We develop inference methods about long†run comovement of two time series. The parameters of interest are defined in terms of population second moments of low†frequency transformations (“low†pass†filtered versions) of the data. We numerically determine confidence sets that control coverage over a wide range of potential bivariate persistence patterns, which include arbitrary linear combinations of I(0), I(1), near unit roots, and fractionally integrated processes. In an application to U.S. economic data, we quantify the long†run covariability of a variety of series, such as those giving rise to balanced growth, nominal exchange rates and relative nominal prices, the unemployment rate and inflation, money growth and inflation, earnings and stock prices, etc.

This paper considers nonstandard hypothesis testing problems that involve a nuisance parameter. We establish an upper bound on the weighted average power of all valid tests, and develop a numerical algorithm that determines a feasible test with power close to the bound. The approach is illustrated in six applications: inference about a linear regression coefficient when the sign of a control coefficient is known; small sample inference about the difference in means from two independent Gaussian samples from populations with potentially different variances; inference about the break date in structural break models with moderate break magnitude; predictability tests when the regressor is highly persistent; inference about an interval identified parameter; and inference about a linear regression coefficient when the necessity of a control is in doubt.

This paper considers estimation and hypothesis testing in linear time series when some or all of the variables have (possibly multiple) unit roots. The motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. Parameters that can be written as coefficients on mean zero, nonintegrated regressors have jointly normal asymptotic distribution, converging at the rate of T(superscript "one-half") In general, the other coefficients (including the coefficient on polynomials in time), and associated t and F test statistics, have nonstandard asymptotic distributions.

We develop a Bayesian latent factor model of the joint long-run evolution of GDP per capita for 113 countries over the 118 years from 1900 to 2017. We find considerable heterogeneity in rates of convergence, including rates for some countries that are so slow that they might not converge (or diverge) in century-long samples, and a sparse correlation pattern (“convergence clubs”) between countries. The joint Bayesian structure allows us to compute a joint predictive distribution for the output paths of these countries over the next 100 years. This predictive distribution can be used for simulations requiring projections into the deep future, such as estimating the costs of climate change. The model's pooling of information across countries results in tighter prediction intervals than are achieved using univariate information sets. Still, even using more than a century of data on many countries, the 100-year growth paths exhibit very wide uncertainty.

Are business cycles mainly the result of permanent shocks to productivity? This paper uses a long-run restriction implied by a large class of real-business-cycle models--identifying permanent productivity shocks as shocks to the common stochastic trend in output, consumption, and investment--to provide new evidence on this question. Econometric test indicate that this common-stochastic-trend / cointegration implication is consistent with postwar U.S. data. However, in systems with nominal variables, the estimates of this common stochastic trend indicate that permanent productivity shocks typically explain less than half of the business-cycle variability in output, consumption, and investment.

A bstract-We discuss the problem of testing for constant versus time varying regression coefficients. Our alternative hypothesis allows the coefficients to follow a stationary AR(1) process with unknown autoregressive parameter. Standard testing procedures are inappropriate since this parameter is identified only under the alternative. We propose a test statistic which is a function of a sequence of Score statistics, and depends only on the regressors and the OLS residuals. The distribution of the test statistic is discussed, power and size are investigated using Monte Carlo methods, and an empirical example investigating stability in the gold and silver markets is presented.

Using factor methods, we decompose industrial production (IP) into components arising from aggregate and sector-specific shocks. An approximate factor model finds that nearly all of IP variability is associated with common factors. We then use a multisector growth model to adjust for the effects of input-output linkages in the factor analysis. Thus, a structural factor analysis indicates that the Great Moderation was characterized by a fall in the importance of aggregate shocks while the volatility of sectoral shocks was essentially unchanged. Consequently, the role of idiosyncratic shocks increased considerably after the mid-1980s, explaining half of the quarterly variation in IP.

We describe how capital accumulation and the network structure of US production interact to amplify the effects of sectoral trend growth rates in total factor productivity and labor on trend GDP (gross domestic product) growth. We derive expressions that conveniently summarize this long-run amplification effect by way of sectoral multipliers. We estimate that sector-specific factors have historically accounted for approximately three-fourths of long-run changes in GDP growth. Trend GDP growth fell by nearly 3 percentage points over the postwar period, with especially significant contributions from the Construction sector in 1950–80 and the Durable Goods sector in 2000–2018. No sector has contributed any steady significant increase to the trend growth rate of GDP in the past 70 years.

The dynamics of a linear (or linearized) dynamic stochastic economic model can be expressed in terms of matrices (A, B, C, D) that define a state space system for a vector of observables. An associated state space system (A, ^ B,C, ^D) determines a vector autoregression for those same observables. We present a simple condition for checking when these two state space systems match up and when they do not when there are equal numbers of economic and VAR shocks. We illustrate our condition with a permanent income example.