The analysis of cointegrating polynomial regressions, i.e, regressions that include an integrated process and its powers as explanatory variables is extended from the time series to the panel case by developing two estimators, a modified and a fully modified OLS estimator. As usual in the cointegration literature, the stationary errors are allowed to be serially correlated and the regressors are allowed to be endogenous. Both individual and time fixed effects are accommodated and the analysis uses an i.i.d. random linear process framework. The modified OLS estimator utilizes the large cross-sectional dimension that allows to consistently estimate and subtract an additive bias term without the need to also transform the dependent variable as required in fully modified OLS estimation. Both developed estimators have zero mean Gaussian limiting distributions and thus allow for standard asymptotic inference. A brief application to the environmental Kuznets curve illustrates the developed methods.

The Hodrick–Prescott (HP) filter is a commonly used tool in macroeconomics to obtain the HP filter trend of a macroeconomic variable. In macroeconomics, the difference between the original series and this trend is called the ‘cyclical component’. In this article, we derive the autocovariance function and the spectrum of the cyclical component of a series that consists of a constant, a linear time trend, a unit root process, and a weakly stationary process. We show that the autocovariance function of the cyclical component of such a series depends on (i) the autocovariance of the innovations of the unit root process; (ii) the autocovariance of the weakly stationary process and; (iii) a component of the weights of the HP filter that is important in the middle of a large sample. The result for the spectrum of the cyclical component matches with earlier results in the literature that were obtained by using an approximate approach. Lastly, we derive the cross‐covariance function and the cross‐spectrum of the cyclical components of two cointegrated series.

This paper derives the limit distribution of the rescaled sum of the absolute value of an integrated process with continuously distributed innovations raised to a negative power less than $-$ 1, and of the analogous statistic that is obtained using the same function of an integrated process but only considering positive values of the integrated process. We show that the limit behavior of this statistic is determined by the values of the integrated process that are closest to 0, and find the limit behavior of the values of the integrated process that are closest to 0.

This article explores a simple property of the Hodrick–Prescott (HP) filter: when the HP filter is applied to a series, the cyclical component is equal to the HP-filtered trend of the fourth difference of the series, except for the first and last two observations, for which different formulas are needed. We use this result to derive small sample results and asymptotic results for a fixed smoothing parameter. We first apply this property to analyze the consequences of a deterministic break. We find that the effect of a deterministic break on the cyclical component is asymptotically negligible for the points that are away from the break point, while for the points in the neighborhood of the break point, the effect is not negligible even asymptotically. Second, we apply this property to show that the cyclical component of the HP filter when applied to series that are integrated up to order 2 is weakly dependent, while the situation for series that are integrated up to order 3 or 4 is more subtle. Third, we characterize the behavior of the HP filter when applied to deterministic polynomial trends and show that in the middle of the sample, the cyclical component reduces the order of the polynomial by 4, while the end point behavior is different. Finally, we give a characterization of the HP filter when applied to an exponential deterministic trend, and this characterization shows that the filter is effectively incapable of dealing with a trend that increases this fast. Our results are compared with those of Phillips and Jin (2015, Business cycles, trend elimination, and the HP filter).

In this note, we consider a location model with an unobserved endogenous dummy variable without regressors and without instrumental variables. We show that it is possible to estimate the location parameter under regularity assumptions. One of these assumptions is a symmetry assumption on the error. However, a lower bound to the asymptotic variance is derived that shows that estimators of the model will inherently possess an extremely large asymptotic variance under sensible assumptions on the error term.

This paper derives the limit distribution of the rescaled sum of the reciprocal of the positive part of a random walk with continuously distributed innovations, and of the rescaled sum of the reciprocal of the absolute value of a random walk with continuously distributed innovations. It also considers this statistic for the case of a simple random walk, and shows that the limit distribution is different for this case.

A class of conditional heteroskedasticity models is introduced and analyzed. This class of models is motivated by the desire to allow the level of a GARCH process to influence the volatility. We show the existence of a unique strictly stationary solution which is β-mixing. The analysis of this model does not rely upon Markov chain methods.

Mixing properties for the dynamic Tobit model are shown for the case of mixing innovations. We show α or β-mixing given mixing assumptions on the errors. Additionally, we show that the results here hold for a more general class of nonlinear time series for which the dynamic Tobit model is a special case.

Many papers in the housing literature treat the intertemporal evolution of the logarithm of US real house prices as a unit root process. They also study the cointegration relationship among the logarithm of real house prices and fundamental economic variables such as income and they apply an error correction specification for modeling and forecasting real house prices. This paper argues that the logarithm of US real house price is not a unit root process. Instead, the evidence from a 120-year national dataset and metro area level and state level panel data sets supports the notion that US house prices are trend stationary. One result of this conclusion is that the validity of analyses of US house prices based on cointegration and error correction models needs to be reconsidered.

The Hodrick-Prescott (HP) filter is a commonly used tool in macroeconomics used to extract a trend component from a time series. In this paper, we derive a new representation of the transformation of the data that is implied by the HP filter. This representation highlights that the HP filter is a symmetric weighted average plus a number of adjustments that are important near the beginning and end of the sample. The representation allows us to carry out a rigorous analysis of properties of the HP filter without using the ARMA-based approximation that has been used previously in the literature. Using this new representation, we characterize the large T behavior of the HP filter and find conditions under which it is asymptotically equivalent to a symmetric weighted average with weights independent of sample size. We also find that the cyclical component of the HP filter possesses weak dependence properties when the HP filter is applied to a stationary mixing process, a linear deterministic trend process, or a process with a unit root. This provides the first formal justification of the use of the HP filter as a tool to achieve weak dependence in a time series. In addition, a large smoothing parameter approximation to the HP filter is derived, and using this approximation, we find an alternative justification for the procedure given in Ravn and Uhlig (2002) for adjusting the smoothing parameter for the data frequency.

This study establishes the analytical results of periodic transformations with weighted unit root processes. This study considers time series regression when the regressors are periodic transformations on a scaled I(1) process. When the transformation is a periodic function on a scaled I(1) process with a symmetric distribution, the convergence rate is slower than that of de Jong (2002). The proposed methods can be useful for developing the periodic transformation of nonlinear regression for nonstationary time series.

This study establishes analytically what the asymptotic behavior of the Dickey-Fuller coefficient tests and the Dickey-Fuller t-statistic tests will be when the true data-generating process is a trigonometric function of an integrated process. Using some recently established limit theorems, it is shown that for such a data generating process, the asymptotic behavior of these unit root tests is reminiscent of that of a stationary series. These results confirm analytically the conclusions of Granger and Hallman in an earlier paper. The results of this paper are applied to the Fourier flexible form methodology.

Yu et al. (2008) establish asymptotic properties of quasi-maximum likelihood estimators for a stable spatial dynamic panel model with fixed effects when both the number of individuals n and the number of time periods T are large. This paper investigates unstable cases where there are unit roots generated by temporal and spatial correlations. We focus on the spatial cointegration model where some eigenvalues of the data generating process are equal to 1 and the outcomes of spatial units are cointegrated as in a vector autoregressive system. The asymptotics of the QML estimators are developed by reparameterization, and bias correction for the estimators is proposed. We also consider the 2SLS and GMM estimations when T could be small. (C) 2011 Elsevier B.V. All rights reserved.

For many time series in empirical macro and finance, it is assumed that the logarithm of the series is a unit root process. Since we may want to assume a stable growth rate for the macroeconomics time series, it seems natural to potentially model such a series as a unit root process with drift. This assumption implies that the level of such a time series is the exponential of a unit root process with drift and therefore, it is of substantial interest to investigate analytically the behavior of the exponential of a unit root process with drift. This paper shows that the sum of the exponential of a random walk with drift converges in distribution, after rescaling by the exponential of the maximum value of the random walk process. A similar result was established in earlier work for unit root processes without drift. The results derived here suggest the conjecture that also in the case when the Dickey-Fuller test or the KPSS statistic is applied to the exponential of a unit root process with drift, these tests will asymptotically indicate stationarity.

The censored regression model and the Tobit model are standard tools in econometrics. This paper provides a formal asymptotic theory for dynamic time series censored regression when lags of the dependent variable have been included among the regressors. The central analytical challenge is to prove that the dynamic censored regression model satisfies stationarity and weak dependence properties if a condition on the lag polynomial holds. We show the formal asymptotic correctness of conditional maximum likelihood estimation of the dynamic Tobit model, and the correctness of Powell’s least absolute deviations procedure for the estimation of the dynamic censored regression model. The paper is concluded with an application of the dynamic censored regression methodology to temporary purchases of the Open Market Desk. This article has supplementary material online.

This paper considers dynamic time series binary choice models. It proves near epoch dependence and strong mixing for the dynamic binary choice model with correlated errors. Using this result, it shows in a time series setting the validity of the dynamic probit likelihood procedure when lags of the dependent binary variable are used as regressors, and it establishes the asymptotic validity of Horowitz’s smoothed maximum score estimation of dynamic binary choice models with lags of the dependent variable as regressors. For the semiparametric model, the latent error is explicitly allowed to be correlated. It turns out that no long-run variance estimator is needed for the validity of the smoothed maximum score procedure in the dynamic time series framework.

This paper shows that the sum of the exponential of a unit root process converges in distribution, after rescaling by the exponential of the maximum value of a BeveridgeNelson approximation to the unit root process. The increments of the unit root process are assumed to be linear processes, and the limit distribution that is derived depends on the structure of the linear process and on its innovation distribution. This analytical result is relevant to empirical work because macroeconomic time series are often assumed to have a unit root in their logarithms, implying that the levels of such series are the exponential of a unit root process. This new piece of econometric methodology is applied to analyzing the limit distribution of the Dickey-Fuller test under the assumption that the data are the exponential of a unit root process, and to the KPSS test. The limit distribution of the KPSS test applied to data that are the exponential of a unit root process turns out to be pivotal, and the KPSS test converges at the same rate as when the data are stationary. The analytical results on the limit behavior of the KPSS test seem to pose an empirical puzzle, since the KPSS test rejects in many instances for the level series when one might expect a unit root in the logarithm of the series.

This article draws out some implications of son targeting fertility behavior and studies its determinants. We demonstrate that such behavior has two notable implications at the aggregate level: (a) girls have a larger number of siblings (sibling effect), and (b) girls are born at relatively earlier parities within families (birth-order effect). Empirically testing for these effects, we find that both are present in many countries in South Asia, Southeast Asia, and North Africa but are absent in the countries of sub-Saharan Africa. Using maximum likelihood estimation, we study the effect of covariates on son targeting fertility behavior in India, a country that displays significant sibling and birth-order effects. We find that income and geographic location of families significantly affect son targeting behavior.

This note concerns an asymptotic distribution result from the literature on nonlinear estimation with integrated variables. It points out a way of strengthening local asymptotic distribution results towards results that hold for the global minimizer of the criterion function.