Detecting structural change with heteroskedasticity

Communications in Statistics - Theory and Methods

ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20

Detecting structural change with heteroskedasticity Mumtaz Ahmed, Gulfam Haider & Asad Zaman To cite this article: Mumtaz Ahmed, Gulfam Haider & Asad Zaman (2016): Detecting structural change with heteroskedasticity, Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2016.1235200 To link to this article: http://dx.doi.org/10.1080/03610926.2016.1235200

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Date: 31 July 2017, At: 05:58

COMMUNICATIONS IN STATISTICS — THEORY AND METHODS , VOL. , NO. , – https://doi.org/./..

Detecting structural change with heteroskedasticity Mumtaz Ahmeda,b , Gulfam Haiderc , and Asad Zamand a Department of Economics, Cornell University, Ithaca, NY, USA; b Department of Management Sciences, COMSATS Institute of Information Technology, Islamabad, Pakistan; c Fast School of Management, FAST National University of Computer and Emerging Sciences, Chiniot Faisalabad Campus, Pakistan; d Pakistan Institute of Development Economics, Islamabad, Pakistan

ABSTRACT

ARTICLE HISTORY

The hypothesis of structural stability that the regression coefficients do not change over time is central to all applications of linear regression models. It is rather surprising that existing theory as well as practice focus on testing for structural change under homoskedasticity – that is, regression coefficients may change, but the variances remain the same. Since structural change can, and often does, involve changes in variances, this is a puzzling gap in the literature. Our main focus in this paper is to utilize a newly developed test (MZ) by Maasoumi et al. (2010) that tests simultaneously for break in regression coefficients as well as in variance. Currently, the sup F test is most widely used for structural change. This has certain optimality properties shown by Andrews (1993). However, this test assumes homoskedasticity across the structural change. We introduce the sup MZ test which caters to unknown breakpoints, and also compare it to the sup F. Our Monte Carlo results show that sup MZ test incurs only a low cost in case of homoskedasticity while having hugely better performance in case of heteroskedasticity. The simulation results are further supported by providing a real-world application. In real-world datasets, we find that structural change often involves heteroskedasticity. In such cases, the sup F test can fail to detect structural breaks and give misleading results, while the sup MZ test works well. We conclude that the sup MZ test is superior to current methodology for detecting structural change.

Received  June  Accepted  September  KEYWORDS

Bootstrap; heteroskedasticity; regime shift. JEL CLASSIFICATION

C; C; C

1. Introduction The hypothesis of structural stability is fundamental to the use of the linear regression model. Maasoumi et al. (2010) provide a brief review of the huge literature on testing for structural change. In the past two decades, attention has focused on detecting structural change while allowing for exceedingly complex error processes with heteroskedasticity and autocorrelation. There are two important gaps in this literature, which are addressed in this paper: (1) Allowing for extremely general heteroskedastic and autocorrelated error structures leads to substantial loss of power in finite samples. CONTACT Mumtaz Ahmed ma@cornell.edu; [email protected] Department of Economics, Cornell University, Ithaca -, NY, USA; Department of Management Sciences, COMSATS Institute of Information Technology, Park Road Chak Shehzad, Islamabad , Pakistan. ©  Taylor & Francis Group, LLC

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(2) Even complex error processes are assumed to remain the same across the breakpoint of structural change. Thus, these models do not embed even the simplest heteroskedastic model of a single break in variances in a standard linear regression model.1 Accordingly, we study a simple problem of central interest which has been largely ignored in the literature. This is to test for structural change when not only the regression coefficients but also the error variances change. It would seem entirely natural that structural change would involve changes in both, but most of the literature and applied work assumes constancy of variances across structural change. Maasoumi et al. (2010) developed a new test (MZ test) of the joint hypothesis that regression coefficients and error variances both change after a fixed and known breakpoint. This paper extends their results to the case of the unknown breakpoint. Our results are confined to the simplest error structure under classical assumption. Extensions to more complex cases should build on the intuitions provided by this basic case. Quandt (1960) first proposed the sup F test as the likelihood ratio test for detecting structural change with an unknown breakpoint. This became popular and widely used after Andrews (1993) proved certain optimality properties and also derived its asymptotic distribution. However, this test, and many others proposed in the literature,2 all assume homoskedasticity. To add to the problem, no test is available to detect unknown break in variance in the existing literature. Maasoumi et al. (2010) develop the MZ test for simultaneous change in regression coefficients and error variances at a fixed and known breakpoint. Based on the likelihood ratio principle, a natural extension of the MZ test to the case of the unknown breakpoint is the sup MZ test – this simply takes the maximum value of MZ assessed over all possible breakpoints. The sup MZ test can be used to test for single unknown simultaneous break in mean and variance. In the existing literature, there exists no other test which tests for break in mean and variance simultaneously. Thus, in this study, we compare the performance of sup MZ with the commonly used sup F test, which tests for break in mean only, to detect potential break points in the data. In particular, we evaluate and compare the size and power of both tests in the presence of heteroskedasticity by performing extensive Monte Carlo simulations. These show that the loss of power incurred by testing for structural change in variance is small, while the gain in power under heteroskedasticity is large. These simulation results are further supported by an empirical example, which shows that breakpoints in actual GDP series often have simultaneous changes in regression coefficients and variances. Accordingly, the sup MZ test has better empirical performance. The rest of the paper is organized as follows. The regression model and tests for parameter instability are provided in next section, while the Monte Carlo design and simulation results are presented in Section 3. An application using real-world data is provided in Section 4, and the last section concludes.

2. Regression model and tests of parameters instability The standard linear regression model (1) is used: yt = xt β + t ,  

t is i.i.d. N(0, σ 2 ),

t = 1, 2, 3 . . . T

(1)

See, for example, Kramer et al. (), Tran (), and Hayashi () among many others which do not cover the case we consider in this paper. See, for example, Brown et al. (), Bai (, a, b), Diebold and Chen (), Banerjee et al. (), Perron and Vogelsang ().

COMMUNICATIONS IN STATISTICS — THEORY AND METHODS

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The stability of all (k + 1) parameters (β, σ 2 ) is central to get valid inferences and better forecasts in empirical models. In the present paper, we consider the case of single unknown break that occurs at t = j, where, k < a ≤ j ≤ b < T − k. This means that we can split the data into two subgroups with T1 and T2 observations, respectively. Andrews (1993) found that structural change too close to the endpoints creates difficulties in obtaining powerful tests. Accordingly, “a” and “b” are time periods chosen so there are sufficient number of observations both before and after the breakpoint to allow for a reasonably powerful test. The model in Equation (1) can be written in matrix form as: Y = Xβ + ,  is i.i.d. N(0, σ 2 IT )

(2)

where Y is T × 1 vector of independent variable, X is T × k matrix of regressors, β is a k × 1 vector of regression coefficients, and  is a T × 1 vector of regression errors. Let βˆ0 = (X  X )−1 X Y and σˆ 02 = ||Y − X βˆ0 ||2 /(T − k) be the usual OLS estimates of the parameters β and σ 2 , respectively, under the null hypothesis of structural stability. Let (βi , σi2 ), i = 1, 2 represent the (k + 1) parameters in each subgroup. The regression model for each subgroup is   Y1 = X1 β1 + 1 , 1 is i.i.d. N 0, σ12 IT1 (3)   2 Y2 = X2 β2 + 2 , 2 is i.i.d. N 0, σ2 IT2 (4) Let βˆi = (Xi Xi )−1 XiYi , i = 1, 2, and σˆ i2 = ||Yi − Xi βˆi ||2 /(Ti − k), i = 1, 2, be the standard OLS estimates for the two models, before and after the structural break. Maasoumi et al. (2010) calculate a test of the null hypothesis of structural stability versus the alternative of simultaneous break in regression coefficients and variance: H0 : β1 = β2 , σ12 = σ22 versus H1 ; β1 = β2 or σ12 = σ22 The MZ statistics, defined below, modifies the LR statistic by adjusting for the degrees of freedom:   (5) MZ = (T − k) log σˆ 02 − (T1 − k) log σˆ 12 + (T2 − k) log σˆ 22 Maasoumi et al. (2010) showed that this adjustment improves the finite sample performance of the test. To adapt the test for use when the breakpoint is unknown, we propose the “sup” version of MZ and label it as sup MZ. When break point is unknown, we calculate MZ for all potential break points j, k < a ࣘ j ࣘ b < T – k and take the maximum value of these MZ j values. This maximum value is defined to be “sup MZ,” calculated as: sup MZ = max MZ j , k < a ≤ j ≤ b < T − k a≤ j≤b

(6)

Because this is asymptotically equivalent to the LR test for a single breakpoint, it shares all the asymptotic optimality properties of the LR test. This differs from the popular and widely used sup F test in that it allows for simultaneous change in both regression coefficients and the variance at the breakpoint. The sup F test assumes constancy of variances throughout (before and after structural change) and is optimal under this assumption. Interestingly, the analysis of Andrews (1993) applies without change to the asymptotic distribution of the sup MZ test, with the replacement of K by K + 1 (denoted as p and p + 1 in the Andrews’ paper). Theorem 3 of Andrews (1993) characterized this distribution as a maximum of a square of a standardized tied-down Bessel process of order p. Theorem 3 characterizes the distribution of the sup MZ statistic with the single change that the dimension of the number of constraints p must be increased by 1 to account for the additional constraint

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M. AHMED ET AL.

of equality of variances being tested. This is because the sup F tests the null hypothesis that the K regression parameters are the same across the breakpoint, while the MZ test adds the hypothesis that the error variance is the same across the regimes.

3. Monte Carlo design and simulation results In existing literature, there exists no other test which tests for break in mean and variance simultaneously. Thus, in this study, we compare the performance of sup MZ with sup F to detect the potential break point in the data. In particular, we evaluate and compare the size and power of both tests in the presence of heteroskedasticity by making use of simulations as well as empirically. We measure degree of heteroskedasticity (H) following Maasoumi et al. (2010) as:     (7) H = log W1 σ12 + W2 σ22 − W1 log σ12 + W2 log σ22 where W1 = T1 /T and W2 = T2 /T are the weights. In simulations, we vary the value of heteroskedasticity varying the value of the standard errors in the two regimes – before and after structural change. The distance (D) between coefficients (β1 and β2 ) of two subgroups is measured via a non centrality parameter, following Massoumi et al. (2010), defined as:  −1  −1 D = (β1 − β )t X1 X1 (8) (β1 − β ) + (β2 − β )t X2 X2 (β2 − β ) The critical values for both sup F and sup MZ tests3 are computed under the null hypothesis of no break in mean (D = 0) and variance (H = 0). More specifically, we consider the standard linear regression model, yt = α1 + α2 xt + t ,

t ∼ N(0, σ 2 ),

t = 1, 2, . . . , T

(9)

We generate regressor (x) as an evenly spaced series ranging from 1 to 50 with a step size of 0.5, (T = 100), which is kept fixed throughout the simulations. All tests under consideration are based on the OLS residuals, which have distribution independent of the regression coefficients. Thus, we set α1 = α2 = 1 and generate errors () from standard normal distribution. The dependent variable (y) is generated from Equation (9). These simulated data are used to get critical values which are obtained by performing 30,000 replications. For our simulation setup, the computed critical values for sup F and sup MZ at 5% significance level are 8.06 and 14.73, respectively, while the asymptotic critical value of sup F test at 5% significance level is 5.99. This means that sup F test rejects the null hypothesis when asymptotic critical values are used. Simulated critical values are essential to make correct finite sample inferences. The power of sup F and sup MZ is calculated by performing 10,000 Monte Carlo simulations under the alternative hypothesis of existence of break in at least one of mean and variance. The break is introduced by taking different combinations of D and H. In addition, we set break at different locations in the data by taking a specified proportion of observations (π j , j = 10%, 20%, 30%, 40%, 50%) before the break point. To compare the performance, we compute the percentage difference in powers (sup MZ minus sup F). The results with break at 30% of the data are presented in Table 1. 

We also compute powers of variants of sup F tests, the Avg F, Exp F proposed by Andrews and Ploberger () and their corresponding “MZ” versions, that is, Avg MZ and Exp MZ. The simulations results show that the performance of these alternatives is not good as compared with sup F and sup MZ tests. These results are not reported here to save space but are available from authors upon request.

D\ H

. . . . . . . . . . . . . .

.

− . − . − . − . − . − . − . − . − . − . − . − . − . − .

.

− . − . − . − . − . − . − . − . 0.33 0.33 0.93 2.77 2.57 2.27 .

− . − . − . 7.80 9.13 13.77 15.93 19.20 14.67 13.87 14.13 14.33 12.43 11.53 .

− . 14.53 19.83 28.03 31.83 34.70 36.90 37.27 34.43 34.53 32.50 31.17 28.13 26.67

Table . % Difference, sup MZ minus sup F with break at % of the data.

.

5.73 28.80 37.57 45.13 50.70 53.97 55.10 56.33 56.77 57.10 54.03 54.63 53.00 49.57 .

31.27 40.97 49.53 55.97 62.67 68.23 71.50 74.07 74.87 73.87 74.87 75.27 72.77 73.27 .

51.47 58.30 65.93 71.43 75.60 79.27 82.87 84.43 86.37 88.30 90.00 90.80 91.73 91.83

.

57.47 66.37 69.50 74.77 78.00 81.30 84.10 86.33 88.40 88.90 89.97 91.03 92.13 92.80

.

67.93 71.90 75.13 80.67 80.57 83.30 85.40 87.63 88.50 90.53 89.77 91.87 91.90 91.93

.

76.17 79.57 80.47 83.83 85.60 88.03 87.03 88.40 89.47 89.30 89.40 91.93 92.27 91.30

COMMUNICATIONS IN STATISTICS — THEORY AND METHODS 5

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M. AHMED ET AL.

In Table 1, an entry of 7.8 means that sup MZ has 7.8% greater power than the sup F test, while an entry of −9.99 indicates that sup MZ has 9.99% less power than sup F. Because the sup F test targets D and ignores H, it has higher power for higher values of D when H is close to 0. Since sup MZ tests for both D and H, it has high power when H is high, while being worse than sup F when H is low. The boldface numbers in Table 1 show the best performances of sup MZ against sup F, while the shaded cells indicate the region where sup F performs better. The sup MZ has a power loss of 30% in its worst-case scenario when H = 0, so that the assumptions of sup F hold. However, since sup F is not designed to detect changes in variance, it performs far worse in structural change scenarios involving both H and D. Similar results are obtained when break is introduced at other places like 10%, 20%, 40%, and 50% of the data. These results are not reported here to save space but are available from authors upon request.

4. An application In any concrete testing situation, choice of sup F or sup MZ is partly an empirical question. That is, is structural change in regression coefficients accompanied by changes in variances or not? Furthermore, our models are better adapted to shorter economic time series, where the simple models under discussions provide adequate fit.4 In order to assess empirical significance of our results, a standard Keynesian consumption function is estimated for several countries by using annual time series data taken from International Financial Statistics (IFS) database (May 2008) on household final consumption expenditure (C, measured in constant 2005 US dollar) and real GDP (Y, measured in constant 2005 US dollar). Only those countries are included in the analysis for which at least 30 years of data are available.5 Ct = a + bYt + t ,

t ∼ N(0, σ 2 ),

t = 1, 2, . . . , T

(10)

Economic theory tells us that these series should be co-integrated and hence superconsistency holds in this regression. This means that the misspecifications caused by ignoring short-run dynamics should not matter in large samples. We plan to test whether or not structural change occurs, and compare the relative efficacy of sup F and sup MZ at detecting structural change. In particular, we apply both sup F and sup MZ tests to detect the (possible) unknown break in the selected data for each country using bootstrap-based critical values.6 The results are summarized in Table 2, which contains country names along with the estimated value of non centrality parameter (D) and the degree of heteroskedasticity (H), under the assumption of a break at the most probable (estimated) location. In addition, it contains the break dates detected by sup F and sup MZ tests. We have also applied Goldfeld– Quandt (GQ) test (see Goldfeld and Quandt, 1965) to test the null of homoskedasticity at the point where sup MZ detects break. This is because GQ is an optimal test to detect break in variance (see Zaman, 1996, pp. 139–140). The outcome of GQ test is represented by either a Yes or a No, where, a Yes indicates that break in variance is detected by GQ test, while a No shows that GQ test is unable to detect break in variance. For ease in comparison, we split Table 2 into two panels. Panel A (on the left) contains the results where degree of heteroskedasticity (H) is either zero or low, while Panel B (on the right) summarizes the results for high values of H. For countries with low degree of heteroskedasticity, the sup F test  



The complex models with HAC error processes are based on asymptotic results more suitable for large time series. The selected countries are Australia, Belgium, Cameroon, Canada, China, Cyprus, Denmark, Egypt, Fiji, Finland, France, Germany, Greece, Hungary, Italy, India, Indonesia, Japan, Jordan, Kuwait, Korea, Libya, Malaysia, Mexico, New Zealand, Norway, Oman, Pakistan, Philippine, Qatar, Saudi Arabia, Singapore, Spain, Sri Lanka, Sweden, Switzerland, and the United States. Bootstrap-based critical values are obtained by performing , replications for each country.

        

    

    

Serial number

D

H

Sup F Sup MZ

Break year

(a) No break in mean and variance Australia   — — China   — — France   — — Singapore   — — US   — — (b) Break in mean only (same break year detected) Denmark .    India     Libya     Oman .    Qatar     (c) Break in both mean and variance (same break year detected) Jordan  .   Germany −. .   Kuwait  .   Sweden . .   Cameroon . .   Canada . .   Fiji . .   Belgium . .   Saudi Arabia . .  

Country

Panel A: Low H case

Yes Yes Yes Yes Yes Yes Yes Yes Yes

No No No No No

No No No No No

GQ test

Table . Performance of sup F and sup MZ in detecting unknown break in real data.

           

     

Serial number

D

H

Sup F

Sup MZ

Break year

(a) Break in both mean and variance (only sup MZ detects break) Egypt  . —  Italy . . —  New Zealand . . —  Sri Lanka  . —  Switzerland . . —  Pakistan . . —  (b) Break in both mean and variance (different break year detected) Cyprus . .   Finland . .   Greece . .   Hungary . .   Indonesia  .   Japan .E+ .   Korea  .   Malaysia  .   Mexico . .   Norway . .   Philippine . .   Spain . .  

Country

Panel B: High H case

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Yes Yes Yes Yes Yes Yes

GQ test

COMMUNICATIONS IN STATISTICS — THEORY AND METHODS 7

Sup F

           

Country

Cyprus Korea Mexico Finland Greece Norway Hungary Japan Spain Malaysia Indonesia Philippines

Break year

           

Sup MZ . . . . . . . . . . . .

H . . . . . . . . . . . .

Log(D) . . . . . . . . . . . .

Sup F

Power to detect change along with H and log(D)

Table . Power of sup F and sup MZ (break detected at different locations).

. . . . . . . . . . . .

H

. . . . . . . . . . − . .

Log(D)

. .  . . . . . . . . . .

Sup F

. . . . . . . . . . . .

Sup MZ

Power to detect change at correct change point along with H and log(D)

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should perform somewhat better. However, as shown in Table 2, several countries have high degree of heteroskedasticity following structural change. For these countries, we expect sup MZ test to perform substantially better than sup F. In Panel A of Table 2, we have the countries with low degree of heteroskedasticity (H), and as expected the two tests provide essentially equivalent results. Both tests detect structural change at the same breakpoint. In contrast, panel B contains those countries where degree of heteroskedasticity (H) is high. For countries at positions 20–25, sup F fails to detect break in contrast with sup MZ. This is due to large change in degree of heteroskedasticity (H) indicating that when there is break in variance, sup F may lead to wrong conclusion regarding the structural change. In addition, panel B contains some countries at positions 26–37, where break is detected by both sup F and sup MZ but at different locations, while GQ test detects break at the point where it is indicated by sup MZ. This suggests that if the degree of heteroskedasticity (H) is large, then sup F test will detect wrong break point even if the value of non centrality parameter D is high. To provide further support to our claim, we reconsider the countries at positions 26–37 in Panel B of Table 2, for which both tests detect break at different locations. We take the breakpoint suggested by sup MZ as an alternative hypothesis of structural change and calculate power of both sup F and sup MZ via bootstrapping. In particular, for each country, we split the data into two parts at the point where break is suggested by sup MZ. We estimate the consumption function in Equation (10) for each part separately. From the estimated parameters, we generate simulated consumption series (C) while keeping income series (Y) fixed. The power of sup F and sup MZ is calculated by performing 5000 bootstrap replications. In addition, we calculate power of sup F to detect change, that is, the power of sup F is calculated at the break point suggested by sup F. The power analysis is provided in Table 3. From the results in Table 3, we can see that power of sup F test is low against sup MZ; here, loss of power is due to failure to detect the correct breakpoint. These results reveal an interesting problem with the sup F test. If D is high, the sup F test will detect the change, but the change point detected will be wrong when H is also high. This is shown in Table 3, where the power of sup F to detect the correct breakpoint is almost zero even though D is large, while the sup MZ picks up both the existence of a structural break and its location correctly. There is an easy intuitive explanation of this finding. When there is high variance after the structural change, the sup F test is working with a wrong estimate of the variance on both sides of the change. This means that it can often misclassify the trend followed by observations on either side of the structural change, leading to loss of accuracy in detecting the point of change. This adds weight to our recommendation that we should use sup MZ in preference to sup F in empirical applications which roughly match the assumptions of our model.

5. Conclusion In this paper, we consider the case of structural change at a single unknown breakpoint. In particular, we propose a new test, the sup MZ, which simultaneously tests for the potential break in regression coefficients as well as in variance. A closely related test is the Andrews’ sup F test which tests for break in regression coefficients only under the assumption of no break in variance. These tests are not directly comparable, because they test for different types of structural change. We can compare the “regret” as follows. When the breakpoint occurs at the middle of the sample, the maximum gain of sup F over sup MZ is only 12.5% in the

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case of homoskedasticity, which is most favorable to the sup F test.7 But when we switch from homoskedastic case to heteroskedastic, the most favorable case of sup MZ, the maximum gain of sup MZ is 85%. Thus, sup MZ is favored from the point of view of minimax regret. Since it is partly an empirical question as to whether structural change occurs with homoskedasticity or heteroskedasticity, we apply both tests to detect break in real data. In particular, we estimated the consumption function for those countries for which data on consumption and real GDP is available for 30 years. In a large percentage of countries, structural change appears to occur with simultaneous change in regression coefficients and variances. In these cases, the sup F performed poorly, as in most of the cases it fails to detect break, while in some cases it detects break at wrong location due to presence of heteroskedasticity. Thus, our results strongly support the use of sup MZ test over the sup F to test for structural instability. There are obvious extensions to the case of multiple breakpoints which could be the target of future research.

References Andrews, D.W.K. (1993). Test for parameter instability and structural change with unknown change point. Econometrica. 61(4):821–856. Andrews, D.W.K, Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica. 62:1383–1414. Bai, J. (1994). Least square estimation of a shift in linear processes. J Time Ser Analys. 15(5):453–472. Bai, J. (1997a). Estimation of a change point in multiple regression models. Review Econ Stat. 79(4):551– 563. Bai, J. (1997b). Estimating multiple break one at a time. Econ Theo. 13(3):315–352. Banerjee, A., Lumsdaine, R.L., Stock, J.H. (1992). Recursive and sequential test of the unit-root and trend-break hypothesis: Theory and international evidence. J Business Econ Stat. 10(3):271–287. Brown, R.L., Durbin, J, Evans, J.M. (1975). Techniques for testing the constancy of regression relationships over time. J Royal Stat Soc. 37(2):149–192. Diebold, F.X., Chen, C. (1996). Testing structural stability with endogenous break point: A size comparison of analytic and bootstrap procedures. J Econom. 70:221–241. Goldfeld, S.M., Quandt, R.E. (1965). Some tests for heteroskedasticity. J Am Stat Assoc. 60:539–547. Hayashi, N. (2005). Structural changes and unit roots in Japan’s macroeconomic time series: Is real business cycle theory supported? Jpn World Econ. 17:239–259. International Financial Statistics (IFS) database. (May 2008). International Monetary Fund. Kramer, W., Ploberger, W., Alt, R. (1988). Testing for structural change in dynamic models. Econometrica. 56(6):1355–1369. Maasoumi, E., Zaman, A., Ahmed, M. (2010). Tests for structural change, aggregation, and homogeneity. Econ Model. 27(6):1382–1391. Perron, P., Vogelsang, T.J. (1992). Testing for unit root in a time series with a changing mean: Corrections and extensions. J Business Econ Stat. 10(4):467–470. Quandt, R. (1960). Tests of the hypothesis that a linear regression obeys two separate regimes. J Am Stat Assoc. 55(290):324–330. Tran, K.C. (1999). Testing for structural change in the dynamic adjustment model with autoregressive errors. Empirical Economics. 24:61–76. Zaman, A. (1996). Statistical Foundations for Econometric Techniques. San Diego, CA: Academic Press Inc.

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Table  presented earlier has breakpoint at % of the sample, where this minimax regret increases to %. This is about the maximum power loss possible from using sup MZ test in the worst case of zero heteroskedasticity over all the positions (%, %, %, %, and %) that we tested in our model.