The second-order bias and mean squared error of estimators in time-series models

Abstract

We develop the analytical results on the second-order bias and mean squared error (MSE) of estimators in time series. These results provide a unified approach to developing the properties of a large class of estimators in the linear and nonlinear time series models and they are valid for both the normal and non-normal sample of observations, and where the regressors are stochastic. The estimators included are the generalized method of moments, maximum likelihood, least squares, and other extremum estimators. Our general results are applied to a wide variety of econometric models. Numerical results for some of these models are presented.

Bias in the estimation of the mean reversion parameter in continuous time models

“…We have presented two alternative expressions for approximating the bias of the mean reversion estimator in a continuous time di¤usion model, based on the method proposed by Bao and Ullah (2007). The simpler expression mimics the bias formula derived by Marriott and Pope (1954) for the discrete time AR model and corresponds to the bias formula derived independently by Tang and Chen (2009) for the same model but with unknown mean.…”

Bias-Correction in Vector Autoregressive Models: A Simulation Study

“…Neither Pope nor Tjøstheim and Paulsen refer to Yamamoto and Kunitomo [27] who, on the other hand, do not refer to Tjøstheim and Paulsen [26]. As noted by Bao and Ullah [23], the bias expression they derive is consistent with the bias expression in Nicholls and Pope [28] and Pope [29] but they do not refer to Yamamoto and Kunitomo [27].…”

“…In particular, the bias in the simple univariate AR(1) model has been analyzed in many papers over the years using both analytical expressions, numerical computations, and simulations, e.g., Orcutt and Winokur [19], Sawa [20], MacKinnon and Smith [21], Patterson [22], and Bao and Ullah [23]. In a multivariate context analytical expressions for the finite-sample bias in estimated vector autoregressive (VAR) models have been developed by Tjøstheim and Paulsen [26], Yamamoto and Kunitomo [27], Nicholls and Pope [28], Pope [29], and Bao and Ullah [23]. However, there are no detailed analyses of the properties of these multivariate analytical bias formulas.…”

Moments of the estimated Sharpe ratio when the observations are not IID

“…For the proof and applications of the above results, see, e.g., Bao and Ullah (2003) and Ullah (2004). Given the above results and the fact that ι Mι = ι MΣι = ι MΣMι = ι (MΣ) 2 ι = ι (MΣ) 2 Mι = ι (MΣ) 3 ι = ι (MΣ) 3 Mι = 0, we can derive the expectations λ ij = E[(y My) i (ι y) j /σ 2i+j (T − 1) i T j ] for y ∼ N(u, σ 2 Σ) as follows λ 10 = m 1 /(T − 1), λ 02 = ι Σι/T 2 + S 2 , λ 11 = Sm 1 /(T − 1), λ 20 = m 2 1 + 2m 2 /(T − 1) 2 , λ 12 = (m 1 ι Σι + 2ι ΣMΣι)/T 2 (T − 1) + S 2 m 1 /(T − 1), λ 21 = S m 2 1 + 2m 2 /(T − 1) 2 , λ 30 = m 3 1 + 6m 1 m 2 + 8m 3 /(T − 1) 3 , λ 22 = m 2 1 + 2m 2 ι Σι + 4m 1 ι ΣMΣι + 8ι (ΣM) 2 Σι /T 2 (T − 1) 2 + S 2 m 2 1 + 2m 2 /(T − 1) 2 , λ 31 = S m 3 1 + 6m 1 m 2 + 8m 3 /(T − 1) 3 , λ 40 = m 4 1 + 12m 2 1 m 2 + 12m 2 2 + 32m 1 m 3 + 48m 4 /(T − 1) 4 .…”