Number Of Time Series Observations Research Articles

Testing Beta-Pricing Models Using Large Cross-Sections

Abstract We propose a methodology for estimating and testing beta-pricing models when a large number of assets is available for investment but the number of time-series observations is fixed. We first consider the case of correctly specified models with constant risk premia, and then extend our framework to deal with time-varying risk premia, potentially misspecified models, firm characteristics, and unbalanced panels. We show that our large cross-sectional framework poses a serious challenge to common empirical findings regarding the validity of beta-pricing models. In the context of pricing models with Fama-French factors, firm characteristics are found to explain a much larger proportion of variation in estimated expected returns than betas. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

CCE in fixed‐T panels

SummaryThe presence of unobserved heterogeneity and its likely detrimental effect on inference has recently motivated the use of factor‐augmented panel regression models. The workhorse of this literature is based on first estimating the unknown factors using the cross‐section averages of the observables, and then applying ordinary least squares conditional on the first‐step factor estimates. This is the common correlated effects (CCE) approach, the existing asymptotic theory for which is based on the requirement that both the number of time series observations, T, and the number of cross‐section units, N, tend to infinity. The obvious implication of this theory for empirical work is that both N and T should be large, which means that CCE is impossible for the typical micro panel where only N is large. In the current paper, we put the existing CCE theory and its implications to a test. This is done by developing a new theory that enables T to be fixed. The results show that many of the previously derived large‐T results hold even if T is fixed. In particular, the pooled CCE estimator is still consistent and asymptotically normal, which means that CCE is more applicable than previously thought. In fact, not only do we allow T to be fixed, but the conditions placed on the time series properties of the factors and idiosyncratic errors are also much more general than those considered previously.

Generalized fixed‐T panel unit root tests

AbstractPanel data unit root tests, which can be applied to data that do not have many time series observations, are based on very restrictive error and deterministic component specification assumptions. In this paper, we develop a new, doubly modified estimator, based on which we propose a panel unit root test that allows for multiple structural breaks, linear and nonlinear trends, heteroscedasticity, serial correlation, and error cross‐section heterogeneity, when the number of time series observations is finite. The test has the additional perk that it is invariant to the initial condition.

Tests of Alternative Asset Pricing Models Using Individual Security Returns and a New Multivariate F-Test

This paper examines relative performance of alternative asset pricing models using individual security returns. The standard multivariate test used in studies comparing the performance of asset pricing models requires the number of stocks to be less than the number of time series observations, which requires grouping stocks into portfolios. This results in a loss of disaggregate stock information. We apply a different statistical test to overcome this problem and to investigate relative performance of alternative asset pricing models using individual security returns instead of portfolio returns. Our findings suggest that a parsimonious six-factor model that includes the momentum and orthogonal value factors outperforms all other models based on a number of measures as well as the average [Formula: see text]-test. Unlike the standard multivariate test, we find that the average [Formula: see text]-test has superior power to discriminate among competing models and does not reject all tested models.

Rank based cointegration testing for dynamic panels with fixed T

In this paper, we show that the cointegration testing procedure of Binder et al. (Econom Theory 21:795–837, 2005) for Panel Vector Autoregressive model of order 1, PVAR(1) is not valid due to the singularity of the hessian matrix. As an alternative we propose a method of moments based procedure using the rank test of Kleibergen and Paap (J Econom 133:97–126, 2006) for a fixed number of time series observations. The test is shown to be applicable in situations with time-series heteroscedasticity and unbalanced data. The novelty of our approach is that in the construction of the test we exploit the “weakness” of the Anderson and Hsiao (J Econom 18:47–82, 1982) moment conditions. The finite-sample performance of the proposed test statistic is investigated using simulated data. The results indicate that for most scenarios the method has good statistical properties. The proposed test provides little statistical evidence of cointegration in the employment data of Alonso-Borrego and Arellano (J Bus Econ Stat 17:36–49, 1999).

A heteroskedasticity robust Breusch–Pagan test for Contemporaneous correlation in dynamic panel data models

This paper proposes a heteroskedasticity-robust Breusch–Pagan test of the null hypothesis of zero cross-section (or contemporaneous) correlation in linear panel data models, without necessarily assuming independence of the cross-sections. The procedure allows for either fixed, strictly exogenous and/or lagged dependent regressor variables, as well as quite general forms of both non-normality and heteroskedasticity in the error distribution. The asymptotic validity of the test procedure is predicated on the number of time series observations, T, being large relative to the number of cross-section units, N, in that: either (i) N is fixed as T→∞; or, (ii) N2/T→0 as both T and N diverge, jointly, to infinity. Given this, it is not expected that asymptotic theory would necessarily provide an adequate guide to finite sample performance when T/N is “small”. Because of this we also propose, and establish asymptotic validity of, a number of wild bootstrap schemes designed to provide improved inference when T/N is small. Across a variety of experimental designs, a Monte Carlo study suggests that the predictions from asymptotic theory do, in fact, provide a good guide to the finite sample behaviour of the test when T is large relative to N. However, when T and N are of similar orders of magnitude, discrepancies between the nominal and empirical significance levels occur as predicted by the first-order asymptotic analysis. On the other hand, for all the experimental designs, the proposed wild bootstrap approximations do improve agreement between nominal and empirical significance levels, when T/N is small, with a recursive-design wild bootstrap scheme performing best, in general, and providing quite close agreement between the nominal and empirical significance levels of the test even when T and N are of similar size. Moreover, in comparison with the wild bootstrap “version” of the original Breusch–Pagan test (Godfrey and Yamagata, 2011) our experiments indicate that the corresponding version of the heteroskedasticity-robust Breusch–Pagan test appears reliable. As an illustration, the proposed tests are applied to a dynamic growth model for a panel of 20 OECD countries.

Testing Beta-Pricing Models Using Large Cross-Sections

Building on the Shanken (1992) estimator, we develop a new methodology for estimating and testing beta-pricing models when a large number of assets N is available but the number of time-series observations is small. We show empirically that our large N framework can change substantially common empirical findings regarding estimated risk premia and validity of beta-pricing models. We generalize our theoretical results to the more realistic case of unbalanced panels. The practical relevance of our findings is confirmed via Monte Carlo simulations.

Fixed-Effects Panel Data Models: To Cluster or Not to Cluster

Fixed effects panel data models are not immune to problems caused by nonstationarity. Using Monte Carlo experiments, we find that OLS standard errors invariably fail to generate significance tests with correct size when either or both of the regressors are I(1). Clustered standard errors generate correct standard errors if the number of groups is 50 or more and the number of time series observations are 25 or more. With respect to unbalanced models in which an I(1) variable is regressed on an I(0) variable or vice-versa, clustering the standard errors will generate correct standard errors, but not for small values of N and T. We find that neither OLS nor clustered standard errors generate reliably sized significance tests in the case of heterogeneous panels in which the dependent variable, the independent variable, or both are mixes of stationary and nonstationary groups. This could be important given the well-known weakness of unit root tests and the fact that rejecting the null hypothesis in a panel unit root test implies only that one or more of the groups are stationary. First differencing is a simple solution, in which case OLS standard errors are preferred. However, first differences generate a short run model. For estimation in levels, clustered standard errors for relatively large N and T and a simulation or bootstrap approach for smaller samples appears to be the best method for significance tests in fixed effects models in the presence of nonstationary time series.

Bias-corrected estimation of panel vector autoregressions

We derive a bias-corrected least-squares estimator for panel vector autoregressions with fixed effects. The estimator is straightforward to implement and is asymptotically unbiased under asymptotics where the number of time series observations and the number of cross-sectional observations grow at the same rate. This makes the estimator particularly well suited for most macroeconomic data sets.

Testing for predictability in panels of any time series dimension

The few panel data tests for the predictability of returns that exist are based on the prerequisite that both the number of time series observations, T, and the number of cross-section units, N, are large. As a result, it is impossible to apply these tests to stock markets, where lengthy time series of data are scarce. In response to this, the current paper develops a new test for predictability in panels where N is large and T≥2 can be either small or large, or indeed anything in between. This consideration represents an advancement relative to the usual large-N and large-T requirement. The new test is also very general, especially when it comes to allowable predictors, and is easy to implement. As an illustration, we consider the Chinese stock market, for which data are available for only 17 years, but where the number of firms is relatively large, 160.

Improving the finite sample performance of autoregression estimators in dynamic factor models: A bootstrap approach

ABSTRACTWe investigate the finite sample properties of the estimator of a persistence parameter of an unobservable common factor when the factor is estimated by the principal components method. When the number of cross-sectional observations is not sufficiently large, relative to the number of time series observations, the autoregressive coefficient estimator of a positively autocorrelated factor is biased downward, and the bias becomes larger for a more persistent factor. Based on theoretical and simulation analyses, we show that bootstrap procedures are effective in reducing the bias, and bootstrap confidence intervals outperform naive asymptotic confidence intervals in terms of the coverage probability.

Testing for a Structural Break in Dynamic Panel Data Models with Common Factors

This paper develops a method for testing for the presence of a single structural break in panel data models with unobserved heterogeneity represented by a factor error structure. The common factor approach is an appealing way to capture the effect of unobserved variables, such as skills and innate ability in studies of returns to education, common shocks and cross-sectional dependence in models of economic growth, law enforcement acts and public attitudes towards crime in statistical modelling of criminal behaviour. Ignoring these variables may result in inconsistent parameter estimates and invalid inferences. We focus on the case where the time frequency of the data may be yearly and thereby the number of time series observations is small, even if the sample covers a rather long period of time. We develop a Distance type statistic based on a Method of Moments estimator that allows for unobserved common factors. Existing structural break tests proposed in the literature are not valid under these circumstances. The asymptotic properties of the test statistic are established for both known and unknown breakpoints. In our simulation study, the method performed well, both in terms of size and power, as well as in terms of successfully locating the time at which the break occurred. The method is illustrated using data from a large sample of banking institutions, providing empirical evidence on the well-known Gibrat's `Law'.

A fixed-[formula omitted] version of Breitung’s panel data unit root test

We extend Breitung’s (2000) panel data unit root test to the case of fixed time (T) dimension while still allowing for heteroscedastic and serially correlated error terms. The analytic local power function of the new test is derived assuming that only the cross section dimension of the panel grows large. It is found that, if the errors are serially correlated, the test has non-trivial power. Monte Carlo experiments show that the suggested test is more powerful when the number of cross section units is moderate or large, regardless of the number of time series observations.

Discussion of “Forecasting macroeconomic variables using collapsed dynamic factor analysis” by Falk Bräuning and Siem Jan Koopman

Factor models have become an important tool in the modern forecaster’s toolkit. They provide a means of producing forecasts when the number of indicator series, N , exceeds the number of time series observations, T . A further perceived benefit is that they relieve the pressure on the forecaster to select the preferred indicator(s), and in turn indicator-based forecasting model, from a potentially large set.1 Various different factor models have been proposed, from principal components to parametric dynamic factor models and spectral density methods, with the applied literature tending to conclude that factor models are helpful for the production of short-term point forecasts. Brauning and Koopman (2014, BK henceforth) build on this now considerable body of literature (recently reviewed by Stock & Watson, 2011) by proposing a new dynamic factor model – the Collapsed (Dynamic) Factor Model (CFM) – for short-term forecasting. This includes nowcasting, where the distinguishing feature is the use of mixed-frequency data and jagged or ragged edges. BK’s proposed model (see their Eq. (9)) can be seen as a generalisation of the widely used Stock-Watson approach, which augments the forecasting model with principal components. This is because BK’s model accommodates measurement error by treating the principal components as noisy estimates of the true factors. However, principal

The economic environment and the growth in reserves from known gas fields in the Gulf of Mexico

Abstract The vast proportion of yearly oil and gas reserve additions are accounted for by reserve changes in known oil and gas fields. For example, in 2008 approximately 96% of the dry natural gas reserve additions in the United States were accounted for by reserve changes in known fields. The increase in proved ultimate recovery that results from these reserve changes is known as reserve appreciation or reserve growth. Ideally, reserve growth should be measured by comparing the current known petroleum volumes equal to cumulative production plus remaining reserves of fields discovered in say, 1950, with their initial discovery sizes. Unfortunately, implementing the ideal is often not possible since the initial discovery sizes of fields discovered before 1975, or thereabouts, are many times unknown. Up to now, the standard solution to the problem has been based largely on the work of Arrington (1960) who developed a method to estimate reserve growth when initial discovery sizes are unknown and the number of time series observations is small. While the contribution of the economic environment to the growth process has been acknowledged by geologists, the application of the Arrington method presumes that growth is independent of the economic environment. In our view, this creates a bias. For example, if, as we suspect, the economic environment affects reserve growth, then an Arrington-based supply analysis will underestimate the response of supplies to higher prices. In this paper, we employ a rich database from the former United States Minerals Management Service (MMS) to evaluate the contribution of the economic environment to the reserve growth process. In contrast to Forbes and Zampelli (2009) which examined reserve growth at the individual field level, the analysis in this paper aggregates the field level data based on the year of first production. One advantage of this approach is that it has the potential to yield insights about the resource potential of a particular geologic province. The results strongly suggest that age is not the sole factor in explaining a field’s annual reserve growth. In particular, we find that the annual growth rate in the known petroleum volumes of a field is affected by the economic environment as proxied by price. Incorporation of this effect into the modeling process has the potential to improve the accuracy of resource assessments.

Robust Two-Pass Cross-Sectional Regressions: A Minimum Distance Approach

We examine the asymptotic and finite-sample properties of the two-pass (TP) cross-sectional regressions estimators when factors and asset returns are conditionally heteroskedastic and/or autocorrelated. Using a minimum distance approach, we derive the heteroskedasticity- and/or autocorrelation-consistent (HAC) standard errors and the optimal TP estimator. A HAC model specification test statistic is also derived. Our Monte Carlo simulation results reveal the importance of controlling for autocorrelation. The HAC standard errors produce the most reliable inferences under autocorrelation. The HAC specification test is a viable alternative if the number of asset returns is small and the number of time-series observations is large. (JEL: C12, C13, C3) Copyright The Author, 2012. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

GMM estimation of the number of latent factors: With application to international stock markets

Abstract We propose new generalized method of moments (GMM) estimators for the number of latent factors in linear factor models. The estimators are appropriate for data with a large (small) number of cross-sectional observations and a small (large) number of time series observations. The estimation procedure is simple and robust to the configurations of idiosyncratic errors encountered in practice. In addition, the method can be used to evaluate the validity of observable candidate factors. Monte Carlo experiments show that the proposed estimators have good finite-sample properties. Applying the estimators to international stock markets, we find that international stock returns are explained by one strong global factor. This factor is highly correlated with the Fama–French factors from the U.S. stock market. This result can be interpreted as evidence of market integration. We also find two weak factors closely related to markets in Europe and the Americas, respectively.

ECONOMETRIC ANALYSIS OF PANEL DATA: EDITORIAL INTRODUCTION

During 2007, the Singapore Economic Review invited submissions of research papers for consideration to be collected in a Special Issue on: The Theory and Practice of the Econometric Analysis of Panel Data. More in particular, papers were solicited advancing the understanding of the accuracy and adequacy of econometric inference obtained from panel data analysis, or producing such inference relating to economic or general social issues by analyzing panel data preferably (but not necessarily) on South-East Asia at the country, sub-regional, or regional level. From the submitted papers, a selection was made covering many of the divergent aspects of advanced panel data analysis. These papers were thoroughly refereed and this is the resulting special issue. It contains contributions with applications addressing issues in macro, micro, and financial economics and some focus also on the further development and understanding of the techniques of panel data analysis. Three of the eight papers analyze data on South-East Asia. The majority of the articles in this issue concern the analysis of standard panel data, constituting repeated observations in time on a cross-section of subjects. For some of these studies, the time-series sample size is relatively large or about equal in size to the cross-section sample, whereas others focus on the typical micro panel in which the number of time-series observations is very small and the cross-section sample size is relatively large. Mostly, the available panel data sets are unbalanced, meaning that not all cross-section units have been observed over exactly the same time period. In two papers, the panel nature of the analyzed data is less standard. They concern the series of bids during a large number of auctions, and the individual members of a large number of households. It seems fair to say that this collection of papers demonstrates abundantly how ubiquitous and diverse panel data are, how challenging and rewarding their

Estimation of Nonstationary Heterogeneous Panels

We introduce a new Stata command, xtpmg, for estimating nonstationary heterogeneous panels in which the number of groups and number of time-series observations are both large. Based on recent advances in the nonstationary panel literature, xtpmg provides three alternative estimators: a traditional fixed-effects estimator, the mean-group estimator of Pesaran and Smith (Estimating long-run relationships from dynamic heterogeneous panels, Journal of Econometrics 68: 79–113), and the pooled mean-group estimator of Pesaran, Shin, and Smith (Estimating long-run relationships in dynamic heterogeneous panels, DAE Working Papers Amalgamated Series 9721; Pooled mean group estimation of dynamic heterogeneous panels, Journal of the American Statistical Association 94: 621–634).

Are European Financial Markets Integrated? An Approximate Factor Model Investigation

This paper proposes an alternative to extract European common factors and to measure the level of European equity markets integration. Following Heston, Rouwenhorst and Wessels [1995], we use an approximate factor model (Connor 1984) to perform European integration tests. The key feature of this approach is to work with T, the number of time series observations which is typically far smaller than the number of assets, N. The use of the Jones [2001] procedure allowing for heteroskedasticity in time-series residuals is fundamental because the explanatory power of integration tests strongly depends on the pervasive common factors extraction method. Our findings can be summarised as follows. First, European financial markets have been highly integrated over the last 5 years. Second, being a member of the EMU is not sufficient to be integrated to the European capital markets. Third, we demonstrate that the factor structure of European stock returns has changed in the last few years.