Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

Censored regression quantiles with endogenous regressors

This paper develops a semiparametric method for estimation of the censored regression model when some of the regressors are endogenous (and continuously distributed) and instrumental variables are available for them. A “distributional exclusion” restriction is imposed on the unobservable errors, whose conditional distribution is assumed to depend on the regressors and instruments only through a lower-dimensional “control variable,” here assumed to be the difference between the endogenous regressors and their conditional expectations given the instruments. This assumption, which implies a similar exclusion restriction for the conditional quantiles of the censored dependent variable, is used to motivate a two-stage estimator of the censored regression coefficients. In the first stage, the conditional quantile of the dependent variable given the instruments and the regressors is nonparametrically estimated, as are the first-stage reduced-form residuals to be used as control variables. The second-stage estimator is a weighted least squares regression of pairwise differences in the estimated quantiles on the corresponding differences in regressors, using only pairs of observations for which both estimated quantiles are positive (i.e., in the uncensored region) and the corresponding difference in estimated control variables is small. The paper gives the form of the asymptotic distribution for the proposed estimator, and discusses how it compares to similar estimators for alternative models.     

Can child care policy encourage employment and fertility?: Evidence from a structural model

We develop a structural model of female employment and fertility which accounts for intertemporal feedback effects between these two outcomes. To identify the effect of financial incentives on employment and fertility we exploit variation in the tax and transfer system, which differs by employment state and number of children. Specifically, we simulate in detail the effects of the tax and transfer system, including child care costs. The model provides estimates of the structural preferences of women that can be used to study the effect of various policy reforms. Results show that increasing child care subsidies, conditional on employment, increases the labor supply of all women as well as the fertility rates of the childless and highly educated women.     

Spurious regressions in technical trading

This paper investigates the spurious effect in forecasting asset returns when signals from technical trading rules are used as predictors. Against economic intuition, the simulation result shows that, even if past information has no predictive power, buy or sell signals based on the difference between the short-period and long-period moving averages of past asset prices can be statistically significant when the forecast horizon is relatively long. The theoretical analysis reveals that both ‘momentum’ and ‘contrarian’ strategies can be falsely supported, while the probability of obtaining each result depends on the type of the test statistics employed.     

Estimation and model selection of semiparametric multivariate survival functions under general censorship

We study estimation and model selection of semiparametric models of multivariate survival functions for censored data, which are characterized by possibly misspecified parametric copulas and nonparametric marginal survivals. We obtain the consistency and root-n$n$ asymptotic normality of a two-step copula estimator to the pseudo-true copula parameter value according to KLIC, and provide a simple consistent estimator of its asymptotic variance, allowing for a first-step nonparametric estimation of the marginal survivals. We establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application is provided.     

Unit root testing under a local break in trend

Recent approaches to testing for a unit root when uncertainty exists over the presence and timing of a trend break employ break detection methods, so that a with-break unit root test is used only if a break is detected by some auxiliary statistic. While these methods achieve near asymptotic efficiency in both fixed trend break and no trend break environments, in finite samples pronounced “valleys” in the power functions of the tests (when mapped as functions of the break magnitude) are observed, with power initially high for very small breaks, then decreasing as the break magnitude increases, before increasing again. In response to this problem, we propose two practical solutions, based either on the use of a with-break unit root test but with adaptive critical values, or on a union of rejections principle taken across with-break and without-break unit root tests. These new procedures are shown to offer improved reliability in terms of finite sample power. We also develop local limiting distribution theory for both the extant and the newly proposed unit root statistics, treating the trend break magnitude as local-to-zero. We show that this framework allows the asymptotic analysis to closely approximate the finite sample power valley phenomenon, thereby providing useful analytical insights.     

Testing for a change in persistence in the presence of non-stationary volatility

In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0)$I\left(0\right)$ to I(1)$I\left(1\right)$ behaviour or vice versa, of, inter alia, [Kim, J., 2000. Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116]. We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical illustration using US price inflation data is provided.     

Unit root log periodogram regression

Log periodogram (LP) regression is shown to be consistent and to have a mixed normal limit distribution when the memory parameter d=1$d=1$. Gaussian errors are not required. The proof relies on a new result showing that asymptotically infinite collections of discrete Fourier transforms (dft's) of a short memory process at the fundamental frequencies in the vicinity of the origin can be treated as asymptotically independent normal variates, provided one does not include too many dft's in the collection.     

Predictive regression under various degrees of persistence and robust long-horizon regression

The paper proposes a novel inference procedure for long-horizon predictive regression with persistent regressors, allowing the autoregressive roots to lie in a wide vicinity of unity. The invalidity of conventional tests when regressors are persistent has led to a large literature dealing with inference in predictive regressions with local to unity regressors. Magdalinos and Phillips (2009b) recently developed a new framework of extended IV procedures (IVX) that enables robust chi-square testing for a wider class of persistent regressors. We extend this robust procedure to an even wider parameter space in the vicinity of unity and apply the methods to long-horizon predictive regression. Existing methods in this model, which rely on simulated critical values by inverting tests under local to unity conditions, cannot be easily extended beyond the scalar regressor case or to wider autoregressive parametrizations. In contrast, the methods developed here lead to standard chi-square tests, allow for multivariate regressors, and include predictive processes whose roots may lie in a wide vicinity of unity. As such they have many potential applications in predictive regression. In addition to asymptotics under the null hypothesis of no predictability, the paper investigates validity under the alternative, showing how balance in the regression may be achieved through the use of localizing coefficients and developing local asymptotic power properties under such alternatives. These results help to explain some of the empirical difficulties that have been encountered in establishing predictability of stock returns.     

Testing for a break in trend when the order of integration is unknown

Harvey, Leybourne and Taylor [Harvey, D.I., Leybourne, S.J., Taylor, A.M.R. 2009. Simple, robust and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029] develop a test for the presence of a broken linear trend at an unknown point in the sample whose size is asymptotically robust as to whether the (unknown) order of integration of the data is either zero or one. This test is not size controlled, however, when this order assumes fractional values; its asymptotic size can be either zero or one in such cases. In this paper we suggest a new test, based on a sup-Wald statistic, which is asymptotically size-robust across fractional values of the order of integration (including zero or one). We examine the asymptotic power of the test under a local trend break alternative. The finite sample properties of the test are also investigated.     

Testing for jumps in noisy high frequency data

This paper proposes a robustification of the test statistic of Aït-Sahalia and Jacod (2009b) for the presence of market microstructure noise in high frequency data, based on the pre-averaging method of Jacod et al. (2010). We show that the robustified statistic restores the test’s discriminating power between jumps and no jumps despite the presence of market microstructure noise in the data.     

A simple,robust and powerful test of the trend hypothesis

In this paper we develop a simple test procedure for a linear trend which does not require knowledge of the form of serial correlation in the data, is robust to strong serial correlation, and has a standard normal limiting null distribution under either I(0)$I(0)$ or I(1)$I(1)$ shocks. In contrast to other available robust linear trend tests, our proposed test achieves the Gaussian asymptotic local power envelope in both the I(0)$I(0)$ and I(1)$I(1)$ cases. For near-I(1)$I(1)$ errors our proposed procedure is conservative and a modification for this situation is suggested. An estimator of the trend parameter, together with an associated confidence interval, which is asymptotically efficient, again regardless of whether the shocks are I(0)$I(0)$ or I(1)$I(1)$, is also provided.     

A semiparametric cointegrating regression: Investigating the effects of age distributions on consumption and saving

We consider a semiparametric cointegrating regression model, for which the disequilibrium error is further explained nonparametrically by a functional of distributions changing over time. The paper develops the statistical theories of the model. We propose an efficient econometric estimator and obtain its asymptotic distribution. A specification test for the model is also investigated. The model and methodology are applied to analyze how an aging population in the US influences the consumption level and the savings rate. We find that the impact of age distribution on the consumption level and the savings rate is consistent with the life-cycle hypothesis.     

Estimating deterministic trends with an integrated or stationary noise component

We propose a test for the slope of a trend function when it is a priori unknown whether the series is trend-stationary or contains an autoregressive unit root. The procedure is based on a Feasible Quasi Generalized Least Squares method from an AR(1) specification with parameter α$\alpha$, the sum of the autoregressive coefficients. The estimate of α$\alpha$ is the OLS estimate obtained from an autoregression applied to detrended data and is truncated to take a value 1 whenever the estimate is in a T^−δ$T^{-\delta }$ neighborhood of 1. This makes the estimate “super-efficient” when α=1$\alpha =1$ and implies that inference on the slope parameter can be performed using the standard Normal distribution whether α=1$\alpha =1$ or |α|<1$\left|\alpha \right|<1$. Theoretical arguments and simulation evidence show that δ=1/2$\delta =1/2$ is the appropriate choice. Simulations show that our procedure has better size and power properties than the tests proposed by [Bunzel, H., Vogelsang, T.J., 2005. Powerful trend function tests that are robust to strong serial correlation with an application to the Prebish–Singer hypothesis. Journal of Business and Economic Statistics 23, 381–394] and [Harvey, D.I., Leybourne, S.J., Taylor, A.M.R., 2007. A simple, robust and powerful test of the trend hypothesis. Journal of Econometrics 141, 1302–1330].     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.     

Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses

Perron [Perron, P., 1989. The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401] introduced a variety of unit root tests that are valid when a break in the trend function of a time series is present. The motivation was to devise testing procedures that were invariant to the magnitude of the shift in level and/or slope. In particular, if a change is present it is allowed under both the null and alternative hypotheses. This analysis was carried under the assumption of a known break date. The subsequent literature aimed to devise testing procedures valid in the case of an unknown break date. However, in doing so, most of the literature and, in particular the commonly used test of Zivot and Andrews [Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251–270], assumed that if a break occurs, it does so only under the alternative hypothesis of stationarity. This is undesirable since (a) it imposes an asymmetric treatment when allowing for a break, so that the test may reject when the noise is integrated but the trend is changing; (b) if a break is present, this information is not exploited to improve the power of the test. In this paper, we propose a testing procedure that addresses both issues. It allows a break under both the null and alternative hypotheses and, when a break is present, the limit distribution of the test is the same as in the case of a known break date, thereby allowing increased power while maintaining the correct size. Simulation experiments confirm that our procedure offers an improvement over commonly used methods in small samples.