RAM specification using arrow-and-lag notation
Each line of the RAM specification for make_dsem_ram consists of four (unquoted) entries,
separated by commas:
- 1. Arrow specification:
This is a simple formula, of the form
A -> B or, equivalently, B <- A for a regression
coefficient (i.e., a single-headed or directional arrow);
A <-> A for a variance or A <-> B for a covariance
(i.e., a double-headed or bidirectional arrow). Here, A and
B are variable names in the model. If a name does not correspond
to an observed variable, then it is assumed to be a latent variable.
Spaces can appear freely in an arrow specification, and
there can be any number of hyphens in the arrows, including zero: Thus,
e.g., A->B, A --> B, and A>B are all legitimate
and equivalent.
- 2. Lag (using positive values):
An integer specifying whether the linkage
is simultaneous (lag=0) or lagged (e.g., X -> Y, 1, XtoY
indicates that X in time T affects Y in time T+1), where
only one-headed arrows can be lagged. Using positive values to indicate lags
then matches the notational convention used in package dynlm.
- 3. Parameter name:
The name of the regression coefficient, variance,
or covariance specified by the arrow. Assigning the same name to two or
more arrows results in an equality constraint. Specifying the parameter name
as NA produces a fixed parameter.
- 4. Value:
start value for a free parameter or value of a fixed parameter.
If given as NA (or simply omitted), the model is provide a default
starting value.
Lines may end in a comment following #. The function extends code copied from package
`sem` under licence GPL (>= 2) with permission from John Fox.
Simultaneous autoregressive process for simultaneous and lagged effects
This text then specifies linkages in a multivariate time-series model for variables \(\mathbf X\)
with dimensions \(T \times C\) for \(T\) times and \(C\) variables.
make_dsem_ram then parses this text to build a path matrix \(\mathbf{P}\) with
dimensions \(TC \times TC\), where element \(\rho_{k_2,k_1}\)
represents the impact of \(x_{t_1,c_1}\) on \(x_{t_2,c_2}\), where \(k_1=T c_1+t_1\)
and \(k_2=T c_2+t_2\). This path matrix defines a simultaneous equation
$$ \mathrm{vec}(\mathbf X) = \mathbf P \mathrm{vec}(\mathbf X) + \mathrm{vec}(\mathbf \Delta)$$
where \(\mathbf \Delta\) is a matrix of exogenous errors with covariance \(\mathbf{V = \Gamma \Gamma}^t\),
where \(\mathbf \Gamma\) is the Cholesky of exogenous covariance. This
simultaneous autoregressive (SAR) process then results in \(\mathbf X\) having covariance:
$$ \mathrm{Cov}(\mathbf X) = \mathbf{(I - P)}^{-1} \mathbf{\Gamma \Gamma}^t \mathbf{((I - P)}^{-1})^t $$
Usefully, computing the inverse-covariance (precision) matrix \(\mathbf{Q = V}^{-1}\) does not require inverting \(\mathbf{(I - P)}\):
$$ \mathbf{Q} = (\mathbf{\Gamma}^{-1} \mathbf{(I - P)})^t \mathbf{\Gamma}^{-1} \mathbf{(I - P)} $$
Example: univariate first-order autoregressive model
This simultaneous autoregressive (SAR) process across variables and times
allows the user to specify both simutanous effects (effects among variables within
year \(T\)) and lagged effects (effects among variables among years \(T\)).
As one example, consider a univariate and first-order autoregressive process where \(T=4\).
with independent errors. This is specified by passing sem = "X -> X, 1, rho \n X <-> X, 0, sigma" to make_dsem_ram.
This is then parsed to a RAM:
| heads | to | from | paarameter | start |
| 1 | 2 | 1 | 1 | <NA> |
| 1 | 3 | 2 | 1 | <NA> |
| 1 | 4 | 3 | 1 | <NA> |
| 2 | 1 | 1 | 2 | <NA> |
| 2 | 2 | 2 | 2 | <NA> |
| 2 | 3 | 3 | 2 | <NA> |
| 2 | 4 | 4 | 2 | <NA> |
Rows of this RAM where heads=1 are then interpreted to construct the path matrix \(\mathbf P\), where column "from"
in the RAM indicates column number in the matrix, column "to" in the RAM indicates row number in the matrix:
$$ \mathbf P = \begin{bmatrix}
0 & 0 & 0 & 0 \\
\rho & 0 & 0 & 0 \\
0 & \rho & 0 & 0 \\
0 & 0 & \rho & 0\\
\end{bmatrix} $$
While rows where heads=2 are interpreted to construct the Cholesky of exogenous covariance \(\mathbf \Gamma\)
and column "parameter" in the RAM associates each nonzero element of those
two matrices with an element of a vector of estimated parameters:
$$ \mathbf \Gamma = \begin{bmatrix}
\sigma & 0 & 0 & 0 \\
0 & \sigma & 0 & 0 \\
0 & 0 & \sigma & 0 \\
0 & 0 & 0 & \sigma\\
\end{bmatrix} $$
with two estimated parameters \(\mathbf \beta = (\rho, \sigma) \). This then results in covariance:
$$ \mathrm{Cov}(\mathbf X) = \sigma^2 \begin{bmatrix}
1 & \rho^1 & \rho^2 & \rho^3 \\
\rho^1 & 1 + \rho^2 & \rho^1 (1 + \rho^2) & \rho^2 (1 + \rho^2) \\
\rho^2 & \rho^1 (1 + \rho^2) & 1 + \rho^2 + \rho^4 & \rho^1 (1 + \rho^2 + \rho^4) \\
\rho^3 & \rho^2 (1 + \rho^2) & \rho^1 (1 + \rho^2 + \rho^4) & 1 + \rho^2 + \rho^4 + \rho^6 \\
\end{bmatrix} $$
Which converges on the stationary covariance for an AR1 process for times \(t>>1\):
$$ \mathrm{Cov}(\mathbf X) = \frac{\sigma^2}{1+\rho^2} \begin{bmatrix}
1 & \rho^1 & \rho^2 & \rho^3 \\
\rho^1 & 1 & \rho^1 & \rho^2 \\
\rho^2 & \rho^1 & 1 & \rho^1 \\
\rho^3 & \rho^2 & \rho^1 & 1\\
\end{bmatrix} $$
except having a lower pointwise variance for the initial times, which arises as a "boundary effect".
Similarly, the arrow-and-lag notation can be used to specify a SAR representing
a conventional structural equation model (SEM), cross-lagged (a.k.a. vector autoregressive)
models (VAR), dynamic factor analysis (DFA), or many other time-series models.