hosted in a permanent repository at the Cornell Institute for Social and
Economic Research (CISER) at Cornell University
(https://doi.org/10.6077/fb1a-c376).

###  4.1 Aggregation of point weather data

As indicated in section 2, the most basic form of weather data comes from
weather stations. These are point datasets where weather observations are geo-
referenced. Relying on these direct observations may seem appealing but they
present some clear challenges for the non-specialist. The first inconvenience
is that the spatial distribution of weather stations may be sparse (see Fig.
1). This means that the researcher may not be able to accurately obtain
weather information in the precise locality under study. The second issue is
that weather data are not always quality-controlled. That means that the
researcher needs to spend considerable effort “cleaning” the data for
suspicious or implausible outliers. The third challenge is that weather
observations may be missing in some time periods. These issues of sparsity,
data quality and attribution, pose empirical challenges related to measurement
error and attrition bias which have important econometric consequences.

In cases when all units of observations are very close to weather stations,
say in the example of geo-referenced farm-level data, it makes sense to rely
on the closest weather station data. In cases when the researcher is
interested in weather conditions that fall between weather stations, some form
of interpolation may be necessary. The most basic type of interpolation is an
inverse-distance weighting approach.

Performing your own interpolations, however, is generally not ideal. There are
numerous high-quality datasets created by meteorologists and climate
scientists based on sophisticated interpolations or other model based
approaches (e.g. interpolation accounting for orography or reanalysis). Those
third-party products typically implement cutting-edge approaches and undergo
careful quality control procedures by dedicated trained professionals. Trying
to replicate similar work for an economic research project is risky because
these procedures are error-prone. Readers and reviewers are rightfully
skeptical of whether the researchers were able to perform interpolation
adequately. This requires researchers to allocate non-negligible amount of
effort describing and validating their interpolation approach.

The comparative advantage of economists does not lie on environmental data
interpolation. We are thus generally better off relying on datasets developed
by climate scientists unless there is a strong reason not to do so. For
instance, performing one’s own interpolation may be justified for study
regions where existing gridded datasets have been found to be unreliable.

In the accompanying R code and data, I provide a simple introduction to basic
forms of spatial interpolation (see 1_weather_data.R). To illustrate these
techniques I rely on weather station data from the GHCN database falling over
the lower 48 US states. I also select maximum temperature (Tmax) on August 16,
2020 as the example. Figure 3A shows Tmax on that day over more than 6,000
weather stations across the US. We can see some regions of the country have a
a relatively sparse network of weather stations.

Notes: All maps show maximum temperature (Tmax) on August 16 of 2020. A: Tmax
over 6,785 weather stations from the GHCN database falling over CONUS. B:
Assigns value of weather station closest to a county’s centroid. C: Inverse-
distance spatial interpolation based on the 5 nearest stations to the county’s
centroid. D: Inverse distance interpolation based on stations located within
1° of the a county’s centroid.

Figure 3: Spatial interpolation of weather station data.

Suppose we are interested in creating a county-level dataset based on these
weather stations. This means converting these 6,000 station-level observations
to a little more than 3,000 counties. Perhaps the most basic type of mapping
is to simply assign to a county the value of the weather station that is
nearest to the centroid of a county. This is precisely what is shown in Fig.
3B. This approach may work better in areas with homogeneous landscapes and no
mountain ranges. But this approach fully reflects any local noise stemming
from a particular weather station.

An alternative approach is to perform a weighted average of nearby weather
stations. Figure 3C shows this procedure for the $k$ nearest stations to a
county’s centroid (with $k=5$). In computing these weighted averages, the
weights are proportional to the inverse of the distance to the county’s
centroid. This way stations that are closer to the county are weighted more
heavily. Naturally, these weights must add to unity. This approach smooth out
local noise from any particular weather station. The downside, obviously, is
that it may smooth out temperature too much. especially over location with
major spatial climatic differences over short distances (e.g. California). In
addition, this approach may be problematic if weather stations are extremely
sparse. In such cases this approach could potentially over-smooth weather
conditions. One potential fix is to reduce the number $k$ of neighboring
stations used in the interpolation. In general, this approach appears more
suitable to situations with relatively dense or homogeneous network of weather
stations.

In other contexts, relying on all the weather stations within a certain
distance of a county could be preferable. That is the strategy used in
generating Fig.3D. In that interpolation, I chose all stations within 1° (~111
km) and relied on an inverse-distance weight. This procedure may be
appropriate when the station distribution is highly sparse and heterogeneous.
If the distance cutoff is too large, this approach may also over-smooth
weather conditions in regions with contrasting climates over short distances
(e.g. California). However, setting a distance cutoff that is too small may
lead to missing observations if station density is sparse in certain parts of
the study region.

I should note that interpolating temperature variables is prone to smaller
errors than precipitation. Temperature is a smoother field and is more
amenable to these procedures. Precipitation, on the other hand, is spatially
discontinuous, particularly at finer temporal scales (e.g. daily), so
interpolation can introduce substantial error at fine spatial scales.

In some circumstances a researcher might be interested in harnessing high
spatial resolution from one dataset with the higher temporal resolution of
another. For instance, the PRISM data is available at a daily scale starting
in 1981. But the monthly PRISM data starts in 1896. What if we were interested
in deriving a daily PRISM-like dataset prior to 1981 based on the PRISM grid?
That is precisely what Schlenker and Roberts (2009a) did, where they harnessed
the high spatial resolution of the monthly PRISM data over 1950-2005, and
combined with with daily weather station data over the same period. Their goal
was to derive the daily distribution of temperature at the fine spatial scale
of the PRISM grid.

Here I illustrate how to do this for daily maximum temperature (Tmax) and
precipitation (ppt) using the monthly 4-km dataset from PRISM and daily but
relatively sparse weather station data from the GHCN database. I focus our
attention on weather conditions over CONUS for August 16, 2020 and present the
procedure step by step.

The first step is to interpolate daily weather station to the PRISM grid. The
goal of this step is to “infuse” the PRISM grid cells with the temporal
variation from the weather station data. Note that the PRISM grid has 872,505
cells of which 481,631 fall over land, whereas the GHCN database has maximum
temperature data for 6,028 stations and precipitation data for 14,492
stations.444The weather stations counts concern balanced weather stations for
the month of August of 2020. This means that this interpolation has a larger
“target” dataset (PRISM grid) than the “source” dataset (stations). This will
expectedly lead to fairly smoothed out patterns at a very fine scale and might
not respect fine-scale spatial differences arising due to orography or
proximity to the coasts. Specifically, I select the 5 nearest weather stations
for each PRISM grid cell and compute an inverse-distance weighted average of
each variable. This is repeated for each PRISM grid cell, noting that the
weather station population varies by variable. This step leads to 31 daily
interpolated layers for each variable for the month of August of 2020.

The second step consists in ensuring that the daily interpolated PRISM data
are consistent with the monthly PRISM data. That is, we constrain the daily
data so that when aggregated it matches exactly the monthly PRISM data. To do
this for temperature, I start by subtracting monthly average from the daily
interpolated data layers. Essentially, this only leaves the local anomalies in
the interpolated data. I then proceed to add the monthly average from the
reference PRISM grid to these daily interpolated anomalies. This reference
monthly file has a finer and more accurate rendering of spatial climatic
differences. For precipitation, I scale the daily precipitation variables
based on the ratio of the total precipitation from the daily interpolated
files and the reference monthly PRISM data layer. This procedure ensures that
I do not create “ghost” rainfall in places and days where it did not rain.

Notes: All maps show maximum temperature (Tmax) on August 16 of 2020. A: Tmax
over 3,431 weather stations from the GHCN database. B-C: Inverse-distance
spatial interpolation based on the stations located within 0.5 and 1° of the
each county’s centroid. D: Inverse-distance interpolation based on the 5
nearest stations to each county’s centroid.

Spatial interpolation of weather station data.

I summarize the results of this interpolation in figure 4.1. Fig. 4.1A shows
the resulting interpolated maximum temperature for August 16, 2020. Notice
that this interpolation technique preserves very fine-scale spatial variation
close to mountains and coasts. As previously mentioned, PRISM provides daily
data since 1981 so we can contrast our interpolation with theirs, shown in
panel B. The spatial patterns are very similar, but not identical as indicated
in panel C showing the difference between panels A and B. The PRISM team
likely operates a more sophisticated technique than what I employ here.

Figure 4.1D shows the interpolation for daily precipitation for the same day.
The spatial patterns of precipitation appear slightly less realistic than the
daily PRISM data shown in panel E, but the differences are surprisingly small
as indicated in panel F. However, note that differences can be very large
(e.g. shows rain in panel D and no rain in panel E) in certain places due to
differences in interpolation techniques.

While simple interpolation techniques like this appear to work, I invite the
reader to exhibit extreme caution when performing these calculations. The
output from these procedures is not validated which means that we have a very
limited understanding of the type of the underlying measurement error.
Advanced users are invited to seek additional resources in kriging and more
advanced geospatial statistic tools.

###  4.2 Aggregation of gridded weather data

Here we turn our attention on how to efficiently aggregate gridded weather
data by administrative units (e.g. counties). As discussed in section 2.3,
gridded or raster data are matrices with data on a particular variable (e.g.
precipitation) linked to information on how the cells of the matrix map into
into space and time. These are essentially geographically and temporally
referenced matrices. This characterization is important because it means that
manipulating such data can be greatly improved by relying on matrix algebra
which are performed efficiently by programming software like R. Weather raster
files can come in a variety of formats but perhaps the most common is NetCDF.
These are typically multi-layer raster files in which each layer corresponds
to a time period. For instance, it is common to see daily weather datasets
provided as multi-layer annual files (1 file with 365 layers).

The way that each grid cell in a raster is mapped into a particular location
on Earth is determined by a few parameters, including the extent, spatial
resolution, and projection. The extent defines the “boundaries” of the raster.
Because rasters are rectangular, the extent is simply a matrix providing
information on the southern, northern, western and eastern limits of the
raster. These limits are typically provided in degrees (but not always). A
global dataset can thus range from -180° W to +180° E, for rasters centered in
Europe, and from -180° S to 180°N, although it is rare to see rasters extend
to the poles. The spatial resolution of the raster simply indicates the size
of a grid cell in degrees, although sometime also in meters. Finally, the
projection is the rule by which the surface of the Earth is flatten for the
purpose of visualization. Gridded weather datasets are generally “unprojected”
using a simple latitude-longitude (or lat-lon) projection. This portrays the
gridded surface of the earth as a flat rectangle on a computer screen.
Importantly, changing the projection of a raster can be time consuming (at
least in R) because the boundaries of each grid cell need to be reprojected.
As a result, it is often preferable to change the re-project polygon data
(e.g. administrative boundaries) to match that of the raster data, and not the
other way around.

The key point to efficiently map gridded data to administrative units is to
recognize gridded data as a matrix. A multi-layer raster containing daily
information for $N$ grid cells and $T$ time periods can be through of as an
$N\times T$ matrix, which I denote $G$ (for gridded). The target aggregated
dataset would be a $n\times T$ matrix, which I denote $A$ (for aggregated or
administrative). We usually have $n\ll N$, but not always (more on this
later). The “trick” is to consider the transformation of $G$ into $A$ as a
matrix multiplication of the form:

$\underset{n\times T}{A}=\underset{n\times N}{P}\times\underset{N\times T}{G}$

Matrix $P$ is a transformation or projection matrix that converts the gridded
data $G$ into aggregate data matrix $A$. That transformation is simply a
weighted average of rows in $G$ to obtain aggregate matrix $A$. To clarify
this further, the first row of $P$ corresponds to the first administrative
unit in the aggregated dataset (e.g. a country, state, county, etc.). The
first column of $P$ corresponds to the first grid cell in the gridded dataset.
Thus the first row corresponds to a vector of weights for computing a weighted
average of grid cells in $G$ in order to obtain the data for the first
administrative unit, which is located on the first row of $A$.Values in each
row in $P$ must add to 1, and only grid cells falling within the
administrative unit should have non-zero positive values.

That means that $P$ is mostly full of zeros and could be adequately
represented as a sparse matrix on a computer. For a sparse matrix, a computer
only holds the values and location of non-zero values in the memory. This is a
memory efficient way of storing large matrices and for computing fast matrix
multiplications.

The projection matrix $P$ can be created in various ways. Perhaps the most
basic approach is to give equal weight to every grid cell falling within each
administrative boundary. However, as we saw in Fig. 2, giving the same weight
to every location within a county or state may be misleading because it does
not convey information about the locations within the administrative units
where the economic activity of interest operates or relies upon. This is
particularly critical for large administrative areas like countries, or large
geographically diverse states, like in the Western parts of the US.

A more common approach of constructing the projection $P$ matrix is for it to
reflect weights that correspond to areas where relevant economic activities
take place. For instance, in the case of economic studies exploring the effect
on overall economic activity, it is common to see gridded weather data
aggregated over areas where population are located within the administrative
units. In the case of agriculture, it is common to rely on cropland, or
cropland and pastures, as the underlying land cover information to construct
such weights. For global studies one can rely on global datasets describing
the global distribution of agricultural lands (e.g. Ramankutty et al., 2008).
For more regional studies, one can rely on finer scale land cover datasets. In
the US context, it is common to rely on either the USDA NASS Cropland Data
Layer (CDL) or USGS National Land Cover DataBase (NLCD). The CDL is an annual
product with 30m resolution and is differentiated by crop. The NLCD is also
30m but has broad land cover categories as it pertains to agriculture (e.g.
cropland or pasture).

The advantage of performing spatial aggregation in this manner is that the
researcher only needs to compute $P$ once. Matrix $P$ is essentially a matrix
of aggregation weights. The spatial aggregation can be done in a fraction of a
second by performing the matrix multiplication above because these matrix
operations can be done very fast with current computers. The main bottleneck
in the spatial aggregation is reading matrix $G$ into the memory and perhaps
writing matrix $A$ to the disk. This workflow presents a substantial gain in
speed when aggregating very large datasets with either high spatial or
temporal resolution. The alternative typically consists of using canned
functions that perform overlays of raster and polygons for each raster layer.
Such strategies have a substantial overhead that make the rapid aggregation of
large datasets infeasible.

Notes: The cropland, pasture and grassland shares are computed based on 30m
land cover data from the National Land Cover Database (NLCD) for 2016.

Figure 4: Fraction of cropland, pasture and grassland in each PRISM gridcell.
4

The R code included with this chapter provides an illustration of how to
compute the matrix $P$ based specific types land cover falling within each US
county to aggregate PRISM data (see 1_weather_data.R). The first step is to
create a new land cover raster that matches the target PRISM raster grid. This
aggregation can be done by computing “zonal statistics” which is simply
computing the frequency of small 30m land cover pixels falling within each
PRISM grid cell. Fig. 4 shows the fraction of each PRISM grid cell covered by
cropland, pasture and grassland. As expected, areas in the Midwest and the
Central Plains are mostly allocated to these classes. The second step is to
derive a vector of weights that sum to 1 within each county, based on the land
cover fractions of the grid cell falling within each county. The third step is
to store these aggregation weights in a sparse matrix $P$. The final step is
to perform the matrix multiplication to perform the aggregation.

Importantly, note that the interpolation of weather station data described in
subsection 4.1 can also be conducted using sparse matrices. In fact, the
implementation shown in the R code provided does precisely that. In other
words, performing averages of observations over nearby weather stations does
not require a loop. Doing a loop would be tremendously inefficient and
possibly make the project intractable for large datasets. The aggregations
described here are simply linear transformations of matrices, whether the
source matrix represents raster or point data, and where the transformation
matrix $P$ is a sparse matrix with aggregation weights.

###  4.3 Estimating non-linear effects

Now that we have covered the basics of how to aggregate weather data to
administrative units, I focus our attention to the estimation of nonlinear
effects of temperature. Specifically, I illustrate how to estimate the semi-
parametric model introduced in Schlenker and Roberts (2009a). Rather than
estimating the effect of temperature on crop yields, that study examined the
effect of exposure to different levels of temperature on crop yields. I
emphasize the term exposure, because the underlying variables in the analysis
are actually measures of the amount of time spent at various temperature
intervals or “bins”.

The motivation behind this approach is that temporal averaging of temperature
conceals the exposure to extreme temperatures. Two temperature sequences may
have the exact same average, but one may exhibit considerably more exposure to
high temperature. The underlying hypothesis is that exposure to various levels
of temperature affect agricultural outcomes (e.g. crop yield) very
differently, so capturing the varying effects of exposure to the entire
temperature distribution may yield deeper insights.

Capturing such linearities requires information on how much time is spent at
each temperature level or “bin”. These bins are typically 1°C wide, ranging
from say 0° to 40°C. Intra-daily (e.g. hourly) data is often impossible to
obtain in a reliable fashion so deriving these intra-daily distributions often
requires making assumptions about the temperature-time path. A common
assumption is that temperature follows a sine curve passing between the
minimum and the maximum temperature of each day. This naturally requires data
on daily minimum and maximum temperature.

Notes: The figure represent minimum (Tmin) and maximum (Tmax) temperature data
from a random location in the lower 48 states from the PRISM dataset during
the month of August of 2020. The grey dots describing a sine curve between
consecutive Tmax and Tmin are obtained via interpolation every 15 minutes.
Temperature intervals of 1°C are highlighted in dashed lines over the entire
month. The green distribution on the right shows the underlying temporal
distribution of temperature throughout the month. It is essentially a
histogram of the 15-min points that I just described. The distribution adds to
the total number of hours (or days) in the month.

Figure 5: Illustration of the construction of temperature bin exposures from
daily minimum and maximum temperature.

The R code provided illustrates how to build exposure data from a sequence of
daily minimum and maximum temperature. The results for a random location in
the US is shown in Fig. 5. The figure shows the daily sequence of Tmax and
Tmin with in red and blue points, respectively. To this I add an interpolated
intra-day temperature trajectory. The code generates a series of points at
15-minute intervals based on a double sine curve that passes through Tmax and
Tmin of consecutive days. Computing the exposure bins simply consists in
determining the frequency of these 15-minute interval points throughout the
month. This distribution is shown in green on the right side of the figure.
Note that the support of this distribution is time and is measured in hours or
days, not °C. The distribution describes how much time is spent in each
temperature interval. By construction, summing over all the bins over a month
adds to the number of hours or days in that month. Importantly, note that the
temperature “bins” described here are not counts of days in which the average
temperature falls in a particular interval.

It is important to emphasize that these bins we are discussing are exposure
bins. They are not dummy variables that count whether the average temperature
fell within a particular bins. That approach would not capture within intra-
day variation in temperature. This is a common point of confusion. In
addition, note how averaging temperature over time, say over the entire month,
would conceal exposure to extreme temperatures within the month.

We can perform the “binning” exercise for all grid cells located over the US
and for each month of the year. In the code provided I do this for every month
over the 1981-2020 period for bins ranging from $-10$ to $50$°C in 1°C
intervals. For each month, the code creates a file with 61 layers (one per
bin) corresponding to the amount of time spent in each bin. I illustrate the
exposure above 30°C for August of 2020 in Fig. 6. This is essentially showing
the sum of all the exposure bins above 30°C in August (depicted for one grid
cell in Fig. 5) over all PRISM grid cells. In the map you can appreciate the
importance of latitude and orography in explaining cross-sectional differences
in temperature exposure. For instance, certain parts of the Southwest
experienced more than 720 hours above $30$°C. August has 31 days or 744 hours,
meaning that some of these regions experienced temperatures above $30$°C over
virtually every single moment of the month. In contrast, other regions of the
country experienced little to no exposure beyond $30$°C (dark blue).

Notes: The derivation of this exposure relied on daily data from PRISM.

Figure 6: Map of exposure above 30°C in August of 2020.

Note that these exposure bins for temperature are related to the degree-day
concept discussed in subsection 2.5. In fact, degree days can be computed
directly from these exposure bins. Specifically, computing degree-days between
a low threshold $\underline{h}$ and a high threshold $\bar{h}$ can be written
as:

$DD_{\underline{h}\textrm{ to
}\bar{h}}=\sum_{k=\underline{h}}^{\bar{h}-1}z^{k}\times\left(\underline{h}-30+1\right)$

where $z^{k}$ is the exposure (e.g. in days) to the $k$-th temperature bin.
For instance, when computing growing degree-days between 8 and 32°C, we would
set set $\underline{h}=8$ and $\bar{h}=32$. When computing “extreme” degree-
days, say above 30°C, we would set $\underline{h}=30$ and $\bar{h}$ at a very
high level that is never reached (e.g. 60°C). In essence, the expression above
approximates the area under the temperature curve and between a lower and an
upper threshold.

Now let’s move to how to represent the effect of temperature exposure on crop
yield following the approach laid out in Schlenker and Roberts (2009a). The
underlying data generating process presented in that study is of the form:

$y_{it}=\intop
g(h)\phi_{it}(h)d(h)+p_{it}+p_{it}^{2}+\psi(t)+\alpha_{i}+\epsilon_{it}$

where $y_{it}$ is the log of yield in location or county $i$ and year $t$,
$p_{it}$ is growing-season precipitation, $\psi(t)$ is a time trend,
$\alpha_{i}$ is a county fixed effect and $\epsilon_{it}$ is the error term.
The first term represents the effect of temperature on crop yield, where
$g(h)$ is the marginal effect of temperature $h$ on yield, and $\phi_{it}(h)$
is the growing-season density at $h$ in that location and year. The integral
simply means that the product of marginal effect and exposure is summed over
the entire temperature range, given that the integrating variable is $h$. This
continuous representation is theoretical and is not tractable for estimation.
However, that integral can be approximated in various ways empirically. In the
original study, the authors provided 3 ways of approximating this function,
using a piece-wise linear function, a step function and a Chebyshev polynomial
of degree 8.

In this chapter and in the included R code I illustrate how to estimate these
models using step functions, natural cubic splines, and Chebyshev polynomials
(see 2_nonlinear_effects.R). I focus on US corn yields for the 1981-2020
period east of the 100th meridian West in order to focus on mostly rainfed
counties. I set a growing season ranging from August to September. I also
bottom and top code the exposure data so that bins range from 0 to 38°C. This
avoid having too little exposure at the tails of the temperature distribution,
which makes the estimation noisier.

Perhaps the most basic and intuitive approximation is to use a step function.
If the temperature range during the growing season ranges from say 0 to 40°C,
then the approximation with eight 5°C steps can be represented as:

$y_{it}=\sum_{k=1}^{8}\beta_{k}z_{it}^{k}+p_{it}+p_{it}^{2}+\psi(t)+\alpha_{i}+\epsilon_{it}$

where $z_{it}^{k}$ is the amount of time spent in the $k$-th step or
temperature interval. This means that we essentially estimate separate
coefficients for each temperature interval. The regression coefficients are to
be interpreted as the effect on yield of spending an additional hour (or day)
in that particular bin.

Figure 7 illustrates the marginal effects of models with step functions of
different widths. In each panel the blue line represents the estimates for
each step together with 95 and 99 percent confidence intervals in blue. The
green distribution underneath the response function describes the growing-
season temporal distribution of temperature exposure in the sample. By
construction, that distribution sums to 183 days between April and September.

The first panel of Fig. 7 shows the response function with 1°C steps. This
leads to a fairly noisy response function, particularly toward high levels of
temperature. As a result, although the point estimates suggest exposure to
temperature above 30°C appear detrimental, these effects are not statistically
different from zero. This imprecision likely results from the high degree of
collinearity between neighboring bins. Indeed, the time spent on any given
year between 31 and 32°C is always very correlated to the time spent between
32 and 33°C, and so on for other bins. This is problematic and thus model like
this are better avoided. The role of bin size has also been discussed in
Carter et al. (2018).

One natural way to avoid collinearity is to aggregate bins into wider steps or
intervals. The second and third panels in Fig. 7 show the response functions
for models based on steps that are 3° and 7°C wide, respectively. As expected,
the resulting response functions are more precisely estimated. Both model also
how that temperature exceeding 30°C are detrimental to corn yield. For
instance, the last model indicates that an additional day of exposure to the
last bin over the growing season (183 days) reduces crop yield by 0.05 log
points, or about 5%. That is a substantial reduction.

Notes: Standard errors are clustered at the state and year level. The data
covers US corn yields over the 1981-2020 period east of the 100th meridian
West. Precipitation variables were included in the regression but coefficients
are not shown. The blue colored bands around the response function corresponds
to 95 and 99 percent confidence intervals. The green histogram represents the
growing-season of exposure to all temperature bins.

Figure 7: Effects of temperature exposure on corn yields based on step
functions of varying widths.

Note that the step function approach assumes that the marginal effects between
neighboring steps are unrelated. That is, the model does not impose any
structure on how smooth the response function could be, which could render the
estimation unnecessarily noisy.

One way to address this is to allow marginal effects to vary smoothly across
neighboring temperature bins. This can be implemented with a natural cubic
spline or a Chebyshev polynomial. Both of these approaches involve a basis
matrix $B$ which is used to project the temperature bins (say $J$ bins) into a
smaller space (say of size $K$) and thus reduce dimensionality prior to
estimation ($K<J$). That is, we are able to estimate only $K$ parameters to
represent the marginal effects of $J$ individual bins.

To illustrate, the basis matrix $B$ of a natural cubic spline with $J$ degrees
of freedom evaluated over $K$ temperature bins has $K$ rows and $J$ columns.
The basis matrix maps a $nT$-by-$K$ matrix $Z$ of data, with columns
representing temperature bin exposures, into a $nT$-by-$J$ matrix $X$ of
transformed variables used in the regression analysis. In matrix form, this
mapping can be represented as:

$\underset{nT\times J}{X}=\underset{nT\times K}{Z}\times\underset{K\times
J}{B}$

This essentially reduces the underlying binned data with $K$ bins to $J$
regressors. Naturally, we select $J\ll K$ in order to substantially reduce the
dimensionality of the temperature space. We can now write the regression model
in the following form:

$\displaystyle y_{it}$
$\displaystyle=\sum_{j=1}^{J}\sum_{k=1}^{K}\left(\gamma_{j}B_{j}^{k}z_{it}^{k}\right)+p_{it}+p_{it}^{2}+\psi(t)+\alpha_{i}+\epsilon_{it}$
$\displaystyle=\sum_{j=1}^{J}\gamma_{j}\underbrace{\sum_{k=1}^{K}B_{j}^{k}z_{it}^{k}}_{x_{it}^{j}}+p_{it}+p_{it}^{2}+\psi(t)+\alpha_{i}+\epsilon_{it}$

where $B_{j}^{k}$ is the element in the $k$-th row and $j$-th column of the
basis matrix $B$. The term $z_{it}^{k}$ corresponds to one row (observation
$it$) and the $k$-th column of $Z$, and $x_{it}^{j}$ corresponds to one row
(observation $it$) and the $j$-th column of $X$. That is, rather than
estimating $K$ separate coefficients for each individual bin $z^{k}$, we end
up with only $J$ regressors $x^{j}$.

After the estimation, one can recover the marginal effects evaluated at each
of the $K$ bins by pre-multiplying the vector $\hat{\Gamma}$ (containing the
$J$ estimated coefficients) by the basis matrix:

$\underset{K\times 1}{\hat{\beta}}=\underset{K\times
J}{B}\times\underset{J\times 1}{\hat{\Gamma}}$

This operation returns a vector $\hat{\beta}$ of temperature effects evaluated
at each one of the $K$ original temperature bins in matrix $Z$. One can also
easily derive an estimate of the variance of these temperature effects as
follows:

$\underset{K\times K}{Var(\hat{\beta})}=\underset{K\times
J}{B}\times\underset{J\times J}{Var(\hat{\Gamma})}\times\underset{J\times
K}{B^{{}^{\prime}}}$

Obtaining standard errors for the marginal effects consists in selecting the
squared root of the diagonal elements of $Var(\hat{\beta})$.

Notes: Standard errors are clustered at the state level. The data covers corn
yields over 1981-2020 east of the 100th meridian West. Precipitation variables
were included in the regression but coefficients are not shown. The blue
colored bands around the response function corresponds to 95 and 99 percent
confidence intervals. The green histogram represents the growing-season of
exposure to all temperature bins.

Figure 8: Effects of temperature on corn yields based on natural cubic
splines.

To fix ideas, I illustrate this estimation technique using cubic natural
splines with different degrees of freedom in Fig. 8. Each panel in the top row
shows the the columns of the basis matrix $B$ for splines with 3, 7 and 12
degrees of freedom. With temperature bins defined over the 0 to 38°C, the
basis matrices have sizes of $39\times 3$ , $39\times 7$ and $39\times 12$,
respectively. The top row shows these basis matrices performs linear
transformation of neighboring bins that are mapped into a reduced number of
regressors. This means that each regressor roughly reflects the effects of
fluctuations in temperature exposure occurring in neighboring bins.
Importantly, these regressors can be transformed “back” to the original
support, as previously mentioned and as illustrated in the second row of the
figure.

Each panel in the bottom row of Fig. 8 shows the response functions (the
vector $\hat{\beta}$ derived above) along with 95 and 99 percent confidence
bands (based on $Var(\hat{\beta})$) corresponding to these splines with
varying degrees of freedom. I also depict the underlying distribution of
temperature exposure over the growing season in green.

The first panel of Fig. 8 corresponds to a spline with 3 degrees of freedom.
This response function clearly exhibits limited flexibility. This is evident
by the symmetric response function shown on the lower left panel. Note,
however, how the spline with 7 degrees of freedom allows for a more flexible
estimation of the response function. This response function shows that
exposure above 30°C are clearly detrimental to crop yields.

Moving to the most flexible specification with 12 degrees of freedom, shows a
response function that is a bit more unstable but still shows a distinct
pattern where exposure to temperature exceeding 30°C are clearly detrimental.
Note that the response function because more imprecise at high levels of
temperature. This is likely driven by the high flexibility of the model over a
temperature range with relatively little variation.

Notes: Standard errors are clustered at the state level. The data covers corn
yields over 1981-2020 east of the 100th meridian West. Precipitation variables
were included in the regression but coefficients are not shown. The blue
colored bands around the response function corresponds to 95 and 99 percent
confidence intervals. The green histogram represents the growing-season of
exposure to all temperature bins.

Figure 9: Effects of temperature on corn yields based on Chebyshev
polynomials.

It is also possible to estimate these temperature effects based on a
Chebyschev polynomial. The procedure is very similar to a spline given this
strategy also involves a basis matrix. The first row of Fig. 9 shows the
columns of the associated basis matrix for Chebyshev polynomials of degree 3,
7 and 12. Unlike for the spline, note that the values are not locally defined.
What that means is that each regressor carries information regarding exposure
to all bins of the temperature distribution. As a result polynomials tend to
be a bit less table around the extremes than splines. The bottom row of the
figure shows the response functions and the underlying distribution of
temperature exposure. Similar to the spline, we find that exposure above 30°C
appears detrimental to corn yields. Note how allowing too much flexibility in
the polynomial lead to much noisier effects around the right tail of the
distribution.

Overall, this section shows that allowing too much flexibility in the semi-
parametric response function leads to relatively noisy effects at the extreme
end of the temperature distribution. We also see that not allowing for enough
flexibility tends to understate the effects of extreme temperatures.

I should highlight that there has not been a formal exploration of the
advantages of these semi-parametric approaches. These models tend to fit the
data better than alternative models based on average temperature. But it is
unclear to what extent these approaches could reduce issues related to spatio-
temporal aggregation bias.

To conclude, note that an important assumption of the approach presented here
is that the effects of temperature exposure are additive throughout the
growing season. In other words, the timing of temperature exposure is
irrelevant. This obviously contradicts agronomic conventional wisdom
stipulating that the timing of weather conditions is particularly important
for crop yield determination. I now turn to a generalization of this model to
allow for time-varying effects within the growing season.

###  4.4 Estimating within-season varying effects

The timing of environmental conditions plays a critical role in agriculture
particularly in crop production (Fageria et al., 2006). In the case of many
field crops like cereals and leguminous crops, the ability of plants to store
biomass in useful parts of the plant (e.g. grain) is in large part determined
by the success of their flowering process. If the flowering process falters,
then the plant loses its ability to store biomass in seeds (grain fill).
Importantly, flowering is a delicate stage of plant development that is fairly
vulnerable to environmental stresses. Flowering also occurs over a relatively
short period of time around the middle of the crop cycle in annual crops,
meaning that environmental conditions can have drastically different effects
throughout the growing season.

So how does this affect the estimation of statistical crop yield models? Most
standard statistical yield models do not consider the timing of environmental
conditions in great detail. For instance, the models we explored in the
previous subsection assume additivity of weather effects on yield. That means
that the timing of environmental conditions within the season are irrelevant.

There have been various efforts over the years to account for these within
season time-varying effects. The most basic approach is to simply include
monthly weather variables for various critical period of the growing season.
However, those approaches have been found not to substantially improve model
fit or lead to significantly different conclusions than models that assume
additivity (e.g. see the appendix in Schlenker and Roberts, 2009b). However,
these efforts commonly rely on calendar periods of the year rather than actual
stages of crop development (e.g. Gammans et al., 2017).

Perhaps the earliest economic study exploring climate change impacts on crop
yields that accounts for biophysical features underlying the non-additivity of
weather in the growing season is Kaufmann and Snell (1997). In that study, the
authors define weather variables over periods corresponding to crop
development stages. Many more studies have adopted this approach. For
instance, Ortiz-Bobea and Just (2013) estimates a corn yield model with 3 sub-
seasons matching crop stages to analyze the effectiveness of changing planting
as an adaptation to rising damages from a warming climate. Another example is
Welch et al. (2010) that matches weather conditions to vegetative and ripening
phases of rice. In a recent study, Shew et al. (2020) link weather conditions
to wheat development stages to analyze variations in sensitivity to extreme
heat across cultivars in South Africa.

One potential limitation of previous work is that it assumes that the effects
between neighboring portions of the growing season are independent. That is,
nothing in the modeling approach allows for marginal effects of weather
conditions to vary smoothly within the growing season itself.

To address this limitation Ortiz-Bobea et al. (2019) introduce a bi-
dimensional spline that allows the effects of soil moisture and temperature to
vary smoothly in levels and throughout the growing season. This model is akin
to a bi-dimensional generalization of the crop yield model presented in the
previous subsection. We now not only have exposure bins to various levels of
environmental variables, but time within the season also becomes a “bin”. So
rather than fitting a spline over a vector of binned exposures, this approach
applies a tensor spline to a 2-dimensional set of bins.

That study introduces a conceptual model similar to the following, where crop
yield may be differently affected by the distribution of environmental
conditions throughout the growing season:

$y_{it}=\intop\intop
g(h,p)\phi_{it}(h,p)d(h)d(p)+p_{it}+p_{it}^{2}+\psi(t)+\alpha_{i}+\epsilon_{it}$

where $\phi_{it}(h,p)$ describes the distribution of the environmental
variable $h$ at each level of progress $p$ in the growing season, and $g(h,p)$
describes the marginal effect of the environmental variable throughout the
season. Although he I only show one environmental variable, the study
considers soil moisture and air temperature. The rest of the specification is
analogous to the specification in the previous sub-section.

Similarly, one cannot estimate this model with a double integral, but one can
approximate the $\phi_{it}(h,p)$ with 2-dimensional bins and then employ a
semi-parametric technique to estimate $g(h,p)$. Again the 2 dimensions are
progress in the season and the level of the environmental variable. For
example, in the case of temperature throughout the growing season, we can
define exposure bins over 1°C temperature intervals every week of the growing
season. In this example we thus have 1°C bins in the “temperature” dimension,
and weekly bins in the “season progress” dimension.

Let me illustrate how to construct the tensor spline necessary to reduce the
dimensionality of these 2D bins. Let’s start by considering the basis matrix
for a natural cubic spline for the environmental variable, which constitutes
the first “dimension”. Let’s denote this matrix $B_{1}$. With $J_{1}$ degrees
of freedom evaluated over $K_{1}$ bins, this matrix has a dimension of $K_{1}$
rows and $J_{1}$ columns. These bins can be 1°C interval if we are dealing
with temperature. Similarly, let’s define a basis matrix $B_{2}$ for the
second “dimension” which is the time or progress within the growing season.
With $J_{2}$ degrees of freedom evaluated over $K_{2}$ bins, this matrix has a
dimension of $K_{2}$ rows and $J_{2}$ columns. The associated can be weeks or
similarly short time intervals within the growing season.

The tensor basis matrix is constructed based on the kronecker product of these
two basis matrices such that $B=B_{1}\varotimes B_{2}$ is a matrix with
$K_{1}K_{2}$ rows and $J_{1}J_{2}$ columns. Essentially, our 2D bin space has
$K_{1}K_{2}$ bins ($K_{1}$ in the first “variable” dimension and $K_{2}$ in
the second “season progress” dimension) and the goal of this tensor basis
matrix is to reduce the dimensionality of this space prior to estimation. This
means that our binned data can be stored in a matrix $Z$ with $nT$ and
$K_{1}K_{2}$ columns. Each row of that matrix corresponds to one observation
(e.g. a county-year $it$). Thus the tensor basis matrix $B$ maps a $nT$ by
$K_{1}K_{2}$ matrix $Z$ of 2-dimensional binned data, with columns
representing variable and progress bin exposures, into a $nT$ by $J_{1}J_{2}$
matrix $X$ of transformed variables used in the regression analysis. In matrix
form, this mapping can be represented as:

$\underset{nT\times J_{1}J_{2}}{X}=\underset{nT\times
K_{1}K_{2}}{Z}\times\underset{K_{1}K_{2}\times J_{1}J_{2}}{B}$

One way to visualize how this dimensionality reduction works is to extract one
row of $Z$, let’s call it $z_{it}$ with dimension $1$ by $K_{1}K_{2}$, and
organize it in “two dimensions” in a matrix $z_{it}^{2D}$ of dimension
$K_{1}\times K_{2}$. This essentially re-arranges the bins so that we have
$K_{1}$ rows of environmental variable bins, and $K_{2}$ columns of season
progress bins. What the tensor spline is essentially doing is the following
transformation:

$\underset{J_{1}\times J_{2}}{x_{it}^{2D}}=\underset{J_{2}\times
K_{2}}{B_{1}^{{}^{\prime}}}\times\underset{K_{1}\times
K_{2}}{z_{it}^{2D}}\times\underset{K_{1}\times J_{1}}{B_{2}}$

where $x_{it}^{2D}$ is the transformed variable for observation $it$
rearranged in two dimensions.

The regression analysis is performed with $J_{1}J_{2}$ regressors, rather than
on all the $K_{1}K_{2}$ individual bins. After the estimation, one can recover
the marginal effects evaluated at each of the $K_{1}K_{2}$ bins by pre-
multiplying the vector $\hat{\Gamma}$ (containing a vector with the
$J_{1}J_{2}$ estimated coefficients) by the tensor basis matrix:

$\underset{1\times K_{1}K_{2}}{\hat{\beta}}=\underset{K_{1}K_{2}\times
J_{1}J_{2}}{B}\times\underset{J_{1}J_{2}\times 1}{\hat{\Gamma}}$

This operation returns a vector $\hat{\beta}$ of temperature effects evaluated
at each one of the $K_{1}K_{2}$ original temperature bins in matrix $Z$. To
visualize marginal effects on a 2-dimensional space, one would have to
rearrange $\hat{\beta}$ in 2 dimensions to obtain a matrix $\hat{\beta}^{2D}$
of dimension $K_{2}\times K_{1}$. This would allow showing marginal effects
with season progress bins on the horizontal axis and the variable bins in the
vertical axis. One can also easily derive an estimate of the variance of these
temperature effects as follows:

$\underset{(K_{1}K_{2})\times(K_{1}K_{2})}{Var(\hat{\beta})}=\underset{(K_{1}K_{2})\times(J_{1}J_{2})}{B}\times\underset{(J_{1}J_{2})\times(J_{1}J_{2})}{Var(\hat{\Gamma})}\times\underset{(J_{1}J_{2})\times(K_{1}K_{2})}{B^{{}^{\prime}}}$

Obtaining standard errors for the marginal effects consists in selecting the
squared root of the diagonal elements of $Var(\hat{\beta})$. For more details,
I invite the reader to consult the supplementary data in Ortiz-Bobea et al.
(2019).

This may seem a bit too theoretical, so the attached code and data provides an
implementation for US corn yields (see 3_time-varying_effects.R). In the
example, I consider 1°C temperature bins between 0 and 35°C ($K_{1}=36$) and
monthly temporal bins between April and October ($K_{2}=7$). Ideally, we would
probably want to have finer scale temporal bins (e.g. weeks or pentads) but
the use of monthly bins makes this illustration more tractable as it makes use
of data already generated for other parts of the chapter. I also select
degrees of freedom $J_{1}=6$ and $J_{2}=3$. As a result, our binned matrix $Z$
has $36\times 7=252$ bins. Our tensor basis matrix $B$ has $6\times 3=18$
columns. All these steps are clearly annotated in the R script file.

Notes: Standard errors are clustered at the state and year level. The data
covers corn yields over 1981-2020 east of the 100th meridian West. Panel A
shows marginal effects of additional exposure to temperature bins in different
parts of the growing season. Marginal effects that are statistically different
from zero at a 95% level are indicated with a star (*). Panel B shows the
density (in days) in each one of the $36\times 7=252$ bins. These bins
vertically add to the amount of time in each month. Summing across all months
and temperature levels adds up to the total number of days in the growing
season (183 days).

Figure 10: Time-varying effects of temperature on US corn yields throughout
the growing season.

Figure 10A shows the marginal effects in their 2-dimensional form
($\hat{\beta}^{2D}$). The panel shows that high temperature above 30°C appear
detrimental especially in the months between June and August. Marginal effects
that are statistically different from zero at a 95% level are highlighted with
a star. In contrast, high temperatures appear beneficial early in the season
in April, although I later show that result is barely significant.

Figure 10B shows the underlying density over the $36\times 7=252$ bins. As
expected, the distribution of temperature in April and and October is toward
lower temperature, whereas the distribution is toward higher temperatures in
the summer months. This seasonality will naturally affect the precision of
estimates for temperature bins that exhibit little to no exposure over certain
parts of the year. For instance, there is little exposure above 30°C in the
months of April and October. This should render the estimation of marginal
effects around that time particularly noisy.

To explore this, Fig. 4.4 shows a cross-section of marginal effects evaluated
at each temporal bin, that is, at each month. The marginal effects in April
and October at high levels of temperature are rather noisy and are not
significant at a 99% confidence level. Notice there is little to no density
(green histogram) at high levels of temperature in those months. In contrast,
the marginal effects in July and August are clearly significant at a 99%
confidence level and those effects appear more precisely estimated.

Notes: Standard errors are clustered at the state level. The data covers corn
yields over 1981-2020 east of the 100th meridian West. Precipitation variables
were not included. The blue colored bands around the response function
corresponds to 95 and 99 percent confidence intervals. The green histogram
represents the monthly exposure to all temperature bins.

Time-varying effects of temperature on US corn yields evaluated at each month
of the growing season.

This subsection illustrates how one can tractably harness variation in
environmental variables within the growing season to estimate within-season
time-varying effects.

This type of model seems particularly useful when researchers seek to harness
important within-season differences in crop yield sensitivity. For instance,
this might be particularly useful for modeling changes in planting dates or
growing seasons, or for estimating high-frequency within-season yield
forecasts. Ortiz-Bobea et al. (2019) shows that this approach performs better
out of sample than their traditional counterparts that do not allow for time-
varying effects (presented in the previous subsection). However, the
differences in model fit are not enormous and the magnitude of projected
impacts of climate change (without changes in growing seasons) remain similar.
One potential reason for this is that very high temperature typically occur in
the summer when crops like corn are flowering. This means that the detrimental
effect of season-long exposure to extreme temperature in traditional additive
models could be primarily reflecting the effect of high temperature (and
drought) during the sensitive stages of crop development that happen to
coincide with the warmer summer months.

This model also has some limitations. First, the process of constructing bins
is somewhat more involved especially if constructing temporal bins finer than
a month. However, the code provided with this chapter should alleviate many of
these concerns. Another issue is that marginal effects can be fairly imprecise
around areas with little to no exposure (see 10B). So perhaps this approach
may perform better with standardized variables that exhibit similar
distribution across different points of the growing season. That would ensure
enough density throughout the entire rectangular support of the 2-dimensional
spline.

Potential future research directions may include improvements to this approach
including the implementation of penalized B-splines, also know as P-splines,
to improve model fit. For instance, the approach implemented in Ortiz-Bobea et
al. (2019) to select the flexibility of the tensor spline in both direction
was based on a relatively onerous grid search. There are more advanced
approaches to implementing these techniques. It might also be possible to
explore multi-dimensional splines that allow smooth interactions between
environmental indicators throughout the growing season. However, this is
likely to require large datasets and possibly experimental data to obtain
independent draws of environmental indicators which tend to be highly
correlated in observational settings (e.g. high temperature and drought).

For additional reader regarding the use of high-frequency weather data for
analyzing climate change impacts I invite the reader to consult Ghanem and
Smith (2020). This also relates to a broader literature on model selection
(see Cui et al., 2018).

###  4.5 Spatial dependence

A common characteristic of agricultural data is that neighboring locations
exhibit similar values. For instance, a map of crop yields, farm profits or
farmland values all appear spatially correlated even at large spatial scales.
There are multiple reasons for this. One of them has to do with the spatial
dependence of land characteristics such as soil texture, slope or climate.
These spatially-dependent factors influence the performance of agriculture
which in turns also ends up exhibit spatial dependence. In addition, socio-
economic factors can also play a role including local and regional
regulations, the presence of irrigation or transportation infrastructure, or
the proximity to certain markets or population centers.

In practice, econometric models do not incorporate all these spatially-
dependent drivers of our agricultural outcomes of interest. As a result, these
omitted variables end up in the error term which ends up also exhibiting
spatial dependence. Moreover, weather and climate variables, which are used as
predictors, are also spatially dependent. The combination of spatially-
dependent error terms and regressors leads particular challenges in regression
analysis. For starters, the assumption that errors are independent no longer
holds. So adopting standard errors that are simply robust to
heteroscedasticity is insufficient. Ignoring positive spatial dependence in
the error term in the presence of positively spatially-dependent regressors
leads to overconfident inference. Standard errors in those regressions are
simply wider than they appear.

The challenge of spatial dependence is analogous to issues that arise in the
presence of clustered error terms and regressors (Cameron and Miller, 2015;
Moulton, 1986, 1990). A clustered variable is one that exhibits within-group
(cluster) correlation. In observational settings, this correlation is
typically positive. Ignoring the positive within-cluster correlation also
leads to the estimation of standard errors that are narrower than they truly
are.

One common approach to “correct” for spatial dependence is to cluster standard
errors at regional scales larger than the unit of analysis. For instance, in a
model based on US county data that would be to cluster at the district or
state level. What this approach assumes is that there is a common “shock” to
all locations within that cluster but no correlation between locations across
cluster boundaries. In the case of US counties, that would mean that
neighboring counties exhibit correlated errors or regressors within states,
but that there is no correlation across state lines. Given that the drivers of
many agricultural outcomes are natural in nature and do not follow
administrative borders, this assumption is unlikely to be valid, at least a
priori.

A more conceptually appropriate way to correct for spatial dependence in a
linear model is to correct directly for spatial dependence. This would allow
to account for smoother patterns of correlation between neighboring locations
irrespective of whether locations fall within the same state or district. One
approach to achieve this is the adopting a spatial heteroscedasticity and
autocorrelation consistent (HAC) estimator of the variance covariance matrix
introduced in Conley (1999). The idea is close to a spatial analogue to the
heteroscedasticity and autocorrelation consistent estimator of the covariance
matrix introduced in Newey and West (1986) to address serial correlation. The
key idea behind the spatial HAC estimator is the use of a kernel that weighs
the cross-products in the computation of the covariance matrix based on the
spatial distance between observations. The implementation requires a distance
threshold (say 500 miles) beyond which one assumes there is no correlation
between observations. This is typically implemented via a Bartlett window
which assigns the value of 1 for the observation at hand (distance equal 0)
and decreases linearly down to 0 when one reaches the threshold. Beyond that
point the assigned weight is 0.

The R code that accompanies this chapter provides an implementation of the
spatial HAC standard errors (see 4_spatial_dependence.R). The implementation
allows the specification of multiple distance thresholds as well as that of
various weighting kernels in addition to the more common Bartlett window.

Another approach to account for spatial dependence is to harness the spatial
dependence in the estimation to obtain a more efficient estimator. This can be
achieved with a Spatial Error Model (SEM) which can be estimated via GMM or
MLE (Anselin, 1988). This approach is relatively uncommon and can be found in
a few studies in the literature (e.g. Schlenker et al., 2006). The difference
between the spatial HAC correction above and the SEM is analogous to the
difference between correcting for heteroscedasticity and estimating a
Generalized or Weighted Least Squares. In the former case one seeks to correct
the estimation of standard errors of an otherwise inefficient estimator,
whereas in the latter one tries to harness more information about the
distribution of the error term to derive a more efficient estimator.

One perceived disadvantage of the SEM is that it requires specifying a weight
matrix, imposing some structure on the nature of the spatial dependence
between neighboring observations. However, these concerns are likely overblown
(LeSage and Pace, 2014). Moreover, doesn’t the imposition of a kernel and a
cutoff distance in the estimation of the spatial HAC estimator also impose
some form of structure to the nature of spatial dependence? One approach to
avoid unnecessary criticism by reviewers unfamiliar with the SEM is to present
the SEM results along with results based on OLS (corrected for spatial
dependence). Note however, that the SEM and OLS estimates are consistent in
the absence of an omitted variable, but the presence of an omitted variable
induces different types of biases in these two estimators. This situation
permits the implementation of a spatial Hausman test (Pace and LeSage, 2008).
The R code that accompanies this chapter provides an implementation of the SEM
model for balanced panel models. The implementation allows the specification
of various types of weight matrices.

Notes: The data covers corn yields over 1981-2020 east of the 100th meridian
West. Precipitation variables were included in the regression but coefficients
are not shown. The blue colored bands around the response function corresponds
to 95 and 99 percent confidence intervals for each type of standard error. The
green histogram represents the growing-season of exposure to all temperature
bins. The response function is based on a natural cubic spline with 7 degrees
of freedom and 1°C bins between 0 and 38°C over the April-September growing
season. Note the Spatial Error Model (SEM) is estimated based on a smaller
balanced panel.

Different standard errors and estimators for the effects of temperature
exposure on corn yields.

I illustrate the marginal effects of exposure to various temperature levels
with various standard errors for OLS and for the SEM in Fig. 4.5. Panels A
through F are based on OLS and have the same point estimates. Panel G is based
on the SEM, which is a different estimator, so the response function is
different.

Panel A of Fig. 4.5 shows a confidence band that assumes errors are
independent and identically distributed. Even beforehand, we know this
assumption is incorrect because the error is spatially dependent so these
confidence band is deceptively narrow. The R code shows that the residuals
appear highly correlated and that a Moran’s I test rejects the hypothesis of
no spatial correlation. Note that correcting simply for heteroscedasticity in
B does not change things much.

Interestingly, Fig. 4.5C and D show that clustering the standard error by
state or by state and year leads to much more conservative standard errors
which leads to a much wider confidence band around our response function.
While many of the effects of temperature exposure in the 0-25°C largely
indistinguishable from zero at conventional levels, the effects of exposure
beyond 30°C appear clearly detrimental.

Figure 4.5E is based on the spatial HAC standard errors proposed in Conley
(1999) using a cutoff distance of 500 miles. Panel F shows the confidence band
when the cutoff is increased to 1000 miles. Note these standard errors are of
a similar magnitude to those clustered by state or by state and year. This is
an interesting finding because it suggested that in some cases, clustering may
yield similarly conservative estimates despite not accounting for cross-
cluster correlations.

Finally, Fig. 4.5G shows estimates based on the SEM estimated via Maximum
Likelihood. Note that the implementation in R only allows balanced panels. So
while the OLS models where estimated with an unbalanced panel of 2,248
counties over 1981-2020 and 69,190 observations, the SEM was estimated with a
balanced panel of 599 counties over the same period with only 23,960
observations. And yet, the standard errors –which are conceptually correct–
are much narrower. As explained above, this estimator is more efficient than
OLS because it harnesses information about spatial dependence in the
estimation.

###  4.6 Common robustness and sensitivity checks

It is often common for reviewers in the peer review process to ask authors to
perform additional robustness or sensitivity checks. While these requests can
often feel like an unnecessary burden to authors, these checks can help build
more confidence on the results and clarify to what degree certain modeling
assumptions have an outsized influence. This relates to a large degree to the
classic call in Leamer (1983) for the systematic adoption of sensitivity
checks in empirical research.

Perhaps the most basic type of check is about heterogeneity, which can
generally be manifested either in space or time. When estimating weather
effects on economic outcomes in a panel setting, it is generally a good idea
to test whether coefficients are stable across major regional or temporal
subsets of the data (e.g. with a Wald test). Understanding heterogeneity is
critical and can even become the primary focus of a study. For instance,
Butler and Huybers (2013) find that corn yields in hotter regions in the US
appear less sensitive to extreme heat relative to colder regions, which the
authors suggest is indicative of adaptation to climate change (see Schlenker
et al., 2013 for a response). Analogously, Ortiz-Bobea et al. (2018) explores
temporal heterogeneity in the response function of US agricultural TFP to
document the rising sensitivity of Midwestern agriculture to higher
temperature.

Another aspect that often receives considerable scrutiny in empirical work are
nonlinearities in how weather or climate variables affect agricultural
outcomes of interest. For instance, Mendelsohn et al. (1994) included linear
and quadratic coefficients for monthly temperature and precipitation variables
in the specification. This captures the idea that while certain weather
conditions are optimal for agricultural production, extreme weather tends to
be detrimental. The work of Schlenker and Roberts (2009a) also highlighted the
importance of nonlinearities for temperature by introducing a semi-parametric
approach that flexibly harnesses the influence of daily temperature exposure
on crop yields.

Sometimes, however, regression coefficients are difficult to interpret
directly, particularly when dealing with higher term polynomials or splines.
As a result, a useful approach is to compare the impact of a uniform warming
(e.g. +2°C) or a percentage change in precipitation across alternative
specifications. There is a broad recognition that temperature effects may not
only capture heat stress but also moisture stress (see Lobell et al., 2013 or
Ortiz-Bobea et al., 2019) so the emphasis tends to be justifiably more focused
on projected impacts than on the values of the estimated parameters
themselves. This is perhaps slightly different from other research areas where
the value of estimated parameters are quantities with a clear theoretical
grounding in economics (e.g. a price elasticity). In general, it is advisable
to compare models, not only in terms of their fit (ideally out of sample) but
also in terms of the associated impacts of some specific change in the
distribution of weather or climatic conditions. This facilities comparison
across models.

Another modeling choice that receives some scrutiny is the choice of the
weather dataset and how it is aggregated in space or time. It is not uncommon
to see researchers reporting results based on alternative weather datasets, or
alternative ways of spatially aggregating weather to the unit of analysis. As
indicated earlier in this chapter, one can spatially aggregate gridded weather
data based on a variety of different of weighting schemes (e.g. cropland or
cropland and pasture weights, etc). The same goes for spatially interpolated
weather station data. The choice of seasons and how weather conditions are
aggregated over time, is also a common point of scrutiny. In general, it is
difficult to know a priori the role of these modeling assumptions in our
results. It is thus generally the case that researchers present results based
on alternative definitions of the growing season.

An important aspect of conducting empirical research is appropriately
characterizing the uncertainty around estimated parameters. The assumption
that errors are independent and identically distributed is virtually always
violated in observational settings. When analyzing climate-economy linkages,
weather conditions tend to be strongly spatially correlated across great
distances (see previous subsection). And just like economic outcomes, their
unobservable drivers are also spatially dependent. Ignoring this positive
spatial dependence typically leads to overconfidence by underestimating
standard errors. In such cases, a common solution is to correct for spatial
dependence in a linear model using spatial HAC standard errors (e.g. Conley,
1999). It is also common to see researchers cluster standard errors at
relatively large regional levels (e.g. at the state level in a county-level
panel setting) in order to capture the contemporaneous dependence. However,
such clustering approaches would not account for correlation across clusters
that may be occurring. But as shown in the previous subsection, these
approaches can point to similar standard errors.

It is also not uncommon to see the adoption of weighted regressions, where
observations are weighted based on some metric indicative of each
observation’s variance, like acreage. The idea is that the variance of
observations in locations with small acreage is higher, so these would be
given a smaller weight in a weighted least squares estimation. However, this
approach poses some challenges regarding the interpretation of the results if
the introduction of regression weight drastically alter the estimated
coefficients. This may signal a misspecification or an unaccounted
heterogeneity (see Solon et al., 2015 for guidance on regression weights). The
adoption of spatial HAC errors seems more appropriate, at least conceptually,
because it jointly accounts for heteroscedasticity and spatial dependence.

An increasingly common exercise in empirical studies is the implementation of
“placebo” checks or tests (Eggers et al., 2021). The idea behind a placebo
check is that a treatment should not appear to have an effect on an untreated
unit of analysis. In our context, for instance, it means that a weather shock
in location $i$ at time $t$ should have no effect on the outcome of interest
in a different location or time period. This is a bit tricky because weather
is not independently “assigned”, meaning that similar weather shocks occur in
neighboring locations, so a weather shock in one location may well appear to
have an effect in neighboring locations simply because of such contemporaneous
correlations. We could also make an analogous point in the time dimension in
the presence of serially correlated weather conditions (e.g. in places where
droughts tend to be multi-year phenomena).

A common placebo check in panel settings is to estimate the model with lead
weather. It is reasonable to assume that future weather fluctuations, which
are arguably impossible to predict accurately in advance, should have no
bearing on a current outcome. An analogous spatial version of this check is to
simply randomly reassign weather of other locations to the outcome variable.
In a large enough panel, this “reshuffling” should result in results that are,
on average, insignificantly different from zero. However, note that it is
still possible to get a significant effect by chance. This is a key weakness
of such simple placebo checks, which are not actual statistical tests.

A preferable approach is to conduct a placebo test in the form of a
permutation test (also called randomization test). These are non-parametric
tests for which the researcher constructs the distribution of a test statistic
under the null hypothesis of “no effect”. In a setting estimating the effect
of weather on an economic outcome, the idea is to reshuffle the weather data
across units, re-estimate the model with the permuted weather predictors,
store the coefficients and repeat this process, say, ten thousand times. The
resulting distribution of this spurious coefficients should be centered around
zero. The researcher can then contrast the sample estimate, i.e. the estimate
with the correctly matched data, to this distribution to obtain a p-value.
This procedure seems straightforward when assessing a single coefficient, but
the analysis can be done when estimating a quadratic relationship (e.g. see
Figure 2 in Ortiz-Bobea et al., 2021). Another approach that might help
conducting multiple simultaneous tests is to derive the distribution of the
test statistic as the linear combination of coefficients (e.g. computing the
impact of a 2°C warming at each iteration).

Notes: Standard errors are clustered at the state level. The data covers corn
yields over 1981-2020 east of the 100th meridian West. The models are sorted
by adjusted R2 with the best-fitting model shown on the right.

Estimated impact of a 2°C for 72 alternative specifications.

This section highlights that conducting empirical research in this area
requires making a relatively large number of modeling assumptions along the
way. In order to build confidence in our results, it is critical to justify
these assumptions and, ideally, to show to what extend results are sensitive
to these. However, presenting the results of the paper under many alternative
sets of modeling assumptions can be tedious and make any paper feel
overburdened. For instance, the online appendix in Ortiz-Bobea et al. (2019)
is 74 pages long and includes 6 pages of single-spaced text, 57 supplementary
figures and 7 tables.

One strategy to show robustness checks in a more parsimonious way is to
present the main estimate in a study under alternative modeling assumptions in
a specification chart or curve (Simonsohn et al., 2019). This type of chart
can summarize a large number of estimates in a single visualization, which
helps authors and readers better understand the important of specific modeling
assumptions. Rather than thinking about these charts as a way to show
“robustness”, it is preferable to think about this as an exercise in
transparency and as a way to characterize the potential uncertainty regarding
model selection.

The R code provided with this chapters provides an implementation of a
specification chart with a reproducible example (see 5_robustness _checks.R).
In this example I estimate a statistical crop yield model based on a panel of
corn yields east of the 100th meridian West over the 1981-2020 period. The
dependent variable is log yield and I consider a wide range of variation of
the model specification. Specifically, I explore the role of the temperature
variable (Tmax, Tmean or Tmin), the inclusion of a precipitation variable, the
adoption of a quadratic or cubic functional form for the weather regressors,
the definition of the growing season (March-August, April-September or the
entire year) and wether a quadratic time trend in the regression model is
estimate by state or pooled for all counties. Obviously, we could add more
checks.

It is very common for researchers to conduct analysis and show results based
on a particular “baseline” model. Let’s say that our baseline model is based
on Tmean, includes precipitation, adopts a quadratic response function, adopts
a growing season defined over March-August, and includes a common time trend
for the sample. One way to show how this model compares to alternative models
is to conduct the analysis on an exhaustive combination of these modeling
assumptions. However, comparing coefficients across models with different
temperature variables and functional forms can be intractable. A better
approach is to compare the implied effect of a, say, 2°C warming.

Figure 4.6 shows the effect of a 2°C warming for 72 different specifications.
In this example, I have sorted the models from the lowest to the highest
fitting specification based on the adjusted R2. Models could also be sorted
based on any other criteria including out-of-sample Means Squared Error, their
point estimate or simply preserve the original order in which the models were
stored in the input table. What is interesting here is that the best fitting
models (on the right) tend to point to larger negative impacts. The baseline
model we selected, highlighted in red, in not much different than many other
well-fitting models.

A few things stand out while exploring this specification chart. First, models
based on Tmin tend to point to small impacts and perform relatively poorly in
terms of fit. Second, models based on Tmax tend to point to larger impacts and
fit better, especially when coupled with precipitation variables. Third, there
is no discernible difference between models estimated with a quadratic or a
cubic functional form. Fourth, the shorter March-August season tends to fit
better whereas the full annual season performs more poorly. Finally, models
estimated based on a state-level time trend tend to perform better in terms of
fit, and also put to larger impacts relative to models with pooled time
trends.

This exercise shows that modeling assumption can drastically alter reported
estimates. Being upfront and transparent about how the main baseline results
shown in the paper can change with modeling assumptions should build
confidence in the reader that results were not cherry picked. Figure 4.6 is
fully reproducible with the included R code. The code also includes a function
that allows the creation of such charts with relative ease. I have also
included a reproducible example in spec_chart_reproducible_example.R that
showcases all the capabilities of this function.

##  5 Conclusion

This chapter provides an overview of the economic literature analyzing climate
change impacts and adaptation in agriculture. The first subsection cover some
basic concepts and knowledge necessary to understand the discussions and
debate in the literature. This includes overviews of common sources and the
nature of weather and climate change data.

The following subsection cover a range of topics with an emphasis on methods
for analyzing various aspects of the impacts of climate change on agriculture.
I discuss early studies based on biophysical approaches and their evolution
and subsequent integration with trade models. I then describe the emergence
and motivation behind econometric techniques and the explosion of empirical
studies analyzing impacts of weather fluctuations on agricultural outcomes
based on longitudinal data. I also discuss emerging efforts to link
statistical and biophysical techniques, as well as new methods to combine the
advantages of cross-sectional and longitudinal methods in the empirical
literature. I also spend some time discussing work exploring various mechanism
of adaptation, including irrigation, crop choices, planting and cropping
frequency decisions. I finally provide a very short overview of recent work
analyzing the role of trade in determining climate change impacts. I conclude
with some possible future directions of research and collaborations with non-
economists.

The last subsection is a unique effort to provide a hands on introduction to
new researchers in this literature. The subsection discusses very practical
empirical challenges as well as common tasks necessary to estimate more
sophisticated empirical models in the field. That part also discusses the
estimation of standard errors in the presence of spatial dependence. I also
spend a bit of time discussing common sensitivity checks in the literature and
introduce a parsimonious way of presenting numerous robustness checks without
overwhelming the reader.

All of the figures in this chapter are fully reproducible. This code
represents the entire workflow including downloading some of the weather
datasets, transforming the weather data, spatially aggregating and
interpolating weather data, estimating various types of semi-parametric models
and the construction of various types of standard errors that are relevant in
this literature. It is my hope that this chapter removes some of the barriers
to entry to this field of research. Thank you for reading.

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