Identification of Leverage Points in Principal Component Regression and r-k Class Estimators with AR(1) Error Structure

Abstract

The determination of leverage observations have been frequently investigated through ordinary least squares and some biased estimators proposed under the multicollinearity problem in the linear regression models. Recently, the identification of leverage and influential observations have been also popular on the general linear regression models with correlated error structure. This paper proposes a new projection matrix and a new quasiprojection matrix to determination of leverage observations for principal component regression and r-k class estimators, respectively, in general linear regression model with first-order autoregressive error structure. Some useful properties of these matrices are presented. Leverage observations obtained by generalized least squares and ridge regression estimators available in the literature have been compared with proposed principal component regression and r-k class estimators over a simulation study and a numerical example. In the literature, the first leverage is considered separately due to the first-order autoregressive error structure. Therefore, the behaviours of first leverages obtained by principal component regression and r-k class estimators has been also investigated according to the autocorrelation coefficient and biasing parameter through applications. The results showed that the leverage of the first observation obtained by principal component regression and r-k estimators is smaller than that obtained by generalized least squares and ridge regression estimators. In addition, as the autocorrelation coefficient goes to -1, the leverage of the first transformed observation decreases for PCR and r-k class estimators, while its increases while the autocorrelation coefficient goes to 1.

Keywords

• Autocorrelation
• first-order autoregressive error
• leverages
• multicollinearity
• biased estimators

AR (1) Hata Yapısı ile Temel Bileşen Regresyon ve r-k Sınıf Tahmincilerinde Kaldıraç Noktalarının Belirlenmesi

Abstract

Lineer regresyon modellerinde kaldıraç gözlemlerin belirlenmesi sıklıkla sıradan en küçük kareler ve bazı yanlı tahmin ediciler üzerinden araştırılmıştır. Son zamanlarda kaldıraç ve etkin gözlemlerin belirlenmesi genel lineer regresyon modellerinde de popüler olmuştur. Bu çalışmada birinci dereceden otoregresif hata yapısına sahip genel lineer regresyon modelinde temel bileşenler regresyon ve r-k sınıf tahmin edicileri için sırasıyla yeni bir projeksiyon matrisi ve bir yarı projeksiyon matrisi önerilmektedir. Bu matrislerin bazı yararlı özellikleri sunulmuştur. Literatürde bulunan genelleştirilmiş en küçük kareler ve ridge regresyon tahmin edicilerinden elde edilen kaldıraç gözlemleri, bir simülasyon çalışması ve sayısal bir örnek üzerinden önerilen temel bileşen regresyonu ve r-k sınıfı tahmin edicileriyle karşılaştırılmıştır. Literatürde birinci dereceden otoregresif hata yapısı nedeniyle birinci kaldıraç ayrı olarak ele alınmaktadır. Bu nedenle, temel bileşen regresyonu ve r-k sınıfı tahmin edicileriyle elde edilen ilk kaldıraçların davranışları, uygulamalar aracılığıyla otokorelasyon katsayısına ve önyargı parametresine göre de incelenmiştir. Sonuçlar, temel bileşen regresyonu ve r-k tahmin edicileriyle elde edilen ilk gözlemin kaldıraçlığının, genelleştirilmiş en küçük kareler ve sırt regresyon tahmin edicileriyle elde edilenden daha küçük olduğunu göstermiştir.

Keywords

• Otokorelasyon
• birinci dereceden otoregresif hata
• kaldıraçlar
• çoklu iç ilişki
• yanlı tahmin ediciler

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