Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter _d0$d_0$ are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.     

Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu ( 2009a ) and is based on a Feasible Generalized Least Squares procedure that uses a super‐efficient estimator of the sum of the autoregressive coefficients α when α = 1. The resulting Wald test statistic asymptotically follows a chi‐square distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias‐corrected version of the OLS estimator of α proposed by Roy and Fuller ( 2001 ). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modelling the trend of global and hemispheric temperatures.     

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.     

Analysing inflation by the fractionally integrated ARFIMA–GARCH model

This paper considers the application of long-memory processes to describing inflation for ten countries. We implement a new procedure to obtain approximate maximum likelihood estimates of an ARFIMA—GARCH process; which is fractionally integrated I(d) with a superimposed stationary ARMA component in its conditional mean. Additionally, this long memory process is allowed to have GARCH type conditional heteroscedasticity. On analysing monthly post-World War II CPI inflation for ten different countries, we find strong evidence of long memory with mean reverting behaviour for all countries except Japan, which appears stationary. For three high inflation economies there is evidence that the mean and volatility of inflation interact in a way that is consistent with the Friedman hypothesis.     

The effect of data transformation on common cycle,cointegration, and unit root tests: Monte Carlo results and a simple test

Cointegration, common cycle, and related tests statistics are often constructed using logged data, even without clear reason why logs should be used rather than levels. Unfortunately, it is also the case that standard data transformation tests, such as those based on Box–Cox transformations, cannot be shown to be consistent unless assumptions concerning whether variables I(0)$I(0)$ or I(1)$I(1)$ are made. In this paper, we propose a simple randomized procedure for choosing between levels and log-levels specifications in the (possible) presence of deterministic and/or stochastic trends, and discuss the impact of incorrect data transformation on common cycle, cointegration and unit root tests.     

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.     

Nonlinearity,nonstationarity, and spurious forecasts

Implications of nonlinearity, nonstationarity, and misspecification are considered from a forecasting perspective. Our model allows for small departures from the martingale difference sequence hypothesis by including a nonlinear component, formulated as a general, integrable transformation of the I(1)$I(1)$ predictor. We assume that the true generating mechanism is unknown to the econometrician and he is therefore forced to use some approximating functions. It is shown that in this framework the linear regression techniques lead to spurious forecasts. Improvements of the forecast accuracy are possible with properly chosen nonlinear transformations of the predictor. The paper derives the limiting distribution of the forecasts’ mean squared error (MSE). In the case of square integrable approximants, it depends on the _L2$L_2$-distance between the nonlinear component and approximating function. Optimal forecasts are available for a given class of approximants.     

Regression‐based Tests for a Change in Persistence*

We show that the minimal forward (reverse) recursive unit tests of Banerjee, Lumsdaine and Stock [Journal of Business and Economics Statistics (1992) Vol. 10, pp. 271–288] are consistent against the alternative of a change in persistence from I(0) to I(1) [I(1) to I(0)]. However, these statistics are also shown to diverge for series which are I(0) throughout. Consequently, a rejection by these tests does not necessarily imply a change in persistence. We propose a further test, based on the ratio of these statistics, which is consistent against changes either from I(0) to I(1), or vice versa, yet does not over‐reject against constant I(0) series. Consistent breakpoint estimators are proposed.     

A complete asymptotic series for the autocovariance function of a long memory process

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q$p,d,q$) model. The leading term of the expansion is of the order O(1/k^1−2d)$O(1/k^{1-2d})$, where k$k$ is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^3−2d)$O(1/k^{3-2d})$. The derivation uses Erdélyi’s [Erdélyi, A., 1956. Asymptotic Expansions. Dover Publications, Inc, New York] expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2π}$\{0,2\pi \}$. Numerical evaluations show that the expansion is accurate even for small k$k$ in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k$k$. The approximations are easy to compute across a variety of parameter values and models.     

PITFALLS IN CROSS‐SECTION STUDIES WITH INTEGRATED REGRESSORS: A SURVEY AND NEW DEVELOPMENTS

In cross‐section studies, if the dependent variable is I(0) but the regressor is I(1), the true slope must be zero in the resulting “unbalanced regression.” A spuriously significant relationship may be found in large cross‐sections, however, if the integrated regressor is related to a stationary variable that enters the DGP but is omitted from the regression. The solution is to search for the related stationary variable, in some cases the first difference of the integrated regressor, in other cases, a categorical variable that can take on limited number of values which depend on the integrated variable. We present an extensive survey, new developments, and applications particularly in finance.     

Autocovariance functions of series and of their transforms

We derive a method to link exactly the autocovariance functions of two arbitrary instantaneous transformations of a time series. For example, this is useful when one wants to derive the autocovariance of the logarithm of a series from the known autocovariance of the original series and, more generally, when one wishes to describe the time-series effects of applying a nonlinear transformation to a process whose properties are known. As an illustration, we provide two corollaries and three examples. The first corollary is on the commonly used logarithmic transformation, and is applied to a geometric auto-regressive (AR) process, as well as to a positive moving-average (MA) process. The second corollary is on the tan⁻¹(·) transformation which will turn possibly unstable series into stable ones. As an illustration, we obtain the autocovariance function of the tan⁻¹(·) of an arithmetic AR process. This filter, while always producing a bounded process, preserves the stability/instability distinction of the original series, a feature that can be turned to an advantage in the design of tests. We then present a probabilistic interpretation of the main features of the new autocovariance function. We also provide a mathematical lemma on a general integral which is of independent interest.     

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.     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

Econometric Methods for Modelling Systems With a Mixture of i(1) and i(0) Variables

This paper considers structural models with both I(1) and I(0) variables. The structural shocks associated with either set of variables could be permanent or transitory. We classify the shocks as (P1,P0) and (T1,T0), where P/T distinguishes permanent/transitory, while 1/0 means they are attached to structural equations with either I(1) or I(0) variables as their ‘dependent’ variable. We show that P0 shocks can affect cointegration analysis and provide a formula to compute the permanent component if they are present. Finally, we reformulate a well‐known empirical structural vector autoregression showing the impact of P0 shocks when there are just long‐run parametric and sign restrictions. Copyright © 2015 John Wiley & Sons, Ltd.     

Panels with non-stationary multifactor error structures

The presence of cross-sectionally correlated error terms invalidates much inferential theory of panel data models. Recently, work by Pesaran (2006) has suggested a method which makes use of cross-sectional averages to provide valid inference in the case of stationary panel regressions with a multifactor error structure. This paper extends this work and examines the important case where the unobservable common factors follow unit root processes. The extension to I(1)$I\left(1\right)$ processes is remarkable on two counts. First, it is of great interest to note that while intermediate results needed for deriving the asymptotic distribution of the panel estimators differ between the I(1)$I\left(1\right)$ and I(0)$I\left(0\right)$ cases, the final results are surprisingly similar. This is in direct contrast to the standard distributional results for I(1)$I\left(1\right)$ processes that radically differ from those for I(0)$I\left(0\right)$ processes. Second, it is worth noting the significant extra technical demands required to prove the new results. The theoretical findings are further supported for small samples via an extensive Monte Carlo study. In particular, the results of the Monte Carlo study suggest that the cross-sectional-average-based method is robust to a wide variety of data generation processes and has lower biases than the alternative estimation methods considered in the paper.     

Gamma-admissibility of estimators in the one-parameter exponential family

Summary Admissibility of estimators under vague prior information on the distribution of the unknown parameter is studied which leads to the notion of gamma-admissibility. A sufficient condition for an estimator of the formδ(x)=(ax+b)/(cx+d) to be gamma-admissible in the one-parameter exponential family under squared error loss is established. As an application of this result two equalizer rules are shown to be unique gamma-minimax estimators by proving their gamma-admissibility.     

Beyond panel unit root tests: Using multiple testing to determine the nonstationarity properties of individual series in a panel

Most panel unit root tests are designed to test the joint null hypothesis of a unit root for each individual series in a panel. After a rejection, it will often be of interest to identify which series can be deemed to be stationary and which series can be deemed nonstationary. Researchers will sometimes carry out this classification on the basis of n$n$ individual (univariate) unit root tests based on some ad hoc significance level. In this paper, we suggest and demonstrate how to use the false discovery rate (FDR)$\left(FDR\right)$ in evaluating I(1)/I(0)$I\left(1\right)/I\left(0\right)$ classifications.     

Differencing of random walks and near random walks

The traditional rationale for differencing time series data is to attain stationarity. For a nearly non-stationary first-order autoregressive process—AR (1) with positive slope parameter near unity—we were led to a complementary rationale. If one suspects near non-stationarity of the AR (1) process, if the sample size is ‘small’ or ‘moderate’, and if good one-step-ahead prediction performance is the goal, then it is wise to difference the data and treat the differences as observations on a stationary AR (1) process. Estimation by Ordinary Least Squares then appears to be at least as satisfactory as nonlinear least squares. Use of differencing for an already stationary process can be motivated by Bayesian concepts: differencing can be viewed as an easy way to incorporate non-diffuse prior judgement—that the process is nearly non-stationary—into one's analysis. Random walks and near random walks are often encountered in economics. Unless one's sample size is large, the same statistical analyses apply to either.