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ejpam-2492	Ridley, Dennis	On the Variance of Antithetic Time Series	2015	19	.pdf	application/pdf	7621	308	71	To illustrate the effect of variance, let us consider the results of hypothetical simulations for the lognormal time series X t = exp � Yt � , where Yt = βYt−1 + et , t = 2,3, . . − 2w ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 Var � X t �− 2 (1−w)exp � ln X̂ + s2 2 � σ2/ q Var � X t � p exp(σ2)− 1 Var � X t � =  �w2 − 2w � ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 + 1− 2 (1−w)exp � ln X̂ + s2 2 � σ2/ q Var � X t � p exp(σ2)− 1 + (1−w)2exp � 2ln X̂ + s2 � Var � ln X̂ t � Var � X t � + 2w (1−w) ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � « × exp � ln X̂ + s2 2 � Cov � ln X̂ t , X t−1 � Var � X t � � Var � X t � Appendix C: Per Unit Reduction in MSE Denoting the per unit reduction in MSE due to combining as PMSEc � σ2 � = MSE � σ2 �−MSEc � σ2 � MSE (σ2) REFERENCES 445 Substituting for MSE � σ2 � and MSEc � σ2 � from Appendix A and Appendix B. MSE � σ2 � =  1− ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 Var � X t � MSEc � σ2 � =  �w2 − 2w � ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 + 1− 2 (1−w)exp � ln X̂ + s2 2 � σ2/ q Var � X t � p exp(σ2)− 1 + (1−w)2exp � 2ln X̂ + s2 � Var � ln X̂ t � Var � X t � + 2w (1−w) ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � « × exp � ln X̂ + s2 2 � Cov � ln X̂ t , X t−1 � Var � X t � � Var � X t � From which MSE � σ2 �−MSEc � σ2 � =  1− ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 Var � X t �− ��w2 − 2w � × ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 + 1− 2 (1−w)exp � ln X̂ + s2 2 � σ2/ q Var � X t � p exp(σ2)− 1 + (1−w)2exp � 2ln X̂ + s2 � Var � ln X̂ t � Var � X t � + 2w (1−w) ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � « × exp � ln X̂ + s2 2 � Cov � ln X̂ t , X t−1 � Var � X t � � Var � X t � =  − �w2 − 2w+ 1 � ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � «2 + 2 (1−w)exp � ln X̂ + s2 2 � σ2/ q Var � X t � p exp(σ2)− 1 − (1−w)2exp � 2ln X̂ + s2 � Var � ln X̂ t � Var � X t � − 2w (1−w) ¨ Φ+ Cov � ǫt , X t−1 � Var � X t � « × exp � ln X̂ + s2 2 � Cov � ln X̂ t , X t−1 � Var � X t � � Var � X t � =  −(w−	cache/ejpam-2492.pdf	txt/ejpam-2492.txt