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The Relationships between permanent and transitory movements in U.S. output and the unemployment rate.

Article Excerpt
MANY MACROECONOMIC TIME series can usefully be decomposed into two unobserved components. One unobserved component reflects permanent, or trend, movements in the series, while the other captures transitory movements in the series. There may exist important relationships between the unobserved components of two or more different macroeconomic variables. For example, Okun's Law (Okun 1962) suggests that output and the unemployment rate are related. This paper therefore investigates the relationships between the permanent and transitory movements in U.S. output and the unemployment rate using a bivariate correlated unobserved components model.

The results shed light on a number of important debates. First, regarding the importance of permanent versus transitory movements in real GDP, the results are consistent with the finding by Morley, Nelson, and Zivot (2003, hereafter MNZ) that movements in U.S. real GDP are largely permanent. Including the unemployment rate as an additional variable does not qualitatively change the real GDP results from MNZ's univariate model. This result contradicts the claim of Clark (1987) that using unemployment rate data to help identify transitory movements in real GDP should strengthen the case for large transitory movements in real GDP. Second, this paper contributes to the debate about the variability in the natural rate of unemployment (l) by finding support for a variable permanent component in the unemployment rate.

The model also provides estimates of the different relationships between the unobserved components of output and the unemployment rate. Four correlations are of particular interest in terms of addressing ongoing debates in the literature. The first two correlations are the "within-series correlations," that is, those between the innovations to the permanent and transitory components of the same series. The univariate analysis (MNZ) found that for U.S. real GDP these innovations are significantly negatively correlated. Clark (1989) estimated a restricted bivariate model of output and the unemployment rate and concluded that the assumption of zero correlation for the within-series correlation of real GDP was appropriate. The estimate from the unrestricted bivariate model presented here, however, indicates that the MNZ result is robust and is not just a consequence of univariate analysis. The innovations to the permanent and transitory components of the unemployment rate are also negatively correlated. This suggests that the components of the unemployment rate have a similar relationship as those of real GDP.

The third and fourth important correlations involve cross-series correlations. The correlation between the transitory components of real GDP and the unemployment rate provides an estimate of the coefficient traditionally associated with Okun's Law. Okun (1962) suggested that a 1% decrease in transitory unemployment corresponds to a 3% increase in transitory real GDP. Traditionally, Okun's coefficient has been estimated by first estimating the unobserved components and then estimating the correlation between the estimated components. In this paper, however, the correlation is directly estimated within the model. The estimated coefficient of - 1.4% is smaller in absolute value than is typically found.

Finally, the correlation between the permanent innovations of real GDP and the unemployment rate measures "Okun's coefficient for permanent movements." The coefficient representing the relationship between the unemployment rate and output in the long run is found to be -2.0%, which is closer to what Okun posited for the short run (and equal to modern estimates; Grant 2002). All of these correlations lend support to a theory of the U.S. economy, such as that of Kydland and Prescott (1982), where permanent shocks move the economy while transitory movements primarily reflect the adjustment of variables to their new steady-state values.

This paper proceeds as follows. Section 1 presents the model. Section 2 presents the results of estimating the model with U.S. data on output and the unemployment rate and discusses sensitivity of the estimates. Section 3 concludes.

1. THE MODEL

Output (y) and the unemployment rate (u) can each be represented as the sum of a permanent component and a transitory component. The permanent component ([tau]) is the steady-state level after removing all temporary movements. The transitory component (c) embodies all temporary movements and is assumed to be stationary:

[Y.sub.it] = [[tau].sub.it] + [C.sub.it}, i = y or u. (1)

Each of the trend components is assumed to be a random [walk.sup.2] to allow for permanent movements in the series: (3)

[[tau].sub.it] = [[mu].sub.i] + [[tau].sub.it-1] + [[eta].sub.it].

For output, the model allows for a drift ([[mu].sub.y]) in the permanent component, but the drift for the unemployment rate was insignificant and is not included in the reported models. The final model also includes a one-time structural break in the drift term for real GDP, as discussed below in Section 2.1. (4)

Following MNZ, Clark (1987, 1989), and Watson (1986), each transitory component is modeled as an autoregressive process of order two (AR(2)). (5)

[C.sub.it] = [[psi].sub.1i][C.sub.it-1] + [[psi].sub.2i][C.sub.it-2] + [[epsilon].sub.it]. (3)

The correlated unobserved components model assumes the permanent and transitory innovations ([[eta].sub.it], and [[epsilon].sub.it]) are jointly normally distributed random variables with mean zero and a general covariance matrix (allowing possible correlation between any of the unobserved innovations). The model can be represented in state-space form so that the Kalman filter can be applied for maximum likelihood estimation of the parameters and the components. (6)

2. RESULTS

The data used are the natural log of U.S. real GDP multiplied by 100 (y) and the U.S. civilian unemployment rate (u). The data are quarterly, from 1948:1 to 2005:4. (7) While the estimates...

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