Selected Topics in Applied Econometrics

Just like it is the case with two-dimensional panel data models, multi-dimensional panel data models are estimated under the assumptions of fixed effects and random effects. In application, three-dimensional panel data models with two units and a time dimension and panel data models with more dimensions are generally referred to as multi-dimensional panel data models because they are larger in size than known two-dimensional panel data models.

In empirical studies, multi-dimensional panel data models are mainly composed of two units and a time dimension. Although models with more than three dimensions are examined theoretically and estimation methods are explained, such models are less in number compared to the three-dimensional panel data models.

In the application section, the effects of capital structures and characteristics of the firms which are traded in Borsa Istanbul on the value of firms in question are analyzed using multi-dimensional panel data models. In the literature, there are many studies in which the effects of capital structure and firm characteristics on firm value are investigated by two-dimensional panel data models. The difference of this study in comparison to other studies in the literature is the use of multi-dimensional panel data models. Thus, the unobservable firm and time effects as well as the unobservable sector effect can be included in panel data models.

The study continues with the section which examines multi-dimensional fixed effects models theoretically and explains estimation methods. Then, multi-dimensional random effects models are examined and estimation methods are explained. In the application part of the study, an application has been made using multi-dimensional panel data models obtained from nested units. The study concludes with the conclusion section.

Multi-Dimensional Fixed Effects Models and Estimation Methods

As with two-dimensional panel data models, the unit and time effects in the models are also included in the model as parameters to be estimated in the models of multi-dimensional fixed effects panel data. While unobservable effects are allowed to be correlated with explanatory variables, it is assumed that there are no correlations between the error term and the explanatory variables. Thus, even if there are correlations between explanatory variables and unobservable effects, consistent estimators are obtained, provided that there is no relationship between the error term and the explanatory variables.

The majority of the multi-dimensional fixed effects models, which are estimated under different specifications in applied studies, are given together in Matyas and Balazsi (2012), Balazsi et al. (2018) and Balazsi et al. (2017b) and their estimation methods are explained in detail. In these studies, the details of the three-dimensional panel data studies obtained from the non-nested units are included and the proposed estimation methods for the three-dimensional models are extended to cover the panel data models of more than three dimensions.

Examples of multi-dimensional fixed effects models derived from nested units are the panel data obtained by firms and the grouping of firms within sectors. In this case, three-dimensional panel data models which include the sector, the firms nested or grouped within sectors, and a time dimension can be encountered. There could also be multi-dimensional panel data models with more than three dimensions depending on the application subject, the purpose of the study and the availability of the data in the multi-dimensional nested panel data models which have got a data structure with one unit grouped within another unit.

Below is a nested panel data model in which one unit is grouped within another unit.

(1)

Here M indicates the sectors and N indicates the firms inside the sectors. The number of firms in a sector may not be equal. The model structure that allows for the use of different number of firms in the sectors is below is the unobservable unit effect of the i. sector and is the unit effect of the j. firm in the i. sector.

If the time effect is added to the Model 1 where a unit is nested in the other unit, then the model given below is obtained. Here .

(2)

The within estimation method will be introduced for multi-dimensional unbalanced panel data model, which allows for the number of firms in each sector to be different.

(3)

Here . In order to obtain the two-way fixed effects estimator of the three-dimensional panel data model that does not have the time effect, the transformation matrix is (Baltagi et al., 2001:13). Here . This transformation matrix provides the within transformation below:

Here .

The same transformation is performed for each explanatory variable in the model.

With the addition of the time effect to Model 3 above, a three-way panel data model is obtained and is given below:

(4)

Here . The within transformation matrix used to obtain the within estimator is described below.

This transformation matrix provides the within transformation as follows:

Here in comparison to the transformations of the previous model, the difference is .

The same transformation is performed for each explanatory variable in the model.

.

The fixed effects estimator with the transformed variables is expressed in general below.

Multi-Dimensional Random Effects Models and Estimation Methods

The effects that cannot be observed as part of the component error term, rather than as parameters to be estimated, may be included in the model assuming that it is a random variable. Considering the structure of the multi-dimensional panel data model in fixed effects models, assuming the unobservable effects as parameters to be estimated leads to a loss of degree of freedom. This problem is not encountered in models with random effects. However, for the estimators to be unbiased and consistent, in models with random effects, in addition to the assumption that the error term and the explanatory variables are unrelated, the unobservable effects are assumed to be unrelated with the explanatory variables.

In the case of multi-dimensional random effects panel data models, the models can be grouped as major nested and non-nested models according to the nesting a unit within another unit. Matyas et al. (2012), Balazsi et al. (2017a) examined the multi-dimensional random effects panel data models obtained from non-nested units in their studies. In these studies, the models which are the most preferred and estimated under different specifications are introduced and estimation methods are explained in empirical studies related to multi-dimensional random effects panel data models.

The nested random effects panel data models obtained in the sense of grouping one unit within the other unit can also be expressed under different specifications. As in the case of fixed effects models, the random effects models will be defined through the example of sectors and the firms grouped in these sectors. As was mentioned in the fixed effects models, the nested models with more than three dimensions can also be encountered.

Below is a random effects panel data model with the unit effects in which one unit is nested within the other unit.

(5)

The unbalanced models due to different number of firms in the sectors are defined as in Model 5 M indicates the sectors and N indicates the firms in those sectors. is the unobservable unit effect of the sector and is the j. firm’s unit effect in the sector. If the time effect is added to the above-mentioned Model 5, then the model presented below is obtained.

(6)

Here if the number of firms in the sector is different, then the definition is . In addition to the estimation that for random effect models obtained from nested units, it is estimated that the unobservable effects and the error term by these effects are independent from one another (Baltagi et al., 2001, p.359).

Generalized least squares (GLS) and maximum likelihood estimation (MLE) methods used in known two-dimensional panel data models can be adapted for multi-dimensional random effects models. In the nested multi-dimensional panel data models where one unit is nested within the other unit, the unobservable effect of the nested unit can be considered as a nested effect. In the error components Model 5, is defined as the unobservable nested effect of j. firm in the i. sector. With the effect of nested effect approach, the random effects panel data models in which one unit is nested within the other unit can be analyzed for balanced and unbalanced panel data models.

It was first proposed by Baltagi (1993) that the unit effect of a unit nested in the other unit could be considered as a nested effect, and then Xiong (1995) also contributed to this. The nested random effects approach by Baltagi et al. (2001) and Antweiler (2001) is also adapted for unbalanced panel data models. These models are unbalanced panel data models in terms of the number of firms in different sectors or different numbers of cities in the regions. The estimation methods proposed by Baltagi et al. (2001) and Antweiler (2001) allow for the estimation of unbalanced panel data models.

With the nested effect approach, estimation of multi-dimensional random effects panel data models with GLS and MLE methods can be realized. First, the GLS estimation method and the feasible generalized least squares (FGLS) estimation method, which is based on the estimation of the variance components to apply the GLS estimation method, are explained. The compound error term of the error components of Model 5 is described below.

Here indicates the unobservable sector effect. indicates the nested effect or the unobservable nested effect of the j. firm in the i. sector and it has been allowed to use different number of firms in the sectors. The above-described compound error term is described below with matrices.

Here the dummy variable matrices are and and the variance-covariance matrix of the error term is expressed below (Baltagi et al., 2001, p.359). Here is the ones vector and the sizes are arranged according to indexes.

The variance-covariance matrix has been rewritten below for the i. block with the Wansbeek and Kapteyn (1982) approach.

Here when ve is written and the variance-covariance matrix is organized, the spectral decomposition of the obtained variance-covariance matrix is given below.

The eigenvalues obtained from the spectral separation of the variance-covariance matrix and the corresponding projection matrices are set out below.

, (Baltagi et al., 2001, p.360).