V 5 a 1 a
1
OC 1 a
2
Time 1 a
3
D~t 1 a
4
V
21
1 e, This specification accommodates a gradual impact of parameter shifts at each break
date. For example, a one-time shift in the intercept at time t has an initial impact of Dt
but a long-run impact of Dt 1 2 a
4
.
IV. The Results
We test for three potential sets of breaks: intercept only; intercept and coefficient of opportunity cost; and intercept and coefficients of opportunity cost and time trend. Panel
A in Table 1 presents test statistics and estimated coefficients for the case in which only the intercept is allowed to change. Three break dates are found—1970Q2, 1981Q1, and
1991Q4 — using the Bai–Perron sequential test under the global null. The estimation package provided by Bai and Perron also includes test statistics based on an unconditional
Table 1. Bai-Perron Results for M2 Velocity Models 1964Q1–1998Q4
A. Intercept only varies.
a
Test Statistics Global
SupF
T
1 63.7459
Sequential under the global null SupF
T
2|1 6.6526
SupF
T
3|2 25.9214
SupF
T
4|3 4.4714
Purely sequential SupF
T
2|1 6.6526
Number of breaks selected by: BIC
3 LWZ
1 Parameter estimates using sequential test under the global null
b
M2 vel.t21
5 0.736518 0.0294 T
1
5 1970Q2 1968Q1, 1971Q2 b
time
5 0.000776 0.0001 T
2
5 1981Q1 1979Q3, 1982Q3 b
M2 opp. cost
5 0.008884 0.0012 T
3
5 1991Q4 1991Q2, 1992Q2 d
1
5 0.408665 0.0461 d
2
5 0.388547 0.0454 d
3
5 0.365692 0.0455 d
4
5 0.412684 0.0516 Parameter estimates using purely sequential test
b
M2 vel.t21
5 0.755264 0.0294 T
1
5 1991Q4 1991Q2, 1992Q1 b
time
5 0.000277 0.0001 b
M2 opp. cost
5 0.008946 0.0013 d
1
5 0.380393 0.0459 d
2
5 0.438586 0.0519
a
Time, opportunity cost of M2, and 1-quarter-lagged M2 velocity were used as regressors in all models. Those variables not mentioned explicitly as changing d coefficients were included as constant-coefficient regressors b coefficients. In all
cases, a minimal break fraction of 0.15 was used, corresponding to a minimal regime length of 21 periods. Indicates significance at the 5 significance level.
Indicates significance at the 1 significance level.
Structural Uncertainty and Breakpoint Tests 105
null, a purely sequential test, Bayesian Information Criteria BIC, and Liu, Wu, and Zidek LWZ tests. The BIC test also finds three breaks, but the purely sequential test and the
LWZ test reveal only one break, in 1991Q4. Following Bai and Perron, we feature the results of the sequential test under the global
null. Thus, we conclude that there are three breaks for the case in which only the intercept is allowed to shift. Figure 2A illustrates the estimates of both the long- and short-run
equilibrium trends for M2 velocity evaluated at the mean value of opportunity cost based on the Bai–Perron estimated breakpoint dates. The short-run paths are based on the
estimated AR1 process.
The results for the tests allowing for shifts in both the intercept and coefficient of opportunity cost are provided in panel B. Again, three breaks are found using the
Table 1. Continued
B. Intercept and coefficient of opportunity cost vary.
