Basic strctural models

Defining a BSM model with rjd3sts

Thie package allows several equivalent definitions of a basic strctural model. We present below some of them.

To compare the results (more precisely the likelihood) of the different approaches, it is important to compute the marginal likelihood.

Standard definition, noise in the state

library(rjd3sts)
library(rjd3toolkit)

s<-log(retail$BookStores)

# create the model
bsm<-model()
# create the components and add them to the model
add(bsm, locallineartrend("ll"))
add(bsm, seasonal("s", 12, type="HarrisonStevens"))
add(bsm, noise("n"))
rslt<-estimate(bsm, log(s), marginal=T)

print(result(rslt, "likelihood.ll"))
print(result(rslt, "parameters"))

Standard definition, noise in the measurement

# create the model
bsm<-model()
# create the components and add them to the model
add(bsm, locallineartrend("ll"))
add(bsm, seasonal("s", 12, type="HarrisonStevens"))
  # create the equation (fix the variance to 1)
eq<-equation("eq", 1,T)
add_equation(eq, "ll")
add_equation(eq, "s")
add(bsm, eq)
rslt<-estimate(bsm, log(s), marginal=T)

print(result(rslt, "likelihood.ll"))
print(result(rslt, "parameters"))

components with fixed variances, aggregated with diffuse weights (noise in the state)

# create the model
bsm<-model()
  # create the components, with fixed variances, and add them to the model
add(bsm, locallineartrend("ll", 
                             levelVariance = 1, fixedLevelVariance = TRUE) )
add(bsm, seasonal("s", 12, type="HarrisonStevens", 
                     variance = 1, fixed = TRUE))
add(bsm, noise("n", 1, fixed=TRUE))
  # create the equation (fix the variance to 1)
eq<-equation("eq", 0, TRUE)
add_equation(eq, "ll", .01, FALSE)
add_equation(eq, "s", .01, FALSE)
add_equation(eq, "n")
add(bsm, eq)
rslt<-estimate(bsm, log(s), marginal=TRUE)

print(result(rslt, "likelihood.ll"))
p<-result(rslt, "parameters")
print(p)

cat ('to be noted:')

cat ('p[5]*p[5]=l-var=', p[5]*p[5])
cat ('p[5]*p[5]*p[2]=slope-var=', p[5]*p[5]*p[2])
cat ('p[6]*p[6]=s-var=', p[6]*p[6])

bsm with long term trend and cycle

# create the model
bsm<-model()
  # create the components and add them to the model
add(bsm, locallevel("l", initial = 0) )
add(bsm, locallineartrend("lt", levelVariance = 0, 
                             fixedLevelVariance = TRUE) )
add(bsm, seasonal("s", 12, type="HarrisonStevens"))
add(bsm, noise("n", 1, fixed=TRUE))
  # create the equation (fix the variance to 1)
rslt<-estimate(bsm, log(s), marginal=TRUE)

print(result(rslt, "likelihood.ll"))
print(result(rslt, "parameters"))

ss<-smoothed_states(rslt)
plot(ss[,1]+ss[,2], type='l', col='blue', ylab='trends')
lines(ss[, 2], col='red')