In the course of the German Energiewende (energy transition), the demand for biomass has increased considerably with silage maize being an important plant for high dry matter yields. The share of the total production in agriculture was 18 % in 2014 , with an increasing share of agricultural area used for silage maize from 15.4 % in 2010 to 17.7 % in 2015 . With that in mind, the observed susceptibility of silage maize towards extreme dry conditions during summer time supports the detection of relevant factors for yield variation for instance in 2015;. Knowing the determinants of maize variation can help to mitigate welfare losses. For instance, detrimental effects of soil moisture shortage and abundance can be mitigated by the means of irrigation and drainage and thus are key for targeted and efficient development of adaptation measures .

In general, two different kinds of modeling approaches are employed to assess the impact of weather or climate on the agricultural sector. These are structural (integrated assessment) models and reduced form models . Whilst structural approaches specify the economic behavior based on theoretical models and assumptions and thus have “the ability to make predictions about counterfactual outcomes and welfare” , the advantage of reduced form approaches is “transparent and credible identification” by exploiting the exogenous variation of key parameters . Regression models are used to estimate the variation in the dependent variable within various sectors by the means of damage or dose-response functions . In the agricultural sector, the major explanatory variables are temperature based . The use of temperature as the main explanatory variable is questioned in this study by using reduced form models to identify the impact of different determinants on crop yield.

In the agricultural context, most advances have been made regarding dose-response functions through the development of temperature estimates on high spatial and temporal resolutions . Based on these data, many studies employ a precise term which integrates cumulative exposure to specific temperature ranges over the growing period as major explanatory variable. Those are defined as growing degree days and accumulated measures of extreme heat above a certain threshold, for instance extreme, heat, killing, or damage degree days . showed that the time in which a plant is exposed to a temperature above a threshold during each day of the growing season can explain almost half of its yield variations. For corn, this threshold is estimated to be 29 ∘C. Thus, it is highly recommended to account for nonlinearity in temperature. This is particularly important in the context of climate change, as the likelihood of significant and non-marginal changes in relevant factors increases. Currently, nonlinear measures with thresholds such as extreme degree days (EDD) are considered to be the best predictor of crop yield variation .

Recent research suggests that the main reason of the importance of EDD is the high correlation with measures of cumulative evaporative demand , for instance vapor pressure deficit VPD;. There is evidence that the effect of EDD and measures for evapotranspirative demand is overstated when neglecting proper control for water supply . For instance, soil moisture is considered a major limiting factor to maize growth . Extreme high temperature amplifies the impact of soil moisture deficit because of surface–atmosphere coupling , but the opposite is not necessarily the case as droughts occur independently of heat . highlight the impact of interactive effects between VPD and water supply to further improve model predictability. In Germany, a recent statistical impact assessment of weather fluctuations affecting maize and winter wheat recognizes water shortage as a major limiting factor . These studies employ proxies to control for the primary source of water, such as precipitation and measures for evapotranspirative demand. The water holding capacity of the soil and the persistence of soil moisture is often not considered.

One basic assumption in EDD is that temperature effects are additive substitutable, which means that their impact is constant for all development stages of the plant. This assumption is rejected in both agronomic studies and large-scale empirical analyses . For example, the susceptibility to high temperatures is increased during flowering (i.e., tasseling, silkening, and pollination) and the reproductive period. Similar to heat measurements, the sensitivity to water stress is dependent on the development stage of the plant . For instance, it is shown for climate projections in India that a more uneven distribution of precipitation within a season overturns positive effects of an increase in total precipitation . It is argued to control for intraseasonal-varying weather induced effects on crop yield variation. This issue is amplified for precipitation controls compared to temperature. The distribution of measures such as EDD partially overlaps with the sensitive phase of plant growth see Fig. A14 of, but precipitation, as a control for water supply, is commonly aggregated for the entire growing season among others. These studies do not explicitly account for seasonality of water-supply-related effects. Overall, controls for meteorological effects averaged over the entire season may bias the estimated dose-response function and diminish the predictive power of the models because they do not account for the seasonal interaction between water supply and water demand .

Based on this analysis, it is the main aim of this study to investigate the intraseasonal predictability of soil moisture to estimate silage maize yield in Germany. It is also evaluated how approaches considering soil moisture perform compared to those using meteorological variables. The examined hypotheses are that (a) models with soil moisture are better able to predict yield than meteorology-only approaches and (b) temporal patterns in the seasonal effects of the explanatory variables matter, i.e., there is no additive substitutability. In order to analyze these hypotheses, the intraseasonal effects of soil moisture and meteorological variables for nonirrigated arable land in Germany are examined in this study. In detail, the following research questions are addressed: (1) is there predictability of soil moisture additionally to meteorology? (2) If so, how does it compare to the one by meteorological determinants? (3) Is there temporal pattern in the seasonal effects of all explanatory variables (meteorology and soil moisture)? Along with this analysis we also evaluate (4) how models based on different meteorological determinants perform compared to each other.

To answer these research questions, a reduced form panel approach is employed to examine the nonlinear intraseasonal partial effects of soil moisture anomalies and the meteorological variables temperature, potential evapotranspiration, and precipitation. For this purpose, we use a new data set which is additionally comprised of soil moisture anomaly data. The aim is to evaluate whether soil moisture anomalies have predictive skills and how the effects differ from those using only meteorological variables. Soil moisture and any derived index are highly autocorrelated in time and thus provide an integrated signal of the meteorological conditions in the preceding and subsequent months e.g.,. This persistence does not allow for cumulative measures as those used for temperature, but it avoids the inflation of the error terms. Commonly, the predictive power of models only employing meteorological variables can be improved by accounting for the regional-specific temporal distribution of the phenological stages . The integrated signal of the meteorological conditions provided by any measure derived from soil moisture, however, allows the employment of monthly averages to account for these intraseasonal effects. In our study, it is implicitly controlled for the interaction of both variables controlling for water supply and water demand because these show high correlation on a monthly basis. Different model configurations for each month of the growing season are compared by model selection criteria to allow conclusions about the effect of soil moisture anomalies on the explanatory power of the model and to test the assumption of additive substitutability. Further, the difference in explanatory power of models either using potential evapotranspiration or average temperature is evaluated. The partial effects of all covariates of the best model for each month are examined. For the purpose of a comprehensive examination, we also investigate the effects of wet anomalies.

The main aim of this study is the identification of the monthly effects of soil moisture anomalies on crop yield. The model relates silage maize yield deviation (Y) to a stepwise function of soil moisture anomalies (SMI) and polynomials of the meteorological variables (P, T, E). Also, an error term is included which is composed of the fixed effects (c), a time-invariant categorical administrative district identifier, and the observation-specific zero-mean random-error term, which is allowed to vary over time (ϵ). Each monthly model can be written as Yik=∑n=16αnI(SMIikm∈Cn)+∑j=13βj(Pikm)j+∑j=13γj(Tikm)j+∑j=13δj(Eikm)j+cim+ϵikm.

The index i represents the administrative districts, k the years, and m each month of the growing season, while the superscript j refers to degrees of the polynomials. I(⋅) is the indicator function of the soil moisture categories Cn, which is 1 if the SMI belong to class n and 0 otherwise. The six classes are defined as severe drought (SMI ≤0.1), moderate drought (0.1< SMI ≤0.2), abnormally dry (0.2< SMI ≤0.3), abnormally wet (0.7< SMI ≤0.8), abundantly wet (0.8< SMI ≤0.9), and severely wet (0.9< SMI), respectively. The estimated coefficients of the model are α, β, γ, and δ and are constrained to be the same for all administrative districts. Time-invariant differences between administrative districts are taken into account by the fixed effects. These consist of the districts specific mean values of the individual variables on the right and left sides of the equation.

The explanatory variables are correlated to each other (Table 2). Thus, higher nonorthogonal polynomials induce singularity in the moment matrix which cannot be inverted as required by the ordinary least-squares estimation of the coefficient. The polynomials are limited to a degree of three to avoid this and other detrimental consequences of multicollinearity such as the inflation of the standard errors. Additionally, E and T are treated as mutually exclusive because of the high correlation of E and T (Table 2). The coefficients γ or δ are set to 0, accordingly.

In addition to soil moisture, a meteorological and a fixed effect term are included. The fixed effects potentially reduce omitted variable bias because they take into account the time-variant confounding factors specific to each spatial unit, such as average weather conditions and the water storage capacity of the respective soil. It is also assumed that farmers have optimized the entire production process at their location given their experience at that location. Soil and plant management, such as the choice of varieties, is adapted based on this long-term experience. Therefore, the coefficients of the exogenous variables are determined on the basis of year-to-year variations. By restricting the coefficients to be same in all administrative districts, it is implicitly assumed that the response of plants to interannual stressors is the same across all locations. Differences in the sensitivity to exogenous weather and soil moisture fluctuations implied by the use of different silage maize varieties could thus be neglected by the model. If it is also assumed that these interannual fluctuations in weather and soil moisture are not fully taken into account by the farmer in the cultivation decisions, this corresponds to a randomized allocation of the farmer to a treatment group and can therefore be regarded as a natural experiment . The outlined interpretation of the coefficients is particularly suitable for SMI, because this index, which describes deviations from the median, is per definition an anomaly.

Endogenous variables are not included because these are considered as bad control in frameworks as those defined by . For instance, prices are affected by weather realizations and climate and are thus defined as endogenous . Other studies additionally use annual fixed effects and interaction terms of both time- and entity-specific fixed effects to control for time-specific confounding factors e.g.,. These terms are not used in this study because annual variation should be explicitly accounted for by the weather variation of the exogenous variables. Annual fixed effects would diminish the entity-specific interannual variation of the exogenous variables and thereby potentially amplify measurement errors .

Various estimation approaches are used to evaluate the quality of the models. Models can be distinguished by the explanatory variables they use and the degree of polynomials in the meteorological terms. The maximum number of parameters estimated in a model is 12. The Bayesian information criterion (BIC) is used for model selection in the next section. The BIC is composed of the maximum of the likelihood function for a particular set of variables as well as a penalty term . The latter adjusts the model selection criterion for the number of parameters to account for overfitting. This allows to choose across models with different number of variables. The BIC imposes a higher penalty on overfitting compared to other model selection criteria based on maximum likelihood such as the Akaike information criterion . The penalty particularly affects the soil moisture anomaly term because it always adds six parameters. Overall, the model with the lowest BIC is preferred. To derive the BIC, a generalized linear model is fitted using the glm function .

Additionally, the models are evaluated according to their adjusted coefficient of determination (adj. R2, Sect. 4.2). Ordinary least squares using the lm function are employed with a dummy variable for each administrative districts to explicitly account for the fixed effects. As default, a demeaning framework has been applied to investigate the model performance in terms of R2. The demeaning framework involves converting the data by subtracting the administrative district average from each variable. The estimated coefficients are the same for the least-squares dummy variable regression, a demeaning framework, and maximum likelihood (BIC). This is in accordance with the theory that normal distributed error terms estimators based on maximum likelihood and least squares are the same.

The standard errors of the coefficients are corrected for spatial autocorrelation. For this purpose, the robust covariance matrix estimator proposed by is employed to construct standard errors based on asymptotic formulas. No weights capturing decaying effects in space are used because the administrative districts have different areas and the spatial extent of SMI occurrences is heterogeneous. This can be regarded as comparable to block-bootstrapping at a country level, which has been used in many comparable studies relying on resampling methods e.g.,. Further, serial correlation and heteroscedasticity are also controlled for . Overall, this approach is rather conservative but in alignment with the proposal of to take the largest robust standard error as measure of precision.

In this study, the intraseasonal effects of soil moisture on silage maize yield in Germany are investigated. It is also evaluated how approaches considering soil moisture perform compared to meteorology-only ones. A demeaned reduced form panel approach is applied, which employs polynomials of degree three for variables of average temperature, potential evapotranspiration, precipitation, and a stepwise function for soil moisture anomalies to capture nonlinearities. Potential evapotranspiration and average temperature are mutually exclusive. The model selection is based on the BIC and the adjusted coefficient of determination (R2).

This study provides a proof of concept that (a) soil moisture improves the capability of models to predict silage maize yield compared to meteorology-only approaches and (b) temporal patterns in the seasonal effects of the explanatory variables matter. Results show that soil moisture anomalies improve the model fit in all model configurations according to both the BIC and R2. SMI entails the highest explanatory power in all months but May (most explained by T) and July (most explained by P). This highlights that soil moisture adds different information than meteorological variables. All time-invariant variables show seasonal patterns in accordance to each particular growing stage of silage maize. Furthermore, the dynamic patterns of the SMI effects originate from the seasonality in absolute soil moisture. Those results support the supposition that it is necessary to control for intraseasonal variability in both the index for soil moisture and meteorology to derive valid impact assessments. Also, the comparison of various meteorological effects based on BIC shows that potential evapotranspiration adds no explanatory power compared to average temperature. Further, partial effects of precipitation outweigh those of temperature when controlling for intraseasonal variability.

The temporal resolution for the meteorological and soil moisture data is months. This might be too low to accurately resolve the stage of plant growth. Future improvements will involve the use of daily data from high-resolution remote sensing campaigns which would allow us to determine growing seasons more accurately.

Our results have further implications for climate change impact assessment. First,soil moisture can improve agricultural damage assessment and enrich the climate adaptation discourse in this realm, which is mostly based on temperature measures as major explanatory variable . We recommend controlling for at least those seasonal dependent pathways that affect plant growth presented in our study. Measures of soil moisture should be considered to derive evidence about climate impacts and adaptation possibilities. This particularly concerns climate econometrics, where frequently used reduced form approaches and dose-response functions should also control for soil moisture. For example, derived from a dose-response function only relying on temperature measures that the sensitivity to EDD is lower in southern than northern US counties. Based on these estimates, they concluded that the south is better adapted to hot conditions than the north. Transferring those adaptation potential to future impacts diminishes the estimated losses. However, various issues need to be considered when employing such an approach, such as the costs of adaptation and wrong institutional incentives . Also, argued that higher average humidity levels in the south diminish the correlation between heat and measures based on evapotranspirative demand. Accordingly, it is recommended to directly control for evapotranspirative demand by VPD. As shown in Sect. 4.1, no superior effect of potential evapotranspiration over temperature was found when controlling for either precipitation or both precipitation and SMI. Potential evapotranspiration and VPD both account for the water demand of the atmosphere. Instead, the results of this study show that controlling for water supply by measures of either soil moisture and precipitation avoids biased effects in a humid climate. This study further indicates that it is necessary to account for the seasonal dynamics in both the meteorological and soil moisture effects that constitute the variation in crop yield to employ spatial adaptation as surrogate for future adaptation.

Second, the definition of an index as anomaly has general implications for climate econometrics. Such an index is less prone to systematic errors because any bias associated to the spatial processing and the meteorological or climatological modeling is minimized . Also, the persistence in soil moisture and the resulting smoother distribution in comparison to the meteorological variables might deliver more reliable estimates than climate assessment based on meteorological variables because climate simulations only show robust trends at coarse temporal resolutions . An index can also be interpreted as interannual variability beyond the demeaning framework. Any linear model employing a categorical variable for each spatial unit is equivalent to joint demeaning of both the dependent and the independent variables and thus the source of variation is the deviation from the mean. For instance, anomalies are used within the adaptation discourse to derive implications for short-term measures . Again, in such a setting soil moisture can serve as a more comprehensive measure than the commonly used temperature.

Finally, this study has also several implications for the design of adaptation measures on weather realizations to reduce current welfare losses of climate events . First, indexes derived from soil moisture can be used in risk transfer mechanisms. For instance, insurance schemes based on a particular weather index can be enhanced in both developed and developing countries . Second, the detrimental effects of wet soil moisture anomalies might allow one to extend the risk portfolio of multi-peril crop insurance and thus foster the advancement and implementation of those schemes in Germany . Third, the installation of agricultural infrastructure should be investigated because negative effects of soil moisture anomalies can be mitigated by irrigation and drainage. In 2010, only 2.34 % of the agricultural area used for silage maize was irrigated (own calculation from data provided by ) and the latest numbers about drainage systems in Germany date back to 1993 .

Overall, an index of soil moisture considering intraseasonal variability has relevant implications for current and future damage assessment and adaptation evaluation, which are supposed to gain importance in the course of climate change.