The ZA One-Time Structural Break at Unknown Date

Since the Second World War, the macro series in major European countries is characterised by the presence of several domestic and external shocks, so it is not possible to know exactly when the optimal date of change occurred. Hence, it is appropriate to undertake a test of the unit-root hypothesis allowing endogenously determined one-time structural break. The ZA method provides an estimation procedure for determining the breakpoint in a manner that gives the least favourable weight to the unit-root hypothesis using the test statistics for 𝛼 = 1(𝑖 = 𝐴, 𝐵, 𝐶), where 𝜆 is chosen in such a manner that the one-sided 𝑡 statistics for testing 𝛼 = 1 is minimised. If 𝜆_𝑖𝑛𝑓^𝑖 represents such a minimum value for model 𝑖, and it follows that

𝑡_𝛼𝑖[𝜆_𝑖𝑛𝑓^𝑖 ] = inf

𝜆𝜖Λ𝑡_𝛼𝑖(𝜆) (4.11) where λ is a specified closed subset of (0, 1). ZA’s method of unit-root test involves estimation of the equations given by Perron. The null hypothesis in this method is specified that it does not require inclusion of the dummy variable 𝐷(𝑇𝐵)_𝑡, which is included in Perron’s Model A, and C. Model B is constructed using the intervention outlier (IO) model instead of the two-step additive outlier (AO) model as in Perron. Following ZA, this study treats the structural break as endogenous to determine persistence and transitory innovations in the given series.

The summary results for ZA’ Model 𝐴^𝑍𝐴, 𝐵^𝑍𝐴 and 𝐶^𝑍𝐴 are reported in Table 4.1 and the detailed results in Table 4.4A. The test also finds no evidence of residual serial correlation in the error terms.

The endogenous structural break test in ZA approach is a sequential test, which utilises the full sample and uses a different dummy variable for each possible break date (Narayan and Smyth, 2008). The break date is selected where the t-statistic from the ADF test is at a minimum (most negative). Consequently, a break date is chosen where the evidence is least favourable for the unit root null. The critical values in ZA (1992) are different to the critical values in Perron (1989). The differences are because the selection of the time of the break is treated as the outcome of an estimation procedure, rather than predetermined exogenously as in Perron. For each series the coefficients are estimated using the three ZA models with break function 𝜆 =^𝑇_𝑇^𝐵, ranging from 𝑗 =

2

𝑇 to 𝑗 =^(𝑇−1)_𝑇 . The 𝑡 statistics for 𝛼 = 1 reported in the table are the minimum values over all 𝑇 − 2 regressions. The estimated break years 𝑇_𝐵(= 𝜆𝑇) are the observation corresponding to 𝜆_𝑖𝑛𝑓^𝑖 and the minimum value of 𝑡_𝛼𝑖(𝜆). It can be seen that the break period that minimises the one-sided 𝑡 statistics for 𝛼^𝑖 = 1 does not coincide with the breakpoints chosen exogenously from visual inspection of the time plots of the series.

Table 4.1 below summarises the ZA findings of the MEFT series that possess significant break dates at 1%, 5% and 10% level of significance. Unlike previous studies, this research emphasises

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on MEFT series with significance level (α), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%. However, the MEFT series with 10% obtained from Model A, Model B and Model C are also included to provide further information. If one treats these variables as nonstationary without taking into account the structural breaks, the researcher may use the first or second differences or log differences of the data to achieve stationarity. This leads to loss of long-run cointegrated information as the data are differenced while they are actually break stationary.

Table 4.1 Summary of the MEFT Series with a Break Date Identified by ZA Approach

𝑀𝑜𝑑𝑒𝑙 𝐴^𝑍𝐴 𝑀𝑜𝑑𝑒𝑙 𝐵^𝑍𝐴 𝑀𝑜𝑑𝑒𝑙 𝐶^𝑍𝐴 Significant MEFT series α ≠ 0 ; β = 0 α = 0 ; β ≠ 0 α ≠ 0 ; β ≠ 0 Break Dates

LCPI (F) (-4.7142)*[12] (-5.7586)*** [12] [1973M09] A

[1974M01] C

LEXR (F) (-4.1860)*[3] (-4.8793)*[3] [1981M02] B

[1984M04] C

LLTIR (F) (-5.893)***[12] (-5.8939)*** [12] [1980M07] BC

LIBR (F) (-8.9404)*** [7] (-4.6498)** [7] (-8.0750)*** [7] [2006M10] B [2008M10] AC

LMHE (F) (-4.6836)* [12] (-4.3389)* [2] [1974M03] A

[1982M04] B

LGNP (ME) (-5.5860)*** [3] [1974Q02] C

LGDP (ME) (-4.5897)* [3] (-5.7915)*** [3] [1974Q02] AC

Notes: Figures in parenthesis are t statistics; *, **, and *** denote level of significance of the test of 𝛼^𝑖= 1 (𝑖 = 𝐴, 𝐵, 𝐶) at 10%, 5% and 1%, respectively. The ZA critical values are used to reject the false hypothesis. A, B and C refer to Model A, Model B and Model C, respectively. F = Financial; ME = Macroeconomic. See critical values in Appendix 4.

Source: author’s analysis.

Following the ZA method, the research allows a one-time structural break around the intercept, slope and simultaneous changes in both intercept and slope. The alternative hypothesis is a trend stationary process that allows for a one-time break in the level, the trend or both. The estimated break points where the unit root null is rejected are ranging from 1973 to 2008. The unit root null is rejected at 5% sig. level for only 5 of the MEFT series (additional 2 series at 10% LS) in favour of the alternative break-point stationary series. These MEFT series show long-term permanent shocks with a non-reverting mean but remains with a non-constant long-run mean, which implies that the series have no tendency to return to a long-run deterministic path. Furthermore, the variance of the series becomes time-dependent. Similarly, ZA reject unit root at the five percent sig. level for only three out of 13 variables (23.08% proportion) using the NP data as compared to 23%

proportion of the UK MEFT series that includes all the ZA and NP variables. Re-examining the NP data Lumsdaine and Papell (1997) show more evidence against unit roots than ZA but less than Perron (1989). Using finite-sample critical values, LP reject the unit root null for five series at the 5% level of significance. LS (2003) apply two-break minimum LM unit root test to NP’s (1982) and compared it with LP test. At the 5% level of significance, they reject the null for six series with the LP test and four series with the LM test. Only the unit root null of industrial production and the unemployment rate are rejected in both LP and LM tests.

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To assess the significance of 𝑡_𝛼𝑖(𝜆_𝑖𝑛𝑓), the ZA break dates, the asymptotic estimated break point critical values are reported by ZA (1992, Tables 2, 3 and 4, pp.256 – 257). Allowing a one-time break fraction in the level of the trend function of the crash model and treating the break fraction as the outcome of the estimation procedure defined in Equation (4.4, 4.5 and 4.6), the unit root null is rejected for the total of 7 MEFT series in favour of the one-break alternative. All the series identified as one-point stationary are identified as unit root in the ADF stationarity test. 𝑀𝑜𝑑𝑒𝑙 𝐴^𝑍𝐴 tests the unit root null that an MEFT series has a unit root with a structural break in the intercept (also called sudden crash or a mean shift). The unit root null is rejected for LCPI (at 𝛼 =90%), LIBR (at 𝛼 = 99%), LMHE (at 𝛼 =90%), and LGDP (at 𝛼 = 90%). At 1 − 𝛼 =10% the structural change is considered to be a weak break point (ZA, 1998) for 𝐿𝐶𝑃𝐼, 𝐿𝑀𝐻𝐸 and 𝐿𝐺𝐷𝑃. This leads to the conclusion that the ZA approach rejects the unit root null for highly significant (1%) MEFT series based on Model 𝐶^𝑍𝐴 than Model 𝐴^𝑍𝐴 and 𝐵^𝑍𝐴. These are: 𝐿𝐼𝐵𝑅 (𝑀𝑜𝑑𝑒𝑙 − 𝐴^𝑍𝐴);

𝐿𝐿𝑇𝐼𝑅 (𝑀𝑜𝑑𝑒𝑙 𝐵^𝑍𝐴 & 𝐶^𝑍𝐴) and 𝐿𝐶𝑃𝐼, 𝐿𝐼𝐵𝑅, 𝐿𝑇𝐼𝑅, 𝐿𝐺𝐷𝑃 and 𝐿𝐺𝑁𝑃 (𝑀𝑜𝑑𝑒𝑙 − 𝐶^𝑍𝐴).

The unit root null is rejected for 𝐿𝐿𝑇𝐼𝑅 (at 𝛼 =99%), 𝐿𝐼𝐵𝑅 (at 𝛼 =95%), 𝐿𝐸𝑋𝑅 (at 𝛼 =90%), and 𝐿𝑀𝐻𝐸 (at 𝛼 =90%). 𝐿𝐿𝑇𝐼𝑅 with a break point in 1980𝑀07 and 𝐿𝐼𝐵𝑅 in 2006𝑀10 have highly significant break points with a level stationary process that allows for a one time break in the trend.

The break points are associated with the early 1980s recessions from 1980Q1 to 1981Q1. The main cause for this recession period was the action taken by the monetarist government to reduce inflation.

During this period, company earnings have declined by 35%, unemployment has risen by 125%

from 5.3% of the working population in August 1979 to 11.9% in 1984. The outcomes characterised by the fact that the long-term interest rate and the inter-bank rate categorised by gradual rate of changes during the early 1980s second oil price shock. The case with LIBR, the trend break analysis picks up the first signal when the gradual change began after continuous decline since early 1990s.

This gradual increase was due to banks reaction to the monetary policy shock, where the Bank of England begins to increase interest rate to stabilise the booming housing prices.

Studies show that 𝑀𝑜𝑑𝑒𝑙 𝐶^𝑍𝐴 is more representative than 𝑀𝑜𝑑𝑒𝑙 𝐴^𝑍𝐴 and 𝑀𝑜𝑑𝑒𝑙 𝐵^𝑍𝐴 in the absence of consistency. This is also shown in the results. When some discrepancies arise while using 𝑀𝑜𝑑𝑒𝑙 𝐴^𝑍𝐴 and 𝑀𝑜𝑑𝑒𝑙 𝐵^𝑍𝐴, 𝑀𝑜𝑑𝑒𝑙 𝐶^𝑍𝐴 is considered to be the most reliable approach.

𝑀𝑜𝑑𝑒𝑙 𝐶^𝑍𝐴 tests the presence of the contemporaneous sudden crashes and gradual changes of the financial and macroeconomic series simultaneously. The trend break dates in 1973/74 is associated with the first oil price shock, 1980/81 with the second oil price shocks and the early 1980 recession in the UK. Overall, the results show that all MEFT series have unit root (degree of persistence shocks) both in levels and in trend. The exceptions are the consumer price index 𝐿𝐶𝑃𝐼; the long- term interest rates 𝐿𝐿𝑇𝐼𝑅; the inter-bank rates 𝐿𝐼𝐵𝑅; gross national product 𝐿𝐺𝑁𝑃, and gross

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domestic product 𝐿𝐺𝐷𝑃. The results also show that the weak break point stationary series based on Model 𝐶^𝑍𝐴 is the exchange rate 𝐿𝐸𝑋𝑅 at 𝛼 =90%. These series have a simultaneous level and trend stationary process that allows a one-time break in both the levels and trend.

The results are consistent with NP (1982), ZA (1992) and Narayan and Smyth (2005). ZA rejects the unit root null for GNP in US data and NS rejects GNP in Australian data. The NS data spans from 1960 to 2004 so size of the sample period could be the main reason that they found no statistical significance to reject more series than the data used in this research. The longer the span of the period, the more MEFT series found to be stationary with, at least, a one-time break.

Furthermore, ZAs approach failed to reject the unit root null for 17 out of 22 (77%) of the MEFT series based on 𝑀𝑜𝑑𝑒𝑙 𝐶^𝑍𝐴. Generally, the 5 MEFT series that show a significant break-point stationary property also show high cluster of breaks mainly in 1974 (𝐿𝐺𝐷𝑃,𝐿𝐺𝑁𝑃, and 𝐿𝐶𝑃𝐼)⁶⁴ which corresponds to the world financial crisis that begins after OPEC quadruples the price of oil.

The long-term interest rate series shows a break point stationary with a break point in 1980s, which corresponds to the 1980s global credit crunch that prevents many developing countries from paying their debt due to the bond and equity market crashes. Finally yet importantly is the inter-bank rate (LIBR). This series has been volatile during the recent global financial crisis and post-crisis periods.

It shows a stationary property with a highly significant break in 2008. The period corresponds to the U.S. real estate crisis, which causes the collapse of massive international banks and financial institutions of many industrial countries, including the United Kingdom. This resulted in the collapse of the equity market that led to the perforation of the credit market. Recession in the UK lasted from 2008Q2 until 2009Q3 and gradually extended to 2012, which was the deepest since WW II. Consequently, manufacturing output declined by 7% by the end of 2008 that affected banks and investment firms with many established businesses. These businesses had no other options but to simply declare bankruptcy and ultimately collapsed. Subsequently, the central bank decided to cut the interest rate to historically lowest level of 0.5%. This significant and persistent shock is picked up by the one break ZA analysis for inter-bank rate (LIBR). This variable shows both sudden and simultaneous sudden and gradual changes, which combines the properties of additive outliers (AO) and innovation outliers (IO), respectively.

It is also important to note that four of the five significant break point stationary series belongs to the financial sector which tend to be affected by the volatile of oil price shock but the other two variables with significant break point belongs to the ME series. However, for the rest of the other variables, the ADF unit root test remains the same. Accordingly, the ZA one break test approach

64 The break dates are the combination of mean shift (crash) and changes in slop (growth – gradual change). It is also shown in the above table that some of the breaks were due to simultaneous changes in the form of crash and growth.

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seems to support the alternative hypothesis. At this stage, one can partially conclude that macroeconomic variables are more likely to revert to stable growth path with a minimum level of drift than the financial series. Therefore, it is important to make the necessary allowance by adding indicator variables to permit structural movements when specifying economic models.

As stated in the theoretical and empirical discussions, Model C of the ZA approach that signifies the sudden and gradual shifts of the UK MEFT series identifies the breaks better than the two other models (Model A and Model B). The empirical analysis for the one break investigation highlights that the ZA approach is able to identify the endogenously determined significant breaks for each series that correspond to the events in the UK economy. However, the weakness of this approach is its inability to identify more than one structural breaks. The ADF diagnostics reveal that the UK macroeconomic, financial and monetary sectors are characterised by nonstationarity so one can assume the presence of multiple structural breaks in the form of crashes and gradual changes. A well-recognised defect of the ADF and PP stationarity tests is the potential confusion of structural breaks in the series as evidences of nonstationarity. The results of the empirical analysis show that when one structural break is allowed into the ZA unit root test, the null hypothesis of unit root with a structural break is rejected at 1%, and 5% levels of significance for 5 MEFT series. Unlike the ZA one time break approach, all variables found to have a unit root, according to the ADF test, when a structural break is not introduced. Previous researches also show similar outcomes.

The ZA approach identifies only 20% of the data series as one break stationary, which implies that the power of the stationarity test improves by only 20% as compared to the ADF test without structural break. This leads to the assumption that the other 80% of the series contains no break but are nonstationary, implying that the economic time series are described as nonstationary process.

The estimation of such variables can lead to spurious regression and their economic interpretation will not be meaningful. When a series contains one unit root, the traditional practice in economics research is to transform the series by differencing or log differencing the variables before including them in the model. This incurs a loss of significant amount of important information. On the other hand, if nonstationarity exists among set of variables, regression involving the levels of the variables can proceed without generating spurious results. On this ground, the ZA one time break approach can be challenged because it is not able to detect more than one breaks in some economic time series. It is also important to note that the critical values of ADF-type endogenous tests are derived without assuming break(s) under the null hypothesis. The following graphical break point expositions show the break point stationarity properties identified by the ZA models (see Figure 4.1). The graphs are drawn in relation to Table 4.1. Except LMHE, all the other series show that they have simultaneous sudden and gradual shift differentials.

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Model A and C Model B and C

Model B and C Model A and C

Model C Model A, B and C

-6 -5 -4 -3 -2 -1 0

60 65 70 75 80 85 90 95 00 05 10

Zivot-Andrew Breakpoints (LGNP)

Source: author’s analysis

Figure 4.1 Persistent Shock Plots of Sig. Break Dates based on the ZA one Break Approach

It should be noted that ZA (1992) and Peron (1997) IO and AO approaches only capture one (the most significant) structural break in each MEFT series. The question is -what if there are multiple structural breaks in a series. The argument here is that disregarding the presence of additional shocks could lead to a further loss of information particularly when there is more than one structural break (LP, 1997) in the given series. On this same issue, Ben-David et al. (2003:p304) state,

“…just as failure to allow one break can cause non-rejection of the unit root null by the Augmented Dickey–Fuller test, failure to allow for two breaks, if they exist, can cause non-rejection of the unit root null by the tests which only incorporate one break…”.

-4.8 -4.4 -4.0 -3.6 -3.2 -2.8 -2.4

1980 1985 1990 1995 2000 2005 2010

Zivot-Andrew Breakpoints for LnIBR

Zivot-Andrew Breakpoints (LLTIR) Zivot-Andrew Breakpoints (LGDP)

Zivot-Andrew Breakpoints (LCPI) Zivot-Andrew Breakpoints (LEXR)

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However, detailed and convincing the ZA method appears to be, allowing only one break point is not enough to improve the power of the traditional ADF test. It is also possible that more than one structural breaks could exist in MEFT series (BP, 2003a). Furthermore, the Zivot and Andrews test exhibits size distortion in the presence of a break under the null hypothesis. This leads to the rejection of the null ambiguously. When utilizing the ADF and the ZA tests, researchers may conclude that a time series is stationary with break when in fact the serious is nonstationary with break. When the number of breaks increases, the spurious rejections might occur more often.

Additionally, the one and two break approaches tend to identify the break point prior to the true break (i.e., at 𝑇_𝐵−𝑡 rather than at 𝑇_𝐵). This problem occurs not only in the null but also in the alternative hypothesis (Nunes et.al., 1997; Volgelsang and Perron, 1998; LS, 2001). The BP multiple structural break programming algorithm does not suffer from size distortion and spurious rejection of the hypotheses. It is also able to identify more than one breaks using the global minimiser and sequential algorithms. Therefore, the study proposes the MSB algorithm as an alternative approach utilizing the theoretical and empirical expositions given by BP. Despite its sound empirical approach, there are few studies attempted to investigate MSBs for the UK data using the dynamic programming algorithm of the BP type. Against this backdrop, the following section expands the theoretical and empirical investigation to a multiple structural break.