How can I test the two parameter estimates in the different model? I want to test the two parameter estimates in the different model.
First model:
\begin{equation}
Y_{t,USGDP}=\beta_{0,US}+\beta_{1,US}\times X_{t,USconsumption}+u_{t,US}
\end{equation}
Second model:
\begin{equation}
Y_{t,JapanGDP}=\beta_{0,Japan}+\beta_{1,Japan}\times X_{t,Japanconsumption}+u_{t,Japan}
\end{equation}
where $t$ denotes a time, $Y$ is a dependent variable, $X$ is a dependent variable, $u$ is an error term. Ignore the unit root process in the model here.
Here, I want to test $\beta_{1,US}=\beta_{1,Japan}$.
Please teach me the methodology of the test.

(I’m going to ignore the time component and leave it to you to handle, for instance, an autoregressive error term when you do inference.)
A typical way to test this is with an interaction term between the $X$ variable and an indicator variable about the country.
Instead of having two regression models,
$$\text{(Japan) } y= \alpha_0+\alpha_1X+\epsilon\\
\text{(US) }y=\beta_0+\beta_1X+\epsilon
$$
Use one regression model with $Z$ as an indicator variable for the country.
$$
Y=\gamma_0+\gamma_1X+\gamma_2Z+\gamma_3XZ+\epsilon
$$
In this setup, $\gamma_2$ is the difference between the intercepts for each country, and $\gamma_3$ is the difference between the slopes for each country, both of which you can formally test.
