Stochastic growth: Empirical evidence from the G7 Countries

LUCIO SARNO University of Oxford

Oxford, United Kingdom

Stochastic Growth: Empirical Evidence from the G7 Countries*

This paper provides a methodological contribution to the empirical literature on stochastic economic growth. The augmented Solow-Swan model with stochastic capital accumulation rates and labor force growth implies that the steady-state labor productivity level towards which an economy converges varies over time and across countries, and the adjustment process towards long-run equilibrium follows a nonlinear process. Using postwar data for the G7 countries, we provide strong empirical evidence supporting the e~dstence of the long-run relationship implied by the augmented Solow-Swan model. Moreover, the hypothesis of linearity of the adjustment towards long-run equilibrium is rejected in favor of a logistic smooth transition autoregressive process.

1. Introduction In recent years, a large body of empirical research on economic growth

has developed, largely stimulated by the highly influential work published in the late 1980s and early 1990s by, inter alios, Baumol (1986), De Long (1988), Barro (1991), Mankiw, Romer and Well (1992). One of the issues currently receiving widespread attention by researchers is testing the implications of the augmented version of the Solow-Swan (Solow 1956; Swan 1956) growth model. In particular, a reformulation of the Solow-Swan model which allows for a stochastic, as opposed to a constant, rate of accumulation of capital and rate of labor force growth implies that the steady-state equilibrium level of labor productivity towards which an economy converges is a stochastic process itself (Mankiw, Romer and Well 1992; Durlauf and John- son 1992). Also, this stochastic version of the Solow-Swan growth model leads to a notion of convergence which is different from the definition of convergence traditionally used in the relevant literature, and its implications can be tested using time-series techniques, as opposed to the cross-section

*This paper was begun while the author was a Visiting Scholar at Columbia University. The author is indebted to two anonymous referees for helpful suggestions and constructive criti- cisms, to Roberto Cellini for detailed comments on a previous version oft_his paper, to Richard Clarida, Tullio Jappelli and Xavier Sala-i-Martin for helpful suggestions, and to David Peel and Mark Taylor for discussions on relevant econometric issues. The author is also grateful to the Rotary Foundation for financial support. All remaining errors or omissions are solely the re- sponsibility of the author.

Journal of Macroeconomics, Fall 1999, Vol. 21, No_ 4, pp. 691-712 Copyright © 1999 by Louisiana State University Press 0164-0704/99/$1.50

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analysis which has been typical of the empirical research in this area. In practice, researchers have employed cointegration and error correction techniques to distinguish long-run and short-run in the relationship between labor productivity and the determinants of the stochastic steady-state of labor productivity. In general, however, the empirical evidence suggests the rejection of the augmented Solow-Swan model also using time series techniques (see Cellini 1997).

A possible rationalization of the failure to establish cointegration (at least with plausible estimates of the eointegrating vector) on the fundamental regression implied by the augmented Solow-Swan model may be that the number of observations used is often too low to provide any reasonable degree of test power in the conventional statistical tests for nonstationarity of the residuals of the cointegrating regression. Accordingly, one possible remedy to the problem of lack of power could be to increase the sample period under investigation. However, with annual data, the long samples required to generate a reasonable level of statistical power with standard stationarity tests, may be unavailable for many countries and may also po- tentially be inappropriate because of regime changes or structural breaks, implying that inferences drawn on long sample periods may become unre- liable. Moreover, in investigating the low-frequency characteristics of time series processes, Shiller and Perron (1985) note that the span of the data set, in terms of years, is more important than the number of observations per se (see also Davidson and MacKinnon 1993, chap. 20). Therefore, to shed light on the existence of the long-run relationship implied by the augmented Solow-Swan model, it may be preferable to use statistical inference based on data for the postwar period only.

In addition, recent research also suggests that conventional tests for nonstationarity lose power when the true data generating process is nonlinear (Enders and Granger 1996; Balke and Fomby 1997; Taylor and Peel 1997; Taylor and Sarno 1998). This is particularly relevant in the present context since the steady-state labor productivity level towards which an economy converges according to the augmented Solow-Swan model is governed by an equation which predicts nonlinear adjustment towards long-run equilibrium.

The aim of this paper is to provide an econometric methodology to test the implications of the augmented Solow-Swan model, specifically allowing for nonlinear adjustment towards long-run equilibrium in the fundamental Solow-Swan cointegrating regression. Using postwar data for the G7 countries, our analysis results in the application of a linear cointegration procedure which is relatively more robust to the presence of nonlinearity in the data generating process, while the adjustment towards equilibrium is

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explicitly modeled as a logistic smooth transition autoregressive process of the type popularized by Granger and Terasvirta (1993) and Ter~isvirta (1994).

The remainder of the paper is set out as follows. In Section 2 we briefly outline the augmented Solow-Swan model and some of the relevant empirical literature. In Section 3 we introduce the parametric nonlinear models considered in this paper as well as the econometric strategy followed for linearity testing and for model specification, estimation and evaluation. In Section 4 we describe the data and discuss the empirical results. A final section concludes.

2. The Solow-Swan Model in the Presence of a Stochastic Steady State

Consider a production function with constant returns to scale, for which the Inada (1963) conditions hold:

Y, = ~ H ~ (E&)~, (1)

where Y, K, H and L denote the flow of output, the level of physical capital, the level of human capital and labor respectively; E is a time-varying labor- augmenting technological shift parameter, identical across countries (so that E L may be thought of as the supply of efficiency units of labor); a and 13 denote technology parameters, 3' --- 1 - oL - 13 and t is a time subscript. Physical and human capital accumulation evolve according to the following equations:

d K / d t = SK, t Yt - 5 K t ,

d H / d t = SH,t Yt - 8Pi t , (9.)

where sK and s H denote the fractions of output devoted to the accumulation of physical and human capital respectively, and 5 is a constant depreciation rate, identical for physical and human capital. Assuming that the rate of exogenous technical progress, g, is constant and that labor force grows at a rate nt such that

E/Eo = e ~ , (3)

Lt/Lo = e ~' , (4)

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then, if sK, SH and n are constant, the constant steady-state or equilibrium level for K/EL, H / E L and Y / E L can be derived:

_ _ _ _

/ ,

g g ,

(5)

where ~ - n + g + 8; hats and stars refer to variables defined in efficiency units and to steady-state variables, respectively. Also, the steady-state productivity equation implied by the Solow-Swan model may be derived as

lny* = (lnE0 + gt) + (cdT)lnsk + (13/7)lnsn - [(a + 13)/7] ln~. (6)

If sK, Sn and n are allowed to vary over time as well as across countries, however, the steady-state labor productivity level towards which an economy converges also varies over time and across countries and, in the neighbor- hood of the steady-state path, labor productivity is governed by the equation (see Mankiw, Romer and Weil 1992; Durlauf and Johnson 1992; Cellini 1997):

Alnyt+l = g + (1 - e -°') {(lnE0 + gt) + (odT)lnsK,t

+ (13/7) lnSH,t -- [(a + 13)/7] In ~t -- lny~}

= g + (1 -- e-°0 {lny* -- lny}t, (7)

where A denotes the first difference operator, 0t =- ~t7 and ~t =- (nt + g + 8). Noting that inside the braces of Equation (7) one finds the difference between the steady-state level of labor productivity, determined by sK, t, si_i, t and nt, and the current level of labor productivity, Equation (7) may be interpreted as a nonlinear error correction model such that labor productivity in a given country will rise (fall) if its current level is below (above) its steady-state level. Interestingly, however, the speed of convergence of a country's labor productivity towards its steady state, 0t, varies over time and depends upon the labor force growth rate, nt, implying that the steady-state path is conditional on the current level of physical and human capital accumulation rates as well as on the labor force growth rate. In other words, in this version of the Solow-Swan model, given the stochastic nature of the

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variables SK,t, SH,t and nt, the equilibrium level of labor productivity becomes, by consequence, also stochastic.

Given (7), error correction and cointegration techniques seem to be a natural way of testing the implications of the Solow-Swan model with stochastic steady state. This can be done by testing for cointegration between labor productivity, the propensities to accumulate physical and human capital and the labor force growth rate, provided those variables are found to be integrated processes of order one. Establishing cointegration between labor productivity in a given country and its stochastic determinants implies the stationarity of the difference between the steady-state level of labor productivity and its current level, and, consequently, the stationarity of labor productivity growth.

Notably, Cellini (1997) tests the augmented Solow-Swan model using the two-step Engle-Granger procedure (Granger 1986; Engle and Granger 1987), the Johansen cointegration procedure (Johansen 1988, 1991) and the procedure proposed by Phillips and Loretan (1991), using 29-year annual time series for the U.S., France, Italy and Japan. His empirical results suggest, however, that, although a stochastic trend is generally found in the variables in question, there is no tendency of the current labor productivity towards its equilibrium path in the estimated error correction models. Nev- ertheless, Cellini (1997, 151) admits that "we are aware that additional investigation is required and improvements in this methodology are possible and necessary."

The failure to establish a long-run relationship between labor productivity and its determinants in a given country can be rationalized on the basis of the argument that the span of available data is too short to provide any reasonable degree of test power in the conventional statistical tests for nonstationarity. Moreover, this problem is particularly important if the adjustment towards equilibrium is relatively slow, that is, shocks to the steady- state level of labor productivity are highly persistent, and also if linear cointegration techniques are employed when the true adjustment towards equilibrium is nonlinear (see Enders and Granger 1996; Balke and Fomby 1997; Taylor and Peel 1997; Taylor and Sarno 1998). Equation (7) provides a significant insight into the nature of steady-state labor productivity, clearly showing how the adjustment towards equilibrium implied by the augmented Solow-Swan model may be nonlinear and, therefore, the difference between the current labor productivity and its steady-state level follows a process which reverts toward its mean nonhnearly, t

1King, Plosser, Stock and Watson (KPSW) (1991) present empirical evidence that the Solow- Swan model satisfies a cointegrating relationship in a different perspective, as cointegration arises between labor productivity and total factor productivity. Most tellingly, however, Granger

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The implications of the stochastic version of the Solow-Swan model are also consistent with the definition of convergence proposed by Bernard and Durlanf (1995), who use annual data from 1900 to 1987 for fifteen OECD countries and provide evidence that, although differences in income per capita across OECD countries have diminished over the sample period, there is no strong tendency toward full convergence. 2

The present paper may be seen as an extension of this strand of the literature, in that we use postwar data for the G7 countries and estimate the augmented Solow-Swan model using cointegration techniques while allowing for nonlinearity in the adjustment toward long-rnn equilibrium, which is explicitly modeled as a smooth transition autoregressive process.

3. Econometric Methodology Testing the Implications of Stochastic Growth

Given the reformulation of the Solow-Swan model described in Sec- tion 2, the model can be tested by testing for cointegration between labor productivity, the propensities to accumulate physical and human capital and labor force growth, provided those variables are found to be integrated processes of order one, and therefore, for the model not to be rejected, the deviation from long-run equilibrium has to be a stationary process. In addition, as shown in Section 2, the Solow-Swan model in the presence of stochastic steady-state is consistent with a nonlinear adjustment process.

The characterization of the nonlinear adjustment process we consider in the present context is in terms of a smooth transition autoregressive (STAR) model (see Granger and Ter~isvirta 1993; Teriisvirta 1994, 1996), where adjustment takes place in every period but the speed of adjustment varies with the extent of the deviation from equilibrium. While STAR models have proved to be very successful in modeling macroeconomic time series in various contexts (see Granger and Teriisvirta 1993, chap. 9), it is clear that the nonlinear adjustment theoretically implied by the augmented Solow- Swan model does not provide a compelling rationale for STAR models. In fact, our choice to characterize the nonlinear adjustment process in this context using a STAR model is mainly motivated on statistical grounds, as strong empirical evidence in favor of nonlinearity of the type displayed by STAR models has been provided using output, industrial production and

and Swanson (1996) use an updated version of the data set used by KPSW and are able to detect very significant nonlinearity in the time series examined, which could be modeled using various nonlinear specificalSons yielding a remarkable improvement in both goodness-of-fit statistics and forecasting relative to linear time series methods.

2See also Campbell and Manldw (1989), Cogley (1990), Bernard and Durlauf (1996) and Bernard and Jones (1994; 1996a, 1996b).

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Stochas t ic G r o w t h : E m p i r i c a l E v i d e n c e

other related time series by a number of researchers (notably Granger and Ter~isvirta 1993; Ter/isvirta 1995; Skalin and Ter~isvirta 1996). Inadequate- ness of STAR models to describe the behavior of the equilibrium error from the augmented Solow-Swan model may be expected to show up in the non- rejection of the linearity test statistics employed here--as these are tests of the null hypothesis of linearity against the specific alternative hypothesis of STAR--or in the inability to fit a STAR model with plausible estimates of the parameters. With this caveat in mind, however, the empirical results discussed below suggest that, in modeling deviations from the steady state, STAR models provide a very significant improvement in terms of goodness- of-fit, in particular, yielding a strong reduction of the residual variance, relative to the best-fitting linear autoregressive model.

Hence, let us assume that the deviations from equilibrium in the Solow-Swan model with stochastic steady state can be described by the logistic STAR (LSTAR) model or by the exponential STAR (ESTAR) model given by Equations (8) and (9), respectively:

gt = ~10 -~- ~{wt -4- (~20 + 7~2wt) [1 + exp{-Z L (zt-~ - CL)}] -1 + us, (8)

z~ = ~ho + rt~w, + (g20 + g~wt) [1 - exp{-Z~(zt_d - %)2}] + v~ ; (9)

where zt is a stationary and ergotic process, representing in the present context the equilibrium error from the augmented Solow-Swan model; rq0, gto, rt2o, g2o, CL, Ce are constant terms; ?~j = (•jl, ' ' ' , 7tjp)' and gj = ( ~ l j l ,

. . . . gjp)' forj = 1,2; w t = (z t -1 , • • . , Z t -p) ' ; d is a delay parameter (d > 0); ut ~ iid (0, cry) and vt ~ iid (0, cry); and )~L, )~e > 0. The expressions inside the square brackets in Equations (8) and (9) denote the transition functions FL(Z, -a) = [1 + exp[--)~c(Z~_ d -- CL)}] -~ and Fe(Zt -d ) = [1 -- exp{- )~E(Z~-d -- ce)Z}] which characterize the transition o f zt from one regime to another, and the parameters )~L and Ke determine the speed of the transition between the two regimes for the LSTAR and the ESTAR model respectively. The transition function of the LSTAR is a monotonically increasing function of zt-e and yields asymmetric adjustment towards equilibrium, whereas the transition function of the ESTAR is symmetric about ce although the tendency to move back to equilibrium is stronger the larger the deviation from equilibrium.

Nevertheless, the intuition behind the two STAR models is quite different. The LSTAR approaches a two-regime TAR model when )~L goes to infinity, as its transition function is a step function of Zt-d, which switches from zero to unity at zt-d = CL. When EL goes to zero, however, the LSTAR approaches a linear AR(p) model. Hence, the local dynamics of the LSTAR

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Lucio Sarno

is very different for high and low values of zt-a. For the ESTAR, however, the transition function approaches a linear AR(p) model both when kc goes to infinity and when it goes to zero; the middle regime corresponds to Zt-d = ca when F = 0, while the outer regime corresponds to zt-a = -+ ~ when F = 1. Hence, the ESTAR transition function, being symmetric about ca, is the same for high and low values Of Zt_d, while the mid-range behavior of the variable is different, a

In our empirical analysis, we select the most adequate STAR to model the deviation from equilibrium in the augmented Solow-Swan model on a pure econometric ground, following a selection procedure due to Teriisvirta (1994).

Modeling Nonlinear Adjustment Since the transition function in (8) ((9)) implies that F = 0 when EL

= 0 (EE = 0), the linearity hypothesis may be expressed as Ho: ~'L = 0 (~-E = 0). However, (8) ((9)) is only identified under the alternative hypothesis HI: LL > 0 (Xa > 0) since, if Xz = 0 (La = 0) is true, x20 (~t20), the vector n2 (g2) and CL (cE) can take any value. This problem is overcome deriving a Lagrange Multiplier (LM) test statistic, LM1, under the assumption that the unidentified parameters are fixed, and then the value of the statistic corresponding to sup LM1 is selected. In the case of STAR models, the resulting statistic follows the ~2 distribution under the null hypothesis of linearity. 4 Teriisvirta (1994) derives LM-type tests of linearity against LSTAR or ESTAR models and also suggests a decision rule for choosing between LSTAR and ESTAR. If the delay parameter d is fixed, the linearity test against STAR consists of estimating by ordinary least squares (OLS) the auxiliary regression,

P 3

j = l (10)

and testing the null hypothesis that

HOL: ~qj = K2j = Kaj = 0 {j = 1 , . . . , p) (n)

against the alternative that HOL is not valid.

aSee Granger and Ter~isvirta (1993) or Ter~isvirta (1996) for a more detailed discussion of the behavior of the transition functions of STAR models.

4In practice, the solution involves approximating the transition function by a Taylor series expansion around the equilibrium and reparametefizing in such a way that the identification problem disappears (Saikkonen and Luukkonen 1988; Luukkonen, Saikkonen and Ter~svirta 1988).

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To specify d, the linearity test is repeated for a set of values d = 1,2, . . . . . . D. If linearity is ~jected for more than one value of d, then d is determined as the value (d) that minimizes the P-value of the linearity test (linearity is rejected most strongly when d = d). In practice the ordinary F-test is used as an approximaldon to the LM-type test to improve size and power properties, which is particularly important when the order p of the linear AR model is large and the number of observations is small (eg., Harvey 1990).

The decision rule proposed by Ter/isvirta for choosing between a LSTAR and an ESTAR model is based on a sequence of nested tests within (10). First, the null hypothesis HoL in (11) must be rejected using an ordinary F-test (FL). Then the following hypotheses are tested:

H0a:K~ = 0 5 ' = 1 . . . . . p) ; (12)

H02 : lc2j = 0h% = 0 (] = 1 . . . . , p); (13)

H01 : Klj = 0[K2j = K3j = 0 (] = 1 , . . . , p) ; (14)

Again, an F-test is used, with the corresponding test statistics denoted by Fa, F2 and F1, respectively. The decision rule is as follows: 5 After rejecting (11), the three hypotheses (12)-(14) are tested using F-tests. If the test of (13) has the smallest P-value, an ESTAR model is chosen; otherwise a LSTAR model is selected (see Terasvirta 1994, 210).

Hence, our modeling strategy may be summarized as follows. First, we specify the order p of the linear AR model on the basis of the partial autocorrelation function (PACF) ofzt and other standard criteria such as the Akaike information criterion (AIC), the Schwartz information criterion (SIC) and the method proposed by Campbell and Perron (1991). Second, we specify a set of plausible values (1,2 . . . . , D) for d. For each value old, we test linearity against STAR using F L, the F statistic associated with HOL in (11); if linearity is rejected, we select d = d such that minimizes the marginal significance level of the linearity test. Third, we execute the nested tests (12)-(14) and use the Ter~isvirta rule to choose between a LSTAR and an ESTAR.

5The sequence of tests is based on the relationship between the parameters of (10) and those of the corresponding nonlinear model. In parlScular, based on a third-order Taylor expansion, the restrictions ~caj = 0 (and, if 1~20 = ce = 0, ~qj = 0) for a l l j hold for ESTAR models, whereas for LSTAR models they can only hold if n 2 is a vector of zeros. On the other hand, the restrictions Ic2j = 0 for allj hold for LSTAR models if n20 = CL = 0, whereas for ESTAR models they hold only if btz is a vector of zeros.

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Lucio Sarno

Estimation of STAR is by nonlinear least squares, which provides estimators that are consistent and asymptotically normal (see Klimko and Nelson 1978; Tong, 1990, 299-302). To select a good starting value for ~,i (i = E, L), Ter~isvirta (1994) suggests standardizing the exponent of the transition function by dividing it by the sample variance of zt (£i = 1 is recom- mended as starting value). To check whether a global minimum has been achieved, estimation may be carried out with different starting values of ~.

Model evaluation includes a check of whether the estimates seem reasonable and the computation of the roots of the polynomials corresponding to the lower and upper regimes to investigate the stability properties of the estimated model. Evaluation, of course, also involves an examination of the residuals for autocorrelation, autoregressive conditional heteroskedasticity (ARCH) and normality as well as an examination of goodness-of-fit statistics, such as the coefficient of determination and the ratio of the estimated STAB residual variance to the residual variance of the alternative linear p-th order AR model.

4. Empirical Analysis Data

Annual data for real gross domestic product (GDP) per worker at constant international price level, real GDP per capita at constant international price level, investment share of GDP and population were taken from the Summers-Heston (1991) Penn World Tables. In addition, annual data for the secondary school enrollment rate and population for different age groups were taken from UNESCO sources. All data cover the sample period from 1950 to 1990 for the G7 countries, providing us with 41 observations. Using these data we constructed the data set employed in the empirical analysis: the natural logarithm of real GDP per worker at constant international prices (Y), the natural logarithm of the investment share of GDP divided by 100 (SK), the natural logarithm of the secondary school enrollment for the appropriate age group divided by 1000, used as a proxy for the propensity to human capital accumulation (SH) (see Manldw, Romer and Well 1992, 419, Barro and Lee 1993), and the natural logarithm of the sum of the employ- ment growth rate plus the technological progress rate plus the depreciation rate (Z) (we assumed that g + 8 = 0.05).

While we admit that the number of observations is still relatively lim- ited for certain statistical tests, the span is relatively long and available for all major industrialized countries, and may allow us to capture some of the features of the economic growth processes experienced by the countries examined. Nevertheless, the main focus of the paper remains methodolog-

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ical in the sense of proposing an estimation strategy for empirical investi- gations of economic growth.

Linear Cointegration Tests The Solow-Swan model in the presence of a stochastic steady state

sketched in Section 2 requires that labor productivity and its determinants are nonstationary, I(1) processes and that a long-run relationship exists between them, that is, they cointegrate, and the deviation from long-run equilibrium is stationary.

The empirical results from executing augmented Dickey-Fuller test statistics for nonstationarity on Y, SK, SH and Z, reported in Table 1, did not enable us to reject the null hypothesis of unit root behavior for any of the series in levels, with the exception of SK for the U.S., 6 while we were able to reject the null of nonstationarity for each of the series in first difference, concluding therefore that all series in question are realizations from stochastic processes integrated of order one.

As a preliminary to the tests of linearity of the adjustment towards equilibrium in the Solow-Swan model, we need to establish a series for the deviations from the long-run equilibrium of labor productivity, which we derive as the residuals retrieved from the cointegrating regression involving a deterministic linear trend, Y, SK, SH and Z. The Monte Carlo results provided by Balke and Fomby (1997) suggest that estimation of the long- run linear equilibrium using the Johansen (1988, 1991) cointegration procedure when the true adjustment towards equilibrium is nonlinear does not yield misleading results in terms of significant loss of power or size distortion, and therefore we employ this procedure to test for cointegration and estimate the equilibrium error.

In Table 2 (Panels A-B) we report the results for the maximum eigen- value and the trace statistics from employing the Johansen procedure on a first-order vector autoregression (VAR) with Y, SK, SH and Z, also allowing for a deterministic linear trend. 7 The empirical results clearly suggest that for all countries considered there is a long-run relationship between the variables considered and the cointegrating vector is unique.

6This finding is also consistent with the results provided by Cellini (1997), despite the extension of the sample period in the present paper.

TWe were very careful in adequately selecting the number of lags in the VARs, being aware of the sensitivity of vector autoregression analysis to the lag length in this context. Twice lagged variables were found not statistically significantly different from zero at conventional nominal levels of significance and a lag length of one was also suggested by the AIC and the SIC. In fact, using Monte Carlo simulations, Cheung and Lai (1993) show that for autoregressive processes with no moving average dependencies the AIC and the SIC indicate the right lag order of a VAIl used for testing for cointegration in 99.86% and 99.96% of cases, respectively.

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In Table 2 (Panel C) we also report the estimated cointegrating vector after normalizing the coefficient of Y to - 1. The results are very satisfactory in that the estimated coefficients on SK, SH and Z are correctly signed and the hypothesis that the sum of the coefficient on SK plus the coefficient on SH equals the coefficient on Z could not be rejected for all countries considered, providing therefore strong support to the augmented Solow-Swan model, s'9 Interestingly, the estimated coefficients do not differ widely across G7 countries, yielding sensible implied values of the structural parameters (see also Romer 1987, 1989; Barro and Sala-i-Martin 1992; Durlauf and Johnson 1992; Mankiw, Romer and Well 1992; Cellini 1997). In general, for all countries considered, while human capital appears to play a significant role in the cointegrating regression, the implied value of the physical capital accumulation rate is always higher than the implied value of the human capital accumulation rate (a > 13). The estimated coefficients on SK and SH given in Table 2 (Panel C) imply, for example, that a = 0.4402 and

= 0.2283 for the U.S., whereas c~ = 0.3858 and ]3 = 0.3204 for the U.K. 1° Mso, as the estimated cointegrating vectors are fairly consistent with

our economic priors, we felt justified in using the cointegrating residuals, that is, the restricted equilibrium error (zt) for each of the countries examined, as the basis for our linearity tests and econometric modeling of the nonlinear adjustment towards long-run equilibrium in the augmented Solow-Swan model.

Linearity Tests and STAR Estimation Results Upon examination of the partial autocorrelation function of the equi-

librium errors and other conventional criteria such as the AIC, the SIC and

SNote that Cellini (1997) is able to establish cointegration for the U.S. and France using a shorter sample period, but the estimated parameters of the cointegrating vector are often incorrectly signed or insignificant.

"These results should be taken with caution, however, in light of the fact that the Johansen cointegration procedure is known to be biased towards rejection of the null hypothesis in a small sample (Cheung and Lai 1993). The critical values used here are taken from Osterwald- Lenum (1992). While some of the statistics reported in Table 2 are very large, other stalSstics (especially for the U.S. and Japan) are quite close to the relevant 5% critical value. Nevertheless, in the subsection "Linearity Tests and STAR Estimation Results," we model the cointegrating residuals as nonlinear LSTAR models, which are obviously stationary, hence yielding important complementary evidence supporting the presence of cointegration m the VARs estimated in the present section.

1°We also tested for cointegration using the Engle-Granger procedure, but we were always unable to reject the null hypothesis of no-cointegration. This is not surprising, however, given that the augmented Dickey-Fuller test statistic is known to have low power in small sample In addition, the Engle-Granger procedure does not allow us to investigate the possibility of mul- tiple cointegralfing vectors among the four variables included in the estimated regression.

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TABLE 3. Linearity Tests

Panel A: P-values for the linearity test F c (d = 1, . . . , 4; p = 1) U.S. U.K. Japan Germ. France Canada Italy

d = 1 0.252 0.755E-2 0.177E-4 0.269E-2 0.464 0.068 0.045 d = 2 0.305 0.151 0.094 0.598E-2 0.307 0.375E-2 0.031 d = 3 0.239 0.496 0.890 0.624 0.111 0.720 0.068 d = 4 0.031 0.694 0.478 0.262 0.079 0.543 0.278

Panel B: P-values for the linearity tests F a, F 2 and F 1 (given d) U.S. U.K Japan Germ. France Canada Italy

Fa 0.374E-2 0.021 0.402E-3 0.021 0.081 0.352E-2 0.039 /72 0.017 0.072 0.586E-3 0.047 0.382 0.159E-1 0.050 F1 0.375E-2 0.022 0.714E-3 0.013 0.183 0.367E-2 0.041

NOTES: The statistics FL and Fi (i = 1, 2, 3) are Lagrange multiplier test statistics for linearity constructed as described in the text. The marginal significance levels are calculated using the appropriate F-distribution; d denotes the delay parameter and p the order of the autoregression.

the method proposed by Campbell and Perron (1991), we concluded that one lag was adequate for all countries considered and hence we set p = 1 in the artificial regressions used for the linearity tests on zt.

In Table 3 (Panel A) we report the linearity tests based on F c and the auxiliary regression (10); the results indicate that linearity is rejected for five G7 countries at the i% nominal level of significance, for the U.S. at the 5% level and for France only at the 10% level. Given that nonlinearity is more likely to be found on high-frequency data and tends to be strongly reduced as an effect of aggregation (see Granger and Lee 1993; Sarno 1998), these findings may be regarded, with the exception of the results for France, as strong evidence of nonlinearity. Having considered values of the delay parameter between one and four (d = 1 , . . . ,4), the linearity tests suggested that d = 1 for the U.K., Germany and Japan, while d = 2 for Canada and Italy, and d = 4 for the U.S. and France. We then executed the tests F3, F2 and F 1 and applied the Teriisvirta (1994) rule to select between a LSTAR and an ESTAR. On the basis of the results reported in Table 3 (Panel B), the Ter~svirta rule led us to choose a LSTAR for the equilibrium errors of all countries considered. H'12

nNote, however, that for France the null hypothesis could not be rejected for any of F a,/72 and F1. Although it is a relatively less powerful linearity test statistic (see Luukkonen, Saikkonen and Ter~isvirta 1988), we also computed CUSUM test statistics, which were found significant

705

Lucio Sarno

Given the encouraging results from the linearity tests, we estimated LSTAR models of the form (8) with p = i and d set according to our linearity test results for each of the equilibrium errors. The results, reported in Table 4, indicate that the estimated LSTARs have similar features, and, in particular, similar speed of adjustment coefficients (in the range between - 0.882 for Japan and - 1.395 for the U.K.) with the exception of the French equilibrium error, whose results should be taken with caution since linearity was rejected only at the 10% nominal level of significance.

Goodness-of-fit statistics are also very satisfactory, with the coefficients of determination in the range between 0.769 and 0.908. Most tellingly, we computed the ratio of the residual variance of the estimated LSTAR models to the residual variance of a linear autoregressive process of order one, V; the results show that six out of the seven estimated LSTAR models (except for France) lead to a significant reduction, up to 23% for Germany, of the residual variance relative to the alternative linear autoregressive model. 13

A battery of diagnostics is also reported in Table 4, including a Jarque- Bera test for normality, a test for ARCH, a Ljung-Box test for residual serial correlation and a Durbin-Watson statistic, none of which was found significant at conventional nominal levels of significance.

The analysis of the stability properties of the estimated LSTAR models displayed is very easy since, with p = 1, the characteristic polynomials corresponding to the "lower" regime (when F(Zt-d) = 0) and the "upper" regime (when F(zt-d) = 1) are simple first-order difference equations. The roots of the polynomials for both the lower and the upper regime for all

for six out of seven G7 countries (except for France) at the 5% nominal level, providing important additional evidence of nonlinearity.

l~We addressed thoroughly the question of the robusteness of our lineanty test results. The main concern involves the possibility of a spurious rejection of the linearity hypothesis under the test statistic F1 in Equation (10), given the small number of observations used. We addressed this issue by executing a number of Monte Carlo experiments constructed using 5,000 repli- cations in each experiment, and with identical random numbers across experiments, Our simulations, based on a sample size comprising 40 artificial data points, suggested that the linearity test FL does not tend to over-reject the null hypothesis oflinearitywhen the true data generaiSng process is linear autoregressive, even if nonstationary, and rejection of the null does not occur by chance when the process is even marginally nonlinear LSTAR, thus increasing our confidence in the linearity test results (full details on the Monte Carlo simulations are available on request).

lZIn general, ff one incorrectly rejects linearity, the likely outcome is that an appropriate STAR model is not found. Nevertheless, a nonlinear model could still be "spuriously" fit even if the process is in fact linear; in that case, if a STAR model can be estimated, the residual variance is likely to be very close to the residual variance of the linear autoregressive alternative (Granger and Ter~virta 1993). In the case of our LSTAR model for France, despite the nonlinearity detected not being strong, a LSTAR could be estimated, generating also a slight reduction of the residual variance relative to an AR(1).

706

I d ~ l v v

v i ~ ~

t-.- co

I ~ ~

I I

v , . . ~ 0

N N

• ~ . ~

-~ ~

~ . . ~ ~ ' ~

r.,O . 0 " ~ h D

, . ~ 1~,-~"7 . ~

. ~ ,..~ e ~

~ ~:.~ .~.~

707

Lucio Sarno

estimated LSTAR models are always inside the unit circle, implying that the LSTARs are not explosive under any of the two regimes.

Overall, therefore, the estimation results are extremely satisfactory. We have uncovered evidence of nonlinearity, and are able to model the deviation from equilibrium in the augmented Solow-Swan model as a LSTAR model for all G7 countries. Also, the estimated LSTAR models have similar features, display satisfactory goodness-of-fit statistics and allow a significant reduction of the residual variance relative to the linear autoregressive counterpart. 14

5. Conclusions In this paper, we have examined the implications of the Solow-Swan

model with stochastic, nonstationary capital accumulation rate and labor force growth. In particular, the augmented Solow-Swan model implies that the steady-state labor productivity level towards which an economy converges varies over time and across countries, a long-run relationship exists between labor productivity and the stochastic determinants of the steady state and the adjustment process towards equilibrium follows a nonlinear process.

Using postwar data for G7 countries, we can reject the hypothesis of no-cointegration in the fundamental regression of the stochastic Solow-Swan model, suggesting that physical and human capital accumulation rates and labor force growth, all found to be first-difference stationary processes, generate a stochastic long-run equilibrium level of labor productivity. In addition, we provide uncovered empirical evidence that the hypothesis of linearity of the adjustment towards long-run equilibrium in the model is rejected in favor of a logistic smooth transition autoregressive process.

In general, therefore, our empirical analysis suggests that the augmented Solow-Swan model is consistent with the international evidence from the G7 countries, implying that cross-countries differences in income per capita may be explained primarily by cross-country differences in saving rates, human capital and population growth.

Future research is expected to introduce in the analysis other factors and, in particular, education policies and political stability, in order to assess their role in affecldng cross-country differences in the determinants of the

14The finding that LSTAR well characterizes deviations from the steady state in the augmented Solow-Swan model may also be interpreted as evidence of business cycle asymmetries. In that sense, the LSTAR model describes a situation where the contraction and expansion phases of an economy display different dynamics and the transition from one phase to the other occurs smoothly (see the references in Granger and Tergsvirta 1993, 141~t7).

708


stochastic steady state) 5 Also, it should be interesting to employ the esti- mat_ion strategy suggested in this study on data for developing countries. Most importantly, although the empirical results reported in this paper may be regarded as quite satisfactory, we believe that further insights may be gained by experimenting with alternative nonlinear processes for modeling the adjustment in the augmented Solow-Swan model in a fashion relatively more directly linked to economic theory; a natural extension of this work is to examine the nonlinear error correction models, whose theory is currently being developed, implied by the nonlinearity in the deviations from long- run equilibrium which we have highlighted.

Received: September 1997 Final version: May 1998

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Stochastic growth: Empirical evidence from the G7 Countries