Charles R. Nelson

Trends and random walks in macroeconmic time series: Some evidence and implications

Abstract This paper investigates whether macroeconomic time series are better characterized as stationary fluctuations around a deterministic trend or as non-stationary processes that have no tendency to return to a deterministic path. Using long historical time series for the U.S. we are unable to reject the hypothesis that these series are non-stationary stochastic processes with no tendency to return to a trend line. Based on these findings and an unobserved components model for output that decomposes fluctuations into a secular or growth component and a cyclical component we infer that shocks to the former, which we associate with real disturbances, contribute substantially to the variation in observed output. We conclude that macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may never be successful in explaining a large fraction of output variation and that stochastic variation due to real factors is an essential element of any model of macroeconomic fluctuations.

Parsimonious Modeling of Yield Curves

This paper introduces a parametrically parsimonious model for yield curves that has the ability to represent the shapes generally associated with yield curves: monotonic, humped, and S-shaped. The authors find that the model explains 96 percent of the variation in bill yields across maturities during the period 1981-83. The movement of the parameters through time reflects and confirms a change in Federal Reserve monetary policy in late 1982. The ability of the fitted curves to predict the price of the long-term Treasury bond with a correlation of 0.96 suggests that the model captures important attributes of the yield/maturity relation. Copyright 1987 by the University of Chicago.

A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’☆

Abstract This paper introduces a general procedure for decomposition of non-stationary time series into a permanent and transitory component allowing both components to be stochastic. The permanent component is shown to be a random walk with drift and the transitory or cyclical component is a stationary process with mean zero. The decomposition methodology, which depends only on past data and therefore is computable in ‘real time’, is applied to the problem of measuring and dating business ‘cycles’ in the portwar U.S. economy. We find that measured expansions and contractions are of roughly equivalent duration and that our dating of cyclical episodes tends to lead the traditional NBER dating and, to a lesser extent, the ‘growth cycle’ chronology of Zarnowitz and Boschan (1977).

HAS THE U.S. ECONOMY BECOME MORE STABLE? A BAYESIAN APPROACH BASED ON A MARKOV-SWITCHING MODEL OF THE BUSINESS CYCLE

We hope to answer three questions: Has there been a structural break in postwar U.S. real GDP growth towards stabilization? If so, when? What is the nature of this structural break? We employ a Bayesian approach to identify a structural break at an unknown changepoint in a Markov-switching model of the business cycle. Empirical results suggest a break in GDP growth toward stabilization, with the posterior mode of the break date at 1984:1. Furthermore, we find a narrowing gap between growth rates during recessions and booms that is at least as important as any decline in the volatility of shocks.

The Distribution of the Instrumental Variables Estimator and its T-Ratiowhen the Instrument is a Poor One

When the instrumental variable is a poor one, in the sense of being weakly correlated with the variable it proxies, the small sample distribution of the IV estimator is concentrated around a value that is inversely related to the feedback in the system and which is often further from the true value than is the plim of OLS. The sample variance of residuals similarly becomes concentrated around a value which reflects feedback and not the variance of the disturbance. The distribution of the t-ratio reflects both of these effects, stronger feedback producing larger t-ratios. Thus, in situations where OLS is badly biased, a poor instrument will lead to spurious inferences under IV estimation with high probability, and generally perform worse than OLS.

Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator

New results on the exact small sample distribution of the instrumental variable estimator are presented by studying an important special case. The exact closed forms for the probability density and cumulative distribution functions are given. There are a number of surprising findings. The small sample distribution is bimodal. with a point of zero probability mass. As the asymptotic variance grows large, the true distribution becomes concentrated around this point of zero mass. The central tendency of the estimator may be closer to the biased least squares estimator than it is to the true parameter value. The first and second moments of the IV estimator are both infinite. In the case in which least squares is biased upwards, and most of the mass of the IV estimator lies to the right of the true parameter, the mean of the IV estimator is infinitely negative. The difference between the true distribution and the normal asymptotic approximation depends on the ratio of the asymptotic variance to a parameter related to the correlation between the regressor and the regression, error. In particular, when the instrument is poorly correlated with the regressor, the asymptotic approximation to the distribution of the instrumental variable estimator will not be very accurate.

Mean Reversion in Stock Prices? a Reappraisal of the Empirical Evidence

The paper re-examines the empirical evidence for mean-reverting behaviour in stock prices. Comparison of data before and after World War II shows that mean reversion is entirely a pre-war phenomenon. Using randomization methods to calculate significance levels, we find that the full sample evidence for mean reversion is weaker than previously indicated by Monte Carlo methods under a Normal assumption. Further, the switch to mean-averting behaviour after the war is about to be too strong to be compatible with sampling variation. We interpret these findings as evidence of a fundamental change in the stock returns process and conjecture that it may be due to the resolution of the uncertainties of the 1930s and 1940s.

Business Cycle Turning Points, A New Coincident Index, and Tests of Duration Dependence Based on a Dynamic Factor Model With Regime Switching

The synthesis of the dynamic factor model of Stock and Watson (1989) and the regime-switching model of Hamilton (1989) proposed by Diebold and Rudebusch (1996) potentially encompasses both features of the business cycle identified by Burns and Mitchell (1946): (1) comovement among economic variables through the cycle and (2) nonlinearity in its evolution. However, maximum-likelihood estimation has required approximation. Recent advances in multimove Gibbs sampling methodology open the way to approximation-free inference in such non-Gaussian, nonlinear models. This paper estimates the model for U.S. data and attempts to address three questions: Are both features of the business cycle empirically relevant? Might the implied new index of coincident indicators be a useful one in practice? Do the resulting estimates of regime switches show evidence of duration dependence? The answers to all three would appear to be yes.

SPURIOUS PERIODICITY IN INAPPROPRIATELY DETRENDED TIME SERIES

Econometric analysis of time series data is frequently preceded by regression on time to remove a trent component in the date. The resulting residuals are then treated as a stationary series to which procedures requiring stationarity, such as spectral analysis, can be applied. The objective is often to investigate the dynamics of transitory movements in the systems, for example, in econometric models of the business cycle. When the data does consist of a deterministic function of time plus a stationary error then regression residuals will clearly be unbiased estimates of the stationary component. However, if the data is generated by (possibly repeated) summation of a satisfactory and inevitable process then the series cannot be expressed as a deterministic function of time plus a stationary deviation, even though a least squares trend line and the associated residuals can always be calculated for any given finite sample. In a recent paper, Chan, Hayya, and Ord (1977) hereafter CHO) were able to show that a residuals from linear regression of a realization of a random walk (the summation of a purely random series) on time have autocovariances which for given lag are a function of time and thereafter that the residuals are not stationary. Further, CHO established that the expected sample autocovariance function (the expected autocovariances for given lag averaged over the time interval of the sample) is a function of sample size as well as lag and therefore an artifact of the detrending procedure. This function is characterized by CHO in their figure 1 as being effectively linear in lag (although the exact function is a fifth degree polynomial) with the rate of decay from unity at the origin depending inversely on sample size.

Why Are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP So Different?

This paper reconciles two widely used decompositions of GDP into trend and cycle that yield starkly different results. The Beveridge-Nelson (BN) decomposition implies that a stochastic trend accounts for most of the variation in output, whereas the unobserved-components (UC) implies cyclical variation is dominant. Which is correct has broad implications for the relative importance of real versus nominal shocks. We show the difference arises from the restriction imposed in UC that trend and cycle innovations are uncorrelated. When this restriction is relaxed, the UC decomposition is identical to the BN decomposition. Furthermore, the zero-correlation restriction can be rejected for U.S. quarterly GDP, with the estimated correlation being -0.9.