Wednesday, January 20, 2016

Time-Varying Dynamic Factor Loadings

Check out Mikkelsen et al. (2015).  I've always wanted to try high-dimensional dynamic factor models (DFM's) with time-varying loadings as an approach to network connectedness measurement (e.g., increasing connectedness would correspond to increasing factor loadings...).  The problem for me was how to do time-varying parameter DFM's in (ultra) high dimensions.  Enter Mikkelsen et al.  I also like that it's MLE -- I'm still an MLE fan, per Doz, Giannone and Reichlin.  It might be cool and appropriate to endow the time-varying factor loadings with factor structure themselves, which might be a straightforward extension (application?) of Sevanovic (2015).  (Stevanovic paper here; supplementary material here.)

Maximum Likelihood Estimation of Time-Varying Loadings in High-Dimensional Factor Models

Jakob Guldbæk Mikkelsen (Aarhus University and CREATES) ; Eric Hillebrand (Aarhus University and CREATES) ; Giovanni Urga (Cass Business School)


In this paper, we develop a maximum likelihood estimator of time-varying loadings in high-dimensional factor models. We specify the loadings to evolve as stationary vector autoregressions (VAR) and show that consistent estimates of the loadings parameters can be obtained by a two-step maximum likelihood estimation procedure. In the first step, principal components are extracted from the data to form factor estimates. In the second step, the parameters of the loadings VARs are estimated as a set of univariate regression models with time-varying coefficients. We document the finite-sample properties of the maximum likelihood estimator through an extensive simulation study and illustrate the empirical relevance of the time-varying loadings structure using a large quarterly dataset for the US economy.


  1. Dear Frank Diebold, you might want take a look at my 2007 paper on time varying loadings entitled "Factor stochastic volatility with time varying loadings and Markov switching regimes"
    HF Lopes, CM Carvalho
    Journal of Statistical Planning and Inference 137 (10), 3082-3091, 2007.

    1. Just glanced at it. If I read correctly you have TV loadings, but not factor structure in the TV loadings. Anyway, very nice, and I like your "Our main motivation for time-dependent loadings is to permit changes in covariances that are not exclusively
      associated to changes in the individual factor variances."

  2. I noticed your suggestion of endowing DFM loadings themselves with a factor structure. I think there may be a short argument that this would not be a good approach: commonality in factor loadings is an additional source of commonality in the conditional mean, which should simply appear as an additional factor for the level equation. I would be curious if you think there is anything interesting to say in reconciling this with the Stevanovic result, that shocks to time-varying parameters are likely of low dimension.

    A quick linear example: Suppose a vector Y_t follows a single factor model with factor F_t, time-varying loadings Lambda_t, and idiosynratic shocks epsilon_t. Further suppose the time-varying loadings Lambda_t themselves follow a factor structure, with factor V_t, constant loadings Gamma, and idiosyncratic (white noise) shocks e_t. We can alternatively identify this model by writing Y_t with two factors: F_t, and the second factor Z_t = V_t * F_t. The loadings on F_t will be the time-varying white noise e_t, while the loadings on Z_t will be the constant Gamma.

    This almost suggests an approach where any model with correlated time-varying loadings should instead be written as a multiple-factor model with orthogonal loadings, and perhaps a more complex example will suggest loadings moving at different frequencies?

    1. I don't see the reason for your negative assessment. Call it whichever you want: (1) a single-factor model with TV factor loadings with single-factor structure, or (2) a two-factor model with one factor's loadings fixed and the other factor's loadings varying idiosynchratically. Either way, it may be
      a very precise and novel (and testable!) setup.

  3. Thanks for all these references. Lemke, Eickmeier and Marcellino (2015, A Classical Time Varying FAVAR Model: Estimation, Forecasting, and Structural Analysis, Journal of the Royal Statistical Society, Series A, 178, 493–533) have also estimated a version of DFM with time-varying loadings.

    Regarding Ross concerns, I guess we end up having a reduce-rank regression with (conditional) heteroskedasticity. This is at least what happens in a linear regression with TVPs having a linear factor structure. With a DFM, it gets nasty because factors are latent and Lambda_t is a matrix (so vec(Lambda_t) has a factor structure). In that case, I agree that the conditionnal mean of data is affected by the reduced-rank in loadings. Suppose there are K data factors and q loadings factors. What happens when K>q or K<q? Now, can we recover the time-varying structure by adding more factors? I'm not sure, but this can be verified at least in simulations.

    However, I'm still puzzled by the same question: why loadings are time-varying? Of course, a procedure that allows for TVPs is likely to provide a better in-sample fit, and especially within these Stock and Watson data where so many macro and financial indicators have dynamic correlations that are likely to vary over the business cycles.

    A possibility is that the number of fundamentals, or at least, their pervasiveness in the sample, change over time. Suppose there are K factors during the sample period, but there are times where M<K factors are important (think of different sources of recessions during postwar period, great inflation, great moderation, great recession, etc.). This ends up directly in the loadings matrix. But simply estimating the loadings does not let us know why they are unstable. And I don't even talk about estimating the number of factors, which seems to be a quite difficult problem when loadings are time-varying (I provide some simulation and empirical evidence here:

    Another one: the true model is nonlinear. Hence, it can be approximated by a time-varying parameter linear model. Therefore, the structure of relations between the series is not time varying, it's simply nonlinear and we don't know its functional form.

  4. What do you think of Hedibert's motivation that I quote above?

    1. "Our main motivation for time-dependent loadings is to permit changes in covariances that are not exclusively
      associated to changes in the individual factor variances."

      I think the motivation is very interesting. After all, factor analysis is about explaining the covariances between variables, while PCA takes care (in addition) of variances.

      Let X_it be a data point. Suppose there is one data factor and one TV loading factor. The covariance between X_it and X_jt will depend on the product of variances of both factors as well as on the constant loadings of the TV loadings on their factor.
      Hence, a strong TV loadings factor is necessary an important data factor?


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