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It is not about estimating those equations via PC. There are various methods to estimate the latent factor fth, one of which is principal components. They have asked us to use that. Series(z) in those equations is observed data so we use the estimated fth and observed z to perform the OLS as suggested AR(3) or ARMA(1,0,3) would make the residual series ...


In the paper you cited in the question, the equation (1) is not the equation of state in kalman filter model, but an $AR(3)$ estimated via OLS as shown in Stock & Watson (2002). What the authors estimated in the paper using the Kalman filter is the latent variables $f_t,_h$ and the relative lags through which they estimated both the equation (1) and ...

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