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Suppose I would like to estimate the following model

\begin{align} y_t &= \Lambda f_t + Bx_t + u_t\\ f_t &= A_1f_{t-1} + \cdots + A_pf _{t-p} + \eta_t & \eta_t \sim N(0, I)\\ u_t &= C_1u_{t-1} + \cdots + C_q u_{t-q} + \epsilon_t & \epsilon_t \sim N(0, \Sigma) \end{align}

and suppose I want to forecast housing prices. I have a number of exogenous variables but also want to include a latent variable that serves as a proxy for the 'market development'. So in the model above, $f_t$ would be the market development.

My question is, can you estimate this model for a univariate time series $y_t$ (and is this a correct way to approach the problem)? Because for instance https://www.statsmodels.org/dev/examples/notebooks/generated/statespace_dfm_coincident.html states that

"Factor models generally try to find a small number of unobserved “factors” that influence a substantial portion of the variation in a larger number of observed variables"

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    $\begingroup$ If you want to estimate market development you need to include variables that are affected by it. If you have just a single time series there is no way that your factor can capture the market, because you don’t let your model interact with data. What you usually use is a relatively large vector of time series as observables. If you’re just interested in forecasting one of these series, then you just focus on the predictions for it and ignore the rest. $\endgroup$
    – hejseb
    Commented Apr 3, 2020 at 12:05

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You seem to have a state space model for $y$ and within it, a factor $f$. The model for the factor is not specified in terms of observable variables. If $f$ is a factor derived from some factor model and a set of variables $z_1,\dots,z_k$ that do not coincide with $y_t$ or $x_t$ with errors that are independent of errors in the state space model $\eta$ and $\epsilon$, things should work out fine. If there are some dependencies of overlap between sets of variables, a closer look is needed to determine the feasibility of identification and estimation.

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  • $\begingroup$ This is a bit general. More details on the problem are needed to make it more specific. $\endgroup$
    – Richard Hardy
    Commented Apr 3, 2020 at 10:54
  • $\begingroup$ But does $f$ need to be derived from some set of variables $z_1, \ldots, z_k$? Because the link in the question does not seem to derive the underlying factor from a set of variables but rather just from the multiple time series themselves? I was interpreting the model that I described as follows: Suppose that the housing price has a sort of 'base price' based on some exogenous variables (location, size, etc) and then also a market development part (which would then be term $f$ in the model). Is that a correct way of viewing it? $\endgroup$
    – Whizkid95
    Commented Apr 4, 2020 at 11:53
  • $\begingroup$ @Whizkid95, I do not know so much about state space models, but it seems to me you cannot identify $f$ in the model you have. In your model, it is likely impossible to separate $f$ from $u$. $\endgroup$
    – Richard Hardy
    Commented Apr 4, 2020 at 12:35

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