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The impact of vintage on the persistence of gross domestic product shocks
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Economics Letters 98 (2008) 301 – 308 www.elsevier.com/locate/econbase
The impact of vintage on the persistence of gross domestic product shocks Christian Macaro ⁎ Faculty of Economics, Tor Vergata University, I-00133, Rome, Italy Received 15 September 2006; received in revised form 2 March 2007; accepted 2 May 2007 Available online 10 May 2007
Abstract This paper aims to demonstrate that the data revision process affects the persistence of gross domestic product shocks. Results will contribute to the debate between unit root and linear models. © 2007 Elsevier B.V. All rights reserved. Keywords: Revisions; GDP; Long memory JEL classification: C22; C82; E30
1. Introduction The analysis of shock persistence in macroeconomics became popular after Nelson and Plosser (1982) reported their findings on stochastic trends. In particular, they argue that the real gross domestic product (GDP) follows a unit root model. Rudebusch (1993) claims that the presence of a unit root in the real GDP is uncertain. Diebold and Rudebush (1989), Sowell (1992), Gil-Alana and Robinson (1997) try to identify the presence of strongly persistent shocks. Henry and Zaffaroni (2002) emphasize that the aforementioned researchers have yet to reach an agreement regarding their empirical results. Croushore and Stark (2001) explain that, when researchers compare their results with other empirical studies, they do not consider that results may differ because the quality and definition of data might have changed over time. ⁎ Tel.: +39 0328 379 8332. E-mail address: [email protected]. 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.05.007
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C. Macaro / Economics Letters 98 (2008) 301–308
This study puts Croushore and Stark's insight into practice by presenting a sensitivity analysis of shock persistence in the GDP. In doing so, it benefits from recent developments in fractionally integrated process literature and, particularly, on semiparametric estimation procedures (Robinson, 1995; Shimotsu and Phillips, 2005; Abadir et al., 2005). As pointed out in Gil-Alana and Robinson (1997), the advantage of considering fractionally integrated processes allows for more flexible definitions of non-stationarity, while Nelson and Plosser (1982) consider only the unit root and the linear trend cases. Moreover, the implementation of semiparametric procedures allows for seasonality and other short run dynamics which typically characterize macroeconomic time series. 2. Data and vintages This paper analyzes GNP/GDP1 vintages released between the fourth quarter of 1965 and the fourth quarter of 2004. Since data have been recorded starting with the first quarter of 1947, the vintage released in the fourth quarter of 1965 has 75 observations, and the vintage released in the fourth quarter of 2004 has 231 observations. The resulting real time data set was made available by Croushore and Stark (2001) and by Croushore and Stark (2004) who have been systematically monitoring the U.S. macroeconomic time series. Within the time span considered, the definition of the output has been modified several times. These changes should be taken into account when comparing empirical works which analyze different vintages; if not, the apparent differences could be attributed to the wrong cause. In a report attached to the data set, Croushore and Stark emphasize that, in the first quarter of 1976, the US government made thorough revisions to NIPA 2 data: changes were significant and included modifications to the definitions of variables, as well as new data sources. Similar modifications affected vintages released after the first quarter of 1986. In the first quarter of 1992, the GNP was transformed into the GDP. After the first quarter of 1996, the real data collecting method switched from fixed to chain weighted. Finally, in the fourth quarter of 1999, the BEA3 instituted a comprehensive revision to NIPA data. The relevance of the revisions that occurred in early 1996 and late 1999 is further emphasized by the presence of some missing data in the vintages released from the first quarter of 1996 to the first quarter of 1997 and from the fourth quarter of 1999 to the first quarter of 2000. This suggests that the mechanism used to determine the US output changed dramatically. In particular, Landefeld and Parker (1997) emphasize that the fixed chain method, implemented before 1996, tends to underestimate the growth rate of the real GDP before the base quarter of the deflater (third quarter of 1987), and overestimates the growth rate after the same base quarter. This effect, which increases with temporal distance, has likely inflated the persistence of real GDP shocks in vintages released before 1996. Moreover, Moulton and Seskin (1999) emphasize that in 1999 the BEA classified software expenditures as investments. The influence of this revision was substantial considering that the GDP growth rate was modified by 0.2 percentage (Seskin, 1999). Because of the rapid technological
1 2 3
The performance of a regional economy was initially measured by the gross national product (GNP). NIPA stands for National Income and Product Account. BEA stands for Bureau of Economic Analysis.
C. Macaro / Economics Letters 98 (2008) 301–308
303
innovations in software development, prices tend to change quickly, and this may have reduced the persistence of real GDP shocks. Due to the importance of the revisions which occurred in early 1996 and late 1999, and because the US GNP turned into GDP in early 1992, the rest of this paper will focus on vintages released after the fourth quarter of 1991. 3. Estimation methodology The approach followed here is partially inspired by Nelson and Plosser (1982) and Gil-Alana and Robinson (1997) and consists in the analysis of shock persistence in two rival models. The first model is inspired by the ideas presented in Sowell (1990) and it will be referred to as the fractional unit root model: Yn ¼ en
ð1 LÞ1þd en ¼ un fIðOÞ;
ð1Þ
where I(0) denotes any covariance stationary process with spectral density function that is positive, bounded and bounded away from zero. The second is the linear trend model: ð1 LÞd en ¼ vn fIð0Þ:
Yn b gn ¼ en
ð2Þ
It is worth noting that the nature of shocks depends on the values of the parameters d and δ. More precisely, we have: d b −1, d = −1, −1 b d b −1/2, −1/2 ≤ d b 0, d = 0, d N 0,
δb0 δ=0 0 b δ b 1/2 1/2 ≤ δ b 1 δ=1 δN1
⇒ ⇒ ⇒ ⇒ ⇒ ⇒
overdifferentiated shocks, weakly persistent and stationary shocks, strongly persistent and stationary shocks, strongly persistent and mean-reverting shocks, unit root shocks, explosive shocks.
Robinson (1995) presents the Local Whittle estimator (LW), which can estimate d ∈ (−3/2, −1/2) in model (1). Abadir, Distaso, and Giraitis (2005) present the Fully-Extended Local Whittle estimator (FELW), which estimates δ ∈ (−1/2, 1/2) ⋃ (1/2, 3/2) ⋃ (3/2, 5/2) in the more general framework of model (2). Shimotsu and Phillips (2005) present the Exact Local Whittle estimator (ELW), which can estimate d ∈ (−∞, ∞) in model (1). 4. Notes on the estimators The vintage sample size (N) ranges from 184 to 231 observations. The choice of the bandwidth parameter M = ⌊N / 8⌋ seems appropriate for the LW estimator, because a simulation study by Robinson (1995) shows that for 256 observations the bias is negligible and the standard deviation remains relatively low. A similar value is chosen for the FELW which generalizes the LW. The implementation of the ELW estimator requires more careful handling. Shimotsu and Phillips (2005) show simulation results of this estimator for M = ⌊N 0.65 ⌋, when there is no linear trend and the short run component (v n) is a white noise process. However, analysis of Table 1 seems to
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suggest that the ELW estimator is still consistent, though it may not be asymptotically normally distributed, when it is implemented on trending data, like model (2). Moreover, the analysis of the MSEs shows that the bandwidth M = ⌊N 0.6 ⌋ performs relatively better when the spectral density of the short run component (v n) is characterized by finite peaks at low frequencies, and sufficiently well when the short run component is a simple white noise. 5. Results This section focuses on the measurement of shock persistence (d and δ) in the fractional unit root and in the linear trend model for vintages released between the first quarter of 1992 and the fourth quarter of Table 1 Simulation study for the Exact Local Whittle estimator
Contamination structure
λ = 0.6 Bias(δˆ )
sd(δˆ )
MSE(δˆ )
λ = 0.65 Bias(δˆ )
sd(δˆ )
MSE(δˆ )
White noise AR1 (0.2) AR1 (0.4) AR1 (0.6) AR2 (0.2, 0.1) AR2 (0.4, 0.1) AR2 (0.6, 0.1) AR2 (0.1, 0.2) AR2 (0.1, 0.4) AR2 (0.1, 0.6) Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0,
0.001 0.021 0.062 0.148 0.063 0.126 0.238 0.083 0.193 0.298 − 0.01 0.013 0.070 0.186 0.065 0.152 0.329 0.096 0.250 0.434 − 0.011 0.016 0.069 0.188 0.067 0.157 0.327 0.098 0.251 0.434
0.121 0.122 0.124 0.129 0.121 0.124 0.139 0.124 0.135 0.151 0.13 0.131 0.132 0.134 0.134 0.134 0.125 0.131 0.133 0.087 0.133 0.131 0.134 0.134 0.132 0.132 0.124 0.133 0.131 0.088
0.015 0.015 0.019 0.038 0.019 0.031 0.076 0.022 0.056 0.111 0.017 0.017 0.022 0.053 0.022 0.041 0.124 0.026 0.080 0.196 0.018 0.017 0.023 0.053 0.022 0.042 0.122 0.027 0.080 0.196
0.006 0.036 0.092 0.193 0.090 0.166 0.276 0.113 0.231 0.312 − 0.003 0.033 0.112 0.255 0.103 0.211 0.396 0.139 0.313 0.468 − 0.004 0.036 0.109 0.256 0.104 0.214 0.394 0.141 0.313 0.468
0.103 0.103 0.106 0.119 0.105 0.112 0.136 0.107 0.128 0.149 0.112 0.112 0.114 0.114 0.113 0.116 0.097 0.113 0.112 0.058 0.112 0.112 0.113 0.115 0.113 0.114 0.097 0.114 0.111 0.058
0.011 0.012 0.020 0.052 0.019 0.040 0.095 0.024 0.070 0.120 0.012 0.014 0.025 0.078 0.023 0.058 0.166 0.032 0.110 0.222 0.013 0.014 0.025 0.079 0.023 0.059 0.165 0.033 0.110 0.222
0.2) and white noise 0.2) and AR1 (0.2) 0.2) and AR1 (0.4) 0.2) and AR1 (0.6) 0.2) and AR2 (0.2, 0.1) 0.2) and AR2 (0.4, 0.1) 0.2) and AR2 (0.6, 0.1) 0.2) and AR2 (0.1, 0.2) 0.2) and AR2 (0.1, 0.4) 0.2) and AR2 (0.1, 0.6) 0.4) and white noise 0.4) and AR1 (0.2) 0.4) and AR1 (0.4) 0.4) and AR1 (0.6) 0.4) and AR2 (0.2, 0.1) 0.4) and AR2 (0.4, 0.1) 0.4) and AR2 (0.6, 0.1) 0.4) and AR2 (0.1, 0.2) 0.4) and AR2 (0.1, 0.4) 0.4) and AR2 (0.1, 0.6)
The sample size is N = 200, the bandwidth parameter is M = ⌊Nλ⌋ and the true value of the parameter is δ = 1. The pure fractional integrated process, (1 − L)δ Xn = zn, is contaminated with a linear trend on Xn and with zn ∼ ARi(ϕ1, …, ϕi). Estimates reported in bold characters correspond to the lowest MSEs.
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Table 2 Local Whittle, Fully-Extended Local Whittle and Exact Local Whittle estimates of the shock persistence (d an δ) in the fractional unit root and the linear trend models
Vintage 1992:1 ⋯ 1992:4 1993:1 1993:2 1993:3 1993:4 1994:1 1994:2 1994:3 1994:4 1995:1 1995:2 1995:3 1995:4 1996:1 ⋯ 1997:1 1997:2 1997:3 1997:4 1998:1 1998:2 1998:3 1998:4 1999:1 1999:2 1999:3 1999:4 2001:1 2000:2 2000:3 2000:4 2001:1 2001:2 2001:3 2001:4 2002:1 2002:2 2002:3 2002:4 2003:1 2003:2
Unit root Local Whittle m dˆ
22 23 23 23 23 23 23 23 23 24 24 24
25 25 25 25 25 25 25 25 26 26
26 26 26 26 27 27 27 27 27 27 27 27 28
NA ⋯ NA − 0.06 − 0.03 − 0.04 − 0.05 − 0.05 − 0.06 − 0.06 − 0.06 − 0.06 − 0.04 − 0.05 − 0.05 NA ⋯ NA − 0.16 − 0.16 − 0.16 − 0.16 − 0.16 − 0.17 − 0.17 − 0.17 − 0.15 − 0.15 NA NA − 0.20 − 0.20 − 0.21 − 0.22 − 0.20 − 0.21 − 0.22 − 0.22 − 0.22 − 0.22 − 0.22 − 0.23 − 0.21
sd(dˆ )
P-value
Linear trend Fully-Extended Local Whittle m δˆ sd(δˆ ) P-value
0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.55 0.75 0.73 0.60 0.60 0.57 0.56 0.53 0.53 0.70 0.63 0.63
23 23 23 23 23 23 23 23 24 24 24 24
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.12 0.11 0.11 0.11 0.10 0.09 0.08 0.09 0.13 0.12
25 25 25 25 25 25 25 26 26 26
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09
0.04 0.04 0.04 0.03 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03
26 26 26 27 27 27 27 27 27 27 27 28 28
NA ⋯ NA 0.96 0.96 0.96 0.93 0.93 0.92 0.91 0.90 0.92 0.91 0.91 0.91 NA ⋯ NA 0.84 0.83 0.83 0.83 0.82 0.81 0.81 0.83 0.82 0.82 NA NA 0.74 0.73 0.72 0.74 0.75 0.76 0.77 0.77 0.77 0.77 0.76 0.79 0.78
Linear trend Exact Local Whittle m δˆ sd(δˆ )
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.71 0.69 0.67 0.52 0.49 0.44 0.40 0.35 0.45 0.40 0.39 0.37
23 23 23 23 23 23 23 23 23 24 24 24
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.10 0.10 0.09 0.08 0.08 0.06 0.05 0.08 0.07 0.06
24 24 24 24 24 24 25 25 25 25
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09
0.01 0.01 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02
25 25 25 25 25 25 25 25 26 26 26 26 26
NA ⋯ NA 0.94 0.93 0.93 0.94 0.95 0.94 0.93 0.92 0.91 0.90 0.88 0.90 NA ⋯ NA 0.83 0.83 0.84 0.82 0.82 0.82 0.82 0.80 0.84 0.83 NA NA 0.77 0.78 0.76 0.75 0.74 0.73 0.72 0.73 0.76 0.79 0.80 0.80 0.80
P-value
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.60 0.52 0.48 0.54 0.65 0.56 0.49 0.42 0.39 0.32 0.24 0.31
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.10 0.10 0.11 0.08 0.08 0.08 0.07 0.05 0.12 0.09
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.03 0.04 0.04 0.05
(continued on next page)
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Table 2 (continued)
Vintage
Unit root Local Whittle m dˆ
2003:3 2003:4 2004:1 2004:2 2004:3 2004:4
28 28 28 28 28 28
−0.21 −0.21 −0.22 −0.22 −0.23 −0.23
sd(dˆ )
P-value
Linear trend Fully-Extended Local Whittle ˆδ m sd(δˆ ) P-value
0.09 0.09 0.09 0.09 0.09 0.09
0.03 0.03 0.02 0.02 0.02 0.01
28 28 28 28 28 28
0.78 0.78 0.77 0.76 0.75 0.74
0.09 0.09 0.09 0.09 0.09 0.09
0.02 0.02 0.01 0.01 0.01 0.01
Linear trend Exact Local Whittle ˆδ m sd(δˆ )
P-value
26 26 26 26 26 26
0.05 0.10 0.12 0.02 0.02 0.02
0.81 0.84 0.85 0.77 0.77 0.77
0.10 0.10 0.10 0.10 0.10 0.10
P-values are evaluated for H0: d = 0 in the fractional unit root model and H0: δ = 1 in the linear trend case.
2004. Data are transformed into logarithms, and preliminary estimates show that d ∈ (−0.5, 0.5) and δ ∈ (0.5, 1.5) respectively. The first column of Table 2 shows LW estimates of d when data in the fractional unit root model are differentiated (e.g. Yn − Yn−1). Vintages from the first quarter of 1993 to the fourth quarter of 1995 (hereafter early vintages) accept the unit root hypothesis; vintages from the second quarter of 1997 to the third quarter of 1999 (intermediate vintages) do not reject the unit root hypothesis at the 10% level; vintages from the second quarter of 2002 to the fourth quarter of 2004 (recent vintages) seem to reject the unit root hypothesis at or below the 4% level in favour of lower orders of integration. Fig. 1 summarizes the results of the LW estimator and illustrates that the shock persistence changes dramatically in vintages released after the first quarter of 1996 and the fourth quarter of 1999. The significant estimates observed for recent vintages could indicate that the persistence of real GDP shocks has become statistically weaker than in the unit root case. An equivalent conclusion is evinced from the second column of Table 2, which presents the FELW estimates of ä in the linear trend model. Shocks in recent vintages seem to be strongly persistent and meanreverting (e.g. δ ∈ [0.5, 1)). Moreover, the unit root hypothesis (δ = 1) is always rejected at or below the 2% level. Finally, early and intermediate vintages seem to accept, or equivocally reject, the presence of a unit root.
Fig. 1. Local Whittle estimates of the shock persistence (d ) in the fractional unit root model after data have been differentiated. The bandwidth parameter is M = ⌊N / 8⌋. The circles represent the estimates. The two solid lines represent the confidence interval at 95% (H0: d = 0). The dashed line represents the null hypothesis. The vintages are reported on the X-axis.
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The robustness of the FELW is considered by implementing the ELW estimator of δ in model (2). Even if the ELW estimator is not specifically designed for trending data, as in the case of the FELW estimator, the third column of Table 2 shows that results are still equivalent. It is important to emphasize that all these findings are obtained from semiparametric procedures and, therefore, they are robust to any short run dynamics underlying the data. 6. Conclusions The results for early vintages are consistent with the results of researchers who suggest that the real GDP is a unit root process. In the paper written by Nelson and Plosser (1982), an even earlier vintage is studied. However, it is plausible that their results are similar to those for early vintages. Diebold and Rudebush (1989), Sowell (1992) and Gil-Alana and Robinson (1997) analyze data that belong to early or intermediate vintages. Results reported in this study have indicated that these vintages are characterized by great uncertainty, since it is difficult to unequivocally reject either the unit root or the linear trend. The results for recent vintages reject the unit root hypothesis and accepts the linear trend model with strongly persistent and mean reverting shocks. Based on this analysis, I would suggest that comparisons of results from the empirical literature of fractionally integrated processes and GDP series should be made in reference to the data vintage. In doing so, empirical results by Nelson and Plosser (1982), Diebold and Rudebush (1989), Sowell (1992) and GilAlana and Robinson (1997) seem to be consistent with one another. Acknowledgments The author is grateful to Liudas Giraitis and Franco Peracchi for their valuable support. Helpful comments were provided by Karim Abadir, Paolo Zaffaroni, Gianluca Cubadda, Amelia Perea and an anonymous referee. The standard disclaimer applies. References Abadir, K., Distaso, W., Giraitis, L., 2005. “Semiparametric Estimation and Inference for Trending I(d) and Related Processes,” Discussion Papers 05/15, Department of Economics, University of York. Croushore, D., Stark, T., 2001. A real-time data set for macroeconomists. Journal of Econometrics 105, 111–130. Croushore, D., Stark, T. (2004): http://www.phil.frb.org/econ/forecast/reaindex.html. Diebold, F.X., Rudebush, G., 1989. Long memory and persistence in aggregate output. Journal of Monetary Economics 24, 189–209. Gil-Alana, L., Robinson, P.M., 1997. Testing of unit root and other nonstationary hypotheses in macroeconomic time series. Journal of Econometrics 80, 241–268. Henry, M., Zaffaroni, P., 2002. The long range dependence paradigm for macroeconomics and finance. In: Doukhan, P., Oppenheim, G., Taqqu, M.S. (Eds.), Long Range Dependence: Theory and Applications. Landefeld, J., Parker, R., 1997. BEA's chain indexes, time series and measures of long-term economic growth. Survey of Current Business 58–68 (June). Moulton, B., Seskin, E., 1999. A preview of the 1999 comprehensive revision of the National Income and Product Accounts. Survey of Current Business 6–17 (October). Nelson, C.R., Plosser, C.I., 1982. Trends and random walks in macroeconomic time S. Journal of Monetary Economics 10, 139–162. Robinson, P.M., 1995. Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661. Rudebusch, G., 1993. The uncertain unit root in real GNP. Journal of Econometrics 83 (1), 264–272.
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Seskin, E., 1999. Improved estimates of the National Income and Product Accounts for 1959–98: results of the comprehensive revision. Survey of Current Business 6–17 (December). Shimotsu, K., Phillips, P.C.B., 2005. Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933. Sowell, F., 1990. The fractional unit root distribution. Econometrica 58 (2), 495–505. Sowell, F., 1992. Modeling long-run behaviour with the fractional ARIMA model. Journal of Monetary Economics 29, 277–302.
Economics Letters 98 (2008) 301 – 308 www.elsevier.com/locate/econbase
The impact of vintage on the persistence of gross domestic product shocks Christian Macaro ⁎ Faculty of Economics, Tor Vergata University, I-00133, Rome, Italy Received 15 September 2006; received in revised form 2 March 2007; accepted 2 May 2007 Available online 10 May 2007
Abstract This paper aims to demonstrate that the data revision process affects the persistence of gross domestic product shocks. Results will contribute to the debate between unit root and linear models. © 2007 Elsevier B.V. All rights reserved. Keywords: Revisions; GDP; Long memory JEL classification: C22; C82; E30
1. Introduction The analysis of shock persistence in macroeconomics became popular after Nelson and Plosser (1982) reported their findings on stochastic trends. In particular, they argue that the real gross domestic product (GDP) follows a unit root model. Rudebusch (1993) claims that the presence of a unit root in the real GDP is uncertain. Diebold and Rudebush (1989), Sowell (1992), Gil-Alana and Robinson (1997) try to identify the presence of strongly persistent shocks. Henry and Zaffaroni (2002) emphasize that the aforementioned researchers have yet to reach an agreement regarding their empirical results. Croushore and Stark (2001) explain that, when researchers compare their results with other empirical studies, they do not consider that results may differ because the quality and definition of data might have changed over time. ⁎ Tel.: +39 0328 379 8332. E-mail address: [email protected]. 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.05.007
302
C. Macaro / Economics Letters 98 (2008) 301–308
This study puts Croushore and Stark's insight into practice by presenting a sensitivity analysis of shock persistence in the GDP. In doing so, it benefits from recent developments in fractionally integrated process literature and, particularly, on semiparametric estimation procedures (Robinson, 1995; Shimotsu and Phillips, 2005; Abadir et al., 2005). As pointed out in Gil-Alana and Robinson (1997), the advantage of considering fractionally integrated processes allows for more flexible definitions of non-stationarity, while Nelson and Plosser (1982) consider only the unit root and the linear trend cases. Moreover, the implementation of semiparametric procedures allows for seasonality and other short run dynamics which typically characterize macroeconomic time series. 2. Data and vintages This paper analyzes GNP/GDP1 vintages released between the fourth quarter of 1965 and the fourth quarter of 2004. Since data have been recorded starting with the first quarter of 1947, the vintage released in the fourth quarter of 1965 has 75 observations, and the vintage released in the fourth quarter of 2004 has 231 observations. The resulting real time data set was made available by Croushore and Stark (2001) and by Croushore and Stark (2004) who have been systematically monitoring the U.S. macroeconomic time series. Within the time span considered, the definition of the output has been modified several times. These changes should be taken into account when comparing empirical works which analyze different vintages; if not, the apparent differences could be attributed to the wrong cause. In a report attached to the data set, Croushore and Stark emphasize that, in the first quarter of 1976, the US government made thorough revisions to NIPA 2 data: changes were significant and included modifications to the definitions of variables, as well as new data sources. Similar modifications affected vintages released after the first quarter of 1986. In the first quarter of 1992, the GNP was transformed into the GDP. After the first quarter of 1996, the real data collecting method switched from fixed to chain weighted. Finally, in the fourth quarter of 1999, the BEA3 instituted a comprehensive revision to NIPA data. The relevance of the revisions that occurred in early 1996 and late 1999 is further emphasized by the presence of some missing data in the vintages released from the first quarter of 1996 to the first quarter of 1997 and from the fourth quarter of 1999 to the first quarter of 2000. This suggests that the mechanism used to determine the US output changed dramatically. In particular, Landefeld and Parker (1997) emphasize that the fixed chain method, implemented before 1996, tends to underestimate the growth rate of the real GDP before the base quarter of the deflater (third quarter of 1987), and overestimates the growth rate after the same base quarter. This effect, which increases with temporal distance, has likely inflated the persistence of real GDP shocks in vintages released before 1996. Moreover, Moulton and Seskin (1999) emphasize that in 1999 the BEA classified software expenditures as investments. The influence of this revision was substantial considering that the GDP growth rate was modified by 0.2 percentage (Seskin, 1999). Because of the rapid technological
1 2 3
The performance of a regional economy was initially measured by the gross national product (GNP). NIPA stands for National Income and Product Account. BEA stands for Bureau of Economic Analysis.
C. Macaro / Economics Letters 98 (2008) 301–308
303
innovations in software development, prices tend to change quickly, and this may have reduced the persistence of real GDP shocks. Due to the importance of the revisions which occurred in early 1996 and late 1999, and because the US GNP turned into GDP in early 1992, the rest of this paper will focus on vintages released after the fourth quarter of 1991. 3. Estimation methodology The approach followed here is partially inspired by Nelson and Plosser (1982) and Gil-Alana and Robinson (1997) and consists in the analysis of shock persistence in two rival models. The first model is inspired by the ideas presented in Sowell (1990) and it will be referred to as the fractional unit root model: Yn ¼ en
ð1 LÞ1þd en ¼ un fIðOÞ;
ð1Þ
where I(0) denotes any covariance stationary process with spectral density function that is positive, bounded and bounded away from zero. The second is the linear trend model: ð1 LÞd en ¼ vn fIð0Þ:
Yn b gn ¼ en
ð2Þ
It is worth noting that the nature of shocks depends on the values of the parameters d and δ. More precisely, we have: d b −1, d = −1, −1 b d b −1/2, −1/2 ≤ d b 0, d = 0, d N 0,
δb0 δ=0 0 b δ b 1/2 1/2 ≤ δ b 1 δ=1 δN1
⇒ ⇒ ⇒ ⇒ ⇒ ⇒
overdifferentiated shocks, weakly persistent and stationary shocks, strongly persistent and stationary shocks, strongly persistent and mean-reverting shocks, unit root shocks, explosive shocks.
Robinson (1995) presents the Local Whittle estimator (LW), which can estimate d ∈ (−3/2, −1/2) in model (1). Abadir, Distaso, and Giraitis (2005) present the Fully-Extended Local Whittle estimator (FELW), which estimates δ ∈ (−1/2, 1/2) ⋃ (1/2, 3/2) ⋃ (3/2, 5/2) in the more general framework of model (2). Shimotsu and Phillips (2005) present the Exact Local Whittle estimator (ELW), which can estimate d ∈ (−∞, ∞) in model (1). 4. Notes on the estimators The vintage sample size (N) ranges from 184 to 231 observations. The choice of the bandwidth parameter M = ⌊N / 8⌋ seems appropriate for the LW estimator, because a simulation study by Robinson (1995) shows that for 256 observations the bias is negligible and the standard deviation remains relatively low. A similar value is chosen for the FELW which generalizes the LW. The implementation of the ELW estimator requires more careful handling. Shimotsu and Phillips (2005) show simulation results of this estimator for M = ⌊N 0.65 ⌋, when there is no linear trend and the short run component (v n) is a white noise process. However, analysis of Table 1 seems to
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suggest that the ELW estimator is still consistent, though it may not be asymptotically normally distributed, when it is implemented on trending data, like model (2). Moreover, the analysis of the MSEs shows that the bandwidth M = ⌊N 0.6 ⌋ performs relatively better when the spectral density of the short run component (v n) is characterized by finite peaks at low frequencies, and sufficiently well when the short run component is a simple white noise. 5. Results This section focuses on the measurement of shock persistence (d and δ) in the fractional unit root and in the linear trend model for vintages released between the first quarter of 1992 and the fourth quarter of Table 1 Simulation study for the Exact Local Whittle estimator
Contamination structure
λ = 0.6 Bias(δˆ )
sd(δˆ )
MSE(δˆ )
λ = 0.65 Bias(δˆ )
sd(δˆ )
MSE(δˆ )
White noise AR1 (0.2) AR1 (0.4) AR1 (0.6) AR2 (0.2, 0.1) AR2 (0.4, 0.1) AR2 (0.6, 0.1) AR2 (0.1, 0.2) AR2 (0.1, 0.4) AR2 (0.1, 0.6) Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0, Linear trend (0,
0.001 0.021 0.062 0.148 0.063 0.126 0.238 0.083 0.193 0.298 − 0.01 0.013 0.070 0.186 0.065 0.152 0.329 0.096 0.250 0.434 − 0.011 0.016 0.069 0.188 0.067 0.157 0.327 0.098 0.251 0.434
0.121 0.122 0.124 0.129 0.121 0.124 0.139 0.124 0.135 0.151 0.13 0.131 0.132 0.134 0.134 0.134 0.125 0.131 0.133 0.087 0.133 0.131 0.134 0.134 0.132 0.132 0.124 0.133 0.131 0.088
0.015 0.015 0.019 0.038 0.019 0.031 0.076 0.022 0.056 0.111 0.017 0.017 0.022 0.053 0.022 0.041 0.124 0.026 0.080 0.196 0.018 0.017 0.023 0.053 0.022 0.042 0.122 0.027 0.080 0.196
0.006 0.036 0.092 0.193 0.090 0.166 0.276 0.113 0.231 0.312 − 0.003 0.033 0.112 0.255 0.103 0.211 0.396 0.139 0.313 0.468 − 0.004 0.036 0.109 0.256 0.104 0.214 0.394 0.141 0.313 0.468
0.103 0.103 0.106 0.119 0.105 0.112 0.136 0.107 0.128 0.149 0.112 0.112 0.114 0.114 0.113 0.116 0.097 0.113 0.112 0.058 0.112 0.112 0.113 0.115 0.113 0.114 0.097 0.114 0.111 0.058
0.011 0.012 0.020 0.052 0.019 0.040 0.095 0.024 0.070 0.120 0.012 0.014 0.025 0.078 0.023 0.058 0.166 0.032 0.110 0.222 0.013 0.014 0.025 0.079 0.023 0.059 0.165 0.033 0.110 0.222
0.2) and white noise 0.2) and AR1 (0.2) 0.2) and AR1 (0.4) 0.2) and AR1 (0.6) 0.2) and AR2 (0.2, 0.1) 0.2) and AR2 (0.4, 0.1) 0.2) and AR2 (0.6, 0.1) 0.2) and AR2 (0.1, 0.2) 0.2) and AR2 (0.1, 0.4) 0.2) and AR2 (0.1, 0.6) 0.4) and white noise 0.4) and AR1 (0.2) 0.4) and AR1 (0.4) 0.4) and AR1 (0.6) 0.4) and AR2 (0.2, 0.1) 0.4) and AR2 (0.4, 0.1) 0.4) and AR2 (0.6, 0.1) 0.4) and AR2 (0.1, 0.2) 0.4) and AR2 (0.1, 0.4) 0.4) and AR2 (0.1, 0.6)
The sample size is N = 200, the bandwidth parameter is M = ⌊Nλ⌋ and the true value of the parameter is δ = 1. The pure fractional integrated process, (1 − L)δ Xn = zn, is contaminated with a linear trend on Xn and with zn ∼ ARi(ϕ1, …, ϕi). Estimates reported in bold characters correspond to the lowest MSEs.
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Table 2 Local Whittle, Fully-Extended Local Whittle and Exact Local Whittle estimates of the shock persistence (d an δ) in the fractional unit root and the linear trend models
Vintage 1992:1 ⋯ 1992:4 1993:1 1993:2 1993:3 1993:4 1994:1 1994:2 1994:3 1994:4 1995:1 1995:2 1995:3 1995:4 1996:1 ⋯ 1997:1 1997:2 1997:3 1997:4 1998:1 1998:2 1998:3 1998:4 1999:1 1999:2 1999:3 1999:4 2001:1 2000:2 2000:3 2000:4 2001:1 2001:2 2001:3 2001:4 2002:1 2002:2 2002:3 2002:4 2003:1 2003:2
Unit root Local Whittle m dˆ
22 23 23 23 23 23 23 23 23 24 24 24
25 25 25 25 25 25 25 25 26 26
26 26 26 26 27 27 27 27 27 27 27 27 28
NA ⋯ NA − 0.06 − 0.03 − 0.04 − 0.05 − 0.05 − 0.06 − 0.06 − 0.06 − 0.06 − 0.04 − 0.05 − 0.05 NA ⋯ NA − 0.16 − 0.16 − 0.16 − 0.16 − 0.16 − 0.17 − 0.17 − 0.17 − 0.15 − 0.15 NA NA − 0.20 − 0.20 − 0.21 − 0.22 − 0.20 − 0.21 − 0.22 − 0.22 − 0.22 − 0.22 − 0.22 − 0.23 − 0.21
sd(dˆ )
P-value
Linear trend Fully-Extended Local Whittle m δˆ sd(δˆ ) P-value
0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.55 0.75 0.73 0.60 0.60 0.57 0.56 0.53 0.53 0.70 0.63 0.63
23 23 23 23 23 23 23 23 24 24 24 24
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.12 0.11 0.11 0.11 0.10 0.09 0.08 0.09 0.13 0.12
25 25 25 25 25 25 25 26 26 26
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09
0.04 0.04 0.04 0.03 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03
26 26 26 27 27 27 27 27 27 27 27 28 28
NA ⋯ NA 0.96 0.96 0.96 0.93 0.93 0.92 0.91 0.90 0.92 0.91 0.91 0.91 NA ⋯ NA 0.84 0.83 0.83 0.83 0.82 0.81 0.81 0.83 0.82 0.82 NA NA 0.74 0.73 0.72 0.74 0.75 0.76 0.77 0.77 0.77 0.77 0.76 0.79 0.78
Linear trend Exact Local Whittle m δˆ sd(δˆ )
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.71 0.69 0.67 0.52 0.49 0.44 0.40 0.35 0.45 0.40 0.39 0.37
23 23 23 23 23 23 23 23 23 24 24 24
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.10 0.10 0.09 0.08 0.08 0.06 0.05 0.08 0.07 0.06
24 24 24 24 24 24 25 25 25 25
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09
0.01 0.01 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02
25 25 25 25 25 25 25 25 26 26 26 26 26
NA ⋯ NA 0.94 0.93 0.93 0.94 0.95 0.94 0.93 0.92 0.91 0.90 0.88 0.90 NA ⋯ NA 0.83 0.83 0.84 0.82 0.82 0.82 0.82 0.80 0.84 0.83 NA NA 0.77 0.78 0.76 0.75 0.74 0.73 0.72 0.73 0.76 0.79 0.80 0.80 0.80
P-value
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.60 0.52 0.48 0.54 0.65 0.56 0.49 0.42 0.39 0.32 0.24 0.31
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.10 0.10 0.11 0.08 0.08 0.08 0.07 0.05 0.12 0.09
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.03 0.04 0.04 0.05
(continued on next page)
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Table 2 (continued)
Vintage
Unit root Local Whittle m dˆ
2003:3 2003:4 2004:1 2004:2 2004:3 2004:4
28 28 28 28 28 28
−0.21 −0.21 −0.22 −0.22 −0.23 −0.23
sd(dˆ )
P-value
Linear trend Fully-Extended Local Whittle ˆδ m sd(δˆ ) P-value
0.09 0.09 0.09 0.09 0.09 0.09
0.03 0.03 0.02 0.02 0.02 0.01
28 28 28 28 28 28
0.78 0.78 0.77 0.76 0.75 0.74
0.09 0.09 0.09 0.09 0.09 0.09
0.02 0.02 0.01 0.01 0.01 0.01
Linear trend Exact Local Whittle ˆδ m sd(δˆ )
P-value
26 26 26 26 26 26
0.05 0.10 0.12 0.02 0.02 0.02
0.81 0.84 0.85 0.77 0.77 0.77
0.10 0.10 0.10 0.10 0.10 0.10
P-values are evaluated for H0: d = 0 in the fractional unit root model and H0: δ = 1 in the linear trend case.
2004. Data are transformed into logarithms, and preliminary estimates show that d ∈ (−0.5, 0.5) and δ ∈ (0.5, 1.5) respectively. The first column of Table 2 shows LW estimates of d when data in the fractional unit root model are differentiated (e.g. Yn − Yn−1). Vintages from the first quarter of 1993 to the fourth quarter of 1995 (hereafter early vintages) accept the unit root hypothesis; vintages from the second quarter of 1997 to the third quarter of 1999 (intermediate vintages) do not reject the unit root hypothesis at the 10% level; vintages from the second quarter of 2002 to the fourth quarter of 2004 (recent vintages) seem to reject the unit root hypothesis at or below the 4% level in favour of lower orders of integration. Fig. 1 summarizes the results of the LW estimator and illustrates that the shock persistence changes dramatically in vintages released after the first quarter of 1996 and the fourth quarter of 1999. The significant estimates observed for recent vintages could indicate that the persistence of real GDP shocks has become statistically weaker than in the unit root case. An equivalent conclusion is evinced from the second column of Table 2, which presents the FELW estimates of ä in the linear trend model. Shocks in recent vintages seem to be strongly persistent and meanreverting (e.g. δ ∈ [0.5, 1)). Moreover, the unit root hypothesis (δ = 1) is always rejected at or below the 2% level. Finally, early and intermediate vintages seem to accept, or equivocally reject, the presence of a unit root.
Fig. 1. Local Whittle estimates of the shock persistence (d ) in the fractional unit root model after data have been differentiated. The bandwidth parameter is M = ⌊N / 8⌋. The circles represent the estimates. The two solid lines represent the confidence interval at 95% (H0: d = 0). The dashed line represents the null hypothesis. The vintages are reported on the X-axis.
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The robustness of the FELW is considered by implementing the ELW estimator of δ in model (2). Even if the ELW estimator is not specifically designed for trending data, as in the case of the FELW estimator, the third column of Table 2 shows that results are still equivalent. It is important to emphasize that all these findings are obtained from semiparametric procedures and, therefore, they are robust to any short run dynamics underlying the data. 6. Conclusions The results for early vintages are consistent with the results of researchers who suggest that the real GDP is a unit root process. In the paper written by Nelson and Plosser (1982), an even earlier vintage is studied. However, it is plausible that their results are similar to those for early vintages. Diebold and Rudebush (1989), Sowell (1992) and Gil-Alana and Robinson (1997) analyze data that belong to early or intermediate vintages. Results reported in this study have indicated that these vintages are characterized by great uncertainty, since it is difficult to unequivocally reject either the unit root or the linear trend. The results for recent vintages reject the unit root hypothesis and accepts the linear trend model with strongly persistent and mean reverting shocks. Based on this analysis, I would suggest that comparisons of results from the empirical literature of fractionally integrated processes and GDP series should be made in reference to the data vintage. In doing so, empirical results by Nelson and Plosser (1982), Diebold and Rudebush (1989), Sowell (1992) and GilAlana and Robinson (1997) seem to be consistent with one another. Acknowledgments The author is grateful to Liudas Giraitis and Franco Peracchi for their valuable support. Helpful comments were provided by Karim Abadir, Paolo Zaffaroni, Gianluca Cubadda, Amelia Perea and an anonymous referee. The standard disclaimer applies. References Abadir, K., Distaso, W., Giraitis, L., 2005. “Semiparametric Estimation and Inference for Trending I(d) and Related Processes,” Discussion Papers 05/15, Department of Economics, University of York. Croushore, D., Stark, T., 2001. A real-time data set for macroeconomists. Journal of Econometrics 105, 111–130. Croushore, D., Stark, T. (2004): http://www.phil.frb.org/econ/forecast/reaindex.html. Diebold, F.X., Rudebush, G., 1989. Long memory and persistence in aggregate output. Journal of Monetary Economics 24, 189–209. Gil-Alana, L., Robinson, P.M., 1997. Testing of unit root and other nonstationary hypotheses in macroeconomic time series. Journal of Econometrics 80, 241–268. Henry, M., Zaffaroni, P., 2002. The long range dependence paradigm for macroeconomics and finance. In: Doukhan, P., Oppenheim, G., Taqqu, M.S. (Eds.), Long Range Dependence: Theory and Applications. Landefeld, J., Parker, R., 1997. BEA's chain indexes, time series and measures of long-term economic growth. Survey of Current Business 58–68 (June). Moulton, B., Seskin, E., 1999. A preview of the 1999 comprehensive revision of the National Income and Product Accounts. Survey of Current Business 6–17 (October). Nelson, C.R., Plosser, C.I., 1982. Trends and random walks in macroeconomic time S. Journal of Monetary Economics 10, 139–162. Robinson, P.M., 1995. Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661. Rudebusch, G., 1993. The uncertain unit root in real GNP. Journal of Econometrics 83 (1), 264–272.
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Seskin, E., 1999. Improved estimates of the National Income and Product Accounts for 1959–98: results of the comprehensive revision. Survey of Current Business 6–17 (December). Shimotsu, K., Phillips, P.C.B., 2005. Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933. Sowell, F., 1990. The fractional unit root distribution. Econometrica 58 (2), 495–505. Sowell, F., 1992. Modeling long-run behaviour with the fractional ARIMA model. Journal of Monetary Economics 29, 277–302.