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To what extent can long-differencing capture climate adaptation?
The economic costs of future climate change will depend not only on the trajectory of warming itself, but on the extent to which households, firms, and governments adapt to the changing climate they face. However, there is substantial uncertainty around the degree of existing adaptation and what the limits to adaptation might be. Economists are thus increasingly interested in measuring the extent of adaptation to climate change carleton2024hecc,lemoine2025nber,kolstad2020reep. A common approach to estimating and testing for adaptation uses long differences (LD) dell2012aejm,burke2016aejep to isolate climate variation from transitory weather shocks, comparing multi-year averages to filter out the transitory shocks and reveal how systems adjust to permanent environmental shifts over time. Adaptation is then inferred by comparing LD estimates of long-run responses to climatic variables with standard panel fixed effects (FE) estimates of short-run responses to observed weather. This empirical approach has been used to study adaptation and the long-run effects of climate in a wide range of contexts, including agriculture won2024ajae,cui2025,chen2021jde,yu2021sr,taylor2026jaere, labor liu2023aejep, manufacturing ponticelli2023nber, migration obolensky2024nber,baylis2025jpub, and health obradovich2018pnas,carleton2017pnas. Despite the wide use of LD to FE comparisons, little is known about whether this comparison actually recovers adaptation. The goal of this paper is to formally examine the conditions under which comparisons of FE and LD estimators can be informative about the extent of adaptation. Our results show that neither LD nor FE estimators are consistent for their respective target parameters, and thus resulting tests for adaptation can be substantively underpowered.
We first show that the LD and FE estimands are weighted averages of the long- and short-run response. Intuitively, the weight on the short-run response in the LD estimand depends on the contribution of short-run weather variation that survives the LD transformation. We therefore refer to it as the LD “contamination weight”. The weight on the long-run response in the FE estimand depends on the contribution of climate variation that survives the FE transformation. We refer to it as the FE contamination weight. Under mild conditions, both LD and FE estimands are attenuated towards zero and away from their respective targets.
This formal analysis relies on a standard outcome model used in the literature burke2016aejep to justify long-differencing, but does not require any assumptions on how the climate process itself evolves. Indeed, an important takeaway from our results is that long-differencing is agnostic about the climate process, though not about how the outcome of interest is generated given climate and weather shocks.\footnote{We show how the probability limits are affected by a deviation from the proposed outcome model to allow for response heterogeneity (see Remark (ref)).}
Using the probability limits, we can show that comparing FE and LD estimands constitutes a test by implication of the null hypothesis of no adaptation. The equality of the FE and LD estimands is a necessary but not sufficient condition of the no-adaptation hypothesis. If the LD and FE contamination weights sum to one, then this test would have trivial power regardless of the extent of adaptation. Caution is therefore warranted when interpreting non-rejections of adaptation tests, as with other tests that are based on an implication of the null hypothesis in question (such as identification tests), as opposed to an equivalent condition.
We illustrate the formal analysis numerically using an empirically-calibrated simulation using the temperature dataset used in burke2016aejep. Consistent with our formal results, this simulation design demonstrates that the difference between LD and FE estimators is (in most conventional settings) downwardly biased relative to the true extent of adaptation. This downward bias can range from 30-80% depending on the choice of averaging window used in the LD estimator and can lead to substantive loss in finite-sample power. We also report simulation results for the contamination weights, which underscore that the LD contamination weight tends to be larger in magnitude relative to the FE contamination weight. Furthermore, the LD contamination weight can vary substantially with the choice of the averaging window in the LD estimator.
In order to formally examine the role of the averaging window in the LD contamination weight, we consider two analytical examples as well as numerical examples using temperature data. We analyze the contamination weights analytically for two time series processes, specifically a linearly trending and a unit root case. For both examples, our analysis demonstrates that the relationship between the averaging window and the LD contamination weight is nonlinear and depends on the climate process. We demonstrate the corresponding patterns using the temperature data used in our simulations, where the climate component is defined as the 10- or 30-year normal. This analysis demonstrates that the specific choice of the bias-minimizing averaging window depends on the unobserved climate process and the LD contamination weight can be substantive, even at the minimizing choice of averaging window. In order to assess the magnitude of this weight in practice, researchers would have to consider different climate specifications. The contamination weights estimated using these specifications can also be used to construct confidence intervals for the true extent of adaptation by test inversion. We discuss these implications for empirical practice in Section (ref).
This paper highlights econometric issues with the use of long differencing to measure adaptation. Concerns about the variation the LD method uses to identify long-run response were illustrated in burke2016aejep and discussed in other work kolstad2020reep,lemoine2018nber,lemoine2025nber. We demonstrate that LD and FE are biased away from their respective targets and demonstrate the implications for testing and measuring the extent of adaptation based on their difference.
Our work also contributes to a broad body of methodological work measuring long-term climate impacts and adaptation. Among the methods currently used, including non-linear panel data methods merel2021ajae,deryugina2017nber, multistage models auffhammer2022jeem,butler2013ncc, and partitioning variation approaches bento2023jeem,bilal2024nber,moore2014ncc,gammans2017, LD is a commonly used approach. Our formal analysis provides guidance for interpreting adaptation tests and measures based on differences between LD and FE.
Our analysis is closely related to work on measurement error in panel models (griliches1986errors) where within- and first-difference transformations split regressor variation in distinct ways and therefore produce predictable patterns of bias. In their classical errors-in-variables setting, griliches1986errors decompose observed regressors into a true signal and an i.i.d.\ measurement error. They compare estimators across within and first-difference transformations to characterize attenuation bias and recover the true coefficient. Our setting shares the similar feature of a regressor decomposed into two components with different time-series properties, but our analysis departs in a substantive way: the high-frequency component in our setting is not measurement error but instead a weather shock carrying its own causal short-run response. Differential filtering therefore does not produce attenuation toward zero, but yields estimators that are weighted averages of long- and short-run responses, with biases of opposite signs relative to their respective targets. Whereas griliches1986errors use differential filtering to recover a single coefficient, our analysis shows that a test for climate adaptation attenuates the very difference between long-run and short-run responses that the test is meant to detect. Last but not least, we do not (and cannot) assume that climate and weather shocks are uncorrelated as in the classical measurement error problem, as they are necessarily dependent.
In this section, we demonstrate that the FE and LD estimands are biased away from their respective targets, characterize their contamination weights and show that the adaptation test is a test by implication, highlighting its dependence on the bias of both estimands. We then illustrate this analysis in an empirically-calibrated simulation study.
FE models are used widely to estimate the effect of weather on economic outcomes of interest. The predominant specifications examined in this literature are linear in the parameters, while nonlinear in the higher-frequency (e.g.\ daily) temperature cui2024on
where $i$ denotes the cross-sectional unit, $t$ denotes the time period (commonly year), and $h$ the higher frequency dimension, which is typically daily. The higher-frequency temperature time series is denoted by $\mathcal{W}_{it}\equiv \{W_{it1},\dots,W_{itH}\}$. To simplify notation, we assume that $x_{it}$ is scalar, but the framework allows $x_{it}$ to be multi-dimensional and include bins, degree day measures, splines and other popular specification choices as a special case.
To allow cross-sectional units to respond differently to short- and long-run changes in weather, suppose that observed weather $x_{it}$ is the sum of climate $c^*_{it}$ and weather shocks $w^*_{it}$ that are both latent to the econometrician, $x_{it}=c^*_{it}+w^*_{it}$.\footnote{For instance, if climate only changes the mean as assumed in some climate models burke2016aejep,deryugina2017nber, such that $x_{it}\sim (\mu_{it},\sigma_x^2)$, then climate is given by $c_{it}^*=\mu_{it}$ and weather by deviations from climate $w_{it}^*=x_{it}-\mu_{it}\sim (0,\sigma_w^2)$. The assumption that weather is composed of a low-frequency component representing climate and a high-frequency component representing weather shocks is also present in work including gospodinov2025, bilal2024nber, and bento2023jeem. } The following specification burke2016aejep allows $\theta_S$ to capture the short-run response to $w_{it}^*$ and $\theta_L$ to capture the long-run response to $c_{it}^*$,\footnote{See burke2016aejep Equation (2) in Supplemental Appendix Section A.2.1.}
In the absence of adaptation ($\theta_L=\theta_S$), Eq.\ (ref) simplifies to Eq.\ (ref); that is, the latter is a restricted version of the former.
The setup in Eq.\ (ref) parallels that of griliches1986errors, who study within- and first-difference estimators under classical measurement error. The key distinction is that they treat the second component as a nuisance to be eliminated, whereas we treat it as a weather shock with its own causal effect — which, as shown below, changes the nature of the resulting bias.
To simplify presentation hereinafter, we abstract from covariates and additional fixed effects in Eq.\ (ref) and consider the following model,
We emphasize however that this omission is without loss of generality as the extension of our results to (ref) is immediate by Frisch-Waugh-Lovell Theorem.
In this section, we introduce the FE and LD estimators as well as the adaptation test.
Let $\tilde z_{it}$ represent the within-transformed version of $z_{it}$, formally $\tilde{z}_{it}\equiv z_{it}- \frac{1}{T}\sum_{t=1}^Tz_{it}$. To simplify notation, we use $\sum_t$ to denote $\sum_{t=1}^T$. The FE model is given by the following,
The population analogue of the FE estimator, $\hat{\beta}_{FE}$, as $n\rightarrow\infty$ is denoted by $\beta_{FE}\equiv \dfrac{\bar{E}[\sum_t\tilde{x}_{it}\tilde{y}_{it}]}{\bar{E}[\sum_t\tilde{x}_{it}^2]}$, where $\bar{E}[Z_i]\equiv \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^nE[Z_i]$ and we assume sufficient regularity conditions hold.
The LD approach relies on a different transformation of the outcome and regressor that takes the difference of their average in two equal-sized, non-overlapping windows. The logic is that temporal averaging cancels out transitory weather fluctuations, leaving only climate variation in the differences between averages. The LD-transformed model is given by
where $\Delta \bar z_{i}$ is the long-differenced version of $z_{it}$, formally $\Delta \bar z_{i}\equiv \frac{1}{\tau}\sum_{t=T-\tau+1}^T z_{it}-\frac{1}{\tau}\sum_{t=1}^\tau z_{it}$ for some $1\leq \tau\leq T/2$. Let $\beta_{LD}$ denote the population analogue of the LD estimator $\hat{\beta}_{LD}$, formally $\beta_{LD}\equiv \frac{\bar{E}[\Delta \bar x_i\Delta \bar y_i]}{\bar{E}[\Delta \bar x_i^2]}$, where we assume sufficient moment conditions.\footnote{With a slight abuse of notation, we will use $\Delta \bar z_i^2$ to denote the square of $(\Delta \bar z_i)^2$ to avoid having too many parantheses throughout.} The LD estimand therefore depends on the user's choice of $\tau$: the time span over which the average is computed in the LD transformation. Figure (ref) presents the choices of $\tau$ together with the study period ($T$) in recent articles that have used LD and demonstrates a lack of consensus on the choice of this tuning parameter.
Differences between ${\beta}_{FE}$ and ${\beta}_{LD}$ are attributed to adaptation. The economic reasoning behind this comparison is that the fixed effect estimand $\beta_{FE}$ measures the marginal responses to weather where certain long-run adaptations, such as technology or capital investments, are not possible lemoine2018nber,burke2016aejep.\footnote{See lemoine2018nber for detailed analysis of the margins of adaptation captured by panel first- and long-difference estimators in a dynamic environment where agents maximize expected payoffs that depend on a stock variable.} By contrast, the LD estimand reflects a longer horizon where it is assumed that agents are able to adjust along additional margins. Under this framework, the null hypothesis of no adaptation is given by:
A rejection of $H_0$ implies that the long-run response to climate is statistically different from the short-run response. The measured adaptive behavior depends on the direction of the difference. In the often-studied cases where larger marginal effects of treatment imply adverse outcomes (i.e.\ heat on mortality or crop yields), a long-run response that is smaller than short-run ($|\beta_{LD}|<|\beta_{FE}|$) indicates long-run adaptation.\footnote{It may be worth noting that this test does not speak to the cost of adaptation.} Conversely, a long-run response that is larger ($|\beta_{LD}|>|\beta_{FE}|$) indicates intensification of climate impacts exceeds agents' ability to adapt.
In this section, we show the probability limits of the FE and LD estimators. While the difference between the probability limit of a given estimator and its target parameter is technically an inconsistency, we will often refer to it as bias for simplicity. We provide the proofs for the following propositions in Appendix (ref). In addition to standard moment conditions required for the probability limits to be well-defined, we impose strict exogeneity throughout the paper to focus our analysis on the bias stemming from $\theta_L\neq \theta_S$.
Part (i) of Proposition (ref) shows that the FE estimand is a weighted average of short- and long-run responses to weather and climate, $\theta_S$ and $\theta_L$, respectively. The weight on $\theta_S$, $\omega_S^{FE}$, depends on the within-demeaned weather shocks $\tilde w^*_{it}$ and its covariance with the within-demeaned climate. Intuitively, if climate variation is eliminated by the within-group transformation, this weight equals 1. Part (ii) of Proposition (ref) demonstrates that the weight on the long-run response $\omega_L^{FE}$ biases the FE estimand away from the short-run response and depends on the variation from the unobserved component of climate that survives the within-transformed observed weather ($\tilde x_{it}$). We will therefore refer to $\omega_L^{FE}$ as the FE contamination weight. The resulting bias therefore depends on both the magnitude of $\omega_L^{FE}$ and the difference between $\theta_L$ and $\theta_S$. This resonates with recent work challenging the notion that FE estimators solely capture short-run response since agents can adapt to expectations of change lemoine2018nber,shrader2023wp.
Part (i) of Proposition (ref) shows that the LD estimand is also a weighted average of the short- and long-run parameters. The weight from short-run variation, $\omega_S^{LD}$, depends on the extent of the variation from the unobserved weather shocks $w^*_{it}$ that survives the LD transformation. We will refer to $\omega_S^{LD}$ as the LD contamination weight. Part (ii) of Proposition (ref) demonstrates that the bias of $\beta_{LD}$ depends on both the magnitude of $\omega_S^{LD}$ and the difference between short- and long-run response parameters.
To provide intuition on the weight in $\omega_S^{LD}$, the bias of $\beta_{LD}$ in Proposition (ref), it is helpful to consider the case where the long-differenced climate and weather shocks are uncorrelated, $\bar{E}[\Delta\bar w_i^*\Delta \bar c_i^*]=0$. In this case, $\omega_S^{LD}$ depends inversely on the climate signal-to-noise ratio. To see this, first we simplify $\omega_S^{LD}$ under the zero-covariance assumption and then divide both numerator and denominator by $\bar{E}[\Delta \bar{w}_i^{*2}]$, which yields the following,
where we use $SNR_c\equiv \bar{E}[\Delta \bar{c}_i^{*2}]/\bar{E}[\Delta \bar{w}_i^{*2}]$ to denote the climate signal-to-noise ratio assuming $\bar{E}[\Delta \bar{w}_i^{*2}]> 0$. Clearly, $\omega_S^{LD}$ (and consequently the bias of $\beta_{LD}$) decreases as $SNR_c$ increases. In Section (ref), we examine how the choice of $\tau$, the LD averaging window, affects the magnitude of $\omega_S^{LD}$.
The formulae in Propositions (ref) and (ref) highlight the potential for bias in both estimators of short- and long-run responses. Importantly, the weights and resulting bias depend on components of the data-generating process unobservable to the researcher, which we analyze numerically in Section (ref).
Given the natural connection to measurement error problems, we next examine the conditions under which our setting yields attenuation bias similar to classical measurement error problems. To obtain this result, we impose plausible non-negative covariance assumptions. Specifically, we assume that LD- and FE-transformed weather shocks and climate are non-negatively correlated. The climate literature often considers transitory inter-annual weather variability and long-run climate trends to be physically independent phenomena driven by distinct mechanisms marotzke2015nature,maher2018grl,lehner2023erc,hasselmann1976tf, which imply that the two variables are uncorrelated. While this covariance might temporarily be ambiguous in short panels due to cyclical patterns like El Niño, over longer horizons this correlation dissipates (or is likely positive). Let $Sgn(x)$ denote the function that yields the sign of its argument $x$, specifically $Sgn(x)=1\{x>0\}-1\{x<0\}$.
The sign restriction on the covariances in Corollary (ref) allows us to place bounds on the magnitude and direction of the FE and LD biases relative to their respective targets. The magnitude of the bias of each estimator is weakly smaller than the difference between the short- and long-run population estimands. The third result in Corollary (ref) states that the biases are of opposite signs. The implication is that the biases demonstrated in Propositions (ref) and (ref) attenuate differences between the FE and LD estimands away from their respective targets.
The following corollary building on Propositions (ref) and (ref) characterizes the relationship between $\beta_{LD}-\beta_{FE}$ and the extent of adaptation, $\theta_L-\theta_S$. This characterization has two practical takeaways. First, testing adaptation using the null hypothesis, $H_0: \beta_{FE}=\beta_{LD}$, constitutes a test by implication. Second, to estimate $\theta_L-\theta_S$ consistently, one has to estimate the contamination weights, which require an assumption on the climate process.
Corollary (ref) characterizes the difference between the two estimands, which form the basis of the adaptation test. The difference in estimands is linear in $(\theta_L-\theta_S)$ with a slope equal to $(1 - \omega_S^{LD} - \omega_L^{FE})$. Since $\omega_S^{LD}$ and $\omega_L^{FE}$ depend on the data-generating process of $c_{it}^*$ and $w_{it}^*$, this corollary highlights the consequences of LD and FE comparisons being agnostic about the climate process. It also demonstrates that if researchers are willing to impose assumptions on this process, then they can conduct inference on $\theta_L-\theta_S$ using Corollary (ref). We discuss this implication further in Section (ref).
Corollary (ref) further demonstrates that the test of $H_0: \beta_{LD}=\beta_{FE}$ is based on a necessary, but not sufficient, condition of $H_0:\theta_L=\theta_S$. It therefore controls size, but may have trivial power under the alternative if $\omega_S^{LD}+\omega_L^{FE}=1$. Furthermore, even if $\omega_S^{LD}+\omega_L^{FE}\neq 1$ but close to it, then this can compromise the power of the test as we demonstrate numerically in the following section. Intuitively, the equality $\beta_{FE} = \beta_{LD}$ is implied by the no-adaptation hypothesis $\theta_L = \theta_S$, but it is not equivalent to it: by Corollary (ref) the same equality can arise under adaptation whenever $\omega^{LD}_S + \omega^{FE}_L = 1$. A non-rejection is therefore consistent with either no adaptation or with adaptation masked by contamination. This is the same logic as other tests by implication, such as over-identification tests.
We illustrate the formal analysis in a simulation design calibrated to the empirical setting in burke2016aejep. The authors study the sensitivity of crop yields to temperature, finding negative effects of degree days over 29$^\circ$ Celsius with FE and LD approaches. Using the observed temperature data, we conduct simulations with $c_{it}^*$ defined as a climate normal, $c_{it}^*=\frac{1}{m}\sum_{\ell=1}^mx_{i(t-\ell)}$, to evaluate the performance of various specifications of the studied estimators.
We reconstruct the temperature dataset used in burke2016aejep to measure adaptation of corn yields to extreme heat. Table (ref) presents the details of the data-generating process. We construct a dataset of county-level daily temperature for a longer sample period than the one used in burke2016aejep (1950-2022) (See Appendix (ref)). The additional data allow us to rely on the climate normal specification of ($c_{it}^*$) and compare our formal results across various specification choices for the LD estimator.
Figure (ref) presents the simulation mean of $\hat{\beta}_{LD}-\hat{\beta}_{FE}$ and the simulation rejection probabilities of the adaptation test ($H_0: \beta_{FE}=\beta_{LD}$). We vary $\theta_L$ over the grid $\{-0.07,-0.06, \dots, 0\}$, where we fix $\theta_S=-0.07$. As a result, the range of values we consider for $\theta_L$ starts from the no-adaptation case ($\theta_L=\theta_S$) to the full adaptation case ($\theta_L=0$), where the degree days above 29$^\circ$C are no longer harmful.\footnote{The FE estimate in burke2016aejep is $-0.07$, whereas the LD estimate is $-0.02$ (see Table (ref) for a replication using our constructed sample.)} The results presented in Figure (ref) correspond to the specification of $c_{it}^*=\frac{1}{m}\sum_{\ell=1}^mx_{i(t-\ell)}$ with $m=10$ (10-year climate normal), $T=25$ and $\tau\in\{5,10\}$. Appendix (ref) provides a broader set of simulation statistics for this variant of the simulation design as well as for other variants with different choices of $m$ and $T$.
Panel (a) of Figure (ref) plots the simulation mean $\hat{\beta}_{LD}-\hat{\beta}_{FE}$ as a function of $\theta_L-\theta_S$, the extent of adaptation in the true DGP. Consider the case with $\tau=5$, the simulation mean of $\hat{\beta}_{LD}-\hat{\beta}_{FE}$ is about 20% of $\theta_L-\theta_S$. As per Corollary (ref), this proportion should be explained by $(1-\omega_S^{LD}-\omega_L^{FE})$. Indeed, $\hat{\beta}_{LD}-\hat{\beta}_{FE}$ is linear in $\theta_L-\theta_S$ with a slope of about $0.2$, since simulation means of $\hat{\omega}_S^{LD}$ and $\hat{\omega}_L^{FE}$ equal to 0.825 and -0.0265, respectively (see Table (ref)).\footnote{Note that we can use these results from Table (ref), even though it reports the simulation results for $\theta_L=-0.02$ and $\theta_S=-0.07$, since $\omega_S^{LD}$ and $\omega_L^{FE}$ do not depend on the values of $\theta_L$ and $\theta_S$.} When $\tau$ is increased, however, the slope of the line increases to about 0.7, since the simulation mean of $\hat{\omega}_S^{LD}$ and $\hat{\omega}_L^{FE}$ equal 0.3119 and -0.0055, respectively (see Table (ref)). This simulation study demonstrates that the contamination weight $\omega_S^{LD}$ tends to be larger in magnitude relative to $\omega_L^{FE}$.\footnote{This demonstrates that the assumption made in carter2018arre which suggests that the LD estimator is consistent for $\theta_L$ is implausible for temperature data and realistic climate models.} The asymmetry between the two contamination weights has a simple source. The within transformation differences out unit-specific means and leaves predominantly high-frequency variation, so little climate variation survives and $\omega^{FE}_L$ is small. The LD transformation instead averages within windows to cancel weather, but the variance of the surviving shock falls with $\tau$---$\bar{E}[\Delta \bar{w}^{*2}_i] = 2\sigma^2_w/\tau$, for example, if $w_{it}^*$ are i.i.d.\ shocks---thus in finite windows a non-trivial share of weather remains and $\omega^{LD}_S$ is comparatively large. Averaging is thus a blunter instrument for isolating climate than differencing is for isolating weather shocks.
In addition, our results here demonstrate a case of attenuation bias per Corollary (ref), where both $\hat{\beta}_{LD}$ and $\hat{\beta}_{FE}$ under-estimate $\theta_L$ and $\theta_S$, respectively (see Table (ref)). We note however, that in other variants of our simulation design we find $\tau$ values that lead to over-estimation of $\theta_L-\theta_S$ (see Panel (b) of Figure (ref)).
Panel (b) of Figure (ref) presents the simulation rejection probabilities of the adaptation test ($H_0: \beta_{FE}=\beta_{LD}$). This figure demonstrates that the larger bias when using $\tau=5$ relative to $\tau=10$ can have substantive power consequences. We caution, however, that the choice of $\tau$ may also have an impact on the sampling variability of the LD estimator, which can have power implications.\footnote{For instance, for the case with $c_{it}^*$ as the 30-year normal, using $\tau=5$ leads to a higher rejection probability relative to $\tau=10$ (see Figure (ref)), as the former is associated with a higher simulation standard deviation than the latter.}
In this section, we demonstrate that the extent of contamination from short-run weather variation in the long-differenced climate is inversely related to the climate signal-to-noise ratio and analyze this object and its dependence on the choice of $\tau$ in the long-differencing approach. We do so with the aid of two analytical examples of climate specifications of $x_{it}$, where we can solve for the climate signal-to-noise ratio analytically as a function of $T$, $\tau$, and other parameters of the data-generating process. We also compute the climate signal-to-noise ratio using temperature data.
First, suppose that $c_{it}^*=\mu_i+\delta_it$ and $w_{it}^*|\mu_i,\delta_i\overset{i.i.d.}{\sim} (0,\sigma_w^2)$, then in this case the climate signal-to-noise ratio simplifies to
where $\sigma_{\delta}^2\equiv Var(\delta_i)$. For a formal derivation of this result with all relevant assumptions, see Proposition (ref) and its proof.
The main takeaways from Eq.\ (ref) are that $SNR_c$ is increasing in the time horizon $T$ and $\sigma_\delta^2/\sigma_w^2$. As a result, for $\omega_S^{LD}\rightarrow 0$, either $T\rightarrow \infty$ and/or $\sigma_\delta^2/\sigma_w^2\rightarrow \infty$. Panel (a) in Figure (ref) demonstrates however that for fixed $T$ and $\sigma_\delta^2/\sigma_w^2$, $SNR_c$ depends nonlinearly on $\tau$.\footnote{This nonlinearity stems from a trade-off in how $\tau$ affects the variance of the long-differenced climate and weather shocks, as evidenced by their respective variance formulae derived in Proposition (ref) $\bar{E}[\Delta \bar{c}_i^{*2}]=\sigma_\delta^2(T-\tau)^2$ and $\bar{E}[\Delta \bar{w}_i^{*2}]=\frac{2\sigma_w^2}{\tau}$.} For this specification of $c_{it}^*$, there exists $\tau\in(1,T/2)$ that maximizes $SNR_c$, specifically $\tau^*=T/3$. It is important to note, however, that even if we use the $SNR_c$-maximizing $\tau^*$, this does not mean that $\omega_S^{LD}=0$ and thereby $\beta_{LD}=\theta_L$. Indeed, Figure (ref) demonstrates that the magnitude of $SNR_c$ and subsequently $\omega_S^{LD}$ even at the optimal $\tau^*$ will ultimately depend on the value of $T$ and $\sigma_\delta^2/\sigma_w^2$.
Since climate normals, popular in climate modeling, give rise to a unit-root AR(m), where $x_{it}=\frac{1}{m}\sum_{\ell=1}^mx_{i(t-\ell)}+\eta_{it}$ and $c_{it}^*=\frac{1}{m}\sum_{\ell=1}^mx_{i(t-\ell)}$, we also consider the unit root example. To simplify illustration, we first consider the special case where $m=1$ and thereby $c_{it}^*=x_{i(t-1)}$\footnote{This is an empirically relevant case if economic agents use lagged weather as their expectation of weather (climate).} and consider the more general climate normal case using the temperature data we use in our simulations in Figure (ref).
Similar to the linearly trending case, the unit-root example also demonstrates a trade-off in the choice of $\tau$. In this case, a larger $\tau$ decreases both the variance of $\Delta\bar w_i^*$ and $\Delta \bar c_i^*$. The climate signal-to-noise ratio therefore depends nonlinearly on $\tau$ as demonstrated clearly in Panel (b) of Figure (ref). Since $E[\Delta \bar w_i^*\Delta\bar c_i^*]\neq 0$ in this case, the signal-to-noise ratio is given by
Proposition (ref) and its proof state the assumptions and provided a detailed derivation of the signal-to-noise ratio in the unit root case as well as the choice of $\tau$ that maximizes this ratio and thereby minimizes $\omega_S^{LD}$.
The two analytical examples provide multiple takeaways. First, the choice of $\tau$ presents a trade-off, even when we solely consider consistency as a criterion. To avoid overlapping averaging windows, $\tau$ can take values from 1 to $T/2$. For both analytical examples we consider, the choice of $\tau$ that maximizes the climate signal-to-noise ratio is in the interior of the domain, specifically $T/3$ in the trend-stationary case and $3T/8$ in the unit-root case. A second important takeaway is that, even if one were to maximize the climate signal-to-noise ratio, the magnitude of this maximum depends on the time horizon of the sample, $T$. In both examples, the climate variation is increasing in $T$, whereas the weather variation does not depend on it. For the trend-stationary case, the maximum $SNR_c$ is cubic in $T$, whereas in the unit-root case it is quadratic in $T$.
We next compute the components of $SNR_c$ and $\omega_S^{LD}$ using the temperature data used in our simulation design in Section (ref). Panels (a) and (b) of Figure (ref) plot the sample analogue of the LD-transformed climate variation ($\bar{E}[\Delta\bar c_i^{*2}]$), the LD-transformed weather variation ($\bar{E}[\Delta\bar w_i^{*2}]$) and their covariance ($\bar{E}[\Delta\bar c_i^*\Delta \bar w_i^*]$), where $c_{it}^*$ is a 10- and 30-year normal, respectively. Panel (a) demonstrates that, similar to the analytical examples, we see that a larger $\tau$ decreases the short-run weather variation. The climate variability also declines with the increase in $\tau$, as in the unit-root case, but only marginally relative to short-run weather variability. The covariance term is negative in this example and gets smaller in magnitude as $\tau$ increases. By contrast, when we consider the 30-year climate normal in Panel (b), we find that $\tau$ has hardly any impact on the climate variation and the covariance components, which are small in magnitude, though it vastly reduces the short-run weather variability.
Finally, we examine the implied $SNR_c$ and $\omega_S^{LD}$ in the 10- and 30-year climate normal case. We first note that the $SNR_c$ is more nonlinear than in the analytical examples (Panel (c) of Figure (ref). It increases in $\tau$ up to $\tau=10$, and then decreases and reaches its maximum at $T=12$. For the 30-year normal, however, regardless of the choice of $\tau$, the signal-to-noise ratio is quite low and therefore $\omega_S^{LD}$ is close to one for most values of $\tau\in\{1,\dots,12\}$. As a result, regardless of the choice of $\tau$, the bias of the LD estimator toward the short-run response will be substantive. In practice, researchers can estimate $\omega_S^{LD}$ in their setting in order to assess the magnitude of the bias for various models of $c_{it}^*$ that are plausible in their context. We further discuss how such models can be used in Section (ref).
\noindentFE and LD estimands are weighted averages of the long-run and short-run response. This is a consequence of neither the LD nor the FE transformation being able to isolate the desired variation in temperature directly. The FE transformed temperature might still be contamination with climate variation, whereas the LD transformed temperature might contain variation from short-run weather shocks. As a result, comparisons of LD and FE estimands are not consistent for the extent of adaptation in general. In an empirically-calibrated simulation design using temperature data, we find that the LD contamination weight tends to be larger leading the LD estimator to suffer from greater bias in this design.
\noindentCaution is warranted when interpreting non-rejections of adaptation tests. Adaptation tests based on FE and LD estimators are tests by implication. This is a consequence of long-differencing not imposing assumptions on the climate process. Since they do not based on a null hypothesis equivalent to the no-adaptation hypothesis ($\theta_L=\theta_S$), their non-rejection must be interpreted with caution. Corollary (ref) demonstrates that if the FE and LD contamination weights sum to one, then a test based on a comparison of LD and FE estimators will have trivial power, regardless of the true extent of adaptation ($\theta_L-\theta_S$).
\noindentChoice of $\tau$ may exacerbate the bias of LD estimators. The analytical and numerical examples demonstrate that the choice of averaging window in the LD estimator ($\tau$) can have substantive implications for the bias of the LD estimator and the power of the adaptation test. They also demonstrate that the averaging window that minimizes the bias of the LD estimator depends on the underlying climate data-generating process. For some data-generating processes, even the LD estimator using the bias-minimizing choice of $\tau$ might still be substantively biased if the climate signal-to-noise ratio is relatively low for that choice of $\tau$. This analysis highlights the importance of considering the time span of a study, the underlying climate data-generating process and the associated climate signal-to-noise ratio when comparing LD and FE estimators and interpreting adaptation tests.
Assessing the magnitude of the contamination weights requires specifying climate models. Despite the LD estimator not requiring a specification of the climate model, this is necessary to assess its bias in finite samples. Using different models for climate justified by the empirical context, one can estimate both FE and LD contamination weights in order to assess the extent of the bias of FE and LD. Indeed, one can use such estimates to construct confidence intervals for the extent of adaptation, $\theta_L-\theta_S$, by inverting a test of the equality in Corollary (ref).\footnote{Once researchers specify climate models, direct estimation of $\theta_L$ and $\theta_S$ using the assumed climate model becomes a compelling alternative as proposed in, for example, bento2023jeem. } The test inversion is required in this context, since weak identification concerns would arise when the contamination weights sum to one.
\noindentWhile agnostic about the climate-shock process, the validity of long-differencing rests on a separable outcome model. Our formal analysis demonstrates that the validity of long-differencing and the implied adaptation test requires an outcome model that is separable in the climate and shock components as in Eq.\ (ref) (in addition to covariates). Economic theory and climate science can be used to justify such an outcome model. The presence of response heterogeneity, a deviation from separability between observables and unobservables, further highlights the importance of considering the assumptions on the outcome model when interpreting long-differencing results.
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