Evaluation of panel data estimators under the unbalanced panel data for small data sizes occasioned by missingness

Introduction

There is a growing interest in the investigation of panel data analysis especially small sample panel data set. There is no rule of thumb for choosing a sample size, however, appropriate sample size is required for validity. If the sample size is too small the outcomes will be invalid, moreover, if the sample is too large it will result into waste of time and money. (VanVoorhis and Morgan 2007, pg. 48; Memon et al. 2020, pg. 4) [16,12] believes regression analysis should have at least 10 observations per variable. (Indrayan and Mishra 2021) [10] stress the importance of small data in medical research. Researchers are in a cross-road as to what estimator to be used in the analysis for a small sample size. Ordinary least square (OLS) estimator is believed to have small sample properties. (Santos and Barrios 2011) [14] study small and large sample properties of the within-groups (WG) estimator and the first difference generalized method of moments (FD-GMM) estimator of a dynamic panel data (DPD) model. In their study, different panel data samples were examined; a case of small-time dimensions case with T = 3 and T= 5 for small sample or finite sample properties; and a small size for both the cross-section and time dimensions, say N= 20 were considered. In literatures, the Monte-Carlo studies to investigate the performances of the five different estimators of balance panel data models under unbalanced panel data models show results for the performances for the criteria computed; mean square error (MSE) and mean absolute error (MAE) show that for balance panel models and a respective estimate for unbalanced panel models ranking have the same pattern. That is Between estimator ranked 1 st and First Difference estimator ranked last, while Within estimator, Random (Swamy-Arora's) estimator and Pooling estimator do not show a steady pattern of ranking. (Balogun et al. 2022) [4] .

Using the same procedure and measurement criteria, Balogun and Yahya (2024) [3] developed an estimator, particularly for small sample panel data, that outperformed the trio's between estimator researched. The Between Median estimator (BMd) is an extension of the between estimator. It uses Generalized Least Squares (GLS) in the one-way error component of the between estimator, and the resultant estimation is the regression of the dependent variable Y on the mean of the independent variable(s) X(s). This estimator was modified by regressing the Y dependent variable's median on the median of the X(s) independent variable(s) to obtain a Between Median Estimator. An investigation of the performances of the six various estimators that includes the developed estimator of balance panel data estimators under the Unbalanced panel data where increasing degrees of missingness are added to a panel data size reveals that the BMd estimator outperformed the other estimators under studied. (Balogun et al., 2022) [4] . In general, the Between Median estimator outperformed the other five estimators investigated for panel data estimators under the unbalanced data set for a levels of sample sizes (n), as signal in the study. (Balogun and Yahya 2024) [3] . Similar to (Balogun and Yahya 2024) [3] , Between Median estimator outperformed the estimators used in this investigation.

Material and Methods

The Classical Panel Data Model Consider a general panel data model

where, y it is the response for unit i at time t, α i denotes the individual-specific intercept, vector 𝑋 𝑖𝑡 ′ contains k regressors for unit i at time t, vector β contain regression coefficients to be estimated and μ it is the error component for unit i at time t, 𝑖 = 1,2 … , 𝑛 and t = 1,2 …., T In equation (3.1) considering the panel data model that has two exogenous and one endogenous variable;

where α i = α + 𝜀 𝑖 , the individual-specific intercept (α i ) includes the effects of those variables that are peculiar to the ith individual and that are timevariant. This model becomes

where, y it is the response of an individual i over period t 𝜀 𝑖 is the individual-specific error component and μ it is the combined time-series and cross -section error component with variance 𝛿 𝜀 2 and 𝛿 𝜇 2 respectively.

If w it = 𝜀 𝑖 +μ it, then model (3.3) becomes:

Estimators of Panel Data Model

The most common estimators of panel data models to be fitted with different conditions in criteria and efficiency are discussed in this section. Some of these estimators are as follow.

Pooled Estimator:

This Estimator stacks the data over i and t into one long regression with 𝑛𝑇 observations and estimates of the parameters are obtained by OLS using the model (Greene 2008) [9] ; (Garba et al. 2013) [8] ; (Balogun and Yahya 2024) [3] .

Within Estimator:

This is equivalent to "amemiya". (Amemiya 1971) [1] ; (Matyas and Sevestre 1992) [11] ; (Balogun and Yahya 2024) [3] . This regresses on the deviations from the individual or/and time mean.

Between Estimator (BTW):

This regresses the group means of Y on the group means of X's in a regression of n observations. It uses cross-sectional variation by averaging the observations over period t; (Creel and Tillman 2011) [7] ; (Wooldridge 2012) [17] . Explicitly, it converts all the observations into individual-specific average and performs OLS on the transformed data. (Balogun and Yahya 2024) [3] .

First-Differenced Estimator (FD)

This is the ordinary least squares estimation of the difference between the original model and its one-period-lagged model. (Arellano 2003) [2] ; (Baltagi 2005) [5] ; (Balogun and Yahya 2024) [3] 3.2.5. Random Estimator This is equivalent to "swar" models. (Swamy and Arora 1972) [15] ; (Cottrell 2017) [6] . " According to Hauser, the estimator follows the underlying model expression. (Balogun and Yahya 2024) [3] .

Monte-Carlos Procedure

Monte-Carlos is a mathematical technique based on experiment for evaluation and estimation of problems which are intractable by probabilistic or deterministic approach.

The Data Structure

The panel data model considered is of the form:

𝑡 = 1, … , 𝑇; 𝑖 = 1, … , 𝑛 𝑘 and 𝑘 = 1, … , 5.

i) A total of 𝑘 = 5 subjects (for instance) was studied and simulated over T = 5 times. Thus, a total of n = N (𝑘 × 𝑇) observations were generated. ii) The sample sizes considered are n = (20 and 30); at different sample units 𝑘 = 5, with 𝑇 = 4 and 6 iii) An independent variable 𝑥 𝑖𝑡 was considered and simulated from the Gaussian distribution; 𝑋 𝑖𝑡 ~𝑖𝑖𝑑 𝑁(20, 1). iv) Over fixed time interval of k = 4 and 6 v) The vector of parameters, 𝛽 is set as 𝛽 ̂= (𝛽 ̂0𝑖𝑡 , 𝛽 ̂1𝑖𝑡 ) / = (20, 1) / . vi) The response variable 𝑌 𝑖𝑡 was simulated from (1) accordingly using all the above definitions. vii) Unbalance time interval was infused into the data by randomly removing 𝜔% of the total sample from the data, where 𝜔 take on values 5, 10, 15 and 20. (Balogun et al., 2022) [4] .

Models Assessment

The following model assessment criteria shall be employed to determine the relative Absolute efficiencies of the various estimation considered.

i) Mean Absolute Error (MAE):

Where, n is the number of errors, Σ is the summation symbol, that is, adding individual 𝑖 over 𝑛 and time 𝑡 over 𝑇, |𝛽 𝑖𝑡 -𝛽 | is the absolute errors. iii) Mean Square Error (MSE):

Where, n is the number of errors, Σ is the summation symbol, that is, adding individual 𝑖 over 𝑛 and time 𝑡 over 𝑇, (𝛽 𝑖𝑡 -𝛽) is the square errors. (Robeson and Wilmott 2023) [13] ; (Balogun et al., 2022) [4] . Also, Figure 2 shows the pattern of the MAE of N across various levels of T, as the level of missingness grows, the MAE converge throughout the N values.

Result and Discussion

Conclusion

Finding in this work shows the Between Median estimator (BMd) outperform other panel data estimators considered for small sample size of unbalanced data, when N (20 and 30).

Therefore, it is concluded that the Between estimator performed best, this aligns with the findings of (Balogun and Yahya 2024) [3] . They show that, for the data structure considered Between Median estimator is the best estimator.

Recommendation

It is advised that the Between Median estimator be used to fit the panel data models when there is noticeable missingness in tiny data with varying sample sizes (20 and 30) with varying percentages of missingness (0, 5, 10, 15, and 20).

Limitation

The scope of this work is restricted to the examination of small panel data sizes when 𝛽 = 1, 𝑜𝑟 a unit slope (beta), further investigation can be made with different values of 𝛽, 𝑜𝑟 𝛼. Also, the size N across T can also be increase to examine the performance of the developed estimator (BMd)

Ethical Approval

This work was approved by the University Ethical Review Committee (UERC) of University of Ilorin, Nigeria.

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