backshift

Backshift is a term used in linguistics and statistics to refer to a backward movement in time or tense within a sentence or data sequence. In linguistics, backshifting (also called reported speech backshift) describes how the tense in subordinate clauses is often shifted back when the content is reported by someone else. For example, direct speech He says, "I am hungry" becomes indirect speech He said that he was hungry. Modals and other auxiliaries may also backshift: will becomes would, can becomes could, may becomes might. The extent of backshifting can vary by language and context; in some cases backshifting is optional or replaced by other constructions.

In time series analysis, backshift refers to the backward shift operator B, defined by B x_t = x_{t-1} for a time series x_t. The operator is used to express differences and lag polynomials compactly. For instance, the difference Δ x_t = x_t - x_{t-1} can be written as (1 - B) x_t. An autoregressive model AR(p) can be written φ(B) x_t = ε_t, with φ(B) = 1 - φ_1 B - ... - φ_p B^p. The backward shift notation simplifies the representation of lag structures and is a standard tool in ARIMA modeling.

Notes: The term may be encountered in different disciplines, and its precise meaning depends on context. In some discussions, backshift may refer to the general idea of moving elements to earlier positions or times.