Stationarity Test with a Structural Break

The auxiliary regression tests for a unit root in 𝑦_𝑡, where 𝑦 refers to the MEFT series, 𝑡 = 1, … , 𝑇, is an index of time, and 𝛥𝑦_𝑡−𝑗 is the lagged first difference to accommodate serial correlation in the errors. Equation (4.2) tests the null of a unit root against a trend-stationary alternative⁵⁷. As in Hall (1994), the lag length is selected through the ‘𝑡 𝑠𝑖𝑔. ’. This approach involves starting with a predetermined upper bound 𝑘𝑚𝑎𝑥 where 𝑘𝑚𝑎𝑥 = 12 ( ^𝑇

100)^1/4(Narayan and Smyth, 2005). If the last included lag is significant, 𝑘_𝑚𝑎𝑥 is chosen. However, if 𝑘 is insignificant, it is reduced by one lag until the last lag becomes significant. If no lags are significant, 𝑘 is set equal to zero. As shown by Ng and Perron (1995) the ‘𝑡 𝑠𝑖𝑔. ’ approach produces test statistics, which has better properties in terms of size and power than when lag length is selected with some information-based criteria⁵⁸. As shown in the above equations, a constant or a constant and a trend is included in the ADF test regression. For either case, Elliot et al. (1996) propose a simple modification of the ADF approach to construct DF-GLS test, in which the time series are de-trended so that explanatory variables are

"taken out" of the data prior to running the test regression. Phillips and Perron (1988, PP hereafter) propose an alternative (nonparametric) method of controlling for serial correlation when testing for a unit root. The PP method estimates the non-augmented DF test equations (see Equation 4.1 and 4.2) without ∑^k_j=1d_jΔy_t−j term on the rhs, and modifies the t-ratio of the α coefficient so that serial correlation does not affect the asymptotic distribution of the test statistic.

4.2.2 Stationarity Test with a Structural Break

The major downside of unit root tests is that it assumes 𝑑_𝑗 (Equation 4.1 and 4.2) is correctly specified. Perron (1989, 1990) notes that if there is a break in the deterministic trend 𝑑_𝑗, then unit

57 In Equation (4.1) and Equation (4.2) the null and alternative hypotheses for a unit root in 𝑦𝑡 are 𝐻0: 𝛼 = 0 and 𝐻1: 𝛼 < 0 .

58 Akaike Information Criterion (AIC) or the Schwartz Bayesian Criterion (SBC).

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root tests lead to a misleading conclusion in favour of a unit root, when in fact there is not. Since Perron’s seminal paper, the debate on the effect of trend breaks on unit root tests and his assumption of known break point attracted a growing criticism. It was argued that if the break point is treated as endogenous, then Perron's conclusions are likely to be reversed (Maddala and Kim, 1998). The issue of endogenous or exogenous treatment of the break point is still inconclusive (Narayan and Popp, 2013). Since the seminal paper of NP (1982), macroeconomists have been interested in unit roots and the source of model misspecification. Using the standard unit root test approach without SBs for a data spanning over 100 years, NP could not reject the null hypothesis of an autoregressive unit root for 13 out of 14 US macro time series. The existence of a unit root was interpreted as having important implications for the theory of business cycles and the persistence of the effect of real shocks to the economy. Cochrane (1991a and 1991b) notes that the evidence on unit roots is empirically ambiguous and irrelevant to the question of persistence of the effect of real shocks.

To overcome the problem with the traditional ADF, Perron (1989) proposes a method that allows for a known or exogenous structural break in the Augmented Dickey-Fuller (ADF) tests. Following this development, many authors including ZA (1992) and Perron (1997), determine the break point endogenously from the data. Lumsdaine and Papell (1997) extend the ZA (1992) one-break approach to two breaks approach. However, these endogenous break point tests are criticised on the grounds of their treatment of breaks under the null hypothesis. Given the breaks are absent under the null hypothesis of unit root, there may be some tendencies for tests to suggest evidence of stationarity with breaks (LS, 2003). LS suggest a two break minimum Lagrange Multiplier (LM) unit root test in which the alternative hypothesis unambiguously implies that the series is trend stationary.

Exogenous and Endogenous Changes

A year after Phillip-Perron (Phillips and Perron, 1988) seminal paper on the unit root test approach, studies have emerged due to the consequence of spurious results and the power problem of ADF and PP tests. In order to accommodate structural changes in the data series, Perron proposes three characterisations of the trend break alternative models: (i) a model that allows for a one-time structural break in the intercept of the trend; (ii) a model that allows for a one-time structural break in the slope of the trend function, and (iii) a model that allows for a one-time structural break in the intercept and slope of the trend function. In his attempt to demonstrate how structural breaks in a series can lead to spurious results, Perron (1989) uses the idea of exogenously determined breaks informed by prior knowledge. Such exogenous assumptions have effects on the timing and properties of the critical values that are used to compare with the test results. ZA (1992), on the other hand, allowed for endogenously determined breaks chosen based on particular statistical

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criteria in an economically atheoretical way. Their critical values are likewise affected by the testing methods and in the original form; the number of breaks permitted is limited. The main differences in the testing procedures, proposed by Perron (1989), and subsequently by ZA (1992) over the original Dickey–Fuller approach, involve the addition of various dummy variables to Equations (4.1 and 4.2). The aim is to capture changes in the intercept and/or trend using the recursive estimation method as shown in Equation (4.3):

𝑦_𝑡 = 𝜇 + 𝜌𝑦_𝑡−1+ 𝛽𝑡 + 𝛾𝐷𝑇 + 𝜃𝐷𝑈 + ∑ 𝜑∆𝑦_𝑡−𝑖+ 𝑢_𝑡

𝑝

𝑖=1

(4.3)

where 𝐷𝑇 = 1, 𝑖𝑓 𝑡 > 𝑇𝐵 and 0 otherwise and 𝐷𝑈 = 1, 𝑖𝑓 𝑡 > 𝑇𝐵, 0 otherwise. 𝑇𝐵 refers to the time of the break. 𝐷𝑈 and DT are included to capture the possibility of ‘crashes’ (DU), trend or gradual changes (DT) and joint crashes and trend changes (DU and DT). However, Perron’s known assumption of the break date was criticised, most notably by Christiano (1992) as simply a process of ‘data mining’. Christiano argues that the data based procedures are typically used to determine the most likely location of the break. Since then, several studies have developed using different methodologies to endogenously determine the break date (see also Banerjee et al., 1992; ZA, 1992;

Perron and Vogelsang, 1992; Perron 1997; Lumsdaine and Papell, 1997). These studies have shown that bias in the usual unit root tests can be reduced by endogenously determining the time of SBs.

If the data-generating process involves more than one break as might be expected in the long time series, the same problem persists as in the original approach. Vogelsang (1997, 2012) shows the loss of power that ensures when using a one-break model in a world of two breaks. Empirically, Ben-David and Papell (1998) present evidence of more than one break and Lumsdaine and Papell (1997) consider a generalisation of the endogenous break point procedure using the ZA (1992) approach.

Regarding the dating of structural breaks, studies show that the potential break is assumed to be known a priori. Test statistics are then constructed by adding dummy variables to represent crashes and gradual structural changes. This extends the standard Dickey-Fuller procedure (Perron 1989) to a break date stationarity. However, Christiano (1992) and Bai and Perron (2003a) argue that this approach further invalidates the distribution theory underlying the conventional testing. Following this criticism, a number of studies have developed various methods to determine endogenously determined dates rather than a known a priori approach. The latter approach includes ZA (1992), LP (1997), Perron and Vogelsand (1992) and Bai and Perron (2003a). They have shown that the expected bias in the usual mean reversion tests can be reduced by endogenously determining the time of structural breaks and accounting for the breaks in the testing process. However, limiting the number of breaks to either one or two can reduce the power of ADF test so bias still exists.

147 Econometrics of Structural Breaks

Structural breaks are commonly present in many macroeconomic and financial time series (e.g.

Stock and Watson, 1996; Ang and Bekaert, 2002) and are one of the major reasons for misspecification and poor performance of macroeconomic and financial models (Clements and Hendry, 1998a). ZA propose a variant of Perron’s original test in which they assume that the exact time of the break point is unknown. Instead, a data dependent algorithm is used to proxy Perron’s subjective procedure to determine the break points. Following Perron’s characterisation, ZA proceed with three models: (1) model A, permits a one-time change in the level of the series; (2) model B, allows for a one-time change in the slope of the trend function, and (3) model C, combines one-time change in the level and the slope of the trend function of the series. Hence, to test for a unit root against the alternative of a one-time structural break, ZA employ the following regression equations corresponding to the above three models:

Model A: Δ𝑦_𝑡 = 𝑐 + 𝛼𝑦_𝑡−1 + 𝛽_𝑡+ 𝛾𝐷𝑈_𝑡+ ∑ 𝑑_𝑗Δ𝑦_𝑡−1+ 𝜀_𝑡 (4.4)

𝑘

𝑗=1

Model B: Δ𝑦_𝑡 = 𝑐 + 𝛼𝑦_𝑡−1 + 𝛽_𝑡+ 𝜃𝐷𝑇_𝑡+ ∑ 𝑑_𝑗Δ𝑦_𝑡−1+ 𝜀_𝑡 (4.5)

𝑘

𝑗=1

Model C: Δ𝑦_𝑡 = 𝑐 + 𝛼𝑦_𝑡−1 + 𝛽_𝑡+ 𝛾𝐷𝑈_𝑡+ 𝜃𝐷𝑇_𝑡+ ∑ 𝑑_𝑗Δ𝑦_𝑡−1+ 𝜀_𝑡 (4.6)

𝑘

𝑗=1

where 𝐷𝑈_𝑡 is an indicator dummy variable for a mean shift occurring at each possible break-date (TB) while 𝐷𝑇_𝑡 is the corresponding trend shift variable. Formally,

𝐷𝑈_𝑡= { 1 … … . 𝑖𝑓 𝑡 > 𝑇𝐵,

0 … … . … . . 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒𝑎𝑛𝑑 𝐷𝑇_𝑡 = {𝑡 − 𝑇𝐵 … … 𝑖𝑓 𝑡 > 𝑇𝐵

0 … … . … 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑎𝑛𝑑

the null hypothesis in all the three models is 𝛼 = 0, which implies that the series {yt} contains a unit root with a drift that excludes any structural break, while the alternative hypothesis α < 0 implies that the series is a trend-stationary process with a one-time break occurring at an unknown point in time. The ZA method regards every point as a potential break-date (TB) and runs a regression for every possible break-date sequentially. From amongst all possible break-points (TB), the method selects the date which minimises the one-sided 𝑡 −statistics for testing α̂ (= α − 1) = 1 as its choice of break date (TB). According to ZA, the presence of the end points cause the asymptotic distribution of the statistics to diverge towards infinity. Therefore, some region must be chosen so that the end points of the sample are not included. ZA suggest the ‘trimming region’ to be specified as (0.15𝑇, 0.85𝑇). According to Perron, most economic time series can be adequately modelled using either model A or C. Subsequent literature primarily applied model A and/or model

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C. Criticising Perron, Sen (2003) argues that if one uses model A when in fact the break occurs according to model C then there will be a substantial loss of power. However, if break is characterised according to model A, but model C is used then the loss in power is minor. This suggests that model C is superior to model A. As there is no conclusive agreement which method to use in order to investigate structural breaks since Perron’s seminal paper, this study employs the three models to identify the single and representative break point(s). In a similar setting, Lumsdaine and Papell (1997) extend the ZA test of Equation (4.4 to 4.6), by adding dummy variables for intercept and slope changes in a single model:

𝑦_𝑡 = 𝜇 + 𝜌𝑦_𝑡−1+ 𝛽𝑡 + 𝜃𝐷𝑇₁+ 𝛾𝐷𝑈₁+ 𝜃𝐷𝑇₂+ 𝛾𝐷𝑈₂+ ∑ 𝜑∆𝑦_𝑡−𝑖+ 𝑢_𝑡

𝑝

𝑖=1

(4.7) as in the ZA single break test algorithm, three types of models can be considered but there are more variations including two breaks in the intercept and two breaks in the slope. Being a variation of a standard unit root test, the 𝑡 − 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 on 𝜌 is compared to the relevant critical value found in Lumsdaine and Papell (1997). Lee and Strazicich (2001, 2003) employ a similar approach to Lumsdaine and Papell (1997) but their test statistic uses a minimum Lagrange Multiplier (LM) test criteria. This approach is based on the results from Schmidt and Phillips (1992) on the potential for unit root tests to report spurious rejections when the null includes a genuine structural break. The Lee and Strazicich LM-based test assumes that the null hypothesis has permanent structural shift with up to two breaks. In the LS approach, the 𝑡 − 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 to test the null arises via a LM principle based on scoring methods. This test procedure was also utilised by Greasley et al. (2010) to analyse the empirics of long-run growth. The ability to permit up to and more than two breaks in the null and two breaks in the level or slope of the alternative makes the approach particularly flexible and realistic.

The two important issues stemmed from the above variety of approaches are: (1) the issue raised due to the trade-off between the power of the test and the amount of information incorporated with respect to the choice of break point (Perron 1997:p378); (2) these tests only capture the single most significant break in each series but disregard the presence of more than one or two SBs. The question here is - what if there are multiple breaks in each individual series? (Maddala and Kim, 2003).

Studies show that assuming a single endogenously or exogenously determined structural break is insufficient and leads to a loss of information when more than one breaks exist (Lumsdaine and Papell, 1997). Accordingly, LP argue that permanent structural shift tests that account for two significant structural breaks are more powerful than those that allow for a single break. However, limiting structural shifts as one or two break points rather than making an open assumption for the presence of multiple structural breaks is likely to contribute to the loss of information (BP, 2006;

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Clemente et al., 1998). One of the reasons why the BP methods are not frequently used could be due to the complex nature of the DGP and the iterative sequential algorithm methods that tests one break against more than one successive breaks. Because of this, there are few attempts made to investigate the presence of MSBs based on the BP dynamic programming algorithm.

Based on Perron and Vogelsang (1992), Clemente et al. (1998) attempt to investigate multiple break points. Ohara (1999) uses a sequential 𝑡 − 𝑡𝑒𝑠𝑡𝑠 based on Zivot and Andrew’s approach to examine the case on multiple breaks with unknown break dates. Ohara provides evidence and argues that unit root tests with multiple trend breaks are necessary for both asymptotic theory and empirical applications. The next section briefly discusses the development of various approaches since NP’s (1982) seminal paper and converses the MSB approach from theoretical, methodological and empirical perspectives.