Suppose the true relationship in data is driven by AR(1) process as follows: $$X_t=\rho X_{t-1}+\epsilon_t\hbox{ , }|\rho|<1$$ and $\epsilon$ is a white noise of $(0,1)$ expectation and variance. It is easy to show that, when we run OLS on the following misspecified model: $$X_{t+1}=\beta X_{t}+\nu_t$$ we will still have consistency of our estimate, i.e. $$\hat\beta \overset{p}{\to} \rho$$ The question is about the intuition of the result - suppose the initial equation describes the labour force in an economy that gets halved every period (i.e. $\rho=0.5$). Surely my ''correct'' estimate for $\beta$ is $2$, given my mis-specified process is moving in the opposite direction?
The only way I can think of this that doesn't contradict anything is that I know that my process is covariance stationary (I need this to show consistency too). Given that, my meaning of $\rho$ is something else - it just tells me how much of my current population value comes from last period's value (i.e. a value of $0.5$ does not mean population gets halved every period - only that half of the current value can be explained by the past value). Then, indeed, it seems that tomorrow's population and yesterday's population carry just as much information about my population today, and the two estimates should be the same. Is this correct?