11.2 Theoretical Background
11.2.1 The Dynamic Path Model DPLS
The dynamic form of the structural model can be transformed into
the exterior shape of the ``normal" PLS model:
 |
(11.4) |
where
 |
(11.5) |
is a matrix containing the lag operator
with
. On what we call now the dynamic PLS model (DPLS)
 |
(11.6) |
the original PLS algorithm is applicable. Initially, Boolean
design matrices
,
and
corresponding to the unlagged and lagged dependencies in the inner
model (11.6) and to the outer model,
i.e., to the zero restrictions for the coefficient matrices
,
and
, must be fixed. The inner model can be
illustrated by a path diagram, including additional arrows for the
lagged relationships. The inner design matrix
contains the digit one where there is a connection
between two LVs in the path model and consists of zeros elsewhere.
Similarly, the lag design matrix
consists of ones and
zeros corresponding to whether or not there is assumed to be first
order lagged (auto)regression between latent variables.
is the outer design matrix
corresponding to whether or not a variable
of
belongs to the block of a certain latent variable, i.e., a row
of
.
11.2.2 PLS Estimation with Dynamic Inner Approximation
For simplicity, the symbols for the empirically estimated LVs and
coefficients will not be distinguished from those for the
corresponding theoretical quantities in this section. In order to
estimate the weight matrix
, the following steps will be
repeatedly executed:
- Initial representation of the latent variables as components of the
manifest variables with chosen starting values for the matrix
 |
(11.7) |
- Standardization of the LVs to unit variance
 |
(11.8) |
where
 |
(11.9) |
is the
matrix of all time scores of
for
. Elementwise multiplication of matrices is
denoted by
.
- Calculation of ``neighbourhood'' variables corresponding to the inner
path model:
 |
(11.10) |
 |
(11.11) |
that means
 |
(11.12) |
where
and
are suitable inner weighting
matrices, e.g.:
 |
(11.13) |
 |
(11.14) |
with
and
being the design matrices for the
inner model.
 |
(11.15) |
 |
(11.16) |
are the correlation matrix and the first order autocorrelation
matrix of LVs, respectively, with
 |
(11.17) |
 |
(11.18) |
- New values of the weight matrix
are gained by OLS
estimation:
where
is the outer design matrix.
- The estimated coefficients
are substituted for the
previous elements of the weight matrix
Using this new weight matrix we continue the procedure by
repeating step 1.
The iteration process is stopped when subsequent estimations of the LVs
in step 2 do not relevantly differ from the previous
ones.
Then the coefficient matrices
and
of the inner model
(11.4)
 |
(11.20) |
can be estimated by a suitable method for dynamic models, such as
OLS, GLS, Cochrane-Orcutt, ECM etc. The loadings
of the outer
model (11.2) are estimated by simple OLS.
11.2.3 Prediction and Goodness of Fit
By substituting (11.4) for
in
(11.2) we obtain
Then substituting (11.6) for
, we have
 |
(11.22) |
or
 |
(11.23) |
Using this prediction formula, we can construct a goodness-of-fit
criterion.
From (11.21) it follows that the predictable part of
is
. Let
denote the whole predicted data matrix,
the matrix of the LVs,and
the empirical correlation or covariance matrix of
.
Then the empirical covariance of these predictions is
with
being the first order autocorrelation matrix. The
inconsiderable inaccurracy of the last relationship arises from
tiny differences that might occur between the covariances of the
latent variables
and those of the lagged LVs
.
It is easy to see that G* contains in its diagonal the
variances of the predictable part, or what Lohmöller (1989)
calls the ``redundant" part, of the MVs.
Following Lohmöller (1989) again we calculate the ratio of two
diagonal matrices
 |
(11.26) |
where
denotes the empirical covariance matrix of
the manifest variables. The entries in the diagonal of G are
ratios expressing to what extent the variance of each manifest
variable is reproduced by the variance of the predictable part,
i.e., by the model.
The average of these measures
 |
(11.27) |
is the redundancy coefficient or average redundancy and is used
for the evaluation of the goodness of fit of the model.