LM cointegration tests allowing for an unknown number of breaks: implications for the forward rate unbiasedness hypothesis

ABSTRACT

This article extends the Lagrange multiplier (LM) cointegration test proposed by Westerlund and Edgerton (WE 2007) by allowing for an unknown number of breaks. Monte Carlo simulations provide two main results. First, a loss of power in the LM cointegration tests is detected when potential multiple breaks are ignored. Second, the modified testing procedures do not affect the asymptotic distribution and major properties of the tests of WE under the null, but noticeably increase their testing power in presence of multiple breaks. We also provide empirical applications of the proposed tests for the forward rate unbiasedness hypothesis (FRUH). The results reveal that the FRUH does hold when the effects of the multiple structural breaks are taken into account.

KEYWORDS:

• Cointegration
• structural changes
• forward rate unbiased hypothesis

JEL CLASSIFICATION:

• C12
• C15
• C32
• F31

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ For example, see Campos, Ericsson, and Hendry (1996), Gregory and Hansen (1996), Leybourne and Newbold (2003).

² This property can be also found in LM test-based unit root tests with structural changes. See Lee and Strazicich (2003) for more detail.

³ There are some more studies which extend the conventional tests by allowing multiple breaks. These contributions include Hansen (2003), Hatemi-J (2008), Maki (2012).

⁴ To denote statistics for each model, we use subscripts N, C, and S. We use subscript N to denote the model with no break (Model A). The subscript C and S stand for Model B and Model C, respectively. For example, ϕ and t-statistic for the model with level shift (model C) can be denoted by ϕ_c and t_c, respectively.

⁵ Alternatively, the location of break can be estimated by minimizing the individual test statistics as done in Gregory and Hansen (1996). However, Westerlund and Edgerton (2007) reports that searching the break point based on the procedure results in serious size distortion of the LM cointegration test.

⁶ Also, see the discussion in Barnhart and Szakmary (1991) and Zivot (2000b).

⁷ The I(1) behaviour of forward and spot rates has been confirmed extensively in the literature (see, for example, Meese and Singleton 1982; Baillie and Bollerslev 1989; Mark 1990; Liu and Maddala 1992; Crowder 1994; Clarida and Taylor 1997). Given the widely accepted fact, we do not present the results of the unit root tests but they are available upon request.

⁸ One can test the FRUH by prespecifying the cointegrating vector as $(1,-1)$. However, the power of such procedures is guaranteed only when the prespecified cointegrating vector is correct (see, for example, Horvath and Watson 1995; Zivot 2000a). Moreover, prespecifying cointegrating vector of $(1,-1)$ might not be appropriate in the LM testing procedure. Since the variable ${\tilde{S}}_t$ which is used to compute the LM testing regression in (4) is based on the OLS estimates from the regression of differenced variables rather than the level specification as in (13) and (14), we estimate the cointegration parameters instead of prespecifying them.

⁹ Considering five breaks for macro time series seem to be enough. See other examples from Sakoulis, Zivot, and Choi (2010) and Maki (2012).

¹⁰ Critical values for the tests can be found in Schmidt and Phillips (1992).