Journal of Probability and Statistics

Research Article

The Failure of Orthogonality under Nonstationarity: Should We Care About It?

Table 1

Finite-sample behavior of ̂ ‌ 𝛽 − 𝛽 𝑦 : mean and standard deviation.
DGP parametersSample size
𝛽 𝑦 𝜇 𝑥 Statistic50100200500700
−0.9−1.5Mean 𝟏 . 𝟓 ⋅ 𝟏 𝟎 − 𝟐 𝟑 . 𝟏 ⋅ 𝟏 𝟎 − 𝟑 𝟕 . 𝟖 ⋅ 𝟏 𝟎 − 𝟒 𝟖 . 𝟔 ⋅ 𝟏 𝟎 − 𝟓 𝟓 . 𝟔 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 1 . 8 ⋅ 1 0 − 3 ) ( 1 . 7 ⋅ 1 0 − 4 ) ( 1 . 8 ⋅ 1 0 − 5 ) ( 1 . 1 ⋅ 1 0 − 6 ) ( 3 . 9 ⋅ 1 0 − 7 )
−0.75Mean 𝟏 . 𝟓 ⋅ 𝟏 𝟎 − 𝟐 𝟑 . 𝟑 ⋅ 𝟏 𝟎 − 𝟑 𝟕 . 𝟔 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟎 ⋅ 𝟏 𝟎 − 𝟒 𝟓 . 𝟑 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 1 . 9 ⋅ 1 0 − 3 ) ( 1 . 6 ⋅ 1 0 − 4 ) ( 1 . 8 ⋅ 1 0 − 5 ) ( 1 . 1 ⋅ 1 0 − 6 ) ( 3 . 9 ⋅ 1 0 − 7 )
0.75Mean 𝟏 . 𝟒 ⋅ 𝟏 𝟎 − 𝟐 𝟑 . 𝟐 ⋅ 𝟏 𝟎 − 𝟑 𝟖 . 𝟎 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟎 ⋅ 𝟏 𝟎 − 𝟒 𝟓 . 𝟖 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 1 . 7 ⋅ 1 0 − 3 ) ( 1 . 6 ⋅ 1 0 − 4 ) ( 1 . 8 ⋅ 1 0 − 5 ) ( 1 . 0 ⋅ 1 0 − 6 ) ( 3 . 9 ⋅ 1 0 − 7 )
1.5Mean 𝟏 . 𝟓 ⋅ 𝟏 𝟎 − 𝟐 𝟑 . 𝟎 ⋅ 𝟏 𝟎 − 𝟑 𝟕 . 𝟐 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟎 ⋅ 𝟏 𝟎 − 𝟒 𝟓 . 𝟖 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 1 . 8 ⋅ 1 0 − 3 ) ( 1 . 6 ⋅ 1 0 − 4 ) ( 1 . 8 ⋅ 1 0 − 5 ) ( 1 . 1 ⋅ 1 0 − 6 ) ( 3 . 9 ⋅ 1 0 − 7 )
0.9−1.5Mean 𝟐 . 𝟐 ⋅ 𝟏 𝟎 − 𝟑 𝟔 . 𝟐 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟒 ⋅ 𝟏 𝟎 − 𝟒 𝟐 . 𝟏 ⋅ 𝟏 𝟎 − 𝟓 𝟏 . 𝟏 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 2 . 4 ⋅ 1 0 − 4 ) ( 2 . 8 ⋅ 1 0 − 5 ) ( 3 . 4 ⋅ 1 0 − 6 ) ( 2 . 1 ⋅ 1 0 − 7 ) ( 8 . 0 ⋅ 1 0 − 8 )
−0.75Mean 𝟐 . 𝟒 ⋅ 𝟏 𝟎 − 𝟑 𝟔 . 𝟎 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟓 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟗 ⋅ 𝟏 𝟎 − 𝟓 𝟏 . 𝟕 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 2 . 3 ⋅ 1 0 − 4 ) ( 2 . 8 ⋅ 1 0 − 5 ) ( 3 . 4 ⋅ 1 0 − 6 ) ( 2 . 2 ⋅ 1 0 − 7 ) ( 7 . 9 ⋅ 1 0 − 8 )
0.75Mean 𝟐 . 𝟒 ⋅ 𝟏 𝟎 − 𝟑 𝟓 . 𝟖 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟑 ⋅ 𝟏 𝟎 − 𝟒 𝟐 . 𝟐 ⋅ 𝟏 𝟎 − 𝟓 𝟓 . 𝟒 ⋅ 𝟏 𝟎 − 𝟔
Stand.dev. ( 2 . 3 ⋅ 1 0 − 4 ) ( 2 . 9 ⋅ 1 0 − 5 ) ( 3 . 5 ⋅ 1 0 − 6 ) ( 2 . 2 ⋅ 1 0 − 7 ) ( 8 . 0 ⋅ 1 0 − 8 )
1.5 Mean 𝟐 . 𝟐 ⋅ 𝟏 𝟎 − 𝟑 𝟓 . 𝟔 ⋅ 𝟏 𝟎 − 𝟒 𝟏 . 𝟐 ⋅ 𝟏 𝟎 − 𝟒 𝟐 . 𝟏 ⋅ 𝟏 𝟎 − 𝟓 𝟏 . 𝟏 ⋅ 𝟏 𝟎 − 𝟓
Stand.dev. ( 2 . 4 ⋅ 1 0 − 4 ) ( 2 . 9 ⋅ 1 0 − 5 ) ( 3 . 4 ⋅ 1 0 − 6 ) ( 2 . 2 ⋅ 1 0 − 7 ) ( 7 . 9 ⋅ 1 0 − 8 )