The asymptotic behaviour of the residual sum of squares in models with multiple break points

Figures & data

Table 1. Critical values for test based on $F_{\lambda }\left(\bar{\lambda }\right)$.

m	10%	5%	1%	m	10%	5%	1%
1	5.9415	7.3581	10.5845	6	25.2819	27.9196	33.3415
2	10.2164	11.9835	15.8540	7	28.8528	31.6485	37.3694
3	14.1666	16.2043	20.5591	8	32.3859	35.3227	41.3215
4	17.9626	20.2202	24.9877	9	35.8842	38.9584	45.2137
5	21.6575	24.1175	29.2265	10	39.3541	42.5614	49.0649

Critical values at the 10, 5, and 1% significance level of the limiting distribution of $F_{\lambda }\left(\bar{\lambda }\right)$ in Corollary 2, for models with m number of breaks.

Figure 1. Power of $F_{\lambda }\left(\bar{\lambda }\right)$ statistics for $H_0:{\bar{\lambda }}_1=0.5+\kappa$. The true break is at ${\lambda }_1^0=0.5$. Power is shown for κ = 0,0.02,…,0.2 where κ = 0 corresponds to the true break and higher values to null hypotheses that are further away from the true break. The tests are conducted at the 5% significance level. Higher α values correspond to smaller magnitudes of the break determined by $0.3\times 50^{\alpha }∕T^{\alpha }$.

Table 2. Estimated monetary policy rules.

	1968Q4–2005Q4	1968Q4–1980Q3	1980Q4–1985Q3	1985Q4–2005Q4
A. Estimated coefficients
πt+1|t	0.41 (1.10)	0.80 (3.63)	1.68 (10.53)	0.16 (1.13)
${\tilde{x}}_{t+1|t}$	0.23 (0.77)	0.94 (2.77)	0.01 (0.04)	0.09 (0.96)
rt−1	0.65 (1.64)	0.37 (1.08)	-0.20 (1.27)	1.33 (8.54)
rt−2	0.11 (0.31)	0.03 (0.17)	0.09 (0.53)	-0.43 (3.17)
c	0.12 (0.21)	1.03 (1.04)	3.75 (2.91)	0.09 (0.59)
B. Implied steady-state monetary policy responses
πt+1|t	1.74 (1.01)	1.33 (1.53)	1.51 (5.73)	1.59 (0.74)
${\tilde{x}}_{t+1|t}$	1.00 (0.93)	1.56 (1.12)	0.01 (0.04)	0.91 (0.43)

Breaks in the monetary policy rule (51) are detected using the BIC and HQIC information criteria with m(p+3) effective parameters, a maximum of five breaks and a minimum of 10% of sample observations are required to be in each estimated monetary policy regime. Inflation and unemployment gap forecasts are treated as endogenous in the monetary policy rule, with breaks detected separately for each reduced form equation. Figures in parentheses are t-ratios. The implied steady-state responses of monetary policy to inflation and the unemployment gap shown in Panel B are obtained from the estimated coefficients assuming constant short-term interest rates ($r_t=r_{t-1}=r_{t-2}$).

Figure 2. 99% break fraction confidence set for monetary policy application. The confidence set shows the break fraction pairs (λ₁, λ₂) for which the statistic $F_{\lambda }\left(\bar{\lambda }\right)$ does not reject the corresponding joint null hypothesis at the 1% level, when applied to each permissible null hypothesis subject to a 15 observation minimum segment (𝜖 = 0.10). The λ₁ = 0.32 break fraction corresponds to 1980Q3 and is the only date of a first break that does not reject the null while λ₂ can take any value from 0.42 to 0.62, or 1984Q2 to 1991Q4.