aaaffiliationtext: _University of Arizona, Tucson, USA_ bbaffiliationtext:
_Michigan State University, East Lansing, USA_ ccaffiliationtext: _Purdue
University, West Lafayette, USA_

# Risk, Agricultural Production, and Weather Index Insurance in Village India

Jeffrey D. Michler Corresponding author email<EMAIL_ADDRESS>We
greatly appreciate the helpful comments and suggestions from Philip Abbott,
Hans Binswanger-Mkhize, Michael Roberts, and Neda Trifkovic along with those
from seminar participants at the Econometric Society World Congress in
Montreal, the Midwest International Economics Development Conference in
Madison, the Agricultural and Applied Economics Association Annual Meeting in
San Francisco, and the International Conference of Agricultural Economists in
Milan. This paper and our thinking have been shaped by conversations with
Steve Boucher, Garth Holloway, Mario Miranda, and Robert Townsend. We are
solely responsible for any errors or misunderstandings. Frederi G. Viens
Gerald E. Shively

(March 2021)

###### Abstract

We investigate the sources of variability in agricultural production and their
relative importance in the context of weather index insurance for smallholder
farmers in India. Using parcel-level panel data, multilevel modeling, and
Bayesian methods we measure how large a role seasonal variation in weather
plays in explaining yield variance. Seasonal variation in weather accounts for
19-20 percent of total variance in crop yields. Motivated by this result, we
derive pricing and payout schedules for actuarially fair index insurance.
These calculations shed light on the low uptake rates of index insurance and
provide direction for designing more suitable index insurance.

_JEL Classification_ : C11, D81, G22, O12, O13, Q16, Q12
_Keywords_ : Weather Risk; Agricultural Production; Index Insurance; Bayesian
Analysis; Multilevel Models; India

##  1 Introduction

Agricultural production is complex and risky. Weather is just one of several
potential causes of yield variability. Other determinants include the quantity
and quality of inputs, the agronomic characteristics of farmed parcels, the
inherent or learned abilities of farmers, the policy environments in which
farmers operate, and changes in technology (Hardaker et al., 1997). In this
paper we use parcel-level panel data from India to measure the sources of
variability in agricultural production and assess their relative importance.
Ascertaining just how large a role weather plays in determining yields is
vital to estimating the projected effects of climate change on a variety of
outcomes, including aggregate economic activity (Schlenker et al., 2006;
Deschênes and Greenstone, 2007; Burke et al., 2015). Understanding weather
induced variation in yields is also essential when designing and implementing
weather index insurance, an increasingly popular micro-level risk management
strategy in developing countries. However, recent pilot projects and
randomized control trials in Asia have found limited farmer uptake of index
insurance (Hazell et al., 2010; Giné et al., 2012; Cole et al., 2013; Cai et
al., 2015) Using a multilevel modeling approach and Bayesian estimation we
find that a relatively small fraction of the variability in yields can be
attributed to seasonal variation in weather. While weather variability does
not contribute much to yield variability, on average, our Bayesian methodology
reveals the presence of infrequent but potentially costly extreme weather
events. This result motivates our reassessment of the pricing of rainfall
insurance and may partly explain the low uptake of weather index insurance by
farmers.

The goal of weather index insurance is to assist farmers in managing covariate
risk. In developing country agriculture, covariate risk is particularly
difficult to informally insure against since, by its very nature, it affects
neighboring households and limits the effectiveness of traditional risk
sharing mechanisms. Several early studies looked at the role of weather risk
on agricultural production and insurance in India. Townsend (1994) found that
household consumption moves with village-level consumption and is not affected
much by idiosyncratic shocks, which households cover via borrowing, gifts, and
asset sales. He concludes that while households can insure against individual
risks, covariate risk remains a problem. Along similar lines, Rosenzweig and
Binswanger (1993) focus on the role of weather risk on production and asset
portfolios. They find that village-level rainfall variables explain only a
small proportion of household-level profit variability. And, similar to
Townsend (1994), they find households insuring against non-covariate risk. In
part, the recognition that households are less successful in insuring against
covariate weather risk than in insuring against non-covariate risk motivates
recent attempts to design and implement weather-based index insurance for poor
farmers.

This apparent need for weather index insurance has been met by surprisingly
little uptake of the product by farmers in India (Giné et al., 2008; Giné et
al., 2012). There are many potential barriers to uptake of insurance,
including trust in the insurance company, household liquidity constraints, and
lack of financial literacy of household decision makers (Cole et al.,
2013).111For example, Banerjee et al. (2014) find no demand among Indian
households for health insurance bundled with microfinance, even among those
for whom there was clear value. They attribute this low uptake to poor
understanding of the insurance product and poor support for enrolling by
insurance underwriters. In addition to these consumer-centric explanations for
low uptake, flaws in the product itself may be a cause, specifically the
imperfect correlation between crop yield and farm profit, the outcome of
ultimate interest to farmers (Binswanger-Mkhize, 2012). This uninsured
exposure, or basis risk, can be a significant deterrent to purchasing
insurance by reducing the utility gain for households. There are two potential
sources of basis risk. One is the loss caused by an event not measured by the
index. This could include, for example, low temperatures that retard crop
growth or high winds that cause crop damage, when the insurance index is based
on rainfall. Another potential source of basis risk is low correlation between
the index and covered losses. This can occur, for example, when measurement
stations that are used to trigger payouts are spatially distant from insured
crops and provide poor congruence with local conditions. A less common example
is potentially insured crops that are sufficiently resilient to the phenomena
which forms the basis for the insurance, such as drought-tolerant varieties
covered by rainfall-based insurance or pest-resistant cotton to pest
infestation (Liu, 2013). In this paper we focus on these production-oriented
sources of basis risk and empirically estimate the amount of variability in
yield coming from seasonal weather variability.

The most common type of weather index insurance in India is rainfall insurance
(Barnett and Mahul, 2007; Giné et al., 2007; Akter et al., 2009). However,
surprisingly little empirical research has been conducted to confirm the
underlying relationship between rainfall or seasonal weather variability and
yield variability. Barnett and Mahul (2007) suggest that rainfall variability
accounts for $50$ percent of yield variability but do not support this figure
with data or a source. Giné et al. (2012) assert that as much as 90 percent of
variation in Indian crop production is driven by rainfall volatility, citing
Parchure (2002) as their source. But Parchure relies on a 1976 report by the
Indian National Commission on Agriculture (NCA), the details of which are
somewhat opaque.222According to Parchure (2002), the $90$ percent variation is
actually for cotton and groundnuts while variation in yield due to rainfall
variability is $45$ and $47$ percent for wheat and barley, respectively.
Parchure (2002) actually cites the 1976 NCA report as stating “that rainfall
variations accounted for $50\%$ of the variability in agricultural yields.”
This may be where Barnett and Mahul (2007) get their number. Examining the
report, specifically pages 47-8, these numbers were obtained from a linear
regression “with yield as the dependent variable and total rainfall during the
five crop growth phases as the independent variable” (National Commission on
Agriculture, 1976). The $50$ percent number is apparently the R2 on the
regression. Even less data exist on the relationship between yield variability
and other measures of covariate risk, such as temperatures, wind speed, or the
occurrence of natural disasters. Finally, to our knowledge, no data exist on
covariate risk’s share in explaining variance in crop yield, which is where we
focus our attention. Previous studies that have attempted such measurements
were constrained by data, econometric techniques, and computing power so that
they were unable to net out measurement error and fully describe the sources
of output variability. We attempt to rectify these shortcomings and fill this
research gap.

We examine the different sources of yield variability using a
multilevel/hierarchical regression framework. This approach more fully
accounts for the covariance structure of the data than a standard regression
framework and allows us to control for inputs at the parcel-level and also to
isolate the amount of yield variance due to parcel-level effects, household-
level effects, seasonal weather effects, village-level effects, and time.
Using Bayesian methods, we draw the underlying distribution of the random
error term corresponding to different sources of variability, thereby
providing a quantitative measure of potentially insurable risk. Bayesian
methods are particularly useful because the underlying distribution for
several of the error terms turn out to be highly skewed and non-normal.
Considering all sources of yield variance, we find that seasonal variation in
weather accounts for 19-20 percent of total variance in agricultural
production.

While we conclude that weather variability accounts for a much smaller share
of yield variability than has been previously argued, our results should not
be interpreted as implying that smallholder farmers do not need weather
insurance. Rather, what they need is better insurance - specifically,
insurance that reduces the basis risk that results from a low correlation
between the index and crop loss. Furthermore, skewness in the distribution for
seasonal weather variability highlights the need for access to affordable
risk-management tools. Indeed this skewness is a sign that, despite the low
share of seasonal weather variation in yield variability, extreme weather
events, while rare, are nevertheless potentially costly.

Motivated by these findings, in the second part of the paper we shift our
focus to the closely related concern of proper pricing of index insurance
contracts. We provide a brief quantitative reassessment of the index insurance
contracts studied by Giné et al. (2007), Giné et al. (2012), and Cole et al.
(2013). We conclude that these contracts are overpriced. This further
underscores that flaws in the product are a proximate cause of low uptake and
that farmers may rationally prefer to focus their risk management choices on
minimizing other sources of risk.

To tie our results as closely as possible to the issue of weather risk in
agricultural production, we use the International Crops Research Institute for
the Semi-Arid Tropics (ICRISAT) household survey data from the same Indian
villages studied by Townsend (1994) and Rosenzweig and Binswanger (1993).
However, as a further extension, we use an expanded data set that covers 24
newly added villages. In our analysis we use a panel of 10 cropping seasons
from 2009 through 2013. This time horizon has both advantages and
disadvantages. The primary benefit is that over a short time frame parcel and
household characteristics, along with policy and technology, are unlikely to
have significantly changed. By controlling for these “fixed” effects, along
with parcel-level variable inputs in each time period, we can measure the
variance in crop output resulting from weather risk. The primary shortcoming
of our dataset is that the 10 seasons of production we observe contains only
one large-scale weather-related natural disasters: the 2009 drought. This
means that our measure of weather’s share in total yield variability may be an
underestimate.333That said, India has experienced only two large-scale
droughts in the last 25 years (2002 and 2009). The Indian Meteorological
Department considers a drought to have occurred when seasonal rainfall
deficiency falls is $26$ percent below average rainfall. Nevertheless, our
analysis sheds new empirical light on the potential value of weather insurance
during typical production periods. Our five year time horizon is also likely
to mimic the typical time scale under which farmers in village India make
investment decisions.

By using a multilevel approach to measuring the roles of idiosyncratic and
covariate risk in agricultural production in village India we contribute to
three separate streams of research. The first is the literature focused on
production risk and insurance in the developing world. Our aim is to connect
two strands of this research. The first, originating with Townsend (1994),
Rosenzweig and Binswanger (1993), Rosenzweig and Wolpin (1993), and Kochar
(1999) has focused on the theoretical and empirical relationship between
production, risk, and insurance. The second focuses on interventions to help
insure production against risk (Barnett and Mahul, 2007; Giné et al., 2007;
Skees et al., 2007; Giné et al., 2008; Akter et al., 2009; Gaurav et al.,
2011; Clarke et al., 2012; Chantarat et al., 2013; Cole et al., 2013). We
empirically estimate the role different sources of variability play in
determining the mean and variance of crop output. We then use this information
to inform the value and pricing of index insurance products.

Our second contribution is to the nascent economics literature that applies
multilevel models to household data. Although we introduce no new methods, per
se, in the paper, we do contribute a methodological innovation by expanding
the use, understanding, and adaptability of multilevel modeling, Bayesian
inference, and Gibbs sampling, all in the context of household data.

The third body of literature to which we contribute is the empirical
estimation of production functions. Until recently it has been computationally
difficult to estimate production functions with multiple nested levels of
data. While panel data fixed effects methods are equivalent to a multilevel
model with a single level, few alternatives exist for models with a large
number of nested levels. Production data are often collected at the farm or
factory level, which suggests at minimum a two-level model. Adding additional
levels to account for and distinguish among temporal and/or spatial variation
can provide more efficient estimation of production by more accurately
modeling the data generating process. We contribute to this strand of
literature by comparing production function parameter estimates obtained by
standard ordinary least squares to multilevel regression estimates obtained by
maximum likelihood and Bayesian estimation techniques.

##  2 Data

To conduct our empirical analysis, we use household data from villages in
India. These data were collected as part of the Village Level Studies/Village
Dynamics Study of South Asia (VDSA, 2015). The data set combines high and low
frequency household data from 30 Indian villages. The villages include the
three studied by Townsend (1994) and the ten studied by Rosenzweig and
Binswanger (1993). However, while much of the previous research has relied on
the low frequency data, we utilize a newly available high frequency data set
covering the years 2009 through 2013. These data include monthly household
observations on input purchases and labor expenditure for on-farm activities
and crop production. The added value of a high frequency data set is that,
with multiple crops on multiple parcels for multiple seasons in a single year,
it provides much more detailed and accurate farm production information than
the low frequency data.

We utilize monthly parcel-level data aggregated to the seasonal level. We
focus on five crops: paddy rice, sorghum, wheat, maize, and cotton. Together
these crops account for 60 percent of total crop observations in the VDSA and
cover 78 percent of the total parcel-level observations.444With the exception
of pigeon pea, a perennial crop, the five crops used in our analysis are the
five most frequently observed crops in the data set. We include cotton instead
of pigeon pea for two reasons. First, pigeon pea is a perennial crop and
therefore may be treated by farmers differently than annual crops when
considering insurance. Second, cotton is one of the crops considered by Clarke
et al. (2012) in their product design and ratemaking for insurance contracts
in Gujarat. Thus, despite it being a non-food crop, we include it to bring our
analysis more closely in line with existing literature. This provides us with
11,942 parcel-level observations. Rice is the most common crop, accounting for
46 percent of total observations. Next most common is sorghum, accounting for
23 percent of observations. Wheat constitutes 14 percent of observations while
maize and cotton account for the remaining 9 and 8 percent of observations
respectively.

The locations of VDSA villages provides us with a great deal of heterogeneity
in climate, weather, crop choice, and cultivation practices. Statistics
describing these data, by crop, are presented in Table 1. Twenty-seven
villages cultivate at least two crops, generally a single crop in each of the
two growing seasons. Only two villages cultivate all five crops. Cotton is the
most input intensive crop, using more labor, fertilizer, mechanization, and
pesticide than any other crop. Sorghum is the least labor intensive while
wheat uses the least fertilizer, mechanization, and pesticide. In total, our
11,942 parcel-level observations come from 5,100 unique parcels operated by
1,079 distinct households, in 30 villages, farming across 10 seasons. We
exploit this nested data structure in our empirical analysis.

One final source of variation in the data structure, which plays an important
role in our analysis, is the number of time observations for each crop. Three
crops (rice, wheat, and maize) are grown in both the _Kharif_ monsoon season
and the post-monsoon _Rabi_ season. Sorghum is only grown in _Rabi_ while
cotton is only grown in _Kharif_. The limited time series for these crops may
affect our estimation of seasonality’s effects on yield variability, albeit in
an unknown way.

##  3 Econometric Framework

###  3.1 Ordinary Least Squares

We begin by estimating a simple linear regression for yield. Let $y_{is}$
denote the log of yield for parcel $i$ in season $s$. We estimate

$y_{is}=X_{is}\beta+\alpha_{s}+\epsilon_{is}$ (1)

where $X_{is}$ is a matrix of data and $\beta$ is a vector of regression
coefficients associated with various crops. In order to account for the
spatial dimension and allow our proxy for weather to vary across both location
and time, the seasonal fixed effect $\alpha_{s}$ is an indicator for each time
period $(t=10)$ in each village $(v=30)$. We assume the error term is
$\epsilon_{is}\sim\mathcal{N}(0,\sigma^{2})$ so that
$y_{is}\sim\mathcal{N}(X_{is}\beta+\alpha_{s},\sigma^{2})$, where $\beta$ and
$\sigma^{2}$ are regression and variance parameters which are season
independent while $\alpha_{s}$ depends on the season.

We note several drawbacks associated with this linear estimation of the
production function. First is that it precludes one from discerning the role
of weather in yield variability. Equation (1) can estimate the impact of
parcel-level inputs on yield at each point in time as well as the impact of a
seasonal weather dummy on yield. But because these variables impact mean
yield, the specification does not allow us to measure the share of yield
variability $(\sigma^{2})$ represented by seasonal variability in weather. A
second drawback is that OLS limits our ability to control for additional
clustered effects, such as parcel, household, village, or temporal effects.
While changes in season clearly impact the effectiveness of parcel-level
inputs, equally relevant effects may exist at these other levels. Some
households may be more efficient in their application of labor compared to
others, while some parcels may be of better quality, resulting in less need
for, say, fertilizer. Some households may live in villages in states with
favorable agricultural policy resulting in better access to inputs. Finally,
while we do not expect much technological change between 2009 and 2013, the
quality of inputs continues to evolve, which may impact yields over time. Even
if it were computationally feasible to estimate season-specific, household-
specific, and parcel-specific parameters using OLS, such grouped data would
violate the assumption of independence for all data (Corrado and Fingleton,
2012). A third drawback, and perhaps the most important for us, is that OLS
requires the variance terms to be normally distributed. While in many
applications this assumption is appropriate, we should not expect weather
events to be normally distributed. If time-specific effects have a highly
skewed distribution, OLS will produce biased parameter estimates and estimates
of risk severity may ignore extreme events.

###  3.2 The Multilevel Model

A multilevel or hierarchical modeling strategy addresses the first two
drawbacks associated with the standard linear approach to estimation.555Gelman
and Hill (2007) provide an introduction to multilevel analysis. First,
multilevel models offer a natural way to assess the role of seasonal changes
in weather on variation in yields by explicitly modeling the variance, not
just the mean of the data. This allows us to measure the different sources of
variance in yields. In our case, a multilevel approach also allows us to
disaggregate total variance in yields into its multiple sources, so as to
measure the relative contribution of seasonal weather risk in production.
Second, a multilevel approach allows us to control for each grouping of the
data without adding to the computational burden and without violating
independence assumptions.

For expository purposes we start with an illustrative example of a simple two-
level model in which realizations of yields are grouped within seasons. Let
$y_{ns}$ denote the log of an observed yield, $n$, realized in season $s$. We
estimate

$\displaystyle y_{ns}$ $\displaystyle=X_{n}\beta+\alpha_{s}+\epsilon_{ns}$
(2a) $\displaystyle\alpha_{s}$ $\displaystyle=\mu+\nu_{s}$ (2b)

where $X_{n}$ and $\beta$ are as previously defined, and $\alpha_{s}$ is a
seasonal effect that is a function of an overall mean, $\mu$, and a random
disturbance term, $\nu_{s}$. We assume that
$\epsilon_{ns}\sim\mathcal{N}(0,\sigma_{\epsilon}^{2})$,
$\nu_{s}\sim\mathcal{N}(0,\sigma_{\nu}^{2})$, and $\epsilon_{ns}$ is
independent of $\nu_{s}$.

In order to make our parameter of interest explicit, we can rewrite equations
(2a) and (2b) in terms of a probability distribution so that

$y_{ns}\sim\mathcal{N}(X_{ns}\beta+\mu,u_{n}),$ (3)

with $u_{n}\equiv\sigma_{\nu}^{2}+\sigma_{\epsilon}^{2}$. The above
distribution is obtained by substituting (2b) into (2a) and using the
independence of $\epsilon_{ns}$ and $\nu_{s}$. Defining the regression
equation in this way highlights the very specific dispersion structure of the
residual, which is where our interest lies. It also allows us to easily define
the intraclass correlation coefficient (ICC),

$\rho=\frac{\sigma_{\nu}^{2}}{\sigma_{\nu}^{2}+\sigma_{\epsilon}^{2}},$ (4)

which is similar to the proportion of explained variance in an OLS regression.

The value of a multilevel model becomes obvious as we add additional levels.
In our analysis we can view each observation on yield, $y_{n}$, as coming from
a parcel group $i$; each parcel group as being nested within a household, $h$;
each household as observed during a specific season, $s$; each season
occurring in a unique village, $v$; and each village existing at a given time,
$t$.666Note that $i$, $h$, $s$, $v$, and $t$ can be understood as functions of
$n$ so that each unique data point corresponds to the index $n$, each data
point can be identified with a unique parcel $i$, each parcel can be
identified with a unique household $h$, each household experiences a unique
season $s$, each season occurs in a unique village $v$, and each village
exists at unique time $t$. We can write the multilevel model as:

$\displaystyle\textbf{Level 0
}\mathrm{(yields},n=11,942):y_{n}=X_{n}\beta+\alpha_{ihsvt}+\epsilon_{n}$ (5a)
$\displaystyle\textbf{Level 1
}\mathrm{(parcels},i=5,100):\alpha_{ihsvt}=\alpha_{hsvt}+\nu_{ihsvt}$ (5b)
$\displaystyle\textbf{Level 2
}\mathrm{(households},h=1,079):\alpha_{hsvt}=\alpha_{svt}+\nu_{hsvt}$ (5c)
$\displaystyle\textbf{Level 3
}\mathrm{(seasons},s=240):\alpha_{svt}=\alpha_{vt}+\nu_{svt}$ (5d)
$\displaystyle\textbf{Level 4
}\mathrm{(villages},v=30):\alpha_{vt}=\alpha_{t}+\nu_{vt}$ (5e)
$\displaystyle\textbf{Level 5 }\mathrm{(time},t=10):\alpha_{t}=\mu+\nu_{t}$
(5f)

where again $X_{n}$ is a matrix of input data and $\beta$ is a vector of
regression coefficients associated with the various crops.

At level 0 the model estimates the log of yield as a function of inputs,
similar to that in equation (1), with yields being a function of a specific
parcel $i$ and an idiosyncratic error term
$\epsilon_{n}\sim\mathcal{N}(0,\sigma^{2})$, where $\sigma^{2}$ is a constant
variance parameter which we assume does not depend on $i$, $h$, $s$, $v$, or
$t$. Each $n$ observation comes from a parcel cluster $i$ which we assign a
unique intercept, $\alpha_{ihsvt}$. This parcel-level intercept allows the
relationship between inputs and yield to differ across parcels depending on
parcel-level characteristics. While some parcel characteristics can be
observed, many are difficult to measure or costly to observe. Such
characteristics include soil micro-nutrients, grade, and aeration or
composition. By including a unique intercept term for each parcel we can
control for these parcel characteristics.

Level 1 of the model groups parcels within households. Here parcel intercepts,
$\alpha_{ihsvt}$, are a function of household characteristics,
$\alpha_{hsvt}$, and a random disturbance term,
$\nu_{ihsvt}\sim\mathcal{N}(0,\sigma_{1}^{2})$, where $\sigma_{1}^{2}$ is a
constant variance parameter. We assume that the $\nu_{ihsvt}$ terms are
independent of each other and that the vectors $\epsilon_{n}$ and
$\nu_{ihsvt}$ are independent. The household-level intercept allows variation
in parcel-level production to be dependent on household characteristics. In
most applications, the analyst attempts to control for unobserved household
ability through proxy variables such as age or education. The multilevel
approach allows us to control for any unobserved household-level
characteristics by assigning each household a unique intercept term without
the need to rely on proxies. The use of a single disturbance term for all data
points $n$ corresponding to a given parcel group further enhances this control
by imposing a covariance structure which is consistent with variation at the
parcel group level.

Level 2 of the model groups households within seasons. Here household-level
intercepts, $\alpha_{hsvt}$, are a function of season, $\alpha_{svt}$, and a
random disturbance term, $\nu_{hsvt}\sim\mathcal{N}(0,\sigma_{2}^{2})$, where
$\sigma_{2}^{2}$ is a constant variance parameter.777As in the case of the
level 1 disturbance terms, the terms $\nu_{hsvt}$ are assumed to be
independent of each other. The corresponding vector $\nu_{hsvt}$ is assumed to
be independent of $\epsilon_{n}$ and $\nu_{ihsvt}$. The same group-level
independence assumptions hold for all levels. The season-level intercept
allows variation in household-level efficiency to depend on seasonal weather
events. While household ability is often viewed as time invariant, it will be
time dependent if household ability is diminished or enhanced by changes in
weather (Kochar, 1999). Households experienced in dealing with droughts may
find their ability diminished by flooding or cyclones. _A priori_ , there is
no reason to assume that seasonality or changes in weather have a constant or
stationary effect on household characteristics. By allowing household-level
intercepts to vary across season, we are relaxing the assumption that
household characteristics are either time invariant or affected by weather in
the same way each season.

Level 3 of the model groups seasons within villages. Here season-level
intercepts, $\alpha_{svt}$, are a function of a village, $\alpha_{vt}$, and a
random disturbance term, $\nu_{svt}\sim\mathcal{N}(0,\sigma_{3}^{2})$, where
$\sigma_{3}^{2}$ is a constant variance parameter. The village-level intercept
allows for season-on-season variation of average yields within each village.
By making seasonality a function of both village and time, we are able to
account for both the spatial and temporal nature of weather. This allows our
model to account for the fact that not all crops are grown in each village and
in each time period.

Level 4 of the model groups villages within time. Here village-level
intercepts, $\alpha_{vt}$, are a function of time, $\alpha_{t}$, and a random
disturbance term, $\nu_{vt}\sim\mathcal{N}(0,\sigma_{4}^{2})$, where
$\sigma_{4}^{2}$ is a constant variance parameter. The time-level intercept
allows variation in village-level effects to depend on time. While our data
cover only five years, we cannot rule out changes in policy, technology, or
both during this period. In India, the introduction of new technologies and
the implementation of new policies frequently occur at the state level and
thus will not uniformly effect all villages. By allowing our village-level
intercepts to vary across time we can control for unobservable technological
and political changes.

Level 5 of the model defines the temporal intercepts as a function of an
overall mean, $\mu$, and a random disturbance term,
$\nu_{t}\sim\mathcal{N}(0,\sigma_{5}^{2})$, where $\sigma_{5}^{2}$ is a
constant variance parameter. Having controlled for all other potentially
relevant sources of yield variability, we interpret seasonal variation as
coming solely from weather events.

The model represented above by equations (5a)-(5f) and its
dependence/independence structures can be summarized as a single model in
terms of a probability distribution with a special error structure which is a
sum of independent disturbance terms with a nested dependence on indexes:

$y_{n}\sim\mathcal{N}(X_{n}\beta+\mu,u_{n}),$ (6)

where $u_{n}$ is a specific covariance matrix which is the sum of six
covariance matrices corresponding to the disturbance vectors $\epsilon_{n}$,
$\nu_{ihsvt}$, $\nu_{hsvt}$, $\nu_{svt}$, $\nu_{vt}$, and $\nu_{t}$ from each
level. Considered as a single error term, this error structure is non-trivial,
though the expression shows how it works intuitively. Equation (6) makes clear
that our parameters of interest are not the additive non-interacting scale
terms (the $\alpha$’s) but the components of the error term $(\sigma^{2}$,
$\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$, $\sigma_{4}^{2}$,
$\sigma_{5}^{2})$. In particular, our model, whether represented by equations
(5a)-(5f) or by equation (6), shows that the main goal in uncertainty
quantification is to estimate the disturbance terms. Indeed, this is the key
to evaluating the share of variance in yield corresponding to each level in
the hierarchy.

The ICC for the five-level model is the percentage of the total variance that
is explained by the variance within clusters of groups. So, the correlation
between realizations of yield within the same parcel is

$\rho(parcel)=\frac{\sigma_{1}^{2}}{\sum_{i=0}^{5}\left(\sigma_{i}^{2}\right)}.$
(7)

The correlation between realizations of yield within the same household is

$\rho(household)=\frac{\sigma_{2}^{2}+\sigma_{1}^{2}}{\sum_{i=0}^{5}\left(\sigma_{i}^{2}\right)}.$
(8)

The correlation between realizations of yield within the same season is

$\rho(season)=\frac{\sigma_{3}^{2}+\sigma_{2}^{2}+\sigma_{1}^{2}}{\sum_{i=0}^{5}\left(\sigma_{i}^{2}\right)}.$
(9)

The correlation between realizations of yield within the same village is

$\rho(village)=\frac{\sigma_{4}^{2}+\sigma_{3}^{2}+\sigma_{2}^{2}+\sigma_{1}^{2}}{\sum_{i=0}^{5}\left(\sigma_{i}^{2}\right)}.$
(10)

And, the correlation between realizations of yield during the same time period
is

$\rho(time)=\frac{\sigma_{5}^{2}+\sigma_{4}^{2}+\sigma_{3}^{2}+\sigma_{2}^{2}+\sigma_{1}^{2}}{\sum_{i=0}^{5}\left(\sigma_{i}^{2}\right)}.$
(11)

By construction the ICC increases as we move to higher levels of aggregation.
Thus, we also calculate each level’s contribution to total variance in the
model. This is simply the variance at each level divided by $u_{n}$, the total
variance in yields.

###  3.3 Bayesian Estimation of the Multilevel Model

While multilevel models address the first two drawbacks of OLS estimation,
they still rely on the standard, though potentially unsupported, assumption of
normality of the disturbance term at each level. We can verify if the
normality assumption placed on each of the random disturbance terms is
reasonable using a likelihood ratio (LR) test. We visualize the results of
these tests using zeta profile plots, which plot the sensitivity of the model
fit to changes in values of particular parameters. While these plots are not
equivalent to drawing the underlying distributions of the estimators, they
represent a similar idea and can be interpreted as representing the underlying
distributions. First, we estimate the model in equations (5a)-(5f) using
maximum likelihood. Then we hold a single parameter fixed and vary the other
parameters, assessing the fit of each new iteration compared to the globally
optimal fit using the LR as our comparison statistic. We then apply a signed
square root transformation to the LR statistic and plot the absolute value of
the resulting function, $|\zeta|$, in comparison to the estimated parameter
values. The zeta profile plots resulting from the model fit to equations
(5a)-(5f) are presented in Figure 1.888Given the large number of parameters
($20$ coefficients and $6$ disturbance terms) we prefer the graphical
representation of the outcomes of the LR tests. Numerical results of the tests
are available from the authors upon request. Parameters with underlying normal
distributions have straight line zeta profile plots. When this is the case
these parameters provide a good approximation to the normal distribution and
standard confidence intervals can be used for inference. When zeta profile
plot lines are not straight, the normal distribution is a poor approximation
of the underlying distribution. For parameter estimates on data, a non-normal
distribution simply requires an adjustment of the relevant test statistics.
But, if the underlying distributions of our disturbance parameters are non-
normal, we are faced with a violation of the normality assumptions of the
error term. In Figure 1, zeta profile plots for the data, the parcel
disturbance term $(\sigma_{1})$, and the idiosyncratic disturbance term
$(\sigma)$ represent good approximations of normality. The zeta profile plot
for the weather disturbance term $(\sigma_{3})$ and the village disturbance
term $(\sigma_{4})$ are slightly skewed. The zeta profile plot for the
household disturbance term $(\sigma_{2})$ and the time disturbance term
$(\sigma_{5})$ are highly skewed and cannot be assumed to follow a normal
distribution. Thus, we cannot assume normality of the individual error terms
nor the composite error term. Since we do not, _a priori_ , know the
distribution of $\epsilon_{n}$, maximum likelihood methods cannot be used to
reliably estimate our model.

We address this drawback to classical estimation of multilevel models by
adopting a Bayesian framework and using Markov Chain Monte Carlo (MCMC)
methods to obtain posterior estimates. For estimation of posterior
distributions we use the Gibbs sampler, which iteratively constructs a
sequence of samples from the univariate random values of each response
variable, and of each model parameter, conditional on all other parameters and
variables. This method allows us to compute all features of the marginal and
joint distributions because the marginal samples are iteratively fed back into
the conditional posterior densities of all other parameters and variables for
each sampling. This allows us to calculate unbiased point estimates and
confidence intervals for all variables without recourse to normality
assumptions.

We define the Bayesian estimator using our illustrative two-level model and
then provide parameter definitions for the specific model defined by equations
(5a)-(5f).999Cameron and Trivedi (2005) and Hanmaker and Klugkist (2011)
outline Bayesian estimation of multilevel models. For a general two level
model, let $y_{km}$ be a unique observation $k=1,...,K$ from the group
$m=1,...,M$. Within each group $m$, the data are distributed according to a
particular distribution $G$ with parameter $\gamma$ such that $y_{km}\sim
G(\gamma_{m})$. We assume that parameter $\gamma_{m}$ comes from a
distribution $L$ with parameter $\lambda$ such that $\gamma_{m}\sim
L(\lambda)$. Finally, we assign a distribution to the hyperparameter $\lambda$
so that $\lambda\sim Q(a,b)$, where $a$ is the mean and $b$ is the variance of
the distribution. The joint posterior distribution of all unknown parameters
is derived using Bayes’ theorem:

$p(\lambda,\gamma|y)\propto p(y|\gamma,\lambda)p(\gamma|\lambda)p(\lambda)$
(12)

where $y=(y_{11},...,y_{1K},...,y_{M1},...,y_{MK})$ is all the data,
$\gamma=(\gamma_{1},...,\gamma_{M})$ is the group level parameter, and
$\lambda$ is the population parameter. The density of the data is obtained
through

$p(y|\gamma,\lambda)=\prod_{m=1}^{M}\prod_{k=1}^{K}p(y_{km}|\gamma_{m},\lambda).$
(13)

This is an independence assumption, and the individual density,
$p(y_{km}|\gamma_{m},\lambda)$, is assumed to be known. Again by the
independence assumption, the prior for the group level effect is assumed to be

$p(\gamma|\lambda)=\prod_{m=1}^{M}p(\gamma_{m}|\lambda).$ (14)

From the joint posterior distribution we can derive the conditional posterior
distributions.101010In terms of the independence assumptions made for the MLE
multilevel model, assumption (13) corresponds exactly to assuming that the
error terms $(\epsilon_{n})$ are independent of each other while assumption
(14) corresponds exactly to assuming that the elements in each group-level
$\nu$ term are independent of each other. The Bayesian multilevel setup in
which each group-level parameter is assumed to have its own prior distribution
is consistent with assuming in the MLE multilevel model that $\epsilon_{n}$
and the $\nu$’s are independent.

The Bayesian estimator for our multilevel model given by equations (5b)-(5e)
can be stated by redefining the model in probability terms:

$\displaystyle\textbf{Level 0}:$ $\displaystyle\ \
y_{n}\sim\mathcal{N}\left(X_{n}\beta+\alpha_{ihsvt},\sigma^{2}\right)$ (15a)
$\displaystyle\textbf{Level 1}:$ $\displaystyle\ \
\alpha_{ihsvt}\sim\mathcal{N}\left(\alpha_{hsvt},\sigma_{1}^{2}\right)$ (15b)
$\displaystyle\textbf{Level 2}:$ $\displaystyle\ \
\alpha_{hsvt}\sim\mathcal{N}\left(\alpha_{svt},\sigma_{2}^{2}\right)$ (15c)
$\displaystyle\textbf{Level 3}:$ $\displaystyle\ \
\alpha_{svt}\sim\mathcal{N}\left(\alpha_{vt},\sigma_{3}^{2}\right)$ (15d)
$\displaystyle\textbf{Level 4}:$ $\displaystyle\ \
\alpha_{vt}\sim\mathcal{N}\left(\alpha_{t},\sigma_{4}^{2}\right)$ (15e)
$\displaystyle\textbf{Level 5}:$ $\displaystyle\ \
\alpha_{t}\sim\mathcal{N}\left(\mu,\sigma_{5}^{2}\right).$ (15f)

The hyperpriors, or the prior distributions of the hyperparameters, are
defined as:

$\displaystyle\textbf{Hyperpriors}:$
$\displaystyle\mu\sim\mathcal{N}\left(a,b\right)$
$\displaystyle\sigma_{4}^{2}\sim IG\left(d_{4},g_{4}\right)$
$\displaystyle\sigma_{3}^{2}\sim IG\left(d_{3},g_{3}\right)$
$\displaystyle\sigma_{2}^{2}\sim IG\left(d_{2},g_{2}\right)$
$\displaystyle\sigma_{1}^{2}\sim IG\left(d_{1},g_{1}\right)$
$\displaystyle\sigma_{0}^{2}\sim IG\left(d_{0},g_{0}\right)$

where $IG$ is the inverse gamma distribution. The assumption that the priors
are distributed inverse gamma is a common assumption in Bayesian econometrics
in order to facilitate the mathematical analysis via the Gibbs sampler. We
select values for the distributions to ensure that priors are uninformative.
We then use a burn-in period of 5,000 iterations and an additional 5,000
iterations to ensure convergence of the iterative simulations and sufficient
mixing of the Gibbs sampler.

##  4 Econometric Results

We present the results from a large complement of regressions in Tables 2 \-
4. Table 2 presents a trio of results from fitting the production function
using OLS, MLE, and Bayes. Table 3 presents estimated variances, ICCs, and
variance shares from the MLE and Bayesian versions of the production function.
Table 4 presents both MLE and Bayesian estimation results from crop-by-crop
production functions.111111In the interests of parsimony we focus on the
residual estimates for the crop specific models. The point estimates on inputs
are presented in Appendix Table A1. All models are estimated using the log of
yield as the dependent variable and log values of inputs as independent
variables.121212Given the prevalence of zero values in the input data, and to
a lesser extent in the output data, we use the inverse hyperbolic sine
transformation to convert levels to logarithmic values. Hence, point estimates
can be read directly as elasticities.

###  4.1 Production Function Results

The three production functions presented in Table 2 rely on the same sample
and contain the same set of inputs. To account for heterogeneous input
response across crops, we allow all slope and intercept estimates to vary by
crop. Model 1 is the classical OLS regression as represented by equation (1).
This regression model contains fixed effects for seasons but does not account
for the multilevel structure of the data. Models 2 and 3 are the MLE and
Bayesian regressions and explicitly account for the clustering of observations
at the parcel, household, season, village, and temporal levels. The point
estimates from the Bayesian regression are based on posterior density
estimates derived from iterating from our uninformative priors. Results from
these regressions point to a fairly robust set of basic patterns that are
repeated with few exceptions. These include (i) positive and significant
production relationships between yields and measured inputs (labor,
fertilizer, mechanization and pesticides), and (ii) diminishing returns to
inputs other than labor. We observe only one instance in which the point
estimate for an input is negative and significant (for pesticides in the case
of wheat). Returns to scale appear to to be increasing for all crops. The
Bayesian estimation of the production function generates point estimates that
are broadly similar in sign, magnitude, and significance to those of the OLS
and MLE multilevel regressions.

Table 3 reports estimated variance parameters (Panel A), ICCs (Panel B) and
variance shares (Panel C) for the multilevel MLE and Bayes regressions. These
statistics establish the key findings that inform our insights into the
potential role of weather index insurance.131313An expanded set of results
from models estimated using different hierarchical structures is provided in
Appendix Tables A2 and A3. We focus attention on Panel C of Table 3, which
compactly summarizes the data expressed in the upper panels of the table.
Reading down the rows of the table allows us to assess the decomposition of
variance and, by extension, the relative importance of each level in
explaining overall variance in yields. We find for the MLE specification
(Model 2 reported in Table 2) that 23 percent of the total variance in yields
comes from between-parcel differences, 19 percent is attributed to between-
season differences, 15 percent comes from between-village differences, three
percent is attributed to differences across time, and 40 percent of the total
residual is idiosyncratic noise. In the Bayesian specification (Model 3
reported in Table 2) 24 percent of total variance in yields comes from
between-parcel differences, 20 percent is attributed to between-season
differences, 15 percent comes from between-village differences, six percent is
attributed to differences across time, and 35 percent is idiosyncratic noise.
An intuitive interpretation of these results is that much of the differences
observed in yields reflects differences between parcels, such as soil quality.
Household or farmer capability, relative to other sources of variability, is
unimportant in explaining differences in yields. In other words, good farmers
cannot make up for bad soil but bad farmers can still prosper if they have
good soil. Similar to Townsend (1994) and Rosenzweig and Binswanger (1993) we
find that idiosyncratic sources of risk play a much larger role in determining
observed yields than covariate sources. Unlike Townsend (1994) and Rosenzweig
and Binswanger (1993), we are able to quantify these differences. Considering
all sources of variance in yields, only 37-41 percent comes from covariate
sources while the remaining 59-63 percent comes from idiosyncratic sources.
Thinking in terms of insurable weather risk, only 19-20 percent of the
variability in crop yield is due to seasonal weather variation. These basic
patterns highlight the relatively small importance of between-season yield
variance compared with other sources of yield variance.141414Note that the
share of variance that remains unexplained is not the appropriate measure of
model fit. The Akaike Information Criterion (AIC) and Deviance Information
Criterion (DIC) reported in Table 2 provides a measure of model fit. In both
the AIC and DIC, lower values correspond to better model fit. The DIC is a
generalization of the more common Bayesian Information Criterion (BIC) and is
used here to facilitate comparison between the ML and Bayesian estimations of
the same models.

Strictly speaking, the non-normality of the error term means that the OLS and
MLE regressions are misspecified. Accordingly, we would expect estimates from
these models to be biased. However, in our application, only the disturbance
terms associated with households and time are highly skewed and non-normal
(recall Figure 1). Furthermore, these two sources of variance make up a very
small share of total variance. Thus, any bias that is introduced through model
misspecification is mitigated by the small impact of the skewed noise term on
the overall estimation. We highlight that this is an artifact of the current
application and data. If non-normal variation accounted for more of the total
variance we would expect to find greater bias in our OLS and MLE estimates.

Figure 2 reports distribution profiles based on the Bayesian regression
results for the associated sources of variance measured by Model 3 in Table 2.
These histograms are constructed from the posterior distributions estimated
using the Gibbs sampler. We can draw several conclusions from this graph.
First, several of the variance terms are not normally distributed, which
violates the normality assumption in OLS and non-Bayesian multilevel
regression. Second, normal confidence interval calculation for the parameters
will be inaccurate. Third, the long right-hand side tails in the distributions
for households and time, and to a lesser extent season, indicates that these
controls by themselves are insufficient to explain the observed yield
variability in a classical setting. Said differently, our model relies on
indicators for household ability, occurrences of weather, and technical change
and does not take into account their severity. The skewed output of our
Bayesian analysis hints that while most households are similar in ability
there are a few outlying farmers with exceptional ability. Similarly, while
weather events account for only 20 percent of variance in yields, severe
weather events remain important in considering tools to mitigate risk. Our
actuarial analysis in Section 5 addresses this limitation in our regression
analysis by separately assessing the importance of weather severity in
determining the potential benefits of weather index insurance for rural
households.

###  4.2 Variance Results by Crop Type

Up to this point, our analysis has accommodated heterogeneity in the input-
response curves of the crops under consideration by allowing slopes and
intercepts to vary across crops. By pooling these data, however, the variance
results reported in Table 3 are implicitly derived under the assumption that
the same variance structure of the hierarchical model applies to all crops
under consideration and that, by extension, the same weather risk profile
applies to each crop. We now relax that assumption and re-estimate the MLE and
Bayesian regressions, in each case using five separate, crop-specific sub-
samples of the data. Our goal is to allow for a more comprehensive assessment
of the variance structure of yields, to gain additional insights into the
nature of weather risk exposure among farmers in the sample. The new samples
vary in size and coverage, which also permits us to limit the influence of
rice in our results, since rice dominates the pooled sample by contributing
roughly half of all observations. In three cases (rice, wheat, and maize) we
observe data across all 10 seasons. In the case of sorghum and cotton we
observe data for only five seasons.

As a complement to the data reported in Table 3, we present in Table 4 a full
set of estimated variance parameters, ICCs, and variance shares for the crop-
specific versions of our regressions.151515We relegate the discussion of the
point estimates for parcel-level inputs to Appendix A and Appendix Table A1.
As in the case of the regressions with the pooled samples, the MLE and
Bayesian regressions explicitly account for the clustering of observations at
the parcel, household, season, village, and temporal levels. The point
estimates from the Bayesian regression are based on posterior density
estimates derived from iterating from our uninformative priors. As before, we
focus attention on Panel C of Table 4. Once again we see a close
correspondence between the MLE and Bayesian results. Reading across the table,
however, we observe substantial variation in the sources and relative
importance of each level in explaining overall variance in crop-specific
yields. For example, over 30 percent of variance comes from between-parcel
differences for rice and maize, compared with less than two percent for
sorghum and cotton. Of particular importance for index insurance, we observe
only moderate variation in the contribution of between-season weather
differences, which range from 14 percent of variance in the case of rice, to
as much as 21 percent in the case of wheat and cotton. We also find more
notable differences between the estimated variances from MLE and Bayesian
regressions, with implications for the role of non-normality and the inability
of the MLE approach to properly account for skewness in several of the
disturbance terms. As noted above, to the extent that non-normal errors
account for more of the total variance one would expect to find greater bias
in the OLS and MLE point estimates vis-à-vis the Bayes estimates. This is
evident when level variance terms tend to be small, resulting in MLE estimates
that collapse to zero while Bayesian estimates are small but non-zero.

##  5 Actuarial Values of Rainfall-Index Insurance Risk

The likely importance of extreme and infrequent weather events in yield
variability, evidenced by the skewness in the posterior distribution of
$\sigma_{3}^{2}$, highlights the need for smallholder farmers to have access
to affordable risk-management tools. In other words, despite the low share of
seasonal weather variation in yield variability, our data still suggest that
potential insurance purchasers need to consider weather risk, since extreme
weather events, while rare, are nevertheless potentially costly over the long
run. This motivates us to ask whether tools for managing this type of risk are
adequately accessible. In this section, we complete our analysis of weather
risk and agricultural production with a brief quantitative assessment which
suggests that rainfall index insurance, as currently marketed in village
India, is overpriced.

Our parcel-level panel data are not adapted to measuring the severity of
weather events. In order to estimate the actuarial cost of these events, we
rely on rain-index insurance contracts which ICICI Lombard offers in the
region and which form the basis for the analysis conducted by Cole et al.
(2013) and Giné et al. (2007). The terms of these contracts are district-
dependent, and are the sum of three individual contracts in the three
successive phases of the monsoon season. The start and finish dates defining
each of the three phases are determined for each village for each year, as a
function of the daily evolution of rainfall for the village’s corresponding
weather station.161616Specifically, Phase I begins on the day when accumulated
rainfall since the traditional monsoon start date exceeds $50$mm. In Eastern
and Central India, monsoon start date is generally considered 1 June, so Phase
I begins for a village when accumulated rainfall since 1 June exceeds $50$mm.
In Western India monsoon start date is figured as 1 July. If accumulated
rainfall in the first month fails to exceed $50$mm, then the first phase
automatically begins on the first of the next month. Phase I and II are each
35 days in length. Phase II begins the day after the close of Phase I. Phase
III begins the day after the close of Phase II. In contrast to the first two
phases, Phase III lasts for 45 days (World Bank, 2011). In Phases I and II,
corresponding to planting and heading, a drought risk exists. In Phase III,
corresponding to harvest, risk arises from excess rain. In Phase I and II the
contract pays zero Indian Rupees (Rs.) if the cumulative rainfall exceeds a
certain “strike” value $k$. It then pays a number of Rs. which decreases
linearly with the cumulative rainfall in that phase (the value reported in the
contract is the slope’s absolute value $m$), and if this rainfall amount
reaches down to or beyond a certain low “exit” number $z$, the contract pays a
maximum value $M$, typically $M=1000$ Rs.171717Typically, the payout amount at
the exit value $z$, which equals $(k-z)m$, is less than the maximum payout $M$
after exit, though it can sometimes be higher; such an odd structure might be
construed as a contract design error, but does occur in commercial practice;
we have encountered one instance of this in our analysis of the ICICI Lombard
contracts. The probability of the rainfall equaling exactly $z$ millimeters is
extremely low, which means that there is usually no ambiguity as to whether to
pay the value $(k-z)m$ or the typically larger value $M$. We have identified
one important exception to this situation, that in which the exit equals $0$.
ICICI Lombard proposes such contracts in relatively arid districts. In that
case, the ambiguity would pose a problem, and the company pays the exit max
value $M$ when rainfall is $0$. We have taken this into account in our
calculations. The contract is therefore structured so as to have a piecewise
linear payout which is typically discontinuous, with a single jump up as
rainfall goes from the exit value $z$ to lower values (See Figure 3). In Phase
III, the contract is structured in an opposite way: the payout equals zero for
cumulative rainfall below the strike value $k$, it equals $M$ (typically
$1000$ Rs.) above the exit value $z$, and it grows linearly with positive
slope $m$ for cumulative rainfall between $k$ and $z$. The linear increasing
or decreasing portions of the contract structure means that its actuarial
value will be a bona fide quantitative measure of the severity of drought
and/or flood risk. This would not be the case if the contract paid only either
zero or a fixed lump sum, since in that case, the actuarial value would only
measure the probability of a payout, not drought/flood severity.

We utilize daily rainfall data from 16 of the VDSA villages selected to
provide a wide geographic distribution of Indian states with semi-arid,
tropical wet-and-dry, and humid subtropical climates from Andhra Pradesh to
Bihar for the same years as our regression data, which includes the 2009
drought year. Our method of calculating the actuarial value of the payouts
differs from that used in previous studies of the same contracts, which
generally take a long view and utilize rainfall data extending as far back as
four decades (Parchure, 2002; Clarke et al., 2012; Cole et al.,
2013).181818Cole et al. (2013) provide a straightforward and representative
explanation of the standard method of calculation. They utilize 36 years of
rainfall data from five villages in Andhra Pradesh and 38 years of rainfall
data from three villages in Gujarat to calculate values for each of the three
contract phases. We instead use five years of rainfall data to determine
rainfall in each of the three phases in 16 villages. Because of the wide
geographic distribution of our data, this averaging encompasses a number of
differing weather conditions, which can be considered as a proxy for following
a specific village over a long period of time, and ensures that a sufficient
number of extreme weather events are included, a particularly useful feature
in this region where droughts are sporadic. Additionally, we believe that the
shorter time horizon utilized in our calculation more closely approximates the
decision horizon for smallholder farmers in India. Given that the average age
of the head of household in our data is 49 years, rainfall information from 38
years ago is not likely to figure into insurance purchase
decisions.191919Another instance of using wide geographic diversity to study
the distribution of rain-indexed insurance in India can be found in Giné et
al. (2007), where available commercial premia, strikes, and exits, were
averaged over a number of different districts/weather stations. See Appendix C
for a detailed justification of this approach.

Using our five year time horizon combined with wide geographic diversity, we
calculate an estimate of the actuarial value of the payouts for the 3-phase
contract by averaging payouts over all $171$ village-year-phase
combinations.202020We have five years of rainfall observations for three
villages in Andhra Pradesh and two villages in Karnataka. Due to data
collection issues the VDSA only reports three years of rainfall data for four
villages in each Bihar and Odisha and three villages each in Jharkhand. Given
that the contracts under analysis have three phases, this gives us 171
rainfall data points. To illustrate the effect of how different strike and
exit values effect the insurance contract’s value, we repeat the calculation
for three contracts reported in Giné et al. (2007) and Cole et al. (2013),
with low, medium, and high payout structures, and report on the payouts’
actuarial values and payout probabilities. The results from this analysis are
given in Table 5.

The three contracts we consider here are representative of products currently
marketed to smallholder farmers in the region and were chosen so as to present
the most conservative analysis possible.212121We have included in Appendix C
another set of contracts, from Cole et al. (2013) as robustness checks to the
values presented in the paper. These contracts have less favorable terms, and
would paint a picture of the options available to micro-insurance customers
which is arguably too extreme. We have presented our analysis of these less
favorably marketed contracts in Appendix Table A4 for comparison purposes and
for the sake of completeness. Under the most generous payout structure (“high
payout”), the probability of payout is 14 percent. This represents, on
average, little more than one payout in one phase over a two-year period.
Judging by the examples of contracts reported in Cole et al. (2013), it is
more likely that smallholders would be offered contracts in the medium or low
payout ranges, where payout probabilities are 12.3 and 8.8 percent
respectively. Farmers who do not look ahead more than three years in planning
their activities would presumably find little incentive to purchase insurance
where, on average, a 12 percent figure means that they would be as likely as
not to see no payout within their planning horizon, and an eight percent
figure would br even less favorable. This infrequency of payout on what
amounts to income insurance is likely to compound the friction caused by slow
diffusion of uptake. For example, studying households in Gujarat, Cole et al.
(2014) find that demand for insurance is highly sensitive to whether payouts
occurred in the village in the prior year. We suspect that in the villages we
consider, only the high payout contract would likely be of interest to
farmers, and then only if they perceived its premium as close to fair. Table 5
reports those actuarially fair premia.

Cole et al. (2013) report expected payouts as percentages of true premia paid
to the ICICI Lombard insurance company for two villages in Andhra Pradesh in
2006. These numbers are equivalent to saying that the premium is computed by
multiplying the actuarially fair premium by a loading factor, $1+\lambda$,
through the formula:

$1+\lambda=\frac{\textrm{paid price}}{\textrm{actuarially fair price}}.$ (17)

Thus $\lambda$ is a proportional transaction cost which, in principle,
reflects the cost to the company of doing business. The loading factors’
transaction cost parameters computed from reported premia ($1.47$, $2.32$,
$3.86$), are very large compared with other weather insurance markets: a
typical value of $\lambda$ in the US and other countries is between $0.1$ and
$0.2$. For instance, the Poverty Global Practice Group at the World Bank
reports values in the US, Burkina Faso, and Senegal of $\lambda=0.10$ (de
Nicola, 2015). While it may not be possible for insurers to offer contracts
with such low values of $\lambda$ in India, our actuarially fair value for the
high-payout contract, $191$ Rs., can be compared directly to the price quoted
by ICICI Lombard in Cole et al. (2013) for the same contract, which is $280$
Rs. In this case, which is the most favorable comparison we can draw with
commercial contract prices, the high-payout contract would thus represent a
loading factor of $1.47$, i.e. a value of $\lambda=0.47$, a substantially
higher value than those observed in other regions, but still much lower than
those inferred from the two villages reported in Cole et al. (2013). Such
excessive loading factors, and not apparent failure of households to recognize
the value of insurance, appear to be a key element in preventing farmers in
village India from accessing insurance as a risk-management resource and
further explains the low uptake rate of weather index insurance.

##  6 Conclusion

Despite long standing evidence that rural households are unable to fully
insure covariate risk, few studies have attempted to measure just how large a
role covariate events, such as weather, play in agricultural yields. We
address this research gap using agricultural production data covering 11,942
parcel level observations from India. Using a multilevel/hierarchical
regression framework, we estimate the different sources of yield variance.
This approach controls for inputs at the parcel-level and also isolate the
amount of yield variability attributed to parcel-level effects, household-
level effects, seasonal weather effects, village-level effects, and time
effects. Adopting Bayesian estimation techniques allows us to account for the
highly skewed distribution of several of the disturbance terms. Overall, we
find that variability in weather makes up only a small share of the total
variance in crop yields. This suggests that basis risk on the production side
(i.e., low correlation between the weather index and yield loss) is
substantial. The majority of variation in yields does not come from seasonal
variability, which is our proxy for rainfall or weather variability. Rather,
the majority of variation in yields comes from differences in parcels and from
the random disturbance term which captures idiosyncratic events which would
be, by definition, not covered by index insurance. Given the long history of
evidence that farmers are effective in using self-insurance and mitigation
measures to protect against idiosyncratic risk and given the small role
covariate seasonal risk plays in crop variability, farmers may rationally
prefer to forgo weather index insurance at any positive price to focus their
risk management choices on minimizing other sources of risk.

Although there are many potential impediments to insurance uptake, this
research provides evidence for an obvious but until now unsupported
explanation: insurance contracts are overpriced from the perspective of the
farming household. Product design and ratemaking of index insurance in India
has generally taken a long view, utilizing rainfall data extending as far back
as four decades to calculate actuarial rates. From the perspective of the
insurance company, such a long time horizon may support the company’s
longevity in managing risk. However, it may not easily align with the planning
horizons of rural households. Motivated by our econometric results, we take a
shorter-term perspective and calculate the actuarial value of payouts using
rainfall data from five seasons, including a drought season, and a large
number of geographically dispersed locations. We find that loading factors on
these contracts are excessive and that this pricing problem undermines the
incentives for smallholder farmers to purchase insurance. Our combines results
suggest that micro-insurance markets could hold limited promise for improving
household risk management, and then only to the extent that future products
can adapt to the realities facing potential purchasers.

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Table 1: Descriptive Statistics by Crop | Rice | Sorghum | Wheat | Maize | Cotton | Total
---|---|---|---|---|---|---
yield (kg/ha) | 3,356 | 2,368 | 1,537 | 2,596 | 1,383 | 2,658
| (3,252) | (1,429) | (4,038) | (2,397) | (870.4) | (2,956)
labor (hr/ha) | 880.1 | 301.6 | 760.5 | 831.7 | 1156.6 | 749.8
| (825.8) | (265.9) | (2,777) | (769.6) | (644.4) | (1,239)
fertilizer (kg/ha) | 215.0 | 289.3 | 174.6 | 252.3 | 333.9 | 239.2
| (213.6) | (191.9) | (724.0) | (224.6) | (210.0) | (333.4)
mechanization (Rs/ha) | 1,362 | 2,891 | 805.6 | 878.7 | 2,927 | 1,716
| (2,781) | (4,887) | (1,289) | (1,755) | (3,778) | (3,369)
pesticide (Rs/ha) | 61.7 | 124.5 | 57.2 | 59.8 | 2,066 | 235.0
| (266.7) | (566.3) | (345.9) | (319.0) | (2,748) | (1,012)
parcel area (ha) | 0.355 | 0.597 | 0.674 | 0.317 | 0.904 | 0.494
| (0.501) | (0.833) | (0.733) | (0.334) | (0.662) | (0.652)
rainfall (mm) | 539.1 | 395.0 | 286.5 | 456.3 | 426.3 | 468.4
| (253.3) | (166.6) | (83.1) | (192.2) | (178.1) | (234.8)
number of observations | 5,572 | 2,720 | 1,625 | 1,073 | 952 | 11,942
number of parcels | 2,678 | 1,636 | 1,151 | 549 | 600 | 5,100
number of households | 612 | 491 | 383 | 268 | 254 | 1,079
number of seasons | 106 | 81 | 69 | 86 | 51 | 240
number of villages | 21 | 20 | 20 | 20 | 12 | 30
number of time periods | 10 | 5 | 10 | 10 | 5 | 10
Note: Table displays means of data by crop with standard deviations in
parenthesis. All monetary values are in real $2010$ Indian Rupees (Rs).
Table 2: Results of Production Function Regressions

| OLS | MLE | Bayes
---|---|---|---
| (1) | (2) | (3)
rice
log labor | $1.635^{***}$ | $1.531^{***}$ | $1.127^{***}$
| $(0.043)$ | $(0.043)$ | $(0.032)$
log fertilizer | $0.598^{***}$ | $0.512^{***}$ | $0.542^{***}$
| $(0.019)$ | $(0.019)$ | $(0.018)$
log mechanization | $0.206^{***}$ | $0.176^{***}$ | $0.147^{***}$
| $(0.010)$ | $(0.009)$ | $(0.009)$
log pesticides | $0.123^{***}$ | $0.093^{***}$ | $0.088^{***}$
| $(0.011)$ | $(0.010)$ | $(0.010)$
sorghum
log labor | $0.585^{***}$ | $0.559^{***}$ | $0.987^{***}$
| $(0.063)$ | $(0.060)$ | $(0.042)$
log fertilizer | $0.087^{**}$ | $0.108^{***}$ | $0.214^{***}$
| $(0.042)$ | $(0.040)$ | $(0.038)$
log mechanization | $0.321^{***}$ | $0.342^{***}$ | $0.491^{***}$
| $(0.030)$ | $(0.029)$ | $(0.026)$
log pesticides | $0.009$ | $0.012$ | $0.009$
| $(0.018)$ | $(0.016)$ | $(0.016)$
wheat
log labor | $0.993^{***}$ | $0.988^{***}$ | $1.217^{***}$
| $(0.054)$ | $(0.053)$ | $(0.036)$
log fertilizer | $0.110^{***}$ | $0.108^{***}$ | $0.073^{***}$
| $(0.021)$ | $(0.020)$ | $(0.036)$
log mechanization | $0.386^{***}$ | $0.350^{***}$ | $0.366^{***}$
| $(0.019)$ | $(0.018)$ | $(0.036)$
log pesticides | $-0.059^{**}$ | $-0.069^{***}$ | $-0.100^{***}$
| $(0.026)$ | $(0.025)$ | $(0.025)$
maize
log labor | $1.126^{***}$ | $1.174^{***}$ | $1.584^{***}$
| $(0.100)$ | $(0.097)$ | $(0.043)$
log fertilizer | $0.048$ | $0.016$ | $-0.005$
| $(0.041)$ | $(0.039)$ | $(0.040)$
log mechanization | $0.111^{***}$ | $0.096^{***}$ | $0.109^{***}$
| $(0.018)$ | $(0.017)$ | $(0.017)$
log pesticides | $0.086^{***}$ | $0.094^{***}$ | $0.098^{***}$
| $(0.033)$ | $(0.030)$ | $(0.029)$
cotton
log labor | $1.047^{***}$ | $1.064^{***}$ | $1.291^{***}$
| $(0.129)$ | $(0.124)$ | $(0.057)$
log fertilizer | $0.220^{***}$ | $0.230^{***}$ | $0.230^{***}$
| $(0.057)$ | $(0.053)$ | $(0.053)$
log mechanization | $0.072^{**}$ | $0.033$ | $0.036$
| $(0.031)$ | $(0.031)$ | $(0.029)$
log pesticides | $0.128^{***}$ | $0.115^{***}$ | $0.095^{***}$
| $(0.029)$ | $(0.028)$ | $(0.027)$
observations | 11,942 | 11,942 | 11,942
R2 | 0.964 |  |
Log Likelihood |  | -22,268 |
Akaike Inf. Crit. |  | 44,598 |
Deviance Inf. Crit. |  | 44,828 | 42,578
Note: Dependent variable is log of yield. All specifications include
statistically significant crop-specific intercept terms as controls. Column
(1) is a classical OLS regression that includes season fixed effects but does
not account for the multilevel structure of the data. Column (2) is the
maximum likelihood estimate of the multilevel model and contains covariates
and data clustered at parcel, household, season, village, and time. Column (3)
is the Bayesian estimation of the multilevel model. Bayesian calculations use
a burn-in period of 5,000 iterations and an additional 5,000 iterations to
ensure convergence. Standard errors are reported in parentheses (∗p$<$0.1;
∗∗p$<$0.05; ∗∗∗p$<$0.01).

Table 3: Estimated Variance, ICCs, and Variance Shares from Multilevel Regressions | MLE | Bayes
---|---|---
|  |
Panel A: Variance Parameter Estimates
|  |
parcel $(\sigma_{1}^{2})$ | $0.933$ | $1.098$
household $(\sigma_{2}^{2})$ | $0.002$ | $0.004$
season $(\sigma_{3}^{2})$ | $0.790$ | $0.903$
village $(\sigma_{4}^{2})$ | $0.623$ | $0.682$
time $(\sigma_{5}^{2})$ | $0.126$ | $0.261$
idiosyncratic $(\sigma^{2})$ | $1.623$ | $1.613$
|  |
Panel B: Intraclass Correlation Coefficients
|  |
parcel | $0.228$ | $0.241$
household | $0.228$ | $0.242$
season | $0.421$ | $0.440$
village | $0.573$ | $0.589$
time | $0.604$ | $0.646$
|  |
Panel C: Shares of Variance From Each Level
|  |
parcel | 23% | 24%
household | 00% | 00%
season | 19% | 20%
village | 15% | 15%
time | 03% | 06%
idiosyncratic | 40% | 35%
observations | 11,942
Note: In Panel A, estimates of the variance parameters on each level’s
residuals come from estimation of models reported in Columns (2) and (3) in
Table 2. Variances $\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$,
$\sigma_{4}^{2}$, $\sigma_{5}^{2}$ represent the variance in crop yield that
comes from the corresponding level. The final variance parameter
$(\sigma^{2})$ corresponds to the idiosyncratic or unexplained portion of the
model. Intraclass correlation coefficients in Panel B are calculated using the
formulas in equations (7)-(11). Panel C decomposes the ICC into percent of
variance accorded to each level. For comparison, Appendix Tables A2 and A3
reports alternative specifications of the models presented here.
Table 4: Estimated Variance, ICCs, and Variance Shares by Crop

| Rice | Sorghum | Wheat | Maize | Cotton
---|---|---|---|---|---
| MLE | Bayes | MLE | Bayes | MLE | Bayes | MLE | Bayes | MLE | Bayes
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
|  |  |  |  |  |  |  |  |  |
Panel A: Variance Parameter Estimates
|  |  |  |  |  |  |  |  |  |
parcel $(\sigma_{1}^{2})$ | $1.858$ | $1.852$ | $0.000$ | $0.002$ | $1.092$ | $1.059$ | $0.564$ | $0.557$ | $0.000$ | $0.018$
household $(\sigma_{2}^{2})$ | $0.000$ | $0.003$ | $0.048$ | $0.046$ | $0.046$ | $0.034$ | $0.000$ | $0.007$ | $0.013$ | $0.010$
season $(\sigma_{3}^{2})$ | $0.832$ | $0.865$ | $0.166$ | $0.174$ | $2.323$ | $2.573$ | $0.234$ | $0.266$ | $0.367$ | $0.401$
village $(\sigma_{4}^{2})$ | $1.098$ | $1.266$ | $0.133$ | $0.156$ | $5.527$ | $6.411$ | $0.185$ | $0.177$ | $0.000$ | $0.041$
time $(\sigma_{5}^{2})$ | $0.203$ | $0.298$ | $0.000$ | $0.011$ | $0.612$ | $0.694$ | $0.000$ | $0.013$ | $0.521$ | $0.689$
idiosyncratic $(\sigma^{2})$ | $1.918$ | $1.918$ | $0.686$ | $0.684$ | $1.598$ | $1.626$ | $0.517$ | $0.518$ | $0.785$ | $0.766$
|  |  |  |  |  |  |  |  |  |
Panel B: Intraclass Correlation Coefficients
|  |  |  |  |  |  |  |  |  |
parcel | $0.314$ | $0.299$ | $0.000$ | $0.002$ | $0.098$ | $0.085$ | $0.376$ | $0.362$ | $0.000$ | $0.009$
household | $0.314$ | $0.299$ | $0.046$ | $0.045$ | $0.102$ | $0.088$ | $0.376$ | $0.366$ | $0.008$ | $0.015$
season | $0.455$ | $0.439$ | $0.207$ | $0.207$ | $0.309$ | $0.296$ | $0.532$ | $0.539$ | $0.225$ | $0.223$
village | $0.641$ | $0.643$ | $0.336$ | $0.352$ | $0.803$ | $0.813$ | $0.655$ | $0.654$ | $0.225$ | $0.244$
time | $0.675$ | $0.691$ | $0.336$ | $0.363$ | $0.857$ | $0.869$ | $0.655$ | $0.663$ | $0.534$ | $0.602$
|  |  |  |  |  |  |  |  |  |
Panel C: Shares of Variance From Each Level
|  |  |  |  |  |  |  |  |  |
parcel | 31% | 30% | 00% | 00% | 10% | 09% | 38% | 36% | 00% | 01%
household | 00% | 00% | 05% | 04% | 00% | 00% | 00% | 00% | 01% | 01%
season | 14% | 14% | 16% | 16% | 21% | 21% | 16% | 17% | 22% | 21%
village | 19% | 20% | 13% | 15% | 49% | 52% | 12% | 12% | 00% | 02%
time | 03% | 05% | 00% | 01% | 05% | 06% | 00% | 01% | 31% | 36%
idiosyncratic | 32% | 31% | 66% | 64% | 14% | 13% | 34% | 34% | 47% | 40%
observations | 5,572 | 2,720 | 1,625 | 1,073 | 952
Note: In Panel A, estimates of the variance parameters on each level’s
residuals come from estimation of models in corresponding columns in Appendix
Table A1. Variances $\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$,
$\sigma_{4}^{2}$, $\sigma_{5}^{2}$ represent the variance in crop yield that
comes from the corresponding level. The final variance parameter
$(\sigma^{2})$ corresponds to the idiosyncratic or unexplained portion of the
model. Intraclass correlation coefficients in Panel B are calculated using the
formulas in equations (7)-(11). Panel C decomposes the ICC into percent of
variance accorded to each level. Bayesian calculations use a burn-in period of
5,000 iterations and an additional 5,000 iterations to ensure convergence.

Table 5: Actuarial Values and Payout Probabilities

| Contract Structure | Summary Statistics
---|---|---
|  |  |  | Actuarially Fair | Probability | Loading | Years until
| Strike | Exit | Max Payout | Premium | of Payout | Factor | Payout
|  |  |  |  |  |  |
Panel A
High-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $70$ | $10$ | $1000$ | 190.9 | 14.0% | 1.47 | 2.38
Phase II | $80$ | $10$ | $1000$ |  |  |  |
Phase III | $375$ | $450$ | $1000$ |  |  |  |
|  |  |  |  |  |  |
Panel B
Medium-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $78$ | $15$ | $1000$ | 129.2 | 12.3% | 2.32 | 2.71
Phase II | $72$ | $12$ | $1000$ |  |  |  |
Phase III | $499$ | $580$ | $1000$ |  |  |  |
|  |  |  |  |  |  |
Panel C
Low-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $50$ | $5$ | $1000$ | 69.6 | 8.77% | 3.86 | 3.80
Phase II | $60$ | $5$ | $1000$ |  |  |  |
Phase III | $560$ | $670$ | $1000$ |  |  |  |
Note: High- and low-payout contracts come from Cole et al. (2013) while the
medium-payout contract comes from Giné et al. (2007). While each contract from
Cole et al. (2013) is designed for a specific village/weather station, we
follow Giné et al. (2007) in utilizing a large representative set of rainfall
data to calculate actuarially fair premia and the probability of payout.
Actuarially fair premia are calculated using standard actuarial principles.
Payout probability is the average occurrence of a payout. Loading factors are
calculated as $1+\lambda=\frac{\textrm{paid price}}{\textrm{actuarially fair
price}}$. Years until payout are calculated by dividing the inverse of the
probability of payout by three, which assumes payout events occur as a Poisson
process. For comparison, Appendix Table A4 reports the same calculations for
the remaining three contracts reported in Cole et al. (2013).

Figure 1: Zeta Profile Plots for Multilevel Model

Note: The figure presents the zeta profile plots, representing the underlying
distribution of the parameters resulting from the model fit to equations
(5a)-(5f). Plots are calculated by first estimating the model. Then, holding a
single parameter fixed, the other parameters are varied to assess the fit of
each new iteration compared to the globally optimal fit. The comparison
statistic is the likelihood ratio test. Finally, a signed square root
transformation is applied to the LR statistic. The plots are the absolute
value of the resulting function, $|\zeta|$, in comparison to the estimated
parameter values. Parameters with underlying normal distributions have
straight line zeta profile plots. The vertical lines delimit $50\%$, $80\%$,
$90\%$, $95\%$, and $99\%$ confidence intervals. For zeta profile plot lines
that are not straight, the normal distribution is a poor approximation of the
underlying distribution.

Figure 2: Histograms of Level Variance from Bayesian Estimation

Note: Histograms are drawn from posterior distributions of variance terms
estimated from Column (3) reported in Table 2.

Figure 3: Structure of Insurance Contract

Note: Figure shows the representative contract structure used in calculating
payouts presented in Table 5. Specifically the figure represents the Phase I
contract for the high-payout case. The Phase II contract structure is
identical to the Phase I contract with the exception of strike and exit
values. The Phase III contract does not differ in structure but has strike and
exit values set to insure against excess rainfall, meaning the contract slope
is positive. Additionally, the contract is representative of contracts on
offer by ICICI Lombard throughout the region of Andhra Pradesh. The contract
in each phase has an upper and lower rainfall threshold. The policy pays zero
above the strike; otherwise the policy pays a fixed amount for each millimeter
of shortfall (excess in the case of Phase III), until the exit is reached. In
the contract represented this is 10 Rs. per mm. At the exit point the contract
is discontinuous and “jumps” to its maximum value. Note that this specific
contract comes from Giné et al. (2007) and is a slight variation on the
contracts presented in Giné et al. (2008); Giné et al. (2012) and Cole et al.
(2013). The contracts presented in these papers share the same structure but
vary in the strike and exit values.
Source: Giné et al. (2007).

##  Appendix A For Online Publication: Results from Estimation of Crop-
Specific Production Functions

In the interests of space we limit the results from the crop-specific
regressions presented in the paper and simply present the variance parameter
estimates. Here we present the full results from the crop-specific
regressions.

For reference, Table 2 presents the results from the varying-slope model using
OLS, MLE, and Bayesian estimation techniques. Table A1 presents the MLE and
Bayesian point estimates from the production function estimates for each crop.

While point estimates change slightly from the full regression to the crop-by-
crop regressions, qualitatively there is no change. For rice, all point
estimates in both specifications are positive and significant. For sorghum,
all point estimates, save for pesticide, in both specifications are positive
and significant. For wheat, all point estimates in both specifications are
positive and significant, save for pesticide, which is negative and
significant. For maize, all point estimates are positive and significant, save
for fertilizer, which is not significant in both specifications. For cotton,
all point estimates in both specifications are positive and significant, save
for mechanization, which is not significant.

The consistency in point estimates across estimation techniques (OLS, MLE,
Bayes) and across divisions of the data demonstrate a remarkably robust set of
results.

##  Appendix B For Online Publication: Alternative Specifications of the
Multilevel Model

In order to verify the robustness of our primary results this appendix
presents a variety of alternative specifications of our model. The goal is to
determine if grouping the data in different ways might have an effect on our
inferences.

First, in Table A2, we test a variety of “null” models in which covariates are
excluded. In each model we add complexity by including a new level effect.
This allows us to see how our estimate of seasonal variability changes as we
account for additional level effects. We start in columns (1) and (2) with
simply the season, village, and time levels and build up from there. Adding a
household level in columns (3) and (4) simply reduces the unassigned variance
in the idiosyncratic term and has no effect on the other sources of variance.
Adding a parcel level in columns (5) and (6) has a small effect on season-
level variance but primarily reduces the amount of variance ascribed to the
idiosyncratic error term. In columns (7) and (8) we add in parcel-level
measured inputs, which is our preferred specification presented in the paper.
Compared to the “null” models, the inclusion of inputs has the effect of
reduce the variance ascribed to parcel-level effects and the idiosyncratic
error term. It also has the effect of increasing the variance due to weather
and variance at the village-level.

Second, we test models which exclude season, village, and temporal levels.
Each regression includes the full contingent of measured inputs which we
refrain from reporting since our focus is on alternative specifications for
the variance structure. The intuition for focusing on the effect of village
and time levels in the multilevel model is that our proxy for weather is an
interaction effect between village and time. Because of this, the village
and/or time levels may be picking up some of the “true” effects of weather.

Table A3 presents results from six specifications estimated using
MLE.222222The Bayesian estimates tell the same story as the maximum likelihood
estimates and so are omitted. Bayesian estimates are available from the
authors upon request. Column (1) reports results from including just a time-
level as a proxy for weather. Variation in time alone would be a valid proxy
for weather if all locations in the study experienced the same weather. Given
the large geographic dispersion it is unsurprising that a weather proxy that
disregards local variation in weather accounts for a vary small share of total
variance. Column (2) replaces the time-level with a village-level. Variation
across villages alone would be a valid proxy for weather if each time period
experienced the same weather. Given that there are two distinct seasons in
India - wet _Kharif_ and dry _Rabi_ \- and that we examine only five years,
this is not unreasonable. We see this in the relatively large share of
variance due to the village-level. However, there may be other spatial
elements that our village term picks up, such as agronomic suitability or
government policy.

Column (3)-(6) include a season-level which is the interaction between village
indicators and time indicators. This is our preferred proxy for weather as it
allows weather to vary across both villages and time. In column (3) we drop
the village and time levels in favor of only the season-level. The result is a
relatively large share of variance ascribed to differences across location-
time. This might be the “true” measure of the effect of weather variability on
yield variability. Or, there might be location specific effects and time
specific effects that are being captured in this term. To check this, in
column (4) we add back in the time-level effect. Having controlled for
seasonality, the time effect will only pick up variation in yields across time
separate from variation in location-time. Such variation would be from
technological change that effects all villages at the same time or,
potentially, a nationwide weather event. Given that our data covers only five
years the time-only effect is small. In column (5) we add back in the village-
level effect instead of the time effect. Having controlled for seasonality,
the village effect will only pick up variation in yields across village
separate from variation in location-time. Such variation would be from
differences in time-invariant agronomic characteristics of the villages or
differences in state policies that have different impacts of villages. Given
that we have 30 villages across eight Indian states the location-only effect
is moderate, relative to other sources of variation. Column (6) is our
preferred specification since it controls for time-only, location-only, and
location-time events.

Compared to the alternate specifications presented in Table A3, the
specification presented in Column (6) and the paper is a better fit for the
data as measured by the log likelihood, Akaike Information Criterion (AIC),
Deviance Information Criterion (DIC), and the size of the idiosyncratic
residual. The better fit and the ability to control for all potential sources
of variance makes this our preferred specification.

##  Appendix C For Online Publication: Details on the Computation of
Actuarially Fair Premia and Payout Probabilities

This appendix provides details on the sources of data for and the computation
of actuarially fair premia and payout probabilities. It also contains several
robustness checks in which we calculate premia and payout probabilities using
data on actual index insurance contracts provided by Giné et al. (2007); Giné
et al. (2012) and Cole et al. (2013).

###  C.1 Source of the Data

Though we have access to daily rainfall data for all 30 villages in the VDSA
data set, we limit our calculation of premia and payouts to a subset of 16
villages within the Indian states of Bihar, Jharkhand, Odisha, Andhra Pradesh,
and Karnataka. This is done for several reasons.

First, the 16 villages are clustered in nine districts, all of which fall
within the semi-arid, tropical wet-and-dry, and humid subtropical climate
zones. These zones all share quantitatively similar rainfall patterns and
variability during the monsoon season (Giné et al., 2009). Despite the shared
climate zones, our village rainfall data exhibit a level of variability, due
to the wide spatial range, which is of the same order of magnitude as the
variability one expects for monsoon rainfall in a single location in the same
general region of Central and Eastern India. The actual rainfall values are
also representative of what is observed for individual, non-VDSA, locations in
the region. In this fashion, our spatial rainfall variability acts as a proxy
for temporal variability at the district level. We can therefore view our data
as a quasi-panel with multiple observations over space instead of the
traditional panel structure of multiple observations over time.

Second, the restriction of analysis to villages from these five states is
consistent with the premium data from commercial insurers ICICI-Lombard and
IFFCO-Tokio that are most frequently utilized in the academic
literature.232323See the various contracts reported in Clarke et al. (2012),
Clarke et al. (2012), Cole et al. (2013), Giné et al. (2007), Giné et al.
(2008), and Giné et al. (2012). Contracts reported therein are largely for
various village-weather station pairs in Andhra Pradesh with similar monsoon
rainfall characteristics to the villages in the five states we consider from
the VDSA.242424Cole et al. (2013) reports contracts from the largest set of
village-weather station pairs: Anantapur, Atmakur, Hindupur, Narayanpet, and
Mahbubnagar. The three VDSA villages in Andhra Pradesh come from the districts
Prakasam and Mahbubnagar.

Third, inclusion of villages in the Indian states of Madhya Pradesh and
Gujarat turn out to be outliers in the humid direction in Phases I and II. By
excluding these outliers, and focusing on the subset of 16 villages, we
preserve the aforementioned proxy property. This allows us to avoid the use of
villages where rainfall-indexed payouts would never occur. While this
obviously biases our results, given that we exclude villages that never see a
payout, the bias is downwards, making the results presented in the paper
conservative in terms of actuarially fair premia and probability of payout for
the corresponding contracts. Had we included these outliers, our conclusions
about the excessively high commercial loading factors would be even more
extreme.

###  C.2 Computation of Actuarially Fair Premia and Payoff Probabilities

The rain-indexed contracts compute payouts uniformly for all contracts in a
given village by using the rainfall values reported from that village’s rain
gauge. We compute the total rainfall in millimeters for each of the three
phases of the monsoon season for that village and insert these phase totals
into the contract payout function.252525See the definition of the start and
stop of each phase in the text. We work with payout functions which are
characterized by a strike, an exit, and a max payout.262626Again, see the
definitions of these terms and of the function in the text. While there is a
variety of contracts on offer these contracts differ not in structure but only
in their strike, exit, payout slope, and maximum payout. We calculate
actuarially fair premia and payoff probabilities for all five contracts from
Andhra Pradesh in Cole et al. (2013) plus the single contract in Giné et al.
(2007). While Clarke et al. (2012) and Giné et al. (2008) present similarly
structured contracts, the third phase in these contracts insure against
deficit rainfall and therefore are not appropriate for villages in the climate
regions where our rainfall data comes from.

We compute actuarially fair premia using standard actuarial principles based
on the empirical data. Namely, we evaluate the payout function for each
contract at each rainfall phase total and then average these empirical payouts
over the 57 data points for each phase of each contract. The total empirical
actuarially fair value for each contract is simply the sum of the three
corresponding values for each phase of the contract. We compute the payout
probability for each contract by averaging the values 1 or 0 corresponding to
the occurrence of a payout or not for each of the 171 village-year-phase
combinations. This way of accounting keeps track of multiple payouts in each
contract year, which is more conservative (providing a higher empirical
probability) than if one only looked for whether there was one or more payouts
in a given contract year.

###  C.3 Actuarially Fair Premia and Payoff Probabilities Under Alternative
Contracts

As explained in Section 5, we presented in the paper only the two most
favorable contracts from Cole et al. (2013) plus the contract from Giné et al.
(2007). In this appendix we present our calculations from the three less
favorable contracts. As can be seen in Table A4, the payout probabilities for
these contracts are $8.8$, $4.1$, and $2.9$ percent, making them very unlikely
to be taken up by farmers. Additionally, the premia determined by ICICI
Lombard are so high in comparison to the actuarially fair prices we have
computed empirically, that the corresponding loading factors all exceed three,
a sign that these contracts are severely overpriced. Besides our desire to
take a conservative approach, another sign that the analysis of these three
contracts is not appropriate for use as part of our main argument is that
their commercial pricing, while uniformly excessive, is not internally
consistent. Indeed, these three contracts come from three weather stations
(near the towns of Hindupur, Narayanpet, and Mahbubnagar) which are all in the
same climate zone, are all geographically near each other, and all share the
same maximum payout and payout slope. Therefore, since they only differ by
strike and exit values, actuarial principles dictate that the most favorable
strikes and exits should have the highest premium. That would imply that the
premium for the Hindupur contract should be lower than for the other two
contracts. In reality, as seen in Cole et al. (2013), it is higher. When we
consider these contracts, in combination with our preceding analysis, we find
it unsurprising that Cole et al. (2013) found little uptake of weather index
insurance contracts in their randomized control trial, except when farmers
were offered a cash discount. Our findings provide a simple explanation for
the experimental results: the contracts hold little value for rural
households.

Table A1: Classical and Bayesian Estimation of Production Functions by Crop
Type

| Rice | Sorghum | Wheat | Maize | Cotton
---|---|---|---|---|---
| MLE | Bayes | MLE | Bayes | MLE | Bayes | MLE | Bayes | MLE | Bayes
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
log labor | $1.549^{***}$ | $1.552^{***}$ | $0.472^{***}$ | $0.473^{***}$ | $1.411^{***}$ | $1.411^{***}$ | $0.869^{***}$ | $0.871^{***}$ | $1.196^{***}$ | $1.201^{***}$
| $(0.054)$ | $(0.054)$ | $(0.039)$ | $(0.041)$ | $(0.071)$ | $(0.070)$ | $(0.073)$ | $(0.077)$ | $(0.080)$ | $(0.081)$
log fertilizer | $0.594^{***}$ | $0.594^{***}$ | $0.123^{***}$ | $0.123^{***}$ | $0.120^{***}$ | $0.120^{***}$ | $-0.025$ | $-0.026$ | $0.249^{***}$ | $0.248^{***}$
| $(0.026)$ | $(0.027)$ | $(0.025)$ | $(0.026)$ | $(0.027)$ | $(0.028)$ | $(0.030)$ | $(0.032)$ | $(0.037)$ | $(0.039)$
log mechanization | $0.196^{***}$ | $0.197^{***}$ | $0.232^{***}$ | $0.232^{***}$ | $0.449^{***}$ | $0.450^{***}$ | $0.042^{***}$ | $0.041^{***}$ | $-0.001$ | $-0.003$
| $(0.013)$ | $(0.013)$ | $(0.019)$ | $(0.020)$ | $(0.024)$ | $(0.024)$ | $(0.014)$ | $(0.015)$ | $(0.023)$ | $(0.023)$
log pesticides | $0.114^{***}$ | $0.114^{***}$ | $0.012$ | $0.011$ | $-0.161^{***}$ | $-0.162^{***}$ | $0.045^{*}$ | $0.045^{***}$ | $0.140^{***}$ | $0.138^{***}$
| $(0.014)$ | $(0.014)$ | $(0.011)$ | $(0.011)$ | $(0.035)$ | $(0.035)$ | $(0.025)$ | $(0.024)$ | $(0.020)$ | $(0.020)$
observations | 5,572 | 2,720 | 1,625 | 1,073 | 952
num. seasons | 10 | 5 | 11 | 10 | 5
Log Likelihood | -11,264 |  | -3,507 |  | -3,183 |  | -1,521 |  | -1,308 |
Akaike Inf. Crit. | 22,551 |  | 7,036 |  | 6,388 |  | 3,064 |  | 2,639 |
Deviance Inf. Crit. | 22,482 | 21,185 | 6,961 | 6,872 | 6,328 | 5,979 | 2,996 | 2,721 | 2,574 | 2,532
Note: Dependent variable is log of yield. All specifications include a
statistically significant intercept term. Columns (1) and (2) are estimates of
the production function for rice. Columns (3) and (4) are estimates of the
production function for sorghum. Columns (5) and (6) are estimates of the
production function for wheat. Columns (7) and (8) are estimates of the
production function for maize. Columns (9) and (10) are estimates of the
production function for cotton. Odd numbered columns are maximum likelihood
estimates while even numbered columns are Bayesian estimates. Bayesian
calculations use a burn-in period of 5,000 iterations and an additional 5,000
iterations to ensure convergence. Standard errors are reported in parentheses
(∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01).

Table A2: Estimated Variance, ICCs, and Variance Shares from Null Multilevel
Regression Models

| MLE | Bayes | MLE | Bayes | MLE | Bayes | MLE | Bayes
---|---|---|---|---|---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
|  |  |  |  |  |  |  |
Panel A: Variance Parameter Estimates
|  |  |  |  |  |  |  |
parcel $(\sigma_{1}^{2})$ |  |  |  |  | $2.226$ | $2.301$ | $0.933$ | $1.098$
household $(\sigma_{2}^{2})$ |  |  | $0.030$ | $0.020$ | $0.000$ | $0.002$ | $0.002$ | $0.004$
season $(\sigma_{3}^{2})$ | $0.557$ | $0.567$ | $0.557$ | $0.564$ | $0.479$ | $0.487$ | $0.790$ | $0.903$
village $(\sigma_{4}^{2})$ | $0.166$ | $0.339$ | $0.165$ | $0.335$ | $0.258$ | $0.402$ | $0.623$ | $0.682$
time $(\sigma_{5}^{2})$ | $0.026$ | $0.100$ | $0.026$ | $0.112$ | $0.035$ | $0.095$ | $0.126$ | $0.261$
idiosyncratic $(\sigma^{2})$ | $3.853$ | $3.920$ | $3.826$ | $3.901$ | $2.068$ | $2.082$ | $1.623$ | $1.613$
|  |  |  |  |  |  |  |
Panel B: Intraclass Correlation Coefficients
|  |  |  |  |  |  |  |
parcel |  |  |  |  | $0.439$ | $0.429$ | $0.228$ | $0.241$
household |  |  | $0.007$ | $0.004$ | $0.439$ | $0.429$ | $0.228$ | $0.242$
season | $0.121$ | $0.115$ | $0.127$ | $0.118$ | $0.534$ | $0.520$ | $0.421$ | $0.440$
village | $0.157$ | $0.184$ | $0.163$ | $0.186$ | $0.585$ | $0.594$ | $0.573$ | $0.589$
time | $0.163$ | $0.204$ | $0.169$ | $0.209$ | $0.592$ | $0.612$ | $0.604$ | $0.646$
|  |  |  |  |  |  |  |
Panel C: Shares of Variance From Each Level
|  |  |  |  |  |  |  |
parcel |  |  |  |  | 44% | 43% | 23% | 24%
household |  |  | 01% | 00% | 00% | 00% | 00% | 00%
season | 12% | 12% | 12% | 12% | 09% | 09% | 19% | 20%
village | 04% | 07% | 04% | 07% | 05% | 07% | 15% | 15%
time | 01% | 02% | 01% | 02% | 01% | 02% | 03% | 06%
idiosyncratic | 84% | 80% | 83% | 79% | 41% | 39% | 40% | 35%
Observations | 11,942 | 11,942 | 11,942 | 11,942 | 11,942 | 11,942 | 11,942 | 11,942
Log Likelihood | -25,231 |  | -25,226 |  | -24,422 |  | -22,268 |
Akaike Inf. Crit. | 50,480 |  | 50,473 |  | 48,867 |  | 44,598 |
Deviance Inf. Crit. | 50,435 | 50,384 | 50,426 | 50,378 | 48,818 | 46,115 | 44,828 | 42,578
Note: Estimates of the variance parameters in Panel A, Columns (1)-(6), come
from estimations of regressions without covariates (so-called “null” models)
that have been estimated at various levels. Estimates of the variance
parameters in Panel A, Columns (7) and (8) come from estimation of models
reported in Table 2. Columns (1) and (2) contain no covariates and data
clustered only at the season-,village-, and time-level. Column (3) and (4)
contain no covariates and add a cluster at the household-level. Column (5) and
(6) contain no covariates and add a cluster at the parcel-level. Column (7)
and (8) contain covariates and data clustered at all levels. Odd numbered
columns are maximum likelihood estimates while even numbered columns are
Bayesian estimates. Bayesian calculations use a burn-in period of 5,000
iterations and an additional 5,000 iterations to ensure convergence. Variances
$\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$, $\sigma_{4}^{2}$,
$\sigma_{5}^{2}$ represent the variance in crop yield that comes from the
corresponding level. The final variance parameter $(\sigma^{2})$ corresponds
to the idiosyncratic or unexplained portion of the model. Intraclass
correlation coefficients in Panel B are calculated using the formulas in
equations (7)-(11). Panel C decomposes the ICC into percent of variance
accorded to each level.

Table A3: Estimated Variance, ICCs, Variance Shares from Alternative
Specifications

| (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---|---
|  |  |  |  |  |
Panel A: Variance Parameter Estimates
|  |  |  |  |  |
parcel | $1.378$ | $1.171$ | $1.000$ | $1.000$ | $0.992$ | $0.933$
household | $0.179$ | $0.000$ | $0.002$ | $0.002$ | $0.002$ | $0.002$
season |  |  | $1.343$ | $1.290$ | $0.912$ | $0.790$
village |  | $0.726$ |  |  | $0.548$ | $0.623$
time | $0.146$ |  |  | $0.031$ |  | $0.126$
idiosyncratic | $1.957$ | $2.059$ | $1.623$ | $1.623$ | $1.626$ | $1.623$
|  |  |  |  |  |
Panel B: Intraclass Correlation Coefficients
|  |  |  |  |  |
parcel | $0.377$ | $0.296$ | $0.252$ | $0.251$ | $0.243$ | $0.228$
household | $0.425$ | $0.296$ | $0.252$ | $0.252$ | $0.244$ | $0.228$
season |  |  | $0.591$ | $0.577$ | $0.467$ | $0.421$
village |  | $0.479$ |  |  | $0.601$ | $0.573$
time | $0.465$ |  |  | $0.592$ |  | $0.604$
|  |  |  |  |  |
Panel C: Shares of Variance From Each Level
|  |  |  |  |  |
parcel | 38% | 30% | 25% | 25% | 24% | 23%
household | 05% | 00% | 00% | 00% | 00% | 00%
season |  |  | 34% | 32% | 22% | 19%
village |  | 18% |  |  | 13% | 15%
time | 04% |  |  | 02% |  | 03%
idiosyncratic | 53% | 52% | 41% | 41% | 40% | 40%
observations | 11,942 | 11,942 | 11,942 | 11,942 | 11,942 | 11,942
log likelihood | -23,423 | -23,323 | -22,296 | -22,294 | -22,274 | -22,268
Akaike Inf. Crit. | 46,904 | 46,704 | 44,650 | 44,649 | 44,608 | 44,598
Deviance Inf. Crit. | 46,599 | 46,400 | 44,351 | 44,350 | 44,310 | 44,828
Note: Estimates of the variance parameters in Panel A come from alternative
specifications of the models presented in Table 2. All models contain crop-
specific covariates and crop-specific intercepts. All six specifications are
estimated using MLE. Column (1) reports results using only a time-level to
proxy for weather while Column (2) uses only a village-level to proxy for
weather. Column (3) drops the village and time levels in favor of the season-
level. Column (4) adds back in the time-level while Column (5) adds back in
the village-level. Column (6) includes season, village, and time levels.
Intraclass correlation coefficients in Panel B are calculated using the
formulas in equations (7)-(11). Panel C decomposes the ICC into percent of
variance accorded to each level.

Table A4: Actuarial Values and Payout Probabilities for Alternative Contracts

| Contract Structure | Summary Statistics
---|---|---
|  |  |  | Actuarially Fair | Probability | Loading | Years until
| Strike | Exit | Max Payout | Premium | of Payout | Factor | Payout
|  |  |  |  |  |  |
Panel A
High-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $45$ | $5$ | $1000$ | 63.67 | 8.77% | 4.55 | 3.80
Phase II | $55$ | $5$ | $1000$ |  |  |  |
Phase III | $500$ | $570$ | $1000$ |  |  |  |
|  |  |  |  |  |  |
Panel B
Medium-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $25$ | $0$ | $1000$ | 81.71 | 4.14% | 3.73 | 8.06
Phase II | $15$ | $0$ | $1000$ |  |  |  |
Phase III | $500$ | $580$ | $1000$ |  |  |  |
|  |  |  |  |  |  |
Panel C
Low-Payout Contract Structure
|  |  |  |  |  |  |
Phase I | $30$ | $5$ | $1000$ | 42.02 | 2.92% | 8.33 | 11.40
Phase II | $30$ | $5$ | $1000$ |  |  |  |
Phase III | $500$ | $575$ | $1000$ |  |  |  |
Note: All contracts come from Cole et al. (2013). While each contract is
designed for a specific village/weather station, we follow Giné et al. (2007)
in utilizing a large representative set of rainfall data to calculate
actuarially fair premia and the probability of payout. Actuarially fair premia
are calculated using standard actuarial principles. Payout probability is the
average occurrence of a payout. Loading factors are calculated as
$1+\lambda=\frac{\textrm{paid price}}{\textrm{actuarially fair price}}$. Years
until payout are calculated by dividing the inverse of the probability of
payout by three, which assumes payout events occur as a Poisson process.