Regression spectrum

The spectrum of a stochastic process occurring in the regression scheme for a stationary time series. Thus, let a stochastic process $y _ {t}$ which is observable for $t = 1 \dots n$ be represented in the form

$$\tag{1 } y _ {t} = m _ {t} + x _ {t} ,$$

where $x _ {t}$ is a stationary stochastic process with ${\mathsf E} x _ {t} \equiv 0$, and let the mean value ${\mathsf E} y _ {t} = m _ {t}$ be expressed in the form of a linear regression

$$\tag{2 } m _ {t} = \sum_{k=1}^ { s } \beta _ {k} \phi _ {t} ^ {(k)} ,$$

where $\phi ^ {(k)} = ( \phi _ {1} ^ {(k)} \dots \phi _ {n} ^ {(k)} )$, $k = 1 \dots s$, are known regression vectors and $\beta _ {1} \dots \beta _ {s}$ are unknown regression coefficients (cf. Regression coefficient). Let $M ( \lambda )$ be the spectral distribution function of the regression vectors $\phi ^ {(1)} \dots \phi ^ {(s)}$( cf. Spectral analysis of a stationary stochastic process). The regression spectrum for $M ( \lambda )$ is the set of all $\lambda$ such that $M ( \lambda _ {2} ) - M ( \lambda _ {1} ) > 0$ for any interval $( \lambda _ {1} , \lambda _ {2} )$ containing $\lambda$, $\lambda _ {1} < \lambda < \lambda _ {2}$.

The regression spectrum plays an important role in problems of estimating the regression coefficients in the scheme (1)–(2). For example, the elements of a regression spectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for $\beta$ by the method of least squares.

References

[1]	U. Grenander, M. Rosenblatt, "Statistical analysis of stationary time series" , Wiley (1957)