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Is there a permanent component in US real GDP
*
Jack Strauss
Department of Economics, St. Louis University, 3674 Lindell Blvd., St. Louis, MO 63108, USA
Received 1 June 1998; received in revised form 21 May 1999; accepted 23 June 1999
Abstract
Using three recent panel unit root methodologies, we strongly reject the unit root hypothesis for real per capita income for the 48 contiguous states from 1929 to 1995. The rate of convergence is sensitive to assumptions of the heterogeneity of the intercept, first order autoregressive coefficients, lag length and accommodating for a trend break after the 1973 oil shock. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Panel unit root tests; Real GDP; US real income
JEL classification: C23; E1
1. Introduction
The controversy concerning the time series properties of real GDP has far reaching implications for modeling and forecasting the economy and for judging the importance and role of macroeconomic stabilization programs. The lack of consensus is often attributed to the low power of unit root tests or the presence of infrequent structural breaks. Using three panel unit root tests that possess considerably more power than univariate ADF tests, this paper examines the time series properties of US real per capita income from 1929 to 1995. The Abuaf and Jorion (1990), Levin and Lin (1993) and Im et al. (1996) panel tests support a trend stationarity process for US real income. We then demonstrate a shift in the intercept which occurred after the oil shock.
Using a restricted GLS multivariate procedure, Abuaf and Jorion demonstrate that Dickey–Fuller unit root tests should be estimated in a system to ‘‘fully exploit the information in cross-equation correlations.’’ Their work demonstrates that restricting the autoregressive component across all economies and pooling the data in a system of univariate autoregressions significantly increases the power of unit root tests. In an alternative approach that emphasizes i.i.d. disturbances, Levin and Lin
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E-mail address: [email protected] (J. Strauss)
also restrict the first-order autoregressive coefficient (LL) and develop the asymptotic properties of panel unit root tests. Their approach emphasizes independence across economies rather than their dependence, and the increased power in this procedure comes not from utilizing SUR to minimize common disturbances, but from pooling N independent economies. Im, Pesaran and Shin (IPS) relax the restriction on the first-order autoregressive process to accommodate heterogeneous adjustment processes and pool the t statistics or P values from univariate independent ADF regressions. Their increase in power is derived from averaging N independent economies to derive a t-bar (average t)
2
statistic that converges to a standardized normal or a P value that follows a x distribution.
2. Panel unit root methodology
To improve the power of the univariate Dickey–Fuller procedures, Abuaf and Jorion (1990) and Jorion and Sweeney (1996) estimate a SUR system of equations. They argue that it is more efficient to impose an identical first-order autoregressive coefficient on all series as estimating the regressions country by country is clearly inefficient given that the null hypothesis is that of values ofr are unity. Suppose y is the log of real per capita income for state i, and jit 51, . . . , p ADF lags:
p
yit5ai1ditime1riyi,t211OfijDyi,t2j1eit i51, . . . N states, t, . . . T (1)
j51
The restriction imposed across all N economies isr15, . . . ,5rn5r. The null hypothesis is that the stochastic process, y , follows a nonstationary unit root process for all states (soit r51). The alternative hypothesis assumes that per capita income follows a deterministic stationary path in all economies (r,1). The Monte Carlo generating procedures assume that d50, and a and u are estimated from the data. Following Abuaf and Jorion, N error terms e are generated jointly from Tt times from a multivariate normal distribution with mean zero and the given historical covariance matrix.
The Levin and Lin panel unit root approach increases the power of the traditional ADF tests by imposing an identical first order autoregressive coefficient on each series in the panel, i.e., r15
. . . ,rN5r.
p
Dyit5ai1ditime1r yi,t211OuijDyi,t2j1eit t51, . . . T (2)
j51
The null and alternative hypotheses for LL arer50 andr,0. The LL approach requires that all
N
¯
series are independent. Following Hsiao (1986), they subtract cross-section averages yt5oi51yit
from the data to induce independence and reduce contemporaneous correlations. We use the LL critical values.
The IPS test relaxes the assumption of identical first-order autoregressive coefficients of LL. The IPS method pools N independent cross-section ADF unit root tests to evaluate H :0 ri5r50 for
]]
N
]
calculate toi51 t where the t ratio for each series, ti i5ri/ Var(
œ
ri), is from T observations. Following IPS, we subtract cross-sectional means to remove common time specific effects.3. Panel results
The paper uses Bureau of Labor Statistics per capita income data. The BLS also gives the aggregate per capita income as well as a price deflator. Data is missing for the early years for Alaska and Hawaii; hence, we consider the contiguous US states. All data are logged and in real terms.
Table 1 presents SUR, LL and IPS panel tests for real per capita income for P50,1,2,3 lags. The last column presents the univariate ADF for real aggregate per capita income. For clarity, results are presented with y as the dependent variable, and the null hypotheses test whethert r51 or ri51. Alternative lag lengths are reported to demonstrate: (1) different tests (Akaike, Perron) indicate different lag lengths; (2) different states possess require different lag lengths (e.g., using the Perron 10% rule, 2, 36, 8, 2 states require 0, 1, 2 and 3 lags, respectively); and (3) the sensitivity of panel tests to different lag specifications. Further, the table reports both common and heterogeneous intercepts andr terms to highlight the sensitivity of panel tests to alternative specifications of growth rates and speeds of adjustment.
While imposition of a common intercept procedure avoids the individual fixed effects bias (Nickell, 1981; Anderson and Hsiao, 1982) and improves the power of the tests (Levin and Lin, 1993) homogeneity lowers the convergence speed (Lee et al., 1997). Hence, the true speed of adjustment should lie between the procedures. Note that different intercepts allow differences in trend growth rates across economies due to underlying variations in technology, government policy and market structure; however the fixed effects approach creates a bias with an upper-bound of 3r/T. This bias is likely to be large (small) for the SUR (LL) approach. And although Evans and Karras (1996) develop
2
x restriction tests for the intercept term using the LL method, no tests for the SUR method have been established.
Although results cannot reject at the 5% level a unit root using the univariate ADF approach, the panel approaches, however clearly reject the null hypothesis and support a trend stationary process.
Table 1
a
Panel unit root stationarity tests and speed of adjustment estimates (US states 1929–1995)
b
(t stat) (19.12) (26.83) (19.16) (17.24) (2.80) (1.72)
** ** ** ** **
P51r 0.853 0.927 0.915 0.934 0.909 0.857
(t stat) (25.70) (25.10) (20.06) (17.56) (2.92) (3.24)
** ** ** ** **
P52r 0.834 0.916 0.923 0.954 0.927 0.850
(t stat) (25.83) (26.12) (15.11) (14.00) (2.23) (3.07)
** ** ** ** **
P53r 0.839 0.912 0.938 0.954 0.927 0.885
(t stat) (26.83) (32.30) (13.42) (13.00) (2.23) (2.56)
a p
The t-statistics for the SUR and LL approaches are substantially above their Monte Carlo 1% critical values, and z-bar is also above its 1% (one-sided) normalized critical values for one and two lags and 5% for three lags. The speed of adjustment to a shock is sensitive to the heterogeneity of the constant andr terms. For a common intercept, r estimates range from 7 to 9% for SUR and 4 to 6% for LL procedures; for a heterogeneous intercept,r estimates range from 14 to 16% for SUR, 6 to 9% for LL and 9 to 14% for the IPS method. As predicted, homogeneity of the intercept lowers the speed of adjustment.
Table 2 investigates the effect of an additive outlier with slope change (dum*r) (Model (3), Perron, 1994). Studies find that US growth decelerated after the oil shock, and this effect may affect estimates of stationarity by biasing the speed of adjustment upward (Perron, 1989). Although the break is chosen exogenously, Table 1 already has shown that real per capita income follows a trend stationary process; hence, here we investigate the use of slope change on the rate of adjustment. Monte Carlo simulations with exogenous trend break support a trend break at 99%.
Results find the slope break strongly affects the speed of convergence for the LL and IPS procedures; e.g., with a heterogeneous intercept and P52, the estimated r increases from 8.5 to 15.1% and 9.1 to 14.8% for the LL and IPS methods, respectively. The coefficients (t statistics) for the dummy intercept are 1.9% (15.8) and 1.8% (19.8), which are large and significant at the 1% level (using either a stationary distribution or Monte Carlo critical values assuming a nonstationary distribution with trend break). The slope break estimation procedures also affect the SUR r estimates (t stats) moderately; the coefficients for the dummy variable are 1–2% and significant assuming a stationary distribution.
Table 2
a
Panel unit root speed of adjustment estimates with trend break (US states 1929–1995 with break after 1973)
Lag SUR LL IPS ADF
length a 5 ai US
a 5 ai a 5 a a 5 ai a 5 a
**
P50r 0.832** 0.923** 0.844 0.919** 0.835** 0.872
(t-stat) (20.95) (22.80) (28.65) (21.34) (4.18) (2.46)
Dum 20.022** 20.011** 20.018** 20.010** 20.020** 20.011
t-stat (215.87) (29.85) (220.43) (221.26) (221.62) (21.90)
P51r 0.838** 0.921** 0.849** 0.923** 0.852** 0.832
(t-stat) (26.89) (23.43) (29.40) (21.26) (4.43) (3.64)
Dum 20.019** 20.101* 20.016** 20.009** 20.012** 20.007
t-stat (215.68) (210.23) (220.47) (211.81) (218.97) (21.64)
P52r 0.845** 0.900** 0.866** 0.933** 0.838** 0.822
(t-stat) (27.10) (27.10) (24.74) (18.64) (4.29) (3.45)
Dum 20.018** 20.012 20.016** 20.01** 20.020** 20.006
t-stat (216.40) (211.92) (219.80) (213.14) (219.32) (21.55)
P53r 0.840** 0.917** 0.871** 0.930** 0.868** 0.669
(t-stat) (26.54) (28.44) (23.71) (19.73) (3.45) (2.76)
Lastly, the paper examines the effect of an exogenous intercept break after 1973; i.e., we ‘‘allow a one-time change in the intercept of the trend function’’ (Model (1) of Perron, 1994). The coefficients (t statistics) for the dummy intercept are 26.0% (18.8) and 26.0% (18.6), which are large and significant at the 1% level and also effect the speed of adjustment. For instance, for P51 and a heterogeneous intercept, the LL, SUR and IPS procedure yieldr (t stats): 0.838 (26.9), 0.868 (28.1), and 0.863 (4.71), respectively. For P52 and a common intercept, SUR and LL methods yield r
(t-stats): 0.912 (26.1) and 0.933 (21.8), respectively. Since results are also significant, it is difficult to differentiate which type of structural break occurred during the 1970s.
One potential factor mitigating the change in speed of adjustment is the finite sample bias that
1
results when deterministic components such as slope or intercept breaks are added. Monte Carlo simulations illustrate that an additional slope break (dumr in Table 2) bias downward the estimates of
ryt21 from 2 to 3% in the SUR, Levin–Lin and IPS cases. Since the estimates in most cases for the Levin and Lin and IPS are 7–8%, the slope change has a significant effect on the speed of adjustment. For SUR, accounting for potential bias implies that no significant difference inryt21 occurs for most lag specifications. However, note that the bias does not affect the magnitude and t distribution of the dummy term, which is significantly negative at 99% for both break specifications. The increase in power associated with the panel procedure shows persuasively that the oil shock does affect the time series properties of the data for all three panel procedures.
4. Conclusion
This paper adopts three panel unit root procedures to test the time series properties of real per capita income for the US 48 contiguous states, 1929–1995. In contrast to ADF results, the Abuaf and Jorion (1990), Levin and Lin (1993) and Im et al. (1996) panel unit root procedures, which possess considerably more power than univariate tests, find that real per capita income is trend stationary. The results are robust to different lag lengths, panel procedures and intercept assumptions. The speeds of adjustment, however, are sensitive to econometric technique, particularly the heterogeneity of the intercept.
Moreover, allowing for a trend break in 1973 substantially increases the speed of adjustment for the LL and IPS method. For instance, with a heterogeneous intercept and P51, the estimates ofr for a slope break increase from 8.5 to 13.2% and 9.1 to 13.7% for the LL and IPS methods, respectively. The coefficients and t statistics for the break are significant at the 1% level. The trend break approach has only a moderate influence on the estimation of the speed of convergence for the SUR procedure. Results further indicate that both intercept and slope breaks are negative after 1973 and significant at the 99% level.
References
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