Measurement Error and Counterfactuals in Quantitative Trade and Spatial Models

Abstract

Counterfactuals in quantitative trade and spatial models are functions of the current state of the world and the model parameters. Common practice treats the current state of the world as perfectly observed, but there is good reason to believe that it is measured with error. This paper provides tools for quantifying uncertainty about counterfactuals when the current state of the world is measured with error. I recommend an empirical Bayes approach to uncertainty quantification, and show that it is both practical and theoretically justified. I apply the proposed method to the settings in Adao, Costinot, and Donaldson (2017) and Allen and Arkolakis (2022) and find non-trivial uncertainty about counterfactuals.

Figures (8)

• Figure 1: Uncertainty quantification for the Armington model. The counterfactual object of interest is the percentage change in welfare (real consumption) after a 10% increase in all bilateral trade costs. The solid blue line denotes the point estimate $g\left(\left\{ \tilde{F}_{ij}\right\} ,5\right)$, the dashed red line denotes the smoothed estimated posterior distribution $\pi^{\mathrm{post}}\left(g\left(\left\{ F_{ij}\right\} ,5\right)|\left\{ \tilde{F}_{ij}\right\} ,\tilde{\vartheta}\right)$, and the dotted black line denotes the median of $\pi^{\mathrm{post}}\left(g\left(\left\{ F_{ij}\right\} ,5\right)|\left\{ \tilde{F}_{ij}\right\} ,\tilde{\vartheta}\right)$.
• Figure 2: Uncertainty quantification for heteroskedastic normal shocks to $\left\{ \log F_{ij,t}\right\}$ for the percentage change in China's welfare due to the China shock. The solid blue line is the estimate as reported in adao2017nonparametric, the dotted light-blue lines denote the intervals accounting for estimation error as reported in adao2017nonparametric, and the dashed red lines denote the intervals based on the estimated posterior distributions $\pi^{\mathrm{post}}\left(g_{t}\left(\left\{ F_{ij,t}\right\} ,\varepsilon\right)|\left\{ \tilde{F}_{ij,t}\right\} ,\tilde{\vartheta}\right)$ for $t=1,...,T$.
• Figure 3: Uncertainty quantification for winsorized heteroskedastic normal shocks to $\left\{ \log F_{ij,t}\right\}$ for the percentage change in China's welfare due to the China shock. The solid blue line is the estimate as reported in adao2017nonparametric, the dotted light-blue lines denote the intervals accounting for estimation error as reported in adao2017nonparametric, and the dashed red lines denote the intervals based on the estimated posterior distributions $\pi^{\mathrm{post}}\left(g_{t}\left(\left\{ F_{ij,t}\right\} ,\varepsilon\right)|\left\{ \tilde{F}_{ij,t}\right\} ,\tilde{\vartheta}\right)$ for $t=1,...,T$.
• Figure 4: Uncertainty quantification for heteroskedastic normal shocks to $\left\{ \log F_{ij,t}\right\}$ for the percentage change in China's welfare due to the China shock. The solid blue line is the estimate as reported in adao2017nonparametric, the dotted light-blue lines denote the intervals accounting for estimation error as reported in adao2017nonparametric, the dashed red lines denote the intervals based on the estimated posterior distributions $\pi^{\mathrm{post}}\left(g_{t}\left(\left\{ F_{ij,t}\right\} ,\varepsilon\right)|\left\{ \tilde{F}_{ij,t}\right\} ,\tilde{\vartheta}\right)$ for $t=1,...,T$, and the dotted-dashed black lines denote the intervals obtained using Algorithm \ref{['alg:EB_UQ_k_ME_EE']}.
• Figure 5: Plot to compare the normalized residuals with the probability density function of a standardized normal distribution to check whether the normality assumption for the prior is reasonable for adao2017nonparametric.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Proposition 1