Heteroskedasticity
Heteroskedasticity is a well-documented and explored concept in scientific applications that involve the use of Ordinary Least Squares. It is defined as a variation in the “non-constant error variance,” or the idea that in the event predictors are included in a regression model, the residual variability is altered as a function of something excluded from the model (Astivia & Zumbo, 2019, p. 1). It is particularly useful in fields such as structural vector autoregressive analysis (Lütkepohl et al., 2021). Heteroskedasticity can be used to construct instruments using estimators in the absence of external instruments (Baum & Lewbel, 2019). This is particularly useful for researchers conducting linear regression models that contain endogenous regressors.
Heteroskedasticity and Small Sample Sizes
Heteroskedasticity is not considered to be a problem in research studies that have small sample sizes for a variety of reasons. For instance, in their assessment of spatial group-wise heteroskedasticity in special regression models, le Gallo et al. (2020) found that the LM-scan tests demonstrated high power even when dealing with small sample sizes. In addition, the researchers noted that the proportion between the sample size and the size of the true cluster does not affect the power of tests (le Gallo et al., 2020). Instances with high heteroskedasticity can be effectively addressed through the application of the bootstrap procedure (Astivia & Zumbo, 2019). This technique can be used to calculate p-values and confidence intervals in instances where a researcher deals with small sample sizes (Astivia & Zumbo, 2019). It is also advisable to conduct the White test for heteroskedasticity in the Gini sense to obtain more power for small samples (Charpentier et al., 2019). Therefore, heteroskedasticity does not preclude the achievement of meaningful results when dealing with small sample sizes.
Autocorrelation
Autocorrelation is an important factor to consider when conducting research. It is defined as a general statistical property of variables assessed across temporal and geographic space, given that observations sampled closely in time or space tend to have numerous similarities (Silva et al., 2022). It is the likeness between samples of a random variable represented as a function of the time difference between them. It can occur due to sluggishness in time series, and the prevalence of specification bias as a result of the exclusion of important variables due to incorrect function forms (Mukherjee & Laha, 2019). In addition, false assumptions of linearity between factors, the application of incorrect variable selection techniques, and the elimination of non-stationary data can precipitate spatial autocorrelation (Gaspard et al., 2019). Correlated observations are often problematic because they go against the basic statistical assumptions made regarding samples, such as the independence of study elements.
Autocorrelation and Small Sample Sizes
The estimation of autocorrelation is often challenging in instances where researchers deal with small sample sizes. The Durbin h test for autocorrelated error terms can be applied in instances where a researcher is dealing with a small sample size (Yin, 2020). The Durbin-Watson test, which is commonly used for first-order autoregressive schemes, can be applied in small sample sizes to guarantee the validity of the collected information (Mukherjee & Laha, 2019). The application of the aforementioned measures ensures that autocorrelation in small sample sizes does not negatively affect a study’s findings. Autocorrelation refers to the general statistical property of variables evaluated across temporal space, and it highlights the fact that observations sampled closely in time or space often have many similarities. Autocorrelation does not adversely affect a study’s findings in the context of a small sample size.
References
Astivia, O. L. O., & Zumbo, B. D. (2019). Heteroskedasticity in multiple regression analysis: What it is, how to detect it and how to solve it with applications in R and SPSS. Practical Assessment, Research, and Evaluation, 24(1), 1-17. Web.
Baum, C. F., & Lewbel, A. (2019). Advice on using heteroskedasticity-based identification. Stata Journal, 19(4), 757–767. Web.
Charpentier, A., Ka, N., Mussard, S., & Ndiaye, O. H. (2019). Gini regressions and heteroskedasticity. Econometrics, 7(1), 1–16. Web.
Gaspard, G., Kim, D., & Chun, Y. (2019). Residual spatial autocorrelation in macroecological and biogeographical modeling: A review. Journal of Ecology and Environment, 43(1), 1–11. Web.
le Gallo, J., López, F. A., & Chasco, C. (2020). Testing for spatial group-wise heteroskedasticity in spatial autocorrelation regression models: Lagrange multiplier scan tests. Annals of Regional Science, 64(2), 287–312. Web.
Lütkepohl, H., Meitz, M., Netšunajev, A., & Saikkonen, P. (2021). Testing identification via heteroskedasticity in structural vector autoregressive models. The Econometrics Journal, 24(1), 1–22. Web.
Mukherjee, A. Kr., & Laha, M. (2019). The problem of autocorrelation in linear regression detection and remedies. International Journal of Multidisciplinary Research and Modern Education, 5(1), 105–110. Web.
Silva, I., Fleming, C. H., Noonan, M. J., Alston, J., Folta, C., Fagan, W. F., & Calabrese, J. M. (2022). Autocorrelation-informed home range estimation: A review and practical guide. Methods in Ecology and Evolution, 13(3), 534–544. Web.
Yin, Y. (2020). Model-free tests for series correlation in multivariate linear regression. Journal of Statistical Planning and Inference, 206, 179–195. Web.