Fit dynamic structural equation model

Fits a dynamic structural equation model

Usage

dsem(
  sem,
  tsdata,
  family = rep("fixed", ncol(tsdata)),
  estimate_delta0 = FALSE,
  control = dsem_control(),
  covs = colnames(tsdata)
)

Arguments

sem: Specification for time-series structural equation model structure including lagged or simultaneous effects. See Details section in make_dsem_ram for more description
tsdata: time-series data, as outputted using ts
family: Character-vector listing the distribution used for each column of tsdata, where each element must be fixed or normal. family="fixed" is default behavior and assumes that a given variable is measured exactly. Other options correspond to different specifications of measurement error.
estimate_delta0: Boolean indicating whether to estimate deviations from equilibrium in initial year as fixed effects, or alternatively to assume that dynamics start at some stochastic draw away from the stationary distribution
control: Output from dsem_control, used to define user settings, and see documentation for that function for details.
covs: optional: a character vector of one or more elements, with each element giving a string of variable names, separated by commas. Variances and covariances among all variables in each such string are added to the model. Warning: covs="x1, x2" and covs=c("x1", "x2") are not equivalent: covs="x1, x2" specifies the variance of x1, the variance of x2, and their covariance, while covs=c("x1", "x2") specifies the variance of x1 and the variance of x2 but not their covariance. These same covariances can be added manually via argument `sem`, but using argument `covs` might save time for models with many variables.

Value

An object (list) of class `dsem`. Elements include:

obj: TMB object from MakeADFun
ram: RAM parsed by make_dsem_ram
model: SEM structure parsed by make_dsem_ram as intermediate description of model linkages
tmb_inputs: The list of inputs passed to MakeADFun
opt: The output from nlminb
sdrep: The output from sdreport
interal: Objects useful for package function, i.e., all arguments passed during the call

Details

A DSEM involves (at a minimum):

Time series: a matrix $\mathbf X$ where column $\mathbf x_c$ for variable c is a time-series;
Path diagram: a user-supplied specification for the path coefficients, which define the precision (inverse covariance) $\mathbf Q$ for a matrix of state-variables and see make_dsem_ram for more details on the math involved.

The model also estimates the time-series mean $ \mathbf{\mu}_c $ for each variable. The mean and precision matrix therefore define a Gaussian Markov random field for $\mathbf X$:

$$ \mathrm{vec}(\mathbf X) \sim \mathrm{MVN}( \mathrm{vec}(\mathbf{I_T} \otimes \mathbf{\mu}), \mathbf{Q}^{-1}) $$

Users can the specify a distribution for measurement errors (or assume that variables are measured without error) using argument family. This defines the link-function $g_c(.)$ and distribution $f_c(.)$ for each time-series $c$:

$$ y_{t,c} \sim f_c( g_c^{-1}( x_{t,c} ), \theta_c )$$

dsem then estimates all specified coefficients, time-series means $\mu_c$, and distribution measurement errors $\theta_c$ via maximizing a log-marginal likelihood, while also estimating state-variables $x_{t,c}$. summary.dsem then assembles estimates and standard errors in an easy-to-read format. Standard errors for fixed effects (path coefficients, exogenoux variance parameters, and measurement error parameters) are estimated from the matrix of second derivatives of the log-marginal likelihod, and standard errors for random effects (i.e., missing or state-space variables) are estimated from a generalization of this method (see sdreport for details).

References

**Introducing the package, its features, and comparison with other software (to cite when using dsem):**

Thorson, J. T., Andrews, A., Essington, T., Large, S. (In review). Dynamic structural equation models synthesize ecosystem dynamics constrained by ecological mechanisms.

Examples

# Define model
sem = "
  # Link, lag, param_name
  cprofits -> consumption, 0, a1
  cprofits -> consumption, 1, a2
  pwage -> consumption, 0, a3
  gwage -> consumption, 0, a3
  cprofits -> invest, 0, b1
  cprofits -> invest, 1, b2
  capital -> invest, 0, b3
  gnp -> pwage, 0, c2
  gnp -> pwage, 1, c3
  time -> pwage, 0, c1
"

# Load data
data(KleinI, package="AER")
TS = ts(data.frame(KleinI, "time"=time(KleinI) - 1931))
tsdata = TS[,c("time","gnp","pwage","cprofits",'consumption',
               "gwage","invest","capital")]

# Fit model
fit = dsem( sem=sem,
            tsdata = tsdata,
            estimate_delta0 = TRUE,
            control = dsem_control(quiet=TRUE) )
#> 1 regions found.
#> Using 1 threads
#> 1 regions found.
#> Using 1 threads
summary( fit )
#>                           path lag           name start parameter       first
#> 1      cprofits -> consumption   0             a1  <NA>         1    cprofits
#> 2      cprofits -> consumption   1             a2  <NA>         2    cprofits
#> 3         pwage -> consumption   0             a3  <NA>         3       pwage
#> 4         gwage -> consumption   0             a3  <NA>         3       gwage
#> 5           cprofits -> invest   0             b1  <NA>         4    cprofits
#> 6           cprofits -> invest   1             b2  <NA>         5    cprofits
#> 7            capital -> invest   0             b3  <NA>         6     capital
#> 8                 gnp -> pwage   0             c2  <NA>         7         gnp
#> 9                 gnp -> pwage   1             c3  <NA>         8         gnp
#> 10               time -> pwage   0             c1  <NA>         9        time
#> 11               time <-> time   0        V[time]  <NA>        10        time
#> 12                 gnp <-> gnp   0         V[gnp]  <NA>        11         gnp
#> 13             pwage <-> pwage   0       V[pwage]  <NA>        12       pwage
#> 14       cprofits <-> cprofits   0    V[cprofits]  <NA>        13    cprofits
#> 15 consumption <-> consumption   0 V[consumption]  <NA>        14 consumption
#> 16             gwage <-> gwage   0       V[gwage]  <NA>        15       gwage
#> 17           invest <-> invest   0      V[invest]  <NA>        16      invest
#> 18         capital <-> capital   0     V[capital]  <NA>        17     capital
#>         second direction    Estimate  Std_Error   z_value       p_value
#> 1  consumption         1  0.19323185 0.08199229  2.356708  1.843776e-02
#> 2  consumption         1  0.08942112 0.08136334  1.099035  2.717530e-01
#> 3  consumption         1  0.79625663 0.03594118 22.154439 9.452934e-109
#> 4  consumption         1  0.79625663 0.03594118 22.154439 9.452934e-109
#> 5       invest         1  0.48138141 0.08740019  5.507785  3.633777e-08
#> 6       invest         1  0.33084616 0.09069261  3.647995  2.642950e-04
#> 7       invest         1 -0.11150752 0.02408258 -4.630214  3.652875e-06
#> 8        pwage         1  0.44041577 0.02921824 15.073314  2.426312e-51
#> 9        pwage         1  0.14476511 0.03370693  4.294817  1.748376e-05
#> 10       pwage         1  0.13029837 0.02882711  4.519995  6.184119e-06
#> 11        time         2  6.05530071 0.93435313  6.480741  9.127318e-11
#> 12         gnp         2 10.32020497 1.58102475  6.527542  6.685791e-11
#> 13       pwage         2  0.69302930 0.10701058  6.476269  9.401826e-11
#> 14    cprofits         2  4.10929046 0.63141034  6.508114  7.610017e-11
#> 15 consumption         2  0.92285232 0.14239896  6.480752  9.126683e-11
#> 16       gwage         2  1.90952975 0.29464666  6.480745  9.127100e-11
#> 17      invest         2  0.91015034 0.14046563  6.479523  9.201284e-11
#> 18     capital         2  9.67950590 1.49358015  6.480741  9.127332e-11
plot( fit )

plot( fit, edge_label="value" )