dsem {dsem}

Fit dynamic structural equation model

Description

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.

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).

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

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) )
summary( fit )
plot( fit )
plot( fit, edge_label="value" )