Lag operator

In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series

:$$X= {X_1, X_2, dots },

then

:$,$L X_t = X_{t-1} for all $;$t > 1,

where "L" is the lag operator. Sometimes the symbol "B" for backshift is used instead. Note that the lag operator can be raised to arbitrary integer powers so that

:$,$L^{-1} X_{t} = X_{t+1},

and

:$,$L^k X_{t} = X_{t-k}.,

Lag polynomials

Also polynomials of the lag operator can be used, and this is a common notation for ARMA models. For example,

:$$varepsilon_t = X_t - sum_{i=1}^p varphi_i X_{t-i} = left(1 - sum_{i=1}^p varphi_i L^i ight) X_t,

specifies an AR("p") model.

A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as

:$$varphi X_t = heta varepsilon_t,

where φ and θ respectively represent the lag polynomials,

:$$varphi = 1 - sum_{i=1}^p varphi_i L^i,

and

:$$heta= 1 + sum_{i=1}^q heta_i L^i.,

An annihilator operator, denoted $$ [ ] _+, removes the entries of the polynomial with negative power (future values).

Difference operator

In time series analysis, the first difference operator $$Delta is a special case of lag polynomial.

:$$egin{array}{lcr} Delta X_t & = X_t - X_{t-1} \ Delta X_t & = (1-L)X_tend{array}

Similarly, the second difference operator

:$$egin{align} Delta ( Delta X_t ) & = Delta X_t - Delta X_{t-1} \ Delta^2 X_t & = (1-L)Delta X_t \ Delta^2 X_t & = (1-L)(1-L)X_t \ Delta^2 X_t & = (1-L)^2 X_tend{align}

The above approach generalises to the "i" 'th difference operator$$Delta ^i X_t = (1-L)^i X_t

See also

* Autoregressive moving average model
* Shift operator
* Z-transform