The threshold models considered in Chapters 4 and 5 have both had sharp cutoffs between the branches. In many cases, this is unrealistic, and the lack of continuity in the objective function causes other problems—you can’t use any asymptotic distribution theory for the estimates, and without changes, they aren’t appropriate for forecasting since it’s not clear how to handle simulated values that fall between the observed data values near the cutoff.
An alternative is the STAR model (Smooth Transition AutoRegression) and more generally, the STR (Smooth Transition Regression). Instead of the sharp cutoff, this uses a smooth function of a threshold variable. One way to write this is:
yt = Xtβ(1)+ Xtβ(2)G(Zt−d, γ, c) + ut (6.1) The transition function G is bounded between 0 and 1, and depends upon a location parameter c and a scale parameter γ.1 The two standard transition functions are the logistic (LSTAR) and the exponential (ESTAR). The formulas for these are:
G(Zt−d, γ, c) = 1 − [1 + exp (γ(Zt−d− c))]−1 for LSTAR 1 − exp −γ(Zt−d− c)2
for ESTAR (6.2)
LSTAR is more similar to the models examined earlier, with values to the left of c generally being in one branch (with coefficient vector β(1)) and those to the right of c having coefficient vector more like β(1)+ β(2). ESTAR treats the tails symmetrically, with values near c having coefficients near β(1), while those far-ther away (in eifar-ther direction) being close to β(1) + β(2). ESTAR is often used when there are seen to be costs of adjustment in either direction. An unre-stricted three branch model (such as in Balke-Fomby in 5.1) could be done by adding in a secondLSTAR branch.
STAR models, at least theoretically, can be estimated using non-linear least squares. This, however, requires a bit of finesse: under the default initial val-ues of zero for all parameters used for NLLS, both the parameters in the tran-sition function and the autoregressive coefficients that they control have zero derivatives. As a result, if you do NLLS with the default METHOD=GAUSS, it can never move the estimates away from zero. A better way to handle this is to
1There are other, equivalent, ways of writing this. The form we use here lends itself more easily to testing forSTAReffects, since it’s just least squares if β(2)is zero.
58
STAR Models 59 split the parameter set into the transition parameters and the autoregressive parameters, and first estimate the autoregressions conditional on a pegged set of values for the transition parameters. With c and γ pegged, (6.1) is linear, and so converges in one iteration. The likelihood function generally isn’t par-ticularly sensitive to the choice of γ, but it can be sensitive to the choice for c, so you might want to experiment with several guesses for c before deciding whether you’re done with the estimation.
Although the likelihood is relatively insensitive to the value of γ, that’s only when it’s in the proper range. As you can see from (6.2), γ depends upon the scale of the transition variable Zt−d. A guess value of something like 1 or 2 times σ−1z for an LSTAR and σz−2 for an ESTAR is generally adequate. In Terasvirta (1994), theLSTARexponent is directly rescaled by (an approximate) reciprocal standard deviation. If you do that, the γ values will have some simi-larities from one application to the next.
For theLSTAR model, do not use the formula as written in (6.2)—for large pos-itive values of Zt−d, the exp function will overflow, causing the entire function to compute as a missing value. exp of a large negative value will underflow (which you could get in the negative tail for anLSTARand either tail for anES
-TAR), but underflowed values are treated as zero, which gives the proper limit behavior. To avoid the problem with the overflows, use the %LOGISTIC func-tion, which does the same calculafunc-tion, but computes in a different form which avoids overflow.2
LSTAR models include the sharp cutoff models as a special case where γ → ∞.
However, where a sharp cutoff is appropriate, you may see very bad behavior for the non-linear estimates fromLSTAR. For instance, theLSTAR doesn’t work well for the unemployment rate series studied in 4.1. The change in the unem-ployment rate takes only a small number of values, with almost all data points being -.2, -.1, 0, .1 or .2. The estimates for γ and c show as:
Variable Coeff Std Error T-Stat Signif
11. GAMMA 220.943152 286157411.494605 7.72104e-007 0.99999939
12. C 0.018804 24356.890066 7.72021e-007 0.99999939
The standard errors look (and are) nonsensical, but this is a result of the like-lihood (or sum of squares) function being flat for a range of values. It’s always a good idea to graph the transition function against the threshold, particularly when the results look odd. In this case, it gives us Figure 6.1. With the ex-ception of a very tiny non-zero weight at zero, this is the same as the sharp transition. Almost any value of c between 0 and 1 will give almost the identical transition function, and so will almost any large value of γ.
21/(1 + exp(x)) = exp(−x)/(exp(−x) + 1), one form of which will have the safe negative exponents.
STAR Models 60
Change in UR
Transition Function
-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00
Figure 6.1: Transition Function in LSTAR for Unemployment Rate
6.1 Testing for STAR
A straightforward test for the absence of a STAR effect in (6.1) won’t have the standard asymptotic distribution because under the null that β(2) = 0, the tran-sition parameters γ and c aren’t identified. Instead, Terasvirta (and colleagues) in a series of papers proposed a battery of LM tests based upon a Taylor expan-sion of G. Under the null that there is no STAR effect, there should be zero coefficients on a set of interaction terms between the regressors and powers of the transition variable Zt−d. These are computed by the procedure @STARTest.
The output for this for the unemployment rate series (using the first lag as the threshold variable) is in Table 6.1.
AR length 4
Delay 1
Test F-stat Signif
Linearity 4.0449411 0.0000 H01 4.0763236 0.0029 H02 3.6460117 0.0060 H03 4.1738770 0.0024 H12 3.9470675 0.0001
Table 6.1: Test for STAR in series DUR
The Linearity test includes all the interaction terms through the 3rd power of the transition variable, and serves as a general test for a STAR effect. H01, H02 and H03 are tests on the single power individually, while H12 is a joint test with the first and second powers only. For anLSTAR model, all of these should be signficant. For an ESTAR, because of symmetry, the 3rd power shouldn’t enter, so you should see H12 significant and H03 insignificant. For the ESTAR,
STAR Models 61 you would also likely reject on the Linearity line, but it won’t have as much power as H12, since it includes the 3rd power terms as well. A common recom-mendation is to choose the delay d on the threshold based upon the test which gives the strongest rejection.
STAR Models 62