1 CORRECTION OF BILLÉ AND ROGNA (2021)

1.1 Time-varying marginal effects

The long-term marginal effects of equation (7) in the paper Billé and Rogna (2021) are the following

$$\begin{array}{c}\begin{array}{cc}\hfill \frac{\partial \mathbb{E}\left({\mathbf{y}}_t\right)}{\partial {\mathbf{x}}_{1,t}}& ={\left[\left(1-\varphi \right)\mathbf{I}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[({\beta }_1+{\beta }_6)\mathbf{I}+2{\beta }_2\text{diag}({\mathbf{x}}_{1,t})\mathbf{I}+{\beta }_{10}\text{diag}({\mathbf{x}}_{2,t-1})\mathbf{I}+{\beta }_{11}\text{diag}({\mathbf{x}}_{3,t-1})\mathbf{I}\right]\hfill \\ \hfill \frac{\partial \mathbb{E}\left({\mathbf{y}}_t\right)}{\partial {\mathbf{x}}_{2,t}}& ={\left[\left(1-\varphi \right)\mathbf{I}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_3+{\beta }_7\right]\hfill \\ \hfill \frac{\partial \mathbb{E}\left({\mathbf{y}}_t\right)}{\partial {\mathbf{x}}_{3,t}}& ={\left[\left(1-\varphi \right)\mathbf{I}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_4+{\beta }_8\right]\hfill \\ \hfill \frac{\partial \mathbb{E}\left({\mathbf{y}}_t\right)}{\partial {\mathbf{x}}_{4,t}}& ={\left[\left(1-\varphi \right)\mathbf{I}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_5+{\beta }_9\right]\hfill \end{array}\hfill \end{array}$$(1)

中文翻译：

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Billé 和 Rogna 的更正（2021 年）

1 BILLÉ 和 ROGNA 的更正（2021 年）

1.1 时变边际效应

Billé 和 Rogna (2021) 论文中等式 (7) 的长期边际效应如下

$$\begin{array}{c}\begin{array}{cc}\hfill \frac{\partial \mathbb{乙}\left({\mathbf{是的}}_{吨}\right)}{\partial {\mathbf{X}}_{1,吨}}& ={\left[\left(1-\phi \right)\mathbf{我}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[({\beta }_1+{\beta }_6)\mathbf{我}+2{\beta }_2\text{诊断}({\mathbf{X}}_{1,吨})\mathbf{我}+{\beta }_{10}\text{诊断}({\mathbf{X}}_{2,吨-1})\mathbf{我}+{\beta }_{11}\text{诊断}({\mathbf{X}}_{3,吨-1})\mathbf{我}\right]\hfill \\ \hfill \frac{\partial \mathbb{乙}\left({\mathbf{是的}}_{吨}\right)}{\partial {\mathbf{X}}_{2,吨}}& ={\left[\left(1-\phi \right)\mathbf{我}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_3+{\beta }_7\right]\hfill \\ \hfill \frac{\partial \mathbb{乙}\left({\mathbf{是的}}_{吨}\right)}{\partial {\mathbf{X}}_{3,吨}}& ={\left[\left(1-\phi \right)\mathbf{我}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_4+{\beta }_8\right]\hfill \\ \hfill \frac{\partial \mathbb{乙}\left({\mathbf{是的}}_{吨}\right)}{\partial {\mathbf{X}}_{4,吨}}& ={\left[\left(1-\phi \right)\mathbf{我}-\left(\rho +\gamma \right)\mathsf{W}\right]}^{-1}\left[{\beta }_5+{\beta }_9\right]\hfill \end{array}\hfill \end{array}$$(1)