## Double-Length Regressions

Up to this point, the number of observations for all the artificial regressions we have studied has been equal to n, the number of observations in the data. In some cases, however, artificial regressions may have 2n or even 3n observations. This can happen whenever each observation makes two or more contributions to the criterion function.

The first double-length artificial regression, or DLR, was proposed by Davidson and MacKinnon (1984a). We will refer to it as the DLR, even though it is no longer the only artificial regression with 2n observations. The class of models to which the DLR applies is a subclass of the one used for GMM estimation. Such models may be written as

f (y, 0) = є t, t = 1,…, n, £t ~ NID(0, 1), (1.47)

where, as before, each ft () is a smooth function that depends on...

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