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My understanding is that the Anderson-Hsiao adjustment, described here as an example, addresses concerns about potential or actual autocorrelation by taking a first difference of the dependent variable and adding an independent instrument which happens to be the twice lagged first difference of the dependent variable.

My question is: Why doesn't this adjustment continue to permit correlation with the once lagged first difference? If this occurs in every period, isn't that a failed correction for autocorrelation?

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