Logistic Smooth Transition AutoRegressive model.
1 2 3 
x 
time series 
m, d, steps 
embedding dimension, time delay, forecasting steps 
series 
time series name (optional) 
mL 
autoregressive order for 'low' regime (default: m). Must be <=m 
mH 
autoregressive order for 'high' regime (default: m). Must be <=m 
thDelay 
'time delay' for the threshold variable (as multiple of embedding time delay d) 
mTh 
coefficients for the lagged time series, to obtain the threshold variable 
thVar 
external threshold variable 
th, gamma 
starting values for coefficients in the LSTAR model. If missing, a grid search is performed 
trace 
should additional infos be printed? (logical) 
include 
Type of deterministic regressors to include 
control 
further arguments to be passed as 
starting.control 
further arguments for the grid search (dimension, bounds). See details below. 
x[t+steps] = ( phi1[0] + phi1[1] x[t] + phi1[2] x[td] + … + phi1[mL] x[t  (mL1)d] ) G( z[t], th, gamma ) + ( phi2[0] + phi2[1] x[t] + phi2[2] x[td] + … + phi2[mH] x[t  (mH1)d] ) (1  G( z[t], th, gamma ) ) + eps[t+steps]
with z the threshold variable, and G the logistic function,
computed as plogis(q, location = th, scale = 1/gamma)
, so see
plogis
documentation for details on the logistic function
formulation and parameters meanings.
The threshold variable can alternatively be specified by:
z[t] = x[t] mTh[1] + x[td] mTh[2] + … + x[t(m1)d] mTh[m]
z[t] = x[t  thDelay*d ]
z[t] = thVar[t]
Note that if starting values for phi1 and phi2 are provided, isn't
necessary to specify mL
and mH
. Further, the user has to specify only
one parameter between mTh
, thDelay
and thVar
for indicating the
threshold variable.
Estimation of the transition parameters th and gamma, as well as the regression parameters phi1 and phi2, is done using concentrated least squares, as suggested in Leybourne et al. (1996).
Given th and gamma, the model is linear, so regression coefficients can be obtained as usual by OLS. So the nonlinear numerical search needs only to be done for th and gamma; the regression parameters are then recovered by OLS again from the optimal th and gamma.
For the nonlinear estimation of the
parameters th and gamma, the program uses the
optim
function, with optimization method BFGS using the analytical gradient.
For the estimation of standard values, optim
is rerun
using the complete Least Squares objective function, and the standard errors are obtained by inverting the hessian.
You can pass further arguments to optim directly with the control
list argument. For instance, the option maxit
maybe useful when
there are convergence issues (see examples).
Starting parameters are obtained doing a simple twodimensional gridsearch over th and gamma.
Parameters of the grid (interval for the values, dimension of the grid) can be passed to starting.control
.
nTh
The number of threshold values (th) in the grid. Defaults to 200
nGamma
The number of smoothing values (gamma) in the grid. Defaults to 40
trim
The minimal percentage of observations in each regime. Defaults to 10% (possible threshold values are between the 0.1 and 0.9 quantile)
gammaInt
The lower and higher smoothing values of the grid. Defaults to c(1,40)
thInt
The lower and higher threshold values of the grid. When not specified (default, i.e NA), the interval are the trim
quantiles above.
An object of class nlar
, subclass lstar
, i.e. a list
with fitted model informations.
Antonio, Fabio Di Narzo
Nonlinear time series models in empirical finance, Philip Hans Franses and Dick van Dijk, Cambridge: Cambridge University Press (2000).
NonLinear Time Series: A Dynamical Systems Approach, Tong, H., Oxford: Oxford University Press (1990).
Leybourne, S., Newbold, P., Vougas, D. (1998) Unit roots and smooth transitions, Journal of Time Series Analysis, 19: 8397
plot.lstar
for details on plots produced for this model
from the plot
generic.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  #fit a LSTAR model. Note 'maxit': slow convergence
mod.lstar < lstar(log10(lynx), m=2, mTh=c(0,1), control=list(maxit=3000))
mod.lstar
#fit a LSTAR model without a constant in both regimes.
mod.lstar2 < lstar(log10(lynx), m=1, include="none")
mod.lstar2
#Note in example below that the initial grid search seems to be to narrow.
# Extend it, and evaluate more values (slow!):
controls < list(gammaInt=c(1,2000), nGamma=50)
mod.lstar3 < lstar(log10(lynx), m=1, include="none", starting.control=controls)
mod.lstar3
# a few methods for lstar:
summary(mod.lstar)
residuals(mod.lstar)
AIC(mod.lstar)
BIC(mod.lstar)
plot(mod.lstar)
predict(mod.lstar, n.ahead=5)

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