Condence intervals of but-for price estimations : an evaluation of the dummy-variable and the forecasting approach for the German cement cartel

Faculty Economics and Business, University of Amsterdam Bachelor: Econometrics and Operational research

Confidence intervals of but-for price estimations

An evaluation of the dummy-variable and the forecasting approach for the German cement cartel

Roelien Timmer - 10432914 Supervisor: M.J.G. Bun

This thesis is an empirical study which evaluates confidence intervals of but-for price estimations. But-for price estimations are crucial to establish the damage and fines in the case of cartels. In this study intervals are determined for the German cement cartel. Besides, the performances of the intervals are evaluated on the basis of a Monte Carlo analysis. This paper regards different techniques to determine but-for prices, namely the dummy-variable and the forecasting approach. This thesis shows that the confidence intervals of the dynamic models are significantly underestimated, whereas the confidence intervals of the static models are reliable. Besides, this research shows that a simplified version of the dummy-variable confidence intervals is not a good estimator due to the significant underestimation.

Content

1. Introduction p. 4

2. But-for price estimation and confidence intervals p. 7

2.1 The but-for price p. 6

2.1.1 The dummy-variable approach p. 6

2.1.2 The forecasting approach p. 8

2.2 Prediction and forecast intervals p. 9

2.2.1 Prediction intervals p. 9

2.2.2 Forecast intervals p. 10

3. Methodology p. 12

3.1 The German cement cartel p. 12

3.2 Prediction and forecast intervals of but-for price estimations p. 14

3.3 Monte Carlo p. 16

4. Results p. 18

5. Conclusion p. 21

Bibliography p. 24

Introduction

In the current economy, competition between firms is an important pillar. It is regarded as a required condition for innovation, efficiency and economic growth. Avoiding this competition attracts many firms every year again and again. Not rarely do firms decide collectively to abstain from the competition by making price agreements with their peer companies. However, establishing those kind of agreements is strictly forbidden by antitrust authorities.

Avoiding the violation of those antitrust laws is an ongoing battle for antitrust authorities. An important prevention tool is the application of big fines for the par-ticipating firms. Besides punishing firms for their behaviour, the compensation for plaintiffs is an important aspect of those fines. All the injured firms want to be com-pensated for their missed income or the higher costs they made. Therefore the fines are based on the damage which the cartel caused.

The European Commission estimates this damage on the basis of the difference between the actual situation and the one which would have existed without the in-fringement (2003, pp. 9-10). This difference is also defined as the overcharge. The European Commission calls the assessment of harm the but-for analysis (2003, p. 10). The main unknown quantity in this analysis is the so called but-for price. Nowadays, there are still a lot of researchers trying to find the best approximation of this but-for price.

There are two main methods to determine the but-for price, namely the dummy-variable and the forecasting approach. The European Commission applies both methods depending on the circumstances (2003, p. 26). The forecasting approach only uses data of the benchmark period to estimate the parameters of the model whereas the dummy-variable approach uses data of the whole available time period (Nieberding, 2016, p. 369).

To determine which one of those two approaches is better it is important to take into account the uncertainty of the but-for price estimation. Especially in the case of setting penalties the antitrust authorities aim at having an estimation which has a low uncertainty. A method which includes the estimate and the uncertainty is the confidence interval. Therefore, the purpose of this thesis is to compare the confidence intervals

of different but-for price estimation techniques. The two techniques evaluated are the dummy-variable and the forecasting approach. A subquestion of thesis is whether a simplified version of the prediction intervals is a good estimation for the dummy-variable approach. The performances of the confidences intervals of the different but-for price estimations are assessed with the aid of a Monte Carlo simulation.

In previous literature the confidence intervals of the but-for price of the dummy-variable and the forecasting approach have never been calculated. Therefore, the con-clusion of this research is very useful for antitrust authorities and courts. This research shows whether confidence interval are underestimated, reliable or overestimated. This paper regards static and dynamic models. An important point to mention is that the term confidence interval is often interchangeable with prediction and forecast intervals. This thesis applies prediction to the dummy-variable approach and forecasting to the forecast approach.

This thesis begins with a literature study where the dummy-variable and the fore-casting approach are discussed with the aid of previous research. This is followed by an evaluation of two methods which establish confidence intervals, namely the prediction and the forecast method. This literature study is followed by the research methodology where first the characteristics of the German cement cartel data are discussed. Sub-sequently, the estimation of the but-for price, the prediction intervals and the forecast intervals is discussed for this case. Afterwards a Monte Carlo analysis is performed with the aid of the estimated parameters of the German cement cartel. The aim of the Monte Carlo analysis is to simulate but-for prices and to check whether this but-for price developments corresponds with the prediction and forecast intervals. In the next chapter the results are given which consist of the estimated models. Besides, the mean confidence interval width and the coverage rates are given. The coverage rate is the ratio that the but-for price falls with the confidence interval. Those results are followed by a chapter which consists of the conclusion, the shortcomings of this research and suggestions for future research. A structured overview of this paper can be found in the table of content on page 3.

2. But-for price estimation and confidence intervals

2.1 The but-for price

This section discusses the two main estimation methods of but-for prices. In paragraph 2.1.1 the dummy-variable approach is evaluated and afterwards the forecasting approach in paragraph 2.1.2.

Before the discussing the dummy-variable and the forecasting approach a definition of the but-for price is required. The European Commission uses a but-for analysis to estimate the damage caused by the cartel (2003, pp. 9-10). They argue that the main determinant in this analysis is the but-for price, which is the price that would have existed in absence of an cartel.

2.1.1 The dummy-variable approach

The dummy-variable approach is introduced by Rubinfeld and Steiner (1983). They describe this approach as an estimation of an econometric model which is performed on all the available data. They also state that this model includes a dummy which is equal to 1 if there is a cartel, and 0 otherwise. If the coefficient of the dummy-variable is significantly different from zero, they argue that this would indicate the existence of a cartel.

Boswijk, Bun and Schinkel (2016, p. 6) described a common autoregressive dummy-variable model which is derived from previous research. They adopted the following the data generating process in their research:

yt = α1+ α2Dt+ xtβ + γyt−1+ t t = 1 ... T. (1)

In this model yt is specified as the product unit price in period t and yt−1 is the

lagged product unit price of one period. xt is defined as a set of control variables and

pp. 6-7) break down as follow:          Dt = 0 t ≤ TB, Dt = 1 TB < t ≤ TE, Dt = 0 t > TE,

where TB is the begin date of the cartel and TE the end date. Besides, three

as-sumptions are made to be able to perform an Ordinary Least Squares (OLS) regression. The first assumption is that the expected value of t given the explanatory variables

is equal to zero. Second the variance of t has to be constant over t which implies

homoscedasticity. The last assumption is the absence of serialcorrelation, so that t

does not correlate with t−j where j 6= 0. Subsequently Boswijk et al. (2016, p. 7)

define the but-for price in period t as:

bf pt= α1+ xtβ + γbf pt−1+ t t∈ TC with Tc= TB+1, ..., TE. (2)

In this model the dummy-variable for the cartel period is switched off which means that Dt = 0. A prediction of this but-for price can be obtained by first performing an

OLS regression on model (1). These obtained estimations of the coefficients has to be substituted in the following equation which gives an estimation of the but-for price:

ˆ

bf pt= ˆα1+ x0tβ + ˆˆ γ ˆbf pt−1. (3)

Boswijk et al. (2016, p. 9) emphasize that it is important that the dummy of the cartel does not affect the control variables to avoid multicollinearity.

Besides the dummy-variable approach discussed in this paragraph, there is another popular technique to estimate the but-for price, namely the forecasting approach. The next paragraph contains a review of this forecasting approach.

2.1.2 The forecasting approach

In the previous paragraph, the dummy-variable approach was described to estimate the but-for price. This paragraph discusses the forecasting approach on the basis of previous research. Because the two approaches are very similar, a comparison is often made in this paragraph.

Rubinfeld (1985, p. 1087) states that a regression model based on a non-cartel period can be used to predict what the but-for price would be in the cartel period, which is called the forecasting approach. Nieberding mentions that the main difference between the forecasting and the dummy-variable approach is the data selection (2006, p. 369). He explains that the dummy-variable approach evaluates the available data of the whole time period: before, during and after the cartel, whereas the forecasting approach only uses data from before or after the cartel. The dummy-variable and the forecasting approach are even equivalent in special cases. Higgins and Johnson state that they are the same when the dummy variable included has the same frequency as the original data has (2003).

Concluding, the forecasting approach is similar to the dummy-variable approach but only selects data of the benchmark period to estimate the regression. Therefore, the dummy-variable is also redundant in the forecasting method. With the purpose of comparing the two approaches it is logical to use similar models which only differ in the necessary aspects. In section 3.2 this is deliberated extensively for the creation of confidence intervals for a but-for price based on forecasting. The next section discusses the creation of confidence intervals.

2.2 Prediction and forecast intervals

This section discusses prediction and forecast intervals for but-for price estimations on the basis of Greene (2002). Paragraph 2.2.1 contains an evaluation of the creation of prediction intervals which is relevant for the dummy-variable approach. Whereas the next paragraph discusses forecast intervals which corresponds to the forecast approach

1_.

1_{The distinction between the forecast and the prediction interval is explained on page 5 of this}

2.2.1 Prediction intervals

This paragraph discusses prediction intervals according to Greene (2002, p. 111). He defines a prediction as:

ˆ

yo = xo0b (4)

where the b the estimated coefficient based on the data y and X. xo _{refers to the}

new explanatory variable on which a new prediction is based. Consequently he defines the prediction variance as follows:

var(e0_{|X, x}o_{) = σ}2_{+ x}o0_[σ2_(X0_X)−1_]xo ₍₅₎

In this formula σ2 is the variance of the error of the original model. This σ2 can be estimated by the variance of the of the sample regression, s2 .

Combining formula (4) and (5) the prediction interval becomes:

P I = ˆy ± tα/2se(e0) (6)

The strength of this intervals is based on some important classical linear regression assumptions. First, the model should have a linear relation between the explanatory and the dependent variables (Greene, 2002, p. 10). Besides, the independent variables should be exogenous, which implies that the expected value of the disturbances cannot be written as a function of the independent variables. Another important assumption is the homoscedasticity of the error terms, so the variance of the error term is constant over time. Additionally the error terms should not contain autocorrelation. At last, the disturbances should have a normal distribution.

The prediction variance does not take into account time in the form of lagged vari-ables (Greene, 2002, p. 111). The variance ignores the possible correlation between current and past values. Greene points out that the forecasting model addresses the explicit role of time.

2.2.2 Forecast intervals

This paragraph evaluates forecast intervals which takes into account time series as mentioned in the previous section. This is relevant for dynamic models.

There is a difference in the creation of a forecast interval for dynamic and a static forecast model. First, the dynamic model is discussed which is the Autoregressive-Distributed lag model, ARDL(p,q). The letter p is the degree of lags of the dependent variable and q refers to the lags of the explanatory variable. Greene defines the ARDL model as (2002, p. 576): yt = µt+ γ1yt−1+ ... + γpyt−p+ t (7) with: µt = µ + Pr j=0βjXt−j+ δwt

The forecast depends on the classical linear regression assumptions which were dis-cussed in section 2.2.1. Besides, the forecast error arises from three sources (Greene, 2002, p. 576). The first source is the uncertainty of the parameter estimation. Besides, there can be a discrepancy between the forecasted value and the actual value due to disturbances. At last, we assume that the forecasted error has an expectation of zero. The forecasting variance of Greene only takes into account the first and the last source of uncertainty (2002). The forecast variance of the h step ahead forecast becomes:

var(ˆyT +H|T) = σ2[1 + Ψ(1)11+ ... + Ψ(H − 1)11] (8)

This Ψ is split up as follow:

Ψ(i) = Ci_jj0_Ci0 ₍₉₎

C=          γ1 γ2 ... γp 1 0 ... 0 0 1 ... 0 .. .. .. .. 0 0 ... 1         

In the matrix j, σ2 _{can be estimated by the sample variance, s}2_{. The matrix C}H

converges to zero when all the absolute values of the roots of the matrix are less than one, which means that the equation is stable.

The forecasting interval for ARDL models models is defined as:

F I =yT +H|Tˆ ± tα/2se(ˆyT +H|T) (10)

This is the formula for dynamic models, whereas static models have a different formula for the forecast interval. Forecasting of a static model could be regarded as a prediction, because of the absence of lagged variables. Therefore, the forecasting interval of the static forecasting model can be estimated by formula (6).

The main difference between the prediction and the forecasting intervals is the concept of time. The prediction intervals do not into account time series in the form of lagged variables. Whereas, Greene explains that the forecast explicitly treats the role of time and lagged variables, whereas prediction does not (2002, p. 111). Besides, the prediction interval is estimated on the basis of the whole period, cartel plus non-cartel. While, the variance used in the forecasting intervals is only based on the non-cartel period, which would suggest a bigger variance, because of the reduction in observations. The last paragraphs discussed the but-for price estimation and their corresponding confidence intervals. On the basis of those theorems the next chapter describes the methodology of this thesis.

3. Methodology

In this chapter the research methodology is described. The content of the last chapter is combined to subsequently define prediction and forecasting intervals for but-for prices estimations. On basis of those definitions the intervals are calculated in the case of the German cement cartel data. Therefore, the German cement cartel data is evaluated extensively in this chapter. Hereafter, the characteristics of this cement data are used to create prediction and forecast intervals. Finally, on the basis of those Monte Carlo analysis the strength of the definitions are tested.

3.1 The German cement cartel

The German cement cartel is well-known because the scale of the punishment. The firms involved in this scandal received the largest fine ever imposed by the Bundeskartellamt, namely 380 million Euros (Bundeskartellamt, 2013). During the 1990s a group of domi-nant cement companies divided the cement market among themselves and increased the cement price during the 1990s. The cement market can be regarded as an attractive environment to launch a cartel. H¨uschelrath et al. (2013) denotes the homogeneity of the product, the few producers and the high entry barriers as important inviting characteristics. In this thesis the data of the German cement cartel is analysed which is retrieved from the Federal Statistical Office of Germany (2010) 2_{. This is a data}

set that includes information about the cement prices, the production, lime prices and electricity prices. The data is seasonally adjusted where it was suitable (U.S. Census Bureau, 2013). H¨uschelrath states that the data is gathered by questionnaires which were filled in by the main German cement cartels (2012, p. 109). He also mentions that it should be taken into account that those questionnaires can be filled in strategically instead of honestly by the cement firms.

Other important aspects to keep in mind is the presence of a price war after the collapse of the cartel according to H¨uschelrath (2013). After the destruction of the cartel a decrease in the cement price is a natural phenomena. The cement producers claim that the price cutback should be partly assigned to a price war that started after

the end of the cartel period (H¨uschelrath, 2013). To take into account this turbulent period a dummy variable is a logical choice, which is denoted as the transition. This is the period between the end of the cartel and the moment where the cement prices were back at the competitive level. Frank and Schliffke (2013) even devoted a whole research to the aftermath of this cartel. They point out that the court declared the commencement of the transition period at August 2002. Besides, they found in their own research that the transition period should at least last to May 2005. In line of their conclusions this thesis employs a transition period of 2002-Q4 till 2005-Q2.

The notation of the variables in this thesis is based on H¨uschelrath et al. (2013) and Frank and Schliffke (2013). The prices of cement, lime and electricity are denoted by pC

t , pLt and pEt subsequently. The dummy variable of the cartel period is dcartt , where

dcart

t = 1 during the cartel period of 91-Q1 till 01-Q4 and dcartt = 0 otherwise. The

dummy of the transition period is dof f_t , where dof f_t = 1 during the transition period and dof f_t = 0 otherwise. The variable of the cement production is cprodt

The order of integration differs per variable. The results of the Augmented Dickey-Fuller test can be found in appendix (4). The cement price, pC

t , is integrated of order

two for the non-cartel period. Besides, the electricity price is for the total and for the non-cartel period integrated of the second order. As in H¨uschelrath et al. (2013) and Frank and Schliffke (2013) those variation in stationarity is ignored in this thesis. The integration of the second degree could be a consequence of the price war which shifted the price levels. This implies that the cement price is integrated of the second order with a break. The development of the cement price is graphically shown in figure 1 on the next page.

3.2 Prediction and forecast intervals of but-for price estimations

This section discusses the creation of confidence intervals of but-for prices. As men-tioned before, the prediction intervals relate to the dummy-variable approach, whereas forecasting intervals refer to the forecasting approach. Besides, for both approaches a distinction is made between a static and a dynamic model. The models are based on the research of Frank and Schliffke (2013). The logarithms are left out in the two models to be able to construct prediction intervals of the prices instead of the logarithmic prices. Different forms of the following auto regressive dummy-variable model are regarded:

pc

t = α +

Pp i=1γip

c

t−i+ β1dcartt + β2dof ft + β3cprodt+ β4plt+ β5pet+ t (11)

The characteristics of those variables can be found in section 3.1 and trepresents the

disturbance term. How many lags are added depends the presence of autocorrelation. A static version of model (11) can be obtained by excluding the lagged price variables. Performing an OLS regression on this model gives estimations of the coefficients. The presence of heteroscedasticity is identified by the White test (1980). Autocorrelation is assessed by the Breusch-Godfrey test and heteroscedasticity by the Breusch-Pagan-Godfrey test. On the basis of the estimated coefficients the but-for price is estimated by setting the dummy variable of the cartel equal to zero. The dummy-variable of the transition period is already zero during the cartel period, therefore dof f_t is left out. The but-for price estimation on the basis of the model (11) for the period of 91-Q1 till 01-Q4 becomes: ˆ bf p_t= ˆα +Pp i=1ˆγibf p c t−i+ ˆβ3cprodt+ ˆβ4plt+ ˆβ5pet (12)

The first p but-for prices are for the dynamic model set equal the long-term equi-librium. This long term equilibrium is given by:

ˆ bf p_t=

α+ ˆ

ˆ

β

d

+ ˆ

β

cprod

+ ˆ

β

p

+ ˆ

β

p

1−

P

ˆ

γ

The following but-for prices are estimated by an iterative process. Hereafter the prediction error variance can be calculated with the aid of formula (5). The vector xo

are the explanatory variables during period t, where dcart

t = 1 is replaced by dcartt = 0.

The matrix X includes all the observations from the period 91-Q1 till 10-Q3. s2 _{is the}

variance of the error term of the OLS regression of model (11). Hereafter the prediction interval can be calculated according to formula (6) of Greene (2002, p. 111).

The but-for prices can also be forecasted for the dynamic and the static model. Model (12) without the cartel dummy, dcart_t , is estimated by an OLS regression on the basis of the non-cartel period which lasts from 02-Q1 till 10-Q3. For the dynamic model the first p but-for prices are estimated by formula (13). Hereafter the forecast interval for the dynamic model can be constructed by formula (10). The static model is again also estimated by model (12), but without the lagged cement price variables and the cartel dummy. The forecast interval is calculated by formula (6) due to the absence of lagged variables. This is the same formula which is applied to the dynamic and static dummy-variable model.

This and the previous chapter described the way to construct prediction and fore-casting intervals for but-for price estimations. Those intervals are constructed on the basis of the German cement cartel data. The next paragraph discusses a Monte Carlo simulation which is performed to evaluate the confidence intervals.

3.3 Monte Carlo

Last sections discussed the creation of confidence intervals. With the aid of the esti-mations based on this cement cartel a Monte Carlo analysis can be performed. This paragraph discusses how this Monte Carlo analysis is performed.

The first step is simulate prices and but-for prices for a dynamic and static data generating process (DGP) according to formula (11). The error term is assumed to be normally distributed, t∼ i.i.d.(0, σ2), and the number of simulations is 10000.

For the dynamic DGP the first p prices are set equal to the actual observed price:

pr

and the first p but-for prices are set equal to the long run-equilibrium: bf pr t =

α+β

d

+β

cprod

+β

p

+β

p

1−

P

γ

For t = (p+1, ..., 79) the prices and but-for prices for the dynamic DGP are simulated by iterations described in formula (13) and (14).

After the simulation of the static and dynamic DGP, the but-for prices are estimated for the dummy-variable and the forecasting approach according the procedure described in section 2.2. Besides, the 80 percent confidence intervals are determined according to a normal distribution with a t-value of 1.282. For the dynamic DGP, for the first p periods there are no intervals calculated. Those intervals are referred to as Not a Number (NaN).

Hereafter the number of times that the simulated but-for prices falls within the confidence intervals are counted per period. This is denoted as the coverage rate. For the dynamic models, the first p but-for prices are neglected is this calculation, because no confidence intervals are available.

4. Results

This section discusses the results of the regressions, the prediction intervals, the forecast intervals and the Monte Carlo analysis.

First, the dynamic DGP is discussed. The OLS regression of the dynamic model showed that for a model with 2 lags of the cement price the autocorrelation is omitted. Because, the Breusch-Godfrey for autocorrelation test has for this model a p-value of 0.2804. But, the Jarque-Bera and the Breusch-Pagan-Godfrey test shows that there is still presence of non-normal distributed errors and heteroscedasticity. Both price lags have a effect at a significance level of 0.01. The variance of the model is approximately 1.3. The exact values of the regression and the tests can be found in appendix 1. The mean regression coefficients of the simulated Monte Carlo models have similar values and standard deviations. Besides, the mean variances of the simulated dummy-variable and forecasting model are also approximately 1.3. Appendix 5 contains a table of the all the mean coefficients and standard deviations. The mean width of the prediction interval is around 11.0 and for the simplified version around 8.6. The mean width of the forecast intervals starts at 19.4 and slowly decreases to 15.0. The mean confidence intervals for all periods can be found in appendix 2. The mean simulated prices, but for prices, prediction and forecasting intervals are graphically shown in figure 2.

The number of times that the simulated but-for price falls within the 80% confidence interval, denoted by the coverage rate. For the dummy-variable model the coverage rate is around 26%. For the simplified prediction intervals the coverage rate is just around 22%. The coverage rate of the forecast intervals is in period 3 approximately 6.7% and increases gradually to 14.2% in period 44. In this computation of the coverage rate for the forecast approach some outliers were left out3.

Next, the static DGP is discussed. The coefficient of the cartel is approximately 21, which implies that the price during a cartel is 21 euro higher than during a non-cartel period. The Jarque-Bera, Breusch-Godfrey and the Breusch-Pagan-Godfrey test are

3_{The outer fences for are computed by Q1 − 3 ∗ (Q3 − Q1) and Q3 + 3 ∗ (Q3 − Q1), where Q1 is}

the first quartile of the estimated but-for price period and Q3 is the third. Besides, estimated but-for prices below zero are assumed to be outliers. The number of outliers found for the dynamic DGP with forecasting are 88, where 68 of the outliers were negative but-for prices and 20 exceeded the upperbound.

all significant at a level of α = 0.01. This implies that the errors are not normally distributed and autocorrelation and heteroscedasticity are present in the error terms. Hereafter, the Monte Carlo simulation is performed. The mean coefficients of the dummy-variable and the forecast model are comparable. The mean variance of those models are both approximately 11.4, which is the same of the original static model. The width prediction intervals of the static model is on average approximately 10.7, for the simplified version of this prediction interval this is around 8.6. The mean width of the forecast intervals is around 11.0. The mean simulated prices, but-for prices, prediction intervals and forecast intervals are shown in figure 3. The coverage rates of the 80% prediction intervals is around 80%. For the first 10 periods the coverage rate is even higher. The coverage rate of the simplified prediction interval is around 70%. The rate of the forecasting interval is for all periods steady around 78%. All the coverage rates can be found in appendix 3.

5. Conclusion

In this thesis different types of confidence intervals have been constructed for the but-for price estimation based on the German cement cartel. The purpose of this thesis is to determine which of confidence intervals is the most reliable. This assessment is done by a Monte Carlo simulation. A distinction between a dynamic and a static DGP is made. For both DGPs the forecasting and the dummy-variable approach were evaluated. Besides, the prediction interval is estimated by a simplified method, namely an estimation of σ2 instead of the prediction variance, to determine if this simplified method is a good approximation.

To be able to compare all the intervals it is important to know whether the estimated models fulfilled the basic assumptions of a linear regression. It turned out that non of the models fulfilled all the requirements. The static models contained a high degree of auto correlation due to the absence of price lags. In the dynamic model heteroscedasticity was present. All the violated assumptions can be found in section 4. Whether the violations had an influence on the strength of the intervals could be evaluated with the aid of Monte Carlo simulations.

By applying Monte Carlo a set of prices and but-for prices paths for a static and a dynamic DGP were simulated. On the basis of those price paths but-for prices and their confidence intervals were estimated for the forecasting and dummy-variable ap-proach. Hereafter the number of times that the but-for prices falls within the estimated confidence interval were estimated, which leads to the coverage rates. The coverages rates of the static DGP corresponds the best to the 80% intervals. Especially the dummy-variable model had coverage rate around 80%. This implies that the predic-tion intervals estimated by the dummy-variable model are very reliable. Besides, the forecasting model of the static DGP had a coverage rate of around 78% which devi-ates slightly of the 80% interval. The coverage rate of the models based on a dynamic DGP were very low. For the dummy-variable model the coverage rate was just around 27% and for the forecasting just around 20%. From those coverage rates can be con-cluded that the static DGP produces the best confidence intervals, despite the presence of non-normal errors, autocorrelation and heteroscedasticity. From the static DGP

interval.

A subquestion in this research was whether the variance of the regression, σ2_{, is a}

good approximation of the prediction variance. The widths of those intervals are by definition lower than the prediction intervals of Greene (2012). It turned out that the coverage rates are 10 percent point lower than the coverage rates of Greene (2012). Therefore, an advice for antitrust authorities is not to apply this simplified prediction interval due to the significant underestimation.

The underestimation of the of the prediction intervals for the dynamic models can be explained by the idea that prediction intervals do not take into the concept of time in the form of lagged variables as explained by Greene (2012). Despite this shortcoming, often researchers ignore this fact for simplicity. This thesis showed that for the dynamic dummy-variable model the consequences of using this formula are big. The estimated prediction turned out to be a way too small according to the Monte Carlo analysis. Therefore, antitrust authorities should take into account that those dynamic prediction intervals for cartels are significantly underestimated.

Another few limitations of the acquired data of the German cement cartel should be taken into account. First of all, the data was retrieved from questionnaires. As H¨uschelrath et al. (2013) already pointed out those questionnaires can be filled in strategically instead of honestly with the intention of lower fines. Besides, the data was quarterly instead of monthly or daily. A more frequent data set could lead to a better fit of the model. Another important limitation is the choice of the transition period. In this thesis the advice of Frank and Schliffke (2013) is followed, which is a minimum transition period of minimal 41 months. In this thesis the transition period lasts from 2002-Q4 till 2005Q2. It can be debated whether this transition period is too long or too short.

Additionally, the cement price was integrated of degree two, whether the other variables were integrated of the first degree. In this thesis the non-stationarity was ignored like in the research of H¨uschelrath et al. (2013) and Frank and Schliffke (2013). It could be argued that the development of price is stationary of the second degree if shift of special events are taken into account. Those changes can be due to the price war which followed after the termination of the cartel. Besides, the endogeneity of

the cement production variable was ignored. But according to H¨uschelrath et al. the effect of the instrumental variables is negligible with respect to the regression outcomes (2013).

Whether the ignorance of the endogeneity and the non-stationarity has a big in-fluence on the performed regression could be evaluated by future research. Besides, a suggestion for further research is to create prediction and forecasting intervals by bootstrapping. Bootstrapping could give a better approximation of the intervals then using the regular formulas. Another suggestion is to evaluate whether other definitions of prediction and forecast intervals deal better with uncertainty of the but-for price estimations for dynamic models. In this thesis the variance of the intervals was highly underestimated for the dynamic models. Additionally, this research is just based one case study, namely the German cement cartel. Whether the conclusion of this research could be used for the entire cartel industry could be debated. Therefore, it would be useful to check those findings for more cartel cases, although the data availability is lim-ited. Besides, having dataset which has more frequent data, for example daily instead of quarterly, could give more precise intervals.

Concluding, further research is needed to learn how to deal with dynamic models in but-for price estimations. If authorities want to have reliable intervals, the static dummy variable model is suggested.

Bibliography

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Appendix 1 - Regression output and tests

Dynamic dummy-variable model

Variable Coefficient Std. Error t-Statistic Prob.

c -7.255380 6.532147 -1.110719 0.2705 pC_t−1 1.206425 0.110371 10.93060 0.0000 pC_t−2 -0.384339 0.098443 -3.904182 0.0002 dcart_t 4.171756 1.119469 3.726549 0.0004 dof f_t -0.270891 0.691173 -0.391929 0.6963 cprodt -0.016746 0.006172 -2.713228 0.0084 pL t 0.186847 0.105724 1.767311 0.0816 pE t 0.119607 0.046419 2.576650 0.0121 R-squared 0.981174 Jarque-Bera 8.983808 (0.011199) S.E. of regression 1.166576 Breusch-Godfrey 1.295899 (0.2804) Sum squared resid 93.90213 Breusch-Pagan-Godfrey 4.636884 (0.0003)

Static dummy-variable model

Variable Coefficient Std. Error t-Statistic Prob.

c -77.62069 15.40069 -5.040080 0.0000 dcart_t 20.79189 2.449169 8.489365 0.0000 dof f_t 2.732339 1.881512 1.452204 0.1507 cprodt 0.013003 0.015894 0.818094 0.4160 pL t 1.628025 0.230840 7.052604 0.0000 pE t 0.309920 0.130332 2.377933 0.0200 R-squared 0.54378 Jarque-Bera 222.2514 (0.00000)

S.E. of regression 2.314025 Breusch-Godfrey 42.31384 (0.0000) Sum squared resid 380.1844 Breusch-Pagan-Godfrey 7.719322 (0.0000)

Appendix 2. Monte Carlo - Confidence interval width

Part 1: Average 80% confidence interval

Static DGP Dynamic DGP

Period DV DV simplified for DV simplified DV for

1 11.5321 8.6291 19.4219 NaN 2.9703 NaN 2 11.4551 8.6291 18.6168 NaN 2.9703 NaN 3 11.3281 8.6291 17.6506 4.1370 2.9703 2.5983 4 11.2065 8.6291 18.0148 3.9934 2.9703 2.5983 5 10.9761 8.6291 16.8318 3.9111 2.9703 2.5983 6 11.0394 8.6291 16.7444 3.9308 2.9703 2.5983 7 10.9927 8.6291 16.3207 3.8953 2.9703 2.5983 8 10.8980 8.6291 16.0738 3.8485 2.9703 2.5983 9 10.7542 8.6291 15.7721 3.7999 2.9703 2.5983 10 10.8878 8.6291 16.5750 3.8352 2.9703 2.5983 11 10.8949 8.6291 16.8979 3.8242 2.9703 2.5983 12 10.7201 8.6291 16.0216 3.7660 2.9703 2.5983 13 10.6061 8.6291 15.7737 3.7308 2.9703 2.5983 14 10.7441 8.6291 16.2015 3.7717 2.9703 2.5983 15 10.7469 8.6291 16.3522 3.7691 2.9703 2.5983 16 10.7147 8.6291 17.0615 3.7599 2.9703 2.5983 17 10.5092 8.6291 15.8203 3.6947 2.9703 2.5983 18 10.6023 8.6291 15.6832 3.7185 2.9703 2.5983 19 10.5994 8.6291 15.9563 3.7147 2.9703 2.5983 20 10.4074 8.6291 15.0018 3.6511 2.9703 2.5983

Part 2: Average 80% confidence interval

Static DGP Dynamic DGP

Period DV DV simplified for DV simplified DV for

21 10.6799 8.6291 11.4345 3.7412 2.9703 2.5983 22 10.7980 8.6291 11.5840 3.7788 2.9703 2.5983 23 10.8082 8.6291 11.4773 3.7764 2.9703 2.5983 24 10.6420 8.6291 10.9964 3.7204 2.9703 2.5983 25 10.5865 8.6291 10.8395 3.7023 2.9703 2.5983 26 10.7480 8.6291 11.2283 3.7629 2.9703 2.5983 27 10.7238 8.6291 11.2014 3.7466 2.9703 2.5983 28 10.6014 8.6291 11.1285 3.7002 2.9703 2.5983 29 10.5534 8.6291 10.9729 3.6849 2.9703 2.5983 30 10.6179 8.6291 11.3409 3.7126 2.9703 2.5983 31 10.6425 8.6291 11.3843 3.7216 2.9703 2.5983 32 10.4612 8.6291 10.8902 3.6565 2.9703 2.5983 33 10.4730 8.6291 10.8514 3.6587 2.9703 2.5983 34 10.2555 8.6291 13.5660 3.5944 2.9703 2.5983 35 10.3639 8.6291 13.2789 3.6271 2.9703 2.5983 36 10.5156 8.6291 11.1830 3.6650 2.9703 2.5983 37 10.5526 8.6291 10.7669 3.6758 2.9703 2.5983 38 10.7588 8.6291 11.1552 3.7485 2.9703 2.5983 39 10.7474 8.6291 11.0509 3.7508 2.9703 2.5983 40 10.5482 8.6291 10.7633 3.6838 2.9703 2.5983 41 10.3279 8.6291 10.7376 3.6410 2.9703 2.5983 42 10.3754 8.6291 10.7494 3.6743 2.9703 2.5983 43 10.3527 8.6291 10.7162 3.6436 2.9703 2.5983 44 10.2819 8.6291 10.7059 3.5988 2.9703 2.5983

Appendix 3. Monte Carlo - Coverage rates

Part 1: Coverage rate 80% confidence interval

Dynamic DGP Static DGP

Period DV DV simplified for DV simplified DV for

1 NaN NaN NaN 0.8457 0.7212 0.7848

2 NaN NaN NaN 0.8401 0.7148 0.7855

3 0.2637 0.1894 0.0666 0.8291 0.7033 0.7879 4 0.2576 0.1895 0.0698 0.8342 0.7168 0.7949 5 0.2525 0.1947 0.0683 0.8235 0.7089 0.7886 6 0.2675 0.2036 0.0704 0.8191 0.7091 0.7909 7 0.2758 0.2123 0.0745 0.8227 0.7056 0.7821 8 0.2711 0.2107 0.0773 0.8103 0.7013 0.7864 9 0.2754 0.2176 0.0783 0.8144 0.7100 0.7895 10 0.2781 0.2196 0.0817 0.8214 0.7086 0.7984 11 0.2794 0.2194 0.0790 0.8063 0.6969 0.7878 12 0.2774 0.2183 0.0789 0.8033 0.7053 0.7872 13 0.2780 0.2209 0.0814 0.8067 0.7100 0.7883 14 0.2771 0.2181 0.0797 0.8118 0.7098 0.7855 15 0.2814 0.2234 0.0834 0.8125 0.7099 0.7873 16 0.2785 0.2238 0.0808 0.8056 0.7049 0.7888 17 0.2678 0.2170 0.0767 0.7994 0.7086 0.7909 18 0.2748 0.2221 0.0802 0.8026 0.7052 0.7868 19 0.2719 0.2209 0.0834 0.7985 0.7018 0.7829 20 0.2610 0.2122 0.0833 0.7933 0.7044 0.7920

4_{The coverage rate is the ratio that the simulated but-for price falls within the 80 percent estimated}

confidence intervals. For the computation of this ratio outliers are left out. More information about the determination of the outliers can be found in footnote 1.

Part 2: Coverage rate 80% interval

Dynamic DGP Static DGP

Period DV DV simplified for DV simplified DV for

21 0.2758 0.2215 0.0969 0.8078 0.7060 0.7945 22 0.2791 0.2234 0.1115 0.8080 0.7037 0.7879 23 0.2734 0.2152 0.1152 0.8102 0.7081 0.7896 24 0.2648 0.2164 0.1232 0.8055 0.7041 0.7938 25 0.2693 0.2214 0.1345 0.8005 0.7090 0.7849 26 0.2777 0.2234 0.1369 0.8133 0.7139 0.7908 27 0.2776 0.2212 0.1353 0.8038 0.6995 0.7858 28 0.2720 0.2188 0.1405 0.7957 0.6999 0.7889 29 0.2679 0.2186 0.1453 0.8012 0.7028 0.7898 30 0.2681 0.2197 0.1419 0.8006 0.6989 0.7856 31 0.2742 0.2213 0.1416 0.8082 0.7050 0.7918 32 0.2656 0.2210 0.1397 0.7985 0.7083 0.7957 33 0.2663 0.2179 0.1404 0.7957 0.7032 0.7910 34 0.2588 0.2174 0.1358 0.7884 0.7064 0.7833 35 0.2628 0.2172 0.1282 0.7949 0.7069 0.7927 36 0.2644 0.2180 0.1312 0.7947 0.7055 0.7868 37 0.2694 0.2206 0.1305 0.8018 0.7065 0.7889 38 0.2693 0.2145 0.1381 0.8086 0.7091 0.7871 39 0.2707 0.2204 0.1397 0.8071 0.7091 0.7872 40 0.2679 0.2185 0.1415 0.7997 0.7031 0.7897 41 0.2632 0.2196 0.1488 0.8021 0.7141 0.7907 42 0.2608 0.2142 0.1416 0.8007 0.7125 0.7888 43 0.2638 0.2155 0.1426 0.7948 0.7109 0.7909 44 0.2636 0.2182 0.1424 0.7974 0.7181 0.7914

Appendix 4 - Augmented Dickey Fuller test

Augmented Dickey Fuller Test (with constant) Total period: 1-79

level first differences

pC_t -0.970231 0.7601 -5.191897 0.0000 pL_t 1.592649 0.9994 -7.653007 0.000 pE t 2.127220 0.9999 -2.689836 0.0806 cprodt -1.253486 0.6469 -5.676588 0.0000 Non-cartel period: 45-79 level first differences

pC_t 0.326065 0.9763 -2.688517 0.0870 pL_t 1.087511 0.9966 -5.091846 0.0002 pE_t -0.470566 0.8837 -1.222049 0.6515 cprodt -1.489546 0.5255 -12.75129 0.0000

5_{The Augmented Dickey Fuller Test is performed for two different periods, namely the total and}

the non-cartel period. This distinction is made because the coefficients of the forecasting approach are only based on the non-cartel period.

Appendix 5 - Monte Carlo. Model estimation

Estimated coefficients static DGP

CC std DV std for std c -77.2554 15.4007 -77.6940 15.5075 -77.6518 58.4913 dcart t 20.7919 2.4492 20.8393 2.4340 dof f_t 2.7323 1.8815 2.7709 1.8698 2.7340 2.0157 cprodt 0.0130 0.0159 0.0130 0.0160 0.0135 0.02760 pL_t 1.6280 0.2308 1.6269 0.2323 1.6267 1.0151 pE_t 0.3099 0.1303 0.3119 0.1299 0.3112 0.4387 variance: 11.4156 11.4238 1.8678 11.4009 2.9100 n: 79 79 35

Estimated coefficients dynamic DGP

Variable CC std DV std for std c -7.2553 6.5321 -9.3983 6.9062 -11.8778 27.4303 pC_t−1 1.2064 0.1103 1.1498 0.1000 1.0173 0.1848 pC_t−2 -0.3843 0.0984 -0.3597 0.0906 -0.2767 0.1480 dcart t 4.1717 1.1194 4.7478 1.1648 dof f_t -0.2708 0.6911 -0.2131 0.7521 -0.2028 0.9717 cprodt -0.0167 0.0061 -0.0159 0.0057 -0.0134 0.0097 pL t 0.1868 0.1057 0.2367 0.1155 0.2792 0.4889 pE t 0.1196 0.0464 0.1275 0.0536 0.1664 0.2047 variance: 1.3609 1.3470 0.2323 1.3029 0.3707 n: 77 77 33