Palgrave Handbook of Econometrics: Applied Econometrics

1030 The Econometrics of Exchange Rates

A researcher might not know which STAR model the data follows and a sensible
first step would be to have a general linearity test that will include both alternative
models. Teräsvirta (1994) proposes a modeling cycle consisting of the following
stages:

1. Specification of a linear model.


2. Testing for linearity,HL,0:γ=0. Combining (22.5) and (22.6) and recombining
   in terms of identified parameters, the regression equation becomes:

yt=β

′

1 ̃yt+β

′

2 ̄ytyt−d+β

′

3 y ̄ty

2

t−d+β

′

4 y ̄ty

3

t−d+ut, (22.7)

wherey ̄t=(yt− 1 ,...,yt−p). The linearity test has null hypothesisHL,0:β

′

2 =

β

′

3 =β

′

4 =0 and the original hypothesis can be tested by applying the Lagrange

multiplier (LM) principle. The appropriate transition variable lag in the STR

model can be determined without specifying the form of the transition function.

We can compute theF-statistic forHL,0for various values ofd(and differentzt

variables) and select the one for which thep-value of the test is smallest.

3. Selecting the transition function. The choice between ESTAR and LSTAR models
   can be based on the following sequence of null hypotheses:

H 03 :β 4 =0,

H 02 :β 3 = 0 |β 4 =0,

H 01 :β 2 = 0 |β 3 =β 4 =0.

If thep-value for the F-test ofH 02 is smaller than that forH 01 andH 03 then we

select the ESTAR family, otherwise we choose the LSTAR family.

While Teräsvirta (1994) uses a third-order Taylor expansion of the logistic func-
tion and a first-order expansion for the exponential function, Escribano and Jordä
(1999) (EJ) augment the regression equation with a second order expansion of the
exponential function:^10

yt=β

′

1 ̃yt+β

′

2 y ̄tyt−d+β

′

3 y ̄ty

2

t−d+β

′

4 ̄yty

3

t−d+β

′

5 y ̄ty

4

t−d+vt. (22.8)

EJ claim that this procedure improves the power of both the linearity test and the

selection procedure test. The null hypothesis of linearity corresponds toH 0 :β

′
2 =
β

′

3 =β

′

4 =β

′

5 =0. Under this null the LM test is asymptoticallyχ

(^2). The selection
procedure between ESTAR and LSTAR is as follows:

1. Test the nullH 0 :β

′

3 =β

′

5 =0 with an F-test(FE).

2. Test the nullH 0 :β

′

2 =β

′

4 =0 with an F-test(FL).

3. If thep-value ofFEis lower thanFLthen select an ESTAR. Choose an LSTAR
   otherwise.

If the errors display heteroskedasticity, Granger and Teräsvirta (1993) suggest
ways of making the testing procedure more robust.^11 However, Lundbergh and