Index Models

Let \(y_t\) be an \((n \times 1)\) covariance stationary, linearly indeterministic stochastic process. The process \(y_t\) is said to satisfy an unobservable index model, in the sense of Sargent and Sims (1977), if it possesses a representation:

(236)\[y_t = \Lambda(L) f_t + D(L) \epsilon_t\]

where \(f_t\) is a \((k \times 1)\) vector white noise, \(\epsilon_t\) is an \((n \times 1)\) vector white noise, \(\Lambda(L)\) is an \((n \times k)\) matrix of square summable polynomials in the lag operator, \(D(L)\) is an \((n \times n)\) diagonal matrix of square summable polynomials in the lag operator, and the white noises \(f_t\), \(\epsilon_t\), satisfy the orthogonality conditions

(237)\[\begin{split}\begin{aligned} E \begin{bmatrix} f_t \\ \epsilon_t \end{bmatrix} \begin{bmatrix} f_{t-s} \\ \epsilon_{t-s}\end{bmatrix}^T &= \begin{bmatrix} \Sigma_f & 0 \\ 0 & \Sigma_{\epsilon}\end{bmatrix}, \qquad s = 0 \\ &= [0] \qquad s \neq 0 \end{aligned}\end{split}\]

where \(\Sigma_f\) and \(\Sigma_{\epsilon}\) are each diagonal matrices. Thus, each component of \(f_t\) and \(\epsilon_t\) is orthogonal to every other component at all times, and to itself at all nonzero leads and lags. The model is restrictive when \(k\) is sufficiently smaller than \(n\). Usually, it has been assumed that \(k=1\) or \(2\) in applied work.

Equations ((236) and (237)) imply that the covariance generating function of \(y_t\) satisfies

(238)\[S_y(z) = \Lambda(z)\Sigma_f \Lambda(z^{-1})^T + D(z)\Sigma_{\epsilon} D(z^{-1})^T\]

where \(T\) denotes transposition. Equation (238) states that the covariance generating function of \(y\) (or the spectral density matrix of \(y\)) is the sum of a matrix of rank \(k\), namely \(\Lambda(z)\Sigma_f \Lambda(z^{-1})^T\), and a diagonal matrix of rank \(n\), namely \(D(z)\Sigma_{\epsilon} D(z^{-1})^T\). In other words, all of the covariance among distinct components of y is mediated through their common dependence on the \(k\) indexes \(f_t\). Equation (238) is a frequency domain version of a “factor analysis” model, being a factor analytic model for each value of \(z\). Geweke and Singleton (1981a,b) call (238) the “dynamic factor” model.

Following T. C. Koopmans (1947), Sargent and Sims (1977) proposed (236) with \(k=1\) as a statistical model that seemed consistent with the conception of the business cycle used by Arthur Burns and Wesley Claire Mitchell (1946) at the National Bureau of Economic Research. With \(k=1\), the covariances across distinct series entirely reflect their common dependence on the one-dimensional shock \(f_t\).

Sargent and Sims (1977) described some macroeconomic models that would lead an index model like (236) to fit well. More recently, several researchers have applied stochastic optimal growth models which have assumed the form of a one unobservable index model. For example, see Kydland and Prescott (1982) and Altug (1985). The one index in these models is the innovation to a technology shock, while the components of \(\epsilon_t\) are interpreted simply as mutually orthogonal measurement errors. Brock (1982) described a specification of preferences and technology in a stochastic growth model which could lead stock prices to exhibit an unobservable index structure.

A simple example of such a model can be obtained by adding mutually orthogonal measurement errors to the true series for \(y_{nt}, c_t, \) and \(k_{t+1} - k_t\), in the linear stochastic growth model of Chapter XII, Section 9, of Macroeconomic Theory. The one index in this model is \(\theta_t\) the innovation to the technology shock.