Computes least‑squares estimates of a linear regression model for a multichannel time series with a specified base channel.

Routine MLSE performs least‑squares estimation of a linear regression model for a multichannel time series with a specified base channel.

Define the multichannel time series X by

where

with n = NOBSX and m = NCHANX. The columns of X correspond to individual channels of a multichannel time series and may be examined from a univariate perspective. The rows of X correspond to observations of an m‑variate time series and may be examined from a multivariate perspective. Note that an alternative characterization of the multivariate time series X considers the columns of X to be observations of an m‑variate time series with the rows of X containing univariate time series. For example, see Priestley (1981, page 692) and Fuller (1976, page 14).

The model is formed by regressing the base series X1 on its previous values and on the remaining channels X2, …, Xm. The differenced form of the model is given by

where θ0 = CNST is the overall constant, dk = NDIFF(k) is the order of differencing Xk, lk = LAG(k) is the amount Xk lags X1,

and pk = NDPREG(k) for k = 1, 2, …, m.

The undifferenced form of the model is given by

for k = 1, 2, …, m. Note that if l1 ≥ d1 ≥ 0, the base series terms Xt−j,1 at lags j = 1, …, (l1 ‑ 1) are omitted from the right‑hand side of the above model when d1 ≥ 1. In the actual computations, these terms are included.

Consider the Wölfer Sunspot Data (Box and Jenkins 1976, page 530) along with data on northern light activity and earthquake activity (Robinson 1967, page 204) to be a three‑channel time series. Routine MLSE is applied to these data to examine the regressive relationship between the channels.

Consider the Gas Furnace Data (Box and Jenkins 1976, pages 532–533) where X1 is the percent CO2 in the outlet gas and X2 is the input gas rate in cubic feet/minute. Application of routine MLSE to these data produces the following results: