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:

$\displaystyle { \eta }_{t}{ =F\eta }_{t}{ +\nu }_{t}$ (11.4)

where

$\displaystyle { F=B+C}L$ (11.5)

is a matrix containing the lag operator $ L$ with $ L{ \eta }_{t}={
\eta }_{t-1}$. On what we call now the dynamic PLS model (DPLS)

$\displaystyle { \eta }_{t}{ =F\eta }_{t}{ +\nu }_{t}$

$\displaystyle { y}_{t}{ =P\eta }_{t}{ +\varepsilon }_{t}
$

$\displaystyle { \eta }_{t}{ =W}^T{ y}_{t}$ (11.6)

the original PLS algorithm is applicable. Initially, Boolean design matrices $ { D}_{B}$, $ { D}_{C}$ and $ { D}_{P}$ 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 $ { B}$, $ { C}$ and $ { P}$, must be fixed. The inner model can be illustrated by a path diagram, including additional arrows for the lagged relationships. The inner design matrix $ { D}_{B}$ 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 $ { D}_{C}$ consists of ones and zeros corresponding to whether or not there is assumed to be first order lagged (auto)regression between latent variables. $ {D}_{P}=\left( d_{mk}\right) $ is the outer design matrix corresponding to whether or not a variable $ y^{m}$ of $ { Y}$ belongs to the block of a certain latent variable, i.e., a row $ { \eta }
^{_{k}}$ of $ { H}$.


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 $ { W}$, the following steps will be repeatedly executed:

  1. Initial representation of the latent variables as components of the manifest variables with chosen starting values for the matrix $ { W}$

    $\displaystyle { \eta }_{t}{ =W}^T{ y}_{t}$ (11.7)

  2. Standardization of the LVs to unit variance

    $\displaystyle { \eta }_{t}:=\sqrt{T}\left( { I*HH}^T\right) ^{-\frac{1}{2}} { \eta }_{t}$ (11.8)

    where

    $\displaystyle { H=}\left( { \eta }_{1},{ \eta }_{2},\dots,{ \eta }_{T}\right) { ,}$ (11.9)

    is the $ K \times T$ matrix of all time scores of $ { \eta }_{t}$ for $ t = 1, \dots, T$. Elementwise multiplication of matrices is denoted by $ *$.
  3. Calculation of ``neighbourhood'' variables corresponding to the inner path model:

    $\displaystyle { \eta }_{t}^{*}{ =F}^{*}{ \eta }_{t}$ (11.10)

    $\displaystyle { F}^{{ *}}={ B}^{*}+{ C}^{*}L+{ C}^{*T}L^{-1}$ (11.11)

    that means

    $\displaystyle { \eta }_{t}^{*}={ B}^{{ *}}{ \eta }_{t}+{ C}^{{ *}}{ \eta }_{t-1}+{ C}^{*T}{ \eta }_{t+1}$ (11.12)

    where $ { B}^{*}$ and $ { C}^{*}$ are suitable inner weighting matrices, e.g.:

    $\displaystyle { B}^{*}=\left( { D}_{B}+{ D}_{B}^{T}\right) *{ R}$ (11.13)

    $\displaystyle { C}^{*}{ =D}_{C}{ *A}$ (11.14)

    with $ { D}_{B}$ and $ { D}_{C}$ being the design matrices for the inner model.

    $\displaystyle { R=HH}^{T}/T$ (11.15)

    $\displaystyle { A=H}\left( L{ H}\right) ^{T}/T$ (11.16)

    are the correlation matrix and the first order autocorrelation matrix of LVs, respectively, with

    $\displaystyle { H=}\left( { \eta }_{1},{ \eta }_{2},\dots,{ \eta }_{T}\right) { ,}$ (11.17)

    $\displaystyle L{ H}=\left( { \eta }_{0},{ \eta }_{1},...,{ \eta }_{T-1}\right)$ (11.18)

  4. New values of the weight matrix $ { W}$ are gained by OLS estimation:

    $\displaystyle y_{t}^{m}=\omega _{mk}\eta _{t}^{k*}+v_{t}^{mk}$   if$\displaystyle \quad d_{mk}$   =1$\displaystyle \qquad \textrm{(Wold's Mode A)}$ (11.19)

    where $ { D}_{P}=(d_{mk})$ is the outer design matrix.
  5. The estimated coefficients $ \omega _{mk}$ are substituted for the previous elements of the weight matrix $ { W}$

    $\displaystyle { W}:=\left( \omega _{mk}\right) .
$

Using this new weight matrix we continue the procedure by repeating step 1. The iteration process is stopped when subsequent estimations of the LVs $ { \eta }_{t}$ in step 2 do not relevantly differ from the previous ones.

Then the coefficient matrices $ { B}$ and $ { C}$ of the inner model (11.4)

$\displaystyle { \eta }_{t}=\left( { B}+{ C}L\right) { \eta}_{t} +{ v}_{t}$ (11.20)

can be estimated by a suitable method for dynamic models, such as OLS, GLS, Cochrane-Orcutt, ECM etc. The loadings $ { P}$ of the outer model (11.2) are estimated by simple OLS.


11.2.3 Prediction and Goodness of Fit

By substituting (11.4) for $ { y}_{t}$ in (11.2) we obtain

$\displaystyle { y}$ $\displaystyle =$ $\displaystyle { P\eta }_{t}+{ \varepsilon }_{t}$ (11.21)
  $\displaystyle =$ $\displaystyle { PF\eta }_{t}{ +P\nu }_{t}{ +\varepsilon }_{t}$  

Then substituting (11.6) for $ { \eta }_{t}$, we have

$\displaystyle { y}_{t}={ PFW}^{T}{ y}_{t}+{ P\nu }_{t}+ { \varepsilon }_{t}$ (11.22)

or

$\displaystyle { y}_{t}=\left( { PBW}^{T}{ y}_{t}+{ PCW}^{T}{ y} _{t-1}\right) +\left( { P\nu }_{t}+{ \varepsilon }_{t}\right)$ (11.23)

Using this prediction formula, we can construct a goodness-of-fit criterion. From (11.21) it follows that the predictable part of $ { y}_{t}$ is $ { y}_{t}^{*}={ PF\eta }_{t}$. Let $ { Y}^{*}=({ y
}_{1}^{*}{ ,...,y}_{T}^{*})$ denote the whole predicted data matrix, $ { H}=({ \eta }_{1}{ ,...,\eta }_{T})$ the matrix of the LVs,and $ { R}$ the empirical correlation or covariance matrix of $ { H}$. Then the empirical covariance of these predictions is
$\displaystyle cov({ Y}^{*})$ $\displaystyle =$ $\displaystyle { P\ F\ }cov({ H}){ \ F}^{T}{ \ P}
^{T}$ (11.24)
  $\displaystyle =$ $\displaystyle { P\ F\ R\ F}^{T}{ \
P}^{T}$  


$\displaystyle cov({ Y}^{*})$ $\displaystyle =$ $\displaystyle { P(B+C}L){ H\ H}^{T}{ \ (B+C}L)^{T}
{ P}^{T}/T$ (11.25)
  $\displaystyle =$ $\displaystyle ({ PBHH}^{T}{ B}^{T}{ P}^{T}+{ PC}(L{ H})
{ H}^{T}{ B}^{T}{ P}^{T}{ +PBH}(L{ H}
)^{T}{ C}^{T}{ P}^{T}$  
    $\displaystyle +{ PC(}L{
H)(}L{ H)}^{T}{ C}^{T}{ P}^{T})/T$  
  $\displaystyle {\approx}$ $\displaystyle { PBRB}^{T}{ P}^{T}{ +PBAC}^{T}
{ P}^{T}{ +PCRC}^{T}{ P}^{T}{ +(PBAC}^{T}
{ P}^{T}{ )}^{T}{ =G}^{*}$  

with $ { A}$ 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 $ { HH}^{T}/{ T}$ and those of the lagged LVs $ { (}L{ H)(}L
{ H)}^{T}/T$.

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

$\displaystyle { G=(I*G}^{*}{ )(I*\Sigma }_{y}{ )}^{-1}$ (11.26)

where $ { \Sigma }_{y}$ 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

$\displaystyle G^{2}=trace\ { G/M}$ (11.27)

is the redundancy coefficient or average redundancy and is used for the evaluation of the goodness of fit of the model.