Estimating Latent Factors with Kalman Smoothing and AR Models
Summary
The discussion distinguishes an autoregressive model for observed or estimated data from the state equations used in a Kalman filter. In the paper being discussed, the AR(3) relationship is estimated with ordinary least squares; it is not itself necessarily the filter’s state equation. The latent factors and their lags are estimated separately, with three lags selected using the Akaike Information Criterion. Principal components are mentioned as another way to estimate latent factors, while the estimated factors and observed series can then be used in the OLS step.
The exchange also points to FKF and KFAS as R tools and cites a general tutorial on Kalman filtering in R. It does not resolve the questioner’s concern about how the residual covariance estimates fit into the proposed smoothing procedure. The answers differ in emphasis, and the available details are not enough to reconstruct the paper’s full state-space specification or establish that its equations are consistent. Readers should consult the cited paper’s model and distinguish factor estimation, regression, and state-space estimation before implementation.
Key ideas
- An AR(3) regression is not automatically the state equation of a Kalman filter.
- Latent factors may be estimated using principal components or a state-space approach.
- The lag order in the cited analysis was chosen using the Akaike Information Criterion.
- The discussion names FKF and KFAS as R packages for Kalman filtering and smoothing.
- The exchange leaves the relationship between OLS residual covariances and the filter’s state model unresolved.
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# Kalman filtering # Kalman filtering Is it possible to the extract the latent factor f from the following equations using kalman smoothing? f is the unobserved state value while z is observed series. From the literature i could read on web mostly the variable in state equation is a function of its previous one lag however here its a function of the last three lags. Please, can you suggest some literature to understand the computation part and also would like to know which packages in R can be useful to implement this problem in particular Link to original paper, refer to page 6 ## Answer by Quantopik (score 2) https://quant.stackexchange.com/a/17570 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 (2). The number of lags is chosen on the basis of the value of Akaike Information Criterion and, in this case, they choose 3 lags. Keep in mind there are other way to estimate the correct number of lags and it depends on several factors as, for instance, the kind of data, the frequency, the underlying model, etc. Moreover, they suggested that such equations can be estimated via PC analysis too. Instead, as regards the R package you need for implementing and replicating this model, there exist 2 main packages available in R: - FKF - KFAS Moreover, for general references look at: > Tusell, Fernando. "Kalman filtering in R." Journal of Statistical Software 39.2 (2011): 1-27. ## Answer by raghu (score 1) https://quant.stackexchange.com/a/17582 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 independent (not auto correlated). So i guess its been used for that reason If the given equations are not the state space equations why have they asked us to estimate the residual covariance matrix for innovations in both equations via OLS and suggested to use the same for kalman filtering/smoothing It doesn't make sense to estimate residual covariance matrix on a AR(3) process and then use the same the AR(1) state equation while performing Kalman smoothing
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