EViews Help: Background

We begin with a slightly modified version of Equation (33.3) defining the discrete TR model:

where

is a (0, 1) indicator for regime that depends on the observed variable

, where

represents one or more thresholds and

is a threshold slope parameter. Note that we have divided the regressors into two groups—

variables whose coefficients vary across the

regimes, and

variables with coefficients that are regime invariant.

Restricting ourselves to

, as using the fact that

for exactly one

, we may rewrite the discrete TR equation as:

To construct the two-regime STR model, we replace the indicator function with a continuous transition function

that returns values between 0 to 1. Then we have

where

has different properties as

, and

, depending on the specific functional form.

The key modeling choices in Equation (36.3) are the choice of the threshold variable

and the selection of a transition function

. For a given

and

, we may estimate the regression parameters

and the threshold values and slope

via nonlinear least squares. Additionally, given a list of candidate variables for

, we can select a threshold variable using model selection techniques.

• The logistic and normal transition functions are monotonically increasing in

so that the two regimes correspond to high and low values of the threshold variable. The threshold value

determines the point at which the regimes are equally weighted, while

controls the speed and smoothness of the transition. As

, the transition function approaches the indicator function and the model approaches the discrete threshold model.

• The exponential transition function is increasing in absolute deviations of

from the threshold

. Furthermore,

when

, and

approaches 1 as

and

. The ESTR model does not nest the discrete TR model since, as

, the specification becomes linear since

approaches a constant function returning 0 or 1.

• The second-order logistic function does nest the discrete TR model. As

with distinct

approaches 1 for

and

, and

approaches 0 for

in-between. Thus, the L2STR model nests a three-regime discrete TR model where the outer regimes have a common linear specification.

For finite

attains its minimum value at

with a non-zero value.

For further discussion of the implications of various choices for

, see van Dijk, Teräsvirta, and Franses (2002) who offer extensive commentary on the properties of these transition functions and provide concrete examples of their use in empirical settings.

Following estimation of the coefficients using nonlinear least squares (van Dijk, TerasvirtTeräsvirta, and Franses, 2002; Leybourne, Newbold and Vougas, 1998), we may estimate the coefficient covariance matrix using conventional methods available for nonlinear least squares (ordinary, heteroskedasticity consistent, heteroskedasticity and autocorrelation consistent).

where

is an estimate of the information,

is the variance of the residual weighted gradients, and

is a scale parameter.

For the ordinary covariance estimator, we assume that

. Then we have

where

is an estimator of the residual variance (with or without degree-of-freedom correction).

We may estimate

using the outer-product of the gradients (OPG) of the error function or the one-half of the Hessian matrix of second derivatives of the sum-of-squares function:

Alternately, we may assume distinct

and

, and employ a White or HAC sandwich estimator for the coefficient covariance as in “Robust Standard Errors”.

Lastly, following specification and estimation of an STR equation, the modeling cycle typically involves performing a number of hypothesis tests using the STR specification and results. Background information for the various tests performed by EViews will be discussed below.