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The distribution of sectoral TFP growth rates: International evidence
Economics Letters 113 (2011) 252–255
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The distribution of sectoral TFP growth rates: International evidence Edoardo Gaffeo ∗ Department of Economics and CEEL, University of Trento, Italy
article
info
Article history: Received 5 December 2007 Received in revised form 4 August 2011 Accepted 10 August 2011 Available online 22 August 2011
abstract This paper investigates the distributional properties of TFP growth rates for countries in the G7 group. Our findings lend support to the hypothesis that multifactor productivity shocks can be plausibly fitted by a symmetric non-Gaussian stable distribution model. This leads to non-negligible implications for business cycle analysis. © 2011 Elsevier B.V. All rights reserved.
Keywords: TFP Stable distributions Business cycle
1. Introduction Modern business cycle theory explains cyclical fluctuations as the general equilibrium response of rationally maximizing individuals to the cumulative effects of shocks repeatedly buffeting the economy. Among them, technological shocks – the difference between the growth rate of output and the share-weighted growth rates of inputs – are alleged to play a key role. Kydland and Prescott (1991), for instance, suggest that about 70% of US GDP volatility during the postwar era can be attributed to exogenous changes in total factor productivity (TFP). Such a figure is consistent with results in Leduc and Sill (2007), who show that TFP shocks account for about 90% of the post-1984 volatility decline in the US aggregate real output. The theoretical importance of TFP growth rates is reflected in the wealth of empirical studies aimed at measuring them. Starting from the conventional residual approach proposed by Solow (1957), several methodological extensions have been advanced in order to control for imperfect competition and non-constant returns to scale (Hall, 1988, 1990), and for cyclical variation in factor utilization rates (Burnside et al., 1995; Basu, 1996). However, far less attention has been devoted to date to assess the distributional properties of estimated TFP growth rates. As noted in Gaffeo (2008), such a neglect might prove to bear important consequences. Besides helping to improve our understanding on the determinants of long-run economic
∗
Tel.: +39 0461282220; fax: +39 0461282222. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.08.004
performances and standards of living, research efforts in this area can yield significant theoretical byproducts. In particular, it can be shown that if TFP shocks at a microeconomic level are non-Gaussian stable – which occurs whenever the characteristic exponent of the TFP shock distribution α is lower than 2 – aggregate fluctuations decay with the number of microeconomic 1
1 units N at the rate α− , that is much slower than N − 2 as α implied by normally distributed shocks. In other terms, if TFP shocks follow a non-Gaussian stable distribution multi-sector DSGE models can replicate aggregate fluctuations with a degree of volatility in line with that observed in real data without necessarily appealing to aggregate shocks. This occurs whenever the density functions of TFP shocks is sufficiently fat-tailed, so that the probability to observe abrupt discontinuities or extreme events in independent increments is much higher than predicted by a Gaussian distribution. In this note we analyze the distributional features of 2-digit manufacturing sectoral TFP growth rates for an international sample of industrialized economies. Our statistical evidence suggests that productivity shocks follow a symmetric non-Gaussian stable distribution as a general rule, regardless of the method employed in estimating TFP growth series. This finding corroborates and extends previous evidence for the US obtained from a different data source (Gaffeo, 2008). The estimated values for the tail weight parameter α differs significantly from country to country, however. We interpret these results as an admonition to economists interested in comparing international macroeconomic performance to pay attention to the microeconomics of productivity dynamics and the stationary distribution emerging from it. The paper proceeds as follows. Section 2 reviews the main properties of stable processes. Section 3 discusses the data and
E. Gaffeo / Economics Letters 113 (2011) 252–255
the empirical methodology we employ. Results are reported in Section 4. Section 5 concludes. 2. Statistical issues Stable distributions were first characterized by Lévy (1925) in a study of normalized sums of independent and identically distributed terms. Formally, ∑ a distribution F (x) is said to be stable n if the distribution of S = k=1 Xk is of the same type as F (x) in the sense that FS (x) = F (ax + b) for some constants a > 0 and b > 0. Provided that stable distributions do not in general possess a closed-form solution for their density, they can be conveniently expressed in terms of their characteristic function. Among the many different available parameterizations, we choose the S0 (α, β, γ , δ) parameterization proposed by Nolan (1997),1 according to which the characteristic function of the random variable X is given by
πα exp −γ α |t |α 1 + iβ tan 2 1−α (sign t )((γ |t |) − 1) + iδ t if α ̸= 1 E exp(itX ) = 2 α α exp −γ |t | 1 + iβ (sign t ) π if α = 1. (ln |t | + ln γ ) + iδ t
(1)
Four parameters govern a stable distribution. The characteristic exponent or index of stability α , which has a range 0 < α ≤ 2, measures the probability weight in the upper and lower tails of the distribution. In general, the pth moment of a stable random variable is finite if and only if p < α . Thus, for α < 2, a stable process possesses a mean equal to the location parameter δ (which in turn indicates the center of the distribution) but it has infinite variance, while if α < 1 even the mean of the distribution does not exist. β , defined on the support −1 ≤ β ≥ 1, measures the asymmetry of the distribution, with its sign indicating the direction of skewness: when β > 0(β < 0), the distribution is skewed to the right (left). Finally, the scale parameter γ , which must be positive, expands or contracts the distribution around the location parameter δ . The stable distribution function nests several well-known distributions, like the Gaussian N (µ, σ 2 ) (when α = 2, β = 0, γ = σ 2 /2 and δ = µ), the Cauchy (α = 1 and β = 0) and the Lévy–Smirnov (α = 0.5 and β = ±1). All stable distributions are unimodal and bell-shaped, and for α < 2 the tails can be shown to be asymptotically Pareto-like. Stable distributions are particular important as they represent an attractor in the functional space of probability density functions. The generalized Central Limit Theorem (Gnedenko and Kolmogorov, 1954) states that the only possible limiting distribution for sums of independently and identically distributed random variables belongs to the stable family. It follows that the conventional Central Limit Theorem is just a special case of the above—a special case which holds true if and only if one imposes the condition that each of the constituent random variables has a finite variance. All other fat-tailed distributions, like the Student-t or the exponential ones, do not possess this attractive feature. Therefore, whenever one expects that changes
1 The parameterization of the characteristic function S is particularly convenient 0 because the density and the distribution functions are jointly continuous in all four parameters.
253
Table 1 Dimensionality of the data-set. Country
Time span
No. of sectors
Sample size
Canada France Germany Italy Japan UK USA
1971–1992 1981–1991 1979–1993 1982–1994 1981–1995 1971–1987 1964–1994
8 12 13 11 13 11 13
176 132 195 143 195 187 403
in an observed aggregate process might result from a large number of independently occurring small (i.e., microeconomic) shocks, as is the case for TFP, the only theoretically appropriate distributional model is a stable (Gaussian or non-Gaussian) model. 3. Data and empirical methodology This work is based on cost-based estimates of the TFP growth rates for manufacturing sectors in the G-7 economies (USA, Canada, France, Italy, Germany, UK and Japan) made available by Malley et al. (2003). In particular, we employ two measures of TFP growth from their dataset: (1) an estimate of multifactor productivity growth which allow for non-constant returns to scale (NCRS), as in Hall (1988, 1990); (2) an estimate of multifactor productivity growth adjusted to accommodate variations in capital and labor utilization over the business cycle (VFU), as in Burnside et al. (1995) and Basu (1996). Due to lack of data, Malley et al. (2003) computed both measures for 5 countries (USA, Canada, France, Germany and UK), the NCRS measure only for Italy, and the VFU measure only for Japan. Table 1 reports the sample period and the number of 2-digit sectors for each economy. We refer to the original reference for a comprehensive discussion of the statistical properties of the data. For our purposes, data are pooled at the country level. The last column of Table 1 reports the sample size we use in estimation for each country. Estimates of the four parameters for the general stable distribution case are obtained by means of the quantile estimator method (McCulloch, 1986) and by maximum likelihood (ML) (Nolan, 2001). In addition, ML estimations for the symmetric stable (β = 0) and the Gaussian (α = 2, β = 0) restrictions are performed. 4. Results Estimates for the NCRS and the VFU measures of TFP growth rates are presented in Tables 2 and 3, respectively. Two results emerge neatly. First, the Non-Gaussian stable distribution is in general a better approximation to the empirical distribution than is the Gaussian one. Log-likelihood tests reported in the last column of Tables 2 and 3 signal that only in two cases, i.e. Germany and France when the TFP figures are adjusted to allow for non-constant returns to scale and market power, the two restrictions leading to a Gaussian model cannot be rejected at standard statistical levels. In all other cases, the distribution of shocks to multifactor productivity is heavy-tailed. In particular, the probability mass in the tails of the distribution – that is, the probability to observe extreme variations in TFP – is so high that the second moment is divergent. In all cases, the hypothesis of symmetry cannot be rejected. As recalled in Section 1, these results have several implications of relevance for business cycle analysis and modeling, which have been discussed in detail in Gaffeo (2008). Most significantly, it implies that macroeconomic fluctuations can be theoretically reconciled with microeconomic volatility without necessarily recurring to aggregate shocks.
254
E. Gaffeo / Economics Letters 113 (2011) 252–255
Table 2 Parameter estimates of TFP growth rates distribution: NCRS estimates. Stability α
Skewness β
Scale γ
Location δ
Log-likelihood ratio χ2
1.1313 1.6109 1.5602 2
−0.2773 −0.3547
0.0513 0.0507 0.0461 0.0762
0.0506 0.0507 0.0461 0.0385
– –
0.0507 0.0447 0.0454 0.0459
0.0362 0.0380 0.0353 0.0349
– –
0.0501 0.0471 0.0398 0.0376
– –
Canada Quantile ML α -stable ML symmetric α -stable ML normal
0 0
1.9124 11.5405*
France Quantile ML α -stable ML symmetric α -stable ML normal
2 1.9422 1.9827 2
0
−0.9900 0 0
0.7226 0.8806
Germany Quantile ML α -stable ML symmetric α -stable ML normal
1.5878 1.8848 1.8916 2
−0.2482 −0.9900 0 0
0.0588 0.0663 0.0664 0.0699
1.5932 1.7232 1.7180 2
0.2405 0.3141 0 0
0.0734 0.0743 0.0742 0.0875
0.0436 0.0477 0.0523 0.0560
– –
1.8392 1.7186 1.7460 2
1 0.4939 0 0
0.0858 0.0857 0.0865 0.1280
0.0174 0.0225 0.0315 0.0446
– –
0.0416
0.0606 0.0693 0.0678 0.0769
0.0460 0.0536 0.0503 0.0474
– –
2.5816 3.3615
Italy Quantile ML α -stable ML symmetric α -stable ML normal
0.7544 8.1451**
UK Quantile ML α -stable ML symmetric α -stable ML normal
2.0856 89.1839*
USA Quantile ML α -stable ML symmetric α -stable ML normal * **
1.4946 1.8524 1.7919 2
−0.3728 0 0
1.0689 18.7778*
Significant at the 1% level. Significant at the 5% level.
Table 3 Parameter estimates of TFP growth rates distribution: VFU estimates. Stability α
Skewness β
Scale γ
Location δ
Log-likelihood ratio χ2
1.0493 1.0438 1.0396 2
0.2613 1.1479 0 0
0.0073 0.0076 0.0075 0.0146
−0.0017 −0.0006 −0.0005
– –
0
0.0133 0.0122 0.0123 0.0152
0.0012 0.0014 0.0010 0.0007
– –
0.0107 0.0112 0.0112 0.0164
−0.0023 −0.0014 −0.0012 −0.0008
– –
−0.0004
– –
0 0
0.0221 0.0247 0.0246 0.0283
0.1367 0.5782 0 0
0.0088 0.0095 0.0094 0.0102
0.0023 0.0012 0.0019 0.0022
Canada Quantile ML α -stable ML symmetric α -stable ML normal
0.0051
1.4994 455.1748*
France Quantile ML α -stable ML symmetric α -stable ML normal
2 1.6789 1.6901 2
−0.1706 0 0
0.2218 11.4789*
Germany Quantile ML α -stable ML symmetric α -stable ML normal
1.4846 1.5072 1.5118 2
−0.0325 0.0612 0 0
0.0836 40.4886*
Japan Quantile ML α -stable ML symmetric α -stable ML normal
1.4725 1.7449 1.7366 2
0.0613
−0.0737
0.0023 0.0019 0.0016
0.0500 5.9906**
UK Quantile ML α -stable ML symmetric α -stable ML normal
1.6239 1.8694 1.8625 2
– – 1.0036 6.6559**
E. Gaffeo / Economics Letters 113 (2011) 252–255
255
Table 3 (continued) Stability α
Skewness β
Scale γ
Location δ
1.6878 1.8558 1.8768 2
−0.1886 −0.4857
0.0043 0.0046 0.0044 0.0002
0.0007 0.0008 0.0000 0.0000
Log-likelihood ratio χ2
USA Quantile ML α -stable ML symmetric α -stable ML normal * **
0 0
– – 2.3001 10.5442*
Significant at the 1% level. Significant at the 5% level.
Secondly, the estimated index of stability α differs significantly across countries, ranging for instance from 1.04 to 1.87 when TFP growth rates are measured taking into consideration variable factor utilization over the cycle. This fact suggests that the statistical properties of the process leading to the introduction and diffusion across sectors of innovations are country-specific. In addition to their relevance for business cycle theory, our findings have interesting implications for the theory of growth. Theoretical predictions about the distribution of productivity levels across industries obtained in multi-sector Schumpeterian growth models (see e.g. Aghion et al., 2001) depend crucially on the (exogenously given) size of innovative steps on a quality ladder. In this literature the arrival of innovations is generally modeled by recurring to a Poisson process. Our evidence suggests that increments should be better modeled according to α -stable jump processes (Samorodnitsky and Taqqu, 1994). 5. Concluding remarks We find compelling evidence of departures from the Gaussian regime in the distribution of sectoral TFP growth rates for the countries belonging to the G7 group. Just in two cases – namely, Hall-type TFP estimates for France and Germany – the Gaussian hypothesis cannot be rejected at standard statistical levels. Furthermore, the estimated tail weight parameter α differs substantially across countries, meaning that the process driving the arrival and diffusion of technological innovations might be country-specific. We suggest our findings can be useful in organizing the theoretical reasoning underpinning multisector and cross-country models of growth and the business cycle.
Acknowledgments The financial support of the Trento Autonomous Province is gratefully acknowledged. Massimo Molinari provided excellent research assistance. References Aghion, P., Harris, C., Howitt, P., Vickers, J., 2001. Competition, imitation and growth with step-by-step innovation. Review of Economic Studies 68, 467–492. Basu, S., 1996. Procyclical productivity: overhead inputs or capacity utilization? Quarterly Journal of Economics 111, 719–751. Burnside, C., Eichenbaum, M., Rebelo, S., 1995. Capital utilization and returns to scale. NBER Macroeconomics Annual 10, 67–110. Gaffeo, E., 2008. Lévy-stable productivity shocks. Macroeconomic Dynamics 12, 425–443. Gnedenko, B., Kolmogorov, A., 1954. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA. Hall, R., 1988. The relation between price and marginal cost in the US industry. Journal of Political Economy 96, 921–947. Hall, R., 1990. Invariance properties of Solow’s productivity residual. In: Diamond, P. (Ed.), Growth/Productivity/Employment. MIT Press, Cambridge, MA. Kydland, F., Prescott, E., 1991. Hours and employment variation in business cycle theory. Economic Theory 1, 63–81. Leduc, S., Sill, K., 2007. Monetary policy, oil shocks, and TFP: accounting for the decline in US volatility. Review of Economic Dynamics 10, 595–614. Lévy, P., 1925. Calcul de Probabilités. Gauthier Villars, Paris. Malley, J., Muscatelli, A., Woitek, U., 2003. Some new international comparisons of productivity performance at the sectoral level. Journal of the Royal Statistical Society: Series A 166, 85–104. McCulloch, H., 1986. Simple consistent estimators of stable distribution parameters. Communications in Statistics—Simulations 15, 1109–1136. Nolan, J., 1997. Numerical calculation of stable densities and distribution functions. Communications in Statistics—Stochastic Models 13, 759–774. Nolan, J., Barndoff-Nielsen, O., Mikosch, T., Resnik, S., 2001. Maximum likelihood estimators of stable parameters. In: Lévy Processes: Theory and Applications. Birkhauser, Boston, pp. 379–400. Samorodnitsky, G., Taqqu, M., 1994. Stable Non-Gaussian Random Processes. Chapman and Hall, New York. Solow, R., 1957. Technical change and the aggregate production function. Review of Economics and Statistics 39, 312–320.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The distribution of sectoral TFP growth rates: International evidence Edoardo Gaffeo ∗ Department of Economics and CEEL, University of Trento, Italy
article
info
Article history: Received 5 December 2007 Received in revised form 4 August 2011 Accepted 10 August 2011 Available online 22 August 2011
abstract This paper investigates the distributional properties of TFP growth rates for countries in the G7 group. Our findings lend support to the hypothesis that multifactor productivity shocks can be plausibly fitted by a symmetric non-Gaussian stable distribution model. This leads to non-negligible implications for business cycle analysis. © 2011 Elsevier B.V. All rights reserved.
Keywords: TFP Stable distributions Business cycle
1. Introduction Modern business cycle theory explains cyclical fluctuations as the general equilibrium response of rationally maximizing individuals to the cumulative effects of shocks repeatedly buffeting the economy. Among them, technological shocks – the difference between the growth rate of output and the share-weighted growth rates of inputs – are alleged to play a key role. Kydland and Prescott (1991), for instance, suggest that about 70% of US GDP volatility during the postwar era can be attributed to exogenous changes in total factor productivity (TFP). Such a figure is consistent with results in Leduc and Sill (2007), who show that TFP shocks account for about 90% of the post-1984 volatility decline in the US aggregate real output. The theoretical importance of TFP growth rates is reflected in the wealth of empirical studies aimed at measuring them. Starting from the conventional residual approach proposed by Solow (1957), several methodological extensions have been advanced in order to control for imperfect competition and non-constant returns to scale (Hall, 1988, 1990), and for cyclical variation in factor utilization rates (Burnside et al., 1995; Basu, 1996). However, far less attention has been devoted to date to assess the distributional properties of estimated TFP growth rates. As noted in Gaffeo (2008), such a neglect might prove to bear important consequences. Besides helping to improve our understanding on the determinants of long-run economic
∗
Tel.: +39 0461282220; fax: +39 0461282222. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.08.004
performances and standards of living, research efforts in this area can yield significant theoretical byproducts. In particular, it can be shown that if TFP shocks at a microeconomic level are non-Gaussian stable – which occurs whenever the characteristic exponent of the TFP shock distribution α is lower than 2 – aggregate fluctuations decay with the number of microeconomic 1
1 units N at the rate α− , that is much slower than N − 2 as α implied by normally distributed shocks. In other terms, if TFP shocks follow a non-Gaussian stable distribution multi-sector DSGE models can replicate aggregate fluctuations with a degree of volatility in line with that observed in real data without necessarily appealing to aggregate shocks. This occurs whenever the density functions of TFP shocks is sufficiently fat-tailed, so that the probability to observe abrupt discontinuities or extreme events in independent increments is much higher than predicted by a Gaussian distribution. In this note we analyze the distributional features of 2-digit manufacturing sectoral TFP growth rates for an international sample of industrialized economies. Our statistical evidence suggests that productivity shocks follow a symmetric non-Gaussian stable distribution as a general rule, regardless of the method employed in estimating TFP growth series. This finding corroborates and extends previous evidence for the US obtained from a different data source (Gaffeo, 2008). The estimated values for the tail weight parameter α differs significantly from country to country, however. We interpret these results as an admonition to economists interested in comparing international macroeconomic performance to pay attention to the microeconomics of productivity dynamics and the stationary distribution emerging from it. The paper proceeds as follows. Section 2 reviews the main properties of stable processes. Section 3 discusses the data and
E. Gaffeo / Economics Letters 113 (2011) 252–255
the empirical methodology we employ. Results are reported in Section 4. Section 5 concludes. 2. Statistical issues Stable distributions were first characterized by Lévy (1925) in a study of normalized sums of independent and identically distributed terms. Formally, ∑ a distribution F (x) is said to be stable n if the distribution of S = k=1 Xk is of the same type as F (x) in the sense that FS (x) = F (ax + b) for some constants a > 0 and b > 0. Provided that stable distributions do not in general possess a closed-form solution for their density, they can be conveniently expressed in terms of their characteristic function. Among the many different available parameterizations, we choose the S0 (α, β, γ , δ) parameterization proposed by Nolan (1997),1 according to which the characteristic function of the random variable X is given by
πα exp −γ α |t |α 1 + iβ tan 2 1−α (sign t )((γ |t |) − 1) + iδ t if α ̸= 1 E exp(itX ) = 2 α α exp −γ |t | 1 + iβ (sign t ) π if α = 1. (ln |t | + ln γ ) + iδ t
(1)
Four parameters govern a stable distribution. The characteristic exponent or index of stability α , which has a range 0 < α ≤ 2, measures the probability weight in the upper and lower tails of the distribution. In general, the pth moment of a stable random variable is finite if and only if p < α . Thus, for α < 2, a stable process possesses a mean equal to the location parameter δ (which in turn indicates the center of the distribution) but it has infinite variance, while if α < 1 even the mean of the distribution does not exist. β , defined on the support −1 ≤ β ≥ 1, measures the asymmetry of the distribution, with its sign indicating the direction of skewness: when β > 0(β < 0), the distribution is skewed to the right (left). Finally, the scale parameter γ , which must be positive, expands or contracts the distribution around the location parameter δ . The stable distribution function nests several well-known distributions, like the Gaussian N (µ, σ 2 ) (when α = 2, β = 0, γ = σ 2 /2 and δ = µ), the Cauchy (α = 1 and β = 0) and the Lévy–Smirnov (α = 0.5 and β = ±1). All stable distributions are unimodal and bell-shaped, and for α < 2 the tails can be shown to be asymptotically Pareto-like. Stable distributions are particular important as they represent an attractor in the functional space of probability density functions. The generalized Central Limit Theorem (Gnedenko and Kolmogorov, 1954) states that the only possible limiting distribution for sums of independently and identically distributed random variables belongs to the stable family. It follows that the conventional Central Limit Theorem is just a special case of the above—a special case which holds true if and only if one imposes the condition that each of the constituent random variables has a finite variance. All other fat-tailed distributions, like the Student-t or the exponential ones, do not possess this attractive feature. Therefore, whenever one expects that changes
1 The parameterization of the characteristic function S is particularly convenient 0 because the density and the distribution functions are jointly continuous in all four parameters.
253
Table 1 Dimensionality of the data-set. Country
Time span
No. of sectors
Sample size
Canada France Germany Italy Japan UK USA
1971–1992 1981–1991 1979–1993 1982–1994 1981–1995 1971–1987 1964–1994
8 12 13 11 13 11 13
176 132 195 143 195 187 403
in an observed aggregate process might result from a large number of independently occurring small (i.e., microeconomic) shocks, as is the case for TFP, the only theoretically appropriate distributional model is a stable (Gaussian or non-Gaussian) model. 3. Data and empirical methodology This work is based on cost-based estimates of the TFP growth rates for manufacturing sectors in the G-7 economies (USA, Canada, France, Italy, Germany, UK and Japan) made available by Malley et al. (2003). In particular, we employ two measures of TFP growth from their dataset: (1) an estimate of multifactor productivity growth which allow for non-constant returns to scale (NCRS), as in Hall (1988, 1990); (2) an estimate of multifactor productivity growth adjusted to accommodate variations in capital and labor utilization over the business cycle (VFU), as in Burnside et al. (1995) and Basu (1996). Due to lack of data, Malley et al. (2003) computed both measures for 5 countries (USA, Canada, France, Germany and UK), the NCRS measure only for Italy, and the VFU measure only for Japan. Table 1 reports the sample period and the number of 2-digit sectors for each economy. We refer to the original reference for a comprehensive discussion of the statistical properties of the data. For our purposes, data are pooled at the country level. The last column of Table 1 reports the sample size we use in estimation for each country. Estimates of the four parameters for the general stable distribution case are obtained by means of the quantile estimator method (McCulloch, 1986) and by maximum likelihood (ML) (Nolan, 2001). In addition, ML estimations for the symmetric stable (β = 0) and the Gaussian (α = 2, β = 0) restrictions are performed. 4. Results Estimates for the NCRS and the VFU measures of TFP growth rates are presented in Tables 2 and 3, respectively. Two results emerge neatly. First, the Non-Gaussian stable distribution is in general a better approximation to the empirical distribution than is the Gaussian one. Log-likelihood tests reported in the last column of Tables 2 and 3 signal that only in two cases, i.e. Germany and France when the TFP figures are adjusted to allow for non-constant returns to scale and market power, the two restrictions leading to a Gaussian model cannot be rejected at standard statistical levels. In all other cases, the distribution of shocks to multifactor productivity is heavy-tailed. In particular, the probability mass in the tails of the distribution – that is, the probability to observe extreme variations in TFP – is so high that the second moment is divergent. In all cases, the hypothesis of symmetry cannot be rejected. As recalled in Section 1, these results have several implications of relevance for business cycle analysis and modeling, which have been discussed in detail in Gaffeo (2008). Most significantly, it implies that macroeconomic fluctuations can be theoretically reconciled with microeconomic volatility without necessarily recurring to aggregate shocks.
254
E. Gaffeo / Economics Letters 113 (2011) 252–255
Table 2 Parameter estimates of TFP growth rates distribution: NCRS estimates. Stability α
Skewness β
Scale γ
Location δ
Log-likelihood ratio χ2
1.1313 1.6109 1.5602 2
−0.2773 −0.3547
0.0513 0.0507 0.0461 0.0762
0.0506 0.0507 0.0461 0.0385
– –
0.0507 0.0447 0.0454 0.0459
0.0362 0.0380 0.0353 0.0349
– –
0.0501 0.0471 0.0398 0.0376
– –
Canada Quantile ML α -stable ML symmetric α -stable ML normal
0 0
1.9124 11.5405*
France Quantile ML α -stable ML symmetric α -stable ML normal
2 1.9422 1.9827 2
0
−0.9900 0 0
0.7226 0.8806
Germany Quantile ML α -stable ML symmetric α -stable ML normal
1.5878 1.8848 1.8916 2
−0.2482 −0.9900 0 0
0.0588 0.0663 0.0664 0.0699
1.5932 1.7232 1.7180 2
0.2405 0.3141 0 0
0.0734 0.0743 0.0742 0.0875
0.0436 0.0477 0.0523 0.0560
– –
1.8392 1.7186 1.7460 2
1 0.4939 0 0
0.0858 0.0857 0.0865 0.1280
0.0174 0.0225 0.0315 0.0446
– –
0.0416
0.0606 0.0693 0.0678 0.0769
0.0460 0.0536 0.0503 0.0474
– –
2.5816 3.3615
Italy Quantile ML α -stable ML symmetric α -stable ML normal
0.7544 8.1451**
UK Quantile ML α -stable ML symmetric α -stable ML normal
2.0856 89.1839*
USA Quantile ML α -stable ML symmetric α -stable ML normal * **
1.4946 1.8524 1.7919 2
−0.3728 0 0
1.0689 18.7778*
Significant at the 1% level. Significant at the 5% level.
Table 3 Parameter estimates of TFP growth rates distribution: VFU estimates. Stability α
Skewness β
Scale γ
Location δ
Log-likelihood ratio χ2
1.0493 1.0438 1.0396 2
0.2613 1.1479 0 0
0.0073 0.0076 0.0075 0.0146
−0.0017 −0.0006 −0.0005
– –
0
0.0133 0.0122 0.0123 0.0152
0.0012 0.0014 0.0010 0.0007
– –
0.0107 0.0112 0.0112 0.0164
−0.0023 −0.0014 −0.0012 −0.0008
– –
−0.0004
– –
0 0
0.0221 0.0247 0.0246 0.0283
0.1367 0.5782 0 0
0.0088 0.0095 0.0094 0.0102
0.0023 0.0012 0.0019 0.0022
Canada Quantile ML α -stable ML symmetric α -stable ML normal
0.0051
1.4994 455.1748*
France Quantile ML α -stable ML symmetric α -stable ML normal
2 1.6789 1.6901 2
−0.1706 0 0
0.2218 11.4789*
Germany Quantile ML α -stable ML symmetric α -stable ML normal
1.4846 1.5072 1.5118 2
−0.0325 0.0612 0 0
0.0836 40.4886*
Japan Quantile ML α -stable ML symmetric α -stable ML normal
1.4725 1.7449 1.7366 2
0.0613
−0.0737
0.0023 0.0019 0.0016
0.0500 5.9906**
UK Quantile ML α -stable ML symmetric α -stable ML normal
1.6239 1.8694 1.8625 2
– – 1.0036 6.6559**
E. Gaffeo / Economics Letters 113 (2011) 252–255
255
Table 3 (continued) Stability α
Skewness β
Scale γ
Location δ
1.6878 1.8558 1.8768 2
−0.1886 −0.4857
0.0043 0.0046 0.0044 0.0002
0.0007 0.0008 0.0000 0.0000
Log-likelihood ratio χ2
USA Quantile ML α -stable ML symmetric α -stable ML normal * **
0 0
– – 2.3001 10.5442*
Significant at the 1% level. Significant at the 5% level.
Secondly, the estimated index of stability α differs significantly across countries, ranging for instance from 1.04 to 1.87 when TFP growth rates are measured taking into consideration variable factor utilization over the cycle. This fact suggests that the statistical properties of the process leading to the introduction and diffusion across sectors of innovations are country-specific. In addition to their relevance for business cycle theory, our findings have interesting implications for the theory of growth. Theoretical predictions about the distribution of productivity levels across industries obtained in multi-sector Schumpeterian growth models (see e.g. Aghion et al., 2001) depend crucially on the (exogenously given) size of innovative steps on a quality ladder. In this literature the arrival of innovations is generally modeled by recurring to a Poisson process. Our evidence suggests that increments should be better modeled according to α -stable jump processes (Samorodnitsky and Taqqu, 1994). 5. Concluding remarks We find compelling evidence of departures from the Gaussian regime in the distribution of sectoral TFP growth rates for the countries belonging to the G7 group. Just in two cases – namely, Hall-type TFP estimates for France and Germany – the Gaussian hypothesis cannot be rejected at standard statistical levels. Furthermore, the estimated tail weight parameter α differs substantially across countries, meaning that the process driving the arrival and diffusion of technological innovations might be country-specific. We suggest our findings can be useful in organizing the theoretical reasoning underpinning multisector and cross-country models of growth and the business cycle.
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