Examining the time trajectory of GDP growth: a multi country study.Abstract
In this study, we first present an extensive review of the literature on the time dynamics of GDP GDP (guanosine diphosphate): see guanine. , leading up to modern times. We then test for the simultaneous presence (and/or absence) of non-linearity and non-stationarity in the quarterly real GDP growth rate series for the Canada, Japan, United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. and the United Kingdom, encompassing the four decades from the 1960's to the present (2004.) Caner-Hansen (CH, 2001) provides a new test which for the first time can simultaneously examine for both non-stationarity and non-linearity (threshold effects).
Existence of a threshold effect In particle physics, the term threshold effect usually refers to small corrections to rough calculations based on the renormalization group that arise from the detailed behavior near the scale where new physics takes place. would imply that policy intervention occurs only when changes in the real GDP either exceed or are less than a socially and politically acceptable threshold level Noun 1. threshold level - the intensity level that is just barely perceptible
intensity, intensity level, strength - the amount of energy transmitted (as by acoustic or electromagnetic radiation); "he adjusted the intensity of the sound"; "they measured the . This is supported by our results, which indicate the existence of a threshold effect for three of the four countries.
Considerable effort has recently been directed towards examining the cause and effect of the observed volatility reduction in the US GDP over time. Indeed the magnitude of fluctuations of the US GDP has decreased (measured in terms of standard deviation (SD) of quarterly growth rates Growth Rates
The compounded annualized rate of growth of a company's revenues, earnings, dividends, or other figures.
Remember, historically high growth rates don't always mean a high rate of growth looking into the future. of real GDP) dramatically since World War II. In the period 1984 onwards (as compared to the post war period 1950-83) the SD has decreased by one half, a significant drop indeed.
This begs an answer to the question as to why this volatility reduction came about. It is also important from the policy maker's point of view. Policy making is crucially dependent on a proper analysis of macroeconomic mac·ro·ec·o·nom·ics
n. (used with a sing. verb)
The study of the overall aspects and workings of a national economy, such as income, output, and the interrelationship among diverse economic sectors. forecasts, since it is these predictions and their projected path, that help in reducing future uncertainty and volatility in the variable under consideration. Studying the movements of aggregate economic variables in a scientific manner will help in better economic planning economic planning, control and direction of economic activity by a central public authority. In its modern usage, economic planning tends to be pitted against the laissez-faire philosophy which developed in the 18th cent. over the future periods. But success in modeling and forecasting GDP depends on a proper analysis of the dynamics of this variable itself, i.e., a credible characterization of its time path and the nonstationarities and nonlinearities inherent in its behavior.
The empirical literature in this particular case has not proceeded in a logical step by step manner, which would be to first test for the presence of nonlinearities in the level / growth rate of the GDP series, and then use an appropriate nonlinear A system in which the output is not a uniform relationship to the input.
nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. model to replicate its time path over business cycles. In most cases than not, nonlinearity in GDP has been assumed and not tested. Here we would like to take a step back, and test for nonlinearities in the rate of growth of the GDP of four major economies (Canada, Japan, USA and UK), using a recently available (and powerful) test, which would account for the first part of the contribution of this exercise.
The second contribution has to do with the standard assumption made in the literature regarding stationarity of the variable under consideration, and then only testing for nonlinearity, in isolation. This fundamental flaw in the econometric e·con·o·met·rics
n. (used with a sing. verb)
Application of mathematical and statistical techniques to economics in the study of problems, the analysis of data, and the development and testing of theories and models. / empirical methodology encompasses all the parametric as well as nonparametric studies. They are all limited by their assumption that the series under consideration is stationary (more specifically trend stationary, and hence using the common solution of simply detrending the data), and therefore the test is for nonlinearity only. The problem here is that even if this assumption is correct, this automatic "detrending" before setting up the actual threshold autoregressive model would bias the results. In our context, it becomes all the more relevant since we have overwhelming empirical evidence of nonstationarity (unit roots) in the level of real output, see for example Nelson and Plosser (1982), Wasserfallen (1986), Watson (1986), Cheung and Chinn (henceforth From this time forward.
The term henceforth, when used in a legal document, statute, or other legal instrument, indicates that something will commence from the present time to the future, to the exclusion of the past. CC, 1996) and Rapach (2002). Whether level stationarity or otherwise (unit roots) is true has an important bearing for the time path of GDP or the trajectory of business cycles. It would also affect our forecasting capability and concomitantly the credibility of macroeconomic stabilization policies.
The empirical evidence overwhelmingly supports non-stationarity of real GDP using both unit root methods (see CC, 1996) or panel unit root tests (see Rapach, 2002). Thus the need is for an empirical test of "linearity" versus "nonlinearity", but without any "a priori a priori
In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience. " assumptions about the (stationarity/nonstationarity) data series under consideration. This is where the recently available Caner-Hansen (henceforth CH, 2001) study comes into play. It is the first rigorous theoretical and empirical treatment Empirical treatment
Medical treatment that is given on the basis of the doctor's observations and experience.
Mentioned in: Enterobacterial Infections of the simultaneous existence of both nonstationarity and nonlinearity. Now we do not have to assume one to test for the other in isolation, because it allows for nonlinear (asymmetric adjustments) dynamics for unit root testing procedure. This procedure is much more powerful than conventional unit root tests, especially if the adjustment procedure is not symmetric, see Enders and Granger (henceforth EG, 1998).
The third contribution is the "comparative study" being undertaken here of the time dynamics (non-stationarity and non-linearity) of the GDP growth rate of four major world economies namely Canada, Japan, UK and USA. It is of additional interest to examine if these characteristics are uniform (or different) across economies.
This paper proceeds as follows. In section 2 we conduct an extensive and up to date literature review of the relevant articles from both economics and finance, followed by a description of the data used, in section 3. In section 4 we briefly describe the Caner-Hansen methodology. This is followed by an analysis and explanation of the results in section 5, with our concluding remarks in section 6.
An examination of the literature provides ample evidence of the fact that the magnitude of the US GDP fluctuations has decreased dramatically over time. Starting from the early eighties onwards, its volatility (measured in terms of standard deviation of quarterly growth rates of real GDP) has reduced by one half, a significant drop indeed. This begs an answer to the question as to why this reduction came about. An examination of the characteristics / stylized facts In social sciences, especially economics, a stylized fact is a simplified presentation of an empirical finding. While results in statistics can only be shown to be highly probable, in a stylized fact, they are presented as true. of the economy and the nature of the stabilization has been the focus of the study by Ahmed et al., (2002), Blanchard and Simon (2001), Chauvet and Potter (2001), Kim et al., (2004), Kim and Nelson (1999 a, b), McConnell and Perez-Quiros (2000), Niemira and Klien (1994), Stock and Watson (2002) and Warnock and Warnock (2000), and among others. The reasons put forward run the entire gamut of possible rationalizations from improved policy making (fiscal and monetary policy) to structural factors (less friction in the inventory management process and hence better synchronization (1) See synchronous and synchronous transmission.
(2) Ensuring that two sets of data are always the same. See data synchronization.
(3) Keeping time-of-day clocks in two devices set to the same time. See NTP. between demand and supply) to random factors (butterfly effects) to sheer good luck (reduction in the internal and external real shocks.) One school of thought contends it is because aggregate real GDP underwent a reduction in volatility (more due to its cyclical components than trend components), while others find the volatility reduction is pervasive over various sectors of the economy, indicating that this stability was brought about more by improvements in inventory management than by better monetary policy. So the question is, "is it one (or more) factors or a combination of factors?" The answer to this is extremely important, also from the policy maker's point of view.
Examining these fluctuations to see if they can be rationalized by social / economic/ demographic and/or technological factors or if they are purely random is also integrally linked to making credible macroeconomic forecasts. Policy making is crucially dependent on a proper analysis of these forecasts, since it is these predictions (and their path) that help in reducing future uncertainty and volatility in the variable
under consideration. Studying the movements of aggregate economic variables in a scientific manner will help in better economic planning over the future periods. But success in modeling and forecasting GDP depends in an integral manner on a proper analysis of the dynamics of this variable itself, i.e., a credible characterization of its time path and the nonstationarities and nonlinearities inherent in its behavior.
Historically GDP growth (i.e., output changes over time) was studied in a linear set up, following the seminal route prescribed by Box and Jenkins (1976). Works in this line are Braun and Zarnowitz (1989) and Diebold and Rudebusch (1991) among others. But the basic assumption of linearity for business cycles implies symmetric troughs and peaks, which could never be justified by their realistic shape, which were always asymmetric. The reason for the asymmetric shape of expansions and contractions being that the growth rate is faster after a trough but slower after a peak. Economic growth is generally characterized by nonlinear trajectories, fluctuating randomly between two successive periods of expansion and contraction. The reduction in GDP growth volatility is not because of better foresight and less structural friction in the economy, but is more so based on the expectations (and acceptance) of the fact that no regime (expansion or contraction) is permanent. Thus there are built in forces (based on human expectation formation and behavior modification behavior modification
1. The use of basic learning techniques, such as conditioning, biofeedback, reinforcement, or aversion therapy, to teach simple skills or alter undesirable behavior.
2. See behavior therapy. ) which turn the tide in the reverse direction. To borrow a terminology from International Finance, "bandwagon effects" in our expectation formation process resulted in lowering the volatility.
Interestingly, the basic reason for assuming that the growth path of the GDP would be nonlinear was outlined way back in 1964 by Milton Friedman Noun 1. Milton Friedman - United States economist noted as a proponent of monetarism and for his opposition to government intervention in the economy (born in 1912)
Friedman , in an NBER NBER National Bureau of Economic Research (Cambridge, MA)
NBER Nittany and Bald Eagle Railroad Company study on business cycles. He showed an asymmetry Asymmetry
A lack of equivalence between two things, such as the unequal tax treatment of interest expense and dividend payments. in the correlations between successive business cycle phases, where the amplitude of a contraction and the next expansion were correlated, but the expansion phase was not correlated to the amplitude of the next contraction. He called it the "plucking" model of business cycles. The economics literature studying the path of GDP growth has confirmed this asymmetry time and time again. The asymmetric behavior of business cycles is also evident from the way the production possibility frontier (PPF PPF Plasma protein fraction, see there ) is set up. Here recessions are situations below the PPF and thus fluctuations in output are asymmetric in the negative direction. The asymmetric behavior was highlighted by Neftci (1984) who statistically documented the pattern of business cycles. Recessions were consistently found to be steeper than recoveries, as in DeLong and Summers (1986), Falk (1986) and Sichel (1993). It is also evident in the Markov switching model setups used by Hamilton (1989) and Lam (1990 and 2004). Sichel (1993, 1994) found the recessions to be deeper than expansions are tall and reported real GDP to have a "peak reverting" behavior. Sichel conceptualizes them as different levels of steepness and deepness in business cycles during contraction and expansion phases. Then we have Diebold and Rudebusch (1990) and also Durland and McCurdy (1994) who examine business cycle dependence models (in a univariate form) while Kim and Nelson (1998) does so (in a multivariate context.) Another evidence of the same is McQueen and Thorley (1993) who find US business cycles characterized by sharp troughs and round peaks.
In Galvao (2002) the author evaluates a large set of nonlinear univariate time series models of US GDP and finds that nonlinearities are evident due to business cycle asymmetries, which according to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. Balke and Wynne (1995) are because business cycles tend to be concave Concave
Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. during expansions and linear during contractions. Here Galvao uses a linear approximation linear approximation
In mathematics, the process of finding a straight line that closely fits a curve (function) at some location. Expressed as the linear equation y = ax + b, the values of a and b of the cycle model to trace the actual trajectory and finds that linearization In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential does a poor job, and hence nonlinear approximations are the way to go. Moreover, this study measures the business cycle asymmetries as the "excess" as defined by Harding and Pagan (2002). Using this matrix, it evaluates 15 nonlinear time series models, and finds this "excess" over the lower and upper quartiles of the distribution to be different.
Alternative variants of the threshold autoregressive model have been used. Among them is the Beaudry and Koop (1993), who use an endogenous endogenous /en·dog·e·nous/ (en-doj´e-nus) produced within or caused by factors within the organism.
1. Originating or produced within an organism, tissue, or cell. switching technique to measure the "depth of the recession" which allows for nonlinearity in the model. They use the impulse response In simple terms, the impulse response of a system is its output when presented with a very brief signal, an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely short in time technique on real GDP and find that negative innovations are less persistent than positive ones. Extensions on these lines have been done by Bradley and Jansen (1997) and Pesaran and Potter (1997). The latter use three endogenous regimes, namely corridor, floor and ceiling, and try to reproduce the asymmetric shape of the business cycle. Pesaran and Potter also report a nonlinear GDP path with linear contractions and concave expansions. After a thorough comparison, they provide concrete evidence of the importance of nonlinearities in business cycles.")
Then we have Breunig and Stegman (2003) who extend the Bodman and Crosby (1999) Markov Switching (henceforth MS) nonlinear model to study the dynamics of Singapore GDP. This nonlinear setup outperforms the linear AR models. They find that GDP growth is best studied under asymmetric "high growth" and "low growth" phases. Thus the regime switching model is so applicable. They also find the MS model is superior to linear alternatives in "one step ahead" predictions of the future path of GDP. They also use "graphical methods This is a list of graphical methods with a mathematical basis. Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.
There is also a list of computer graphics and descriptive geometry topics. " and demonstrate the added value Added value in financial analysis of shares is to be distinguished from value added. Used as a measure of shareholder value, calculated using the formula:
In recent work by DeJong et al., (2004) the authors build a reduced form In social science and statistics, particularlly econometrics, a reduced form equation is a method of dealing with endogeneity. A reduced form equation is defined by James Stock & Mark Watson (2007) in the following way: model of GDP growth, to study its inherent nonlinearity and volatility. They use a regime switching model between accelerating and decelerating GDP growth rates (also see Kim et al., (2004) who try to capture the pace of output growth between recessions and recoveries, called the bounce back effects.) It follows the Filardo and Gordon (1998) outline where GDP growth accelerates or decelerates stochastically sto·chas·tic
1. Of, relating to, or characterized by conjecture; conjectural.
a. Involving or containing a random variable or variables: stochastic calculus. , but there are observable indicators or "tension index" which helps in predicting the probability and time of a regime shift. The authors justify the nonlinearities in the GDP growth path based on its "probable trajectory" and the accompanying rationale for it. At the onset of an expansion, the GDP has a regime drift component that grows/accelerates along an increasing trajectory. This leads to an increase in the "tension index" which enhances the probability of an oncoming regime shift. Once this happens, the process reverses itself, and the drift component begins to decline along a new path. This movement of the drift component between expansion and contraction phases generates a cyclical behavior in the GDP growth rate. Other forms of heterogeneity het·er·o·ge·ne·i·ty
The quality or state of being heterogeneous.
the state of being heterogeneous. observed in the dynamics of the GDP time path is that they are all different. No two business cycles are alike, differing either in the magnitude of the cycles (moderate long term growth versus rapid short bursts of growth.)
Given this preponderance of evidence A standard of proof that must be met by a plaintiff if he or she is to win a civil action.
In a civil case, the plaintiff has the burden of proving the facts and claims asserted in the complaint. favoring nonlinearities in the basic structure of GDP growth, the rational way to proceed would be to first test for nonlinearity. In most cases, nonlinearity in GDP growth has been assumed and not tested. Here we would like to take a step back, and test for nonlinearity in the US GDP, using a recently available and powerful test, which would account for one part of the contribution of this exercise.
The other has to do with the standard assumption made in the literature regarding stationarity of the variable under consideration, and only testing for nonlinearity, in isolation. This fundamental flaw in the econometric / empirical methodology encompasses all the parametric as well as nonparametric studies. They are all limited by their assumption that the series under consideration is stationary (more specifically trend stationary, and hence the common solution is simple detrending of the data), and therefore the test is for nonlinearity only. The problem here is that even if this assumption (of trend stationarity) is correct, this automatic detrending before setting up the actual threshold autoregressive model would bias the results. (2) Then comes the question "what if the series was a unit root process?" In that case the convergence would have to be justified only in terms of a non-linear adjustment, which actually could (and should) be partly explained by the inherent nonstationarity in the data. Along with that, the true but "undiagnosed" nonstationarity (actual unit roots in the series) would lead to incorrect nonlinear test result inferences, since they all have non-standard asymptotic distributions.
Again, whether level stationarity or otherwise (unit roots) is true has an important bearing for the time path of GDP or the trajectory of business cycles. For example, if real output is trend stationary, then shocks will have only a transitory impact on real output, with convergence happening soon enough. But if it is not so, then real factors such as technology shocks will have permanent ramifications ramifications npl → Auswirkungen pl (not fluctuations around a deterministic 1. (probability) deterministic - Describes a system whose time evolution can be predicted exactly.
2. (algorithm) deterministic - Describes an algorithm in which the correct next step depends only on the current state. trend.) Also, it would affect our forecasting capability and credibility of macroeconomic stabilization policies. The empirical evidence overwhelmingly supports non-stationarity of real GDP using both unit root methods (CC, 1996) or panel unit root tests (R, 2002).
Jones (2003) also stressed this flaw but in the context of short term interest rate dynamics. The problem here is that this "a priori" assumption of stationarity imposes restrictions on the possible shape of the drift function, and thus its determined shape itself is suspect. Not only that, but today "nonlinearity in drift" is gaining more and more advocates, mainly because it fits the real life data better. To circumvent cir·cum·vent
tr.v. cir·cum·vent·ed, cir·cum·vent·ing, cir·cum·vents
1. To surround (an enemy, for example); enclose or entrap.
2. To go around; bypass: circumvented the city. these problems, Jones uses a Bayesian approach to study the character of the drift/diffusion process which has exact finite sample properties, even if the process is nonstationary. He finds evidence favoring a nonlinear drift in high frequency data, but concludes that the extent of nonlinearity is dependent on the beliefs / assumptions made about the shape of the drift function. Ahn and Gao (1999) show that a nonlinear drift function improves on fixed income pricing models.
Thus the need for an empirical test of "linearity" versus "nonlinearity", but without any a priori assumptions about the (stationarity/nonstationarity) data series under consideration. This is where the recently available Caner-Hansen (2001) study comes into play. It is the first rigorous theoretical and empirical treatment of the simultaneous existence of both nonstationarity and nonlineraity. Now we do not have to assume one to test for the other in isolation, because it allows for nonlinear (asymmetric adjustments) dynamics for the unit root testing procedure. This procedure is much more powerful than conventional unit root tests, especially if the adjustment procedure is not symmetric, see EG (1998).
The data is from the OECD OECD: see Organization for Economic Cooperation and Development. (Main Economic Indicators Economic indicators
The key statistics of the economy that reveal the direction the economy is heading in; for example, the unemployment rate and the inflation rate. ) data set, available from the Estima Corporation. We have obtained quarterly GDP data for Canada, Japan, United Kingdom, and the United States. We then calculated the annualized annualized
Of or relating to a variable that has been mathematically converted to a yearly rate. Inflation and interest rates are generally annualized since it is on this basis that these two variables are ordinarily stated and compared. growth rates of GDP. The time period is Canada [1981 Q1-2004 Q1], Japan [1980 Q1-2004 Q1], United Kingdom [1960 Q1-2004 Q1], and United States [1960 Q1-2004 Q1].
In all the studies mentioned here and also in the literature examining regime shifts, the maintained assumption is that the data under consideration is ergodic Adj. 1. ergodic - positive recurrent aperiodic state of stochastic systems; tending in probability to a limiting form that is independent of the initial conditions and stationary. The tests then conducted are for data series nonlinearity and its type. CH (2001) is the first rigorous treatment of the simultaneous existence of both nonstationarity and nonlineraity. There are Wald and "t" test for unit roots, and a sequential Wald test The Wald test is a statistical test, typically used to test whether an effect exists or not. In other words, it tests whether an independent variable has a statistically significant relationship with a dependent variable. for threshold effects. This null has two components, one reflecting the unit roots but free from nuisance parameters and the other similar to the stationary case but nuisance parameter dependent. Thus the distributions are nonstandardized, and have to be derived in every case. The unit root Wald tests have an asymptotic null distribution In statistical hypothesis testing, the null distribution is the probability distribution of the test statistic when the null hypothesis is true. that depends on whether or not there is a threshold effect. These tests are superior (more powerful) than conventional ADF (1) (Application Development Facility) An IBM programmer-oriented mainframe application generator that runs under IMS.
(2) (Automatic Document Feeder) A paper stacker that feeds one sheet of paper at a time into the unit. tests, when the true process is indeed nonlinear. (3) A standard TAR model is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (1)
where [e.sub.t] is an i.i.d, error process and [lambda] is an unknown threshold, within the interval [lambda] [member of] [LAMBDA] = [[[lambda].sub.1], [[lambda].sub.2]] where each segment has a significant presence to be dubbed a regime. The i.i.d errors ensure that the first difference of the series [DELTA][y.sub.t] is stationary and ergodic, so that [y.sub.t] is itself integrated of order one. The regimes (i.e, the TAR models) are estimated by least squares.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
The threshold [lambda] is estimated by minimizing [[sigma].sup.2]([lambda]):
[??] = arg mi[begin strike through]n[end strike through] [??]([lambda]), [lambda] [member of] [LAMBDA] (3)
The first difference model:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
is estimated using the standard Wald and "t" statistic. Here the statistics are standard, but the sampling distribution is non standard. The test in eq.(1) is for the presence of threshold effects, under the joint hypothesis [H.sub.0]: [[theta Theta
A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ].sub.1] = [[theta].sub.2], implying no regimes. Under the null of no threshold effects, [lambda] is not identified, hence the testing procedure is nonstandard non·stan·dard
1. Varying from or not adhering to the standard: nonstandard lengths of board.
2. . The null hypothesis null hypothesis,
n theoretical assumption that a given therapy will have results not statistically different from another treatment.
n of [H.sub.0] : [[theta].sub.1] = [[theta].sub.2] = [theta], simplifies the model to
[DELTA][y.sub.t] = [rho][y.sub.t-1] + [mu] + [alpha]' x [??][y.sub.t-1] + [[epsilon].sub.t] (5)
where [??] [y.sub.t-1] = ([DELTA][y.sub.t-1] ... [DELTA][y.sub.t-k])' are two bootstrap See boot.
(operating system, compiler) bootstrap - To load and initialise the operating system on a computer. Normally abbreviated to "boot". From the curious expression "to pull oneself up by one's bootstraps", one of the legendary feats of Baron von Munchhausen. methods, one for stationarity and other for the nonstationary case. In testing for unit roots and nonstationarity CH discuss three possibilities. In eq. 1, [[rho].sub.1] and [[rho].sub.2] are the determinants of stationarity of [y.sub.t]. Under the null hypothesis,
(1) [H.sub.0] [[rho].sub.1] = [[rho].sub.2] = 0, [DELTA][y.sub.t] is stationary, indicating [y.sub.t] is an I(1) process
(2) If [[rho].sub.1] < 0, [[rho].sub.2] < 0 and (1 + [[rho].sub.1]) (1 + [[rho].sub.2]) <1, then the series is stationary and ergodic
(3) [H.sub.1] : [[rho].sub.1] < 0 and [[rho].sub.2] = 0
What if it is the intermediary case of a partial unit root ? Then,
[[rho].sub.1] < 0 and [[rho].sub.2] = 0
[H.sub.2] : or
[[rho].sub.1] = 0 and [[rho].sub.2] < 0
Here [y.sub.t] is stationary in one regime and nonstationary in another. Their test can distinguish amongst the three. The difficulty is that the null of a unit root ([[rho].sub.1] = [[rho].sub.2] = 0) is compatible with both the existence of a threshold ([[theta].sub.1] [not equal to] [[theta].sub.2]) or the nonexistence non·ex·is·tence
1. The condition of not existing.
2. Something that does not exist.
non of a threshold ([[theta].sub.1] = [[theta].sub.2]). But CH determine that the assumptions of these two situations are different and hence we can simultaneously distinguish between nonstationarity and nonlinearity. Using theorems 5 and 6 of CH, the distinction between linearity and nonlinearity lies in the identification of the threshold parameter [lambda]. With no threshold effects, [lambda] is not identified, and so its estimate "[lambda][??]" is random (as always) and so is [R.sub.t] With threshold effects, [lambda] is identified and with no randomness in [R.sub.t], it is equivalent to the case where [[lambda].sub.0] is known. The unidentified threshold bootstrap imposes the restriction [theta] = [[theta].sub.1] = [[theta].sub.2] (no thresholds) and [rho] = 0 (unit root). In this case the bootstrap p-value is the percentage of simulated test statistic [R.sup.b.sub.t] that exceeds [R.sub.t]. The identified threshold bootstrap requires simulation of the TAR process, and calculating [R.sup.b.sub.t]. Again the bootstrap p-value is the percentage of simulated [R.sup.b.sub.t] that exceeds [R.sub.t].(4) Thus we conclude that in the presence of nonlinearity, the CH threshold unit root tests have more power than the standard ADF tests.
NONSTATIONARITY-NONLINEARITY TEST RESULTS
The results for Canada, Japan, United States and for the United Kingdom are in tables I through 4 respectively. First we test for the presence of threshold effects before rejecting a linear model in favor of a non-linear one.
From table 1, for Canada we see the Wald statistic [W.sub.t] for threshold variables of the form [Z.sub.t] = [y.sub.t] - [y.sub.tm] for delay parameters m = 1, ... 4 are significant across most lags (for m=1, 4 at the 5% level and for m=2 at the 10 per cent level) and therefore rejects the null hypothesis of linearity in favor of a threshold model A threshold model in toxicology posits that anything above a certain dose of a toxin is dangerous, and anything below it safe. This model is usually applied to non-carcinogenic health hazards.
Edward J. Calabrese and Linda A. . Since the results are sensitive to the choice of "m" it is necessary to select "m" endogenously en·dog·e·nous
1. Produced or growing from within.
2. Originating or produced within an organism, tissue, or cell: endogenous secretions. . The least square estimate of m is equivalent to determining "m" such that [W.sub.T] is maximized. According to table 1, this corresponds to a value of m = 4. The p-value when m = 4 is significant at the 5% level. Even if the calculation of m is incorporated into the calculation of the bootstrap value, it is still significant at the 5% level, implying that it is likely that the GDP growth rate for Canada can be represented by a nonlinear model. Then we calculate the threshold unit root test statistic [R.sup.1.sub.t], [t.sub.1] and [t.sub.2] for all lags. All the bootstrap p-values of [R.sup.1.sub.t] are significant at the 1 per cent level. Caner and Hansen state that the [R.sub.1] statistic has power both against a stationary and a partial unit root alternative, and we should use the [t.sub.1] and [t.sub.2] statistic to determine which alternative hypothesis alternative hypothesis Epidemiology A hypothesis to be adopted if a null hypothesis proves implausible, where exposure is linked to disease. See Hypothesis testing. Cf Null hypothesis. might apply. At m=4, the bootstrap p-value of [t.sub.1] = 0.0096 and of [t.sub.2] = 0.336, which is evidence in favor of [[rho].sub.1]<0 and [[rho].sub.2] = 0. This means that there is a unit root in the [y.sub.t] series only in the second regime (when change in the GDP growth rate is greater than 0.887). Therefore, the dynamics of the real GDP series growth rate in Canada are different in the two regimes.
From table 2, for Japan we see the Wald statistic [W.sub.t] for threshold variables of the form [Z.sub.t] = [y.sub.t] - [y.sub.tm] for delay parameters m = 1, ... 4 are significant only for m = 4 at the 10 per cent level and this gives weak support for rejecting the null hypothesis of linearity in favor of a threshold model. The endogenously determined value of m is 4 according to table 3. The p-value when m = 4 is significant at the 10% level. However, if the calculation of m is incorporated into the calculation of the bootstrap value, it is no longer significant even at the 10 per cent level, implying that it is highly unlikely that the GDP growth rate for Japan can be represented by a nonlinear model, and that a linear model is probably more appropriate.
From table 3, for the USA we see the Wald statistic [W.sub.t] for threshold variables of the form [Z.sub.t] = [y.sub.t] - [y.sub.t-m] for delay parameters m = 1, ... 4 are significant only for m = 1 and 4 at the 5 per cent level and this gives mixed support for rejecting the null hypothesis of linearity in favor of a threshold model. The endogenously determined value of m is 4 according to table 5 since the Wald statistic is maximized. The p-value when m=4 is significant at the 5% level. Even when the calculation of m is incorporated into the calculation of the bootstrap value, it is still significant at the 5 per cent level, implying that it is likely that the GDP growth rate for the United States can be represented by a nonlinear model. Then we calculate the threshold unit root test statistic [R.sup.1.sub.t], [t.sub.1] and [t.sub.2] for all lags. All the bootstrap p-values of [R.sup.1.sub.t] are significant at the 1 per cent level. At m = 4, the bootstrap p-value of [t.sub.1] = 0.245 and of [t.sub.2]=0.0001, which is evidence in favor of [[rho].sub.1]=0 and [[rho].sub.2]<0. This means that there is a unit root in the [y.sub.t] series only in the first regime (when change in the GDP growth rate is less than -3.55). Thus, the dynamics of the real GDP series growth rate in the United States are different in the two regimes.
From table 4, for UK we see the Wald statistic [w.sub.t] for threshold variables of the form [Z.sub.t] = [y.sub.1] - [y.sub.tm] for delay parameters m =1, ... 4 are significant only for m=2 and 3 at the 5 per cent level and for m = 1 at the 10 per cent level and this gives support for rejecting the null hypothesis of linearity in favor of a threshold model. The endogenously determined value of m is 2 according to table 7 since the Wald statistic is maximized. The p-value when m = 2 is significant at the 5% level. Even when the calculation of m is incorporated into the calculation of the bootstrap value, it is still significant at the 5 per cent level, implying that it is likely that the GDP growth rate for the United Kingdom can be represented by a nonlinear model. Then we calculate the threshold unit root test statistic [R.sup.1.sub.t], [t.sub.1] and [t.sub.2] for all lags. All the bootstrap p-values of [R.sup.1.sub.t] are significant at the 1 per cent level. At m = 4, the bootstrap p-value of [t.sub.1] = 0.0.0083 and of [t.sub.2] = 0.112, which is evidence in favor of [[rho].sub.1] <0 and [[rho].sub.2]=0. This means that there is a unit root in the [y.sub.t] series only in the second regime (when change in the GDP growth rate is greater than -4.61). This means that the dynamics of the real GDP series growth rate in the United Kingdom are also different in the two regimes.
Thus our results indicate nonstationarity in the GDP growth rates for Canada, United States and the United Kingdom, and stationarity in the GDP growth rate for Japan. This implies that there are regime changes in Canada, United States and United Kingdom, and no regime changes in Japan. This is perhaps due to the fact that in Canada, U.S.A and U.K there have been changes in the party in power at the center over the sample period, whereas in Japan, except for a brief period, the same party has been in power over the entire sample period. The lack on any non-linearity in Japan can be explained, at least partially, by the anemic anemic
pertaining to anemia. growth rates of GDP in the 1990s, which obviously imply that there weren't any major structural changes. It has taken the Japanese government a long time to deal with the structural problems of the Japanese economy, and while the previous policies were in place, there were no structural changes and therefore no non-linearities.
Moreover, for Canada and the United Kingdom we find that the first regime (GDP growth rate is less than the threshold) has a unit root (is mean reverting) and the second regime (GDP growth rate is greater than the threshold) is stationary, whereas for the United States we find that the second regime has a unit root and the first regime is stationary. This implies that for Canada and the U.K. the government intervenes when the changes in the GDP growth rates are small, and the government does not intervene when there are large changes in the growth rate. This would imply that the government's objective is to stimulate the economy, and inflation is not a target. However, for the U.S. the government / Federal Reserve intervenes when the changes in the GDP growth rate are high, implying that inflation possibly is a target. For the U.S. the stationary first regime implies no intervention when changes are small.
An alternate explanation for this is the Romer
A Romer or Roamer is a simple device for accurately plotting a grid reference on a map. (1990) growth model, who states that "..technological change ... lies at the hearth of economic growth," and "technological change arises in large part because of intentional actions taken by people who respond to market incentives." (5) An implication of this model is that "an economy with a larger stock of human capital will experience faster economic growth." (6) The faster growth rates of GDP in the United States compared to Canada, U.K. and Japan in the 1990s can be explained by the rapid rate of advancement of technology in the U.S. compared to the other countries. This growth of technology can, at least partially, be explained by the fact that throughout this period the U.S. was the preferred destination for students and skilled workers throughout the world, and this increase in human capital enabled technology to improve and act as a prime accelerant ac·cel·er·ant
Accelerator. for GDP growth in the U.S. Given the larger role played by the private sector in the United States economy, market incentives for technological progress were more significant in the U.S. compared to the other three countries in our sample.
This would also explain the non-linearities in our results. If technology was progressing more rapidly in the United States, perhaps due to market incentives being more significant, the GDP growth rate would also be more rapid and non-linear, since market incentives were not likely to be spread evenly across the economy and over time. When incentives are the greatest, GDP growth rate would also be the greatest as the U.S. economy is driven by the private sector. If the government is concerned about inflation, this would give the government/Federal Reserve a reason to intervene to keep inflation under control. Therefore, in the United States we have an unit root in the second regime (GDP growth rate is greater than the threshold), whereas the first regime is stationary as when GDP growth rates are low the private sector does not have an incentive to invest in research and development and improve technology as the potential returns from doing so are low. In Canada and the United Kingdom the government plays a much bigger role in the economy, and consequently the private sector plays a smaller role. These two countries also have access to less human capital (for a variety of reasons) compared to the United States. Therefore, the pace of technological improvement will be less than in the United States. Due to the smaller size of the private sector, when GDP growth rates are small, the government often intervenes to attempt to stimulate research and development and therefore technological change. This would lead to non-linearities in the GDP growth rates in the first regime in these two countries. When GDP growth rates improve, and the government steps back, the smaller size of the private sector (and of human capital) in these two countries would mean that innovation and growth will not fluctuate a great deal, and are likely to be linear.
To summarize our results, we find significant evidence indicating that the real GDP growth rate in Canada, U.S.A. and U.K. is non-linear, while for Japan it is linear. An implication is that in Japan policymakers react in a similar way without regard to whether the changes in real GDP growth rate are large or small. This may be due to the fact that the same political party has been in power in Japan (with a brief exception) over the entire sample period. Also, the anemic growth rates of GDP in Japan in the 1990s haven't given the private sector much incentive to invest in technological progress, leading to linear changes. Canada, U.S.A. and U.K. have experienced changes in government and therefore changes in policy over this period, and hence have a nonlinear response to changes in the real GDP growth rate. Policymakers in Canada and the U.K. seem to react when changes are below the threshold (small changes). This could be because for political reasons they are targeting growth and not inflation. The larger role played by the government and the consequent smaller role for the private sector (due at least in part to the more limited access to human capital) implies that at smaller growth rates of GDP there is more government intervention, which together with the private sector might lead to non-linear changes in GDP growth rate. At higher growth rates of GDP when the private sector is left alone by the government, the lack of human capital might lead to more gradual changes in GDP. On the other hand, in the United States the access to human capital and the significantly larger private sector means that technology will improve most rapidly at higher growth rates when the private sector has an incentive to invest in research and development, but at lower growth rates the incentives are absent, and therefore the private sector will reduce investment, and the government's role in the economy is not large enough to have a significant impact. This also implies that the U.S. government/Federal Reserve will be primarily concerned with targeting inflation as they know that there is not much they can do with regards to economic growth, which is determined by the private sector. In Canada and the United Kingdom, the governments will be concerned with economic growth as they know that the private sector is unwilling or incapable of stimulating the economy by themselves.
The authors would like to thank Bruce Hansen for making the GAUSS programs available. Dutt would like to thank the State University of West Georgia In recent years, the university has been named by the Princeton Review as one of the Best Southeastern Colleges and one of America's Best Value Colleges. Its 109 programs of study include 60 at the bachelor's level, 45 at the master's and specialist's, two at the doctoral level and two for their Faculty research grant (#: 10000-1014408-12100-11000: 2004-05).
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SWARNA (BASU) DUTT
University of West Georgia, Carrollton
Emporia State University Emporia State University (ESU) is a comprehensive Regents university serving residents of Eastern Kansas. ESU is located in the city of Emporia, in Lyon County. ESU is just east of the Flint Hills and within two hours drive of the three major metropolitan areas of Kansas: Wichita, , Kansas
ADRIAN M. AUSTIN
University of West Georgia, Carrollton
(1.) Though some nonlinear setups cannot replicate the asymmetries well, some regime switching models with changing variances perform much better than constant variance models.
(2.) This paragraph is heavily dependent on Darbha and Patel (2004).
(3.) Tsay (1997) introduces unit root tests in the presence of threshold effects, but the autoregressive lags are constant across regimes (which is not true here) making it a special case of the CH methodology. Also, Gonzalez and Gonzalo (1998) examine a TAR(l) model with nonstationarity, but of a particular type, namely geometrically ergodic.
(4.) CH run Monte Carlo Monte Carlo (môNtā` kärlō`), town (1982 pop. 13,150), principality of Monaco, on the Mediterranean Sea and the French Riviera. simulations to show their relative strength vis-a-vis the conventional Dickey-Fuller (ADF) tests in the presence of thresholds.
Case 1: The condition [rho]1 = [rho]2 is imposed and [DELTA][mu] = 0 (no regimes), the ADF is more powerful than the CH threshold unit root test. But as [DELTA][mu] increases, the [R.sub.1t] and [R.sub.2t] tests are more powerful than the ADF test.
Case 2: This is where [rho]1 = 0, [rho]2 varies and [DELTA][mu] = 0, a partial unit root model. Here [R.sub.1t] and [R.sub.2t] have substantially greater power than the ADF test. The ADF test is particularly weak when [DELTA][mu] is large. Here the t-ratio test is itself enough to distinguish between the pure unit root, the partial unit root and the stationary cases.
Case 3: [[rho].sub.1] is fixed, [[rho].sub.2] varies and [DELTA][mu] = 0, the stationary case. Here [R.sub.1t] is the most powerful test with [R.sub.2t] a close second.
(5,6.) Romer (1990, pp. S72)
(6.) Romer (1990, pp S99)
Table 1: Canada Threshold and Unit Root Tests Unconstrained Model Bootstrap Threshold Test Unit Root Tests, p-Value [R.sub.1T] M [W.sub.T] 1%C.V. p-Value Asym Boot 1 21.7 26.4 0.0400 0.0055 0.0094 2 19.5 26.6 0.0795 0.0022 0.0048 3 16.7 27.1 0.162 0.0003 0.0016 4 25.5 27.0 0.0135 0.0054 0.0096 Bootstrap Threshold Test [t.sub.1] [t.sub.2] M Asym Boot Asym Boot 1 0.961 0.764 0.0023 0.0029 2 0.742 0.384 0.0021 0.0031 3 0.799 0.447 0.0002 0.0005 4 0.0076 0.0052 0.614 0.336 Notes for tables 1, 2, 3 and 4: (1) Bootstrap p-values are calculated from 10,000 replications (2) "M" is the delay parameter (3) [W.sub.T] is the Wald statistic which tests the null hypothesis of linearity against the alternate hypothesis of a threshold effect. (4) The [R.sub.1] statistic tests the null hypothesis of a unit root against the alternative hypothesis that either the first or the second regime is threshold stationary (the asymptotic and bootstrap p- values for the test statistic are reported. (5) The [t.sub.1] and [t.sub.2] test the null hypothesis of an unit root against the alternate hypothesis of stationarity in the first and the second regime respectively (the asymptotic and bootstrap p-values are reported) Table 2 : Japan Threshold and Unit Root Tests Unconstrained Model Bootstrap Threshold Test Unit Root Tests, p-Value [R.sub.1T] M [W.sub.T] 1%C.V. p-Value Asym Boot 1 16.1 27.8 0.188 0.0958 0.0754 2 11.5 27.2 0.530 0.0140 0.0186 3 8.88 27.7 0.783 0.181 0.135 4 18.8 27.5 0.0908 0.0053 0.0102 Bootstrap Threshold Test [t.sub.1] [t.sub.2] M Asym Boot Asym Boot 1 0.768 0.959 0.0448 0.0290 2 0.813 0.948 0.0059 0.0063 3 0.508 0.248 0.311 0.151 4 0.460 0.989 0.0022 0.0022 Table 3 : USA Threshold and Unit Root Tests Unconstrained Model Bootstrap Threshold Test Unit Root Tests, p-Value [R.sub.1T] M [W.sub.T] 1%C.V. p-Value Asym Boot 1 21.4 25.4 0.0396 0.0000 0.000 2 11.1 26.4 0.522 0.0000 0.0002 3 13.5 25.4 0.312 0.0000 0.000 4 26.1 25.6 0.0089 0.0000 0.0001 Bootstrap Threshold Test [t.sub.1] [t.sub.2] M Asym Boot Asym Boot 1 0.0010 0.0020 0.0008 0.0008 2 0.223 0.119 0.0002 0.0006 3 0.0379 0.0282 0.0002 0.0005 4 0.0456 0.245 0.0000 0.0001 Table 4 : UK Threshold and Unit Root Tests Unconstrained Model Bootstrap Threshold Test Unit Root Tests, p-Value [R.sub.1T] M [W.sub.T] 1%C.V. p-Value Asym Boot 1 18.9 27.0 0.0935 0.0000 0.0011 2 43.3 27.3 0.0004 0.0003 0.0018 3 28.0 27.4 0.0101 0.0002 0.0015 4 12.0 26.9 0.427 0.0010 0.0043 Bootstrap Threshold Test [t.sub.1] [t.sub.2] M Asym Boot Asym Boot 1 0.0986 0.0547 0.0003 0.0021 2 0.0642 0.404 0.0098 0.0126 3 0.957 0.743 0.0000 0.0005 4 0.0074 0.0083 0.212 0.112