## Projecting Impacts of Climate Change with Time Series Models

Once a model has been calibrated with time series data, it can be used to predict yield responses to any hypothetical amount of climate change. (Chapter 3 describes approaches for downscaling climate projections for input into crop models.) For example, temperature and precipitation changes from climate model simulations can be used to generate new values of the relevant predictor variables, which the regression model then translates to yields.

There are, however, three extremely important caveats to the use of time series models for simulating yield responses to climate change, even for the analyst who has successfully navigated the issues described in the previous section. The first is common to all cases of statistical model prediction, and relates to the problem of extrapolating the model beyond the range of calibration data. In particular, as the climate warms growing season temperatures may increasingly exceed the warmest year contained in the historical data used to fit the statistical model. Figure 5.5 illustrates this for the current example: as temperatures rise fewer and fewer predictions will reside in the calibration domain.

The simplest approach to avoiding extrapolation errors is to use statistical models only for the relatively near term where the vast majority of years have historical precedents. For example, if we set an arbitrary threshold that no more than 25% of

Fraction of years above historical maximum Fraction of years above historical maximum

Fraction of years above historical maximum Fraction of years above historical maximum

Fig. 5.5 Fraction of years that exceed historical maximum for the US maize example when temperature (left) or precipitation (right) is increased by different amounts

years should be warmer than the warmest year on record, then based on Fig. 3.5 we would only make projections out to 1.3°C. This corresponds roughly to average projections for 2030, which is still a useful period to analyze. However, using time series based models to make projections to 2080 - where climate model projections commonly exceed 3°C of warming - would be misguided, and in that case other approaches would be more appropriate.

Another approach to address extrapolation error is to implement several different methods of extrapolation to gauge whether results are sensitive to predictions made outside the calibration domain. For example, one can contrast a conservative approach of truncating yields to historical extremes, with a more aggressive approach of allowing yields to extrapolate to zero (Lobell et al. 2006). The point at which the two methods diverge provides a measure of when the time series model is on shaky ground.

The second caveat is another common one in statistics and involves the assumption of stationarity - that relationships observed in the past also apply to the future. As crop varieties and management systems change, however, the response of yields to variations in weather may also change. An example already mentioned is when irrigation is introduced into currently rainfed areas. As with extrapolation, the assumption of stationarity becomes more questionable as the time horizon of projections extends further into the future.

The final and perhaps most serious caveat is the use of models based on year-to-year variations in weather to predict responses to gradual changes in climate. An economist would refer to this as equating short-run and long-run effects, which ignores the ability of humans to adapt to system shocks. For agricultural systems, we attribute the difference between weather and climate responses to the ability of farmers to (1) perceive and (2) adapt to a changing climate. Some have gone so far as to argue that the response to climate can be opposite in sign to that for weather (Hansen 1991), while others argue that adaptation will be very difficult and not entirely effective.

A detailed discussion of adaption is presented in Part 3 of this book. The only point made here is that applying time series based models to projection of climate change implicitly assumes that no adaptation will take place. Note that this assumption does not have to be true for the projections to be useful. One goal of projections, for example, can be to identify where the biggest threats are to agriculture if we do not adapt, in order to guide short-term investments in adaptation (Lobell et al. 2008). Also, comparing time series based projections with those that incorporate adaptation can provide a useful measure of the potential impact of adaptation, a point explored further in the next chapter.

### 5.5 Summary

Time series can be an invaluable resource for understanding the aggregate response of crop production to variations in climatic conditions. Models based on time series depend not only on the data, but on several choices faced in the modeling process.

Most prominent among these are choosing the spatial and temporal extent of the time series data, method of detrending, types and temporal resolution of climatic variables, and the specification of the functional relationship between climate and yield. Poor choices for any of these can potentially lead to invalid estimates of climate responses.

Two general principles are especially useful for time series models. First, the analyst should always plot the data at each step, to examine features such as colinearity, heteroskedasticity, and nonlinearities, rather than rely exclusively on model summaries provided by common software packages. Second, when a choice between two alternatives is not apparent, the analysis should be tried both ways and the results compared. This is analogous to using multiple process-based models that have different but equally defensible assumptions to evaluate model uncertainty.

Users of time series models should be keenly aware that adaptation can, in principle, cause fundamentally different responses to weather and climate. As time series models are based on year-to-year variations in weather, their application to future scenarios of climate change embody an assumption of no adaptation. This can be useful in many situations, especially when results are contrasted with estimates of impacts that include adaptation, but the assumption should be kept explicit at all times. A summary of the key points of this chapter are given below.

• Time series models can be extremely useful for projections for the next 20-30 years, when adaptation is likely to be small and climate is not too far from current conditions. Beyond that, the extrapolation of past relationships to the future becomes more tenuous.

• The most pervasive challenge in time series modeling is co-linearity between the major climate variables known to affect crops, namely temperature, precipitation, and solar radiation.

• There is no single best approach to time series modeling, as optimal decisions will depend on location, crop, and scale. Comparison of results from multiple alternative specifications can be a useful measure of uncertainty.

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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