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Dynamic-factor models

Stata’s dfactor estimates the parameters of dynamic-factor models by maximum likelihood. Dynamic-factor models are flexible models for multivariate time series in which the observed endogenous variables are linear functions of exogenous covariates and unobserved factors, which have a vector autoregressive structure. The unobserved factors may also be a function of exogenous covariates. The disturbances in the equations for the dependent variables may be autocorrelated.

We have data on industrial production (ipman), real disposable income (income), weekly hours worked (hours), and the unemployment rate (unemp). We suspect there exists a latent factor that can explain all four of these series, and we conjecture that latent factor follows an AR(2) process.

The first step is to fit our model:

. webuse dfex
(St. Louis Fed (FRED) macro data)

. dfactor (D.(ipman income hours unemp) =, noconstant) (f=, ar(1/2)), nolog

Dynamic-factor model

Sample: 1972m2 - 2008m11                          Number of obs   =        442
                                                  Wald chi2(6)    =     751.95
Log likelihood = -662.09507                       Prob > chi2     =     0.0000
Coef. Std. Err. z P>|z| [95% Conf. Interval]
L1. .2651932 .0568663 4.66 0.000 .1537372 .3766491
L2. .4820398 .0624635 7.72 0.000 .3596136 .604466
f .3502249 .0287389 12.19 0.000 .2938976 .4065522
f .0746338 .0217319 3.43 0.001 .0320401 .1172276
f .2177469 .0186769 11.66 0.000 .1811407 .254353
f -.0676016 .0071022 -9.52 0.000 -.0815217 -.0536816
var(De.ipman) .1383158 .0167086 8.28 0.000 .1055675 .1710641
var(De.income) .2773808 .0188302 14.73 0.000 .2404743 .3142873
var(De.hours) .0911446 .0080847 11.27 0.000 .0752988 .1069903
var(De.unemp) .0237232 .0017932 13.23 0.000 .0202086 .0272378
Note: Tests of variances against zero are one sided, and the two-sided confidence intervals are truncated at zero.

With our model fit, let’s obtain dynamic forecasts for disposable income beginning in December 2008:

. tsappend, add(3)
. predict income_f, dynamic(tm(2008m12))
. tsline D.income income_f if month >= tm(2005m1)


Even more interesting is the path of our unobserved factor. We have hypothesized that all our observed variables follow the unobserved latent factor. We can obtain the one-step predictions of the factor by typing

. predict factor, factor

We can then trace the path of the factor by graphing the result:

. tsline factor


Extracting the latent factor in this manner is sometimes referred to as extracting or estimating an indicator.

dfactor also estimates the parameters of static-factor models, seemingly unrelated regression (SUR) models, and vector autoregressive (VAR) models by maximum likelihood. dfactor allows for constraints on the covariance matrix of the errors in an SUR model and a VAR model.

After estimation, you can predict both the endogenous variables and the unobserved factors. In addition to one-step predictions, dfactor can produce dynamic multistep predictions.

Explore more time-series features in Stata.





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