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Abstract:
Asymmetric behavior has been documented in post-war quarterly U.S. unemployment rates. This suggests that improvement over conventional linear forecasts may be possible through use of nonlinear time series models. In this paper an out-of-sample forecasting competition is carried out for a set of leading nonlinear time series models. It is shown that several nonlinear forecasts do indeed dominate the linear forecast. The results are sensitive, however, to whether a stationarity-inducing transformation is applied to the nonstationary unemployment rate series.

Abstract:
Asymmetric behavior has been documented in post-war quarterly U.S. unemployment rates. This suggests that improvement over conventional linear forecasts may be possible through use of nonlinear time series models. In this paper an out-of-sample forecasting competition is carried out for a set of leading nonlinear time series models. It is shown that several nonlinear forecasts do indeed dominate the linear forecast. The results are sensitive, however, to whether a stationarity-inducing transformation is applied to the nonstationary unemployment rate series.

Abstract:
This paper reviews recent developments in methods for dealing with weak instruments (IVs) in IV regression models. The focus is more on tests (and confidence intervals derived from tests) than estimators.

The paper also presents new testing results under many weak IV asymptotics, which are relevant when the number of IVs is large and the coefficients on the IVs are relatively small. Asymptotic power envelopes for invariant tests are established. Power comparisons of the conditional likelihood ratio (CLR), Anderson-Rubin, and Lagrange multiplier tests are made. Numerical results show that the CLR test is on the asymptotic power envelope. This holds no matter what the relative magnitude of the IV strength to the number of IVs.

Conditional likelihood ratio test, instrumental variables, many instrumental variables, power envelope, weak instruments

Abstract:
During six weeks in late 1937, Wesley Mitchell, Arthur Burns, and their colleagues at the National Bureau of Economic Research developed a list of leading, coincident, and lagging indicators of economic activity in the United States as part of the NBER research program on business cycles. Since their development, these indicators, in particular the leading and coincident indexes constructed from these indicators, have played an important role in summarizing and forecasting the state of macroeconomic activity.

The paper reports the results of a project to revise the indexes of leading and coincident economic indicators using the tools of modern time series econometrics. This project addresses three central questions. The first is conceptual: is it possible to develop a formal probability model that gives rise to the indexes of leading and coincident variables? Such a model would provide a concrete mathematical framework within which alternative variables and indexes could be evaluated. Second, given this conceptual framework, what are the best variables to use as components of the leading index? Third, given these variables, what is the best way to combine them to produce useful and reliable indexes?

The results of this project are three experimental monthly indexes: an index of coincident economic indicators (CEI), an index of leading economic indicators (LEI), and a Recession Index. The experimental CEI closely tracks the coincident index currently produced by the Department of Commerce (DOC), although the methodology used to produce the two series differs substantially. The growth of the experimental CEI is also highly correlated with the growth of real GNP at business cycle frequencies. The proposed LEI is a forecast of the growth of the proposed CEI over the next six months constructed using a set of leading variables or indicators. The Recession Index, a new series, is the probability that the economy will be in a recession six months hence, given data available through the month of its construction.

This article is organized as follows. Section 2 contains a discussion of the indexes and a framework for their interpretation. Section 3 presents the experimental indexes, discusses their construction, and examines their within-sample performance. In Section 4, the indexes are considered from the perspective of macroeconomic theory, focusing in particular on several salient series that are not included in the proposed leading index. Section 5 concludes.

Abstract:
This paper examines the empirical relationship in the postwar United States between the aggregate business cycle and various aspects of the macroeconomy, such as production, interest rates, prices, productivity, sectoral employment, investment, income, and consumption. This is done by examining the strength of the relationship between the aggregate cycle and the cyclical components of individual time series, whether individual series lead or lag the cycle, and whether individual series are useful in predicting aggregate fluctuations. The paper also reviews some additional empirical regularities in the U.S. economy, including the Phillips curve and some long-run relationships, in particular long-run money demand, long-run properties of interest rates and the yield curve, and the long-run properties of the shares in output of consumption, investment and government spending.

Abstract:
This paper examines monetary policy in Rudebusch and Svensson's (1999) two equation macroeconomic model when the policymaker recognizes that the model is an approximation and is uncertain about the quality of that approximation. It is argued that the minimax approach of robust control provides a general and tractable alternative to the conventional Bayesian decision theoretic approach. Robust control techniques are used to construct robust monetary policies. In most (but not all) cases, these robust policies are more aggressive than the optimal policies absent model uncertainty. The specific robust policies depend strongly on the formation of model uncertainty used, and we make some suggestions about which formulation is most relevant for monetary policy applications.

robust control, parameter uncertainty, Brainard uncertainty

Abstract:
This paper examines monetary policy in Rudebusch and Svensson's (1999) two equation macroeconomic model when the policymaker recognizes that the model is an approximation and is uncertain about the quality of that approximation. It is argued that the minimax approach of robust control provides a general and tractable alternative to the conventional Bayesian decision theoretic approach. Robust control techniques are used to construct robust monetary policies. In most (but not all) cases, these robust policies are more aggressive than the optimal policies absent model uncertainty. The specific robust policies depend strongly on the formation of model uncertainty used, and we make some suggestions about which formulation is most relevant for monetary policy applications.

Abstract:
This paper considers tests of the parameter on endogenous variables in an instrumental variables regression model. The focus is on determining tests that have certain optimal power properties. We start by considering a model with normally distributed errors and known error covariance matrix. We consider tests that are similar and satisfy a natural rotational invariance condition. We determine tests that maximize weighted average power (WAP) for arbitrary weight functions among invariant similar tests. Such tests include point optimal (PO) invariant similar tests. The results yield the power envelope for invariant similar tests. This allows one to assess and compare the power properties of existing tests, such as the Anderson-Rubin, Lagrange multiplier (LM), and conditional likelihood ratio (CLR) tests, and new optimal WAP and PO invariant similar tests. We find that the CLR test is quite close to being uniformly most powerful invariant among a class of two-sided tests. A new unconditional test, P*, also is found to have this property. For one-sided alternatives, no test achieves the invariant power envelope, but a new test - the one-sided CLR test - is found to be fairly close. The finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly non-normal errors via weak instrument asymptotics. Strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments.

Abstract:
This paper considers tests of the parameter on endogenous variables in an instrumental variables regression model. The focus is on determining tests that have some optimal power properties. We start by considering a model with normally distributed errors and known error covariance matrix. We consider tests that are similar and satisfy a natural rotational invariance condition. We determine tests that maximize weighted average power (WAP) for arbitrary weight functions among invariant similar tests. Such tests include point optimal (PO) invariant similar tests.

The results yield the power envelope for invariant similar tests. This allows one to assess and compare the power properties of existing tests, such as the Anderson-Rubin, Lagrange multiplier (LM), and conditional likelihood ratio (CLR) tests, and new optimal WAP and PO invariant similar tests. We find that the CLR test is quite close to being uniformly most powerful invariant among a class of two-sided tests. A new unconditional test, P*, also is found to have this property. For one-sided alternatives, no test achieves the invariant power envelope, but a new test - the one-sided CLR test - is found to be fairly close.

The finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly non-normal errors via weak instrument asymptotics. Strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments.

Instrumental variables regression, invariant tests, optimal tests, similar tests, weak instruments, weighted average power

Abstract:
This paper investigates forecasts of U.S. inflation at the 12-month horizon. The starting point is the conventional unemployment rate Phillips curve, which is examined in a simulated out of sample forecasting framework. Inflation forecasts produced by the Phillips curve generally have been more accurate than forecasts based on other macroeconomic variables, including interest rates, money and commodity prices. These forecasts can however be improved upon using a generalized Phillips curve based on measures of real aggregate activity other than unemployment, especially a new index of aggregate activity based on 61 real economic indicators.

Abstract:
Weak instruments can produce biased IV estimators and hypothesis tests with large size distortions. But what, precisely, are weak instruments, and how does one detect them in practice? This paper proposes quantitative definitions of weak instruments based on the maximum IV estimator bias, or the maximum Wald test size distortion, when there are multiple endogenous regressors. We tabulate critical values that enable using the first-stage F-statistic (or, when there are multiple endogenous regressors, the Cragg-Donald (1993) statistic) to test whether given instruments are weak. A technical contribution is to justify sequential asymptotic approximations for IV statistics with many weak instruments.

Abstract:
Often we are interested in the largest root of an autoregressive process. Available methods rely on inverting t-tests to obtain confidence intervals. However, for large autoregressive roots, t-tests do not approximate asymptotically uniformly most powerful tests and do not have optimality properties when inverted for confidence intervals. We exploit the relationship between the power of tests and accuracy of confidence intervals, and suggest methods which are asymptotically more accurate than available interval construction methods. One interval, based on inverting the P(T) or Q(T) statistic, has good asymptotic accuracy and is easy to compute.

Abstract:
This paper considers VAR models incorporating many time series that interact through a few dynamic factors. Several econometric issues are addressed including estimation of the number of dynamic factors and tests for the factor restrictions imposed on the VAR. Structural VAR identification based on timing restrictions, long run restrictions, and restrictions on factor loadings are discussed and practical computational methods suggested. Empirical analysis using U.S. data suggest several (7) dynamic factors, rejection of the exact dynamic factor model but support for an approximate factor model, and sensible results for a SVAR that identifies money policy shocks using timing restrictions.

Abstract:
'Iterated' multiperiod ahead time series forecasts are made using a one-period ahead model, iterated forward for the desired number of periods, whereas 'direct' forecasts are made using a horizon-specific estimated model, where the dependent variable is the multi-period ahead value being forecasted. Which approach is better is an empirical matter: in theory, iterated forecasts are more efficient if correctly specified, but direct forecasts are more robust to model misspecification. This paper compares empirical iterated and direct forecasts from linear univariate and bivariate models by applying simulated out-of-sample methods to 171 US monthly macroeconomic time series spanning 1959-2002. The iterated forecasts typically outperform the direct forecasts, particularly if the models can select long lag specifications. The relative performance of the iterated forecasts improves with the forecast horizon.

Multistep forecasts, VAR forecasts, forecast comparisons

Abstract:
"Iterated" multiperiod ahead time series forecasts are made using a one-period ahead model, iterated forward for the desired number of periods, whereas "direct" forecasts are made using a horizon-specific estimated model, where the dependent variable is the multi-period ahead value being forecasted. Which approach is better is an empirical matter: in theory, iterated forecasts are more efficient if correctly specified, but direct forecasts are more robust to model misspecification. This paper compares empirical iterated and direct forecasts from linear univariate and bivariate models by applying simulated out-of-sample methods to 171 U.S. monthly macroeconomic time series spanning 1959-2002. The iterated forecasts typically outperform the direct forecasts, particularly if the models can select long lag specifications. The relative performance of the iterated forecasts improves with the forecast horizon.

Multistep forecasts, VAR forecasts, forecast comparisons

Abstract:
From 1960-1983, the standard deviation of annual growth rates in real GDP in the United States was 2.7%. From 1984-2001, the corresponding standard deviation was 1.6%. This paper investigates this large drop in the cyclical volatility OF real economic.activity. The paper has two objectives. The first is to provide a comprehensive characterization of the decline in volatility using a large number of U.S. economic time series and a variety of methods designed to describe time-varying time series processes. In so doing, the paper reviews the literature on the moderation and attempts to resolve some of its disagreements and discrepancies. The second objective is to provide new evidence on the quantitative importance of various explanations for this 'great moderation.' Taken together, we estimate that the moderation in volatility is attributable to a combination of improved policy (20-30%), identifiable good luck in the form of productivity and commodity price shocks (20-30%), and other unknown forms of good luck that manifest themselves as smaller reduced-form forecast errors (40-60%).

Abstract:
This paper examines old and new evidence on the predictive performance of asset prices for inflation and real output growth. We first review the large literature on this topic, focusing on the past dozen years. We then undertake an empirical analysis of quarterly data on up to 38 candidate indicators (mainly asset prices) for seven OECD countries for a span of up to 41 years (1959 1999). The conclusions from the literature review and the empirical analysis are the same. Some asset prices predict either inflation or output growth in some countries in some periods. Which series predicts what, when and where is, however, itself difficult to predict: good forecasting performance by an indicator in one period seems to be unrelated to whether it is a useful predictor in a later period. Intriguingly, forecasts produced by combining these unstable individual forecasts appear to improve reliably upon univariate benchmarks.

Abstract:
This paper examines the forecasting performance of various leading economic indicators and composite indexes since 1988. in particular during the onset of the 1990 recession. The primary focus is on an experimental recession index (tile "XRI"). a composite index which provides probabilistic forecasts of whether the U.S. economy will be in a recession six months hence. After detailing its construction, the paper examines the out-of-sample performance of the XRI and a related forecast of overall economic growth. the experimental leading index (XLI). These indexes performed well from 1988 through the summer of 1990 - for example. in June 1990 the XLI model forecasted a .4% (annual rate) decline in the experimental coincident index from June through September. when in fact the decline was only slightly greater,.8%. However. the XLI failed to forecast the sharp declines of October and November 1990. After exploring several possible explanations. we conclude that one important source of the forecast error was the use of financial variables during a recession that was not associated with a particularly tight monetary policy. Financial indicators -- and the experimental index -- were not alone. however. in failing to forecast the 1990 recession, An examination of 45 economic indicators shows that almost all failed to forecast the 1990downturn. and the few that did provided unclear signals before the recessions of the 19705 and 1980s.

Abstract:
Forecasts of the rate of price inflation play a central role in the formulation of monetary policy, and forecasting inflation is a key job for economists at the Federal Reserve Board. This paper examines whether this job has become harder and, to the extent that it has, what changes in the inflation process have made it so. The main finding is that the univariate inflation process is well described by an unobserved component trend-cycle model with stochastic volatility or, equivalently, an integrated moving average process with time-varying parameters; this model explains a variety of recent univariate inflation forecasting puzzles. It appears currently to be difficult for multivariate forecasts to improve on forecasts made using this time-varying univariate model.

Abstract:
This paper considers forecasting a single time series using more predictors than there are time series observations. The approach is to construct a relatively few indexes, akin to diffusion indexes, which are weighted averages of the predictors, using an approximate dynamic factor model. Estimation is discussed for balanced and unbalanced panels. The estimated dynamic factors are (uniformly) consistent, even in the presence of time varying parameters and/or data contamination, and forecasts based on the estimated factors are efficient. In an application to forecasting U.S. inflation and industrial production using 224 monthly time series, these forecasts outperform various state-of-the-art benchmark models.

Abstract:
Previous authors have reached puzzlingly different conclusions about the usefulness of money for forecasting real output based on closely related regression-based tests. An examination of this and additional new evidence reveals that innovations in M1 have statistically significant marginal predictive value for industrial production, both in a bivariate model and in a multivariate setting including a price index and an interest rate. This conclusion follows from focusing on the trend properties of the data, both stochastic and deterministic, and from drawing inferences using asymptotic theory that explicitly addresses the implications of these trends for the distributions of the various test statistics.

Abstract:
Using quarterly macro data and annual state panel data, we examine various explanations of the low rate of price inflation, strong real wage growth, and low rate of unemployment in the U.S. economy during the late 1990s. Many of these explanations imply shifts in the coefficients of price and wage Phillips curves. We find, however, that once one accounts for the univariate trends in the unemployment rate and in the rate of productivity growth, these coefficients are stable. This suggests that many explanations, such as persistent beneficial supply shocks, changes in firms' pricing power, changes in price expectations arising from shifts in Fed policy, and changes in wage setting behavior miss the mark. Rather, we suggest that explanations of movements of wages, prices and unemployment over the 1990s, and indeed over the past forty years, must focus on understanding the univariate trends in the unemployment rate and in productivity growth and, perhaps, the relation between the two.

Abstract:
No abstract is available for this paper.

Abstract:
This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, here modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum- likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%).

Abstract:
This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.

Abstract:
The conventional heteroskedasticity-robust (HR) variance matrix estimator for cross-sectional regression (with or without a degrees of freedom adjustment), applied to the fixed effects estimator for panel data with serially uncorrelated errors, is inconsistent if the number of time periods T is fixed (and greater than two) as the number of entities n increases. We provide a bias-adjusted HR estimator that is (nT)1/2 -consistent under any sequences (n, T) in which n and/or T increase to infinity. The conventional heteroskedasticity-robust (HR) variance matrix estimator for cross-sectional regression (with or without a degrees of freedom adjustment), applied to the fixed effects estimator for panel data with serially uncorrelated errors, is inconsistent if the number of time periods T is fixed (and greater than two) as the number of entities n increases. We provide a bias-adjusted HR estimator that is (nT)1/2 -consistent under any sequences (n, T) in which n and/or T increase to infinity.

Abstract:
Recent developments in macroeconomic theory emphasize that transient economic fluctuations can arise as responses to changes in long run factors -- in particular, technological improvements -- rather than short run factors. This contrasts with the view that short run fluctuations and shifts in long run trends are largely unrelated. We examine empirically the effect of shifts in stochastic trends that are common to several macroeconomic series. Using a linear time series model related to a VAR, we consider first a system with GNP, consumption and investment with a single common stochastic trend; we then examine this system augmented by money and prices and an additional stochastic trend. Our results suggest that movements in the "real" stochastic trend account for one-half to two-thirds of the variation in postwar U.S. GNP.

Abstract:
The volatility of economic activity in most G7 economies has moderated over the past forty years. Also, despite large increases in trade and openness, G7 business cycles have not become more synchronized. After documenting these twin facts, we interpret G7 output data using a structural VAR that separately identifies common international shocks, the domestic effects of spillovers from foreign idiosyncratic shocks, and the effects of domestic idiosyncratic shocks. This analysis suggests that, with the exception of Japan, the widespread reduction in volatility is in large part associated with a reduction in the magnitude of the common international shocks. Had the common international shocks in the 1980s and 1990s been as large as they were in the 1960s and 1970s, G7 business cycles would have been substantially more volatile and more highly synchronized than they actually were.

Abstract:
No abstract is available for this paper.

Abstract:
An experiment is performed to assess the prevalence of instability in univariate and bivariate macroeconomic time series relations and to ascertain whether various adaptive forecasting techniques successfully handle any such instability. Formal tests for instability and out-of-sample forecasts from sixteen different models are computed using a sample of 76 representative U.S. monthly postwar macroeconomic time series, constituting 5700 bivariate forecasting relations. The tests indicate widespread instability in univariate and bivariate autoregressive models. However, adaptive forecasting models, in particular time varying parameter models, have limited success in exploiting this instability to improve upon fixed-parameter or recursive autoregressive forecasts.

Abstract:
Quah's [1993a] transition matrix analysis of world income distribution based on annual data suggests an ergodic distribution with twin peaks at the rich and poor end of the distribution. Since the ergodic distribution is a highly non-linear function of the underlying transition matrix estimated extremely noisily. Estimates over the foreseeable future are more precise. The Markovian assumptions underlying the analysis are much better satisfied with an analysis based on five-year transitions than one-year transitions. Such an analysis yields an ergodic distribution with 72% of mass in the top income category, but a prolonged transition, during which some inequality measures increase. The rosy ergodic forecast and prolonged transition arise because countries' relative incomes move both up and down at moderate levels, but once countries reach the highest income category, they rarely leave it. This is consistent with a model in which countries search among policies until they reach an income level at which further experimentation is too costly. If countries can learn from each other's experience, the future may be much brighter than would be predicted based on projecting forward the historical transition matrix.

Abstract:
We consider both frequentist and empirical Bayes forecasts of a single time series using a linear model with T observations and K orthonormal predictors. The frequentist formulation considers estimators that are equivariant under permutations (reorderings) of the regressors. The empirical Bayes formulation (both parametric and nonparametric) treats the coefficients as i.i.d. and estimates their prior. Asymptotically, when K is proportional to T the empirical Bayes estimator is shown to be: (i) optimal in Robbins' (1955, 1964) sense; (ii) the minimum risk equivariant estimator; and (iii) minimax in both the frequentist and Bayesian problems over a class of nonGaussian error distributions. Also, the asymptotic frequentist risk of the minimum risk equivariant estimator is shown to equal the Bayes risk of the (infeasible subjectivist) Bayes estimator in the Gaussian case, where the 'prior' is the weak limit of the empirical cdf of the true parameter values. Monte Carlo results are encouraging. The new estimators are used to forecast monthly postwar U.S. macroeconomic time series using the first 151 principal components from a large panel of predictors.

Abstract:
No abstract is available for this paper.

Abstract:
A forecasting comparison is undertaken in which 49 univariate forecasting methods, plus various forecast pooling procedures, are used to forecast 215 U.S. monthly macroeconomic time series at three forecasting horizons over the period 1959 - 1996. All forecasts simulate real time implementation, that is, they are fully recursive. The forecasting methods are based on four classes of models: autoregressions (with and without unit root pretests), exponential smoothing, artificial neural networks, and smooth transition autoregressions. The best overall performance of a single method is achieved by autoregressions with unit root pretests, but this performance can be improved when it is combined with the forecasts from other methods.

Abstract:
This paper surveys the literature since 1993 on pseudo out-of-sample evaluation of inflation forecasts in the United States and conducts an extensive empirical analysis that recapitulates and clarifies this literature using a consistent data set and methodology. The literature review and empirical results are gloomy and indicate that Phillips curve forecasts (broadly interpreted as forecasts using an activity variable) are better than other multivariate forecasts, but their performance is episodic, sometimes better than and sometimes worse than a good (not naïve) univariate benchmark. We provide some preliminary evidence characterizing successful forecasting episodes.

Abstract:
Empirical analysis often raises questions of approximation to underlying individual behavior. Closer approximation may require more complex statistical specifications, On the other hand, more complex specifications may presume computational facility that is beyond the grasp of most real people and therefore less consistent with the actual rules that govern their behavior, even though economic theory may push analysts to increasingly more complex specifications. Thus the issue is not only whether more complex models are worth the effort, but also whether they are better. We compare the in-sample and out-of-sample predictive performance of three models of retirement -- "option value," dynamic programming, and probit -- to determine which of the retirement rules most closely matches retirement behavior in a large firm. The primary measure of predictive validity is the correspondence between the model predictions and actual retirement under the firm's temporary early retirement window plan. The "option value" and dynamic programming models are considerably more successful than the less complex probit model in approximating the rules individuals use to make retirement decisions, but the more complex dynamic programming rule approximates behavior no better than the simpler option value rule.

Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Abstract:
No abstract is available for this paper.

Abstract:
This paper provides asymptotic confidence intervals for the largest autoregressive root of a time series when this root is close to one. The intervals are readily constructed either graphically or using tables in the Appendix. When applied to the Nelson-Plosser (1982) data set, the main conclusion is that the confidence intervals typically are wide. The conventional emphasis on testing for whether the largest root equals one fails to convey the substantial sampling variability associated with this measure of persistence.

Abstract:
This paper considers the estimation of the variance of coefficients in time varying parameter models with stationary regressors. The maximum likelihood estimator has large point mass at zero. We therefore develop asymptotically median unbiased estimators and confidence intervals by inverting median functions of regression-based parameter stability test statistics, computed under the constant-parameter null. These estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability. We apply these results to an unobserved components model of trend growth in postwar U.S. GDP: the MLE implies that there has been no change in the trend rate, while the upper range of the median-unbiased point estimates imply that the annual trend growth rate has fallen by 0.7 percentage points over the postwar period.

Abstract:
Proposed changes in the U.S. Social Security provisions include increasing the normal retirement age from 65 to 67 and changing from 3% to 8% the increase in benefits for each year that retirement is delayed after normal retirement. The paper considers the interaction between these changes and the provisions of employer-provided pension plans. For persons with an employer-provided defined benefit plan, the conclusion is that the Social Security changes will have little effect on labor force participation, but that changes in the firm plan - like increasing the early retirement age - would have very large effects on labor force participation.

Abstract:
This paper investigates the precision of conventional and unconventional estimates of the natural rate of unemployment (the 'NAIRU'). The main finding is that the NAIRU is imprecisely estimated: a typical 95% confidence interval for the NAIRU in 1990 is 5.1% to 7.7%. This imprecision obtains whether the natural rate is modeled as a constant, as a slowly changing function of time, as an unobserved random walk, or as a function of various labor market fundamentals; it obtains using other series for unemployment and inflation, including additional supply shift variables in the Phillips curve, using monthly or quarterly data, and using various measures for expected inflation. This imprecision suggests caution in using the NAIRU to guide monetary policy.

Abstract:
An MLE of the unknown parameters of co integrating vectors is presented for systems in which some variables exhibit higher orders of integration, in which there might be deterministic components, and in which the co integrating vector itself might involve variables of differing orders of integration. The estimator is simple to compute: it can be calculated by running GLS for standard regression equations with serially correlated errors. Alternatively, an asymptotically equivalent estimator can be computed using OLS. Usual Wald test statistics based on these MLE's (constructed using an autocorrelation robust covariance matrix in the case of the OLS estimator) have asymptotic x2 distributions.

Abstract:
This paper develops asymptotic distribution theory for generalized method of moments (GMM) estimators and test statistics when some of the parameters are well identified, but others are poorly identified because of weak instruments. The asymptotic theory entails applying empirical process theory to obtain a limiting representation of the (concentrated) objective function as a stochastic process. The general results are specialized to two leading cases, linear instrumental variables regression and GMM estimation of Euler equations obtained from the consumption-based capital asset pricing model with power utility. Numerical results of the latter model confirm that finite sample distributions can deviate substantially from normality, and indicate that these deviations are captured by the weak instrument asymptotic approximations.

Abstract:
This paper catalogs the business cycle properties of 163 monthly U.S. economic time series over the three decades from 1959 through 1988. Two general sets of summary statistics are reported. The first set measures the comovement of each individual time series with a reference series representing real economic activity. These statistics focus on comovements at business cycle horizons. The second set of statistics examines the predictive content of each of the series for aggregate activity, relative to different sets of conditioning (or predictive) variables. These statistics are constructed and presented in a way that facilitates comparisons across series and across conditioning sets. They also provide new lists of leading indicators based on predictive content for overall economic activity. Some of the results confirm previously recognized empirical regularities, while others provide new or different insights into the business cycle properties of various series.

Abstract:
This paper reviews recent developments in methods for dealing with weak instruments (IVs) in IV regression models. The focus is more on tests and confidence intervals derived from tests than on estimators. The paper also presents new testing results under "many weak IV asymptotics," which are relevant when the number of IVs is large and the coefficients on the IVs are relatively small. Asymptotic power envelopes for invariant tests are established. Power comparisons of the conditional likelihood ratio (CLR), Anderson-Rubin, and Lagrange multiplier tests are made. Numerical results show that the CLR test is on the asymptotic power envelope. This holds no matter what the relative magnitude of the IV strength to the number of IVs.

Abstract:
This paper studies the possibility of using the broad monetary aggregate M2 to target the quarterly rate of growth of nominal GDP. Our findings indicate that the Federal Reserve could probably guide M2 in a way that reduces not only the long-term average rate of inflation but also the variance of the annual rate of growth of nominal GDP. An optimal M2 rule, derived from a simple VAR, reduces the mean ten-year standard deviation of annual GDP growth by over 20 percent. Although there is uncertainty about this value because of both parameter uncertainty and stochastic shocks to the economy, we estimate that the probability that the annual variance would be reduced over a ten year period exceeds 85 percent. A much simpler policy based on a single equation linking M2 and GDP is shown to be almost as successful in reducing this annual GDP variance. Additional statistical tests indicate that M2 is a useful predictor of nominal GDP. Moreover, a battery of recently developed tests for parameter stability fails to reject the hypothesis that the M2 - GDP link is stable, but the M1 - GDP and monetary base - GDP relations are found to be highly unstable. This evidence contradicts those who have argued that the M2 - GDP relation is so unstable in the short run that it cannot be used to reduce the variance of nominal GDP growth.

Abstract:
No abstract is available for this paper.

Abstract:
In most data sets of labor force participation of the elderly, an empirical regularity that emerges is that retirement rates are particularly high at age 65. While there are numerous economic reasons why individuals may choose to retire at 65, empirical models that have attempted to explain the age-65 spike have met with limited success. Interpreted another way, while many models would predict a jump in the hazard rate at age 65, the magnitude of the spike indicates excessive response given the economic considerations that retirees typically face. This paper considers the puzzle of why retirement rates are so high at age 65 and explores a variety of explanations.

Abstract:
We use recent research on estimation and testing in the presence of unit roots to argue that Hall`s (1978) t and F tests of whether consumption is predicted by lagged income, or by lags of consumption beyond the first, are asymptotically valid. A Monte Carlo experiment suggests that the asymptotic t and F distributions provide a good approximation to the actual finite sample distribution.

Abstract:
No abstract is available for this paper.

Abstract:
This paper proposes a class of procedures that consistently classify the stochastic component of a time series as being integrated either of order zero (l(0» or one (l(1» for general 1(0) and 1(1) processes. These procedures entail the evaluation of the asymptotic likelihoods of certain statistics under the 1(0)and 1(1) hypotheses. These likelihoods do not depend on nuisance parameters describing short-run dynamics and diverge asymptotically, so their ratio provides a consistent basis for classifying a process as 1(1) or 1(0). Bayesian inference can be performed by placing prior mass only on the point hypotheses "1(0)" and "1(1)" without needing to specify parametric priors within the classes of 1(0) and 1(1) processes; the result is posterior odds ratios for the 1(0) and 1(1) hypotheses. These procedures are developed for general polynomial and piecewise linear detrending. When applied to the Nelson-Plosser data with linear detrending, they largely support the original Nelson-Plosser inferences. With piecewise-linear detrending these data are typically uninformative, producing Bayes factors that are close to one.

Abstract:
The option value model developed in an earlier paper is used to simulate the effect on retirement of changes in a firm's pension plan compared to the effect of changes in Social Security provisions. The provisions of the firm's pension plan have a much greater effect than Social Security regulations on the retirement decisions of the firm's employees. The analysis supports the following conclusions: (1) Increasing the firm's early retirement age from 55 to 60, for example, would reduce by almost 40 percent, from .48 to .30, the fraction of employees that is retired by age 60. (2) The effect of changes in Social Security rules, on the other hand, would be small. Raising the Social Security retirement ages by one year, for example, has very little effect on employee retirement rates. The proportion retired by age 62 is reduced by only about 4 percent. (3) Changes in Social Security provisions that would otherwise encourage workers to continue working can easily be offset by countervailing changes in the provisions of the firm's pension plan. Firm responses, like delaying the Social Security offset to correspond to m later Social Security retirement age, may simply be m logical revision of current firm plan provisions.

Abstract:
It is well known that the distribution of statistics testing restrictions on the coefficients in time series regressions can depend on the order of integration of the regressors. In practice the order of integration is rarely blown. This paper examines two conventional approaches to this problem, finds them unsatisfactory, and proposes a new procedure. The two conventional approaches- simply to ignore unit root problems or to use unit root pretests to determine the critical values for second-stage inference - both often induce substantial size distortions. In the case of unit root pretests, this arises because type I and II pretest errors produce incorrect second-stage critical values and because, in many empirically plausible situations, the first stage test (the unit root test) and the second stage test (the exclusion restriction test) are dependent. Monte Carlo simulations reveal size distortions even if the regressor is stationary but has a large autoregressive root, a case that might arise for example in a regression of excess stock returns against the dividend yield. In the proposed alternative procedure, the second-stage test is conditional on a first-stage "unit root" statistic developed in Stock (1992); the second-stage critical values vary continuously with the value of the first-stage statistic. The procedure is shown to have the correct size asymptotically and to have good local asymptotic power against Granger-causality alternatives.

Abstract:
No abstract is available for this paper.