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# open the dataset
open nile.gdt
# set up the local linear trend model with the Nile data
Nile = ksetup({nile}, {1}, {1}, {1})
Nile.diffuse = 1
Nile.obsvar = {1}
# initialize the parameters via the reduced form
# \Delta y_t = \eta_t + \Delta \epsilon_t
series dnile = diff(nile)
scalar vd = var(dnile)
scalar veps = -cov(dnile,dnile(-1))
scalar s_eps = sqrt(veps)
scalar s_eta = sqrt(vd - 2*veps)
# estimate the variances via MLE
scalar err = 0
mle loglik = err ? NA : Nile.llt
matrix Nile.statevar = s_eta^2
matrix Nile.obsvar = s_eps^2
err = kfilter(&Nile)
params s_eta s_eps
end mle
# extract the smoothed trend
ksmooth(&Nile)
series nile_trend = Nile.state
# plot series with trend and associated 95% confidence band
series nile_trend_se = sqrt(Nile.statevar)
series hi = nile_trend + 1.96 * nile_trend_se
series lo = nile_trend - 1.96 * nile_trend_se
list X = nile nile_trend hi lo
plot X
options with-lines time-series
literal set linetype 3 lc rgb "#00c0c0"
literal set linetype 4 lc rgb "#00c0c0"
literal set key right top
printf "set title \"%s\"", "Nile data and trend with 95\% confidence band"
end plot --output=display
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