Abstract
We consider a computationally simple orthogonal projection method to implement the (Bai and Ng in Econometrica 70:191–221, 2002) information criterion to select the factor dimension for panel interactive effects models that bypasses issues arising from the joint estimation of the slope coefficients and factor structure. Our simulations show that it performs well in cases the method can be implemented.
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Notes
Pesaran (2006) has proposed a common correlated effects (CCE) estimator for \( \varvec{\beta }\) that does not require the knowledge of the factor structure. However, the implementation of the CCE requires the data generating process of \({\mathbf {x}}_{it}\) to satisfy certain conditions such as:
$$\begin{aligned} {\mathbf {x}}_{it}=\varvec{\Xi }_{i}{\mathbf {f}}_{t}+\varvec{\varepsilon }_{it}, \end{aligned}$$(3.6)such that
$$\begin{aligned} {\mathbf {z}}_{it}&=\left( \begin{array}{c} y_{it} \\ {\mathbf {x}}_{it} \end{array} \right) =\left( \begin{array}{c} \varvec{\beta }^{\prime }\varvec{\Xi }_{i}+\varvec{\lambda }_{i}^{\prime } \\ \varvec{\Xi }_{i} \end{array} \right) {\mathbf {f}}_{t}+\left( \begin{array}{c} v_{it}+\varvec{\beta }^{\prime }\varvec{\varepsilon }_{it} \\ \varvec{\varepsilon }_{it} \end{array} \right) \nonumber \\&={\mathbf {C}}_{i}{\mathbf {f}}_{t}+\left( \begin{array}{c} v_{it}+\varvec{\beta }^{\prime }\varvec{\varepsilon }_{it} \\ \varvec{\varepsilon }_{it} \end{array} \right) , \end{aligned}$$(3.7)where
$$\begin{aligned} rank\left( {\mathbf {C}}_{i}\right) =p\text { and plim}_{N\rightarrow \infty } \frac{1}{N}\sum _{i=1}^{N}\varvec{\varepsilon }_{it}={\mathbf {0}}. \end{aligned}$$(3.3)Since not all the data generating processes of \({\mathbf {x}}_{it}\) satisfy these conditions, for instance, \({\mathbf {x}}_{it}\) could be orthogonal to \( {\mathbf {f}}_{t},\) in this paper, we consider identifying factor dimensions without imposing the Pesaran’s (2006) conditions on \({\mathbf {x}}_{it}.\)
It should be noted that the properties of the projection matrix, \({\mathbf {M}} _{T}\) is invariant to the choice of \(\left( {{\tilde{\mathbf{X}}}}^{\prime } {{\tilde{\mathbf{X}}}}\right) ^{-}\) [e.g., Rao and Toutenburg (1999, p. 374)].
Additional simulation results for model with 5 factors are available upon request.
For one replication of Case 1 when \(N=1000,T=40,\) the CPU time for the ABC approach is 12.11 seconds and 83.55 seconds for the orthogonal projection method and recursively iterating method with the maximum number of iteration equals 10, respectively. The computer running these codes is a Dell Precision Tower 7910 with two Intel Xeon E5-2698v4 CPUs and 64 Gb RAM based on Matlab R2019b.
The data can be downloaded from the data archive of the Journal of Applied Econometrics at http://qed.econ.queensu.ca/jae/2013-v28.6/.
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We wish to thank the editor, an associate editor and two anonymous referees for very helpful comments. Research support by China NSF #71631004 and #72033008 to Cheng Hsiao is gratefully acknowledged.
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Hsiao, C., Xie, Y. & Zhou, Q. Factor dimension determination for panel interactive effects models: an orthogonal projection approach. Comput Stat 36, 1481–1497 (2021). https://doi.org/10.1007/s00180-020-01059-y
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DOI: https://doi.org/10.1007/s00180-020-01059-y