## Panel Analysis

A panel analysis recognizes that there are fundamental differences between spatial units of the analysis (e.g., countries) and that it is a difficult task to account for all these differences explicitly in a model. If these influences impact yields in an additive fashion and if they are time invariant, one can use fixed effects to capture them. In defense of the first assumption, the large majority of regression models use a linear specification and hence all factors are assumed to have an additive influence. Only a nonlinear model in parameters would capture non-additive factors. As long as a cross-sectional analysis uses a linear specification in the parameters, the assumption underlying a fixed effect model is no stronger than modeling each effect directly. A linear model in the parameters includes the case where variables are logged, as the logged variable still interacts linearly. The second assumption that all country-specific variables are time-invariant needs some pondering. Countries can change over time, get new governments, or even split in two. In such a case it is advisable to use fixed effects for each distinct temporal subset.

A fixed effect is a dummy or indicator variable that is set to one if observations from a group (country) are included and is set to zero otherwise. A panel requires at least two observations by country as otherwise the indicator variable would absorb all variation in that country. One cannot include a time-invariant variable in a panel model that uses fixed effects as this variable would be collinear with the fixed effects. For this reason one can for example not estimate a panel model of farmland values that uses average weather as explanatory variable combined with fixed effects. Average weather by definition is constant within a group and hence a linear multiple of the indicator variable.

It can be shown that a fixed effect model is equivalent to a joint group-specific demeaning of the dependent as well as all independent variables (Wooldridge 2001). If we subtract group specific averages from both the dependent and all independent variables and run a linear regression, the coefficients will be identical but the standard errors need to be adjusted for the difference in degrees of freedom. For example, a panel model that regresses country-level yields on average temperature during each growing season can be estimated in two ways. First, one can include a dummy for each country. If one also includes a constant, the dummy for one country has to be dropped to avoid perfect mulicolinearity. Second, one can subtract the average yield in each country from the yearly observations of yields and subtract the average climate in a country from each weather outcome in the country. If one were to then run a linear regression of the demeaned yields on the demeaned average weather without any country-specific dummy variables one would obtain identical regression coefficients as in the first specification. While this is a noteworthy statistical artifact, it also has an important interpretation. The regression uses deviations from country-specific averages to identify the parameter of interest. As discussed in further detail below, this is equivalent to fitting a regression line through each country where the slope is forced to be the same for each country but the intercept is allowed to vary by country. All countries are forced to exhibit the same sensitivity to weather fluctuations. For a model that allows for a distribution of weather sensitivities (i.e., distribution of regression slopes) the interested reader is referred to random coefficient models.

If we return to the initial discussion of this chapter, a panel uses variation in weather (i.e., year-to-year fluctuations in weather) as a source of identification and not differences in average weather (climate). Such a model will not incorporate any adaptation to systematic shifts in average weather. Some researchers therefore favor a random effects model which takes a weighted average of the within-group variation (fluctuations in weather) as well as the between-group variation (differences in average weather or climate). The interested reader is referred to any intermediate econometrics or statistics textbook. Intuitively, a random effects model does not include a separate dummy variable for each group but rather assumes that there might be a group-specific additive error term. This group-specific error term will capture time-invariant additive constants. Since omitted variables are included as a special expression in the error term, the estimated coefficients will suffer from an omitted variable bias similar to a cross-sectional analysis.

Panel data analyses have been used with profits (DeschĂȘnes and Greenstone 2007) and yields (Schlenker and Roberts 2009) as the dependent variable. The advantage of profits is that all crops can be aggregated into one single measure instead of modeling each crop separately. At the same time, there is a potential downfall with a profit or net revenue measure: most studies simply take the difference between total agricultural sales and production expenditures in a given year. Such an analysis neglects that the amount sold is different from the amount produced as most commodities are storable. In high-productivity years when yields are above normal, prices are low and farmers have an incentive to put part of the harvest into storage, i.e., the quantity sold is less than the quantity produced. On the flip side, when yields are below normal and prices are high, farmers have an incentive to sell part of the inventory that was harvested in previous years and hence the quantity sold is higher than the quantity produced. As a result, storage smoothes reported sales and makes them smaller than the full amount produced in good years and larger than the full amount in bad years. This will bias the estimated weather coefficients towards zero (Fisher et al. 2009). What one would need is an economic profit measure (value of production minus production cost) instead of the accounting measure (value of sales minus production cost).

In summary, the advantage of a panel is that one does not have to worry as much about omitted variable bias as the fixed effects capture all time-invariant variables. The downside is that a panel might measure something very different from a cross-sectional analysis, e.g., might capture various sets of adaptation possibilities. Some authors argue that the adaptation possibility is always greater in the cross-section, which would be in line with the Le Chatelier's principle that costs in a constrained system are higher than when constraints are relaxed in the long-run (e.g., fixed capital becomes obsolete). This is, however, not necessarily true in agriculture. Sometimes, adaptation possibilities are available in the short-run that could not be sustained forever. For example, a one-time drought might be mitigated by pumping groundwater. The aquifer recharge might be small enough that such groundwater pumping could not be sustained forever if droughts were to become more frequent. Finally, storage might bias a panel of agricultural profits as it smoothes sales between periods. It is an important omitted variable in a panel analysis that uses agricultural sales in a given year (quantity sold multiplied by price) instead of the value of all goods produced (quantity produced multiplied by price).

## 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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