We therefore take a different approach and estimate the dynamic factor model in its static (or stacked) form. The approach here is quasi-MLE, in the sense that the estimator is motivated by making strong parametric assumptions, but the consistency of the estimated factors is shown under weaker nonparametric assumptions given in section 3. To motivate the estimation strategy, we suppose that h=0 so At=AQ, and e-t is i.i.d. N(0,o^) and independent across series. We also diverge from the treatment of Ft in dynamic factor models, in which Ft is modeled as obeying a stochastic process, and instead treat {FJ as a Txr dimensional unknown nonrandom parameter to be estimated. With this notation and under these restrictive assumptions, the maximum likelihood estimator for (Aq, F) solves the nonlinear least squares problem with the objective function.

Efficient computation of (F, A) depends on whether the panel is balanced. If the panel is balanced, then the parameters can be estimated by solving either of two eigenvalue problems. The first eigenvalue problem obtains by subsituting (2.9a) into (2.8) with /-t= 1 to yield the concentrated objective function, V^y(A,F) = (NT)-^ E^ = iXj’Xf(NT)~* Е^=]Х-‘РрХ^ Because of the normalization F’F/T = 1^, minimizing V^j(A,F) is equivalent to maximizing tr{F'(N_1 which is solved by choosing F as the eigenvectors corresponding to the к largest eigenvalues of the TxT matrix N~* E iXjXj- This is the computational strategy used by Connor and Korajczyk (1986, 1993).

The second eigenvalue problem obtains by substituting (2.9b) into (2.8) to yield the concentrated objective function Vj^p(A,F), which is minimized by the к eigenvectors corresponding to the к largest eigenvalues of the N xN matrix T E t = ixtXt* These eigenvectors are the first к principal components of Xr A different approach must be used in an unbalanced panel. In principal it is possible to iterate on the first order conditions (2.9), subject to the normalization condition F’F/T=Ik. This is, however, computationally burdensome for large N. In the unbalanced panel we therefore minimize using the EM algorithm. Continue to assume that At = Aq. Let X*t denote the latent value of X-t, so X*t = X-qFj. + e-t, and X-t = X*t if 7-t = 1 and X-t is unobserved otherwise, and let VjJ^(F,Aq) denote the “complete-data” likelihood, V^p(F,Aq) = -(NT)”1 E ! E ^ i(xif \oFp2* The EM alg°rithm proceeds by iteratively maximizing the expected complete-data likelihood, Q^(F®,A^) = E[Vftr(F®,Aji))|X,F(i-1),Aji-1>], where F® and A^ respectively denote the ‘ft1 iterates of F and Aq.