The Model of Degressive Taxation of Trucks: Case of the Czech Republic

Verification of tax rate degressivity will be carried out using the multidimensional regression analysis described by Hindls et al. (2007), selected independent variables X₁ (number of axles) and X₂ (permissible maximum weight) per dependent variable Y (annual tax rate) by OLS method (ordinary least squares). In order to model the dependence between independent variables and a dependent variable, the method of sequential selection of regressors, particularly its variant referred to as degressive selection, described by Cipra (2013), will be used. The first step in the sequential selection of regressors is the design of a model in which all explanatory variables are contained, i.e. the variables such as a number of axles and permissible maximum weight. For initial estimation of parameters by OLS method, the straight-line model with the functional form described by Hušek (2007) was selected. For testing purposes, 5% significance level was selected. The results of the initial straight-line models can be seen in (Table 1).

Before subsequent modification of the initial straight-line model, it is necessary to verify if the data is not affected by the issue of collinearity. The issue of collinearity, i.e. the dependence between two explanatory variables, is indicated, at a glance, by the high value of determination coefficient accompanied by a large number of insignificant variables. The determination coefficient in the initial model takes the value of R² = 0,914668 (the model explained 91.47% of the dependent variable variability), nevertheless, all independent variables are significant (their p-values are lower than the selected significance level α = 0.05), which means that the collinearity should not make bigger problems in this case. The above-mentioned can also be verified by means of paired correlation coefficients, principal component method or VIF method (factor analysis) described by Hušek (2007). The calculated paired correlation coefficient of the variable X₁ and X₂ (r_X1X2 = r_X2X1 = 0,6146) is lower than 0.8, which is a reference value referring to the issue of collinearity, i.e. the dependence of two independent variables. In the case of VIF method, the issue of collinearity is referred to by VIF( β_j ) > 10. In this initial straight-line model, however, the calculated VIF variable value is X₁ = 1.607, as well as the VIF variable value is X₂ = 1.607. This method did not discover the issue of collinearity either.

Figure 1

Now, it is possible to go to reduction of the model with insignificant variable as it was proved that the data were not affected by the issue of collinearity. To test the insignificance of individual parameters of the model (i.e. the constant of variable X₁ and X₂), regression parameter significance tests (t-test) where hypotheses in econometric practice are formulated most frequently as follows: H₀ = the parameter is statistically insignificant, H₁ = the parameter is statistically significant (Cipra, 2013). In initial estimation, all independent variables, i.e. constant, variable X₁ and variable X₂ have a p-value ( β₀ = 0.0005, β₁ = 2.61e⁻⁰⁶, β₂ = 3.34e⁻⁰²²) lower than the selected significance level for the purpose of testing hypotheses (α = 0.05) and that is why we reject the zero hypotheses on statistical insignificance of parameters. All independent variables are statistically significant and will be left in the model henceforth. To determine a suitable functional form that would lead to a better explanation of the variability of an independent variable by dependent variables, it is possible to use the residual graphs of independent variables X₁ and X₂ in the straight-line model (Hušek, 2007).

The residuals of the independent variable X₁ do not cluster into any functional form that would be omitted in the model. Consequently, we leave the independent variable X₁ in the initial model in the same form of straight line.

Figure 2

In the case of independent variable X₂ (permissible maximum weight), the line model residuals cluster into the parabola shape which was omitted in the line model. The result, in the form of completion of the 2nd degree polynomial of the independent variable X₂ (permissible maximum weight) into the initial straight-line model can be seen in (Table 2). The inclusion of the 2nd degree polynomial of the independent variable X₂ in the initial line model should result in an increase of the determination coefficient.

The resultant linear regression model with the 2nd degree polynomial has explained 92.91% of the variability of the dependent variable. In comparison with the linear straight-line regression model that has explained 91.47% of the variability of dependent variable, there was some improvement. The sign of the independent variable parameter X₁ is negative ( β₁ = −3954.80), i.e. the model confirmed the above-mentioned statement about the degressive determination of annual tax rates in terms of number of truck axles. The regression equation of the resultant model for determination of the road tax rate has the following form:

where:

X₁ = Number of axles,

X₂ = Permissible maximum weight in tons.

The resultant model with the 2nd degree polynomial of the independent variable X₂ needs to be subject to econometric tests that verify classical linear assumptions – econometric verification. Only after these assumptions are met, the estimation of regression parameters by OLS method can be considered as best possible. Assumptions of the classical linear regression model are as follows: (Cipra, 2013):

For testing the first assumption, i.e. the model is linear in parameters, it contains a level constant (in our case β₀ = 8369.13), is specified correctly and has an additively connected error term, it is possible to use the results in Table 2. Specifically, it is the parameter significance t-test of the resultant model (p-value of individual independent variables of the model) and the F-test of overall significance of the model (P-value (F)).

As stated above, in the case of t-test, the hypotheses are formulated as follows: H₀ = the parameter is statistically insignificant, H₁ = the parameter is statistically significant. All parameters of the independent variables remained significant after the second-degree polynomial was added. Their p-value ( β₀ = 1.22e⁻⁰⁵, β₁ = 6.64e⁻⁰⁷, β₂ = 7.29e⁻⁰⁵, β₃ = 0.0067) assumes the values lower than α = 0.05, that is why we reject the zero hypothesis about their insignificance.

The same ways as in the case of t-tests of individual parameters of the model, the hypotheses are formulated equally also in the case of F-test that tests the significance of the entire model. The resultant p-value (p = 5.03e⁻²³) is lower than α = 0.05 therefore we reject the hypothesis about statistical significance of the model as well. So, it is possible to state that the parameters and the model itself are statistically insignificant, i.e. the model is statistically verified.

For verification of correct specification, which is another condition for fulfilment of the first assumption of the classical linear regression model, it is possible to use RESET and LM specification tests. These tests diagnose the specification errors that came into existence owing to omission of substantial variables or erroneous functional form of the model (Hušek, 2007). In the case of these tests, hypotheses are formulated positively, i.e.: H₀ = the model is specified correctly, H₁ = the model is specified incorrectly. Conversely, the goal is the p-value higher than the selected significance level (α = 0.05).

Based on the specification test results, we reject the zero hypothesis about correct specification of the model since the p-values of specification tests presented in Table 3 assume lower values than the selected significance level α = 0.05. The results show that the model with the 2nd degree polynomial is undersized since the variability of the independent variable is only explained by independent variables X₁ and X₂, so besides the two independent variables, the amount of annual road tax rates is explained by other variables which are unknown at the first glance. Since the objective was to explain the amount of annual road tax with the use of the two independent variables (X₁ and X₂), the results of specification tests cannot be evaluated negatively (Hušek, 2007). Another reason for rejection of the null hypothesis may consist in a low number of values (road tax rate), which implies that every little variation may affect the test results. Similarly, a high percentage of explanation of the variability in the independent variable, R² = 0.929158, may imply that the specification tests go wrong. Generally, it can be said that the first assumption of the classical linear regression model is satisfied.

The second assumption, the error term has a zero-average value, provided that estimation of coefficients by means of OLS method is satisfied since one of the features of parameter estimation using OLS method is the zero-average value of the residual component, i.e. the error term (Cipra, 2013). The second assumption is satisfied as well.

Since the paired correlation coefficient of the residual (error term) and independent variables is zero, based on the results of the statistical software Gretl. The third assumption, the independent variables are not correlated with the error term, is satisfied.

The fourth assumption, i.e. non-existence of the error term autocorrelation, is understood as the dependence between the sequence of values of one variable over time, not as the dependence between two or more variables. Since the model works with cross-sectional data, i.e. there are no variables developing over time, it is not necessary to test the resultant model for the fourth assumption, which is only considered in the case of timeline modelling (Hušek, 2007).

The fifth assumption, i.e. constant variance requirement (homoscedasticity), can be verified using the White test and Breusch-Pagan test. Hypotheses for both tests are formulated, according to Hušek (2007), as follows: H₀ = homoscedasticity, H₁ = heteroscedasticity. In the case of White test, the result is p-value = 0.101436, the resultant p-value of Breusch-Pagan test is the p-value = 0.509328. Both tests of the resultant model have the p-value higher than the selected significance level = 0.05, so we do not reject the null hypothesis about homoscedasticity. This assumption is consequently satisfied as well.

The last of the assumptions, which remains to be verified, is the normality of residuals since the issue of collinearity in independent variables was initially excluded. So, the sixth assumption is also satisfied. For normality testing, according to Soltes (2018), it is possible to use a wide range of tests, e.g. Chi-square test, Doornik-Hansen test, Shapiro-WilkW-test or Jarque-Bery test. Even here, the null hypothesis is formulated positively, therefore: H₀ = residuals are distributed normally, H₁ = residuals are not distributed normally. Resultant p-values should take values higher than α = 0.05.

However, the results from Chi-Square, Doornik-Hansen and Shapiro-Wilk W-tests reject the normality since the p-values in these tests are lower than the selected significance level α = 0.05. The reality follows again from a small amount of data where one extreme value results in rejection of the null hypothesis (Cipra, 2013). The main condition for satisfaction of this assumption is the validity of t-tests of model parameters, but also the F-test for overall argumentativeness of the model, which holds true in this case, since the p-value of all independent variables ( β₀ = 1.22e⁻⁰⁵, β₁ = 6.64e⁻⁰⁷, β₂ = 7.29e⁻⁰⁵, β₃ = 0.0067) in the model with the 2nd degree polynomial is lower than the selected significance level α = 0.05, much like the p-value (p = 5.03e⁻²³) of overall significance of the model. Consequently, also the seventh assumption of the classical linear regression model can be considered as satisfied.

The resultant model with the 2nd degree polynomial was successfully verified both statistically and econometrically, thus the estimation of parameters (β₀ = 8369.13, β₁ = −3954.8, β₂ = 848.59, β₃ = 16.39) by OLS method can be considered as the best possible.

The two-axle truck with the weight of 8 tons (vehicle No. 4), is connected to one-axle trailer with the same permissible maximum weight (vehicle No. 3), but their tax rates per ton are different. The operator is obliged to pay a higher rate per ton in the case of trailer, specifically the tax rate is 1 158.62 CZK/t, in the case of truck, the tax rate is lower by 165.52 CZK/t and it is 993.10 CZK/t. The same applies to the amount of overall taxation of these vehicles, i.e. the one-axle vehicle is taxed by a higher tax rate than the vehicle of the same weight with higher number of axles. Similarly, this truck whose tax rate is 993.10 CZK/t will be taxed more per unit of permissible maximum weight than the three-axle truck with the same permissible maximum weight 8 tons (vehicle No. 6) whose rate per ton is 827.59 CZK/t. The above mentioned holds true also in the case of towing vehicles, namely when the towing vehicle with the permissible maximum weight 16 tons pays a lower road tax in the case of higher number of axles, both in the case of rate per ton and overall annual tax rate, than the towing vehicle with the same permissible maximum weight and lower number of axles. Particularly, the towing vehicle with permissible maximum weight 16 tons and two axles (vehicle No. 7), towing a three-axle trailer with the same permissible maximum weight (vehicle No. 10), is taxed by the rate per ton in the amount of 1 146.36 CZK/t, which is more again in comparison with the amount of the tax rate for semi-trailer, which is 825 CZK/t. In terms of overall amount of the annual tax rate, the amount is 700 CZK in the case of towing vehicle or 13 200 CZK in the case of semi-trailer.

Currently fixed annual road tax rates mean that bigger vehicles, from the perspective of a number of axles, pay lower road tax than those with permissible maximum weight, but with a lower number of axles. The above mentioned corresponds with the negative sign for the number of axles parameter, i.e. the variable X₁, in regression equation (1) and confirms the degressivity of tax rates owing to number of axles, which is intended to support trucks, towing vehicles, semi-trailers and trailers with higher number of axles with the same weight. The result is the harmony between determination of road tax rate in the Czech Republic and the priority of the Ministry of Transport of the Czech Republic according to Váchal (2018), which consists in pressure on operation of multi-axle trucks with the same vehicle weight. Because these trucks are more friendly from the perspective of infrastructure wear and tear. The Czech Republic is not an exception in comparison with the mentioned countries and it favours the selected vehicles that are more friendly from the perspective of infrastructure wear and tear. However, it does not go the way of different road tax rates for vehicles with pneumatic/air suspension and the other vehicles, but by favouring the multi-axle trucks in the form of a lower road tax rate.

The result is in the form of comparison of the taxation of model vehicles, as it emerges from the above mentioned, it corresponds, in light of the variable X₁, with the resultant regression model. The independent variable X₁ was economically verified with success.

For economic verification of the independent variable X₂ and namely for better visualisation of its functional form refer to Figure 3. The independent variable X₂ is plotted on X-axis and the dependent variable on Y-axis. Points in Figure 3 represent the median of the given weights range and the tax rate appertained to it.

Figure 3

The curve visible in (Figure 3), for the purposes of economic verification of the variable X₂, will be halved by an imaginary perpendicular to x-axis. This imaginary perpendicular intersects the x-axis at a distance of 12 units (12 tons) from the origin. The reason for this division is given by different development of overall tax rates and the rates per ton for vehicles with the weight up to 12 tons and those with the weight above 12 tons, which is described in paragraphs below and results from Annex 1.

The one-axle vehicles include e.g. trailers, they are considered as separate vehicles liable to tax as per § 1 sect. 1 Road Tax Act even though they are attached to a truck or a towing vehicle. The one axle trailer with permissible maximum weight 3 tons (vehicle No. 1) is taxed by annual tax rate 1 418.18 CZK/t, which is a higher tax rate in comparison with the trailer having the same number of axles, but with a permissible maximum weight of 8 tons (vehicle No. 3) whose amount of its annual tax rate is 1 158.62 CZK/t. From the perspective of tax per ton, the vehicle with the same number of axles and higher weight is taxed less in this case. However, the amount of overall taxation for these vehicles develops the other way around. The one-axle trailer with the weight of 3 tons is taxed by 3 900 CZK, whereas the trailer with the same number of axles but with the weight of 8 tons is taxed by total of 8 400 CZK, i.e. by a higher tax rate. The same result can be achieved by comparing the taxation of a two-axle truck with a permissible maximum weight of 8 tons (vehicle No. 4) and 12 tons (vehicle No. 5) whose rate per ton is 993.10 CZK, or 939.13 CZK. In both cases, the vehicle with permissible maximum weight is taxed less per ton. In light of total tax, it is quite the opposite. This development of tax rates corresponds with the first part of the curve in Figure 3 because these are the vehicles with permissible maximum weight lower than 12 tons. For these vehicles, the increase in permissible maximum weight results in the increase in overall annual tax rate, so the development of rate per ton is contradictory, i.e. the increase in permissible maximum weight results in the decrease in rate per ton.

For two-axle vehicles, an opposite result can be achieved namely when with the maximum permissible vehicle weight, both total annual road tax rate and rate per ton are higher. Again, it is a 12-ton truck (vehicle No. 4), which is taxed by 939.13 CZK/t, or by 10 800 CZK in light of overall taxation and the two-axle towing vehicle, but with the weight of 16 tons (vehicle No. 7), which was taxed by 1 436.36 CZK/t or by 23 700 CZK. The same tax rate development can be observed in vehicles with four axles or in multi-axle vehicles. Vehicle No. 9 has the permissible maximum weight of 16 tons, which corresponds to rate per ton amounting to 466.67 CZK, vehicle No. 11 has the permissible maximum weight of 24 tons, which corresponds to rate per ton amounting to 737.50 CZK, i.e. higher tax rate. In both above given cases, it holds true that the vehicles with the same number of axles are taxed more with increasing weight per unit of weight, but also globally. In this case, these are vehicles with the permissible maximum weight higher than 12 tons.

In both above given comparisons, the vehicles with the same number of axles are taxed with the increasing weight more from the perspective of annual tax rate. The same result was achieved in the linear regression model with the 2nd degree polynomial since the signs of the permissible maximum weight parameter in the linear regression equation are positive. The overall annual tax rate increases with the increase in permissible maximum weight at the same number of axles. In this context, it is possible to talk only about the direction of dependence, not about the resultant annual tax rate because it is necessary to take the number of axles variable into account.

The development of rates per ton is very interesting; it can be seen in Annex 1, at first the rates are decreasing and then increasing within the groups according to number of axles. The exception is the group of one-axle vehicles where rates per ton are only decreasing and, on the contrary, the rates in the group of 4-axle and multiple axle vehicles are only increasing. It appears from the above given that with lower weight of vehicle, the increase in the maximum permissible vehicle weight is in the lead by a unit against the increase in annual tax rate by less than this unit. The result is the increase in overall road tax rate, but the decrease in rate per ton. Such a development of rates can be observed in vehicles whose permissible maximum weight is approx. 12 tons or lower (first part of the imaginarily divided curve in Figure 3). In vehicles with permissible maximum weight higher than 12 tons, the increase in permissible maximum weight is in the lead by a unit, against the increase in rate more than by this unit. The result is not only the increase in overall road tax rate, but also in rate per ton. The relation between the vehicle weight and the annual tax rate amount corresponds to the functional form in the form of the 2nd degree polynomial, alias parabola. The mentioned regression model arrived at the same result.

In addition to statistical and economic verification (tests), the linear regression model with the 2nd degree polynomial was verified also economically, so it can be said that it corresponds to the simplified reality and it is suitable for interpretation.

Annual road tax rates whose amount was described by the above-mentioned regression equation (1), have as their task, as stated previously, to support the road trains with a higher number of axles that are more friendly to infrastructure.

Within the following procedure, the correlation analysis will be carried out and paired correlation and partial correlation coefficients will be calculated. Within the correlation analysis, the paired correlation coefficients and the partial correlation coefficients can be used. For calculation of paired correlation coefficients, Gretl statistical software can be used again. This software offers the possibility to display correlation matrix (2) that can be seen below.

It follows from the correlation matrix that higher dependences between the dependent variable and independent variables are reached by the maximum permissible vehicle weight (r_YX2 = 0.9235). For comparison, the paired correlation coefficient of the independent variable, number of axles, and the dependent variable r_YX1 = 0.3715. Particularly worthy of note is the sign of this correlation coefficient, i.e. the correlation coefficient of the dependent variable, annual tax rate, and the independent variable, number of axles, which should be negative pursuant to the results of the regression equation (1). However, the value of correlation coefficient may be affected by dependence of another independent variable, which can be verified by calculation of partial correlation coefficients. The calculation of these coefficients is not available in Gretl software and so it is necessary to use the formulas defined in publications, Hušek (2007) or Hendl (2015).

Based on the above given results of partial correlation coefficients, it can be stated that the variable X₂ of variable Y and X₁ affect the annual tax rate most because the difference between the paired correlation coefficient r_YX1 (r = 0.3715) and the partial correlation coefficient r_YX1 ^·_X2 (r = −2.14) is maximal. The above mentioned explains why the paired correlation coefficient r_YX1 is positive. Conversely, the variable X₁ of variable Y and X₂ affects the annual tax rate at the least because the difference between the paired correlation coefficient r_YX2 (r = 0.9235) and the partial correlation coefficient r_YX2 ^·_X1 (r = 1.29) is minimal.