Graph A of Fig. 6.1 displays crop yields in Lesotho in light grey squares and South Africa in black triangles for the years 1961-2000. The x-axis displays average temperature during the growing season, while the y-axis displays log yields (note that this variable can be negative as the log of 1 is 0, so any yield less than 1 ton/ha is a negative number). Lesotho is colder than South Africa as all grey squares lie to the left of the black triangles. Average yields are also lower is Lesotho than in South Africa as the average y-value of the grey squares is lower than the average height of the triangles.
Graph A (Scatter Plot)
Graph B (Cross-sectional Analysis)
16 18 20 22 Average Temperature (Celsius)
Graph D (Country-specific Analysis)
Average Temperature (Celsius) Average Temperature (Celsius)
1 Maize yields and average temperatures in Lesotho and South Africa (1961-2000)
Table 6.1 Summary statistics for maize in Lesotho and South Africa (1961-2000)
Lesotho South Africa
Average log yield (ton/ha) Average temperature (Celsius) Fertilizer (kg/ha)
A cross-sectional analysis would link average yields in Lesotho and South Africa to the respective average temperatures. This is done in graph B, where the 40 observations per country are reduced to 1 showing the average yield and average climate in a given country by averaging the 40 yearly observations. The solid line in panel B links the average outcomes in the two countries, which implies that increasing temperatures are beneficial as yields are increasing in average temperature.
One possible concern is that there are many differences between Lesotho and South Africa. Table 6.1 shows that South Africa uses much more fertilizer: 21.7 kg/ ha compared to the 1.8 kg/ha in Lesotho. If each 1 kg/ha were to increase log yields by 0.0038 (i.e., increases yields by roughly 3.8%) than the entire difference in log yields (0.765=0.547 + 0.218) would be explained by the difference in fertilizer use (19.8 = 21.7 - 1.8) as 19.8*0.0038 » 0.765 and the difference in temperatures would have no explanatory effect on the difference in yields. If the effect of fertilizer on yield is greater than 3.8% per kg, than the difference between South Africa and Lesotho after adjusting for fertilizer use implies that average yields are higher in Lesotho. If we link the yield net of fertilizer use again to temperatures, the estimated relationship would be negative as the country with higher average temperatures has lower average yields. Fertilizer is not the only difference between the two countries, and it should be immediately clear that it is empirically very challenging to account for all differences that might be correlated with differences in climate.
The intuition behind a panel analysis is shown in graph C of Fig. 6.1. It captures all time-invariant effects by an additive constant, i.e., a fixed effect. Differences in fertilizer use, institutional differences, access to markets, etc., are captured by a country-specific constant. This implies that each country can have a unique intercept of the regression line, but the slope of all regression lines is forced to be the same. Only variation within a country is used to identify the regression coefficient of interest (the slope of the regression line with respect to temperature): are yields in Lesotho and South Africa higher or lower in years that are warmer than usual? Note that the answer is the opposite of what we obtained in the cross-section (graph B): higher temperatures are worse, not better. However, analogous to the pure time series, it does not capture the full set of adaptation possibilities as we are identifying the parameter by looking at year-to-year fluctuations within a location. The difference between this and a time series is that we force these within country deviations to have the same effect among all countries.
Graph D of Fig. 6.1 shows the case of a pure time series which would estimate a separate regression for each country. The slope of the regression line is no longer the same but differs by country. A panel using fixed effects is closer to a time series model than to a cross-section. An alternative that uses both the variation within countries as well as the variation between country means (a weighted average of the time series and cross-section) is a random effects model (Wooldridge 2001).
Finally, a panel using fixed effects assumes the same effect of temperatures on yields in all countries, but the functional form decides whether this constant effect is in absolute or relative terms. In a linear model, a 1°C increase in average temperature is assumed to have the same absolute impact on yields, e.g., a decrease of 3 bushels/acre. In a log-linear model a 1°C is assumed to have the same relative impact on yields, e.g., a 3% decrease in yields. In a double-log model where both the dependent and independent variables are specified in logs, a relative temperature deviation (e.g., 5% less than normal) is assumed to have the same relative impact on yields, e.g., a 3% decline in yields.
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