A New Method and Model for the Estimation of Residual Value of Agricultural Tractors

: The residual value of a tractor affects the cost of ownership. As there is not much transactional information available for used tractors, nor is there a history of new tractor prices, existing studies struggle to forecast the residual value of agricultural tractors. This is made even more challenging by the emission-regulation-related tractor price increase, low inﬂation in recent decades, and the complexity of the portfolio offerings from manufacturers. Using the new equivalent tractors, grouped by families of similar characteristics, bypasses these challenges and enables us to obtain larger data sets. These large data sets can be forecasted using transparent linear power regressions that offer the lowest root mean squared error (RMSE = 1.5574) and the highest combined, adjusted coefﬁcient of determination (RSqAdj = 0.8457), outperforming all previously tested studies as well as the ensemble, Gaussian process regression, kernel, linear regression, neural network, support vector machine, and decision tree models. The accessibility of the public information required, as well as its processing using mainstream software through a model that is simple to use, yet robust, enables any stakeholder (manufacturers, sellers, ﬁnancers, insurers, and, most of all, users) to reliably determine the residual value of an agricultural tractor, empowering them to make fact-based, cost-of-ownership-optimized decisions.


Introduction
The operating and ownership costs of machines often comprise more than half of the total crop production costs.Minimizing the machinery portion of the production costs requires a routine assessment of the benefits and costs associated with owning, leasing, or renting machinery [1].
Most farm equipment is still acquired under a conventional purchase plan.The capital may come from the purchaser's own funds, a third-party lender, or a company financing plan.However, an increasing number of major machinery items are being leased, via operating lease (in which the user can tax-deduct the payments as the machine belongs to the financer), via finance lease (in which the user owns the machine and is therefore entitled to take depreciation deductions) or by using a rollover purchase (in which the operator purchases a new or nearly new piece of equipment from a dealer with the expectation that it will be exchanged for another model after one year or season) [2].
Whether a tractor was paid for upfront, used equipment was traded as a payment in kind, or the machinery was traditionally financed, leased, or rented, the residual value has a tremendous impact on the finance cost, as the financer will ensure that the loan's lien is below the residual value [3].If the residual value is uncertain, the financer will include a haircut [4] as a safety factor that renders the finance scheme more expensive to the purchaser.

Current Issues
The portfolio offered by manufacturers has grown complex, to the point of offering, with the same engine power, several wheelbases, multiple transmission options and user interfaces, and different shipping and maximum permissible weights with.the same power.These factors have a tremendous impact on selling price (Table 2).As these features result in different productivity, efficiency, maintenance, and repair requirements they enjoy (or suffer) different demands from the market.Consequently, they have different residual values, despite sharing the same engine power.Thereof, a study considering only the engine power might have challenges discerning the residual value between such different tractors sharing the same power.
The European Commission (EC) off-road diesel engine emission regulations [30][31][32][33] have had a tremendous impact on the lifespan of tractor series (Figure 1).As these features result in different productivity, efficiency, maintenance, and re requirements they enjoy (or suffer) different demands from the market.Conseque they have different residual values, despite sharing the same engine power.There study considering only the engine power might have challenges discerning the resi value between such different tractors sharing the same power.
The European Commission (EC) off-road diesel engine emission regulations [30 have had a tremendous impact on the lifespan of tractor series (Figure 1).The on-road diesel emission regulations have had an impact on the cost [34,35].spite the fact that the last European emission regulation has been already implemente is quite likely that new emission regulations will be implemented with their associ The on-road diesel emission regulations have had an impact on the cost [34,35].Despite the fact that the last European emission regulation has been already implemented, it is quite likely that new emission regulations will be implemented with their associated costs [36].The off-road diesel engine emission regulations cost is even higher, as the fixed costs must be distributed amongst a much smaller number of engines (Figure 2).costs [36].The off-road diesel engine emission regulations cost is even higher, as the fix costs must be distributed amongst a much smaller number of engines (Figure 2).

Goal
The goal of this research is to develop a residual value calculation methodology th is accessible to all stakeholders (owners, users, marketeers, financiers, and insurers) a that finds a balance between simplicity of use and accurate results.This methodology w be applied to standard, agricultural cabbed tractors with more than 75 kW of horsepow from the main OEMs (Case IH, Claas, Fendt, John Deere, Massey Ferguson, and New H land) in the main markets of Western Europe [37].

Dataset
Transactional European information does not exist in sufficient numbers to properly analyzed [23].The number, type, and condition of the European-auctioned m chines are not aligned with standard market expectations.Therefore, this study cons ered agricultural tractors with an engine power higher than 75 kW that were manufa tured by Case IH, Claas, Fendt, John Deere, Massey Ferguson, and New Holland and we advertised on https://www.agriaffaires.com/(accessed 15 July 2022) , https://www.ma

Goal
The goal of this research is to develop a residual value calculation methodology that is accessible to all stakeholders (owners, users, marketeers, financiers, and insurers) and that finds a balance between simplicity of use and accurate results.This methodology will be applied to standard, agricultural cabbed tractors with more than 75 kW of horsepower from the main OEMs (Case IH, Claas, Fendt, John Deere, Massey Ferguson, and New Holland) in the main markets of Western Europe [37].

Dataset
Transactional European information does not exist in sufficient numbers to be properly analyzed [23].The number, type, and condition of the European-auctioned machines are not aligned with standard market expectations.Therefore, this study considered agricultural tractors with an engine power higher than 75 kW that were manufactured by Case IH, Claas, Fendt, John Deere, Massey Ferguson, and New Holland and were advertised on https://www.agriaffaires.com/(accessed on 15 July 2022), https://www.mascus.com/(accessed on 15 July 2022), and https://www.tractorpool.com/(accessed on 15 July 2022) by professional retailers (for which the machine is in good condition as the retailer is obliged to provide a legal warranty on the product, and the price realization expectations are delimited by the financial requirements related to their business sustainability) in Austria, Belgium, Denmark, Estonia, Finland, France, Germany, Italy, Latvia, Lithuania, Netherlands, Norway, Poland, Spain, Sweden, and the United Kingdom [37] by professional sellers.The listings needed to feature the working hours, year of manufacture, and price (VAT excluded and price converted into Euros).At least 300 working hours were required, as tractors with less hours advertised from professionals come from demonstration programs or rental programs; hence, there is an outside source of income in which the seller alters the price realization expectations.
The tractor models were aligned with the OEM's official nomenclature (as sellers tend to include features in the product name with the intent of differentiating their offering), and redundant advertisements were eliminated (as it is frequent that the sellers have business systems interfaced with the different websites in order to achieve the largest possible product awareness; thus, more than one website can feature the same offering).
The dataset obtained for this study was composed of 10,303 uniquely categorized, advertised tractor observations (Table 3 and Figure 3)

Data Systematization and Preprocessing
Calculating the residual value (RV) as: presents quite a challenge.As mentioned above, the availability of used tractor transactional information is scarce, and obtaining the tractor retail prices from all 16 countries in the scope of this study since 1998 is quite an endeavor.Hence, a novel approach was taken by means of the new equivalent tractor concept:

Data Systematization and Preprocessing
Calculating the residual value (RV) as: presents quite a challenge.As mentioned above, the availability of used tractor transactional information is scarce, and obtaining the tractor retail prices from all 16 countries in the scope of this study since 1998 is quite an endeavor.Hence, a novel approach was taken by means of the new equivalent tractor concept:

New Equivalent Tractor
A tractor model belongs to a tractor series, with which it shares a wheelbase, mass, and most characteristics, with the key differentiator being its power.As technology evolves, the tractor series are replaced by newer series with enhancements that improve efficacy and/or efficiency.The evolution is such that it is sometimes not possible to find a current replacement model with the same features as a used one, as those features were rendered obsolete (e.g., synchronized transmissions, two-wheel drive, unsuspended front axle, open circuit hydraulic system, or an open operator station).The retail price of the new series' models includes any inflation changes as well as any cost derived from regulations compliance and any additional features deemed necessary by the market (Table 4).
Obtaining the retail price of current models will be much easier for the subject matter experts using the method described in this model, as the prices are available through some manufacturer's websites and/or through a dealer's quote.

Tractor Family
Manufacturers group their similar models in series.In some cases, these series are quite large and can include several wheelbases, whereas other series are split into separated series (e.g., Case IH's Puma Series vs. New Holland's T7 SWB, and T7 LWB or John Deere's 6 R series, which features models ranging from 6500 kg to 9650 kg of shipping mass).Others differentiate their series by the featured transmission (e.g., Case IH's CVX, Claas' CMATIC, Massey Ferguson's Dyna-VT, and New Holland's Auto Command series, which features a continuous variable transmission vs. the stepped transmissions featured by equivalent models; Massey Ferguson and New Holland go a step further and differentiate between their models by featuring partial powershift transmissions such as the Dyna-4 and Dyna-6, Electro Command, and Dynamic Command).Other manufacturers use their model nomenclature to differentiate the specifications level (e.g., John Deere's premium R series vs. the no so premium M series).
In addition, not all series have the same number of sales; thus, the adverts available on the internet are also quite different, allowing for the series to split into different families that share common features and specifications (Table 5).The combination of the new equivalent tractor and the tractor family have been paramount contributors to coalesce a dataset for this study, which is composed of 10,303 tractors.

Data Analysis
As previously stated, one of the goals of this study is to provide an easy-to-use method for residual value stakeholders.With 1.1 billion users (one in eight people on the planet), Microsoft Excel is one of the most ubiquitous software in both professional and domestic environments.Hence, considering Microsoft Excel as the first option was clear.
Microsoft Excel offers functions that allow several models to make multiple variable regressions, enabling the evaluation of the following regressions: Logarithmic (lin-log) Power (log-log) Exponential (log-lin) In order to evaluate alternative regression options, several different models were analyzed with Matlab, including parametric and non-parametric models (Table 6).The regression trees, support vector machines, ensembles of regression trees, Gaussian process regressions, and neural networks were optimized by machine learning.
In the interest of examining the predictive accuracy of the fitted models, regressions were made with 3, 5, 7, and 9 predicting variables and with a 3-, 5-, 7-, and 9-fold cross-over validation.In addition, 5%, 10%, 15%, 20%, and 25% hold-out validation models were used (in one instance, one regression was performed with a 5% training dataset) (Table 7).In regression analysis, the root mean squared error (RMSE) and adjusted R 2 (RSqAdj) metrics were used to evaluate the performance of the different models.
The root of the error was used to obtain an error with the same unit as the outcome variable for easier interpretation purposes.The closer the point is to the regression, the lower the metric value is and the higher the accuracy of the regression model is.When a model is 100% perfect, this metric value will be equal to zero.
The adjusted R 2 is a better evaluation metric than R 2 .The R 2 is a statistical measure that represents the proportion of variation in the dependent variable that is explained by the regression model.The adjusted R 2 considers the number of predictor variables used to predict the dependent variable [38].
As the proposed power regression model is based on tractor families and uses two predictors, the same Matlab regression models seen in Table 6 including regression trees, support vector machines, ensembles of regression trees, Gaussian process regressions, and neural networks optimized by machine learning) were analyzed for the tractor families that obtained the best RMSE results with the proposed power regression model.

Proposed Regression Models
The proposed power regression model (5) offered the best RMSE and R 2 adjusted results (Table 8 and Figure 4).

Fitted Regression Models with Multiple Variables and Validations.
Even if one of the goals of this study is to provide the best possible results with the most accessible tools and methodology, it is indispensable to evaluate more advanced models and tools.Therefore, as previously stated, multiple models were evaluated (Table 6) using different variables and validation methods (Table 7) Models with seven predictors showed better RMSE values when compared to 3, 5, and predictor-tested models.Models with hold-out validation demonstrated better RMSE values than those with cross-out validation.The best overall model was the rational quadratic Gaussian process regression with seven predicting variables, which was validated with a 10% hold-out and an RMSE value of 0.046 (Table 9).

Fitted Regression Models with Multiple Variables and Validations
Even if one of the goals of this study is to provide the best possible results with the most accessible tools and methodology, it is indispensable to evaluate more advanced models and tools.Therefore, as previously stated, multiple models were evaluated (Table 6) using different variables and validation methods (Table 7).
Models with seven predictors showed better RMSE values when compared to 3, 5, and predictor-tested models.Models with hold-out validation demonstrated better RMSE values than those with cross-out validation.The best overall model was the rational quadratic Gaussian process regression with seven predicting variables, which was validated with a 10% hold-out and an RMSE value of 0.046 (Table 9).This model would rank thirteenth when compared with the tractor families with the best RMSEs of the proposed power model (Table 8).
The exponential Gaussian process regression (GPR) demonstrated more consistent RMSE results across all the tested variables and validations (Figure 5).

Fitted Regression Models of Tractor Families.
As the proposed methodology on power regression models is based on two predicting variables of tractor families, it was essential to test more advanced software using more advanced models.
Hence, the tractor families that rendered the best power regression model RMSE value results (Figure 4) were tested using the same fitted models and a 10% hold-out validation to provide data sets (Table 6).
The optimized Gaussian process regressions of the two predictors, validated with a 10% hold-out of the considered tractor families, provided very satisfactory RMSE and RSqAdj results (Table 10).Across most family groups, the best overall model was the optimized Gaussian process regression (OGPR) model (Figure 6).

Fitted Regression Models of Tractor Families
As the proposed methodology on power regression models is based on two predicting variables of tractor families, it was essential to test more advanced software using more advanced models.
Hence, the tractor families that rendered the best power regression model RMSE value results (Figure 4) were tested using the same fitted models and a 10% hold-out validation to provide data sets (Table 6).
The optimized Gaussian process regressions of the two predictors, validated with a 10% hold-out of the considered tractor families, provided very satisfactory RMSE and RSqAdj results (Table 10).Across most family groups, the best overall model was the optimized Gaussian process regression (OGPR) model (Figure 6).
The results of the proposed power regression model of two predictors, grouped by families tested, were better than the most accurate regression performed by Matlab, even if Matlab was optimized by machine learning (Table 11 and Figure 7).The results of the proposed power regression model of two predictors, grouped by families tested, were better than the most accurate regression performed by Matlab, even if Matlab was optimized by machine learning (Table 11 and Figure 7).The results of the proposed power regression model of two predictors, grouped by families tested, were better than the most accurate regression performed by Matlab, even if Matlab was optimized by machine learning (Table 11 and Figure 7).The proposed power regression model seems to follow the different tractor family residual-value behaviors quite precisely.The fact that the second-best tested model was exponential regression, as was found by Witte, Back, Sponagel, and Bahrs [26], proves that these models exhibit better performance than more complex models such as optimized Gaussian regressions (OGPR).

Discussion
The robustness of the proposed power regression model was compared to the following models: 1.
Models referenced by previous studies, which offered sufficient detail to process the dataset (Table 1); 2.
Fitted regression models of the complete data set with multiple variables and validations (Tables 6 and 7); 3.
Fitted regression models of tractor families with the same predictors used in the proposed power linear regression (Figure 4).

Models Referenced by Previous Studies
The complete data set was processed using data from previous studies whenever it was possible (when enough details were provided) (Table 12 and Figure 8).

General
The proposed power regression model provided the best RMSE and RSqAdj of all the tested models (Figure 9).The proposed power regression model (RMSE = 1.5574|RSqAdj = 0.8457) demonstrated more predictive robustness Table 1.shows how previous studies used, in addition to years of age and hours of usage, brand and power in order to predict the residual value behavior.However, Table 2 shows that power is not enough to differentiate residual value behavior, as even similar tractor families from the same brand with the same power can feature different sizes, masses, transmissions, and user interfaces.The proposed model takes these factors into consideration, drilling down to model levels and grouping them in tractor families to lay a better foundation for more robust results.

General
The proposed power regression model provided the best RMSE and RSqAdj of all the tested models (Figure 9).

General
The proposed power regression model provided the best RMSE and RSqAdj of all the tested models (Figure 9).Compared to the previous studies referenced, the proposed power regression model provides better RMSE and RSqAdj values as it considers not only the brand and very similar power and tractor size (wheelbase and mass) but also very similar specification levels (e.g, transmissions and user interfaces), relating these factors to an equivalent new model that provides a precise price reference, including inflation and production costs.These variations yield a better foundation for more robust results Compared to more advanced fitting models that require specific software, the proposed power regression model provides better RMSE and RSqAdj values and a simpler methodology that is applicable using a more mainstream software.Compared to the previous studies referenced, the proposed power regression model provides better RMSE and RSqAdj values as it considers not only the brand and very similar power and tractor size (wheelbase and mass) but also very similar specification levels (e.g., transmissions and user interfaces), relating these factors to an equivalent new model that provides a precise price reference, including inflation and production costs.These variations yield a better foundation for more robust results.
Compared to more advanced fitting models that require specific software, the proposed power regression model provides better RMSE and RSqAdj values and a simpler methodology that is applicable using a more mainstream software.
The model is fed from public and freely available data.Its ease of use by means of widely known software, united with its transparency, provides infinite analysis options that can be easily visualized (Figure 10).The charts created based on the power linear regression model (Figure 10) clearly depict that the more powerful A|Bb tractor family (Table 5) loses value faster than the

Figure 1 .
Figure 1.European off-road diesel engine emission regulations implementation by engine pow

Figure 1 .
Figure 1.European off-road diesel engine emission regulations implementation by engine power.
Agriculture 2023, 13, x FOR PEER REVIEW 5 of

Figure 2 .
Figure 2. Eurozone harmonized index of consumer prices (HICP) and Germany's tractor fam MSRP evolution relative to 1997.

Figure 2 .
Figure 2. Eurozone harmonized index of consumer prices (HICP) and Germany's tractor family MSRP evolution relative to 1997.

Figure 3 .
Figure 3. Dataset size grouped by country and power segment.

Figure 3 .
Figure 3. Dataset size grouped by country and power segment.

Figure 4 .
Figure 4. Power regression results (bubble size represents number of observations).

Figure 4 .
Figure 4. Power regression results (bubble size represents number of observations).

Figure 5 .
Figure 5. Exponential Gaussian process (GPR) results' RMSE for different tested variables and validations (bubble size represents number of observations).

Figure 5 .
Figure 5. Exponential Gaussian process (GPR) results' RMSE for different tested variables and validations (bubble size represents number of observations).

Figure 6 .
Figure 6.RMSE values for optimized Gaussian process results of two predictors, grouped by families tested (bubble size represents number of observations).

Figure 6 .
Figure 6.RMSE values for optimized Gaussian process results of two predictors, grouped by families tested (bubble size represents number of observations).

Figure 8 .
Figure 8. Reults of previously referenced studies (bubble size represents number of observations).

Figure 8 .
Figure 8. Reults of previously referenced studies (bubble size represents number of observations).

Figure 8 .
Figure 8. Reults of previously referenced studies (bubble size represents number of observations).

Figure 9 .
Figure 9. Summary of all regressions considered in this study (proposed power regression in blue,previous studies referenced in orange, fitted multiple variables and validation models in yellow, and fitted tractor family models in gray) (bubble size represents number of observations).

Figure 9 .
Figure 9. Summary of all regressions considered in this study (proposed power regression in blue, previous studies referenced in orange, fitted multiple variables and validation models in yellow, and fitted tractor family models in gray) (bubble size represents number of observations).

Figure 10 .
Figure 10.Dataset and power regression for 500 h per year (HPY) results of top eight RMSE tractor families.Figure 10.Dataset and power regression for 500 h per year (HPY) results of top eight RMSE tractor families.

Figure 10 .
Figure 10.Dataset and power regression for 500 h per year (HPY) results of top eight RMSE tractor families.Figure 10.Dataset and power regression for 500 h per year (HPY) results of top eight RMSE tractor families.

Table 2 .
Manufacturer's suggested retail price (MSRP) for 107 kW from one brand relative to the most economical offering.
* Brand, family, and model are anonymized to avoid any bias.

Table 3 .
Dataset size grouped by country and power segment.

Table 4 .
Model evolution example.

Table 5 .
Family model details.
* Brand, family and model are anonymized to avoid any bias.

Table 7 .
Number of predicting variables and validations evaluated.

Table 8 .
Tractor family power regression results.

Table 9 .
Fitted regression models with multiple variables and validation RMSE results.

Table 10 .
RMSE values of the results of two predictors, grouped by families evaluated.

Table 10 .
RMSE values of the results of two predictors, grouped by families evaluated.

Table 11 .
Regression results for tractor families using two predictors.

Figure 6 .
RMSE values for optimized Gaussian process results of two predictors, grouped by families tested (bubble size represents number of observations).

Table 11 .
Regression results for tractor families using two predictors.

Table 11 .
Regression results for tractor families using two predictors.
Figure 7. Two predictors, grouped by family power regression (blues) and OGPR (grey) results (bubble size represents number of observations).

Table 12 .
Results of previous studies.