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Lobato and Robinson (1998) develop semiparametric tests for the null hypothesis that a series is weakly autocorrelated, or I(0), about a constant level, against fractionally integrated alternatives. These tests have the advantage that the user is not required to specify a parametric model for any weak autocorrelation present in the series. We extend this approach in two distinct ways. First we show that it can be generalised to allow for testing of the null hypothesis that a series is I(ẟ) for any ẟ lying in the usual stationary and invertible region of the parameter space. The second extension is the more substantive and addresses the well known issue in the literature that long memory and level breaks can be mistaken for one another, with unmodelled level breaks rendering fractional integration tests highly unreliable. To deal with this inference problem we extend the Lobato and Robinson (1998) approach to allow for the possibility of changes in level at unknown points in the series. We show that the resulting statistics have standard limiting null distributions, and that the tests based on these statistics attain the same asymptotic local power functions as infeasible tests based on the unobserved errors, and hence there is no loss in asymptotic local power from allowing for level breaks, even where none is present. We report results from a Monte Carlo study into the finite-sample behaviour of our proposed tests, as well as several empirical examples.

Several modified estimation methods of the memory parameter have been introduced in the past years. They aim to decrease the upward bias of the memory parameter in cases of low frequency contaminations or an additive noise component, especially in situations with a short-memory process being contaminated. In this paper, we provide an overview and compare the performance of nine semiparametric estimation methods. Among them are two standard methods, four modified approaches to account for low frequency contaminations and three procedures developed for perturbed fractional processes. We conduct an extensive Monte Carlo study for a variety of parameter constellations and several DGPs. Furthermore, an empirical application of the log-absolute return series of the S&P 500 shows that the estimation results combined with a long-memory test indicate a spurious long-memory process.

We derive the properties of the periodogram local to the zero frequency for a large class of spurious long-memory processes. The periodogram is of crucial importance in this context, since it forms the basis for most commonly used estimation methods for the memory parameter. The class considered nests a wide range of processes such as deterministic or stochastic structural breaks and smooth trends as special cases. Several previous results on these special cases are generalized and extended. All of the spurious long-memory processes considered share the property that their impact on the periodogram at the Fourier frequencies local to the origin is different than that of true long-memory processes. Both types of processes therefore exhibit clearly distinct empirical features.

Several modified estimation methods of the memory parameter have been introduced in the past years. They aim to decrease the upward bias of the memory parameter in cases of low frequency contaminations or an additive noise component, especially in situations with a short-memory process being contaminated. In this paper, we provide an overview and compare the performance of nine semiparametric estimation methods. Among them are two standard methods, four modified approaches to account for low frequency contaminations and three procedures developed for perturbed fractional processes. We conduct an extensive Monte Carlo study for a variety of parameter constellations and several DGPs. Furthermore, an empirical application of the log-absolute return series of the S&P 500 shows that the estimation results combined with a long-memory test indicate a spurious long-memory process.

This thesis contains three essays on spurious long memory with an introduction to the literature in the first Chapter. Chapter 2 presents a multivariate score-type test against spurious long memory. We prove the consistency of the test against the alternatives of random level shifts and smooth trends. The test statistic is based on the weighted sum of the partial derivatives of the multivariate local Whittle likelihood function. To apply the test to fractionally cointegrated series, the test statistic is calculated for the linearly transformed system after estimating the cointegrating matrix. We derive the limiting distribution and show consistency of this procedure. The test is applied to log-absolute returns and log-realized volatilities of the S&P 500, DAX, FTSE, and NIKKEI. Chapter 3 analyzes inflation rates with newly available econometric methods which allow for a consistent estimation of the persistence parameter under low frequency contaminations and con- sistent break point estimation under long memory without a priori assumptions on the presence of breaks. The persistence of inflation rates is of major importance to central banks due to the fact that it determines the costs of monetary policy according to the Phillips curve. In contrast to previous studies, smooth trends are allowed in addition to breaks as a source of spurious long memory. We support the finding of reduced memory parameters in monthly inflation rates of the G7 countries as well as spurious long memory, except for the US. Nevertheless, only a few breaks can be located. Instead, all countries exhibit significant trends at the 5% level with the exception of the US. Finally, Chapter 4 provides an overview and compares the performance of nine semiparametric estimation methods. Among them are two standard methods, four modified approaches to account for low frequency contaminations and three procedures developed for perturbed fractional processes. Several modified estimation methods of the memory parameter have been introduced in the past years. They aim to decrease the upward bias of the memory parameter in cases of low frequency contaminations or an additive noise component, especially in situations with a short-memory process being contaminated. We conduct an extensive Monte Carlo study for a variety of parameter constellations and several DGPs. Furthermore, an empirical application of the log-absolute return series of the S&P 500 shows that the estimation results combined with a long-memory test indicate a spurious long-memory process.

The focus of the volatility literature on forecasting and the predominance of the conceptually simpler HAR model over long memory stochastic volatility models has led to the fact that the actual degree of memory estimates has rarely been considered. Estimates in the literature range roughly between 0.4 and 0.6 - that is from the higher stationary to the lower non-stationary region. This difference, however, has important practical implications - such as the existence or nonexistence of the fourth moment of the return distribution. Inference on the memory order is complicated by the presence of measurement error in realized volatility and the potential of spurious long memory. In this paper we provide a comprehensive analysis of the memory in variances of international stock indices and exchange rates. On the one hand, we find that the variance of exchange rates is subject to spurious long memory and the true memory parameter is in the higher stationary range. Stock index variances, on the other hand, are free of low frequency contaminations and the memory is in the lower non-stationary range. These results are obtained using state of the art local Whittle methods that allow consistent estimation in presence of perturbations or low frequency contaminations.

We consider a large class of structural change processes that generate spurious long memory. Among others, this class encompasses structural breaks as well as random level shift processes and smooth trends. The properties of these processes are studied based on a simple representation of their discrete Fourier transform. We find, that under very general conditions all of the models nested in this class generate poles in the periodogram at the zero frequency. These are of order O(T), instead of the usual O(T2d) for long memory processes and O(T2) for a random walk. This order arises whenever both the mean changes and sample fractions at which they occur are non-degenerate, asymptotically

The persistence of inflation rates is of major importance to central banks due to the fact that it determines the costs of monetary policy according to the Phillips curve. This article is motivated by newly available econometric methods which allow for a consistent estimation of the persistence parameter under low frequency contaminations and consistent break point estimation under long memory without a priori assumptions on the presence of breaks. In contrast to previous studies, we allow for smooth trends in addition to breaks as a source of spurious long memory. We support the fi nding of reduced memory parameters in monthly inflation rates of the G7 countries as well as spurious long memory, except for the US. Nevertheless, only a few breaks can be located. Instead, all countries exhibit signi cant trends at the 5 percent level with the exception of the US.

The focus of the volatility literature on forecasting and the predominance of the conceptually simpler HAR model over long memory stochastic volatility models has led to the fact that the actual degree of memory estimates has rarely been considered. Estimates in the literature range roughly between 0.4 and 0.6 - that is from the higher stationary to the lower non-stationary region. This difference, however, has important practical implications - such as the existence or non-existence of the fourth moment of the return distribution. Inference on the memory order is complicated by the presence of measurement error in realized volatility and the potential of spurious long memory. In this paper we provide a comprehensive analysis of the memory in variances of international stock indices and exchange rates. On the one hand, we find that the variance of exchange rates is subject to spurious long memory and the true memory parameter is in the higher stationary range. Stock index variances, on the other hand, are free of low frequency contaminations and the memory is in the lower non-stationary range. These results are obtained using state of the art local Whittle methods that allow consistent estimation in presence of perturbations or low frequency contaminations.

This paper provides a multivariate score-type test to distinguish between true and spurious long memory. The test is based on the weighted sum of the partial derivatives of the multivariate local Whittle likelihood function. This approach takes phase shifts in the multivariate spectrum into account. The resulting pivotal limiting distribution is independent of the dimension of the process, which makes it easy to apply in practice. We prove the consistency of our test against the alternative of random level shifts or monotonic trends. A Monte Carlo analysis shows good finite sample properties of the test in terms of size and power. Additionally, we apply our test to the log-absolute returns of the S&P 500, DAX, FTSE, and the NIKKEI. The multivariate test gives formal evidence that these series are contaminated by level shifts.

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We have developed a new test against spurious long memory based on the invariance of long memory parameter to aggregation. By using the local Whittle estimator, the statistic takes the supremum among combinations of paired aggregated series. Simulations show that the test performs good in finite sample sizes, and is able to distinguish long memory from spurious processes with excellent power. Moreover, the empirical application gives further evidence that the observed long memory in German stock returns is spurious.