Growth and volatility regime switching models for New Zealand GDP data
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Buckle, Robert A; Haugh, David; Thomson, Peter Working Paper Growth and volatility regime switching models for New Zealand GDP data New Zealand Treasury Working Paper, No. 02/08 Provided in Cooperation with: The Treasury, New Zealand Government Suggested Citation: Buckle, Robert A; Haugh, David; Thomson, Peter (2002) : Growth and volatility regime switching models for New Zealand GDP data, New Zealand Treasury Working Paper, No. 02/08, New Zealand Government, The Treasury, Wellington This Version is available at: https://hdl.handle.net/10419/205483 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Growth and volatility regime switching models for New Zealand GDP data Robert A Buckle, David Haugh and Peter Thomson N EW Z EALAND T REASURY W ORKING P APER 02/08 J UNE 2002
NZ TREASURY WORKING PAPER 02/08 Growth and volatility regime switching models for New Zealand GDP data MONTH / YEAR June 2002 AUTHORS Robert A Buckle The Treasury, PO Box 3724, Wellington, New Zealand Email Telephone Fax [email protected] 64-4-471-5252 64-4-499-0992 David Haugh The Treasury, PO Box 3724, Wellington, New Zealand Email Telephone Fax [email protected] 64-4-471-5252 64-4-499-0992 Peter Thomson Statistics Research Associates Ltd, PO Box 12 649, Thorndon, Wellington, New Zealand Email Telephone Fax [email protected] 64-4-934-4753 64-4-475-4206 ACKNOWLEDGEMENTS The authors are grateful for comments and suggestions received from participants at the New Zealand Econometrics Study Group meeting held at the University of Auckland in March 2002, the 2nd International Conference on Financial Engineering and Statistical Finance held at Hitotsubashi University, Tokyo, in March 2002, the Australasian Macroeconomics Workshop held in Wellington in April 2002, the NZ Association of Economists’ Conference held in Wellington in June 2002 and the Econometric Society Annual Meetings held at Queensland University of Technology, Brisbane, in July 2002. Particular thanks are due to participants at a New Zealand Treasury seminar held in January 2002 and to Florin Citu, Iris Claus, John Creedy, Viv Hall, Inchi Hu, Adrian Pagan, Peter Phillips and Chris Plantier for specific comments and suggestions on earlier drafts of this paper. NZ TREASURY New Zealand Treasury PO Box 3724 Wellington 6008 NEW ZEALAND Email Telephone Website [email protected] 64-4-472 2733 www.treasury.govt.nz DISCLAIMER The views expressed in this Working Paper are those of the author(s) and do not necessarily reflect the views of the New Zealand Treasury. The paper is presented not as policy, but with a view to inform and stimulate wider debate.
W P 0 2 / 0 8 | G r o w t h a n d v o l a t i l i t y r e g i m e s w i t c h i n g m o d e l s f o r N e w Z e a l a n d G D P d a t a Abstract This paper fits hidden Markov switching models to New Zealand GDP data. A primary objective is to better understand the utility of these methods for modelling growth and volatility regimes present in the New Zealand data and their interaction. Properties of the models are developed together with a description of the estimation methods, including use of the Expectation Maximisation (EM) algorithm. The models are fitted to New Zealand GDP and production sector growth rates to analyse changes in their mean and volatility over time. The paper discusses applications of the methodology to identifying changes in growth performances, and examines the timing of growth and volatility regime switching between production sectors. Conclusions to emerge are that, in contrast to the 1980s, New Zealand GDP growth experienced an unusually long period of time in high growth and low volatility regimes during the 1990s. The paper evaluates sector contributions to this 1990s experience and discusses directions for further development. J E L C L A S S I F I C A T I O N C22 Time series models E23 Production E32 Business fluctuations, cycles O47 Measurement of economic growth K E Y W O R D S Hidden Markov models; regime switching; growth; business cycles; volatility; production sectors; GDP.
Contents 1 Introduction 1 2 Specifying the Markov regime switching model 2 2.1 The McConnell and Perez–Quiros (MPQ) model . . . . . . . . . . . . . . . . 5 2.2 The proposed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Examples ..................................... 8 3 Fitting the model 9 4 HMM models for New Zealand GDP growth 13 4.1 Aggregate real GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Services sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Government and Community Services sector . . . . . . . . . . . . . . . . . . 21 4.4 Manufacturing sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 Primary sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.6 Construction sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Growth and volatility regimes across sectors 32 6 Conclusions 36 Appendix 38 A Mean and autocovariance structure 38 B Recursions and likelihood 40 C Additional figures 42 References 48
List of Tables 1 1–1 mapping of the state labels for to those for and . .......... 5 2 Summary of fitting procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Industry composition of the five production sectors. . . . . . . . . . . . . . . . 13 4 Parameter estimates for the HMM models fitted to GDP and sector growth rates. 14 5 Dates when GDP and selected production sectors were in the high growth regime. 32 List of Figures 1 Simulated quarterly GDP and its growth rates. . . . . . . . . . . . . . . . . . . 4 2 Simulated US growth rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 The Hamilton model fitted to quarterly GDP growth rates. . . . . . . . . . . . 16 4 A 3–2 model fitted to quarterly GDP growth rates: growth regimes. . . . . . . . 17 5 A 3–2 model fitted to quarterly GDP growth rates: volatility regimes. . . . . . . 18 6 The MPQ model fitted to quarterly Services growth rates: growth regimes. . . . 20 7 The MPQ model fitted to quarterly Services growth rates: volatility regimes. . . 21 8 A 2–2 model fitted to quarterly Government and Community Services growth rates: growth regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 9 A 2-2 model fitted to quarterly Government and Community Services growth rates: volatility regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 10 The Hamilton model fitted to quarterly Manufacturing growth rates. . . . . . . 24 11 The MPQ model fitted to quarterly Manufacturing growth rates: growth regimes. 25 12 TheMPQmodelfittedto quarterly Manufacturinggrowthrates: volatilityregimes. 26 13 A 3–1 model fitted to quarterly Manufacturing growth rates. . . . . . . . . . . 27 14 A 3–2 model fitted to quarterly Primary growth rates: growth regimes. . . . . . 29 15 A 3–2 model fitted to quarterly Primary growth rates: volatility regimes. . . . . 30 16 A 3–1 model fitted to quarterly Construction growth rates. . . . . . . . . . . . 31 17 Probability of GDP and its production sectors being in a high growth regime. . 33 18 Probability of GDP and its production sectors being in a high volatility regime. 35 19 Plots of the for a 3–2 model fitted to quarterly GDP growth rates. 42 20 Plots of the for an MPQ model fitted to quarterly Services growth rates......................................... 43
21 Plots of the for a 2–2 model fitted to quarterly Government and Community Services growth rates. . . . . . . . . . . . . . . . . . . . . . . . . 44 22 Plots of the for an MPQ model fitted to quarterly Manufacturing growth rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 23 Plots of the for a 3–1 model fitted to quarterly Manufacturing growth rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 24 Plots of the for a 3–2 model fitted to quarterly Primary growth rates. 47
Growth and volatility regime switching models for New Zealand GDP data 1 Introduction Interpretation of New Zealand’s trend economic growth during the 1990s has been a central issue in recent debate concerning New Zealand’s growth potential, its growth performance relative to that achieved in other developed economies and debate surrounding the impact of economic reforms. One of the difficulties is deciding on the interpretation that should be placed on a series of observed higher or lower growth rates. When should such a sequence be interpreted as a change in the mean growth rate or a change in volatility? One purpose of this study is to obtain more timely and sensitive measures of changes in New Zealand’s economic growth performance and to develop methods for the identification of shifts in growth and volatility regimes. If successful, this will enhance interpretation of current data and policy analysis. These are important objectives given the data limitations that confront researchers measuring real economic growth in New Zealand and the relatively volatile nature of these data in comparison with those for large–scale developed economies such as the United States, Japan, and the larger European economies. Our focus is on quarterly growth rates of New Zealand GDP and those of its production sectors. Growth rates are defined as the first differences of the logarithms of quarterly GDP and are assumed to be stationary. Although linear stationary time series models are commonly fitted to both growth rates and the original GDP data itself, we adopt a simple non–linear stationary model to explain the salient features of the growth rates. There is considerable evidence to suggest that departures from linearity are an important feature of many key macroeconomic series. International evidence includes the documentation of asymmetries in the phases of business cycles by Neftci (1984), Burgess (1992) and Sichel (1993), although no such evidence was found for New Zealand by Giles (1997). A growing body of complementary research shows that real output responds asymmetrically to nominal demand shocks (Cover, 1988; de Long and Summers, 1988; Morgan, 1993; Karras, 1996) and that inflation can induce an asymmetric real output response to changes in demand (for US evidence see Rhee and Rich, 1995; for Australian evidence see Olekalns, 1995). New Zealand evidence includes papers by Buckle and Carlson (1998, 2000) who focus on the impact of cost and demand shocks on the manufacturing and wholesale sector, and Wallace and Evans (1985) who examine the impact of climate on farm production and profit. Such findings have prompted the development of time series models for GDP that assume that the growth rates follow a non–linear stationary process. An important development in this regard is theHamilton (1989)model of the US businesscycle. Hamilton assumesUS GNP growth switches between a finite number of regimes that are discrete episodes of time over which the dynamic behaviour of the series is markedly different. His approach is to use the Goldfeld and Quandt (1973) Markov switching regression to characterise changes in the parameters of an autoregressive process. The economy may be in a fast growth or slow growth phase with the switch between the two governed by the outcome of a Markov process. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 1
Since Hamilton’s model of the US business cycle, the Markov switching autoregressive model has become increasingly popular for the empirical characterisation of macroeconomic series. While not without its critics (see Harding and Pagan, 2001), several researchers have found this framework to be a useful approach for characterising business cycles including, for the US business cycle, Lam (1990), Boldin (1994), Durland and McCurdy (1994), Filardo (1994), Diebold and Rudebusch (1996), Kim (1994) while Krolzig (1997) has also found it a useful tool for investigating the business cycles of Australia, Canada, France, Germany, Japan and the United Kingdom. Regime switching models such as these have also been heavily used in many other disciplines including finance (Hamilton and Susmel, 1994), meteorology (Zucchini and Guttorp, 1991) and speech recognition (Rabiner, 1989) to name but a few. This paper develops and estimates Markov regime switching models for New Zealand real GDP growth and the growth of its component production sectors. The aim is to better understand how these models can be used to identify changes in growth and volatility in a small scale open economy with relatively short time spans of data. The success with which these types of models have been used to identify changes in growth and volatility in larger economies suggests they are worth exploring for New Zealand. Another principal reason for developing regime switching models is to explore the merits of a different way of thinking about how an economy’s growth rate evolves and the interpretation to be placed on changes in the growth and volatility of real output. In effect, these models block the data into periods of time (regimes comprising a number of consecutive quarters) whose time evolution is directly modelled, inaddition tothe quarter–to–quarterevolution within regimes. Thus the various time scales in the data are separately modelled within a simple, open framework that should allow enhanced economic and policy analysis. Because of its readily understood structure, this type of analysis can also be used as an exploratory tool to help guide appropriate specification of other model based methods. The remainder of the paper is structured as follows. Section 2 describes the hidden Markov switching model (HMM model) that we have fitted to New Zealand GDP growth data together with its specification and properties. Section 3 discusses issues concerning the estimation and fitting of HMM models. The results of fitting the HMM models to New Zealand real GDP data and to five production sectors are discussed in Section 4. Section 5 compares the timing of growth and volatility regimes in production sectors and total GDP. Conclusions are drawn and directions for future research are discussed in Section 6. 2 Specifying the Markov regime switching model We assume that we have available observations ( ) on some stationary macroeconomic time series where follows the general model (1) and represents the growth rates of GDP or one of its production sectors. The stochastic process is an unobserved stationary finite Markov chain that takes the values , which index the states of the system. Thus the level and the volatility ! switch between the WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 2
Other Many other reduced models are possible. Such models are referred to informally in Section 4 as – models where the numbers and refer to the number of distinct mean parameters and volatility parameters respectively. For example setting and * , * is an example of a 2–2 model where the growth and volatility regimes coincide with each regime having its own mean and standard deviation. Setting and retaining 4 levels for the / is an example of a 4–1 model. In the latter case the volatility is constant and the 4 states can be allocated to two or more growth regimes. 3 Fitting the model Givenobservations our strategyis to fit the model (1) using maximum likelihood and the Expectation Maximisation (EM) algorithm (see Dempster, Laird and Rubin, 1977) with the choice of model orders guided by the Bayesian Information Criterion (BIC). The latter selects the model that minimises 4 log likelihood ( with respect to the model order ( . As in the case of the AIC or Akaike Information Criterion, the BIC criterion trades model fit against model complexity. The EM algorithm can be used to obtain exact maximum likelihood estimates for certain models. However, in almost all cases we use it to explore the likelihood surface and obtain approximate maximum likelihood estimates which, in turn, are further refined using direct maximum likelihood. In the latter case we take advantage of the EM algorithm’s relative insensitivity to choice of initial values. Issues such as the determination of the standard errors of the parameters and the extraction of the trend and volatility 4 from the data are also considered. Given the density of is given by 4 ! .2 " # ! $ % (7) where /4 The density of the , or equivalently the , , is given by & ' ! .2 )( ( +* ! .2 ( +* * -,.( /* 0/ ! .2 * 0/ / -, (8) WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 9
where the , , are as defined in Section 2.3 and are functions of the 4 parameters ( , ( , * , * . Thus the log–likelihood of and is given by ! & (9) where 4 4 4 4 4 .2/ 4 .2 /4 & .2 The vector in (9) denotes the model parameters ( +*+ ( ), ( ) and so that has dimension 13. In keeping with EM terminology we call the log–likelihood of the complete data . However it is the likelihood of (the incomplete data) that we must determine since this is the only data we have available. The likelihood of is given by & (10) where 1 is over all possible realisations of . It is or that should ideally be optimised with respect to to determine the maximum likelihood estimator of . The more complicated structure of makes it a more difficult function to directly optimise by comparison to . This and other reasons lead us to consider the EM algorithm. If the states were known, then it is the relatively simple complete log–likelihood that would be optimised to determine estimators of . Given only the observations the best (quadratic loss) predictor of is (11) where the expectation operator is with respect to the true distribution indexed by . Given an initial estimate of a new estimate can be found by maximising with respect to the parameters . The new estimate can, in turn, be used for and so on. This recursion forms the basis of the EM algorithm where the determination of the conditional expectation is referred to as the E–step and its maximisation with respect to the M–step. Under certain general conditions it can be shown that the sequence of estimates constructed in this way yields monotonically increasing values of and converges to the maximum likelihood estimator for the incomplete data. Thus the EM algorithm provides an alternative method of maximising the log–likelihood . The computational efficiency of the EM algorithm is greatly enhanced if the E and M steps are readily evaluated, particularly the M step where simpleclosed form solutions are desired. In this case the algorithm is particularly easy to implement. In practice the EM algorithm is often more robust tothe choice of initial starting values thandirect maximum likelihood which, ifnumerical WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 10
optimisation procedures are used, tends to converge to a local rather than a global maximum. However, although better at identifying the region containing the global maximum, the EM algorithm can often be slow to converge in the vicinity of the global maximum. One reason for this is that the EM criterion is essentially a smoothed form of a log–likelihood and so the algorithm is less likely to converge to a local maximum than direct maximum likelihood, but more likely to converge slowly near the maximum due to a flattened log–likelihood surface. These observations and design objectives underpin the development that follows. From (9) and (11) we obtain 4 ! 4 4 4 32/ 4 4 32/ .2/ 4 32/ 2/ .2/ .-/4 4 4 32/ 32/ 2/ .2/ where .-/ 2/ The probabilities and are functions only of the initial parameters , the data , but not the parameters . They need to be determined prior to evaluating and optimising . Efficient recursive algorithms are given in Appendix B for evaluating the and the . An important by–productof these recursions is the evaluation of the exact likelihood given by (10). Thus we now have an appropriate computational framework in place for calculating maximum likelihood estimates by direct maximum likelihood (using numerical optimisation routines) as well as by the EM algorithm. However the and are also useful in their own right to extract estimates of stochastic parameters such as the trend and volatility . For example the best (quadratic loss) estimate of given the data is 32/ (12) and the best (quadratic loss) estimate of given the data is 4 4 32/ (13) These estimates of the time varying mean and variance of are used as informal diagnostic graphical measures in the applications sections. Equally importantly, the and also provide useful measures for identifying and classifying the most likely regimes. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 11
Despite the relatively simple structure of as a function of , analytic solutions for the value of that maximises will only exist in certain situations and then only if certain approximations are made. An important special case is where and all other parameters are distinct. Then, retaining only those terms of order in , the estimates of the parameters that maximise are given by 1 .2/ 1 .2/ 1 .2/ /4 1 .2/ (14) and ( 1 .2/ .-/ 1 .2/ 1 .2/ 1 32/ 1 2/ 1 .2/ 1 32/ ( 1 .2/ .-/ 1 .2/ 1 .2/ 1 32 1 2 1 .2/ 1 32 (15) with the analogous expressions for * , * involving instead of . Equations (14) and (15) provide the required EM recursions which will converge to the maximum likelihood estimate of the parameters in this case where is constrained to be zero and the , are distinct. Although there are other cases where analytic EM recursions can be found, this particular case was used to explore the log–likelihood surface to identify suitable initial estimates for direct maximum likelihood using numerical optimisation procedures. 1. Use the EM recursions (14) and (15) to explore the log–likelihood surface and obtain a range of suitable initial estimates for the maximum likelihood estimate . 2. Starting from these initial estimates, use numerical optimisation procedures to directly maximise the log–likelihood subject to parameter constraints ( , , !"#$ for %'&(*)+ , and , .-/ for %&0)+121+13)54 ). Here the exact log–likelihood is evaluated using the recursions given in Appendix B. 3. Determine the standard errors of the maximum likelihood estimates from the information matrix obtained from the Hessian provided by the optimisation procedure. 4. Examine the resulting estimates, BIC values etc and suitable graphical diagnostics to assess goodness of fit. 5. Identify and classify the hidden states into growth and volatility regimes. Table 2: Summary of fitting procedure. Table 2 provides a summary of the fitting procedure adopted in the applications given in the following sections. Using these methods and strategies, we now fit the various models considered to New Zealand GDP data. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 12
4 HMM models for New Zealand GDP growth This section identifies shifts in mean growth rates and volatilities by fitting Markov switching models to growth rates for total GDP and five production sectors that make up total GDP. The five sectors are Services,Government and Community Services,Manufacturing,Primary, and Construction as defined in Table 3. This work complements and builds on Buckle, Haugh and Thomson (2001) which attempts to identify the evolution of local means and volatility of quarterly growth rates for New Zealand real GDP and its production sectors using simple moving average techniques. The GDP seriesused inthis paperare quarterly realseasonally adjustedchain–linkedproduction GDP for the period 1978:1 to 2000:4. The series are Statistics New Zealand’s new official quarterly chain series from 1987:2 onwards appended to a calibrated chain series for the period back to 1978:1. The calibration procedure is explained in Haugh (2001) and the same GDP series are used in Buckle, Haugh and Thomson (2001). The calibration procedure exploits the statistical relationship between the period of overlapping official chain–linked and ex–official fixed–weight series (1987:2 to 2000:2) which is then used to derive series for each production sector and for total real GDP for the period from 1978:1 to 1987:1. These calibrated series are intended to approximate the chain–linked series over this period and are combined with the respective 1987:2 to 2000:4 chain–linked series available from Statistics New Zealand to form consistent time series data for each sector over the period 1978:1 to 2000:4. The choice of models to fit to GDP and its sectors was informed by the analysis of growth levelsand volatilityreported inBuckle, Haugh and Thomson (2001), includingvisualinspection of quarterly growth rates, and moving averages and standard deviations of GDP and each of the sectors. Examination of the moving averages can be very useful in determining which local means a series appears to move around and the number of means to include in the HMM model. The moving standard deviations can be used similarly to determine the local standard deviations and how many volatility regimes there might be in the data. However, since the standard deviation is dependent on where the mean is placed it is not always as straightforward Sector name Chain linked industries included in the sector Services Communications + Electricity, Gas & Water + Combined Wholesale Trade + Transport & Storage + Finance, Insurance, Business Services & Real Estate + Owner Occupied Dwellings Government and Personal and Community Services + Central Govt & Defence + Community Services Local Govt Services Primary Agriculture + Fishing + Forestry + Primary Food Manufacturing Manufacturing Textiles + Wood & Paper Products + Printing & Publishing + Petroleum etc + Non–Metallic Mineral Products Manufacturing + Basic Metals + Machinery & Equipment + Other Manufacturing + Other Food Manufacturing Construction Construction Table 3: Industry composition of the five production sectors. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 13
Data GDP GDP Ser Gov Man Man Man Pri Con Model Ham 3–2 MPQ 2–2 Ham MPQ 3–1 3–2 3–1 0.85 0.94 0.87 0.97 0.81 0.72 0.98 0.94 0.65 0.75 0.48 0.70 0.98 0.76 0.72 0.56 0.98 0.65 1.00 0.83 0.98 0.98 1.00 0.97 0.73 0.85 0.00 0.00 0.97 0.98 0.92 0.00 0.72 0.75 0.73 0.86 0.15 1.23 0.68 0.28 -0.73 -1.01 -1.16 -1.74 -10.29 0.15 0.25 0.04 0.28 -0.73 -3.07 1.59 4.88 -1.28 1.27 2.06 1.25 0.81 1.89 1.74 4.76 0.97 2.66 1.27 2.06 1.67 0.81 1.89 3.83 4.76 0.97 2.66 0.98 0.22 0.66 0.53 1.77 1.23 1.45 1.89 3.98 0.98 1.01 0.86 1.08 1.77 1.89 1.45 1.89 3.98 0.98 0.22 0.66 0.53 1.77 1.23 1.45 3.30 3.98 0.98 1.01 0.86 1.08 1.77 1.89 1.45 3.30 3.98 -0.03 -0.08 -0.26 -0.35 -0.14 -0.40 -0.29 -0.11 -0.26 -140.13 -135.02 -127.74 -95.01 -201.44 -197.66 -199.21 -241.38 -280.65 AIC 292.26 290.04 277.48 208.02 414.88 417.32 416.42 502.76 579.3 BIC 307.39 315.26 305.22 230.72 430.01 445.06 439.12 527.98 602 Parameters 6 10 11 9 6 11 9 10 9 Table 4: Parameter estimates for the HMM models fitted to GDP and sector growth rates. The sectors are Services (Ser), Government and Community Services (Gov), Primary (Pri), Manufacturing (Man) and Construction (Con) whose composition is given in Table 3. The models fitted include the Hamilton model (Ham) and are otherwise as indicated. as it may seem. In other words, a change in the series may be interpreted as a shift in the local mean or a change in the standard deviation around a constant mean. The HMM is a tool that can be used to more fully understand whether various features of the data are shifts in local means or a change in volatility. Visual inspection of GDP and sector quarterly growth rates suggest that the properties vary markedly across sectors and that allowing for different HMM models with varying numbers of states and varying means and standard deviations is appropriate. An initial model for each sector is selected for fitting based on prior analysis of means and standard deviations using centred moving average estimates of mean quarterly growth. These results are then used to inform any changes in the model being fitted. For example, if a four mean and two standard deviation model (4–2 model) is estimated, but two of the four fitted means are almost identical, a three mean and two standard deviation model (3–2 model) is fitted. This general to specific approach is supplemented by fitting simpler models with fewer parameters, such as the Hamilton two mean and single standard deviation model (2–1 model), to some sectors to obtain more robust estimates of the means which are then compared against the means estimated by more complicated models. The AIC and BIC model selection criteria were used to help select between competing models. This was supplemented by the criterion that the fitted model exhibit persistence in the sense that most regimes would be expected to last for a number of consecutive quarters before a switch takes place. An economy is unlikely to switch between high growth and low growth regimes every quarter because of the underlying economic process, which tends to show ongoing reinforcing behaviour that lasts more than one quarter. For example, in the high growth regime firms may be increasing investment, which leads to increased aggregate output and inWP 02/08 Growth and volatility regime switching models for New Zealand GDP data 14
come, which in turn leads to more demand and so on. On this basis, the preferred model for a series that oscillates between extreme values every quarter for example, would be a constant mean with high volatility rather than two means at each extreme value even if the AIC and BIC favoured the latter model. Table 4 describes the preferred estimated HMM models for GDP and each production sector, and the parameter estimates for each of these models. 4.1 Aggregate real GDP The Hamilton model, originally fitted to the US GNP growth rates, appears successfully to capture the dynamics of New Zealand real GDP. This model has also been successfully fitted to real GDP dynamics for several other countries (see for example Krolzig, 1997). The top panel of Figure 3 shows the quarterly GDP series with the trend estimated from (12) and also from an 11–quarter triangular moving average for comparison. The second panel of Figure 3 plots the probability of being in the high growth regime. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the panel at the bottom of Figure 3. The low–growth mean is estimated to be 0.15 percent per quarter and the high–growth mean is estimated to be 1.27 percent per quarter. The Hamilton model indicates the New Zealand economy has experienced five switches from low to high mean growth between 1978 and 2000, where the economy is regarded as being in a high growth regimewhen the probabilityof being in that stateis 50 percent or greater (otherwise it is defined as being in the low growth regime). The periods in the high growth regime are 1978:2–1978:4, 1981:2–1982:1, 1983:3–1984:2; a period of sustained high growth from 1992:3 to 1996:1; and another high growth regime at the end of the sample period (1999:1–2000:2). The Hamilton model also picks out 1986:2 as a period when GDP was in the high growth regime, but this was probablythe effect of increased spending in anticipation of the introduction of GST on 1 October 1986. With the exception of this mid–1986 spike, the economy was in the low growth regime from 1984:3 to 1992:3. Evidence of a decline in the standard deviation of New Zealand real GDP growth provided in Buckle, Haugh andThomson(2001) suggestshoweverthata richer HMMmodel withmore than one standard deviation may be more appropriate. As a first step we fitted the MPQ model which has four mean growth rates and two standard deviations. This model was used by McConnell and Perez–Quiros (2000) to show evidence of breaks in US GDP volatility and allows the means in both growth regimes of the cycle to vary according to the level of volatility. Fitting the MPQ model indicated two GDP volatility states in NZ real GDP growth, but only three distinct mean growth rates. Two of the estimated four mean growth rates (the two high means) were almost equal. On the basis of this evidence, a three means and two standard deviations model was fitted to NZ real GDP data (3–2 model). In contrast to the Hamilton model which has two states (high and low mean growth states with a common standard deviation), the fitted HMM 3–2 model has four states. Of these, three ( = 1, 3, 4) are classified as belonging to the high growth regime and one ( = 2) is classified as belonging to the low growth regime. The classification of states to regimes is shown in the bottom panel of Figure 4. The high growth regime has estimated mean growth rates of 1.23 percent per quarter and 2.06 percent per quarter. The latter picks out two short duration periods WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 15
GDP Hamilton model Growth rate 1980 1985 1990 1995 2000 −3 −2 −1 0 1 2 3 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1, 2 0.15 0.98 Low growth Constant volatility 3, 4 1.27 0.98 High growth Constant volatility Figure 3: The Hamilton model fitted to quarterly GDP growth rates. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. in 1984 and 1994 when quarterly real GDP growth rates were unusually high. The other high– growth mean and the low–growth mean are close to those for the Hamilton model. Here the probability of a high growth regime occurring is and this is plotted in the middle panel of Figure 4. This results in the identification of four switches from low growth to high growth regimes, one less than the number identified by the Hamilton model (excluding the 1986 GST spike). The periods of high growth regimes were as follows: 1981:3–1982:1, 1983:3–1984:1, 1992:4–1995:3, and 1999:3–1999:4. In comparison to the Hamilton model, the 1978 period is no longer regarded as a switch to a growth regime and the 1986 spike is clearly not present either. Instead, these periods are reWP 02/08 Growth and volatility regime switching models for New Zealand GDP data 16
GDP 3−2 model Growth rate 1980 1985 1990 1995 2000 −3 −2 −1 0 1 2 3 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1 1.23 0.22 High growth Low volatility 2 0.25 1.01 Low growth High volatility 3 2.06 0.22 High growth Low volatility 4 2.06 1.01 High growth High volatility Figure 4: A 3–2 model fitted to quarterly GDP growth rates: growth regimes. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated / . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. garded as periods of high volatility around a low mean. The 1999–2000 and 1992–1996 regimes are also shorter than those determined by the Hamilton model. The top panel of Figure 5 plots the squared deviations of the GDP growth rates from both the 11–quarter moving average trend and the HMM trend which is based on the entire dataset. Both methods clearly identify the mid 1990s as the lowest volatility period since 1978. Buckle, Haugh and Thomson (2001) suggest that the low volatility during this period was driven particularly by a temporary fall in the covariance across the sectors, which appears to cycle with WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 17
GDP 3−2 model Squared deviations from trend 1980 1985 1990 1995 2000 0 2 4 6 8 Probability of high volatility Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Figure 5: A 3–2 model fitted to quarterly GDP growth rates: volatility regimes. The top panel plots the squared deviations (grey dotted line) of the growth rates from their 11–quarter triangular moving average trend, and the squared deviations (solid grey line) of the growth rates from the HMM trend. The estimated volatility (black solid line) obtained from (13) and the triangular 11–quarter moving sample variance (black dashed line) are also plotted. The second panel plots the probability of being in the high volatility regime with the grey horizontal reference line equal to 0.5. no apparent trend. Interestingly, both periods of low volatility of New Zealand real GDP are periods when the economy switched to the high growth regime, but not all high growth regimes are associated with low volatility. The 1992:4 to 1995:3 period stands out however as a distinct period of nirvana, a period of high growth with low volatility. The second panel of Figure 5 plots the probability of being in the high volatility regime and shows two periods during which the standard deviation switches from high to low volatility regimes. The estimated 3–2 model classifies most of the period between 1978 and 2000 as high volatility, with the possible exception of a short period from 1981:3–1982:1 and almost certainly a longer period from 1992:4 to 1995:3. As has been found for the United States (see Kim and Nelson, 1999; McConnell and Perez–Quiros, 2000; Shaghil, Levin and Wilson, 2001) and several other developed economies including Australia (see Blanchard and Simon, 2001; Simon, 2001), there is clear evidence of a switch to lower volatility of New Zealand real WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 18
Manufacturing MPQ model Growth rate 1980 1985 1990 1995 2000 −4 −2 0 2 4 6 8 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1 -1.01 1.23 Low growth Low volatility 2 -3.07 1.89 Low growth High volatility 3 1.74 1.23 High growth Low volatility 4 3.83 1.89 High growth High volatility Figure 11: The MPQ model fitted to quarterly Manufacturing growth rates: growth regimes. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. 1980:4–1982:1, 1983:2–1984:4, 1991:3–1992:1 and 1992:4–1995:1, 1999:1–1999:4. There are also short switches to high growth in 1986–1987, 1989, 1996 and 1997, but the probabilities only just exceed 50 percent and last only one quarter. The Hamilton model is parsimonious with only 6 parameters and is superior to both the MPQ and 3–1 models on the basis of both the AIC and BIC model selection criteria. However, it does not seem to be as appropriate for the early 1980s period. This period is different probably because of the “Think Big” capital expenditure program which boosted activity in the ManuWP 02/08 Growth and volatility regime switching models for New Zealand GDP data 25
Manufacturing MPQ model Squared deviations from trend 1980 1985 1990 1995 2000 0 10 20 30 Probability of high volatility Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Figure 12: The MPQ model fitted to quarterly Manufacturing growth rates: volatility regimes. The top panel plots the squared deviations (grey dotted line) of the growth rates from their 11–quarter triangular moving average trend, and the squared deviations (solid grey line) of the GDP growth rates from the HMM trend. The estimated volatility (black solid line) obtained from (13) and the triangular 11–quarter moving sample variance (black dashed line) are also plotted. The second panel plots the probability of being in the high volatility regime with the grey horizontal reference line equal to 0.5. facturing sector, notably the Machinery and Equipment Manufacturing industry (see Buckle, Haugh and Thomson (2001) who identify the Machinery and Equipment manufacturing industry as the key source of the strong surge in volatility in this sector during the early 1980s). Including the early 1980s period in the estimation of the Hamilton model is likely to result in an upwards bias for the estimated means for the subsequent periods. Estimates using the MPQ and the 3–1 models suggest that this is the case. Using theMPQ model, Manufacturingcan be characterised by the four states and corresponding regimes shown in the table at the bottom of Figure 11. Furthermore, Figure 12 implies that Manufacturing is in the low volatility regime with a standard deviation of 1.23 percent per quarter for most of the sample period except for the period 1982:1– 1984:1 when it switches to the high volatility state with a standard deviation of 1.89 percent per quarter. For this period the overall volatility of the data is also higher since the mean growth rates of 3.83 percent and -3.07 WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 26
Manufacturing 3−1 model Growth rate 1980 1985 1990 1995 2000 −4 −2 0 2 4 6 8 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1 -1.16 1.45 Low growth Constant volatility 2 1.59 1.45 High growth Constant volatility 3, 4 4.76 1.45 High growth Constant volatility Figure 13: A 3–1 model fitted to quarterly Manufacturing growth rates. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated / . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. percent per quarter produce the most extreme deviations about the constant overall mean. This is consistent with the triangular moving sample variance which is also at its highest around this period and at a clearly higher level than the rest of the sample. Outside this period of high volatility the high and low mean growth rates of 1.74 percent and -1.01 percent per quarter are closer to those estimated using the Hamilton model. The MPQ values are however slightly lower which leads to more regime switching because there is now a lower threshold before growth switches to the high regime. As a result the MPQ model suggests that there are nine rather than five switches from low to high growth regimes, some of WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 27
which even the 11–quarter triangular moving average is unable to track. These regimes last for periods of between two and ten quarters which makes it difficult for any one moving average to detect all of them. The timing of the regime switches identified by the MPQ model are generally very close to those identified by the Hamilton model. However the MPQ model more clearly identifies switches to the high growth regime in 1986–1987, 1996 and 1997 and clearly identifies another switch to the high growth regime beginning in 2000:3. The 3–1 model (illustrated in Figures 13) also isolates the early 1980s period as different from the rest of the sample, but uses a third mean level to do this. The three means estimated by this model are -1.16 percent, 1.59 percent and 4.78 percent per quarter, all with a standard deviation of 1.45 percent per quarter. The high growth state with a mean of 4.78 percent growth is realised once, from 1983:3 to 1984:1. During the rest of the sample period Manufacturing switches between the low mean of -1.16 percent and the high mean of 1.59 percent per quarter, which are also slightly lower than the two Hamilton means. The timing of switches to the 1.59 percent high growth state is very similar to the timing of switches derived using the MPQ model and, as with the MPQ model, the 3–1 model has more switches to the high growth regime than the Hamilton model. The 3–1 model (where nine parameters are estimated) is superior to the MPQ model on the basis of the AIC and BIC criteria because it has fewer parameters to estimate than MPQ (eleven parameters). Overall, the manufacturing sector has had frequent regime switches with varying lengths. Interpretation of the number of switches depends on how the early 1980s period is treated. There appears to be no permanent change in volatility in this sector, with the possible exception of a brief spike in volatility in the mid 1980s which is likely to be associated with the “Think Big” capital expenditure programmes. Again plots of the probabilities of being in each of the four states given the data ( for ) are given in Figures 22 and 23 in Appendix C for both the MPQ and 3–1 models. 4.5 Primary sector A model with three mean growth rates and two standard deviations (a 3–2 model) seems to provide a good characterisation of the Primary sector. Further details are given in Figures 14, 15 and Figure 24 where the latter is given in Appendix C. Although the MPQ model also fits the data well, two of the estimated four means were very close to each other (0.80 percent and 1 percent) and, with one less parameter to estimate, a 3–2 model with three states had the better BIC value. Figure 15 shows that, throughout the sample period, the Primary sector was predominantly in the high volatility regime with a 0.97 percent mean quarterly growth rate. This state prevailed from 1978:1 to 1988:1 and from 1994:1 to 2000:4. This tendency to remain in one regime for very long periods distinguishes the Primary sector from other sectors and aggregate GDP, with the exception of the Government and Community Services sector. The Primary sector also illustrates how an informal analysis of the raw quarterly growth rate data could be misleading. The high volatility in the data may make it tempting to infer from a new highgrowthobservation that there has been a shift in the mean and a transition to a new part of the cycle. The 3–2 model WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 28
Primary 3−2 model Growth rate 1980 1985 1990 1995 2000 −5 0 5 10 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1 -1.74 1.89 Low growth Low volatility 2 4.88 1.89 High growth Low volatility 3, 4 0.97 3.30 High growth High volatility Figure 14: A 3–2 model fitted to quarterly Primary growth rates: growth regimes. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. suggests this would often be an incorrect interpretation for this sector. During the middle of the sample period from 1988:2–1994:1 the Primary sector switches to the low volatility regime and also switches between the lowest (-1.74 percent per quarter) and highest (4.88 per cent per quarter) mean growth states. During this period shifts in the mean growth rates, rather than changes in the standard deviations, drive changes in the overall volatility of Primary output. In summary, the Primary sector temporarily switches to low volatility in the late 1980s and early WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 29
Primary 3−2 model Squared deviations from trend 1980 1985 1990 1995 2000 0 20 40 60 Probability of high volatility Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Figure 15: A 3–2 model fitted to quarterly Primary growth rates: volatility regimes. The top panel plots the squared deviations (grey dotted line) of the growth rates from their 11– quarter triangular moving average trend, and the squared deviations (solid grey line) of the GDP growth rates from the HMM trend. The estimated volatility (black solid line) obtained from (13) and the triangular 11–quarter moving sample variance (black dashed line) are also plotted. The second panel plots the probability of being in the high volatility regime with the grey horizontal reference line equal to 0.5. 1990s and returns to the high volatility regime with a constant mean from 1994 onwards. This implies that the Primary sector has not made a permanent contribution to the decline in GDP volatility in agreement with Buckle, Haugh and Thomson (2001). 4.6 Construction sector The Construction sector is the most volatile of New Zealand’s five production sectors and was a difficult sector to model and characterise. A 3–1 model was preferred over other possible model structures. The three mean quarterly growth rates are -10.29 percent, -1.3 percent and 2.66 percent and the standard deviation is 3.98 percent per quarter. The state with -10.29percent mean growth rate (state ) is, ineffect, an outlier statesince itpicks off extremelowgrowth rates. (Other outlier models might also be considered for this data since there also appear to be a WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 30
Construction 3−1 model Growth rate 1980 1985 1990 1995 2000 −15 −10 −5 0 5 10 15 Probability of high growth Time Probability 1980 1985 1990 1995 2000 0.0 0.4 0.8 Regime classification 1 -10.29 3.98 Outlier low growth Constant volatility 2 -1.30 3.98 Low growth Constant volatility 3, 4 2.66 3.98 High growth Constant volatility Figure 16: A 3–1 model fitted to quarterly Construction growth rates. The top panel shows the growth rates (grey line) with the trend (solid line) estimated from (12) and also from an 11–quarter triangular moving average for comparison (dashed line). The grey horizontal lines represent the estimated . The second panel plots the probability of being in the high growth regime with the grey horizontal reference line equal to 0.5. Estimated mean growth rates and standard deviations for each state, and the classification of states to regimes are shown in the bottom panel. few extreme high growth rates.) For most of the sample period the Construction sector switches between a low–growth mean of -1.3 percent mean and a high–growth mean of 2.66 percent. Figure 16 implies that the Construction sector switches frequently between high and low growth regimes. If the quarters1985:1, 1993:4and1995:3 are disregarded asswitches tothe lowgrowth state (the probabilities of these quarters being in the low growth state are only 0.54, 0.54 and 0.67 respectively) there are 10 switches from the low to the high growth regime during the 23 years of the sample (there are nine if the one quarter duration high growth regime in 1992:1 is ignored). WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 31
5 Growth and volatility regimes across sectors The estimation of Markov switching models for production sectors provides a basis for interpreting the temporal relationship between growth and volatility regimes across the sectors and their contribution to the timing of regime switching evident in total GDP. Although this issue could be more rigorously pursued using a vector Markov switching model of the type suggested by Kontomelis (2001) for example, some useful insights can nevertheless be gained by an informal analysis of the relationship between growth and volatility regimes across sectors and their relationship with total GDP growth and volatility. Figure 17 plots and compares the probability of each production sector and total GDP being in a high growth regime for each quarter from 1978:1 to 2000:4. The vertical lines on Figure 17 represent the dates that total GDP switches from a low to a high growth regime and vice versa. Table 5 dates the periods when total GDP, Services, Manufacturing, and Construction were in the high growth regimes. The Primary sector and Government and Community Services sector are not included in Table 5 because they display far fewer growth regime switches. The Primary sector is in the high growth regime during 1978:1 to 1988:1, 1990:1 to 1990:4 and 1993:1 to 2000:4 (that is, the only periods of low growth in the Primary sector are 1988:2–1989:4 and 1991:1–1992:4). The Government and Community Services sector only shows one switch from low to high growth in 1992:3 and remained continuously in the high growthregime from 1992:3 to 2000:4. Data Model Dates of high growth regimes GDP Ham 1978:2–1978:4 1981:2–1982:1 1983:3–1984:2 1992:4–1996:1 1999:1–2000:2 Ser MPQ 1978:2–1978:4 1981:2–1981:3 1983:2–1984:2 1993:1–1995:3 1998:4–1999:3 1985:4–1986:3 Man Ham 1978:1–1979:2 1980:4–1982:1 1983:2–1984:4 1991:3–1992:1 1999:1–1999:4 1992:4–1995:1 Con 3–1 1978:3–1978:4 1981:2–1982:2 1983:2–1985:4 1992:4–1997:1 1999:1– 2000:1 1979:4–1980:2 1987:3–1988:1 1989:1–1989:2 Table 5: Dates when GDP and selected production sectors were in the high growth regime. The sectors considered are Services (Ser), Manufacturing (Man) and Construction (Con). The models fitted include the Hamilton model (Ham) and are otherwise as indicated. Figure 17 illustrates that there is a close association between the timing of switches to the high growth regimes by total GDP growth and by growth in the Services and Manufacturing sectors and to a lesser extent the Construction sector. The Primary and the Government and Community Services sectors display very different regime switching tendencies compared to total GDP and the other three sectors. The lack of any obvious relationship between the timing of Government and Community Services sector regime switching and GDP regime switching is consistent with earlier research by Kim, Buckle and Hall (1994) who found, using growth cycle methods, no significant contemporaneous correlation between cycles in real GDP and cycles in WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 32
GDP 3−2 model Prob 1980 1985 1990 1995 2000 0.0 0.6 Services MPQ model Prob 1980 1985 1990 1995 2000 0.0 0.6 Manufacturing Hamilton model Prob 1980 1985 1990 1995 2000 0.0 0.6 Primary MPQ model Prob 1980 1985 1990 1995 2000 0.0 0.6 Construction 3−1 model Prob 1980 1985 1990 1995 2000 0.0 0.6 Government 2−2 model Time Prob 1980 1985 1990 1995 2000 0.0 0.6 Figure 17: Probability of GDP and its production sectors being in a high growth regime. The vertical grey lines represent the dates that total GDP switches from a low to a high growth regime and vice versa. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 33
real government spending. This observation is also consistent with findings for other countries (see Backus and Kehoe, 1992). The lack of any obvious relationship between the timing of Primary sector regime switching and GDP regime switching is perhaps more surprising. The Manufacturing sector switches to the high state before the other sectors and total GDP during the 1981-1982 and the 1992–1996 high growth regimes. It also switches back to the low growth regime ahead of Services and ahead of GDP after the 1992–1996 high growth regime. However, Manufacturing does not always switch first. The Services sector was the first sector to transit to the 1999:1–2000:2 high growth regime and was the first to subsequently switch back to the low growth regime. Construction has more high growth regimes than the other sectors and was the last sector to switch to the low growth regime after mid 1990s high growth regime. It switched about one year after GDP and two years after Manufacturing switched back to a low growth regime. There is considerable evidence that growth in many developed economies has become less volatile since the 1980s. McConnell and Perez–Quiros (1999) and Kim and Nelson (1999) provide compelling evidence of a sustained switch to lower volatility in US real GDP growth in early 1984. Their conclusions are based on applications of hidden Markov switching models. Using deviations from a moving average measure of trend growth, Blanchard and Simon (2001) provide evidence of similar declines in Canada, United Kingdom, France, and Germany. Simon (2001) shows that Australia’s GDP volatility also declined during the mid 1980s. Figure 18 shows the probability of New Zealand GDP and the five production sectors being in the high volatility regime over the period 1978:1 to 2000:4. New Zealand GDP growth is continuously in the high volatility regime throughout the late 1970s and the 1980s, but switches to the low volatility regime during the early to mid 1990s and remains in that state until 1996. However, in contrast to the findings for other developed countries, the switch of New Zealand’s GDP to the low volatility regime appears to be temporary. New Zealand GDP has remained in the high volatility regime since 1996. The only production sector that switches to lower volatility in the early 1990s is the Services sector. This switchoccurred in 1992:1, slightly earlier than the switch to lower volatility by total GDP. Moreover, in contrast to GDP and all other sectors, the decline in Services sector volatility is a sustained change to lower volatility. Nevertheless, despite contributing approximately 50 percent of total GDP, this is not sufficient to permanently change GDP volatility. This is because GDP volatility is affected by the covariance between the sectors growth rates which has a larger influence on overall GDP volatility than the volatility of the Services sector. Buckle, Haugh and Thomson (2001) show that low covariance between sector growth rates was a significant reason for the temporary decline in the GDP volatility during the 1990s. Furthermore, the switch to low volatility in the Services sector is also partially offset by switches to high volatility in the Primary sector and Government and Community Services sector after 1995. There is no evidence of volatility switching in the Manufacturing and Construction sectors as constant standard deviation models have been fitted to these sectors. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 34
holds for all . Furthermore, using similar conditioning arguments, 32/ 32/ (22) where is a Gaussian density with mean 4 and variance . Values for the are now determined from the forward recursions (22) with 32/ 2/ and where is given by (20). A similar development for the yields the backward recursions 2/ .-/ .-/ 2/ .-/ .-/ .-/ .-/ 4 (23) with for all . Also .-/ .-/ .-/ .-/ .-/ (24) holds for 4 . These computationally efficient recursions provide all the elements needed to determine the log–likelihood of the data from(19), the from (21) and the from (24). WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 41
C Additional figures Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 GDP 3−2 model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 4 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 19: Plots of the for a 3–2 model fitted to quarterly GDP growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 42
Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Services MPQ model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 4 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 20: Plots of the for an MPQ model fitted to quarterly Services growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 43
Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Government 2−2 model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 4 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 21: Plots of the for a 2–2 model fitted to quarterly Government and Community Services growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 44
Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Manufacturing MPQ model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 4 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 22: Plots of the for an MPQ model fitted to quarterly Manufacturing growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 45
Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Manufacturing 3−1 model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 23: Plots of the for a 3–1 model fitted to quarterly Manufacturing growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 46
Prob( S(t) = 1 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Primary 3−2 model Prob( S(t) = 2 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Prob( S(t) = 3 ) 1980 1985 1990 1995 2000 0.0 0.4 0.8 Time Figure 24: Plots of the for a 3–2 model fitted to quarterly Primary growth rates. WP 02/08 Growth and volatility regime switching models for New Zealand GDP data 47
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