Fitting ARIMA with replicates of non-time varying exogenous variables

For simplicity, let's consider an AR(1). The model I want to fit is \begin{align*} Y_{t} = \mu + \beta X + \phi Y_{t-1} + \epsilon_t \end{align*} where $X$ does not vary with time. So how is $\beta$ even identifiable? Well, I have $K$ replicates of these time series, each at different levels of $X$, so ultimately I have something like \begin{align*} Y_{t,k} = \mu + \beta X_k + \phi_k Y_{t-1, k} + \epsilon_{t,k} \end{align*} So ultimately, my response matrix $\{Y_{t, k}\}$ is $T\times K$, and my exogenous predictor $X$ is $1 \times K$. I know the standard fitting in R uses

arima0(y, order = c(1, 0, 0), xreg = X)


But this assumes $X$ itself is time varying, while here I just have different values of $X$ on $K$ independent time-series chains. If it helps, I'm willing to assume $\phi_k = \phi$ constant.

The setting this problem came up is rather trying to predict, say, $Y_{t_\text{new}, k}$ for a previously-observed $k$, I'm trying to predict $Y_{t, k_\text{new}}$ at a previously-observed $t$, but new predictor $X_k$. If it helps, in economic intuition, $Y_{t, k}$ may represent GDP at time $t$ in a country with population $k$, and I retrospectively want to estimate $Y_{t, k'}$ at time $t$ had a country had population $k'$.

Bonus Question: Is there a way to fit \begin{align*} Y_{t,k} = \mu + \beta X_k + \phi Y_{t-1, k} + \psi Y_{t+1, k}+ \epsilon_{t,k} \end{align*} So clearly, the above is no longer a valid time series. But again, I'm not trying to predict in the $t$ direction, but rather in the $k$ direction, so I already know the values of $Y_{t, k}$ at $t-1, t+1$.

• Isn't this a standard panel data problem? Except for the fact that $\beta X_k$ is not identifiable separately because we can always take $\beta'X'_k=\frac{\beta}{c}cX_k$. Also, $\mu$ is not identifiable separately due to an analogous argument. – Richard Hardy Mar 3 '17 at 18:25
• Why wouldn't $\beta$ be identifiable? I have different values of $X_k$, so there should be some way to pool together the different chains. The panel data examples I found do something like $Y_{t, k} = \mu_k + \beta X_k + \phi t$, while I want to replace $t$ with $Y_{t-1}$. Programatically, I'm not sure how to do this in R. – Tom Chen Mar 3 '17 at 18:45
• Hmm, perhaps you are right... I would have to think more. I had a somewhat related question where I write down an estimator, maybe it could be of some help: Panel data model with two-way fixed effects and individual-specific slopes. – Richard Hardy Mar 3 '17 at 18:58
• I'm lost. Let $\mu^*_k = \mu_k + \beta X_k$. I can see how $\mu^*_k$ is identified, but I can't see how $\beta$ is identified? You're trying to estimate $\beta$ by variation across different $k$s but $\mu_k$ also varies across the $k$s! Maybe you want a single constant $\mu$ (instead of a different $\mu_k$ for each $k$)? – Matthew Gunn Mar 3 '17 at 21:07
• Ah, thank you. Yes, let's fix $\mu_k = \mu$ constant. I've updated in the post. – Tom Chen Mar 3 '17 at 22:05