Publications - Morten Ø. Nielsen

Testing for the Appropriate Level of Clustering in Linear Regression Models

MacKinnon, J. G., Nielsen, M. Ø., Webb, M. D. — Tue, 01 Aug 2023 20:22:24 +0200

The overwhelming majority of empirical research that uses cluster-robust inference assumes that the clustering structure is known, even though there are often several possible ways in which a dataset could be clustered. We propose two tests for the correct level of clustering in regression models. One test focuses on inference about a single coefficient, and the other on inference about two or more coefficients. We provide both asymptotic and wild bootstrap implementations. The proposed tests work for a null hypothesis of either no clustering or “fine” clustering against alternatives of “coarser” clustering. We also propose a sequential testing procedure to determine the appropriate level of clustering. Simulations suggest that the bootstrap tests perform very well under the null hypothesis and can have excellent power. An empirical example suggests that using the tests leads to sensible inferences.

Fast and Reliable Jackknife and Bootstrap Methods for Cluster-Robust Inference

MacKinnon, J. G., Nielsen, M. Ø., Webb, M. D. — Tue, 01 Aug 2023 20:22:24 +0200

We provide computationally attractive methods to obtain jackknife-based cluster-robust variance matrix estimators (CRVEs) for linear regression models estimated by least squares. We also propose several new variants of the wild cluster bootstrap, which involve these CRVEs, jackknife-based bootstrap data-generating processes, or both. Extensive simulation experiments suggest that the new methods can provide much more reliable inferences than existing ones in cases where the latter are not trustworthy, such as when the number of clusters is small and/or cluster sizes vary substantially. Three empirical examples illustrate the new methods.

Inference on the dimension of the nonstationary subspace in functional time series

Nielsen, M. Ø., Seo, W., Seong, D. — Thu, 01 Jun 2023 20:22:24 +0200

We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

Truncated sum-of-squares estimation of fractional time series models with generalized power law trend

Hualde, J., Nielsen, M. Ø. — Sat, 01 Jan 2022 20:22:24 +0100

We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.

Cluster-Robust Inference: A Guide to Empirical Practice

Mackinnon, J. G., Nielsen, M. Ø., Webb, M. D. — Wed, 01 Feb 2023 20:22:24 +0100

Methods for cluster-robust inference are routinely used in economics and many other disciplines. However, it is only recently that theoretical foundations for the use of these methods in many empirically relevant situations have been developed. In this paper, we use these theoretical results to provide a guide to empirical practice. We do not attempt to present a comprehensive survey of the (very large) literature. Instead, we bridge theory and practice by providing a thorough guide on what to do and why, based on recently available econometric theory and simulation evidence. To practice what we preach, we include an empirical analysis of the effects of the minimum wage on labor supply of teenagers using individual data.

Adaptive Inference in Heteroscedastic Fractional Time Series Models

Cavaliere, G., Nielsen, M. Ø., Robert Taylor, A. M. — Sat, 01 Jan 2022 20:22:24 +0100

We consider estimation and inference in fractionally integrated time series models driven by shocks which can display conditional and unconditional heteroscedasticity of unknown form. Although the standard conditional sum-of-squares (CSS) estimator remains consistent and asymptotically normal in such cases, unconditional heteroscedasticity inflates its variance matrix by a scalar quantity, λ > 1, thereby inducing a loss in efficiency relative to the unconditionally homoscedastic case, λ = 1. We propose an adaptive version of the CSS estimator, based on nonparametric kernel-based estimation of the unconditional volatility process. We show that adaptive estimation eliminates the factor λ from the variance matrix, thereby delivering the same asymptotic efficiency as that attained by the standard CSS estimator in the unconditionally homoscedastic case and, hence, asymptotic efficiency under Gaussianity. Importantly, the asymptotic analysis is based on a novel proof strategy, which does not require consistent estimation (in the sup norm) of the volatility process. Consequently, we are able to work under a weaker set of assumptions than those employed in the extant literature. The asymptotic variance matrices of both the standard and adaptive CSS (ACSS) estimators depend on any weak parametric autocorrelation present in the fractional model and any conditional heteroscedasticity in the shocks. Consequently, asymptotically pivotal inference can be achieved through the development of confidence regions or hypothesis tests using either heteroscedasticity-robust standard errors and/or a wild bootstrap. Monte Carlo simulations and empirical applications illustrate the practical usefulness of the methods proposed.

To infinity and beyond

Nielsen, M. O., Noel, A. L. — Sat, 01 May 2021 20:22:24 +0200

This article provides an exact algorithm for efficient computation of the time series of conditional variances, and hence the likelihood function, of models that have an ARCH(infinity) representation. This class of models includes, for example, the fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model. Our algorithm is a variation of the fast fractional difference algorithm of Jensen, A.N. and M.o. Nielsen (2014), Journal of Time Series Analysis 35, 428-436. It takes advantage of the fast Fourier transform (FFT) to achieve an order of magnitude improvement in computational speed. The efficiency of the algorithm allows estimation (and simulation/bootstrapping) of ARCH(infinity) models, even with very large data sets and without the truncation of the filter commonly applied in the literature. In Monte Carlo simulations, we show that the elimination of the truncation of the filter reduces the bias of the quasi-maximum-likelihood estimators and improves out-of-sample forecasting. Our results are illustrated in two empirical examples.

Semiparametric tests for the order of integration in the possible presence of level breaks

Iacone, F., Nielsen, M. Ø., Taylor, R. — Fri, 01 Apr 2022 20:22:24 +0200

Lobato and Robinson developed 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 generalized to allow for testing of the null hypothesis that a series is (Formula presented.) for any δ 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 unmodeled level breaks rendering fractional integration tests highly unreliable. To deal with this inference problem, we extend the Lobato and Robinson 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 behavior of our proposed tests, as well as several empirical examples.

Truncated sum of squares estimation of fractional time series models with deterministic trends

Hualde, J., Nielsen, M. Ø. — Sat, 01 Aug 2020 20:22:24 +0200

We consider truncated (or conditional) sum of squares estimation of a parametric model composed of a fractional time series and an additive generalized polynomial trend. Both the memory parameter, which characterizes the behavior of the stochastic component of the model, and the exponent parameter, which drives the shape of the deterministic component, are considered not only unknown real numbers but also lying in arbitrarily large (but finite) intervals. Thus, our model captures different forms of nonstationarity and noninvertibility. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to nonuniform convergence of the objective function over a large admissible parameter space, but, in addition, our framework is substantially more involved due to the competition between stochastic and deterministic components. We establish consistency and asymptotic normality under quite general circumstances, finding that results differ crucially depending on the relative strength of the deterministic and stochastic components. Finite-sample properties are illustrated by means of a Monte Carlo experiment.

Wild Bootstrap and Asymptotic Inference With Multiway Clustering

MacKinnon, J. G., Nielsen, M. Ø., Webb, M. D. — Fri, 01 Jan 2021 20:22:24 +0100

We study two cluster-robust variance estimators (CRVEs) for regression models with clustering in two dimensions and give conditions under which t-statistics based on each of them yield asymptotically valid inferences. In particular, one of the CRVEs requires stronger assumptions about the nature of the intra-cluster correlations. We then propose several wild bootstrap procedures and state conditions under which they are asymptotically valid for each type of t-statistic. Extensive simulations suggest that using certain bootstrap procedures with one of the t-statistics generally performs very well. An empirical example confirms that bootstrap inferences can differ substantially from conventional ones.

The cointegrated vector autoregressive model with general deterministic terms

Johansen, S., Nielsen, M. Ø. — Mon, 01 Jan 2018 20:22:24 +0100

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model X-t = gamma Z(t)+Y-t, where Z(t) belongs to a large class of deterministic regressors and Y-t is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are x(2)-distributed. (C) 2017 Elsevier B.V. All rights reserved.

Fast and wild

Roodman, G. D., MacKinnon, J. G., Nielsen, M. Ø., Webb, M. D. — Tue, 01 Jan 2019 20:22:24 +0100

The wild bootstrap was originally developed for regression models with heteroskedasticity of unknown form. Over the past 30 years, it has been extended to models estimated by instrumental variables and maximum likelihood and to ones where the error terms are (perhaps multiway) clustered. Like bootstrap methods in general, the wild bootstrap is especially useful when conventional inference methods are unreliable because large-sample assumptions do not hold. For example, there may be few clusters, few treated clusters, or weak instruments. The package boottest can perform a wide variety of wild bootstrap tests, often at remarkable speed. It can also invert these tests to construct confidence sets. As a postestimation command, boottest works after linear estimation commands, including regress, cnsreg, ivregress, ivreg2, areg, and reghdfe, as well as many estimation commands based on maximum likelihood. Although it is designed to perform the wild cluster bootstrap, boottest can also perform the ordinary (nonclustered) version. Wrappers offer classical Wald, score/Lagrange multiplier, and Anderson-Rubin tests, optionally with (multiway) clustering. We review the main ideas of the wild cluster bootstrap, offer tips for use, explain why it is particularly amenable to computational optimization, state the syntax of boottest, artest, scoretest, and waldtest, and present several empirical examples.

Asymptotic theory and wild bootstrap inference with clustered errors

Djogbenou, A. A., MacKinnon, J. G., Nielsen, M. Ø. — Tue, 01 Oct 2019 20:22:24 +0200

We study inference based on cluster-robust variance estimators for regression models with clustered errors, focusing on the wild cluster bootstrap. We state conditions under which asymptotic and bootstrap tests and confidence intervals are asymptotically valid. These conditions put limits on the rates at which the cluster sizes can increase as the number of clusters tends to infinity. We also derive Edgeworth expansions for the asymptotic and bootstrap test statistics. Simulation experiments illustrate the theoretical results and suggest that alternative variants of the wild cluster bootstrap may perform quite differently. The Edgeworth expansions explain the overrejection of asymptotic tests and shed light on the choice of auxiliary distribution and whether to use restricted or unrestricted estimates in the bootstrap data-generating process.

Improved likelihood ratio tests for cointegration rank in the VAR model

Boswijk, H. P., Jansson, M., Nielsen, M. Ø. — Thu, 01 Jan 2015 20:22:24 +0100

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but as usual the asymptotic results do not require normally distributed innovations. Our tests differ from existing tests in two respects. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott et al. (1996). Second, our tests incorporate a “sign” restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Johansen, S., Nielsen, M. Ø. — Mon, 01 Jul 2019 20:22:24 +0200

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non-stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non-fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi-squared-test.

Testing the CVAR in the Fractional CVAR Model

Johansen, S., Nielsen, M. Ø. — Thu, 01 Nov 2018 20:22:24 +0100

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and show that the likelihood ratio test statistic for the usual CVAR model is asymptotically chi-squared-distributed. Because the usual CVAR model lies on the boundary of the parameter space for the fractional CVAR in Johansen and Nielsen (2012a), the analysis requires the study of the fractional CVAR model on a slightly larger parameter space so that the CVAR model lies in the interior. This in turn implies some further analysis of the asymptotic properties of the fractional CVAR model.

Economic significance of commodity return forecasts from the fractionally cointegrated VAR model

Dolatabadi, S., Narayan, P. K., Nielsen, M. Ø., Xu, K. — Thu, 01 Feb 2018 20:22:24 +0100

We model and forecast commodity spot and futures prices using fractionally cointegrated vector autoregressive (FCVAR) models generalizing the well-known (non-fractional) CVAR model to accommodate fractional integration. In our empirical analysis to daily data on 17 commodity markets, the fractional model is statistically superior in terms of in-sample fit and out-of-sample forecasting. We analyze economic significance of the forecasts through dynamic (mean-variance) trading strategies, leading to statistically significant and economically meaningful profits in most markets. We generally find that the fractional model generates higher profits on average, especially in the futures markets.

Forecasting daily political opinion polls using the fractionally cointegrated vector auto-regressive model

Nielsen, M. Ø., Shibaev, S. S. — Mon, 01 Jan 2018 20:22:24 +0100

We examine forecasting performance of the recent fractionally cointegrated vector auto-regressive (FCVAR) model. We use daily polling data of political support in the UK for 2010–2015 and compare with popular competing models at several forecast horizons. Our findings show that the four variants of the FCVAR model considered are generally ranked as the top four models in terms of forecast accuracy, and the FCVAR model significantly outperforms both univariate fractional models and the standard cointegrated vector auto-regressive model at all forecast horizons. The relative forecast improvement is higher at longer forecast horizons, where the root-mean-squared forecast error of the FCVAR model is up to 15% lower than that of the univariate fractional models and up to 20% lower than that of the cointegrated vector auto-regressive model. In an empirical application to the 2015 UK general election, the estimated common stochastic trend from the model follows the vote share of the UK Independence Party very closely, and we thus interpret it as a measure of Euroscepticism in public opinion rather than an indicator of the more traditional left–right political spectrum. In terms of prediction of vote shares in the election, forecasts generated by the FCVAR model leading to the election appear to provide a more informative assessment of the current state of public opinion on electoral support than the hung Parliament prediction of the opinion poll.

Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form

Cavaliere, G., Nielsen, M. Ø., Taylor, A.M. R. — Sun, 01 Jan 2017 20:22:24 +0100

We consider the problem of conducting estimation and inference on the parameters of univariate heteroskedastic fractionally integrated time series models. We first extend existing results in the literature, developed for conditional sum-of-squares estimators in the context of parametric fractional time series models driven by conditionally homoskedastic shocks, to allow for conditional and unconditional heteroskedasticity both of a quite general and unknown form. Global consistency and asymptotic normality are shown to still obtain; however, the covariance matrix of the limiting distribution of the estimator now depends on nuisance parameters derived both from the weak dependence and heteroskedasticity present in the shocks. We then investigate classical methods of inference based on the Wald, likelihood ratio and Lagrange multiplier tests for linear hypotheses on either or both of the long and short memory parameters of the model. The limiting null distributions of these test statistics are shown to be non-pivotal under heteroskedasticity, while that of a robust Wald statistic (based around a sandwich estimator of the variance) is pivotal. We show that wild bootstrap implementations of the tests deliver asymptotically pivotal inference under the null. We demonstrate the consistency and asymptotic normality of the bootstrap estimators, and further establish the global consistency of the asymptotic and bootstrap tests under fixed alternatives. Monte Carlo simulations highlight significant improvements in finite sample behavior using the bootstrap in both heteroskedastic and homoskedastic environments. Our theoretical developments and Monte Carlo simulations include two bootstrap algorithms which are based on model estimates obtained either under the null hypothesis or unrestrictedly. Our simulation results suggest that the former is preferable to the latter, displaying superior size control yet largely comparable power.

A fractionally cointegrated VAR model with deterministic trends and application to commodity futures markets

Dolatabadi, S., Nielsen, M. Ø., Xu, K. — Fri, 01 Jan 2016 20:22:24 +0100

We apply the fractionally cointegrated vector autoregressive (FCVAR) model to analyze the relationship between spot and futures prices in five commodity markets (aluminium, copper, lead, nickel, and zinc). To this end, we first extend the FCVAR model to accommodate deterministic trends in the levels of the processes. The methodological contribution is to provide a representation theory for the FCVAR model with deterministic trends, where we show that the presence of the deterministic trend in the process induces both restricted and unrestricted constant terms in the vector error correction model. The consequences for the cointegration rank test are also briefly discussed. In our empirical application we use the data from Figuerola-Ferretti and Gonzalo (2010), who conduct a similar analysis using the usual (non-fractional) cointegrated VAR model. The main conclusion from the empirical analysis is that, when using the FCVAR model, there is more support for the cointegration vector (1, − 1)’ in the long-run equilibrium relationship between spot and futures prices, and hence less evidence of long-run backwardation, compared to the results from the non-fractional model. Specifically, we reject the hypothesis that the cointegration vector is (1, − 1)’ using standard likelihood ratio tests only for the lead and nickel markets.

The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models

Johansen, S., Nielsen, M. Ø. — Fri, 01 Jan 2016 20:22:24 +0100

Asymptotics for the Conditional-Sum-of-Squares Estimator in Multivariate Fractional Time-Series Models

Nielsen, M. Ø. — Thu, 01 Jan 2015 20:22:24 +0100

This article proves consistency and asymptotic normality for the conditional-sum-of-squares estimator, which is equivalent
to the conditional maximum likelihood estimator, in multivariate fractional time-series models. The model is parametric and
quite general and, in particular, encompasses the multivariate non-cointegrated fractional autoregressive integrated moving
average (ARIMA) model. The novelty of the consistency result, in particular, is that it applies to a multivariate model and
to an arbitrarily large set of admissible parameter values, for which the objective function does not converge uniformly in
probability, thus making the proof much more challenging than usual. The neighbourhood around the critical point where
uniform convergence fails is handled using a truncation argument

Numerical distribution functions of fractional unit root and cointegration tests

Mackinnon, J. G., Nielsen, M. Ø. — Wed, 01 Jan 2014 20:22:24 +0100

The impact of financial crises on the risk-return tradeoff and the leverage effect

Christensen, B. J., Nielsen, M. Ø., Zhu, J. — Thu, 01 Jan 2015 20:22:24 +0100

We investigate the impact of financial crises on two fundamental features of stock returns, namely, the risk-return tradeoff and the leverage effect. We apply the fractionally integrated exponential GARCH-in-mean (FIEGARCH-M) model for daily stock return data, which includes both features and allows the co-existence of long memory in volatility and short memory in returns. We extend this model to allow the financial parameters governing the volatility-in-mean effect and the leverage effect to change during financial crises. An application to the daily U.S. stock index return series from 1926 through 2010 shows that both financial effects increase significantly during crises. Strikingly, the risk-return tradeoff is significantly positive only during financial crises, and insignificant during non-crisis periods. The leverage effect is negative throughout, but increases significantly by about 50% in magnitude during financial crises. No such changes are observed during NBER recessions, so in this sense financial crises are special. Applications to a number of major developed and emerging international stock markets confirm the increase in the leverage effect, whereas the international evidence on the risk-return tradeoff is mixed.

A fast fractional difference algorithm

Jensen, A. N., Nielsen, M. Ø. — Wed, 01 Jan 2014 20:22:24 +0100

We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T², where T is the length of the time series. Our algorithm allows calculation speed of order TlogT. For moderate and large sample sizes, the difference in computation time is substantial.

A fractionally cointegrated VAR analysis of economic voting and political support

Jones, M. E. C., Nielsen, M. O., Popiel, M. K. — Wed, 01 Jan 2014 20:22:24 +0100

We use a fractionally cointegrated vector autoregressive model to examine the relationship between Canadian political support and macroeconomic conditions. This model is well suited for the analysis because it allows multiple fractional time series and admits simple asymptotic inference for the model parameters and tests of the hypotheses of interest. In the long-run equilibrium, we find that support for the Progressive Conservative Party was higher during periods of high interest rates and low unemployment, while support for the Liberal Party was higher during periods of low interest rates and high unemployment. We also test and reject the notion that party support is driven only by relative (to the United States) economic performance. Indeed, our findings suggest that US macroeconomic variables do not enter the long-run equilibrium of Canadian economic voting (political opinion poll support) at all.

A necessary moment condition for the fractional functional central limit theorem

Johansen, S., Nielsen, M. Ø. — Fri, 01 Jun 2012 20:22:24 +0200

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

Johansen, S., Nielsen, M. Ø. — Thu, 01 Nov 2012 20:22:24 +0100

Nearly efficient likelihood ratio tests of the unit root hypothesis

Jansson, M., Nielsen, M. Ø. — Sat, 01 Sep 2012 20:22:24 +0200

Local polynomial Whittle estimation of perturbed fractional processes

Frederiksen, P., Nielsen, F. S., Nielsen, M. Ø. — Sun, 01 Jan 2012 20:22:24 +0100

Likelihood inference for a nonstationary fractional autoregressive model

Johansen, S., Ørregård Nielsen, M. — Fri, 01 Jan 2010 20:22:24 +0100

This paper discusses model-based inference in an autoregressive model for fractional processes which allows the process to be fractional of order d or d-b. Fractional differencing involves infinitely many past values and because we are interested in nonstationary processes we model the data X1,...,X_{T} given the initial values X_{-n}, n=0,1,..., as is usually done. The initial values are not modeled but assumed to be bounded. This represents a considerable generalization relative to all previous work where it is assumed that initial values are zero. For the statistical analysis we assume the conditional Gaussian likelihood and for the probability analysis we also condition on initial values but assume that the errors in the autoregressive model are i.i.d. with suitable moment conditions.
We analyze the conditional likelihood and its derivatives as stochastic processes in the parameters, including d and b, and prove that they converge in distribution. We use the results to prove consistency of the maximum likelihood estimator for d,b in a large compact subset of {1/2

A Vector Autoregressive Model for Electricity Prices subject to Long Memory and Regime Switching

Haldrup, N., Nielsen, F., Nielsen, M. Ø. — Fri, 01 Jan 2010 20:22:24 +0100

A powerful test of the autoregressive unit root hypothesis based on a tuning parameter free statistic

Nielsen, M. Ø. — Thu, 01 Jan 2009 20:22:24 +0100

Udgivelsesdato: December

Long Memory in Stock Market Volatility and the Volatility-in-Mean Effect: The FIEGARCH-M Model

Christensen, B. J., Nielsen, M. Ø., Zhu, J. — Fri, 01 Jan 2010 20:22:24 +0100

Udgivelsesdato: June

The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets

Busch, T., Christensen, B. J., Nielsen, M. Ø. — Sat, 01 Jan 2011 20:22:24 +0100

The Effect of Long Memory in Volatility on Stock Market Fluctuations

Christensen, B. J., Nielsen, M. Ø. — Mon, 01 Jan 2007 20:22:24 +0100

Recent empirical evidence demonstrates the presence of an important long memory component in realized asset return volatility. We specify and estimate multivariate models for the joint dynamics of stock returns and volatility that allow for long memory in volatility without imposing this property on returns. Asset pricing theory imposes testable cross-equation restrictions on the system that are not rejected in our preferred specifications, which include a strong financial leverage effect. We show that the impact of volatility shocks on stock prices is small and short-lived, in spite of a positive risk-return trade-off and long memory in volatility.

Asymptotic Normality of Narrow-Band Least Squares in the Stationary Fractional Cointegration Model and Volatility Forecasting

Christensen, B. J., Nielsen, M. Ø. — Sun, 01 Jan 2006 20:22:24 +0100

A regime switching long memory model for electricity prices

Haldrup, N., Nielsen, M. Ø. — Sun, 01 Jan 2006 20:22:24 +0100

Directional Congestion and regime switching in a long memory model for electricity prices

Haldrup, N., Nielsen, M. Ø. — Sun, 01 Jan 2006 20:22:24 +0100

Seasonality in Economic Models

Brendstrup, B., Hylleberg, S., Nielsen, M. Ø., Skipper, L., Stentoft, L. — Thu, 01 Jan 2004 20:22:24 +0100

Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration

Nielsen, M. Ø., Frederiksen, P. H. — Sat, 01 Jan 2005 20:22:24 +0100

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.