## Abstract

This paper extends the Baltagi et al. (J Econom 202:108–123, 2018; Advances in econometrics, essays in honor of M. Hashem Pesaran, Emerald Publishing, Bingley, 2021) static and dynamic \(\varepsilon \)-contamination papers to dynamic space–time models. We investigate the robustness of Bayesian panel data models to possible misspecification of the prior distribution. The proposed robust Bayesian approach departs from the standard Bayesian framework in two ways. First, we consider the \(\varepsilon \)-contamination class of prior distributions for the model parameters as well as for the individual effects. Second, both the base elicited priors and the \(\varepsilon \)-contamination priors use Zellner (Bayesian inference and decision techniques: essays in honor of Bruno de Finetti. Studies in Bayesian econometrics, vol 6, North-Holland, Amsterdam, pp 389–399, 1986)’s *g*-priors for the variance–covariance matrices. We propose a general “toolbox” for a wide range of specifications which includes the dynamic space–time panel model with random effects, with cross-correlated effects *à la* Chamberlain, for the Hausman–Taylor world and for dynamic panel data models with homogeneous/heterogeneous slopes and cross-sectional dependence. Using an extensive Monte Carlo simulation study, we compare the finite sample properties of our proposed estimator to those of standard classical estimators. We illustrate our robust Bayesian estimator using the same data as in Keane and Neal (Quant Econ 11:1391–1429, 2020). We obtain short-run as well as long-run effects of climate change on corn producers in the USA.

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## Notes

We thank Yuehua Wu for the helpful discussions on this issue. Unfortunately, their bias correction method is ineffective when \(T<50\) irrespective of

*N*.“We consider the most commonly used method of selecting a hopefully robust prior in \(\Gamma \) (the \(\varepsilon \)-contamination class of prior distributions), namely choice of that prior \(\pi \) which maximizes the marginal likelihood \(m ( y | \pi )\) over \(\Gamma \). This process is called Type II maximum likelihood by Good (1965)” (Berger and Berliner 1986, p. 463).

Yu et al. (2008) observed that \(y_t\) can have some nonstationary components if \(\phi + \rho + \delta = 1\) but, as underlined by Parent and LeSage (2011), stationarity does not require that \(| \phi | + | \rho | + | \delta | < 1\). LeSage et al. (2019) recall that the dependence parameters \(\phi \), \(\rho \) and \(\delta \) associated with stable processes require \(\phi + \rho +\delta <1 \) and, for cases where \(\rho - \delta > 0\), it requires that \(\phi - \rho + \delta > -1 \). See also Parent and LeSage 2011).

The literature generally recommends using the unit information prior (UIP) to set the

*g*-priors (see Sect. 4.1).A one-step estimation of the ML-II posterior distribution is possible but hardly feasible. This is because the probability density functions of

*y*and that of the base prior \(\pi _{0}\left( \theta , b ,\tau | g_{0},h_{0}\right) \) need to be combined to get the predictive density. The resulting expression is highly complex and its integration with respect to \(\left( \theta ,b,\tau \right) \) is quite involved.\( \varepsilon =0.5\) is an arbitrary value. We implicitly assume that the amount of error in the base elicited prior is \( 50\%\). In other words, \( \varepsilon =0.5\) means that we elicit the \(\pi _{0} \) prior but feel we could be as much as \(50\%\) off (in terms of implied probability sets).

We chose: \(\theta _{0}=0,b_{0}=0\) and \(\tau =1.\)

See section C in the supplementary material. For the MCMC Gibbs sampling, we explicitly introduce uniform distributions for \(\phi \), \(\rho \) and \(\delta \). We use 1000 draws and a warmup of 500 burn-in draws.

We use our own R codes for the Bayesian two-stage two-step model (B2S2S) and the MCMC Gibbs sampling and the “xtdpdqml” Stata command for the QML estimator. We use the same DGP set under R and Stata environments to compare the three methods.

The simulations were conducted using R version 3.3.2 on a MacBook Pro, 2.8 GHz core i7 with 16Go 1600 MGz DDR3 ram.

For the sake of brevity, we will henceforth write B2S2S_mixt and B2S2S_boot when referring to the B2S2S estimators with mixtures of

*t*-distributions and with block resampling bootstrap, respectively.Strictly speaking, we should mention “posterior means" and “posterior standard errors” whenever we refer to Bayesian estimates and “coefficients" and “standard errors” when discussing frequentist ones. For the sake of brevity, we will use “coefficients” and “standard errors” in both cases.

The “nse,” often referred to as the Monte Carlo error, is equal to the difference between the mean of the sampled values and the true posterior mean. As a rule of thumb, as many simulations as necessary should be conducted so as to ensure that the Monte Carlo error of each parameter of interest is less than approximately \(10\%\) of the sample standard error. As shown in the table, the estimated nse easily satisfy this criterion. The “cd” compares means calculated from the first \(10\%\) and last \(40\%\) draws of the Markov chain. Under the null hypothesis of no difference between these means, \(cd \sim N(0,1)\) and indicates that a sufficiently large number of draws have been taken. See Koop (2003) and Koop et al. (2007).

Recall that we use only \(BR=20\) individual block bootstrap samples. Fortunately, the results are very robust to the value of

*BR*. For instance, increasing*BR*from 20 to 200 in the random effects world increases the computation time tenfold but yields practically the same results.For the \(N=63\) census tract rook-style and queen-style contiguities within Syracuse city, the non-sparsity rates are, respectively, \(8.72\%\) and \(7.76\%\) while that of the inverse distance weighting matrix is \(98.41\%\).

The with 4-nearest and 10-nearest neighbors weighting matrices have non-sparsity rates of \(6.35\%\) and \(15.87\%\), respectively.

In a time series: \(x_t = \phi x_{t-1} + u_t \text {, }t=1,\ldots ,T\), \(x_t\) is said to be local-to-unit-root from the explosive side (LTUE) if \(\phi = 1 + 1/T\). \(x_t\) is said to be mildly explosive (ME) if \(\phi = 1 + (T^{\alpha })/T\), with \(\alpha =0.1\) or 0.3 and \(x_t\) is said to be explosive (EX) if \(\phi >1\). When

*T*is large, \(\phi _{\text {LTUE}}< \phi _{\text {ME}} < \phi _{\text {EX}}\) which is not necessarily the case when*T*is small (see for instance Phillips 1987; Phillips and Magdalinos 2007; Tao and Yu 2020)As \(\phi =1.05\), \(\rho =0.8\), \(\delta =-0.84\), \(\varpi _{\min } = -0.0963 \) and \(\varpi _{\max }=1\) where \(\varpi _{\min }\) and \(\varpi _{\max }\) are the minimum and maximum eigenvalues of the spatial weights matrix \(W_N\), we cannot respect one of the two stationarity conditions (4) in footnote 4:

$$\begin{aligned} \left\{ \begin{array}{lllll} \phi + \left( \rho + \delta \right) \varpi _{\min }< 1 &{} \text { if } &{} \rho + \delta < 0 &{} \rightarrow &{} 1.0538 \nless 1, \\ \phi - \left( \rho - \delta \right) \varpi _{\max }> -1 &{} \text { if } &{} \rho - \delta \ge 0 &{} \rightarrow &{} -0.59 > -1. \\ \end{array} \right. \end{aligned}$$We only used 1000 draws and 500 burn-in draws for each replication, which is small for MCMC. Despite this, 1000 replications with \(N = 63\), \(T = 10\) (resp. \(N = 120\), \(T = 20\)) require more than one hour of CPU time (resp. almost 5 hours). Had we used 10, 000 draws and 1000 burn-in draws, it would have taken 8 (resp. 34) hours for \(N = 63\), \(T = 10\) (resp. \(N = 120\), \(T = 20\)). The computation times of B2S2S and QMLE are considerably shorter. For instance, in Table 1 the respective computation times are 3min and 7min for \(N = 63\), \(T = 10\) and 12 min and 20 min for \(N = 120\), \(T = 20\). When using mixtures of

*t*-distributions, the B2S2S requires as little as 15 s for \(N = 63\), \(T = 10\) and 52 sec for \(N = 120\), \(T = 20\).We do not provide simulations for other combinations of \(\phi \), \(\rho \) and \(\delta \) for the sake of brevity.

With Monte Carlo simulations for a SAR model with i.i.d errors, Yang (2021) shows that the biases (resp. RMSEs) (\(\times 100\)) of \(\rho (=0.4)\) for 2SLS are smaller (resp. close) to those of GMM: 0.05 (resp. 1.58) for 2SLS and \(-\,0.64\) (resp. 1.52) for GMM when \(N=50\), \(T=30\) and 0.01 (resp. 0.81) for 2SLS and \(-\,0.31\) (resp. 0.75) for GMM when \(N=100\), \(T=50\). Similar results are obtained for the coefficient \(\beta \).

See section E in the supplementary material for more details on the 2SLS estimator of Yang (2021) extended to the dynamic space–time case. We use our own R codes for our Bayesian estimator and the 2SLS estimator.

For \(N=63\), \(T=30\) (resp. \(T=50\)), the gain factor is 1.4 (resp. 3.2) and for \(N=120\), \(T=30\) (resp. \(T=50\)), the gain factor is 3.3 (resp. 7.8).

The growing season is generally defined as ranging from April 1 to September 30 in the literature. More specifically, it starts at sowing and lasts approximately 150 days.

As pointed out by Keane and Neal (2020), this may involve the use of more heat-tolerant hybrids, improved water retention in fields, irrigation, adjustment of sowing rates, etc. This adaptation includes all sources of covariation between heat and heat sensitivity of agricultural yields. It implies the active adaptation of farmers to temperature for growing techniques, as well as any other factors (not controlled by farmers) that make yields less sensitive to heat in warmer conditions.

The threshold for corn is \(29^{\circ }\)C.

Indeed, the MO-OLS estimation on the static model

$$\begin{aligned} \log y_{ti} = \beta _{1,ti} gdd_{ti} + \beta _{2,ti} kdd_{ti} + \beta _{3,ti} prec_{ti} + \beta _{4,ti} prec^{2}_{ti} + c_{ti} + u_{ti},\quad i=1,\ldots ,N,\quad t=1,\ldots ,T, \end{aligned}$$implies a nonlinear relation between \(\hat{\beta }_{1,ti}\) and \(\log gdd_{ti}\) and between \(\hat{\beta }_{2,ti}\) and \(\log kdd_{ti}\) (see Table H.4 and Figures 10 and 11 in the supplementary material).

Their yield data came from the US Department of Agriculture (USDA) National Agricultural Statistics Service. Temperatures and precipitations data were drawn from Schlenker and Roberts (2009).

Enlargements of these maps are reported in Figures 6 to 8 of the supplementary material.

See the supplementary material for additional maps and descriptive statistics, as well as data on the distribution of the Köppen–Geiger climate classification across counties.

This estimation is significantly better than that obtained by MO-OLS using the static non-spatial model which yields an \(R^2= 0.793\) and a residual variance \(\sigma ^2_u = 0.071\). See Table H.4 in the supplementary material.

We note that it is not possible to separate out the time from space and space–time diffusion effects in this model except if we constrain \(\delta \) to be equal to \(\delta = - \phi \rho \).

The derivation of the dynamic multipliers is given in section H.2 in the supplementary material.

Enlargements of these maps are reported in Figures 12 to 14 of the supplementary material.

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This paper is written in honor of Peter Schmidt for his many contributions to econometrics, in particular his influential contributions to dynamic panel data models. We are grateful to Robin Sickles, Subal C. Kumbhakar and two anonymous referees for their helpful comments and suggestions. The usual disclaimers apply.

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Baltagi, B.H., Bresson, G., Chaturvedi, A. *et al.* Robust dynamic space–time panel data models using \(\varepsilon \)-contamination: an application to crop yields and climate change.
*Empir Econ* (2022). https://doi.org/10.1007/s00181-022-02348-9

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DOI: https://doi.org/10.1007/s00181-022-02348-9

### Keywords

- Climate change
- Crop yields
- Dynamic model
- \(\varepsilon \)-Contamination
- Panel data
- Robust Bayesian estimator
- Space–time

### JEL Classification

- C11
- C23
- C26
- Q15
- Q54