All panel data models—whether they are fixed effect or random effect ones—assume slope homogeneity for the parameter of interest. Disregarding slope heterogeneity could lead to inaccurate findings, so it should be taken into account [1]. The F-test, which compares the sum of squared residuals from OLS regressions with cross-sectional data, can be used to assess the homogeneity of the slope [2]. The later test's underlying problem is its homoskedastic error variance assumption. Due to its reliance on fixed "N" assumptions, the F-test has been demonstrated in Bun [3] to perform poorly unless T is greater than N. Empirical studies rarely utilise panels when T exceeds N since they are so uncommon.

Through the use of huge cross sections and time series, this paper conducts a variety of tests with three-dimensional panel data models that have never been used before in the literature. [4] developed the delta test based on a modified version of Swamy's test, and the test statistic was distributed normally under the null hypothesis of homogeneous slope. This test compares cross-sectional unit-specific regressions with pooled fixed effects regressions. The Delta test permits residual heteroskedasticity since the unit-specific standard errors are used as the weight for the difference. Additionally, they offered an extension of the Blomquist and Westerlund's (HAC) test, which looked at heteroskedasticity and autocorrelation. Cross-sectional dependence can also be evaluated using CSD test. [5] And [6] both recommended the later test, which removed cross-sectional averages from the findings.

In the econometrics literature, cross-sectional dependence has been extensively addressed. It is defined as the interaction between cross-sectional units (such as households, enterprises, and states, etc.). It makes sense to think of reliance over "space" as the antithesis of serial correlation in time series. It may result through social interactions between people, such as consumer imitation and learning within a community, or businesses operating in the same sector. In game theory and industrial organization, this has received a lot of attention. It might also be the result of widely prevalent macroeconomic shocks or unobservable shared variables. Cross-sectional dependence among people is a problem in recent literature when n is large. The cross-sectional of dependence and correlation invalidates traditional t-tests and F-tests that employ standard variance-covariance estimators and leads to efficiency loss for least squares, just like serial correlation in time-series analysis. In a regression model with seemingly unrelated regressions, [7] suggests a cross-sectional dependence (CD) test utilizing the pairwise average of the off-diagonal sample correlation coefficients. For many cross-sectional units (n) measured across T time periods, the CD test is applicable. The CD test is viewed as a test for weak cross-sectional dependence in Pesaran [8].

In econometrics, large-dimensional panel data models have received a lot of attention. [5] Developed and derives the asymptotic normal distributions of the common correlated effects (CCE) estimators for heterogeneous panels under reasonably general constraints. Bai [9] investigates the asymptotic properties of principle component analysis estimators and showed that  are consistent, where N and T denote individual and time series dimensions, respectively. Using quasi-maximum likelihood estimation (QMLE) for dynamic linear panel data models with interactive fixed effects, Moon and Weidner [10] reexamine Bai's [9] PCA estimation and discover two sources of asymptotic bias: the first is caused by serial correlation or heteroskedasticity of the idiosyncratic error term, and the second is caused by the presence of predetermined regressors. Additionally, Moon and Weidner [11] explore the reliability of PCA estimation for panel data models when the number of elements to be included as interaction fixed effects is unknown and must be decided based on specific informational criteria. [12] Investigated panel data model estimation with a multifactor error structure and spatial error correlations and discovered that Pesaran's CCE approach continued to give consistent and asymptotically normal estimates of the slope coefficients.

For large cross-section panels, “Pesaran and Yamagata [13]” modified Swamy's “slope homogeneity test”. Models with only exogenous regressors and normally distributed errors exhibit a typical normal distribution. Monte Carlo simulations showed that in panels with only exogenous regressors, the test had the proper size and power. According to the proposed test, autoregressive (AR) models with moderate root values fared well. However, a bias-corrected bootstrapped variant of the test was suggested for AR models with roots close to unity, and it performs well even when N is big relative to T. Using PSID data, they examined individual income autoregressive models with homogeneous slopes. Even when people with comparable educational backgrounds are taken into account as subgroups, they discovered that the findings demonstrate statistically substantial evidence of earnings dynamics slope variability.

When panel data models with high dimensions and factor error structures are used, testing for slope homogeneity was a challenge that Ando and Bai [14] looked into. Interaction fixed effects were integrated using the Swamy-type test. The asymptotically normal proposed test statistic could link the explanatory variables to the unobserved factors, factor loadings, or both. According to “Monte Carlo simulations”, the suggested test offers adequate size control and power.

By incorporating the heterogeneity tests and estimating techniques employed in two-dimensional models, Tatoglu and Icen [15] expanded the assumption of homogeneity in “multi-dimensional panel data models”. To do this, the multidimensional model was modified with the F and Swamy's tests for parameter homogeneity. When the heterogeneity was found, the four-dimensional panel data model was estimated using both the mean group and random coefficients estimators that were used in two dimensional case. They tested the application of Okun's Law in European nations. They discovered that the countries influenced the parameter heterogeneity. The results show that Okun's Law holds true in European nations, but that Okun's coefficient differs from one country to another.

Objectives of the study

• Applying CD test to detect if there any cross sectional dependence.
• Applying Delta, HAC tests for testing slope heterogeneity.
• Estimating heterogonous slopes using MG, CCEMG and AMG methods.
• Showing the difference between MG and CCEMG estimates for each country.

The paper is structured in this manner. Section 2 reviews and discusses the econometric theory that underlies the various tests and estimation techniques. Section 3 presents the outcomes of applying these tests and estimation techniques to the actual data from UNIDO. A conclusion marks the end of the paper.

For Full Article, Please go through the Link: https://www.pubtexto.com/pdf/?heterogeneous-threedimensional-panel-data-models