Modeling Land Use and Land Cover Change: Using a Hindcast to Estimate Economic Parameters in gcamland v2.0

The world has experienced a vast increase in agricultural production since the middle of the last century. Agricultural land area has also increased at the expense of natural lands over this period, though at a lower rate than production. Future changes in land use and cover have important implications not only for agriculture but for energy, water use, and climate. However, these future changes are driven by a complex combination of uncertain socioeconomic, technological, and other factors. Estimates of future land use and land cover differ significantly across economic models of agricultural production, and 10 efforts to evaluate these economic models over history have been limited. In this study, we use an economic model of land use, gcamland, to systematically explore a large set of model parameter perturbations and alternate methods for forming expectations about uncertain crop yields and prices. We run gcamland simulations with these parameter sets over the historical period in the United States to explore model fitness and to identify combinations that improve fitness. We find that an adaptive expectation approach minimizes the error between simulated outputs and observations, with parameters that suggest that for 15 most crops landowners put a significant weight on previous information. Interestingly, for corn, where ethanol policies have led to a rapid growth in demand, the resulting parameters show that a larger weight is placed on more recent information. We conclude with the observation that historical modeling exercises such as this study are valuable both for understanding real world drivers of land use change and for informing modeling of future land use change.

area in the United States has remained relatively constant between 1975 and 2015. Instead, there has been a shift in crop 30 distribution, with an increasing share of corn and soybeans and a decreasing share of wheat and other grains (Figure 1, (FAO, 2020a;Taheripour and Tyner, 2013)). Future changes in land use and land cover have implications for agricultural production, energy production, water use, and climate. For example, changes in land cover can alter albedo, resulting in changes in local and global temperature and precipitation (Brovkin et al., 2013;Jones et al., 2013;Manoli et al., 2018). Similarly, changes in land use and land cover have implications for water withdrawals and water scarcity (Bonsch et al., 2016;Chaturvedi et al., 2013;Hejazi et al., 2014aHejazi et al., , 2014bMouratiadou et al., 2016). However, there is significant uncertainty in the future evolution of land use and land cover, due to 40 uncertainties in future socioeconomic conditions (e.g., population, income, diet) (Popp et al., 2017;Stehfest et al., 2019), technological change (Popp et al., 2017;Tilman et al., 2011;, climate (Calvin et al., 2020;, and incentives for bioenergy, afforestation, reforestation (Calvin et al., 2014;Hasegawa et al., 2020;Popp et al., 2014Popp et al., , 2017.

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Economic models are widely used to estimate future agricultural production and land use, and estimates of future land use and land cover also differ significantly across such models (Alexander et al., 2017;Von Lampe et al., 2014;Popp et al., 2017).
These models use economic equilibrium, statistical, agent-based, machine learning, and hybrid approaches (Engström et al., 2016;National Research Council, 2014). Even within each category, there are differences across models, both in terms of structure and parameters. For example, among economic equilibrium models of land use change (the approach most commonly 50 used in integrated energy-water-land-climate models), some models use constrained optimization (e.g., GLOBIOM), while other models use a non-linear market equilibrium approach (e.g., GCAM) .
Efforts to evaluate land use models over the historical period are limited. Baldos and Hertel (2013) compare the net change in cropland area, agricultural production, average crop yield, and crop price between 1961 and 2006 simulated by the SIMPLE 55 model to observed changes. Their model matches observations better at the global scale than at the regional scale; additionally, they find that "even knowing yields with certainty does not allow us to predict cropland change accurately over this historical period." Bonsch et al. (2013) compare simulated land-use change CO2 emissions from MAgPIE to observations, finding that the choice of observation dataset matters for how well the model performs. Calvin et al. (2017) and Snyder et al. (2017) compare agricultural production and land area simulated by the GCAM model to observations, finding that the model does 60 better for trends than annual values and that some region/crop combinations are better than others. The authors test the use of expectations about yield using a linear forecast as a driver of land use change instead of observed yield, finding that simulations using expected yield better match observations than those using observed yield. Engstrom et al. (2016) use a Monte Carlo approach to sample parameters in PLUM, simulating agricultural production and land area over the historical period and comparing results to observations. The authors find the model performs better at larger regional aggregations, but the observed 65 grassland and cereal land area falls outside the full range of their ensemble results. However, most land use models outside of these have not used historical simulations for evaluation/validation.
Only a few studies have attempted to draw land use modeling parameters from econometric estimates of land supply elasticity (Ahmed et al., 2009;Lubowski et al., 2008). However, there is usually no fixed relationship between the land supply elasticities 70 and land use modeling parameters in equilibrium models (Zhao et al., 2020a) and, more importantly, empirically estimated elasticities only provide a limited coverage of regions and land use categories (Barr et al., 2011;Lubowski et al., 2008). Thus, the parameters used in land use models are often based on heuristics . For example, Taheripour and Tyner (2013) group regions into four categories based on historical land use change and assign substitution parameters based on those categories.  choose model parameters to replicate empirically estimated parameters; however, there is no 75 unique mapping between the empirical parameter (constant elasticity of land transformation) and the model parameter (logit exponent). While there are many examples of studies exploring sensitivity to drivers of land use change or sensitivity across models, most studies exclude sensitivity to parameters. The small number of studies that do test alternative parameters find that it could significantly alter land use change (Engström et al., 2016;Taheripour and Tyner, 2013;Zhao et al., 2020b).

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In this paper, we use a large perturbed parameter ensemble to determine the model expectation configuration and parameter set that best replicate historical land use and land cover within the United States. Section 2 describes the methodology used in https://doi.org/10.5194/gmd-2020-338 Preprint. Discussion started: 1 December 2020 c Author(s) 2020. CC BY 4.0 License. this study. Section 3 introduces the scenarios included in the study. The primary results and sensitivity analyses are discussed in Sections 4 and 5, respectively. Section 6 includes the discussion and conclusions. We use the gcamland v2.0 software package in this study (Calvin et al., 2019a). gcamland separates the land allocation mechanism in GCAM (Calvin et al., 2019b) into an R package. The model calculates land allocation over time; changes in land use and land cover are driven by changes in commodity prices, yields, costs, and subsidies, all of which are inputs into 90 gcamland. gcamland includes all land use and land cover types, with crops aggregated into 12 commodity groups (see (Calvin et al., 2020) for a mapping). gcamland can be run in several different modes, including hindcast and future scenario options and single and multiple ensemble options. For this paper, we utilize the ensemble and hindcast options, generating large ensembles of hindcast simulations. gcamland can be run for any of the 32 geopolitical regions within GCAM, but for this study we focus on the United States. 95

Economic approach in GCAM and gcamland
Land allocation in gcamland (and GCAM) is determined based on relative profitability, using a nested logit approach (McFadden, 1981;Sands, 2003;. The logit land supply is presented in equation (1). All else equal, an increase in the rental profit rate ( ! ) of one land type will result in an increase in the land area ( ! ) allocated to that land type. The magnitude of the land supply response is dependent on the positive logit exponent ( ) and share-weight parameters ( ! ). is 100 the total land supply, i.e., ∑ ! ! = . The logit formulation assumes that there is a distribution of profit rates for each land type, and the resulting land allocation for a given land type is the probability that land type has the highest profit (Zhao et al., 2020b).
The logit share-weights (the scale parameters in the distribution) are calibrated to perfectly reproduce the data in a base year.
The logit exponent (the shape parameter in the distribution which governs the magnitude of land transformation given relative profit shocks) is one of the parameters of interest in our study. 105 The logit approach is advantageous compared with the constant elasticity of transformation (CET) approach widely used in GGE models as it can directly provide traceable physical land transformation. But like the CET function, the logit land sharing function is parsimonious and nested structure can be used. In gcamland, all crops are nested under cropland. Cropland is nested with forest and then pasture, see Figure 1 in Calvin et al. (2019a). Thus, there are three logit exponent parameters for governing 110 land transformation in the three-level nesting structure. https://doi.org/10.5194/gmd-2020-338 Preprint. Discussion started: 1 December 2020 c Author(s) 2020. CC BY 4.0 License.

Means of forming expectations
There are multiple means of forming expectations in the literature. With perfect foresight, the expected value of a given variable is equal to its realized value: (2) 115 In an adaptive expectation approach (Nerlove, 1958), the expected value is a linear combination of the previous expectation and the new information acquired, with being the coefficient of expectations: 120 Finally, a linear expectation approach uses a linear extrapolation of previous information to form the expectation: where is a fixed number of previous years considered in forming the expectation, ( ) and ( ) are vectors of the variable and year index, respectively, with historical information from year − 1 to − . That is, instead of using all available historical information, forward-looking producers are assumed to rely on only information of the most recent years. 125 In our study, we combine these basic approaches into four different expectation types, specifying the means of calculating expected price and expected yield (Table 1). 1 The expected prices and yield would affect farmers' expected rental profits and, thus, land use decisions.  (Féménia and Gohin, 2011;Lundberg et al., 2015;Mitra and Boussard, 2012;Roberts and Schlenker, 2013) Linear Linear expectations Linear expectations Snyder et al., 2017) Hybrid Linear

Adaptive
Adaptive expectations Linear expectations Tested in this paper

Perturbed parameter ensemble
Within this study, we vary a total of nine parameters (Table 2), including three logit exponents, the coefficient of expectations ( ) for the Adaptive expectation and the number of years (n) used in the Linear expectation. In addition, we allow and n to 135 vary across commodity groups, resulting in three separate realizations for each parameter. We group the commodities to minimize the number of free parameters. The first group includes Corn and OilCrop, which are used for biofuels in the United States and have had shifts in the demand over time as a result of biofuels policies. The second group includes the other two large commodities produced in the United States, Wheat and OtherGrain. The third group includes all other crops. The range of values spanned in the ensemble was chosen to cover all plausible values of each parameter but avoid potential numerical 140 instabilities. Those ranges and their justification are described in Table 2. We use a Latin Hypercube Sampling strategy to generate the ensembles, with 10,000 ensemble members per expectation type and model configuration.

Evaluating model performance
Following the analysis of previous hindcasting experiments with GCAM (Snyder et al., 2017), different measures of model performance are used to select parameter sets that optimize different aspects of model performance.
Normalized root mean square error (NRMSE) considers all deviations between simulated and observed values, and places them in the context of the variance seen in the observational data. For crop i, 150 https://doi.org/10.5194/gmd-2020-338 Preprint. Discussion started: 1 December 2020 c Author(s) 2020. CC BY 4.0 License.
One benefit of this measure is that it includes a natural benchmark of acceptable model performance. While = 0 corresponds to perfect model performance, any < 1, is considered acceptable model performance. This measure puts the deviations between simulation and observation for each crop in the context of that crop's historical variance.

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We also consider the root mean square error (RMSE), and bias, 160 These un-normalized measures make no distinction between different crops; a bias or RMSE of 200 km 2 means exactly the same for Corn as it does for Rice, despite the fact that Corn represents a larger proportion of harvested area in the United States in the historical period. While RMSE is concerned with all deviations between observation and simulation for a crop, bias simply compares the means between observation and simulation. While these means tend to be determined more by the smoothed trend in a time series than variations about the trend, it is important to note that bias specifically does not penalize 165 volatility the way that RMSE and other measures may.
Finally, the Kling-Gupta Efficiency score (Knoben et al., 2019) is also implemented for each crop, for correlation coefficient r, standard deviation , and mean ). While a perfect simulation (NRMSE=RMSE=bias=0) would 170 by definition give perfect KGE (KGE=1), KGE is obviously more defined by penalties between different time series summary statistics, as opposed to the penalties based on simple deviations between simulation and observation at each time point in the other error metrics considered here.
For a given error measurement, the metric is calculated for each crop in each ensemble member. The average metric value 175 across crops is then minimized to select the ensemble member with the most optimal parameters for matching observation. For bias, it is the average across crops of the magnitude of bias that is minimized, to avoid cancellation of errors between crops.
For KGE, it is the average across crops of the quantity 1 − ! that is minimized so that the average across crops of ! is optimized as needed. As an additional sensitivity, the actual land types included in this average metric can be adjusted to include all crops, simply one individual crop, or any combination of crops of interest. 180

Initialization data
To initialize gcamland in this study, we started from the GCAM v4.3 agriculture and land use input data. The GCAM data processing reconciles land use data from FAO with land cover data, ensuring that total areas do not exceed the amount of land in a region. Thus, we chose to use this reconciled data instead of using FAO data directly. We have made two changes to the 185 GCAM v4.3 initialization data.
First, since GCAM has a five-year time step, it uses five-year averages of land use and agricultural production for initialization.
For this study, we have updated the input data to remove the averaging since we are primarily focused on annual time steps; that is, the initialization data in gcamland for a particular year is the data for that year only and not a five-year average around 190 that year as it is in GCAM.
Second, GCAM models land use and land cover at the subnational level (v4.3 used Agro-Ecological Zones; v5.1 and subsequent versions use water basins). However, much of the comparison data is provided at national level. For this study, we aggregate the initialization data to the national level, representing the USA as a single region. The qualitative insights in this 195 paper would not change if we disaggregated to subnational level, but the exact quantitative results would.
Third, GCAM uses constant costs over time. For this study, we have updated the costs to use time-evolving cost data (see next section).

Scenario data
We use data for price and yield from the U.N. Food and Agricultural Organization (FAO, 2018a(FAO, , 2018b(FAO, , 2020b, with data available for all commodities for 1961-2018. Data was aggregated from individual crops to the GCAM/gcamland commodity groups, weighting crops by their production quantity. In some cases, data prior to 1961 are required to generate expectations for the model years ; in these cases, we assume that prices and yields prior to 1961 are held constant at their 1961 205 values. For cost, we use data provided by the U.S. Department of Agriculture (USDA, 2020a), with data available for major crops from 1975-2018. We only include the variable costs as reported by USDA and exclude the allocated overhead costs. We use a representative crop from USDA for each GCAM/gcamland commodity group, as data does not exist for all crops (i.e., we use soybean cost from USDA as a proxy for the cost of OilCrop in gcamland). For subsidies, we combine two different data sets from USDA: the Federal government direct payments (USDA, 2020c) and the farm business income (USDA, 2020b). 210 We only include direct payments from these two reports; thus, our subsidy data is missing many other forms of payment.
Additionally, we only have data for a subset of crops and the categories reported change over time across the two data sets. Because this data is inconsistent and incomplete, we only use it as a sensitivity in this paper and do not include it in the primary analysis.

Observation data 215
We compare model outputs to observation data to evaluate the performance of gcamland. Ideally, the observation data would be completely independent of the model. However, due to limited availability of data sets, 3 we use the FAO harvested area for crops as the observational dataset, despite the fact that it is used to calibrate gcamland. For calibration, we only use a single year of data, so the comparison to the FAO time series is still valid. For land cover, an independent data set is available for use in gcamland; specifically, we use satellite data from the European Spatial Agency (ESA) Climate Change Initiative (CCI), as 220 reported by the FAO (FAO, 2020a) and aggregated to the gcamland land cover classes. Due to differences in definitions and classifications, the grassland and shrubland reported by CCI differ substantially from the gcamland areas even in the calibration year.

Simulations and sensitivities
The default ensemble analyzed in this paper uses 1990 as a calibration year, runs annually through 2015, excludes subsidies, 225 and differentiates the expectation parameters (a and n) by crop groups. To test the sensitivity of the results to each of these assumptions, we re-run the ensemble for with alternative specifications for each assumption (Table 3). Over the last several decades, yields have increased in the United States; prices and profits are more variable ( Figure S1).
Changes in the area of a particular crop, however, are not always correlated with in year profit ( Figure S2, S3). There are several potential reasons for this: 1. Farmers do not know the profit at the time of planting and instead are basing their planting decisions on expectations.

2.
The profit calculated here is missing some other factor (e.g., a government subsidy). 235

3.
Profit relative to another commodity may be better predictor (e.g., if two crops have increases in profit, a farmer might shift to the one with faster increases, resulting in a decline in land area for the other despite its increase in profit).

4.
Different crops may have undergone very different improvements in yields over time.

5.
Other non-economic factors (e.g., distance to markets) might drive land use decisions.
We explicitly test the first two explanations in this paper. The third and fourth are captured in all of our simulations. The fifth 240 is implicitly captured in the calibration routine in gcamland but we do not vary this over time.

Results
This section describes the results from the default gcamland ensemble. This ensemble assumes a calibration year of 1990, an annual time step, subsidies are excluded, and the parameter sets are chosen to minimize the average NRMSE across all crops.
Sensitivity to each of these assumptions is presented in the next section. Note that throughout the results and sensitivity sections 245 the default configuration, with the numerically optimal parameter set and expectation type are shown in thick magenta lines for consistency.

Parameter Sets that Minimize NRMSE in gcamland
NRMSE varies across expectation types, ranging from 1.399 with Adaptive expectations to 1.874 with Linear expectations.
The parameters that minimize NRMSE vary by expectation type (see Figure 2), including the ordering of the logit exponents. 250 In the Adaptive expectations, the logit exponent dictating substitution among crops is larger than the logit exponents determining substitution between crops and other land types. This rank ordering of logit exponents is consistent with the intuition from historical trends in USA land allocation ( Figure 1); specifically, the larger changes in crop mix than total crop area in the observations suggest that the logit for the cropland nest should be larger than the other logits. In all models with imperfect expectations, expected profits are heavily weighted toward previous information, as evidenced by the large values 255 for the share of past information and the number of years in the linear forecast. However, these values vary across crop groups.

Comparing modeled land area to observations
The full ensemble of gcamland simulations results in a large range of land allocated to crops, covering +/-100% of the observed 265 area. The parameter sets that minimize NRMSE in gcamland replicate total harvested cropland area over time in the United

Sensitivity Analysis 280
In this section, we describe the sensitivity of the results above to several different assumptions, including those related to the configuration of the model, the calibration year, the model timestep, and the objective function used. For the model configuration, calibration year, and timestep sensitivities, we generate new ensembles of gcamland results with the appropriate assumption altered. For the sensitivity to objective function, we filter the original ensemble using different criteria to determine the numerically optimal parameter sets. 285

Sensitivity to Model Assumptions
First, we test the sensitivity of the analysis to two different assumptions: (1) whether subsidies are included in the expected profit for crops, and (2) whether the expectation-related parameters differ across crops. For all three sets of assumptions, Adaptive expectations minimizes NRMSE. Varying these assumptions results in differences in parameters (Table S2) and cropland area (Figure 4). Including subsidies increases the NRMSE (from 1.399 in the Default case to 1.46 with subsidies). 290 This is due to the quality of the subsidy data. Including all factors that affect profit should improve the model; however, the subsidy data is incomplete (only direct payments were included for crops where these were reported) and inconsistent (reporting changed over time). However, previous studies have shown that direct payments have little effect on crop production https://doi.org/10.5194/gmd-2020-338 Preprint. Discussion started: 1 December 2020 c Author(s) 2020. CC BY 4.0 License. or land area in the United States (Weber and Key, 2012), suggesting that better subsidy data may not change land allocation decisions substantially. 295 Using the same expectation parameters across commodity groups increases NRMSE (from 1.399 in the Default case to 1.531 with uniform parameters). There are several reasons why different crops could require different parameters. First, one would expect differences between annual and perennial crops due to the lag between planting and harvesting and the multi-year investment required by perennial crops. Second, some crops (e.g., Corn and OilCrop) have had shifts in policy or demand over 300 time (e.g., for biofuels). Such shifts may lead landowners to prioritize newer information. Finally, there could be differences in how markets are structured (e.g., futures contracts) or region-specific differences. These effects are difficult to disentangle in gcamland. Perennial crops are all included in the "All other crops" group. This group is a mix of both perennial and annual, but we do see higher shares of past information in this group than in the other commodity groups in the Default model. Corn and OilCrop rely more heavily on new information when parameters vary, which is consistent with the market shifting 305 hypothesis.

Sensitivity to the Objective Function
The analysis above uses the average NRMSE across all crops as an indicator of "goodness of fit", but other objective functions are possible. In this section, we discuss alternative measures of "goodness of fit", including bias, RMS, and KGE. Additionally, we examine the implications of minimizing NRMSE for an individual crop as opposed to the full set of crops.

Optimizing for different objective functions 315
The parameter sets (Table S3)  The parameter set that minimizes bias, however, is fundamentally different. The logit exponents dictating the substitution between crops and other land types are large (2.17 for the Arable Land nest; 1.37 for the Ag, Forest, and Other nest). The 325 parameter set that minimizes bias also includes the lowest Cropland nest logit value of any objective function (0.28). The resulting simulations for bias exhibit large volatility in land area. Given that bias simply compares the model mean across time to the observation mean across time, this volatility is not penalized in the bias metric, whereas it is penalized for KGE, RMSE and NRMSE. For example, the parameter sets that minimize bias result in an average observed Corn area of 306 and an average simulated Corn area of 307, resulting in a bias of less than 1 thous km 2 . This bias is much lower than the bias for Corn in the 330 other objective functions (NRMSE and RMSE have a bias of 6 thous km 2 ; KGE has a bias of 16 thous km 2 ). Bias is effectively assessing whether the model is correct on average and not whether it captures the trends or volatility; such an objective function is less useful in systems where trends are significant or where the goal is to capture the volatility. From a mechanistic perspective, we hypothesize that the difference in the bias cropland area volatility is due to the differences in the Ag, Forest, and Other logit. 335  Figure 6 shows the difference in the best models when you optimize for a particular set of land types or crops. As seen in this figure, gcamland can track land area for any given crop very well when the ensemble with optimal parameters is chosen specifically for that crop. However, matching all crops at once is more challenging. For example, the parameter sets that 345 minimize NRMSE for Corn result in an excellent match between observations and model output for Corn; however, those parameters result in an overestimation of Wheat land by 250 thous km 2 in 2015 (or ~1/2 of the actual area). The insights from this figure are also confirmed numerically.

Different calibration years
The calibration routine in gcamland calculates share weight parameters ( ! in equation 1) to ensure that the land area is exactly replicated in the specified base year. Those parameters are held constant in all subsequent periods. Changing the base year could result in different calibration parameters and thus different land allocation, even if all other parameters are the same. In 360 this section, we test this sensitivity, using 1975 and 2005 as alternative base years. Figure 7 shows the difference in cropland area for the parameter sets that minimize average crop NRMSE for each calibration year. Those parameter sets are shown in Table S4. The resulting parameters and land use are relatively similar between variants with base years of 1990 (the default described above) and 2005. The logit exponents are small for all three nests, with the largest value over the cropland nest. Both models use more past information for All Other Crops than for Corn and OilCrop, but they differ in the degree of past 365 information used for Wheat and OtherGrain. The variant with a 1975 calibration year, however, has large differences in parameters and behavior from those with 1990 and 2005 calibration years. We hypothesize two reasons for these differences.
First, we have a limited time series prior to 1975, which results in erroneous estimates of expected price and expected yields for parameter sets with large reliance on past information. Second, there is a discrepancy between FAO harvested area and the land cover data sets used in GCAM in 1975 (this discrepancy exists but is much smaller from 1990 onwards). In particular, 370 FAO harvested area is larger than the physical crop area. We correct for this in gcamland by assuming that some areas are planted more than once in a year. However, this results in larger annual yields in gcamland than the harvest yield provided by FAO. This results in higher profit rates that could affect the land allocation. Note that this issue is not a problem in future simulations, like those typically run in GCAM, since the calibration information used in future periods is the information calculated from a more recent year without these data challenges (2010 or 2015 depending on the version of GCAM). 375

Short-run versus long-run parameters
Finally, we examine the sensitivity of the results to the time step. Most studies using GCAM use a five-year time step with perfect expectations. However, we are increasingly interested in quantifying the implications of climate variability and change on agriculture, land use, and the coupled human-Earth system, which requires higher temporal resolution. For purposes of this comparison, we focus on RMSE instead of NRMSE. NRMSE and RMSE differ in that NRMSE is normalized by standard 385 deviation; however, the inclusion of standard deviation introduces inconsistencies when comparing across time steps. For the Default model, the choice of RMSE or NRMSE has no effect on results, but for the five year time step it does. We note any differences that would emerge from using NRMSE in this discussion.
Our hypothesis was that longer time steps would result in larger logit exponents since farmers would have more time to make 390 adjustments and that expectations would matter less with longer time steps. Using RMSE, the former is true, but the latter is not. 5 The five-year timestep results in higher logit exponents, particularly in the Arable Land nest and the Cropland nest; the expectation parameters are similar though ( Figure S11). However, the Hybrid Linear Adaptive expectations minimizes RMSE in the five-year time step model (Table 4), suggesting that expectations are still important for longer time steps. 6 We find that the one-year time step results in a lower RMSE than the five-year time step model, even when the differences in comparison 395 years are taken into account (Table 4): the RMSE computed over five-year increments in the one-year model is still lower than the RMSE in the five-year model. In the five-year time step model, farmers use five-year averages of price and yield when forming expectations. As a result, the five-year time step model will produce different expectations ( Figure S14) and different land allocation results ( Figure S15) than the one-year time step model even when the same parameters are used. The fact that annual time steps reduces RMSE suggests that interannual variability may have a noticeable influence on expectations and the 400 resulting land allocation; that is, farmers consider not just the trend in yield and price but also the variability around that trend. This is particularly true for Corn and OilCrop where more recent information has a larger effect on expectations. In this paper, we have used a perturbed parameter ensemble of simulations of land use and land cover over the historical period to guide the selection of parameters for an economic model. We find that adaptive expectations minimize the error between simulated outputs and observations, consistent with empirical evidence (Mitra and Boussard, 2012). The resulting parameters suggest that for most crops, landowners put a significant weight on previous information. For Corn and OilCrop, however, a 410 large weight is placed on more recent information. This is consistent with an observation by Kelley et al. (2005): "In the case of agriculture, anecdotal evidence suggests that some farmers are more myopic, weighing recent information more than is efficient." The optimal set of parameters is sensitive to the choice of objective function, with differences emerging either when the 415 mathematical formulation of the error is altered or when the set of land types included in the calculation of error is changed.
For the former, we find that using bias as an objective function leads to the largest volatility in annual land allocation. While GCAM has historically performed better at capturing overall trend behavior than annual variations and this has been considered acceptable model behavior Snyder et al., 2017), the results of this study highlight the importance of penalizing variations about the trend as well. For the latter, it is possible to significantly improve the performance for the model 420 for any single crop by optimizing for that crop; however, the resulting parameters may lead to a larger error for a different crop. For example, the parameter sets that minimize NRMSE for Corn result in an excellent match between observations and model output for Corn; however, those parameters result in an overestimation of Wheat land by approximately 250 thous km 2 in 2015.

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We hypothesize that limitations of data affect the performance of some variants of the model. For example, the variant that explicitly excludes subsidies outperforms the one with subsidies, likely due to the poor quality of the subsidy data. Similarly, the variant with 1975 as a calibration year is fundamentally different from the variant with 1990 or 2005 as a calibration year, likely due to discrepancies between harvested and physical area in 1975 and limited availability of the data prior to 1975 that is needed to form expectations. Similarly, the land cover data provides little constraint on the model due to the short time series 430 and difference in definitions of land category. Future work could include improvements in the data and the addition of new data sets to constrain the model. Additionally, we have focused on the United States, using national level data; future work could replicate this analysis for subnational regions or for other countries around the world. Our expectation is that we would find different parameters best replicate observations in other countries, similar to what is asserted in Taheripour and Tyner (2013). 435 Other potential research directions include testing other assumptions in the model (e.g., the nesting structure), new explanatory variables (e.g., crop insurance, speculative storage), alternative decision-making frameworks (e.g., non-logit approaches), or additional behavioral processes (e.g., learning, diffusion). For the nesting structure, we have only tested the default GCAM nesting structure here. Taheripour and Tyner (2013) test an alternative nest and find that it has implications for the share of 440 forest cover (14% vs. 3% depending on the nest). For explanatory variables, studies have indicated that some programs, like crop insurance, are likely to have a direct impact on area planted and production (Young and Westcott, 2000). For alternative decision-making frameworks, Zhao et al. (2020b) demonstrate that the resulting change in land use due to a shock differs depending on the combination of functional form (logit, constant elasticity of transformation, constrained optimization) and parameter value. 445 In this paper, we have focused on the historical period. However, these models and parameter estimates could be used in a simulation of future land use and land cover change to better understand their implications. Additionally, given the connections between energy, water, land, and climate, using these parameters, and the uncertainty around them, in the fully coupled GCAM would be useful in the future. 450