## Cross Sectional Analysis

A cross-sectional analysis of a specific crop would incorporate how a farmer switches to other crop varieties of the same crop (e.g., corn varieties). The idea is to compare corn yields in Iowa with corn yields in warmer states like Arkansas. The problem is that there are other differences between Iowa and Arkansas besides differences in climate. For example, soil quality varies a great deal between states. A cross-sectional analysis would have to account for all covariates to correctly identify the effect of climate on corn yields.

If one is interested in how farmers switch crops with changing climates, a multinomial regression of crop choice on climatic variables, again accounting for all other confounding differences across climate zones explicitly, would identify such switching using cross-sectional data. In a multinomial regression, various crop choices are coded as separate categories. For example, outcome 1 could be growing maize, outcome 2 growing millet, and outcome 3 growing sorghum. A farmer will pick the most profitable crop for a given climate. Methodologically, each crop yield is modeled as a function of the climate variables as well as other controls and an error term. If the error terms follow an extreme value distribution, the probability for choosing each possible outcome has a closed form solution that is used in a multinomial logit regression. If the error terms are normal, the probability of choosing various alternatives has no closed form solution and can only be solved numerically (Maddala 1986). The multinomial logit technique has been applied to crop choices in South America by Seo and Mendelsohn (2008).

One can even go a step further and use farmland values as the dependent variable to implicitly incorporate crop switching without limiting the analysis to certain crop types. Farmland values reflect the value of land if it is put to its most profitable use, whatever that may be. An example might illustrate this point. New York City's Mayflower Hotel On the Park was a medium sized hotel on Central Park West at 61st Street. In the early 2000s it sold for an astonishing 400 million dollars. Why would anybody pay 400 million dollars for a medium-sized hotel? The first thing the new owner did was to knock down the old hotel and build a new luxury condominium (15 Central Park West) that reported apartment sales exceeding 1 billion dollars a few years later. Investors saw a higher value in demolishing the old hotel and putting the parcel to better use.

The idea behind farmland values is the same. If an investor were to look at a farm in Iowa and realize that growing corn is no longer optimal, but growing cotton is, she would be calculating the discounted amount of profit that could be made by farming cotton and should be willing to pay up to that price for the parcel. In equilibrium, the price of farmland, like the price of any type of land, should reflect the best use to which it can be put. A hedonic or Ricardian model hence uses a multi-variate regression that estimates a statistical relationship between farmland values and climatic variables, as well as other controls (Mendelsohn et al. 1994).

Other controls are necessary as other variables that impact farm profitability will also be reflected in farmland values. If a farm has more productive soil with higher yields, the additional yield boost will be priced into the farmland. Ask yourself: if you could buy an acre of land that gives you 120 bushels/acre versus the average 100 bushels/acre, you could sell an extra 20 bushels at 3 dollars per bushel for an additional 60 dollars/year (assuming the production cost are the same between the two plots). Discounting an annuity of 60 dollars using a 5% discount rate would give a net present value of 60/0.05 = 1,200. Since one parcel of land offers an additional discounted value of 1,200 more than the comparison plot at comparable cost, a rational market participant would be willing to pay up to 1,200 dollars extra.

It should be clear that there are some crucial assumptions underlying a hedonic analysis. First, the hedonic analysis assumes that prices are fixed (it is a partial equilibrium analysis). The above example assumes a constant corn price of 3 dollars per bushel. If climate change alters the productivity of entire regions, the demand and supply of various goods will change and prices will adjust accordingly. Second, there might be significant transaction cost in buying/selling farmland (capital as well as labor have to move), which might not give an efficient market outcome. Third, all other confounding variables that are correlated with climate have to be correctly accounted for, e.g., differences in soils, labor costs, or access to markets. If a variable that is correlated with climate is omitted from the analysis, its effect will incorrectly be attributed to climate.

Omitting uncorrelated variables would not bias the coefficient. This is the important fact that one uses in randomized experiments like a medical trial where half the sample is given a drug whose effect the researcher is testing. The control group is usually given a sugar pill. A test of the difference in the outcome of treated patients with the outcome in the control group gives the average treatment effect. While other controls like age or sex might impact the effectiveness of the drug, they will not bias the estimate of the average treatment effect. There should be as many people of various ages in both the treatment and control group if the sample size is large enough and people were assigned randomly to the treatment and control group. In other words the treatment and control group are "balanced." The problem is that in most cases of a cross-sectional analysis the treatment and control group are not balanced. Soil quality, access to markets, and agricultural institutions (extension service) all vary greatly among countries or even on a sub-country level.

Unfortunately, there is no direct test for omitted variables. One possible sensitivity check is to systematically include and exclude various control variables and see whether the coefficient of interest changes. Robust coefficients are reassuring in the following way: omitted variables would have to be correlated with the variable of interest but not the variables that are included and excluded from the analysis. This follows from the fact that the results did not change when various other controls were included and excluded. If the omitted variable were correlated with some of these other controls, the coefficient should have changed. For example, Schlenker et al. (2006) estimate a hedonic regression of farmland values in the Eastern United States on climatic variables as well as other controls (soil measures as well as socio-economic measures like per-capita income). When these other controls are included/excluded, the coefficients on the climatic variables remain robust. This is at least partially reassuring as any omitted variable that biases the coefficient would have to be correlated with climate, but be uncorrelated with various soil measures and socio-economic variables. While such variables might still exist, the set of possible candidates seems to be at least smaller.

The hedonic analysis has been extensively applied in World Bank studies using both farmland values as well as net revenue (a profit measure) as the dependent variable. Since farmland values are the discounted sum of all future net benefits that can be obtained from a piece of land if it is put to the best use, the two are closely related. It should, however, be noted that the cross-sectional analysis is linking farmland values or average profit measures to average weather variables. It is questionable to link profit in one particular year to average weather variables, as random weather outcomes would induce considerable noise and could severely bias the analysis. Citrus trees in California and Florida are usually highly profitable. Yet, in a year when there is a late freeze that kills the harvest and results in very low profits, linking profits from that particularly year to average weather outcomes where freezes only occur infrequently could be very misleading. The next section about panel models discusses whether annual profit measures can be linked to annual weather outcomes.

Since most countries have a narrower climate range than the United States, a cross-sectional analysis is impossible to estimate. Consider a country as small as Lichtenstein with a uniform climate that makes it impossible for a researcher to compare farms in warmer climates to farms in colder climates. A crucial requirement for any cross-sectional study is climate variation across space. One potential solution is to pool data from various countries (Seo and Mendelsohn 2007). On the other hand, the potential downside is that it becomes more difficult to account for all other confounding differences. Not only soils differ between countries but also institutional variables such as political stability and access to credit. Recall that omitting variables that impact farmers that are correlated with climate will bias the coefficients of the climatic variables. For example, if hotter countries were less politically stable and accordingly exhibited lower investments in agriculture, a cross-sectional analysis would wrongfully attribute these politically economy outcome to temperature differences if they were not accurately modeled. More recent studies have therefore sometimes broken down the analysis by agro-ecological zones with comparable farmland and included region or country fixed effects (Seo et al. 2008), which are further discussed in the next section.