Candelon, B.

Dupuy, Arnaud

Gil-Alana, L.

English

Research Papers in Economics

Appendix of: The Nature of Occupational Unemployment Rates
in the United States: Hysteresis or Structural?∗
B. Candelon† , A. Dupuy‡ , and L. Gil-Alana§
June 26, 2008
1 A Simulation Study
In this section, we present some Monte Carlo experiments describing the performance of the procedure
of Section 3. Initially, we consider that the true data generating process is given by Equation ??,w i t h
Dt = I(t>T b); Tb = T/2, β =1 0 ,a n dd1 = d2 =1 . In other words, the true model is:
ut =1 0 Dt + xt,ρ (L;d)xt = wt,ρ (L;d)=( 1− L12)(1 − L) (1)
with white noise wt,a n dT =1 2 0 ,240,360,480,600 and 720. For this purpose, we generate Gaussian series
using the routines GASDEV and RAN3 of Press, Flannery, Teukolsky and Wetterling (1986),? and 1,000
r e p l i c a t i o n sa r eu s e di ne a c hc a s e .
∗We thank participants at a seminar at Maastricht University. Luis Gil-Alana thanks the ﬁnancial support from the Ministerio
de Ciencia y Tecnologia (SEJ2005-07657/ECON, Spain).
†Department of Economics, Maastricht University, Maastricht, the Netherlands.
‡Corresponding author. Research Centre for Education and the Labour Market, (ROA) and Department of Economics,
Maastricht University, P.O. Box 616, MD 6200 MD, Maastricht, the Netherlands. E-mail: a.dupuy@roa.unimaas.nl. Tel: +31
43 3883735.
§Department of Economics, University of Navarre, Pamplona, Spain.
1Across Table 1 we report the probabilities of correctly determining the time of the break and the fractional
diﬀerencing parameters, using a grid of values for the time break T∗ = T/10,T/10 − 1,...,(1),...,9T/10 − 1
and 9T/10,a n d(d10,d 20)-values from 0 to 2 with 0.2 increments. The most noticeable feature observed
in this table is that the procedure accurately determines the break date in all cases, and we ﬁnd zero-
probabilities for all values of d10 and d20 if T∗ is diﬀerent from the true time of the break. Thus, we only
report across the table the values of d10 and d20 where we observe a non-zero probability. We see that
if T =1 2 0 , higher probabilities are obtained at other values than the true ones (e.g. d10 = d20 =0 .8),
however, if T>120, the highest probabilities are obtained in all cases at d10 = d20 =1 ,a n di fT =6 0 0 ,
the probabilities corresponding to the true parameters are higher than 0.9. Note that these probabilities are
based on the grid employed for the orders of integration and hence the probabilities becomes lower as the
range of values for the increments in the d’s is reduced. On the other hand, larger increments produce larger
probabilities of detecting the true values.
(Tables 1 - 3 about here)
We also performed the experiment with other values for the time break and the fractional diﬀerencing
parameters. Table 2 displays the results for Tb = T/4,d 10 =0 .4 and d20 =0 .8 (i.e. stationarity in the ﬁrst
subsample but nonstationarity after the shock), while Table 3 refers to the case of Tb =3 T/4, d10 =0 .8
and d20 =0 .4 (i.e. stationarity only after the break). The results are similar in both cases, and they are
completely in line with those given in Table 1. Thus, if T =6 0 0 , the probability of correctly determining
the true parameters exceeds 0.9 in all cases.
The accuracy in the estimation of the break date in the results presented so far might be a consequence
of the coeﬃcient used for the break dummy in equation (7). Thus, in Table 4, we examine the probability of
correctly determining the break for diﬀerent coeﬃcients for the dummy variable. We now assume that the
break date takes place at T/2,w i t hd1 = d2 =1 , and look at the probability of detecting the true break date
2for a grid of values (T/10,T/10 + 1,...,9T/10 − 1,9T/10),u s i n ga sc o e ﬃcients for the deterministic break,
β =1 0 ,5,3 and 1.
(Table 4 about here)
We observe in this table that if β =1 0or 5, the procedure correctly determines the break date in the
100% of the cases even for a sample size of T =1 2 0 . However, reducing the magnitude of the coeﬃcient, the
probabilities are very small in some cases, especially if the sample size is small.
2 Robinson (1994) Score Test for Seasonal and Long-run Frac-
tional Integration
The set-up in Robinson (1994) is the model given by equations (3) − (5), i.e, It is supposed that wt has
spectral density given by:
ut = β
0zt + xt,, (1 − L12)d1(1 − L)d2xt = wt, (2)
and suppose that the wt above has a spectral density given by:
f(λ;τ)=
σ2
2π
g(λ;τ),−π<λ<π ,
where the scalar σ2 is known and g is a function of known form, which depends on frequency λ and the
unknown (q × 1) parameter vector τ.
Unless g is a completely known function (e.g., g ≡ 1,a sw h e nwt is white noise), we have to estimate the
nuisance parameter τ, for example by ˆ τ = argminτ∈T∗σ2(τ) ,w h e r eT ∗ is a suitable subset of Rq Euclidean
space, and:
3σ2(τ)=
2π
T
T−1 X
s=1
g(λs;τ)−1Ie w(λs)
where I ˆ w is the periodogram of ˆ wt =( 1− L12)d1(1 − L)d2ˆ ut − ˆ β
0
st,
ˆ β =(
T X
t=1
s2
t)−1
T X
t=1
st(1 − L12)d10(1 − L)d20yt,
st =( 1− L12)d10(1 − L)d20zt,
evaluated at the discrete frequencies:
λs =
2πs
T
,
which is given by
I ˆ w(λs)=|(2πT)−1/2
T X
t=1
b wteiλst|2.
Note that the tests are purely parametric, requiring speciﬁc modelling assumptions regarding the short
memory speciﬁcation of wt. Thus, for example, if wt is an AR process of form: φ(L)wt =  t,t h e ng =
|φ(eiλ)|−2,w i t hσ2 = V ( t), so that the AR coeﬃcients are a function of τ.
The test statistic, which is derived via Lagrange Multiplier (LM) principle, adopts the form:
ˆ R =
T
ˆ σ
4ˆ a0 ˆ A−1ˆ a, (3)
where T i st h es a m p l es i z e ,a n d
4ˆ σ2 = σ2(ˆ τ)=
2π
T
T−1 X
s=1
ψ(λs)g(λs;ˆ τ)−1I(λs);
ˆ a =
−2π
T
∗ X
s
ψ(λs)g(λs;ˆ τ)−1I(λs);
ˆ A =
2
T
(
∗ X
j
ψ(λs)ψ(λs)0
−
∗ X
j
ψ(λs)ˆ  (λs)0(
∗ X
j
ˆ  (λs)ˆ  (λs)0))−1
∗ X
j
ˆ  (λs)ψ(λ
0
s))
ψ(λs)0 =[ ψ1(λs),ψ2(λs)];
ψ1(λs)=log|2sin
λs
2
|;
ψ2(λs)=log|2(sinλs/2| + log(2cos(λs/2))
+log|2cosλs| + log|2(cosλs − cos(π/3)|
+log|2(cosλs − cos(2.π/3)|
+log|2(cosλs − cos(π/6)| + log|2(cosλs − cos(5.π/6)|;
ˆ  (λs)=
δ
δτ
logg(λs;ˆ τ)
a n dt h es u m m a t i o no n*i nt h ea b o v ee x p r e s s i o n si so v e rλ ∈ M where M = λ : −π<λ<π , λ *
(ρk − λ1,ρ k + λ1), k =1 ,2,...,s such that ρk,k=1 ,2,...,s are the distinct poles of ψ(λ) on (−π,π].
3 Asymptotic Theory of the Test of Robinson (1994) in the Pres-
ence of a Structural Break
It is straightforward from Robinson (1994) to show that the test statistic has a standard null limit behavior.
For simplicity we just concentrate here on the case of white noise wt. For this purpose we need to rely on the
5deﬁnitions 1, 2 and 3 referring respectively to the classes F, G and H in Appendix 2 in Robinson (1994). The
class F imposes a martingale diﬀerence assumption on the disturbances wt, that is substantially weaker than
the Gaussianity assumptionused in motivating the test. The class G imposes a mild lack of multicollinearity
on the diﬀerenced series for zt,w h i c hi ss a t i s ﬁed by the dummy variables employed in the paper. The class
H refers to some technical restrictions required to approximate integrals by sums. Note that under the null
hypothesis of d = d0, the model under analysis in (3), (4) and (5) becomes:
ρ(L;d)ut = βρ(L,d0)Dt + wt,t =1 ,...,T
where wt is assumed to be I(0) and thus, standard theory applies.
We call Wt = ρ(L,d0)Dt and D =
PT
t=1 WtW0
t. Then, it can be easily seen that EkD1/2(ˆ β − β)k2 =
o(1) as T →∞ ,w h e r eˆ β is the OLS estimation of β. The only requirement is that D,d e ﬁne as above
must be a positive deﬁnite matrix for suﬃciently large T, and this condition is satisﬁed by the dummy
variables employed in the paper. With respect to the test statistic, we ﬁrst decompose ˆ a as described
in Appendix 1 into (ˆ a − a)+( a − a∗)+a∗,w h e r ea∗ = −2π
T
P∗
s ψ(λs)Ie w(λs) and a = −
Pt−1
k=1 ψkCw(k),
Cw(k)= 1
t−k
PT−k
t=1 wtwt+k. It follows from Theorem 1 in Robinson (1994) that (ˆ a − a)=op(T−1/2),
(a−a∗)=op(T−1/2) and a∗ →d N(0,σ4Ψ),w h e r eΨ = 1
2π
R π
−π ψ(λ)ψ(λ)0dλ. On the other hand, noting that
w2
t −σ2 are stationary martingale diﬀerences, and that Ce w−Cw →p 0,t h e nCw(0) →p σ2 and thus, it follows
that ˆ σ
2 →p σ2. Finally, ˆ A → A by Lemma 3 in Robinson (1994). Similar arguments can be developed with
respect to the local eﬃciency power property of the tests (see Theorem 2 in Robinson, 1994) and with the
extension to the weak autocorrelation for the I(0) disturbances wt (Theorem 3).
6Table 1: Break at T/2 with d1=1 and d2=1.
d1 (Seasonality) d2 (Long run) T=120 T=240 T=360 T=480 T=600 T=720
0.6 0.0 0 . 0 1 0 –––––
0.6 0.8 0 . 0 1 0 –––––
0.8 0.8 0.234 0.043 0.010 – – –
1.0 0.8 0.085 0.141 0.087 0.021 0.014 0.007
1.2 0.8 0 . 0 1 0 –––––
1.0 0.2 0.021 0.010 – – – –
0.6 1.0 0 . 0 1 0 –––––
0.8 1.0 0.106 0.032 0.032 0.043 0.023 0.006
1.0 1.0 0.127 0.293 0.500 0.641 0.905 0.972
1.2 1.0 0.010 0.054 – 0.010 – –
1.4 1.0 0 . 0 1 0 –––––
0.8 1.2 0.053 0.065 0.021 0.011 – –
1.0 1.2 0.074 0.217 0.250 0.271 0.058 0.015
1.2 1.2 0.010 0.021 0.010 – – –
0.6 1.4 0 . 0 2 1 –––––
0.8 1.4 0.148 0.065 0.043 – – –
1.0 1.4 0.063 0.065 0.011 – – –
1.2 1.4 0 . 0 1 0 –––––
Note: _ means 0−probability
7Table 2: Break at T/4 with d1=0.8 and d2=0.4.
d1 (Seasonality) d2 (Long run) T=120 T=240 T=360 T=480 T=600 T=720
0.6 0.2 0.106 0.174 0.043 0.010 – –
0.8 0.2 0.159 0.281 0.173 0.065 0.043 0.020
1.0 0.2 0.021 0.010 – – – –
0.4 0.4 0 . 0 6 3 –––––
0.6 0.4 0.393 0.141 0.109 0.066 0.054 0.018
0.8 0.4 0.256 0.359 0.663 0.847 0.902 0.962
1.0 0.4 0.011 0.032 0.011 0.011 – –
Note: _ means 0−probability
8Table 3: Break at 3T/4 with d1=0.4 and d2=0.8.
d1 (Seasonality) d2 (Long run) T=120 T=240 T=360 T=480 T=600 T=720
0.0 0.4 0.085 0.010 – – – –
0.2 0.4 0 . 0 2 2 –––––
0.0 0.6 0.095 0.043 – – – –
0.2 0.6 0.297 0.163 0.087 0.019 – –
0.4 0.6 0.127 0.271 0.163 0.043 0.044 0.021
0.6 0.6 0.023 0.010 – – – –
0.8 0.6 – 0 . 0 1 1 ––––
0.2 0.8 0.138 0.032 0.076 0.054 0.055 0.011
0.4 0.8 0.128 0.380 0.597 0.837 0.891 0.960
0.6 0.8 0.009 0.009 0.010 0.010 – –
0.2 1.0 – 0 . 0 0 9 ––––
0.4 1.0 0.020 0.021 – 0.010 – –
0.6 1.0 – 0.010 0.043 0.021 0.010 0.008
0.2 1.2 0.053 0.020 0.021 – – –
Note: _ means 0−probability
9Table 4: Probability of detecting the true break fraction for diﬀerent break coeﬃcients, beta.
β T=120 T=240 T=360 T=480 T=600 T=720
10 100 100 100 100 100 100
5 100 100 100 100 100 100
3 83.33 92.30 100 100 100 100
1 10.71 13.09 17.98 20.5 33.39 33.57
10

Appendix of: The Nature of Occupational Unemployment Rates in the United States: Hysteresis or Structural?.

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Appendix of: The Nature of Occupational Unemployment Rates in the United States: Hysteresis or Structural?.

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