2 Background

2.2 Theoretical framework

Based on recent work by Gardner and Hendrickson (Reference Gardner and Hendrickson2018) and Mense (Reference Mense2018), in this section we present a specific application of the real-options framework that illustrates why it may be optimal for individuals to postpone their migration response to the impacts of climate change in the face of uncertainty. Consider a representative household who chooses whether to stay in their home country or migrate abroad. The quality of climatic conditions $c(t)$ with $c(0)=c_{0}$ evolves according to a geometric Brownian motion (GBM):

(1)\begin{equation} dc=c\left( \mu dt+\sigma dz\right) , \end{equation}

where $\mu <0$ is a drift parameter capturing the long-term trend of the GBM, $\sigma$ is the standard deviation per unit of time and $dz$ is the increment of a Wiener process of the form $z(t)=\varepsilon (t)\sqrt {dt}$ with $\varepsilon (t)\sim \mathcal {N}(0,\,1)$. Eqn (1) implies that $c$ changes gradually over time without discrete “jumps”, which may apply quite well to a range of slow-onset climatic events such as temperature increase, drought and sea level rise (Mense, Reference Mense2018; Cattaneo et al., Reference Cattaneo, Beine, Fröhlich, Kniveton, Martinez-Zarzoso, Mastrorillo, Millock, Piguet and Schraven2019). The GBM is characterized by a negative long-term trend (indicated by $\mu <0$) while there is uncertainty as to how $c$ evolves in the short term, i.e., although climatic conditions are deteriorating on average over time, there is a positive probability that they will improve in the next period (Gardner and Hendrickson, Reference Gardner and Hendrickson2018).

We assume that the household obtains a perpetual constant dividend $w$ if they choose to emigrate. $w$ could be thought of as an exogenous outside opportunity, reflecting the idea that households may lack accurate information about the prospective destination country (Burda, Reference Burda1995). Migration is also associated with a fixed upfront cost $M > 0$, which encompasses both monetary and psychological cost.

The household's Bellman equation can be expressed as

(2)\begin{equation} rV(c)=c+\frac{1}{dt}\mathbb{E}[dV], \end{equation}

where $r$ is the real interest rate (Dixit and Pindyck, Reference Dixit and Pindyck1994: 101–105). If we apply Ito's Lemma, substitute the right-hand side of equation (1 ) and take expectations, we obtain the following second-order partial differential equation:

(3)\begin{equation} \frac{1}{2}\sigma ^{2}c^{2}V^{^{\prime \prime }}(c)+\mu cV^{^{\prime }}(c)-rV(c)+c=0. \end{equation}

The solution of this differential equation is given by

(4)\begin{equation} V(c)=A_{1}c^{\gamma _{1}}+A_{2}c^{\gamma _{2}}+V_{p}(c), \end{equation}

with $A_{1}c^{\gamma _{1}}$ and $A_{2}c^{\gamma _{2}}$ as homogenous solutions and $V_{p}(c)$ as the particular solution. $A_{1}$ and $A_{2}$ are constants, and $\gamma _{1}$ and $\gamma _{2}$ are the solutions of the characteristic equation $( \sigma ^{2}/2) \gamma ^{2}+( \mu -\sigma ^{2}/2) \gamma -r=0$:

(5)\begin{equation} \begin{aligned} \gamma _{1} & =\frac{1}{\sigma ^{2}}\left[ -\left( \mu -\frac{\sigma ^{2}}{2} \right) -\sqrt{\left( \mu -\frac{\sigma ^{2}}{2}\right) ^{2}+2\sigma ^{2}r} \right] <0 \\ \gamma _{2} & =\frac{1}{\sigma ^{2}}\left[ -\left( \mu -\frac{\sigma ^{2}}{2} \right) +\sqrt{\left( \mu -\frac{\sigma ^{2}}{2}\right) ^{2}+2\sigma ^{2}r} \right] >0. \end{aligned}\end{equation}

The first two terms in equation (4) represent the option value of migrating abroad, whereas $V_{p}(c)$ represents the value of staying in the home country.

Following Gardner and Hendrickson (Reference Gardner and Hendrickson2018), we impose two boundary conditions on this solution in order to solve the model. The first condition requires that the option value of migrating be reduced to zero as $c$ tends to infinity, i.e., there is no incentive to migrate if the quality of climatic conditions is sufficiently high:

(6)\begin{equation} \lim_{c\rightarrow \infty }\left[ V(c)-V_{p}(c)\right] =0. \end{equation}

Since $\gamma _{2}$ is positive, equation (6) implies that $A_{2}=0$. The second condition, also known as the “value-matching” condition, then requires that the value function $V(c)$ be equal to the present value associated with migrating abroad $w/r$ less the cost of migration $M$:

(7)\begin{equation} \begin{aligned} V(c^{{\ast} }) & =A_{1}{c^{{\ast} }}^{\gamma _{1}}+V_{p}(c^{{\ast} })=\frac{w}{r}-M \\ & \Longleftrightarrow A_{1}=\left( c^{{\ast} }\right) ^{-\gamma _{1}}\left[ \frac{w}{r}-M-V_{p}(c^{{\ast} })\right] , \end{aligned}\end{equation}

where $c^{\ast }$ denotes the quality of climatic conditions at which it is optimal to migrate. Put differently, this condition implies that migration is optimal when the household is indifferent between migrating and staying. Substituting $A_{1}$ back into equation (4), we obtain

(8)\begin{equation} V(c)=\left( \frac{c}{c^{{\ast} }}\right) ^{\gamma _{1}}\left[ \frac{w}{r} -M-V_{p}(c^{{\ast} })\right] +V_{p}(c). \end{equation}

Choosing a lower value of $c^{\ast }$, which implies that the household will on average have to wait longer before migrating, lowers the value of $V_{p}(c^{\ast })$, thus increasing the present value of the net benefit of migration $V(c)$. On the other hand, a lower $c^{\ast }$ increases the stochastic discount factor $(c/c^{\ast })^{\gamma _{1}}$, thus reducing $V(c)$. The household therefore chooses $c^{\ast }$ so as to maximize $V(c)$.

The first-order condition is given by

(9)\begin{equation} -\gamma _{1}\left[ \frac{w}{r}-M-V_{p}(c^{{\ast} })\right] =c^{{\ast} }V_{p}^{\prime }(c^{{\ast} }). \end{equation}

As previously noted, the particular solution represents the present discounted value of staying in the home country, i.e.,

(10)\begin{equation} V_{p}(c)=\frac{c}{r-\mu }. \end{equation}

$c^{\ast }$ is thus given by

(11)\begin{equation} c^{{\ast} }=\left( \frac{\gamma _{1}}{\gamma _{1}-1}\right) (r-\mu )\left[ \frac{w}{r}-M\right] . \end{equation}

The model provides two clear predictions: first, migration will only occur once the quality of climatic conditions falls below $c^{\ast }$, that is, once adverse impacts of climate change have exceeded certain thresholds. Second, $c^{\ast }$ negatively depends on migration costs $M$. This implies that the higher the cost of migration, the lower the quality of climatic conditions the household is willing to endure before migrating. The intuition behind this prediction is the following (Mense, Reference Mense2018; Ginbo et al., Reference Ginbo, Di Corato and Hoffmann2021): because migration is costly and to a certain extent irreversible and information about climatic conditions is revealed gradually, it is valuable to postpone the migration decision; this incentive is greater the higher the costs of migration are. However, once climatic conditions deteriorate past a critical level, the household will no longer be better off waiting and will choose to migrate instead.

A number of limitations of the model should be noted. First, as mentioned above, due to the nature of the GBM, the model only applies to gradual changes in climatic conditions and may thus not be suited to capture the effects of fast-onset climatic events such as storms, flooding or extreme heat on international migration. Such events may be modeled more appropriately using Poisson processes (as done, e.g., by Abadie et al. (Reference Abadie, Sainz de Murieta and Galarraga2017)). Incorporating this type of process would present an interesting extension of the model; in the current paper, however, we will instead focus on slow-onset climatic events such as drought and temperature increase and leave these considerations for future research.

Second, the fact that climate-induced international migration appears to be relatively uncommon may also be consistent with a number of alternative explanations. In particular, as noted by Cattaneo and Peri (Reference Cattaneo and Peri2016), this result may be generated by liquidity constraints faced by households in their countries of origin, i.e., households may simply lack the resources to finance the costs of migration. For now, the available data does not allow us to determine the exact mechanism behind a potential threshold effect. However, if households do make their migration decisions according to an option value of waiting rule, we should observe the corresponding threshold to be lower in low-income countries than in middle-income countries since liquidity constraints are likely more relevant in the former. Similarly, equation (11) implies that $c^{\ast }$ is greater the lower the costs of migration, and thus we should expect the threshold level of quality of climatic conditions to be higher for migration to close destinations than to more distant ones.

3 Data

Data on bilateral international migration flows is taken from Abel and Sander (Reference Abel and Sander2014). Based on international migrant stock tables published by the United Nations’ Department of Economic and Social Affairs (United Nations, 2013), the dataset provides information on bilateral migration flows between 196 countries over five-year intervals from 1990 to 2010. The dataset thus allows us to include middle- and low-income countries as both origins and destinations, which is an advantage over other datasets such as Ortega and Peri (Reference Ortega and Peri2013), Vezzoli et al. (Reference Vezzoli, Villares-Varela and de Haas2014) and Wesselbaum and Aburn (Reference Wesselbaum and Aburn2019) which only include OECD countries as destinations.

Information on GDP per capita is obtained from the World Development Indicators (World Bank, 2021), Penn World Tables (Feenstra et al., Reference Feenstra, Inklaar and Timmer2015) and the World Economic Outlook Database (International Monetary Fund, 2021). The data on country population is taken from the World Development Indicators (World Bank, 2021). Drawing from the migration data, we compute bilateral migration rates as the ratio between the migration flow from origin country i to destination country j during five-year period t and the population of i at the beginning of t. Figure A1 in the online appendix shows the kernel density estimation for bilateral migration rates, which reveals that the distribution of this variable is heavily right-skewed.

Following Cattaneo and Peri (Reference Cattaneo and Peri2016) and Beine and Parsons (Reference Beine and Parsons2017), we only include non-OECD countries as countries of origin and distinguish between middle-income and low-income countries, where low-income countries are defined as those countries in the bottom quartile of the distribution of (purchasing power parity-adjusted) GDP per capita in the year 1990. The resulting final sample includes 138 countries of origin, 34 of which are classified as low-income countries according to the above definition, while the remaining 104 are classified as middle-income countries (see the lists of low- and middle-income countries in the online appendix).

Data on monthly mean temperature and precipitation is taken from version 4.05 of the gridded climate dataset created by the Climatic Research Unit of the University of East Anglia (Harris et al., Reference Harris, Osborn, Jones and Lister2020). The original data are gridded to a 0.5$^{\circ }$ latitude by 0.5$^{\circ }$ longitude grid and are then aggregated to area-weighted country-level averages. As argued by Beine and Parsons (Reference Beine and Parsons2015, Reference Beine and Parsons2017), using absolute levels of temperature and precipitation is not appropriate because these variables would not adequately capture how individuals respond to deviations from standard climatic conditions. We follow their suggested approach and instead calculate standardized anomalies of temperature and precipitation as deviations from the respective long-run mean, divided by the respective long-run standard deviation.

However, in order to identify thresholds in the climate-migration relationship at a more granular level and also account for seasonal volatility, we first compute standardized monthly anomalies of temperature and precipitation and then take the average of these over the five-year periods, i.e.,

(12)\begin{equation} C_{i,t}=\frac{1}{60}\sum_{k=1}^{5}\sum_{m=1}^{12}\frac{C_{level,i,t,k,m}-\mu _{i,m}^{LR}(C_{level})}{\sigma _{i,m}^{LR}(C_{level})}, \end{equation}

where $C_{level,i,t,k,m}$ is the level of temperature or precipitation in origin country $i$ in five-year period $t$, year $k$ and month $m$. $\mu _{i,m}^{LR}(C_{level})$ is the 1901–1970 mean of temperature or precipitation of origin country $i$ for month $m$, and $\sigma _{i,m}^{LR}(C_{level})$ is the 1901–1970 standard deviation of temperature or precipitation of origin country $i$ for month $m$. As an alternative measure of climatic anomalies, used as a robustness check, we follow Nawrotzki and Bakhtsiyarava (Reference Nawrotzki and Bakhtsiyarava2017) and Nawrotzki and Bakhtsiyarava (Reference Nawrotzki and Bakhtsiyarava2017) and calculate the share of months of five-year period $t$ in which mean temperature was more than one standard deviation above and mean precipitation was more than one standard deviation below the 1901–1970 mean.

Figure A2 in the online appendix presents kernel density estimations for the standardized temperature and precipitation anomalies. Most notably, figures A2a and A2c indicate that temperature anomalies are not centered around zero in both low- and middle-income countries, and few countries experienced negative temperature anomalies in either sample. Moreover, the distributions of temperature anomalies in low-income countries and precipitation anomalies in middle-income countries appear to be characterized by some extreme positive and negative outliers, respectively. This finding is corroborated by respective excess kurtosis values of 0.38 and 1.14, suggesting that extreme anomalies occur more frequently than would be predicted by an otherwise equivalent normal distribution.

Across the sample period, five low-income countries experienced extreme positive temperature anomalies of more than two standard deviations above the low-income sample mean,Footnote ¹ while three countries experienced extreme negative precipitation anomalies of less than two standard deviations below the sample mean.Footnote ² These observations correspond to 5.88 and 2.21 per cent of the low-income sample, respectively. Likewise, among middle-income countries, extreme positive temperature anomalies were experienced by eight countries,Footnote ³ and eight countries experienced extreme negative precipitation anomalies,Footnote ⁴ corresponding to 3.13 and 1.92 per cent of the middle-income sample, respectively. We conduct robustness checks in section 6 in order to assess whether these outliers may affect the estimation results.

Table 1 presents summary statistics. We observe that middle-income countries have a higher average emigration rate than low-income countries. Consistent with previous literature (e.g., Oezden et al., Reference Oezden, Parsons, Schiff and Walmsley2011), the majority of migration from non-OECD countries can be attributed to migration flows to other non-OECD countries, which account for 76.4 per cent of the migration flow volume between 1990 and 2010. In addition, although migration flows between neighboring countries comprise only 1.4 per cent of observations in the sample, they account for 28.8 per cent of the migration flow volume.Footnote ⁵ This pattern is also reflected in the average migration rates to neighboring countries, which are an order of magnitude larger than migration rates to non-neighboring countries.

Table 1. Summary statistics.

4 Empirical strategy

To identify the nonlinear effects of climate on international migration implied by the real-options framework, we follow Burda et al. (Reference Burda, Härdle, Müller and Werwatz1998) and Basile and Lim (Reference Basile and Lim2006, Reference Basile and Lim2017) and estimate generalized additive models (GAMs). First developed by Hastie and Tibshirani (Reference Hastie and Tibshirani1986), GAMs are an extension of generalized linear models (GLMs) and provide a flexible empirical framework that is particularly well-suited for estimating nonlinear relationships. Unlike GLMs, in a GAM the explanatory variables are specified in terms of smooth nonparametric functions, thus requiring no restrictive a priori assumptions about any parametric functional form (Abe, Reference Abe1999; Ferrini and Fezzi, Reference Ferrini and Fezzi2012). Instead, the degree of nonlinearity is determined directly from the data using an automated smoothing selection criterion. Despite their flexibility, however, GAMs retain the interpretability of GLMs, and results can be interpreted in a straightforward manner using graphical representations of the estimated relationships.

Like GLMs, GAMs require the specification of the distribution of the response variable. In our specific case, online appendix figure A1 indicates a heavily right-skewed distribution of bilateral migration rates, and thus we choose to follow Basile and Lim (Reference Basile and Lim2017) and estimate a GAM with Gamma distribution:

(13)\begin{equation} g(\mathbb{E}(y_{ijt}))=\log (\mathbb{E}(y_{ijt}))=\beta _{0}+s_{1}(T_{it})+s_{2}(P_{it})+\phi _{i}+\phi _{j}+\phi _{t},\end{equation}

where $y_{ijt}$ is the bilateral migration rate from origin country $i$ to destination country $j$ in five-year period $t$ and is assumed to follow a Gamma distribution. To deal with the issue of zero observations in the bilateral migration rates, we apply a common solution used in the literature (e.g., Ortega and Peri, Reference Ortega and Peri2013; Cai et al., Reference Cai, Feng, Oppenheimer and Pytlikova2016; Grimes and Wesselbaum, Reference Grimes and Wesselbaum2019) and add one to all migration flows before computing migration rates. $g(\mathbb {E}(y_{ijt}))=\log (\mathbb {E}(y_{ijt}))$ is the canonical log-link function, which relates the expected value of $y_{ijt}$ to the explanatory variables. $s_{1}(T_{it})$ and $s_{2}(P_{it})$ are unknown smooth functions of the temperature and precipitation anomaly in origin country $i$ in five-year period $t$, respectively, which are estimated using penalized cubic regression splines (Wood, Reference Wood2017). As suggested by Wood (Reference Wood2011), smoothing parameters for the estimated functions $\hat {s}_{1}(T_{it})$ and $\hat {s}_{2}(P_{it})$ are selected using the restricted maximum likelihood (REML) method, which is implemented in the R package mgcv (Wood, Reference Wood2001). $\phi _{i}$ and $\phi _{j}$ are sets of origin and destination country fixed effects, respectively, in order to control for unobserved heterogeneity at the origin and destination country level, and $\phi _{t}$ is a set of time fixed effects.

Following recent literature (Dell et al., Reference Dell, Jones and Olken2014; Cattaneo and Peri, Reference Cattaneo and Peri2016; Beine and Parsons, Reference Beine and Parsons2017; Cattaneo and Bosetti, Reference Cattaneo and Bosetti2017), we choose a parsimonious specification that includes fixed effects but no additional control variables such as GDP per capita, population, quality of institutions or probability of conflicts. As argued by these authors, those variables are likely themselves affected by changes in climatic conditions, and thus including them in the regression may result in an over-controlling problem, leading to biased estimates of the effects of climate on migration.

6 Robustness checks

For our empirical analysis we followed Cattaneo and Peri (Reference Cattaneo and Peri2016) and Beine and Parsons (Reference Beine and Parsons2017) in defining countries in the bottom quartile of the GDP per capita distribution as “low-income countries”. Nevertheless, this delineation inevitably involves some arbitrariness since there is no clear definition of what a low-income country is, and thus a potential concern is that varying the threshold between low- and middle-income countries may yield differing results. To address this concern, we repeat our analysis using the 20th and 30th percentile of the GDP per capita distribution as alternative thresholds.

The results are presented in online appendix table A3. For both alternative thresholds, the edf estimated by the GAM are of similar magnitude compared to the main results, and effects remain statistically significant. Turning to the plots of the estimated effects in online appendix figures A7 and A8, we find very similar relationships compared to the main results when using the 30th percentile of the income distribution as the threshold between low- and middle-income countries. The relationships estimated using the 20th percentile threshold are generally similar to the main results as well.

Another potential concern is that the results may be affected by the choice of the smoothing parameter selection method. While likelihood-based methods such as REML tend to exhibit faster convergence of smoothing parameters to their optimal values than prediction error-based methods such as generalized cross validation (GCV) (Wood, Reference Wood2011), they have also been shown to have a tendency to undersmooth, i.e., to choose a too complex model (Wahba, Reference Wahba1985; Kauermann, Reference Kauermann2005). Therefore, we reestimate equation (13) using GCV rather than REML. As shown in online appendix table A4 and figure A9, the results are very similar to our main findings.

Additional robustness checks are presented in the online appendix. In table A5 and figure A10, we test whether the negative effect of temperature anomalies on international migration in low-income countries is in fact driven by declines in agricultural productivity. For this purpose, we follow Cai et al. (Reference Cai, Feng, Oppenheimer and Pytlikova2016: 149–150) and Cattaneo and Peri (Reference Cai, Feng, Oppenheimer and Pytlikova2016: 134–135) and interact our measures of climatic anomalies with a factor variable indicating origin countries’ quartile in the distribution of agricultural value added as a share of GDP. As shown in figure A10, we observe a negative effect of high positive temperature anomalies for all but the least agriculturally dependent countries. In fact, the results suggest that the higher the level of agricultural dependence, the lower the threshold beyond which the relationship becomes negative. This effect is particularly pronounced for the most agriculturally dependent countries (shown in figure A10 g), which exhibit a negative effect of temperature anomalies on migration across much of the data range. While these findings do not definitively prove that our results are driven by the existence of liquidity constraints, they do provide corroborative evidence of the transmission channel of agricultural productivity postulated by Cattaneo and Peri (Reference Cattaneo and Peri2016).

Furthermore, to address potential concerns over omitted variable bias regarding our parsimonious main specification, in table A6 we include a number of control variables identified as important determinants of international migration in the literature (e.g., Ortega and Peri, Reference Ortega and Peri2013; Beine and Parsons, Reference Beine and Parsons2015). More specifically, we control for the log of the ratio of GDP per capita in origin and destination countries, whether origin-destination pairs share a common language, log distance between origin-destination pairs and the number of years per five-year period in which origin countries experienced civil war.Footnote ⁶ Parametric estimates for the control variables are statistically significant and have the expected signs, and edf of smooth terms are similar in magnitude to the main results. Turning to the estimated relationships between climatic anomalies and international migration shown in figure A11, including the control variables appears to smooth out some of the “wiggliness” present in the relationships in figure 1. Even so, we continue to observe a strong negative effect of high temperature anomalies on international migration for low-income countries.

Next, in table A7 and figure A12, we follow Nawrotzki and Bakhtsiyarava (Reference Nawrotzki and Bakhtsiyarava2017) and Nawrotzki and Bakhtsiyarava (Reference Nawrotzki and Bakhtsiyarava2017) and use five-year period shares of heat and drought months as alternative measures of climatic anomalies (see section 2). Again, edf of smooth terms are similar in magnitude to our main results, and effects remain statistically significant. For low-income countries, we find a hump-shaped relationship between heat month shares and international migration similar to figure 1(c). For middle-income countries, figure A12c suggests a slightly positive effect of high shares of heat months on international migration, but confidence intervals are quite large for this part of the sample.

Finally, to assess whether the results are driven by extreme outliers, we exclude observations with temperature anomalies more than two standard deviations above or precipitation anomalies more than two standard deviations below the respective sample mean. The results are presented in table A8 and figure A13. Compared to the main results, excluding outliers moderates the estimated relationships to an extent. Most notably, the negative effect of negative precipitation anomalies observed in figure 1(f) disappears. However, the remaining relationships are qualitatively similar to the main results.