Measuring predictability: theory and macroeconomic applications

We propose a measure of predictability based on the ratio of the expected loss of a short‐run forecast to the expected loss of a long‐run forecast. This predictability measure can be tailored to the forecast horizons of interest, and it allows for general loss functions, univariate or multivariate information sets, and covariance stationary or difference stationary processes. We propose a simple estimator, and we suggest resampling methods for inference. We then provide several macroeconomic applications. First, we illustrate the implementation of predictability measures based on fitted parametric models for several US macroeconomic time series. Second, we analyze the internal propagation mechanism of a standard dynamic macroeconomic model by comparing the predictability of model inputs and model outputs. Third, we use predictability as a metric for assessing the similarity of data simulated from the model and actual data. Finally, we outline several non‐parametric extensions of our approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Spurious Regression and Trending Variables*

This paper analyses the asymptotic and finite‐sample implications of different types of non‐stationary behaviour among the dependent and explanatory variables in a linear spurious regression model. We study cases when the non‐stationarity in the dependent and explanatory variables is deterministic as well as stochastic. In particular, we derive the order in probability of the t‐statistic in a spurious regression equation under a variety of empirically relevant data generation processes, and show that the spurious regression phenomenon is present in all cases when both dependent and explanatory variables behave in a non‐stationary way. Simulation experiments confirm our asymptotic results.     

Testing for Predictability in panels with General Predictors

The difficulty of predicting returns has recently motivated researchers to start looking for tests that are either more powerful or robust to more features of the data. Unfortunately, the way that these tests work typically involves trading robustness for power or vice versa. The current paper takes this as its starting point to develop a new panel‐based approach to predictability that is both robust and powerful. Specifically, while the panel route to increased power is not new, the way in which the cross‐section variation is exploited also to achieve robustness with respect to the predictor is. The result is two new tests that enable asymptotically standard normal and chi‐squared inference across a wide range of empirically relevant scenarios in which the predictor may be stationary, moderately non‐stationary, nearly non‐stationary, or indeed unit root non‐stationary. The type of cross‐section dependence that can be permitted in the predictor is also very general, and can be weak or strong, although we do require that the cross‐section dependence in the regression errors is of the strong form. What is more, this generality comes at no cost in terms of complicated test construction. The new tests are therefore very user‐friendly. Copyright © 2016 John Wiley & Sons, Ltd.     

Variable selection via composite quantile regression with dependent errors

We propose composite quantile regression for dependent data, in which the errors are from short‐range dependent and strictly stationary linear processes. Under some regularity conditions, we show that composite quantile estimator enjoys root‐n consistency and asymptotic normality. We investigate the asymptotic relative efficiency of composite quantile estimator to both single‐level quantile regression and least‐squares regression. When the errors have finite variance, the relative efficiency of composite quantile estimator with respect to the least‐squares estimator has a universal lower bound. Under some regularity conditions, the adaptive least absolute shrinkage and selection operator penalty leads to consistent variable selection, and the asymptotic distribution of the non‐zero coefficient is the same as that of the counterparts obtained when the true model is known. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.     

Modelling Inflation Volatility

This paper discusses estimation of US inflation volatility using time‐varying parameter models, in particular whether it should be modelled as a stationary or random walk stochastic process. Specifying inflation volatility as an unbounded process, as implied by the random walk, conflicts with priors beliefs, yet a stationary process cannot capture the low‐frequency behaviour commonly observed in estimates of volatility. We therefore propose an alternative model with a change‐point process in the volatility that allows for switches between stationary models to capture changes in the level and dynamics over the past 40 years. To accommodate the stationarity restriction, we develop a new representation that is equivalent to our model but is computationally more efficient. All models produce effectively identical estimates of volatility, but the change‐point model provides more information on the level and persistence of volatility and the probabilities of changes. For example, we find a few well‐defined switches in the volatility process and, interestingly, these switches line up well with economic slowdowns or changes of the Federal Reserve Chair. Moreover, a decomposition of inflation shocks into permanent and transitory components shows that a spike in volatility in the late 2000s was entirely on the transitory side and characterized by a rise above its long‐run mean level during a period of higher persistence. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonparametric methods and local‐time‐based estimation for dynamic power law distributions

This paper introduces nonparametric econometric methods that characterize general power law distributions under basic stability conditions. These methods extend the literature on power laws in the social sciences in several directions. First, we show that any stationary distribution in a random growth setting is shaped entirely by two factors: the idiosyncratic volatilities and reversion rates (a measure of cross‐sectional mean reversion) for different ranks in the distribution. This result is valid regardless of how growth rates and volatilities vary across different economic agents, and hence applies to Gibrat's law and its extensions. Second, we present techniques to estimate these two factors using panel data. Third, we describe how our results imply predictability as higher‐ranked processes must on average grow more slowly than lower‐ranked processes. We employ our empirical methods using data on commodity prices and show that our techniques accurately describe the empirical distribution of relative commodity prices. We also show that rank‐based out‐of‐sample forecasts of future commodity prices outperform random‐walk forecasts at a 1‐month horizon.     

Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

Income distribution and income dynamics in the United Kingdom

In this paper, we propose a model of income dynamics which takes account of mobility both within and between jobs. The model is a hybrid of the mover‐stayer model of income dynamics and a geometric random walk. In any period, individuals face a discrete probability of ‘moving’, in which case their income is a random drawn from a stationary recurrent distribution. Otherwise, they ‘stay’ and incomes follow a geometric random walk. The model is estimated on income transition data for the United Kingdom from the British Household Panel Survey (BHPS) and provides a good explanation of observed non‐linearities in income dynamics. The steady‐state distribution of the model provides a good fit for the observed, cross‐sectional distribution of earnings. We also evaluate the impact of tertiary education on income transitions and on the long‐run distribution of incomes. Copyright © 2001 John Wiley & Sons, Ltd.     

Cross‐sectional Dependency and Size Distortion in a Small‐sample Homogeneous Panel Data Unit Root Test*

In this paper, we investigate the effects of cross‐sectional disturbance correlation in a homogeneous panel data unit root test. As reported by other authors, the unit root test has incorrect size in the presence of cross‐sectional correlation. We suggest that a previously known estimator can be used to reduce the size distortions. We supply response surface estimates for critical values and study the size characteristics of the proposed test. We find that the suggested estimator performs well in small‐sample homogeneous panel data unit root tests. The reduction in size distortion comes at a small cost of lower power against a stationary alternative.     

Keeping off the grass? An econometric model of cannabis consumption in Britain

This paper presents estimates of a dynamic individual‐level model of cannabis consumption, using data from a 1998 survey of young people in Britain. The econometric model is a split‐population generalization of the non‐stationary Poisson process, allowing for separate dynamic process for initiation into cannabis use and subsequent consumption. The model allows for heterogeneity in consumption levels and behavioural shifts induced by leaving education and the parental home. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling and forecasting of stock index volatility with APARCH models under ordered restriction

This article examines volatility models for modeling and forecasting the Standard & Poor 500 (S&P 500) daily stock index returns, including the autoregressive moving average, the Taylor and Schwert generalized autoregressive conditional heteroscedasticity (GARCH), the Glosten, Jagannathan and Runkle GARCH and asymmetric power ARCH (APARCH) with the following conditional distributions: normal, Student's t and skewed Student's t‐distributions. In addition, we undertake unit root (augmented Dickey–Fuller and Phillip–Perron) tests, co‐integration test and error correction model. We study the stationary APARCH (p) model with parameters, and the uniform convergence, strong consistency and asymptotic normality are prove under simple ordered restriction. In fitting these models to S&P 500 daily stock index return data over the period 1 January 2002 to 31 December 2012, we found that the APARCH model using a skewed Student's t‐distribution is the most effective and successful for modeling and forecasting the daily stock index returns series. The results of this study would be of great value to policy makers and investors in managing risk in stock markets trading.     

Steady‐state priors for vector autoregressions

Bayesian priors are often used to restrain the otherwise highly over‐parametrized vector autoregressive (VAR) models. The currently available Bayesian VAR methodology does not allow the user to specify prior beliefs about the unconditional mean, or steady state, of the system. This is unfortunate as the steady state is something that economists usually claim to know relatively well. This paper develops easily implemented methods for analyzing both stationary and cointegrated VARs, in reduced or structural form, with an informative prior on the steady state. We document that prior information on the steady state leads to substantial gains in forecasting accuracy on Swedish macro data. A second example illustrates the use of informative steady‐state priors in a cointegration model of the consumption‐wealth relationship in the USA. Copyright © 2009 John Wiley & Sons, Ltd.     

Local block bootstrap inference for trending time series

Resampling for stationary sequences has been well studied in the last couple of decades. In the paper at hand, we focus on nonstationary time series data where the nonstationarity is due to a slowly-changing deterministic trend. We show that the local block bootstrap methodology is appropriate for inference under this locally stationary setting without the need of detrending the data. We prove the asymptotic consistency of the local block bootstrap in the smooth trend model, and complement the theoretical results by a finite-sample simulation.     

Further Evidence on the Statistical Properties of Real GNP*

The well‐known lack of power of unit‐root tests has often been attributed to the short length of macroeconomic variables and also to data‐generating processes (DGPs) departing from the I(1)–I(0) models. This paper shows that by using long spans of annual real gross national product (GNP) and GNP per capita (133 years), high power can be achieved, leading to the rejection of both the unit‐root and the trend‐stationary hypothesis. More flexible representations are then considered, namely, processes containing structural breaks (SB) and fractional orders of integration (FI). Economic justification for the presence of these features in GNP is provided. It is shown that both FI and SB formulations are in general preferred to the autoregressive integrated moving average (ARIMA) [I(1) or I(0)] formulations. As a novelty in this literature, new techniques are applied to discriminate between FI and SB. It turns out that the FI specification is preferred, implying that GNP and GNP per capita are non‐stationary, highly persistent but mean‐reverting series. Finally, it is shown that the results are robust when breaks in the deterministic component are allowed for in the FI model. Some macroeconomic implications of these findings are also discussed.     

Simulation‐based tests of forward‐looking models under VAR learning dynamics

In this paper we propose a simulation‐based technique to investigate the finite sample performance of likelihood ratio (LR) tests for the nonlinear restrictions that arise when a class of forward‐looking (FL) models typically used in monetary policy analysis is evaluated with vector autoregressive (VAR) models. We consider ‘one‐shot’ tests to evaluate the FL model under the rational expectations hypothesis and sequences of tests obtained under the adaptive learning hypothesis. The analysis is based on a comparison between the unrestricted and restricted VAR likelihoods, and the p‐values associated with the LR test statistics are computed by Monte Carlo simulation. We also address the case where the variables of the FL model can be approximated as non‐stationary cointegrated processes. Application to the ‘hybrid’ New Keynesian Phillips Curve (NKPC) in the euro area shows that (i) the forward‐looking component of inflation dynamics is much larger than the backward‐looking component and (ii) the sequence of restrictions implied by the cointegrated NKPC under learning dynamics is not rejected over the monitoring period 1984–2005. Copyright © 2010 John Wiley & Sons, Ltd.     

Distribution approximations for cointegration tests with stationary exogenous regressors

The distribution of a functional of two correlated vector‐Brownian motions is approximated by a Gamma distribution. This functional represents the limiting distribution for cointegration tests with stationary exogenous regressors, but also for cointegration tests based on a non‐Gaussian likelihood. The approximation is accurate, fast and easy to use in comparison with both tabulated critical values and simulated p‐values. The proposed procedure is applied to a UK model investigating purchasing power parity. Copyright © 2005 John Wiley & Sons, Ltd.     

Demographics and Business Cycle Volatility: A Spurious Relationship?

This paper replicates the estimation results of three studies on the impact of the age composition of the labor force on business cycle volatility and investigates whether they signal a meaningful long‐run relationship. We show that both the volatile‐age labor force share variable and the business cycle volatility measure exhibit non‐stationary behavior but find no robust evidence of cointegration. Hence the estimation results reported in the literature may be spurious. This conclusion is further supported by the finding that the strong relationship (i) disappears when cross‐sectional dependence is accounted for using the CCEP estimator and (ii) is highly sensitive to small changes in the composition of the sample, to data revisions, and to the exact definition of the volatile‐age labor share. Copyright © 2016 John Wiley & Sons, Ltd.     

Break Date Estimation for Models with Deterministic Structural Change

In this article, we consider estimating the timing of a break in level and/or trend when the order of integration and autocorrelation properties of the data are unknown. For stationary innovations, break point estimation is commonly performed by minimizing the sum of squared residuals across all candidate break points, using a regression of the levels of the series on the assumed deterministic components. For unit root processes, the obvious modification is to use a first differenced version of the regression, while a further alternative in a stationary autoregressive setting is to consider a GLS‐type quasi‐differenced regression. Given uncertainty over which of these approaches to adopt in practice, we develop a hybrid break fraction estimator that selects from the levels‐based estimator, the first‐difference‐based estimator, and a range of quasi‐difference‐based estimators, according to which achieves the global minimum sum of squared residuals. We establish the asymptotic properties of the estimators considered, and compare their performance in practically relevant sample sizes using simulation. We find that the new hybrid estimator has desirable asymptotic properties and performs very well in finite samples, providing a reliable approach to break date estimation without requiring decisions to be made regarding the autocorrelation properties of the data.     

A Semi‐Parametric Analysis of Two‐Sided Markets: An Application to the Local Daily Newspapers in the USA

This paper considers an empirical semiparametric model for two‐sided markets. Contrary to existing empirical literature on two‐sided markets, we specify network effects and probability distribution functions of net benefits of the two sides nonparametrically. We then estimate the model by nonparametric instrumental variables regression for local daily newspapers from the USA. We show that semiparametric specification is supported by the data and the network effects are neither linear nor monotonic. With a numerical illustration we demonstrate that the mark‐up of the newspaper on each side changes drastically with the nonlinearly specified network effects from the case with linear network effects. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamics of the federal funds target rate: a nonstationary discrete choice approach

We apply a discrete choice approach to model the empirical behaviour of the Federal Reserve in changing the federal funds target rate, the benchmark of short‐term market interest rates in the US. Our methods allow the explanatory variables to be nonstationary as well as stationary. This feature is particularly useful in the present application as many economic fundamentals that are monitored by the Fed and are believed to affect decisions to adjust interest rate targets display some nonstationarity over time. The chosen model successfully predicts the majority of the target rate changes during the time period considered (1994–2001) and helps to explain strings of similar intervention decisions by the Fed. Based on the model‐implied optimal interest rate, our findings suggest that there is a lag in the Fed's reaction to economic shocks during this period. Copyright © 2004 John Wiley & Sons, Ltd.