Functional form refers to the type of relationship specified between a predictor variable, X, and a yield response variable, Y. The form could be a polynomial relationship, such as Eqs. (5.1) and (5.2), or an exponential relationship such as Eq. (5.3).

Several other classes of equations could also be used, such as regression trees, neural networks, or Mitscherlich equations. The most common forms used for modeling yield responses to weather are the linear model of Eq. (5.1) and the quadratic model of Eq. (5.2).

A useful way to determine the appropriate functional form is to examine a scatter plot of the data, such as in Fig. 5.3. One can also use statistical tests to determine whether a squared term significantly improves the model. A squared term can be very useful when there is an optimum temperature or precipitation amount that falls within the observed data. Thompson (1986), for instance, found that yields were reduced for departures from average June temperatures in five Corn Belt states, whether the departures were towards cooler or warmer weather.

Adding a squared term does not always help, however, as it is adds to the model complexity and can lead to overfitting and lower predictive skill. In general, we have found that higher order terms are more useful as the range of temperatures or precipitation that the crop experiences becomes wider. The reason is illustrated in Fig. 5.4: although no weather variable ever has a truly linear effect,3 a linear approximation can be appropriate over a limited range of the weather variable. In this example, yield exhibits a nonlinear response to temperature, but this response is well approximated

Y =ß0 + ß,X Y =ß0 + ß,X +ß2X: Log(Y) = ß0 + ß,X

3 Extremely low and high values are nearly always bad for crops, so that the optimum value is found somewhere between.

Temperature

Fig. 5.4 A hypothetical relationship between temperature and yield. The range of temperatures experienced in a region will determine whether a linear approximation is appropriate

Fig. 5.4 A hypothetical relationship between temperature and yield. The range of temperatures experienced in a region will determine whether a linear approximation is appropriate in regions 1 and 3 by a linear fit, since they are always on the cool and warm side of the optimum, respectively. In contrast, region 2 includes both temperatures where warming is strongly beneficial and temperatures where warming is quite harmful. Thus, a non-linear function would be necessary in region 2 but not the others.

It follows from the fact that the appropriateness of linear approximations depends on the range of weather experienced that the appropriateness will vary with the choice of model scale, since averages over large regions will show less variation from year to year than will averages over smaller areas. Linear models are therefore usually more appropriate when looking at national or regional time series than when looking at individual counties or states. As a case in point, the relationship between temperature and yield in Fig. 5.3 appears roughly linear even though at the state scale maize yields can exhibit strong nonlinear relationships with weather (Thompson 1986; Schlenker and Roberts 2006). Thus, as with the previous issues, the best choice for functional form will vary with the particular crop, location, and scale of interest.

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