This paper builds on methodology that corrects for irregular spacing between realizations of unevenly spaced time series and provides appropriately corrected estimates of autoregressive model parameters. Using these methods for dealing with missing data, we develop time series tools for forecasting and estimation of autoregressions with cyclically varying parameters in which periodicity is assumed. To illustrate the robustness and flexibility of the methodology, an application is conducted to model daily temperature data. The approach helps to uncover cyclical (daily as well as annual) patterns in the data without imposing restrictive assumptions. Using the Central England Temperature (CET) time series (1772 - present) we find with a high level of accuracy that temperature intra-year averages and persistence have increased in the later sample 1850-2020 compared to 1772 - 1850, especially for the winter months, whereas the estimated variance of the random shocks in the autoregression seems to have decreased over time.
Considerable evidence in past research shows size distortion in standard tests for zero autocorrelation or zero cross-correlation when time series are not independent identically distributed random variables, pointing to the need for more robust procedures. Recent tests for serial correlation and cross-correlation in Dalla, Giraitis, and Phillips (2022) provide a more robust approach, allowing for heteroskedasticity and dependence in uncorrelated data under restrictions that require a smooth, slowly-evolving deterministic heteroskedasticity process. The present work removes those restrictions and validates the robust testing methodology for a wider class of innovations and regression residuals allowing for heteroscedastic uncorrelated and non-stationary data settings. The updated analysis given here enables more extensive use of the methodology in practical applications. Monte Carlo experiments confirm excellent finite sample performance of the robust test procedures even for extremely complex white noise processes. The empirical examples show that use of robust testing methods can materially reduce spurious evidence of correlations found by standard testing procedures.
Considerable evidence in past research shows size distortion in standard tests for zero autocorrelation or cross-correlation when time series are not independent identically distributed random variables, pointing to the need for more robust procedures. Recent tests for serial correlation and cross-correlation in Dalla, Giraitis, and Phillips (2022) provide a more robust approach, allowing for heteroskedasticity and dependence in un-correlated data under restrictions that require a smooth, slowly-evolving deterministic heteroskedasticity process. The present work removes those restrictions and validates the robust testing methodology for a wider class of heteroskedastic time series models and innovations. The updated analysis given here enables more extensive use of the methodology in practical applications. Monte Carlo experiments confirm excellent finite sample performance of the robust test procedures even for extremely complex white noise processes. The empirical examples show that use of robust testing methods can materially reduce spurious evidence of correlations found by standard testing procedures.
Commonly used tests to assess evidence for the absence of autocorrelation in a univariate time series or serial cross-correlation between time series rely on procedures whose validity holds for i.i.d. data. When the series are not i.i.d., the size of correlogram and cumulative Ljung-Box tests can be significantly distorted. This paper adapts standard correlogram and portmanteau tests to accommodate hidden dependence and non-stationarities involving heteroskedasticity, thereby uncoupling these tests from limiting assumptions that reduce their applicability in empirical work. To enhance the Ljung-Box test for non-i.i.d. data a new cumulative test is introduced. Asymptotic size of these tests is unaffected by hidden dependence and heteroskedasticity in the series. Related extensions are provided for testing cross-correlation at various lags in bivariate time series. Tests for the i.i.d. property of a time series are also developed. An extensive Monte Carlo study confirms good performance in both size and power for the new tests. Applications to real data reveal that standard tests frequently produce spurious evidence of serial correlation.
Time series models are often fitted to the data without preliminary checks for stability of the mean and variance, conditions that may not hold in much economic and financial data, particularly over long periods. Ignoring such shifts may result in fitting models with spurious dynamics that lead to unsupported and controversial conclusions about time dependence, causality, and the effects of unanticipated shocks. In spite of what may seem as obvious differences between a time series of independent variates with changing variance and a stationary conditionally heteroskedastic (GARCH) process, such processes may be hard to distinguish in applied work using basic time series diagnostic tools. We develop and study some practical and easily implemented statistical procedures to test the mean and variance stability of uncorrelated and serially dependent time series. Application of the new methods to analyze the volatility properties of stock market returns leads to some unexpected surprising findings concerning the advantages of modeling time varying changes in unconditional variance.
A limit theory is established for autoregressive time series that smooths the transition between local and moderate deviations from unity and provides a transitional form that links conventional unit root distributions and the standard normal. Edgeworth expansions of the limit theory are given. These expansions show that the limit theory that holds for values of the autoregressive coefficient that are closer to stationarity than local (i.e. deviations of the form = 1 + (c/n), where n is the sample size and c < 0) holds up to the second order. Similar expansions around the limiting Cauchy density are provided for the mildly explosive case.