Model Estimation Based on Incomplete Panel Data

Before, we considered complete panel data. Now, we permit data incompleteness arising from the inclusion of mixed-frequency and missing information. For this purpose, the linear relation in Definition 2.2.1 offers a comfortable way for modeling returns, yields, spreads, growth rates, etc. This comes from the fact that many data transformations can be addressed by (2.9). In this context, we first modify the reconstruction formula of Stock and Watson (1999a, 2002b) for ESFMs with an isotropic error model.

Lemma 3.1.7 (Reconstruction of Panel Data From Observations)

Assume the ESFM in Definition 2.1.3 with isotropic error model, whereX ∈R^T×N,F = [F₁, . . . ,F_T]⁰∈ R^T×K and= [1, . . . ,T]⁰∈R^T×N collect complete panel data, hidden factors and idiosyncratic shocks.

With respect to Definition 2.2.1, for1≤i≤N let the vectorXⁱ_obs∈R^T(i),1≤T(i)≤T,summarize the actually observed values of Xⁱ∈R^T, which is the i-th column ofX. Then, the vectorXⁱ givenF,Xⁱ_obs and the parametersΘ ={W, σ²} is multivariate Gaussian with mean and covariance matrix as follows:

EΘ

Xⁱ|F,Xⁱ_obs

= FW⁰_i+µi1T

+Q⁰_i(QiQ⁰_i)⁻¹

Xⁱ_obs−Qi FW⁰_i+µi1T

, VarΘ

Xⁱ|F,Xⁱ_obs

=σ²i

h

IT −Q⁰_i(QiQ⁰_i)⁻¹Qi

i ,

whereWi,µiandⁱ denote thei-th row ofW, thei-th element ofµand thei-th column of, respectively.

Proof:

Rearranging (2.3) in matrix form and focussing on thei-th time series provide for (2.3) and (2.9):

Xⁱ =FW⁰_i+µi1T +ⁱ, Xⁱ_obs =QiFW⁰_i+Qiµi1T +Qiⁱ.

Because of_t∼ N(0_N,Σ) iid, for all 1≤i≤N we getⁱ∼ N(0_T, σ²iI_T) resulting in:

Xⁱ Xⁱ_obs

! _F,Θ

∼ N

FW⁰_i+µi1T

QiFW⁰_i+Qiµi1T

! , σ²i

IT Q⁰_i Qi QiQ⁰_i

!!

.

Finally, the conditional mean and covariance matrix of the multivariate Gaussian distribution (Greene,

2003, pp. 871-872, Theorem B.7) yield the statement. 2

The matrixQ⁰_i(QiQ⁰_i)⁻¹∈R^T×T(i)in Lemma 3.1.7 represents the unique Moore-Penrose Inverse of the matrixQ_i(Rao and Toutenburg, 1999, pp. 372-373, Definition A.64) and satisfies:Q_iQ⁰_i(Q_iQ⁰_i)⁻¹=I_T(i). Regarding the EM, which we will later on develop, the uniqueness eliminates undesired degrees of freedom, while the conditionQiQ⁰_i(QiQ⁰_i)⁻¹=I_T(i)ensures that the linear relation in Definition 2.2.1 holds, when the EM terminates. Note, Lemma 3.1.7 requires the idiosyncratic shocks to be iid such that the presented isotropic error model is just a special case of a more general result.

In a next step, we combine Lemmata 3.1.6 and 3.1.7 to obtain a comprehensive solution for deriving com-plete panel data from observations, estimated factors and parameter estimates. The matrix representation for the reconstruction of the panel dataX ∈R^T×N in (3.8) is given by:

X^rec=







 (X^rec₁ )⁰

... (X^rec_T )⁰









=h

X^rec,1, . . . ,X^rec,Ni

= X−

1T ⊗µˆ⁰X

Wˆ

Wˆ⁰Wˆ−1

Wˆ⁰+

1T⊗µˆ⁰X

, (3.9)

and so, supports the following update formula.

Corollary 3.1.8 (Update Complete Panel Data)

Assume the setting in Lemma 3.1.7, where Θˆ comprises the parameter estimates in Theorem 3.1.3, and let the matrixX^rec∈R^T×N be the reconstructed panel data in (3.9). Then, for1≤i≤N the conditional expectation of vectorXⁱ∈R^T in Lemma 3.1.7 can be estimated for observationsXⁱ_obs∈R^T(i), T(i)≤T, factor meansµFˆ |X = [ ˆµF1|X1, . . . ,µFˆ T|XT]⁰∈R^T×K and parameter estimatesΘˆ as follows:

EΘˆ

h

Xⁱ|Xⁱ_obs,µFˆ |X i

=X^rec,i+Q⁰_i(QiQ⁰_i)⁻¹

Xⁱ_obs−QiX^rec,i

. (3.10)

Proof:

Replace FW⁰_i+µ_i1T

byX^rec,iin Lemma 3.1.7. 2

At this point, we briefly explain what we have so far. Theorem 3.1.3 provides parameter estimates for the ESFMs in Definition 2.1.3 with an isotropic error model. For model selection the criterion in Definition 3.1.4 is used. In a next step, Corollary 3.1.8 updates the artificial, complete panel data, before the overall procedure starts again. Hence, two questions arise: On the one hand, how to initialize artificial, complete panel data? On the other hand, when does the total approach stop? Regarding the initialization of panel data, gaps can be filled with random numbers, interpolations, zeros, etc. At the beginning, the linear relation in Definition 2.2.1 does not necessarily have to hold, since this will be automatically reached by the updates in Corollary 3.1.8. With respect to the second question, we define a termination criterion.

Similar to Doz et al. (2012) and Ba´nbura and Modugno (2014), the termination of the updates in Corollary 3.1.8 relies on the change in the log-likelihood function L(Θ|X). To be more accurate, letL( ˆΘ_(l)|X_(l)) be the log-likelihood function gained from the estimated model parameters ˆΘ_(l)and data sampleX_(l)of

loop (l) of the total routine. Then, our updates stop as soon as it holds:

abs L

Θˆ_(l)|X_(l)

− L

Θˆ_(l−1)|X_(l−1)

1 2

abs

L

Θˆ_(l)|X_(l)

+abs L

Θˆ_(l−1)|X_(l−1) < ξ, (3.11)

withabs(·) denoting the absolute value of a real number. We stop in (3.11), when the absolute value of the relative change in the log-likelihood function is smaller than a prespecified limitξ >0. By contrast, Doz et al. (2012) and Ba´nbura and Modugno (2014) omit the absolute value in the numerator of (3.11), that is, they consider the relative improvement inL(Θ|X). On the one hand, this reflects the theoretical convergence properties of the EM in Wu (1983), since each EM delivers a non-decreasing sequence of log-likelihood functions. On the other hand, numerical inaccuracies in case of real-world data may cause a few tiny declines, before the nearest local optimum is reached. Then, the approach of Doz et al. (2012) and Ba´nbura and Modugno (2014) might stop too early.

As in Section 3.1.1, we summarize all steps as algorithm. For the initialization ofX₍₀₎diverse approaches provide a first set of complete panel data. In general, an EM detects a local optimum of the log-likelihood function. Therefore, the application of various initialization methods, which offers different starting values, improves the chance of reaching a global maximum. In opposition to the choice ofXⁱ₍₀₎, the shape of the matricesQ_i,1≤i≤N,matters, since Algorithm 3.1.2 does not adjust it later on.

Until Algorithm 3.1.2 stops, the estimated factor dimensionK may change several times. To prevent its termination behavior from changes inK, the criterionξcontrols changes inL(Θ|X) instead of changes in Θ. For instance, the convergence criterion in Schumacher and Breitung (2008) does the latter. Moreover,ξ checks for relative changes instead of absolute ones to ensure that neither the dimension of the parameter space nor the sample size have any impact on the termination of the overall routine.

The construction of complete panel data from the latest parameter estimates and observations in (3.10) guarantees that the equality in (2.9) holds at convergence. In each loop, the second term on the right-hand side of (3.10) punishes any deviations from the observed signals. For all complete time series we have Qi=IT and thus, the simplified version of (3.10) is given by:Xⁱ=Xⁱ_obs. That is, these times series are kept in total without any adjustments. For any time series with T(i)< T, we benefit from the unique Moore-Penrose InverseQ⁰_i(QiQ⁰_i)⁻¹satisfying:QiQ⁰_i(QiQ⁰_i)⁻¹=I_T(i).

The advantages of the presented framework are as follows: First, the underlying FM reduces the dimension from the panel data spaceN to the factor span K. In our empirical studies, we have KN causing a significant dimension reduction. Second, the distribution of the factors in Lemma 3.1.5 takes uncertainties arising from lost variation into account. This will be important, when we construct empirical prediction intervals for returns of future periods. Third, Algorithm 3.1.2 admits the inclusion of mixed frequencies and the estimation of high-frequency analogs for low-frequency time series (nowcasting).

Algorithm 3.1.2:Estimate ESFMs with isotropic errors based on incomplete panel data

### Initialization

Define level of variationζ >0 to be covered;

Choose termination criterion ξ >0;

Set loop index (l) = 0;

fori= 1to N do

InitializeXⁱ_(l) (if necessary, fill gaps);

Specify matrixQ_i; end

Estimate ESFM with X_(l)using Algorithm 3.1.1 for variation levelζ and store parameters ˆΘ_(l); Determine log-likelihoodL( ˆΘ(l)|X(l)) in (3.3);

fori= 1to N do

Derive updated panel dataXⁱ_(l+1)from (3.10) and model parameters ˆΘ_(l); end

Estimate ESFM with X_(l+1)using Algorithm 3.1.1 for var. levelζ and store parameters ˆΘ_(l+1); Determine log-likelihoodL( ˆΘ(l+1)|X(l+1)) in (3.3);

### Alternating reconstruction and reestimation while ^abs(^L(^Θˆ(l+1)|X_(l+1))^−L(^Θˆ(l)|X_(l)))

1

2(^abs(^L(^Θˆ(l+1)|X_(l+1)))^+abs(^L(^Θˆ(l)|X_(l)))) > ξ do Set loop index (l) = (l+ 1);

fori= 1to N do

Derive updated panel dataXⁱ_(l+1)from (3.10) and model parameters ˆΘ(l); end

Estimate ESFM withX_(l+1) using Algorithm 3.1.1 for var. levelζ and store parameters ˆΘ_(l+1); Determine log-likelihoodL( ˆΘ_(l+1)|X(l+1)) in (3.3);

end

Im Dokument Estimation of factor models with incomplete data and their applications  (Seite 38-41)