11.1 Introduction

In addition to the well known estimation and confirmation approach for path models, LISREL by Jöreskog and Sörbom (1987) the partial least squares (PLS) algorithm by Wold (1973) has gained popularity as an instrument of analysis and forecasting in sociometrics and econometrics during recent years. The PLS approach to path models is data oriented and mostly descriptive or explorative, the model being defined purely by an algorithm.

The traditional PLS model involves $ M$ observable, manifest variables (MV) $ { y
}^{m}$ $ (m=1,\dots,M)$ and $ K<M$ latent, i.e., not directly observable variables (LV) $ { \eta }^{k}$ $ (k=1,\dots,K)$. The latter are assumed to be certain constructs composed from those MVs. Furthermore, the LVs are assumed to be connected with each other by the linear inner or structural model:

$\displaystyle { \eta }_{t}{ =b}_{0}{ +B\eta }_{t}{ +\nu }_{t}$ (11.1)

where $ { \eta }_{t}=\left( \eta _{t}^{1}{ ,}\eta _{t}^{2}{
\dots},\eta _{t}^{K}\right) ^T$ is the column vector of the scores of all latent variables $ { \eta }_{{}}^{1},\dots,{ \eta
}_{{}}^{k}$ for one case $ t$ $ (t = 1,\dots, T)$, $ B$ is a triangular $ K\times K$ matrix of path coefficients with zero diagonal, and $ b_{0}$ is a location parameter vector usually set equal zero. The error term $ { \nu }_{t}$ has zero expectation.

What is called the outer or measurement model describes the assumed linear relations between the MVs and the LVs:

$\displaystyle { y}_{t}{ =p}_{0}{ +P\eta }_{t}{ +\varepsilon }_{t}\qquad \textrm{(loadings relation)}$ (11.2)

with a block diagonal $ M\times K$ matrix $ P$ of path coefficients and a zero expectation disturbance term $ { \varepsilon }_{t}$. Again the location parameters $ { p}_{0}$ are usually transformed to zero. The latent variables are taken to be weighted sums of manifest variables with a block diagonal weight matrix $ { W}$:

$\displaystyle { \eta }_{t}{ =W}^{\prime }{ y}_{t}\qquad \textrm{(weight relations)}$ (11.3)

The iterative estimation of the weights $ { W}$ is the main aim of Wold's PLS algorithm. The procedure is to start with more or less arbitrarily chosen weights $ { W}$. The next stage involves the calculation of proxies $ { \eta }^{k*}$ for the latent variables $ { \eta }^{k}.$ $ { \eta }^{k*}$ is a weighted sum of all LVs directly connected with $ { \eta }^{k}$ .

Finally new weights $ { W}=\left( { \omega }_{mk}\right) $ are estimated by OLS regression between the MVs and the approxies. The scores of the LVs being approximately known after stopping the iteration process, we can easily estimate the parameter matrices $ { B}$ and $ { P}$ by OLS, using model equations (11.1) and (11.2) respectively.

More details about PLS are given by Lohmöller (1984) who is the author of the PLS computer programme LVPLS.

In Section 11.2 of this paper a PLS-like approach to a class of dynamic models with latent variables will be suggested. The way to use these models for prediction will be shown, and a measure for goodness of fit will be deduced. Then, a computer programme for dynamic partial least squares modelling will be presented. The final section will show a small five-block model with an first order autoregressive distributed lag relation between the LVs.