ASYMPTOTIC INFERENCE FOR AR MODELS WITH HEAVY-TAILED G-GARCH NOISES
研究了自回归模型在重尾G-GARCH噪声下最小二乘估计的相合性和极限分布，发现当噪声尾指数小于2时估计量不一致，这对经济和金融领域的实证分析有重要警示。
It is well known that the least squares estimator (LSE) of an AR( p ) model with i.i.d. (independent and identically distributed) noises is n 1/ α L ( n )-consistent when the tail index α of the noise is within (0,2) and is n 1/2 -consistent when α ≥ 2, where L ( n ) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR( p ) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n 1–2/ α -consistent if α ε (2,4), log n -consistent if α = 2, and n 1/2 / log n -consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.