Autoregression and Redundant Instruments
研究了零均值平稳自回归模型中，当使用滞后长度大于阶数的工具变量时，哪些工具变量是冗余的，并证明了一个更一般的命题。
Consider a zero mean stationary autoregressive model of order k with IID innovations having variance σ: yt = ρ1yt−1 + ρ2yt−2 + · · ·+ ρkyt−k + et. It is well known that the efficient GMM estimator of ρ = (ρ1 ρ2 · · · ρk) ′ based on the instrumental vector zt = (yt−1 yt−2 · · · yt−k yt−k−1 · · · yt−`) consisting of the last ` > k lags of yt, effectively exploits information in the most recent k lags of yt (see, for example, Kim, Qian and Schmidt, 1999). In other words, the instruments yt−k−1, · · · , yt−` are redundant (see Breusch, Qian, Schmidt and Wyhowski, 1999) given yt−1, · · · , yt−k. Prove the following more general proposition: when one uses the instrumental vector zt = (yt−p yt−p−1 · · · yt−p−k+1 yt−p−k · · · yt−p−`+1) for p ≥ 1, the instruments yt−p−k, · · · , yt−p−`+1 are redundant given yt−p, · · · , yt−p−k+1.