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Nonlinear temperature effects indicate severe
damages to U.S. crop yields under climate change
Wolfram Schlenker
a,1
and Michael J. Roberts
b
a
Department of Economics and School of International and Public Affairs, Columbia University, New York, NY 10027; and
b
Department of Agricultural
and Resource Economics, North Carolina State University, Raleigh, NC 27695
Communicated by V. Kerry Smith, Arizona State University, Tempe, AZ, July 1, 2009 (received for review October 13, 2008)
The United States produces 41% of the world’s corn and 38% of the
world’s soybeans. These crops comprise two of the four largest
sources of caloric energy produced and are thus critical for world
food supply. We pair a panel of county-level yields for these two
crops, plus cotton (a warmer-weather crop), with a new fine-scale
weather dataset that incorporates the whole distribution of tem-
peratures within each day and across all days in the growing
season. We find that yields increase with temperature up to 29° C
for corn, 30° C for soybeans, and 32° C for cotton but that tem-
peratures above these thresholds are very harmful. The slope of
the decline above the optimum is significantly steeper than the
incline below it. The same nonlinear and asymmetric relationship
is found when we isolate either time-series or cross-sectional
variations in temperatures and yields. This suggests limited his-
torical adaptation of seed varieties or management practices to
warmer temperatures because the cross-section includes farmers’
adaptations to warmer climates and the time-series does not.
Holding current growing regions fixed, area-weighted average
yields are predicted to decrease by 30 – 46% before the end of the
century under the slowest (B1) warming scenario and decrease by
63– 82% under the most rapid warming scenario (A1FI) under the
Hadley III model.
agriculture
兩
panel analysis
兩
time series
兩
cross section
兩
farmer adaptation
W
ith evidence accumulating that greenhouse gas concen-
trations are warming the world’s climate, research focuses
increasingly on estimating impacts that may occur under differ-
ent warming scenarios and how economies might adapt to
changing climatic conditions. Agriculture is a key focus because
of its direct connection to climate. Although agriculture com-
prises a small share of GDP in the United States, the U.S. persists
in being the world’s largest agricultural producer and exporter of
agricultural commodities, so impacts in the U.S. could have
broad implications for food supply and prices worldwide. At the
same time, debate continues about whether warming will be a net
gain or loss for agriculture in currently temperate climates like
that of the United States (1– 8).
In this paper, we estimate the link between weather and yields
for the three crops with the largest production value in the
United States: corn, soybeans, and cotton. Corn and soybeans,
the nation’s most prevalent crops, are the predominant source of
feed grains used in cattle, dairy, poultry, and hog production.
Corn is also the main source of U.S. ethanol. Cotton is the
fourth-largest crop in acres planted, but more valuable on a
per-acre basis and more suited to warmer climates than are corn
and soybeans. Estimating the correct relationship between
weather and yields for these major crops is a critical first step
before more elaborate models can be used to examine how
crop-planting choices, food and fiber supply, and prices will
ultimately shift in response to climate change.
Our data are comprised of new fine-scale weather outcomes
merged together with a large panel of crop yields that spans most
U.S. counties from 1950 to 2005. The new weather data include
the length of time each crop is exposed to each one-degree
Celsisus temperature interval in each day, summed across all
days of the growing season, all estimated for the specific
locations within each county where crops are grown. The new
fine-scale weather data facilitate estimation of a f lexible model
that can detect nonlinearities and breakpoints in the effect of
temperature on yield. If the true underlying relationship is
nonlinear (e.g., increasing and then decreasing in temperature),
averaging over time or space dilutes the true temperature
response. For example, similar average temperatures may arise
from two very different days, one with little temperature vari-
ation and one with wide temperature variation. Holding the
average temperature constant, days with more variation will
include more exposure to extreme outcomes, which can critically
inf luence yields (9). The empirical challenge is to map an entire
season of widely varying temperature outcomes to each year’s
yield. Accurate estimation of nonlinear effects is particularly
important when considering large, nonmarginal changes in
temperatures, now expected with climate change.
Results
Regression analyses, which control in various ways for precipi-
tation, technological change, soils, and location-specific unob-
served factors, all show a similar nonlinear relationship between
temperature and yields. Additional results and sensitivity checks
are given in the
SI Appendix
.
Yield Growth Increases Gradually with Temperature up to 29 –32°
Celsius, Depending on the Crop, and then Decreases Sharply for All
Three Crops.
Estimates and standard errors of temperature effects
for all three crops are displayed in Fig. 1. The figure has three
frames, where each column represents one crop. Different
specifications of the link between temperature and yield growth
are indicated by the color of the estimated curve. The blue line
shows the most f lexible specification, a step function that fits a
separate growth rate for each three-degree temperature range.
The black line shows the specification where yield growth is
modeled as an eighth-order polynomial function of temperature
and the 95% confidence band after adjusting for spatial corre-
lation is added in gray. The red line shows a piecewise linear
specification that follows the agronomic concept of degree days.
Each specification shows the same characteristic shape, increas-
ing modestly up to a critical temperature and then decreasing
sharply. For corn, the critical threshold temperature is 29° C; for
soybeans it is 30° C; and for cotton it is 32° C.
The vertical axis in each figure marks the log of yield (bushels
per acre for corn and soybeans and bales per acre for cotton)
with the exposure-weighted average predicted yield normalized
to zero. The horizontal axis is temperature. In comparing two
points on any curve, a vertical difference of 0.01 indicates an
Author contributions: W.S. and M.J.R. designed research, performed research, contributed
new reagents/analytic tools, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
1
To
whom
correspondence
should
be
addressed.
E-mail:
wolfram.schlenker@
columbia.edu.
This article contains supporting information online at
www.pnas.org/cgi/content/full/
0906865106/DCSupplemental
.
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approximately 1% difference in average yield for the year. For
example, the blue line frame A (the f lexible model for corn),
substituting a full day (24 h) at 29° C temperature with a full day
at 40° C temperature results in a predicted yield decline of
⬇
7%,
holding all else the same. The green histogram shows the average
exposure to each one-degree Celsius interval during the growing
season (March–August for corn and soybeans and April–
October for cotton).
Coefficients on other explanatory variables (precipitation,
squared precipitation, county fixed effects, and state-specific
quadratic time trends) are not reported here. Precipitation has
a statistically significant inverted-U shape with an estimated
yield-maximizing level of 25.0 inches for corn and 27.2 inches for
soybeans in the f lexible step-function specification in Fig. 1. The
precipitation variables are not statistically significant for cotton,
which is not surprising given that 58% of the crop is irrigated.
Fixed effects control for time-invariant heterogeneity (like soil
quality) and state-specific quadratic time trends control for
technological change. With wide geographic variation in average
yields and a three-fold increase in yields over the sample period,
these controls have strong statistical significance.
The pattern of temperature effects is quite robust to specifi-
cation and controls. The same nonlinear temperature effect
emerges whether or not any of the controls, or any subset of
controls, are included in the regressions. We also find the
estimated temperature effects to be very similar if we instead
control for technology and time effects by using year-fixed
effects rather than state-specific quadratic time trends.
Holding Current Growing Regions Fixed, Area-Weighted Average
Yields Are Predicted to Decrease by 30 – 46% Before the End of the
Century Under the Slowest Hadley III Warming Scenario (B1), and
Decline by 63– 82% Under the Most Rapid Warming Scenario (A1FI).
For comparison, a linear model that uses the average growing-
season temperature as an explanatory variable gives predicted
impacts of
⫺
16% to
⫹
3% (B1) and
⫺
30% to
⫹
6% (A1FI)
among the three crops. Yield predictions are summarized in Fig.
2. Frame A shows predictions for the medium term (2020 –2049)
and frame B for the long term (2070 –2099). Predictions are for
changes in total production under four climate scenarios in the
Hadley III climate model. Across all scenarios, model specifi-
cations, and crops, the aggregate impacts show marked declines,
even though yields in some individual counties are projected to
increase. The driving force behind these large and significant
predicted impacts is the projected increase in frequency of
extremely warm temperatures.
Out-of-Sample Model Predictions Are More Accurate than Previous
Statistical Models.
The new regression models were compared
with other specifications in the literature by using the root-mean
squared error (RMS) of out-of-sample predictions. Each model
is estimated 1,000 times, randomly choosing 48 years of our
56-year history of yields. The estimates are then used to predict
yield outcomes for the remaining eight years (
⬇
14%) of each
sample. We randomly sample whole years and not observations
because yields are spatially correlated in any given year. We
compare our own three specifications of temperature effects
(step function, polynomial, and piecewise linear) with three
alternative specifications: (
i
) a model with average temperatures
for each of four months (1); (
ii
) an approximation of growing-
degree days based on monthly average temperatures (Thom’s
formula) (4); and (
iii
) a measure of growing-degree days,
calculated by using daily mean temperatures (8), that does not
include a separate category for extremely warm temperatures.
As a baseline, we consider a model with county-fixed effects
0
5
10
15
20
25
30
35
40
−0.075
−0.05
−0.025
0
0.025
Temperature (Celsius)
Log Yield (Bushels)
Corn
0
10
5
Exposure (Days)
Step Function
Polynomial (8th−order)
Piecewise Linear
0
5
10
15
20
25
30
35
40
−0.075
−0.05
−0.025
0
0.025
Temperature (Celsius)
Log Yield (Bushels)
Soybeans
0
10
5
Exposure (Days)
Step Function
Polynomial (8th−order)
Piecewise Linear
0
5
10
15
20
25
30
35
40
−0.075
−0.05
−0.025
0
0.025
Temperature (Celsius)
Log Yield (Bales)
Cotton
0
10
5
Exposure (Days)
Step Function
Polynomial (8th−order)
Piecewise Linear
A
B
C
Fig. 1.
Nonlinear relation between temperature and yields. Graphs at the top of each frame display changes in log yield if the crop is exposed for one day to
a particular 1° C temperature interval where we sum the fraction of a day during which temperatures fall within each interval. The 95% confidence band, after
adjusting for spatial correlation, is added as gray area for the polynomial regression. Curves are centered so that the exposure-weighted impact is zero.
Histograms at the bottom of each frame display the average temperature exposure among all counties in the data.
−100
−80
−60
−40
−20
0
Impacts by 2070−2099 (Percent)
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Corn
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Soybeans
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Cotton
−100
−80
−60
−40
−20
0
Impacts by 2020−2049 (Percent)
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Corn
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Soybeans
B1
B2
A2
A1
B1
B2
A2
A1
B1
B2
A2
A1
Cotton
Step Function
Polynomial (8th−order)
Piecewise Linear
A
B
Fig. 2.
Predicted climate-change impacts on crop yields under the Hadley III
climate model. Graphs display predicted percentage changes in crop yields
under four emissions scenarios. Frame A displays predicted impacts in the
medium term (2020 –2049) and frame B shows the long term (2070 –2099). A
star indicates the point estimates, and whiskers show the 95% confidence
interval after adjusting for spatial correlation. The color corresponds to the
regression models in Fig. 1.
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and quadratic state-level trends, but no weather variables. Such
a model simply forecasts the average yield trend for a county in
each year. We derive the percent reduction in the RMS if
weather variables are also included. Results are summarized in
Fig. 3. A reduction by 100% would imply that the model produces
no error and gives a perfect forecast, whereas a percent reduc-
tion of 0% implies that the weather variables have no additional
explanatory power.
The new models reduce RMS by between 40% and 360% more
than other specifications. Welch tests find no significant differ-
ence between our three models, but strongly significant differ-
ences are found when compared with other specifications in the
literature.
The Same Nonlinear Relationship Between Yields and Temperature Is
Observed in both the Cross-Section of Counties and the Aggregate
Year-to-Year Time Series.
The results described in
Yield Growth
Increases Gradually with Temperature up to 29 –32° Celsius
are
from regression models with county-fixed effects, which identify
temperature effects on yields using within-county time-series
weather variation. Although random variation is useful from a
statistical standpoint, such analysis accounts only for grower
adaptation in response to current-year weather (e.g., additional
use of irrigation in a dry year), and not for systematic crop- or
variety-switching in anticipation of a different climate. In con-
trast, a cross-sectional analysis identifies temperature effects by
using only variations between counties with different climates.
Both the cross-section and the time series give results compa-
rable with the baseline model that pools weather and climate
variations. In particular, the cross-sectional and time-series
estimates show the same characteristic nonlinear temperature
relationship, similar optimal temperatures, and predict nearly
equal yield impacts under Hadley III climate-change scenarios.
The Nonlinear Relationship Between Yield and Temperature Observed
in Cooler Northern States Is Similar to the One Observed in Warmer
Southern States.
To explore how the temperature-yield relation-
ship varies over different regions of the country, we divided our
sample into three mutually exclusive geographical regions cor-
responding to the most northern (and coolest) states, the most
southern (and warmest) states, and those in the middle. We focus
on corn because it is grown over the widest geographic area and
has been by far the most valuable crop grown in the United
States. The interesting feature is the relative stability of the
estimated temperature relationship across the three subregions.
The Nonlinear Relationship Between Yield and Temperature Observed
Between 1950 and 1977 Was the Same as the One Observed Between
1978 and 2005.
To explore how the temperature-yield relationship
has changed over time, we split the sample into two time periods
of equal length, 1950 –1977 and 1978 –2005. The critical thresh-
old when temperatures become harmful is rather robust over
time. The similarity of the temperature-yield relationship across
subsamples is particularly interesting given that, because of
technological change, average yields in the more recent sample
are about twice those in the earlier sample.
Greater Precipitation Partially Mitigates Damages from Extreme High
Temperatures.
We explore interactions between temperature and
precipitation outcomes by dividing the sample into quartiles of
total precipitation during the months of June and July. These
estimates have a shape similar to those of the pooled sample up
to the critical temperature threshold. The decline above the
threshold, however, is less steep for corn subsamples with greater
precipitation.* However, omitting temperature-rainfall interac-
tions will not bias predictions of average effects of temperature
and rainfall, as we do not find a significant correlation between
temperature outcomes and precipitation outcomes in the raw
daily data.
The Estimated Climate-Change Impacts Are Insensitive to the Speci-
fied Growing Season and Consistent with Time Separability.
The
baseline model uses temperature and precipitation measures for
the months of March through August for corn and soybeans. In
practice, northern regions tend to plant later than southern
regions, and planting dates may vary from year-to-year depend-
ing on weather conditions. The
SI Appendix
shows results from
eight alternative specifications of the growing season, all with
comparable results. When we limit the growing season to
two-month intervals or estimate separate temperature response
functions for July (when corn f lowers) and other months, we find
qualitatively similar temperature-response functions for each
subperiod. Although F tests give significantly different coeffi-
cient estimates in July compared to the pooled remaining
months, the fit (R
2
) hardly increases at all when breaking the
growing season into various subperiods and predicted climate-
change impacts that are not significantly different. For example,
a corn-yield model with separate temperature coefficients for
July only slightly increases the R
2
from 0.7654 to 0.7698 under
the step function and changes overall predicted long-run yield
impact from
⫺
43% to
⫺
44% under the slow-warming scenario
(B1) and from
⫺
79% to
⫺
74% under the fast-warming scenario
(A1FI).
Discussion
Many studies, spanning several disciplines and employing dif-
ferent methods, have linked weather and climate to agricultural
outcomes such as yields, land values, and farm profits, each with
their own set of strengths and weaknesses.
Agronomic studies are the predominant tool used to evaluate
potential effects from climate change. Examples include refs.13–
17, but there are many others. These studies emphasize the
dynamic physiological process of plant growth, seed formation,
and yield. The process is understood to be quite complex and
*We estimated models with interactions between temperature and rainfall on a shorter
time scale, but these models do not predict out-of-sample significantly better than the
additively separable model reported above. It is possible that the relatively poor predictive
power of precipitation, in comparison with temperature, stems from greater measure-
ment error in the precipitation variable as spatial interpolation is more difficult.
0
5
10
15
Percent Reduction in RMS
Step
Function
Polynomial
(8th order)
Piecewise
Linear
Monthly
Averages
Degree Days
(Thom)
Degree Days
(Daily Mean)
Corn
Soybeans
Cotton
Fig. 3.
Out-of-sample prediction comparison for various model specifica-
tions. Bar charts display the percent reduction in the root-mean-squared
prediction error (RMS) for each model in comparison with a baseline model
with no weather variable. Each model is estimated 1,000 times, where each
replication randomly selects 48 of the 56 years in our full sample. Relative
performance is measured according to the accuracy of each model’s prediction
for the omitted eight years of the sample (
⬇
14%). We sample years instead of
observations because year-to-year weather fluctuations are random, but
there is considerable spatial correlation across counties within each year. Step
function, polynomial (eighth order), and piecewise linear are the models
developed in this paper; the other models are from the existing literature.
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dynamic in nature and thus not easily estimated in a regression
framework. Instead, these studies use a rich theoretical model to
simulate yields given daily and subdaily weather inputs, nutrient
applications, and initial soil conditions. In some cases, simulated
yields are compared with observed yields with some success. We
are not aware of any agronomic study that has tested a simulation
model by using data different from what was used to calibrate it.
Current versions of models developed for many crops are
maintained by the Decision Support System for Agrotechnology
Transfer (www.icasa.net/dssat/).
A strength of simulation models is that they fully incorporate
plant-growth theory. These models also incorporate the whole
distribution of weather outcomes over the growing season. This
differs markedly from earlier regression-based approaches that
typically use average weather outcomes or averages from par-
ticular months and thus give biased estimates of nonlinear
temperature effects. Potential weaknesses of simulation ap-
proaches are their complexity, uncertainty about the structure of
the physiological process, and the large number of parameters.
Some agronomists seem to worry about possible misspecification
and omitted variables biases (10 –12). These models also take
production systems and nutrient applications as exogenous:
There is no account of farm operators’ decisions.
Several earlier economic studies use hedonic models to asso-
ciate land values to land characteristics, including climate, by
using reduced-form linear regression models (e.g., refs. 1, 4, and
6). A strength of this approach is that it accounts for the whole
agricultural sector, not just a single crop at a time. It can also
account for farm operator behavior and adaptation. Cooler
areas are likely to become more like warmer areas, with crop
choices, management practices, and land values changing in
accordance with current geographic variation in climate.
The overarching concern with the hedonic approach, and with
cross-sectional studies generally, is omitted variables bias. Cli-
mate variables (e.g., average temperature) and other variables,
such as soil types, distance to cities, and irrigation systems, are
all spatially correlated. If critical variables correlated with cli-
mate are omitted from the regression model, the climate vari-
ables may pick up effects of variables besides climate and lead to
biased estimates and predictions. Cross-sectional studies may be
strongly suggestive, but do not carry the same empirical weight
as a physical or natural experiment. Indeed, earlier work shows
how omission of heavily subsidized irrigation systems strongly
inf luences predicted climate impacts (18).
Recently, Auffhammer, Ramanathan, and Vincent (7) used a
panel of yields and weather outcomes in India to model the role
of brown clouds. Similarly, Desche
ˆnes and Greenstone (8) linked
agricultural profits in the United States to yearly weather
variations by using a four-year county-level panel from the
Census of Agriculture. Whereas both studies aggregate weather,
our model, like simulation models, incorporates the whole
distribution of temperature outcomes in a f lexible way. Unlike
simulation models, it is amenable to estimation by using standard
regression analysis. Like refs. 7 and 8, we consider specifications
with county-fixed effects to identify parameters by using year-
to-year weather variations that serve as a natural experiment.
Like ref. 7, but unlike ref. 8, we focus on yields rather than
profits. Potential problems with the use of profits are discussed
in ref. 19.
Our study aims to combine the strengths of previous ap-
proaches. We use fine-scale weather data and combine it with
f lexible regression models. The findings are notable for the
consistency of the estimated nonlinear temperature effects
across time, locations, crops, and the many sources of variation
in temperature and precipitation considered.
Perhaps most interesting is the consistency of the estimated
relationship when comparing estimates based on year-to-year
weather variations and cross-sectional climate variations. This
finding suggests limited historical adaptation to extreme heat for
any given crop. This implication follows from the fact that
cross-sectional variations include farmers’ adaptations to
warmer climates whereas estimates using time-series variations
do not. The first approach is akin to hedonic models that link
land values to average weather outcomes, except that it does not
account for crop switching.
There are, of course, many other possible adaptations that this
study cannot address. The simplest form of adaptation would be
to change the locations or seasons where and when crops are
grown.
†
Understanding the scope for this kind of change would
require more careful analysis of potential yield effects on a global
scale. Furthermore, if climate change were anticipated to induce
severe yield impacts on a global scale, then anticipated increases
in commodity prices would likely encourage greater investments
in new seed varieties, irrigation systems, and other technological
changes. Thus, although historical data show the same heat
tolerance in the first and second half of our sample, greater heat
tolerance still may be possible if greater returns for such
innovation arise. Recently, a National Science Foundation-
funded study completed a draft sequence of the corn genome,
which might make it easier to develop new corn varieties with
greater heat tolerance (see http://monsanto.mediaroom.com/
index.php?s
⫽
43&item
⫽
576 for more information).
An important caveat concerns our inability to account for CO
2
concentrations. Plants use CO
2
as an input in the photosynthesis
process, so increasing CO
2
levels might spur plant growth and
yields. Yield declines stemming from warmer temperatures
therefore may be offset by CO
2
-fertilization. Although higher
CO
2
concentrations may boost yields, the magnitude of the effect
is still debated. Long, et al. (12, 20) recently stressed that existing
laboratory studies and field experiments might overestimate this
effect. We cannot account for CO
2
effects in regression analysis
of observed yields because CO
2
concentrations quickly dissipate
throughout the atmosphere, leaving only a gently increasing time
trend, which is impossible to statistically disentangle from tech-
nological change.
Methods
Data.
Yields for corn, soybeans, and cotton for the years 1950 –2005 are
reported by the U.S. Department of Agriculture’s National Agricultural Sta-
tistical Service. These yields equal total county-level production divided by
acres harvested. We limit the analysis to counties east of the 100° meridian
(excluding Florida) for corn and soybeans because cropland in the West often
relies on heavily subsidized irrigation systems.
‡
For cotton, we use all counties
that report cotton yields because there are fewer observations and a larger
share of cotton is grown in the West. In the
SI Appendix
we report results for
all three crops if we split them into Eastern and Western subsets. Results
differ substantially for corn and soybeans, but to a lesser degree for cotton,
and we hence pool observations from the East and West for the latter. For
results reported here, we have 105,981 observations with corn yields,
82,385 observations with soybeans yields, and 31,540 observations with
cotton yields.
Construction of the fine-scale weather data are briefly described here and
in more detail in ref. 21 and the
SI Appendix
. The basic steps are as follows. We
first develop daily predictions of minimum and maximum temperature on a
2.5
⫻
2.5-mile grid for the entire United States. We then derive the time at
which a crop is exposed to each one-degree Celsius interval in each grid cell in
each day. These predictions are merged with a satellite scan that allows us to
select only those grid cells with cropland. We then aggregate the whole
distribution of outcomes for all days in the growing season in each county. To
preserve within-county temperature variation, it is important to derive the
time each grid cell is exposed to each one-degree Celsius interval before
†
In the
SI Appendix
, we show that shifting the growing season one month earlier does little
to mitigate yield loss.
‡
Because access to subsidized water rights is correlated with climate, omitting these
variables, which vary on the subcounty level of irrigation districts, will result in biased
coefficient estimates on the climatic variables in a cross-sectional analysis (18). Across all
counties, 58% of cotton acres, 18% of corn acres and 9% of soybean acres are irrigated.
Schlenker and Roberts
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aggregating to obtain the county-level distribution.
§
The average historical
weather distribution and its variability across time and space are summarized
in the
SI Appendix
.
To examine whether corn varieties have been adapted to warmer temper-
atures, we divide the United States into three regions, the northern, the
interior, and the southern, all east of the 100° meridian.
¶
Climate-change predictions are drawn from the Hadley III model (www.
metoffice.com/research/hadleycentre/). We obtain a monthly model output
for both minimum and maximum temperatures under four major emissions
scenarios (A1FI, A2, B1, and B2) for the years 1960 –2099. The B1 scenario
assumes the slowest rate of warming over the next century and the A1FI
scenario assumes continued use of fossil fuels, which results in the largest
increase in CO
2
-concentrations and temperatures. The B2 and A2 scenarios fall
between the B1 and A1F1 scenario. We find the predicted difference in
monthly mean temperature for 2020 –2049 (medium-term), 2070 –2099 (long-
term), and historic averages (1960 –1989) at each of 216 Hadley grid nodes
covering the United States. Predicted changes in monthly minimum and
maximum temperature at each 2.5
⫻
2.5-mile PRISM grid are calculated as the
weighted average of the monthly mean change in the four surrounding
Hadley grid points, where the weights are proportional to the inverse squared
distance and forced to sum to one. In a final step, we add predicted absolute
changes in monthly minimum and maximum temperatures at each PRISM grid
to the observed daily time series from 1960 to 1989. In other words, we shift
the historical distribution by the mean monthly change predicted by each
climate scenario. An analogous approach was used for precipitation, except
that we use the relative ratio of future predicted rainfall to historic rainfall
instead of absolute changes. The predicted changes in temperatures under
various scenarios are shown in the
SI Appendix
.
Regression Models.
We assume temperature effects on yields are cumulative
over time and that yield is proportional to total exposure. This implies tem-
perature effects are additively substitutable over time. Specifically, plant growth
g
(
h
) depends nonlinearly on heat
h
. Log yield,
y
it
, in county
i
and year
t
is
y
it
⫽
冕
h
h
g
共
h
兲
it
共
h
兲
dh
⫹
z
it
␦ ⫹
c
i
⫹
⑀
it
[1]
where
it
(
h
) is the time distribution of heat over the growing season in county
i
and year
t
. We fix the growing season to months March through August for
corn and soybeans and the months April through October for cotton. Ob-
served temperatures during this time period range between the lower bound
h
and the upper bound
h
. Other control factors are denoted
z
it
and include a
quadratic in total precipitation as well as quadratic time trends by state to
capture technological change. Finally,
c
i
is a time-invariant county-fixed effect
to control for time-invariant heterogeneity, such as soil quality. We allow the
error terms
it
to be spatially correlated by using the nonparametric routine
used by ref. 24.
By using data on exposure to each 1° C temperature interval, we approx-
imate the above integral with
y
it
冘
h
⫽⫺
5
49
g
共
h
⫹
0.5
兲关⌽
it
共
h
⫹
1
兲
⫺
⌽
it
共
h
兲兴
⫹
z
it
␦ ⫹
c
i
⫹
⑀
it
[2]
where
⌽
it
(
h
) is the cumulative distribution function of heat in county
i
and
year
t
.
We consider three specifications for
g
(
h
): A step function with a different
growth rate in each 3° C temperature interval; an eighth order Chebychev
polynomial; and a piecewise linear function. Details are provided in the
SI
Appendix
.
Perhaps the strongest assumption of the regression models is time separa-
bility of temperature effects. This assumption is partly rooted in agronomy. In
crop simulation models, however, temperature effects can vary over the life
cycle of the plant. In the
SI Appendix
, we report results from a model that uses
various definitions of the growing season and allows the coefficients to be
time-varying over the growing season.
ACKNOWLEDGMENTS.
We thank Spencer Banzhaf, Larry Goulder, Jim Mac-
Donald, Mitch Renkow, Bernard Salanie, Kerry Smith, and Wally Thurman, as
well as seminar participants at Arizona State University, Dartmouth University,
Georgia State University, Harvard University, North Carolina State University,
Stanford University, U.C.–Berkeley, U.C.–Davis, University of Illinois, University
of Maryland, University of Nebraska, and University of Wyoming, and two
anonymous referees for useful comments.
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§
An alternative method to approximate the distribution of daily temperatures from the
distribution of monthly average temperatures is developed by ref. 23. This method
appears appropriate for predicting the average frequency that a certain weather outcome
will be realized but less appropriate in predicting a specific frequency of a weather
outcome in a particular year. Thom’s method works well in a cross-sectional analysis where
the dependent variable is tied to expected weather outcomes (for example, the link
between land values and climate). The method is less well suited to our analysis where the
dependent variable (yield) linked to specific weather outcomes.
¶
The northern subset includes counties in Illinois, Indiana, Iowa, Michigan, Minnesota, New
Jersey, New York, North Dakota, Ohio, Pennsylvania, South Dakota, and Wisconsin.
Interior counties are in Delaware, Kansas, Kentucky, Maryland, Missouri, Nebraska, Vir-
ginia, and West Virginia. The southern counties are in Alabama, Arkansas, Georgia,
Louisiana, Mississippi, North Carolina, Oklahoma, South Carolina, Tennessee, and Texas.
15598
兩
www.pnas.org
兾
cgi
兾
doi
兾
10.1073
兾
pnas.0906865106
Schlenker and Roberts
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