of a time series i.e. you could have a monthly time series with 10 years worth of data (120 months) where only 50 of those months having promotions (the most common type of special event in operational forecasting); 35 in the first 7 years (the timings randomly chosen) and 15 in the last three years that were held out for evaluation. Thus you have 70 periods with non-promotial activity and 50 with (35 and 15 within and out-of-sample respectively). It is out of the scope of this study to measure the accuracy in periods where special events are taken place, We do take a decomposition stance and try to estimate/isolate the additive impact of the special event in the baseline level of the series. So a partial forecast is the target, which is exactly the isolated impact of the special event (figure 2).

[Insert Figure 2 The second factor, Strike, has a negative effect and is described via two attributes: a) Percentage, ranging from 20% to 100% in steps of 5% (representing the % participation of employees in the strike), b) Duration, ranging from 1 to7 days.

The rational behind the building of the special events comes from the marketing literature [START_REF] Delvecchio | The effect of sales promotion on post-promotion brand preference: A meta-analysis[END_REF][START_REF] Martsnez-Ruiz | Using daily store-level data to understand price promotion effects in a semiparametric regression model[END_REF][START_REF] Laroche | A model of consumer response to two retail sales promotion techniques[END_REF][START_REF] Kotler | Marketing Management 11 th Edition[END_REF];

where typically a positive event has a positive bell-shaped impact (figure 3a), fading after a certain number o periods (vice versa for a negative one). Sometimes the promotion creates a skewed bell-shaped impact when if forces sales faster (figure 3b) or slower; furthermore sometimes after some periods there is a negative effect in the sales as the product has been oversold during the promotional activity (figure 3c). For the sake of simplicity and to reduce the degrees of freedom of this multiple hypothesis setup, I only investigate special events with impact in one period only -where that impact has a pulse shape so in an additive model it is just a straight addition (or deduction) to the regular sales (regular standing for an equivalent period without promotional activity i.e. last year's sales for the same period in the year)

[Insert Figure 3 about here]

Linearity

For the construction of the test events we used the following generating function: Forecasting with quantitative methods the impact of special events in time series 13 Two levels of Gaussian noise N(µ,U 2 ) were generated: Low V(0,10) and High V(0,30)

Impact = f(

Runs

We ran 50 simulations for each of the four combinations of Linearity and Noise, resulting in 200 simulations.

Simulation Results

The 200 datasets were analyzed, each one consisting of 50 events. 35 events were used to estimate the models' weights and 15 were used for testing. Thus the overall accuracy calculation was based on a sample of 3000 events. The results are presented in tables 1 to 4.

[Insert Table 1 about here]

Overall, EXPRT1 was the better approach, though its performance was not much better than using ANN for all events. Overall ANN was the best among the first four standard approaches and it was significantly better than MLR. This is indicative of the strong effect of the simulated nonlinearities. Both expert approaches were better than all the other methods.

The use of the MdAPE, rather than the MAPE, did not alter the ranking of the methods.

[Insert Table 2 about here]

In table 2 the effect of linearity is presented. As expected, for the linear subset, MLR performed best, while for the non-linear one ANN was the top performer. EXPRT1 and EXPRT2 were close to the best methods in each case. Overall, EXPRT1 was the better approach. However its performance did not differ much from ANN. As before, the use of the MdAPE did not alter these results.

[Insert Table 3 about here]

In table 3 the effect of noise is examined. For all levels of noise the EXPERT1 and EXPRT2 approaches were better than the conventional methods.(Again, the use of the MdAPE made little difference).

[Insert Table 4 provides the computation times for each model, a criterion critical nowadays where retailers have to produce forecasts for thousand of products every day. As expected, ANN and, inevitably, the expert approaches required significantly more processing time.

An application to real-life data

Data from a real-life case study are used to test the aforementioned models and refer to television (TV) audience ratings from 1996 to 2000 in Greece (Nikolopoulos et. al. 2007b).

TV audience ratings are measured as a percentage (%) of people watching a specific programme; when the TV programme broadcasts a special event, such as a major sport event the audience level is radically increased. Impact refers to the additive increase in the percentage of the total audience above the average of the last four relative periods (same time zone within day, same day of week) for a given channel.

AGB Hellas S.A. provided for this study real TV audience ratings data, relating to the target group of men aged 25-44 in Greece. The supplied data included 46 sport programmes. Each event is described by three categorical parameters: Importance, Competition and Time Zone.

The parameters are quantified on scales from 1 to 3 or 1 to 5 in order to make statistical forecasting models easily applicable.

From the 46 programmes for which data was provided, 34 were used for the fitting or training the models and 12 were used as a hold-out sample to evaluate forecasting performance. The values for the three independent variables have been estimated by an expert in audience analysis in Greece. A subset of this data is given in table 5.

[Insert Table 5 about here]

The performance of MLR, BN, B3N, ANN was tested on these real life data and accuracy was measured in terms of Mean Absolute Percentage Error. There was no need for a Percentage/scaled error in this case study as all cases was within the same scale; thus the simplest of unsigned error metrics was chosen; [Insert Table 6 about here]

In this real-life example ANN performs better indicating strong non-linearities in the data.

Nearest Neighbor Approaches follow while MLR comes last; no expert approaches were separately reported in this table as the selection protocols EXPRT1 and EXPRT2 were selecting in all 12 cases the ANN as to perform extrapolation, thus the results were identical to the ANN one (in this real-life example the ANN is trained only once in the available past 34 cases).

Conclusions and Future Research

This study has examined the relative performance of ANN, MLR and NN in estimating the impact of expected irregular future events. The simulation results suggest that the use of two expert approaches should be considered, based on selection protocols between ANN and MLR.

There is strong evidence that, overall, the expert approaches do perform significantly better.

However, the simpler the problem (linear, low noise) the greater the advantage for regression approaches. For more complex problems ANN has a critical advantage. Thus, it is no surprise that, overall, a selection protocol outperformed all other approaches as long as nonlinearities could be successfully detected. For this detection two different approaches were tested, EXPRT1 and EXPRT2. Although EXPRT1 performed slightly better than EXPRT2, we do consider EXPRT2 to be a more promising approach as it is not dependent on the preset threshold of 95% that might have given EXPRT 1 an advantage with the specific dataset that was examined in this study.

Further results from an application of these approaches to real-life data of forecasting TV ratings of major sport events showed promising evidence but at the same highlight strong non-linearities and thus revealing how much more difficult is to forecast real-life data (contrasting to simulating data where always an underlying stochastic mechanism creates the datasets). Future research should focus on constructing an expert model that both detects nonlinearities more efficiently and also makes it selection from a larger set of forecasting methods. Also focus should be given as to how to reduce the computational intensity and thus the required time to setup, optimize, train and run the Artificial Neural Network approaches as to make the applicable in real-life setup. Lastly some more effort should be given on the improvement of the performance of Nearest Neighbor approaches as there are quite intuitively appealing.

Concluding, it should be noted that the severe impact and frequency of irregular events in practical forecasting contexts means that there is a widespread need for models to support judgmental forecasters in their task of adjusting baseline forecasts to take into account the effects of these events. The search for effective modeling procedures is therefore likely to be a fertile research field for many years to come and the synergy required from the

  parameters are fed into the network with the form of an input signal. For each parameter in the input signal that does not appear in the future event, a zero value is assigned. Signal flow through the network progresses in a forward direction, from left to right and on a layer-by-layer basis reaching the output node. The outcome is the potential impact of the future event.

  . We therefore propose a selective algorithm which chooses between MLR and ANN according to whether nonlinearities are detected.How can we detect whether the relationship: Impact = f(attribute values) is linear or not?

  considered. The first one, Promotion, has a positive effect and is described via three attributes: a) Budget, ranging from 50 to 150 in steps of 10 (each step representing 1000€), b) Duration, ranging from 1 to 14 days, c) Media used, with values: 1 (Paper), 2 (Paper + Radio), 3 (Paper + Radio + TV)

  Factor attribute values) + Noise For the function f, two formulae were used: Linear and Non-linear as shown below: Promotion/Linear : 0.7 x Budget + 5 x Duration + 20 x Media Strike/Linear : -100 x Percentage -2 x Duration Promotion/Non Linear : (0.5 x Budget) x ((Duration/14) + Media) Strike/Non Linear : -50 x Percentage x ((Duration/7)+1)

Table 4 about here]

	F o r
	P
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	w

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	Submitted Manuscript
	TABLES & FIGURES
	F o r F o r F o r F o r F o r F o r F o r
	P P P P P P P
	e e e e e e e
	e r e r e r e r e r e r e r
	R e v R e v R e v R e v R e v R e v R e v
	i e i e i e i e i e i e i e
	w w w w w w w

Information Technology is rather crucial (via Data Mining and Information Management techniques and algorithms -Fildes and Nikolopoulos 2006).

Figure 1. Multilayer perceptron (two hidden layers) Figure 2. Promotional periods in a time series

Page 23 of 30 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Figure 3. Impact of promotions/special events in a time series baseline value

	Submitted Manuscript
	F o r
	P
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	R e v
	i e
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	Submitted Manuscript
	F o r
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Table 1 .

 1 Simulation Results, All dataForecasting with quantitative methods the impact of special events in time series 26

	Methods	MAPE	MdAPE
	MLR	51.18%	8.51%
	BN	88.05%	21.51%
	B3N	113.72%	22.36%
	ANN	28.03%	7.29%
	EXPRT1	25.91%	5.75%
	EXPRT2	26.43%	6.21%

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

Table 2 .

 2 Simulation Results, Effect of LinearityForecasting with quantitative methods the impact of special events in time series 27

		Submitted Manuscript		Page 26 of 30
		Linear	Non-linear
	Methods
		MAPE	MdAPE	MAPE	MdAPE
	MLR	9.83%	3.25%	92.53% 25.39%
	BN	70.72%	17.40% 105.37% 27.55%
	B3N	87.85%	17.39% 139.60% 29.76%
	ANN	19.93%	5.61%	36.12%	9.51%
	EXPRT1	9.94%	3.26%	41.88% 11.44%
	EXPRT2	14.34%	3.97%	38.51% 10.08%

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

Table 3 .

 3 Simulation Results, Effect of Noise

	Page 27 of 30	Submitted Manuscript
		High Noise	Low Noise
	Methods
		MAPE	MdAPE	MAPE	MdAPE
	MLR	50.65%	10.13%	51.72%	6.62%
	BN	72.95%	21.51% 103.15% 21.53%
	B3N	85.00%	22.37% 142.45% 22.36%
	ANN	31.75%	8.88%	24.30%	5.87%
	EXPRT1	27.35%	7.01%	24.47%	4.70%
	EXPRT2	29.48%	7.85%	23.38%	4.79%

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

Table 4 .

 4 Simulation Results, Processing Time

	Instant	1-2 Hours
	MLR	)
	BN	)
	B3N	)
	ANN		)
	EXPRT1		)
	EXPRT2		)
	CPU : Intel P4 1.7Ghz, RAM: 512 MB

Table 6 .

 6 Accuracy performance on Real-life Special Events Data (adopted fromNikolopoulos et. al. 2007b)

		MAE
	Methods	(Mean Absolute Error)
	ANN,
	EXPRT1,
	EXPRT2	9.90
	B3N	12.68
	BN	12.89
	MLR	14.54

Page 30 of 30 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

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

Acknowledgements

The Manchester Business School (MBS) Research Support fund 2006/7 funded the work for this paper as part of a one year interdisciplinary project on the development of an innovative forecasting methodology for television ratings. The author would like to thank especially Mr.

Konstantinos

Giannelos for providing the software for running the experiments with the ANNs, a software developed in July 2004 in the Forecasting Systems Unit in National Technical University of Athens for his Diploma dissertation entitled "Irregular Events Impact Estimation Methodology and Information System", under the supervision of Dr K Nikolopoulos, Assistant Professor in MBS and Professor Vassilis Assimakopoulos, Special Secretary for Digital Planning in the Ministry of Economics and Finance in Greece. The author would like to thank Professor Paul Goodwin of the School of Management at the University of Bath for his suggestions in the first submission of this research as well as the Editor of AE and an anonymous referee for their suggestions in the revision of this article.