İstatistiksel model, bir örneklem üzerinden gözlem veya deney sonucu elde edilen bilgiler doğrultusunda, verilerin bütününü ve değişkenler arasındaki ilişkileri açıklamaya, tahmin etmeye veya sonuçları yorumlamaya yarayan matematiksel bir yapıdır. Görünürde ilişkisiz regresyon (Seemingly Unrelated Regressions-SUR) modeller, birden fazla regresyon denklemi birlikte analiz etmek için kullanılan istatistiksel bir model türüdür. Karmaşık sistemlerdeki değişkenler arasındaki ilişkileri tahmin etmek için kullanılan SUR modellerde, bireysel olarak ilişkisiz olmalarına rağmen hata terimleri ilişkili olabilen çoklu regresyon denklemleri ele alınmaktadır. İstatistiksel modellerde tahmin edicilerin seçimi sonuçların tahmini üzerinde büyük bir etkiye sahiptir. Her tahmin edici, kendine özgü avantajları ve dezavantajları olan ve analizin gereksinimlerine göre farklı şekillerde uygun olabilecek özelliklere sahiptir. Bu nedenle, tahmin edici seçimi, uygulamanın amaçları ve veri yapısına dayanır. Farklı tahmin edicilerin karşılaştırılması ve en doğru tahmin edicinin belirlenmesi, modelin performansını artırmak ve verilerin en doğru şekilde yorumlanması amacıyla yapılan bir araştırmadır. Bu tez çalışmasında, SUR modeller altında ön tahmin ve tahmin problemi göz önüne alınmıştır. Ele alınan modellerde bilinmeyen parametrelerin ön tahmin ve tahmin edicilerinin bazı istatistiksel özelliklerine ilişkin sonuçlar verilmiştir. Blok matrisler ve rank özellikleri kullanılarak ön tahmin edicilerin eşitlikleri ve toplamsal ayrışımları üzerine bazı sonuçlar elde edilmiştir. Matris rankı ve inertia formülleri vasıtasıyla ön tahmin edicilerin kovaryans matrislerinin karşılaştırılmasına ilişkin bazı eşitlik ve eşitsizlikler elde edilmiştir. Yedi bölümden oluşan çalışmanın giriş bölümünde, SUR modeller tanıtılmış ve literatür bilgisine yer verilmiştir. İkinci bölümde çalışmada ele alınan konuya ilişkin temel kavram, teorem ve özellikler verilmiştir. Üçüncü bölümde, SUR modeller detaylı şekilde tanıtılmış, ele alınan modellerde ön tahmin ve tahmin edilebilme özellikleri ile ön tahmin ve tahmin edicilerin ifadeleri gösterilmiştir. Çalışmanın ana sonuçları üç ayrı başlık altında incelenmiştir. SUR modeller altında ön tahmin ve tahmin edicilerin eşitlikleri dördüncü bölümde, toplamsal ayrışımları beşinci bölümde ve kovaryans matrislerinin karşılaştırılması ise altıncı bölümde verilmiştir. Ön tahmin ve tahmin edicilerin eşitlikleri, toplamsal ayrışımları ile kovaryans matrislerinin karşılaştırılmasına ilişkin eşitlik ve eşitsizlikler, ayrıca bazı özel durumlara ilişkin sonuçlar blok matris, rank özellikleri ve inertia formülleri kullanılarak elde edilmiştir. Son bölümde ise sonuç ve öneriler verilmiştir.
A statistical model supplies a mathematical framework to analyze, interpret, and make predictions about the relationships between variables based on observed data or experimental results obtained from a sample. It provides a systematic approach to understanding the underlying approaches and associations in the data, allowing researchers to infer meaningful conclusions and make informed decisions. Seemingly Unrelated Regressions (SUR) models are a type of statistical model used to analyze multiple regression equations together. In SUR models, which are used to predict relationships between variables in complex systems, multiple regression equations with seemingly unrelated individual relationships are considered, even though their error terms can be correlated. SUR models were first proposed in 1962 by Arnold Zellner. In this study, Zellner suggested estimating the regression effects with a single equation instead of estimating separately. This suggestion established the basis of SUR models and showed that more effective results were obtained with this method. Instead of estimating the equations that are related to each other separately, SUR models provide to analyze these equations together by bringing them together. Thus, the relationships between the variables can be determined better and the predictions can be more accurate. With this approach, the use of SUR models became widespread and the importance of considering the regression equations together in statistical analysis was emphasized. SUR models have been developed by many researchers and used in various application areas. These models have become an important tool in estimating associated regression equations and analyzing interactions between variables and have had a significant influence on the effectiveness of regression analysis in economics, finance, business and many other disciplines. The choice of estimators in statistical models has a great influence on the estimation of results. Each estimator has properties that have their advantages and disadvantages and may be appropriate in different ways according to the needs of the analysis. Therefore, the choice of the estimator is based on the purposes of the application and the data structure. Comparing different estimators and determining the most accurate estimator is a research to increase the performance of the model and to interpret the data in the most accurate way. The estimation techniques used and researching the properties of estimators are one of the main problems in statistical analysis. The classical methods used to predict and estimate parameters in linear regression models in statistical analysis are Ordinary Least Squares Predictor (OLSP) and Ordinary Least Squares Estimator (OLSE), respectively. OLSP is the prediction of parameters using the least squares method in cases where the relationship of the dependent variable with more than one explanatory variable is considered in the linear regression model. On the other hand, OLSE is the estimation of parameters by minimizing the sum of squared errors between the dependent variable and the predicted parameters in the linear regression model. In linear regression models, other popular methods used in predicting and estimating parameters, except the classical methods, are the best linear unbiased predictor (BLUP) and the best linear unbiased estimator (BLUE), respectively. These are the unbiased predictors and estimators with the smallest covariance according to the Löwner order. For SUR models, which are an extension of linear regression models and where error terms are associated between regression equations, the choice of predictor/estimator can have an important impact on the estimation of results. It is important to compare different predictors/estimators to determine which is appropriate for the situation. Prediction problems are commonly encountered and of great importance in statistical analysis using SUR models. Obtaining accurate predictions significantly influences the model's success and the interpretability of the results. Therefore, conducting a detailed analysis of prediction problems and utilizing appropriate prediction methods enable the effective use of SUR models. Prediction problems are one of the main issues in the statistical analysis of SUR models and have practical significance. The ability to obtain accurate predictions directly impacts the performance of the model and the interpretability of the results. Thus, it is crucial to thoroughly analyze prediction problems and use suitable prediction techniques to ensure the effective utilization of SUR models. Covariance matrix comparison is often used as a criteria for evaluating the performance of predictors and estimators. Additionally, another method for exploring the relationships between predictors/estimators involves establishing equalities between them and deriving additive decomposition expressions for these predictors/estimators. These expressions consisting of equations enable to specify the contributions of partial unknown vectors in linear models, as the models themselves can be related within the system. In this thesis study, the problem of estimation and prediction under SUR models has been considered. Results regarding the statistical properties of unknown parameters' estimators and predictors have been presented under the considered models. By utilizing block matrices and rank properties, some results on the equalities and decomposition of predictors have been obtained. Furthermore, by using matrix rank and inertia formulas, some equalities and inequalities concerning the comparison of covariance matrices of predictors have been derived. By employing rigorous statistical methodologies and carefully evaluating the performance of different predictors within the SUR framework, this study aims to enhance the accuracy of predictions and facilitate a more comprehensive interpretation of the data. This study consists of seven chapters. SUR models are introduced, and overview of the literature is provided in the introduction chapter. The second chapter presents the fundamental concepts, theorems, and properties related to the topic addressed in the study. In the third chapter, SUR models are introduced in detail, and the expressions for estimation and predictability properties, as well as the expressions for estimators and predictors, are presented under the considered models. The main results of the study are examined under three separate headings. The equalities of predictors and estimators under SUR models are presented in the fourth section, the additive decompositions in the fifth section, and the comparison of covariance matrices in the sixth section. Additionally, some results for specific cases are obtained using block matrices, rank properties, and inertia formulas. The final chapter concludes the study by presenting the results and providing recommendations for future research. This study has contributed to a better understanding and increased utilization of SUR models by considering the statistical properties of estimators and predictors under these models. The obtained results provide researchers and practitioners with more insights into the effectiveness of SUR models in regression analysis. Furthermore, the recommendations of this study can serve as a guide for future research and contribute to the development of the literature in this field.
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