Illustrating extraordinary shocks causing trend breaks

Economic Modelling 29 (2012) 1045–1052

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Illustrating extraordinary shocks causing trend breaks Kosei Fukuda ⁎ Faculty of Commerce, Chuo University, 742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan

a r t i c l e

i n f o

Article history: Accepted 25 March 2012 JEL classification: C22 E32 Keywords: Extraordinary shock Hodrick–Prescott filter Trend break

a b s t r a c t Structural breaks in a trending variable have been specified as changes in the drift parameter in the trend component, but extraordinary shocks causing these breaks have not been explicitly formulated. In this paper, the Hodrick–Prescott filter is extended by assuming two kinds of variance for the system noise driving the trend component: the larger one adopted in a point of time causing a trend break, and the smaller one adopted for remaining sequences. The number and location of structural breaks are determined by information criteria. In the proposed method, extraordinary shocks themselves can be illustrated. A Monte Carlo study shows the efficacy of the proposed model. Empirical results suggest that except for the UK, extraordinary shocks in quarterly time series of industrial production are detected for remaining six developed countries. Finally, it is shown that the proposed method considerably outperforms the other competing methods in correctly detecting business cycles. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Statistical detrending methods have been applied widely in empirical macroeconomic studies, even though they are based on no explicit theoretical foundation (see Canova, 1998, for a comprehensive discussion). The most popular method is even now the Hodrick and Prescott (1980, 1997; henceforth, HP) filter even though there are other newly developed filters (Baxter and King, 1999; Christiano and Fitzgerald, 2003) and HP involves the following drawbacks: spurious cycle (Cogley and Nason, 1995; Harvey and Jaeger, 1993; Pedersen, 2001); asymmetry of cycle (Hamilton, 1989; Pasaradakis and Sola, 2003); endpoint problem (Mise et al., 2005); and the invalidity of the smoothness parameter (Fukuda, 2010), among others. The purpose of this paper is to present a new detrending method allowing for extraordinary shocks causing trend breaks by directly extending the HP framework, and to provide empirical evidence on postwar output time series for the G7 countries. In the conventional HP filter, trend breaks cannot be considered. In actual economies, however, extraordinary shocks, such as the oil shock in October 1973, seem to have caused trend breaks in output time series. For example, Perron (1997) analyzed postwar quarterly time series of GDP for the G7 countries and concluded that a structural change occurred in a point of time between 1971 and 1976 (for a similar result, see Kim and Perron, 2009). Neglecting such a shock seems to have made the HP filter provide smoothly breaking trends, while a sudden break occurred in the actual economies.

⁎ Tel.: + 81 42 674 3644; fax: + 81 42 674 3651. E-mail address: [email protected]. 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2012.03.022

A novelty of the paper is to be able to illustrate time-series fluctuations of extraordinary shocks themselves. Hence, the timing and magnitude of these shocks can be visually compared with ordinary shocks. As discussed by Kapetanios and Tzavalis (2010), in the conventional time-series econometrics, structural breaks have been considered to be parameter changes, and the origins of trend breaks have not been explicitly formulated. 1 In the proposed method, the HP filter is first specified in the unobserved components (UC) model (see Gersch and Kitagawa (1983), Harvey (1985), and Watson (1986) for earlier works). In this specification, two kinds of the system noise sequence are explicitly formulated: one called permanent shock, driving the trend component of the output time series; and another called transitory shock, driving the cycle component. Next, two kinds of variance for the system noise driving the trend component are considered: the larger one is adopted for a particular point of time and causes a trend break; and the smaller one is adopted for the remaining sequences. In the proposed method, the number and location of structural breaks are determined by information criteria. This method has robustness to structural breaks which is another advantage to the HP and band-pass filters, because the latter are moving averaged methods influenced directly by structural breaks. On the other hand, this method can model structural breaks explicitly and remove their influence from observed time series.

1 For example, Kim et al. (2005) considered structural breaks in Markov switching models but did not explicitly formulate extraordinary shocks causing structural breaks. More recently, Perron and Wada (2009) introduced a parameter change in the trend component but did not explicitly formulate the innovation causing such a change.

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K. Fukuda / Economic Modelling 29 (2012) 1045–1052

The efficacy of the proposed method is examined by Monte Carlo simulations and empirical applications. First, typical data generating processes are assumed and artificial time series are generated by the assumed process. The proposed method is applied to artificial time series and the frequency count of selecting the correct model is examined. Next, the proposed method is applied to quarterly time series of industrial production from the G7 countries. The two-break model is selected for Japan, and the one-break model is selected for Canada, France, Germany, Italy, and the US. Finally, the no-break model is selected for the UK. Regarding the first break, the earliest break date of 1967q1 is detected in Italy, and the latest break date of 1974q3 is detected in France. Finally, the accuracy of detecting business cycles is examined. Based on official business cycle dates for each country, the accuracy is determined based on the frequency of incorrect detections of business cycles, such as over-detection and non-detection. The over-detection is defined as detecting a spurious business cycle which is not listed in official business cycle dates. The non-detection is defined as not detecting an official business cycle. Empirical examinations suggest that the proposed method considerably outperforms the other alternative methods, such as the HP filter, the Baxter and King (1999) filter, and the Christiano and Fitzgerald (2003) filter. The rest of the paper is organized as follows: in the next section, the new detrending method is presented. In Section 3, the efficacy of the proposed method is examined via Monte Carlo simulations. In Section 4, the proposed method is applied to quarterly time series of industrial production obtained from the G7 countries. Section 5 concludes. 2. Method 2.1. Modeling As shown by Harvey and Jaeger (1993) and King and Rebelo (1993), the HP filter for postwar quarterly output time series yn(n = 1, …, N) can be specified in the UC model:

yn ¼ T n þ C n ; T n ¼ 2T n−1 −T n−2 þ un ; C n ¼ vn ;

where un and vn are Gaussian white noise sequences with variance σu2 and σv2, respectively. When the smoothness parameter (λ) in the HP filter is set to 1600 for the quarterly time series, it is obtained that λ = σv2/σu2 = 1600. This model can be represented by the state space form, and estimation can be easily done using the Kalman filter. In the proposed method, the smoothness parameter is estimated from data, based on the maximum likelihood principle. Furthermore, it is unnatural to consider that the cyclical component should be specified as the white noise sequence. In the present paper, the cyclical component is specified as the second-order autoregressive process,

C n ¼ ϕ1 C n−1 þ ϕ2 C n−2 þ vn ;

where all the roots of the equation 1 − ϕ1z − ϕ2z 2 = 0 lie outside the unit circle. This specification is very simple but has been widely applied to the UC model (see Morley et al., 2003, for an influential and recent example). Furthermore, the covariance (σuv) for un and vn is considered.

Finally, in addition to un, another disturbance term (wn) with 2 some larger variance (σw ) is introduced to represent the magnitude of an extraordinary shock in the trend component.

T n ¼ 2T n−1 −T n−2 þ un ; if n≠nB : T n ¼ 2T n−1 −T n−2 þ wn ; if n ¼ nB :

In the case of an extraordinary negative shock in n = nB, for example, the forecasted value of TnB, given the information obtained up to nB − 1, is decreased by wnB(b 0) T^ nB ¼ 2T^ nB −1 −T^ nB −2 þ wnB :

After the break (n > nB), the trend in movement of Tn returns to that which was true before the extraordinary shock occurred. Given the break point (nB), the maximum likelihood estimation can be carried out for this model (see Hamilton, 1994; Harvey, 1989), and the corresponding information criterion is stored. The information criterion (IC) is obtained as IC ¼ −2ðmaximum log likelihoodÞ þ mp; where m is the number of parameters and p is the penalty term. In 2 this model, parameters are σu2, σv2, σuv, σw , ϕ1 and ϕ2. The penalty term is different among information criteria. That is, p = 2 (Akaike's information criterion, AIC); p = 2 log(log(N)) (Hannan–Quinn criterion, HQC); p = log(N) (Schwartz information criterion, SIC). 2.2. Search of break points In fact, the number and location of break points are unknown, and therefore a search procedure is adopted in this paper. First, the maximum number of break points (Bmax) is considered. Since many break points need huge computational time, a reasonable number is set. In the Monte Carlo study, it is assumed that Bmax = 2, because the true number of break points is assumed to be 0 or 1 in the simulation. In the empirical application, it is assumed that Bmax = 3, because the postwar time series of industrial production is analyzed. Given the value of Bmax, firstly the no-break model is estimated and the corresponding information criterion is stored. Next, the onebreak model is considered. Since the break point is unknown, all possible break points are considered under the following condition. L≤n1 ≤N−L; where L is assumed that L = 0.15N, as suggested by Bai and Perron (2003). Given the break point, the model is estimated and the corresponding information criterion is stored. In the class of the onebreak model, the best model is selected from all possible one-break models, based on the information criterion. The same thing is repeated for other model classes with more than one breaks, and the best model is selected in each model class. Then, the following additional condition is considered. niþ1 −ni ≥L; where ni is the ith break point. Finally, given the value of Bmax, the overall best model is selected. Hence, the number and location of structural breaks are determined via information criterion.

K. Fukuda / Economic Modelling 29 (2012) 1045–1052

1047

1200

20

US (+700)

1100 15

UK(+600) 1000 10 Japan (+500) 900 5 Italy (+400)

0

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193

800

DGP1

DGP2

DGP3

DGP4

Germany (+300) 700

DGP5

Fig. 1. Artificial time series generated from 5 GDPs. France (+200) 600

3. A Monte Carlo study 500

In this section, the efficacy of the proposed method is examined via Monte Carlo simulations. In order to compare the performances of the three information criteria and to investigate the usefulness in trend-cycle decomposition of macroeconomic time series, the following data-generating process (DGP) is considered.

Regarding the trend component Tn, the following five DGPs are considered. In all DGPs, it is assumed that T− 1 = 0.0 and T0 = 0.1. DGP1 (no-break): Tn = 2Tn − 1 − Tn − 2 + un, un ~ N(0, 10 − 6) DGP2 (early and moderate break): DGP1 if n ≠ N/4, and Tn = 3/2Tn − 1 − Tn − 2/2 if n = N/4. DGP3 (late and moderate break): DGP1 if n ≠ N/2, and Tn = 3/2Tn − 1 − Tn − 2/2 if n = N/2.

AIC

DGP1 DGP2 DGP3 DGP4 DGP5 DGP1 DGP2 DGP3 DGP4 DGP5 DGP1 DGP2 DGP3 DGP4 DGP5

HQC

BIC

Fig. 2. Quarterly time series of industrial production for the G7 countries.

DGP4 (early and drastic break): DGP1 if n ≠ N/4, and Tn = Tn − 1 if n = N/4. DGP5 (late and drastic break): DGP1 if n ≠ N/2, and Tn = Tn − 1 if n = N/2. Regarding the cyclical component, the following DGP is assumed considering the cyclical periodicity of typical business cycles in actual economies:

Table 1 Frequency count of the selected model class and RMSE. DGP

400

300 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

yn ¼ T n þ C n ; n ¼ 1; …; N:

Information criterion

Canada (+100)

Model class

RMSE

No-break model

One-break model

Two-break model

0.93 0.39 0.25 0.05 0.01 0.98 0.70 0.57 0.21 0.08 1.00 0.93 0.90 0.61 0.23

0.07 0.57 0.72 0.93 0.91 0.02 0.30 0.43 0.78 0.91 0.00 0.07 0.10 0.39 0.77

0.00 0.04 0.03 0.02 0.08 0.00 0.00 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00

– 17 12 14 10 – 10 12 14 10 – 9 11 11 11

Note: Bold-face letter indicates the frequency count of selecting the correct model class.

C n ¼ 1:6C n−1 −0:7C n−2 þ vn ; vn ∼Nð0; 0:04Þ For example, in the Baxter and King (1999) filter, it is assumed that the length of a business cycle is from 6 quarters to 32 quarters. In this simulation, the assumed length is set as 20 quarters. Furthermore, for simplicity, it is assumed that the covariance between un and vn is zero. Considering empirical studies, the sample size is set as N = 200. Fig. 1 shows artificial time series generated from the above five DGPs. It seems difficult to select the correct model in the case of DGP2 or DGP3. The simulation is implemented as follows. First, one DGP is selected from the five alternatives. Next the disturbance term is generated, substituted into the selected DGP, and artificial time series is obtained. Three alternative model classes — no-break model, onebreak model, and two-break model — are applied to this obtained time series. As discussed in Section 2.1, for example, in the case of 2 one-break model, free parameters to be estimated are σu2, σv2, σuv, σw ,

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K. Fukuda / Economic Modelling 29 (2012) 1045–1052

Table 2 Estimation results for G7 countries. Country

AIC

Parameter estimates for the selected model

No break

One break

Two breaks

Three breaks

σ^ 2u

σ^ 2v

σ^ 2w1

Canada

735.1

733.7

735.7

France

946.2

941.2

943.2

Germany

789.6

788.0

790.0

Italy

932.7

931.5

933.5

Japan

751.2 b1990q3 > 790.1

751.4

US

747.7

748.6

750.6

0.00001 [0.00001] 0.0001 [0.0005] 0.0001 [0.0007] 0.0053 [0.0030] 0.0003 [0.0001] 0.0010 [0.0001] 0.00001 [0.00006]

2.1141 [0.2219] 5.5601 [0.5595] 2.7077 [0.2754] 5.2418 [0.5309] 2.0779 [0.1590] 2.7149 [0.2363] 2.1690 [0.2175]

0.4920 [0.7226] 1.1750 [1.6791] 0.9624 [1.3963] 0.6863 [1.1155] 5.2048 [1.5078]

UK

761.1 b1971q2 > 788.0

732.9 b1973q2 > 940.3 b1974q3 > 786.9 b1970q3 > 929.1 b1967q1 > 754.5 789.4 746.7 b1969q4 >

792.1

σ^ 2w2

0.7138 [0.1602]

0.2119 [0.3171]

σ^ uv

^ ϕ 1

^ ϕ 2

0.0006 [0.0416] − 0.0094 [0.0444] 0.0140 [0.0700] 0.1662 [0.0580] 0.0239 [0.2897] 0.0511 [0.1739] 0.0025 [0.0252]

1.3492 [0.0244] 0.7206 [0.0515] 1.1326 [0.0275] 0.9101 [0.0397] 1.5444 [0.0195] 1.0412 [0.0183] 1.4242 [0.0217]

− 0.3928 [0.0242] 0.1288 [0.0510] − 0.1898 [0.0280] 0.0028 [0.0396] − 0.6434 [0.0193] − 0.0810 [0.0184] − 0.4843 [0.0216]

Note: The figure in brackets presents standard error. The boldface letter indicates the AIC value of the selected model.

ϕ1 and ϕ2. Finally, the best model is selected via each IC. The number of replications is 1000. The frequency count of selecting the correct model class is computed and the root mean square error (RMSE) is calculated for the selected correct model class.

Table 1 compares the frequency count of selecting the correct model class among the three information criteria and the RMSE. First consider simulation results obtained by the AIC. In the case of DGP1, the frequency count of selecting the correct model class (no-break

(2) France

(1) Canada 470

Original

Original

Trend

Trend

450 420

370 400 320

270 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

20

Cycle

350 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

10

10

0

Cycle

0

-10

-10

-20

-20 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

-30 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Growth

Growth 1.5

1.5

1

1

0.5

0.5

0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.2

Shock

0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.5

0

Shock

0

-0.2 -0.5 -0.4 -1

-0.6 -0.8 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

-1.5 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Fig. 3. Trend-cycle decomposition of quarterly time series of IIP.

K. Fukuda / Economic Modelling 29 (2012) 1045–1052

(3) Germany 480

Original

1049

(4) Italy

Trend

Original

Trend

450 430 400 380

350

330 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

15

Cycle

300 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

10

10

Cycle

5

5

0

0 -5

-5 -10

-10

-15 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

-15 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Growth

Growth 1.5

2.5 2

1

1.5 1

0.5

0.5 0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Shock

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.5

Shock

0 -0.5 -1 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Fig. 3 (continued).

model) is 0.93. In the cases of DGPs 2 and 3 (moderate break), this frequency is 0.57 and 0.72, respectively. Finally, in the cases of DGPs 4 and 5 (drastic break), this frequency is 0.93 and 0.91, respectively. The performances of the correct model selection using the AIC are reasonably good. The values of the RMSE range from 10 to 17 and are reasonably small. On the other hand, performances presented by the HQC and the BIC are very poor, particularly in the cases of DGPs 2 and 3 (moderate break). As discussed above, the penalty is too large to correctly detect the one-break model. Since performances presented by the HQC and the BIC are worse than those presented by the AIC, the AIC is used in the empirical studies. 4. Empirical results In the present paper, the quarterly time series of index of industrial production (IIP) is used rather than GDP for the following two reasons. First, a long time series can be obtained2. Second, IIP is one of the most important business cycle indicators and is widely used. The data set was obtained from the International Financial Statistics for the G7 countries. The data are seasonally adjusted, indexed as 2005 = 100, log transformed, and multiplied by 100. The sample period is 1957q1-2007q43.

2 For example, quarterly time series data on Japan's GDP cannot be obtained for the periods before 1980, since the version of the national account was discontinued. 3 Although more recent data are available, these data was neglected because a drastic financial shock, named the Lehman shock, was detected in September 2008.

Fig. 2 plots time series for the G7 countries. For the purpose of clear presentation, transformed data are added by 100 for each country. From a visual inspection of these graphs, the following three findings are obtained. First, with reference to all countries of the G7, it cannot be concluded that the oil shock of 1973 caused no trend break. Second, the turmoil in exchange rate system, such as the Nixon shock of 1971, seems to have caused a trend break in Germany, Italy, Japan, and the US. Finally, except for Japan, there seems to be no need to consider the second break point. Table 2 presents estimation results for the G7 countries. On the left side of this table, the AIC values of the four models—the no-break model, the one-break, the two-break model, and the three-break model—are presented. The boldface letter indicates the AIC value of the selected model. Except for Japan and the UK, the one-break model was selected using the minimum AIC procedure. The earliest break date of this model is 1967q1 in Italy, and the latest break date is 1974q3 in France. In Japan, the second break occurred in 1990q3 when the burst of the financial bubble (a sudden and drastic decrease in stock price) considerably decreased economic activity. Fig. 3 depicts trend-cycle decomposition of quarterly time series of the IIP for the G7 countries. In Japan, for example, growth of the trend component (trend growth) had decreased from 3.2% to 0.9%. The Nixon shock of 1971 decreased the trend growth at 2.3% points. The magnitude of this extraordinary shock is clearly shown in a timeseries plot of the system noise sequence (un). Furthermore, the trend growth had decreased since the bursting of the financial bubble of 1990, and has been stable at about 0%.

1050

K. Fukuda / Economic Modelling 29 (2012) 1045–1052

(6) UK

(5) Japan Original

Trend

480

Original

Trend

450 400 350

430

300 250 200 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Cycle

15 10 5 0 -5 -10 -15 -20 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

380 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

10 5 0 -5 -10

-15 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Growth

Growth 4

0.8

3

0.6

2

0.4

1

0.2

0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.5

Cycle

Shock

0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.2

0

Shock

0.1

-0.5

0

-1 -1.5

-0.1

-2 -2.5 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

-0.2 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Fig. 3 (continued).

These graphs indicate the following two advantages of the proposed method. First, in conventional time series models, trend breaks have been formulated as parameter changes, but extraordinary shocks causing trend breaks have not been explicitly specified. In the proposed method, these extraordinary shocks can be illustrated and the magnitude of these shocks can be compared with that of ordinary shocks. Second, trend growth obtained by the proposed method seems to be more convincing than that obtained by conventional methods. In unit-root econometrics, the trend component of the output time series is specified as time trend with breaks or as I(1) trend with breaks. The trend growth is presented as a straight line in the case of time trend or as a too volatile line in the case of I(1) trend. In the proposed method, trend growth is not fixed but changing smoothly, except in structural breaks. Finally, the accuracy of detecting business cycles is examined. Four trend-cycle decomposition methods are considered: the HP filter, the Baxter and King (1999) filter, the Christiano and Fitzgerald (2003) filter, and the proposed method. The Harding and Pagan (2002) dating method, originated by Bry and Boschan (1971), is applied to the cyclical component obtained by alternative detrending methods4. In the present paper, the accuracy is determined based on the frequency of

4

All computation programs in these exercises are obtained from the following Stata commands: bking (Baxter-King), cfitzrw (Christiano-Fitzgerald), and sbbq (Herding and Pagan).

incorrect detections of business cycles. Official business cycle dates for each country are obtained from the website of the government or nonprofit organization5. Incorrect detections are separated into the following two categories: over-detection and non-detection. The overdetection is defined as detecting a spurious business cycle which is not listed in official business cycle dates. The non-detection is defined as not detecting an official business cycle. Table 3 indicates the frequency count of incorrect detections for four alternative detrending methods and for individual countries. This table clearly shows that the proposed method outperforms the other methods. 5. Conclusion In the conventional analysis of structural breaks in a trending variable, changes in the drift parameter in the trend component have been considered, but extraordinary shocks causing trend breaks have not been explicitly formulated. In this paper, these shocks are explicitly formulated and illustrated with ordinary shocks. The proposed method is obtained by extending the HP filter and by assuming two kinds of variance for the system noise driving the trend component: the larger 5 Business cycle dates for each country are obtained from the following website: the National Bureau of Economic Research for the US (http://www.nber.org/), the Cabinet Office for Japan (http://www.esri.cao.go.jp/), the Organization of Economic Corporation and Development for remaining countries (http://www.oecd.org/).

K. Fukuda / Economic Modelling 29 (2012) 1045–1052 Table 3 Frequency count of incorrect detections.

(7) US 470

Original

1051

Trend

420

370

320

270 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Country

Canada France Germany Italy Japan UK US Average

Four alternative methods HP

BK

CF

Proposed

7 5 4 6 3 5 10 5.7

6 4 5 6 4 9 8 6.0

7 5 3 8 4 5 9 5.9

7 3 3 4 2 3 6 4.0

Note: HP: Hodrick–Prescott; BK: Baxter–King; CF: Christ iano–Fitzgerald.

Cycle

15 10 5 0 -5 -10 -15 -20 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Growth 1.2 1 0.8 0.6 0.4 0.2 0 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

0.1

Shock

developed. Finally, considering the standard error of the variance of the stochastic trend, the null hypothesis of zero variance cannot be rejected for four countries (Canada, France, Germany, and the US). For these countries, a deterministic or stationary trend model with structural breaks should be considered. However, this leads to another problem of selecting between the stationary trend model and the stochastic trend model integrated of order two, allowing for unknown structural breaks in both cases, which has not been resolved via an asymptotic theory. This point is left for future research. Acknowledgment I am grateful to an anonymous reviewer for very useful comments and suggestions.

0 -0.1

References

-0.2 -0.3 -0.4 -0.5 1957q1 1962q1 1967q1 1972q1 1977q1 1982q1 1987q1 1992q1 1997q1 2002q1 2007q1

Fig. 3 (continued).

one adopted at a point of time causing a trend break, and the smaller one adopted for remaining sequences. In the proposed method, the number and location of structural breaks are determined via the information criterion. The results by Monte Carlo simulations suggested that the AIC selects the correct model class more often than the other information criteria, and that the estimates of break points are reasonably well. The proposed method was applied to postwar quarterly time series of industrial production for the G7 countries. Comparing the AIC values, the best model is selected from the four alternatives. The no-break model was selected for the UK, the two-break model was selected for Japan, and the one-break model was selected for the remaining five countries. The first break date ranges from 1967q1 (Italy) to 1974q3 (France). These dates are between the turmoil in exchange rate system and the oil shock. In Japan, the second break was detected in 1990q3 when the burst of the financial bubble occurred. However, there are some limitations in the present paper 6. First, compared to the other filters considered here, the proposed method is very time consuming. Hence, Monte Carlo simulations were not very comprehensive, and the possibility of data snooping cannot be rejected. Second, there is no asymptotic theory to support the efficacy of the proposed method. Regarding time series with second-order integration and with structural breaks, however, the usual asymptotic theory cannot be applied. Hence, a new asymptotic theory should be

6

Suggestions indicated by an anonymous reviewer are gratefully acknowledged.

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