Spatial-temporal Model with Heterogeneous Random Effects

Q: How to estimate scale effect in conditional quantile estimation?

To estimate scale effect in conditional quantile estimation, first obtain an observable approximate using the QMLE z from the first stage. Replace z by z in model (2.3) to get T random approximates of th for each i. Take the average of T random approximates as the final approximate for th. The pseudo random effect follows by th = X ps + th. Apply conditional quantile estimation to model (3.1) with the latent variable th replaced by its observable pseudo scale effect th. Consider the WCQE given by ph(t) = argmin phPh N i=1 1 x ' ai b c r t (th i - x ' ai ph), t T, where Ph is the parameter space of ph(t), T is a closed subinterval of (0, 1), and b c is a consistent estimator of b 0. Theorem 3 verifies the validity of the proposed hybrid estimation procedure, ensuring that replacing th with th does not affect the theoretical properties of the WCQE ph(t).

Q: What is the efficient estimator of b?

The efficient estimator of b is obtained by optimally combining the information of ph* across K quantile levels, denoted as t k = k/(K + 1) for 1 <= k <= K, where K is a fixed integer. This estimator is represented as b(p opt,K) = K k=1 p k,opt ph(t k). The optimal weight p opt,K is calculated by plugging in H and q. The asymptotic covariance of the optimal WQAE b(p opt,K) is U(p opt,K) = (q' H^-1 q)^-1 D^-1 0. This estimator ensures the accuracy of the asymptotics in Section 3, validating the accuracy of the asymptotics.

Q: What are the impacts of socio-economic factors on air quality in China?

The empirical analysis of air quality in China reveals that socio-economic factors, such as Gross Regional Product (GRP) and secondary industry share in GRP, have significant impacts on air quality. The study found that both GRP and secondary industry share have positive impacts on the conditional mean and quantiles of air quality index (AQI). However, the impact of GRP on AQI is insignificant, suggesting that China may be near the peak of the environmental Kuznets curve, where the adverse impact of economic development on the environment diminishes gradually. On the other hand, the secondary industry share has a statistically significant positive impact on the conditional location of AQI, indicating that it may contribute to the deterioration of air quality in China. The study also observed heterogeneity in AQI due to the secondary industry share, with a significant impact at the lower quantile level but no significant effect at the higher level. These findings highlight the importance of considering socio-economic factors in understanding and improving air quality in China.

Q: What potential research topics exist in the proposed model framework?

The proposed model framework presents several potential research topics. Firstly, the asymptotic covariance of the QMLE is infeasible to compute, and the limiting distributions of the WCQE and WQAE are challenging to estimate due to unknown density functions. A new method that simultaneously perturbs individual and time levels could improve bootstrapping efficiency and accuracy. Secondly, missing values in spatial panel data are common, and an estimation procedure for response with missing values is beneficial. Lastly, a location-scale model is specified, and a more general quantile regression model can be considered with quantile dependent coefficients.

Question

1. What is the proposed DSAR model and its contributions?

2. What is the location-scale structure in the DSAR model and how does it combine the advantages of fixed and random effects?

3. How to estimate thi in two-stage hybrid estimation?

4. What are the conditions and assumptions required for the theoretical properties of conditional quantile estimation?

Accepted Answer

The proposed Dynamic Spatial Autoregressive (DSAR) model captures individual heterogeneity in the presence of spatial-temporal effects. It assumes a location-scale structure for the random effect, which can be influenced by space-specific covariates. This model allows for the exploration of the influence of space-specific factors on different quantile levels of the response after removing spatial-temporal effects. The contributions of this paper are threefold: (a) proposing a DSAR model with possibly heterogeneous random effects, (b) providing a new framework to capture individual heterogeneity in the presence of spatial-temporal effects, and (c) exploring the influence of space-specific covariates on response at different quantile levels. The model is the first of its kind to consider a location-scale structure for the random effect that can be affected by space-specific covariates. Additionally, an R package implementing the proposed two-stage hybrid estimation procedure is available for further analysis.

Accepted Answer

The location-scale structure in the DSAR model combines the advantages of fixed and random effects. Specifically, the location u i depicts the difference among individuals on the conditional mean and can be arbitrarily related to time-variant regressors {z lit }, while the scale effect th i allows for conditional heteroscedasticity and reflects the influence of {x li } on the fluctuation. This structure enables the model to flexibly depict the possible influence of space-specific covariates on the individual random effect. The location-scale structure combines the advantages of fixed and random effects by allowing for the inclusion of time-variant regressors and capturing the influence of space-specific covariates on the individual random effect. This provides a more comprehensive and flexible approach to modeling spatial autoregressive effects and individual random effects in the DSAR model.

Accepted Answer

In two-stage hybrid estimation, thi is estimated by replacing it with its observable approximate based on reasonable estimates of l, a, and g. The observable approximate of thi is obtained by model (2.3). The pseudo scale effects, denoted by thi's, are constructed based on l and ph. A quantile regression model is built using Xa as the regressor, and a weighted quantile average estimator (WQAE) is constructed for ph(t). The asymptotics for the proposed two-stage hybrid estimation procedure are established under specific assumptions for T and N. Assumption 3.1 allows T to be fixed or goes to infinity as N goes to infinity. Assumption 3.2 restricts T at a rate not greater than N when it goes to infinity. Assumption 3.3 requires both T and N to go to infinity. The consistency of QMLE in the first stage is established under Assumption 3.1, while the asymptotic normality of QMLE is obtained under Assumption 3.2. The validity of replacing th with th in quantile estimation is ensured by Assumption 3.3. The asymptotic results of WCQE and WQAE in the second stage are established using Assumption 3.3. These steps ensure accurate estimation of thi in two-stage hybrid estimation.

Accepted Answer

The conditions and assumptions required for the theoretical properties of conditional quantile estimation are as follows: 1. Condition 3.8: For 1 <= i <= N, th i R has a bounded conditional density a.s., denoted as sup thR f th i |x i (th) < K for some K > 0 a.s., where R = (-, ), and f th i |x i (th) is bounded away from zero at the point x ' ai ph - r i. 2. Condition 3.9: For the function P i (t, ph, r i ) = E 1 x ' ai b (t - I(th i < x ' ai ph - r i )) x ai with (t, ph, r i ) T x Ph x R, the Jacobian matrices ph ' P i (t, ph, r i ) and r i P i (t, ph, r i ) for each 1 <= i <= N are continuous and have full rank, uniformly over T x Ph x R. These conditions are needed to establish the uniform convergence of ph(t) and are used for weak convergence of ph(t); see Assumption R3 of Chernozhukov and Hansen (2006) and Assumption 4.1(c) of Canay (2011).

Accepted Answer

To estimate scale effect in conditional quantile estimation, first obtain an observable approximate using the QMLE z from the first stage. Replace z by z in model (2.3) to get T random approximates of th for each i. Take the average of T random approximates as the final approximate for th. The pseudo random effect follows by th = X ps + th. Apply conditional quantile estimation to model (3.1) with the latent variable th replaced by its observable pseudo scale effect th. Consider the WCQE given by ph(t) = argmin phPh N i=1 1 x ' ai b c r t (th i - x ' ai ph), t T, where Ph is the parameter space of ph(t), T is a closed subinterval of (0, 1), and b c is a consistent estimator of b 0. Theorem 3 verifies the validity of the proposed hybrid estimation procedure, ensuring that replacing th with th does not affect the theoretical properties of the WCQE ph(t).

Accepted Answer

The efficient estimator of b is obtained by optimally combining the information of ph* across K quantile levels, denoted as t k = k/(K + 1) for 1 <= k <= K, where K is a fixed integer. This estimator is represented as b(p opt,K) = K k=1 p k,opt ph(t k). The optimal weight p opt,K is calculated by plugging in H and q. The asymptotic covariance of the optimal WQAE b(p opt,K) is U(p opt,K) = (q' H^-1 q)^-1 D^-1 0. This estimator ensures the accuracy of the asymptotics in Section 3, validating the accuracy of the asymptotics.

Accepted Answer

The empirical analysis of air quality in China reveals that socio-economic factors, such as Gross Regional Product (GRP) and secondary industry share in GRP, have significant impacts on air quality. The study found that both GRP and secondary industry share have positive impacts on the conditional mean and quantiles of air quality index (AQI). However, the impact of GRP on AQI is insignificant, suggesting that China may be near the peak of the environmental Kuznets curve, where the adverse impact of economic development on the environment diminishes gradually. On the other hand, the secondary industry share has a statistically significant positive impact on the conditional location of AQI, indicating that it may contribute to the deterioration of air quality in China. The study also observed heterogeneity in AQI due to the secondary industry share, with a significant impact at the lower quantile level but no significant effect at the higher level. These findings highlight the importance of considering socio-economic factors in understanding and improving air quality in China.

Accepted Answer

The proposed model framework presents several potential research topics. Firstly, the asymptotic covariance of the QMLE is infeasible to compute, and the limiting distributions of the WCQE and WQAE are challenging to estimate due to unknown density functions. A new method that simultaneously perturbs individual and time levels could improve bootstrapping efficiency and accuracy. Secondly, missing values in spatial panel data are common, and an estimation procedure for response with missing values is beneficial. Lastly, a location-scale model is specified, and a more general quantile regression model can be considered with quantile dependent coefficients.

Spatial-temporal Model with Heterogeneous Random Effects

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Most frequently asked questions

1. What is the proposed DSAR model and its contributions?

2. What is the location-scale structure in the DSAR model and how does it combine the advantages of fixed and random effects?

3. How to estimate thi in two-stage hybrid estimation?

4. What are the conditions and assumptions required for the theoretical properties of conditional quantile estimation?

5. How to estimate scale effect in conditional quantile estimation?

6. What is the efficient estimator of b?

7. What are the impacts of socio-economic factors on air quality in China?

8. What potential research topics exist in the proposed model framework?

Citations

Estimation and bootstrapping under spatiotemporal models with unobserved heterogeneity

Spatial Dynamic Panel Models with Missing Data

References

On the Pooling of Time Series and Cross Section Data

Consistent Estimates Based on Partially Consistent Observations

Economic growth and environmental degradation: The environmental Kuznets curve and sustainable development

A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model

Matrix Mathematics: Theory, Facts, and Formulas

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