Lecture 15 Threshold Autoregressions: III

Lecture 15 – Threshold Autoregressions: III

 Inference on the threshold parameters, γ  Testing for TAR effects and # of regimes As noted in the previous lecture, standard inference procedures regarding the intercept and slope parameters in the TAR model are asymptotically valid, so long as they are estimated using a consistent estimator of the threshold parameter vector γ. [Estimate γ as above; for that γ, estimate β by OLS; apply OLS inference methods to construct CIs for and test hypotheses concerning the components of β.]

Suppose that we believe that yt follows a SETAR(p,2,d) process, where p and d can be known or unknown. Following the procedures discussed earlier, we estimate β, γ (and, if they are unknown, p and d).

 How do we test H0: γ = γ0? (e.g., γ0 = 0)  How do we construct interval estimates for γ?  How do we test the null of a one-regime TAR (i.e., a linear AR) against the alternative of a two-regime TAR?  How do we test the null of a two-regime TAR against the alternative of a three- regime TAR? Testing H0: γ = γ0

Hansen (1997) showed that under the null hypothesis, the sequence of LR-like statistics SSR ( )  SSR(ˆ) LR ( )  T ( T 0 ) T 0 SSR(ˆ) converges in distribution to a r.v., ξ. The distribution of ξ can be derived by simulation methods (HOW?). Simulated percentiles of that distribution are provided in Hansen’s paper (Table 1, row 2).

So, for example, consider testing H0: γ = 0 for the two-regime TAR model of real GDP growth rates. First, we estimate the model treating γ as unknown to get USSR. Second, we estimate the model treating γ as known and equal to 0 to get RSSR. Third, compute LR and compare its value to the percentiles in Hansen’s Table 1. E.g., the 95th percentile of the distribution of ξ is 7.35. So reject H0: γ=0 at the 5-percent level if LR > 7.35. Constructing CIs for γ

Asymptotically valid confidence intervals for γ can be constructed by “inverting the LR statistic.”

That is, to construct, e.g., a 95-percent CI for γ, find the set

Г0.95 = {γ: LRT(γ) < ξ0.95}

= {γ: LRT(γ) < 7.35}

where ξ0.95 means the 95-th percentile of the distribution of ξ.

1. Go through the set of candidate γ’s: y[τT], y[τT]+1,…,y[(1-τ)T]. For each of these candidate γ’s compute LR(γ) to determine if that γ is or is not in Г0.95 (j) (j) (j+1) 2. If, say, y ε Г0.95, then place γε[y ,y ) in Г0.95. One unappealing aspect of this construction is that the resulting CIs can be the union of a collection of disconnect intervals. E.g.,

Г0.95 = [0.20,0.60] U [0.80, 0.90]

Hansen suggests that define use the more conservative procedure of using the “convexified” interval

c 0.95  [ˆ1,ˆ2 ] where

ˆ1  min 0.95 and ˆ2  max 0.95

So, if Г0.95 = [0.20,0.60] U [0.80, 0.90], then c 0.95  [0.20,0.90] .

(Why is this called a “conservative procedure?)

Notes 1. In a paper that Walt Ender, Pierre Siklos, and I are revising for publication in Studies in Nonlinear Dynamics and Econometrics we use Monte Carlo methods to evaluate the finite sample coverage properties of several methods of constructing CIs for the threshold parameter.

- Use normal approximation {(γ-hat + 2*S.E.) for 95% CI} - Use Hansen’s approach - Use bootstrap methods (t and percentile methods)

Our conclusion – none of these procedures have good coverage properties in samples of size 100. As part of our revision, we are considering sample size 250. 2. The asymptotic distribution of β-hat is the same (and normal) whether γ is known or replaced by a consistent estimator. Hansen points out (and my work with Walt and Pierre confirms) that when γ is unknown, this approach does not tend to work well in finite samples (even samples as large as 500-1000!). In particular, the CIs generated using the normal approximation tend to be much too small (too “liberal”). E.g., actual coverage probabilities for nominal 95% intervals are often on the order of 50%-80%!

Hansen suggests an approach that takes the union of CIs for β’s using a set of γ’s around γ- hat rather than simply the CI implied by γ = γ- hat. This provides more conservative intervals, but the choice of an appropriate set of γ’s that does not lead to overly conservative intervals is problematic.

The message – Much work remains to be done on developing reasonably reliable inference methods for the parameters in TAR models. Testing H0: r = 1 (vs. HA: r =2)

The idea: Fit the model with r = 1 and r = 2 and see how much the SSR increases under the restriction that r = 1, i.e., construct the test statistic

F12 = T*(SSR1-SSR2)/SSR2

2 If γ is known, F12 is asymptotically χ (k), where k is the number of parameters that are allowed to vary across regimes (usually, p or p+1).

The problem:

When γ is unknown, the nuisance parameter, γ, is not identified under the null. (Where have we seen this problem before?) So, standard distribution theory will not apply to this (or other natural) statistic(s). [Applying the χ2(k) distribution in this case will lead to rejecting the null too often. I.e., the usual chi-square test’s actual size will be much larger than its nominal size.] Hansen’s solution (1996, Econometrica; see also Hansen 1999, Journal of Economic Surveys for a simpler exposition):

Hansen showed that under the null F12 does converge in distribution to a random variable, say G, under appropriate side condition on {yt}. However, the distribution G depends on the moments of the data, so that asymptotic critical values and p-values would have to be tabulated for each and every application. A “bootstrap-like” algorithm to compute these values is provided in Hansen (1996, 1997).

An alternative, that requires about the same amount of effort and may work better is to simply use a bootstrap procedure.

“Although there is no Monte Carlo study comparing the bootstrap and asymptotic approximations in the context of testing for SETAR models, Diebold and Chen (1996) present an analogous study of the Andrews structural change test in AR(1) model. They find that the bootstrap yields and excellent approximation, a certain improvement over the asymptotic distribution.”

 How would the bootstrap distribution of F12 be constructed in this setting?

 Testing H0: r = 2 (vs. HA: r = 3) is a similar problem. The natural test statistic is F23, F23 = T*(SSR2-SSR3)/SSR3 The asymptotic distribution of F23 is dependent on the data in a complicated way. Bootstrapping F23’s may be the better approach.  Both interval estimation and hypothesis testing in the TAR framework remain open and active areas of research in econometrics.