Editing Lag operator (section)

==Conditional expectation==
It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let <math>\Omega_t</math> be all information that is common knowledge at time ''t'' (this is often subscripted below the expectation operator); then the expected value of the realisation of ''X'', ''j'' time-steps in the future, can be written equivalently as:

:<math>E [ X_{t+j} | \Omega_t] = E_t [ X_{t+j} ] .</math>

With these time-dependent conditional expectations, there is the need to distinguish between the backshift operator (''B'') that only adjusts the date of the forecasted variable and the Lag operator (''L'') that adjusts equally the date of the forecasted variable and the information set:

:<math>L^n E_t [ X_{t+j} ] = E_{t-n} [ X_{t+j-n} ]  ,</math>
:<math>B^n E_t [ X_{t+j} ] = E_t [ X_{t+j-n} ]  .</math>