Publications - Niels Haldrup

A Parametric Factor Model of the Term Structure of Mortality

Haldrup, N., Rosenskjold, C. P. T. — Fri, 01 Mar 2019 10:20:20 +0100

The prototypical Lee–Carter mortality model is characterized by a single common time factor that loads differently across age groups. In this paper, we propose a parametric factor model for the term structure of mortality where multiple factors are designed to influence the age groups differently via parametric loading functions. We identify four different factors: a factor common for all age groups, factors for infant and adult mortality, and a factor for the “accident hump” that primarily affects mortality of relatively young adults and late teenagers. Since the factors are identified via restrictions on the loading functions, the factors are not designed to be orthogonal but can be dependent and can possibly cointegrate when the factors have unit roots. We suggest two estimation procedures similar to the estimation of the dynamic Nelson–Siegel term structure model. First, a two-step nonlinear least squares procedure based on cross-section regressions together with a separate model to estimate the dynamics of the factors. Second, we suggest a fully specified model estimated by maximum likelihood via the Kalman filter recursions after the model is put on state space form. We demonstrate the methodology for US and French mortality data. We find that the model provides a good fit of the relevant factors and, in a forecast comparison with a range of benchmark models, it is found that, especially for longer horizons, variants of the parametric factor model have excellent forecast performance.

A Parametric Factor Model of the Term Structure of Mortality

Haldrup, N., Rosenskjold, C. P. T. — Mon, 15 Jan 2018 10:20:20 +0100

The prototypical Lee-Carter mortality model is characterized by a single common time factor that loads differently across age groups. In this paper we propose a factor model for the term structure of mortality where multiple factors are designed to influence the age groups differently via parametric loading functions. We identify four different factors: a factor common for all age groups, factors for infant and adult mortality, and a factor for the "accident hump" that primarily affects mortality of relatively young adults and late teenagers. Since the factors are identified via restrictions on the loading functions, the factors are not designed to be orthogonal but can be dependent and can possibly cointegrate when the factors have unit roots. We suggest two estimation procedures similar to the estimation of the dynamic Nelson-Siegel term structure model. First, a two-step nonlinear least squares procedure based on cross-section regressions together with a separate model to estimate the dynamics of the factors. Second, we suggest a fully specified model estimated by maximum likelihood via the Kalman filter recursions after the model is put on state space form. We demonstrate the methodology for US and French mortality data. We find that the model provides a good fitt of the relevant factors and in a forecast comparison with a range of benchmark models it is found that, especially for longer horizons, variants of the parametric factor model have excellent forecast performance.

Spikes and memory in (Nord Pool) electricity price spot prices

Proietti, T., Haldrup, N., Knapik, O. — Wed, 15 Nov 2017 10:20:20 +0100

Electricity spot prices are subject to transitory sharp movements commonly referred to as spikes. The paper aims at assessing their effects on model based inferences and predictions, with reference to the Nord Pool power exchange. We identify a spike as a price value which deviates substantially from the normal price, where the latter is defined as the expectation arising from a model accounting for long memory at the zero and at the weekly seasonal frequencies, given the knowledge of the past realizations. Hence, a spike is associated to a time series innovation with size larger than a specified threshold. The latter regulates the robustness of the estimates of the underlying price level and it is chosen by a data driven procedure that focuses on the ability to predict future prices. The normal price is computed by a modified Kalman filter, which robustifies the inferences by cleaning the spikes, i.e. shrinking an observation deviating substantially from the normal price towards the one-step-ahead prediction. Our empirical application illustrates the effects of the spikes on the estimates of the parameters governing the persistence of the series; moreover, a real time rolling forecasting exercise is used to establish the amount of cleaning for optimizing the predicting accuracy at different horizons.

Long memory, fractional integration, and cross sectional aggregation

Haldrup, N., Vera-Valdés, E. — Sun, 01 Jan 2017 10:20:20 +0100

Space-time modeling of electricity spot prices

Abate, G. D., Haldrup, N. — Sun, 01 Jan 2017 10:20:20 +0100

Using data for the Nord Pool power grid, we derive a space-time Durbin model for electricity spot prices with both temporal and spatial lags. Joint modeling of temporal and spatial adjustment effects is necessarily important when prices and loads are determined in a network grid. By using different spatial weight matrices, statistical tests show significant spatial dependence in the spot price dynamics across areas. In fact, estimation of the model shows that the spatial dependence is as important as the temporal dependence in describing the spot price dynamics. We decompose price impacts into direct and indirect effects and demonstrate how price effects transmit to neighboring markets and decline with distance. A forecasting comparison with a non-spatial model shows that the space-time model improves forecasting performance for 7 and 30 days ahead forecasts. A model with time-varying parameters is estimated for an expanded sample period and it is found that the spatial correlation within the power grid has increased over time. We interpret this to indicate an increasing degree of market integration within the sample period.

CREATES Annual Report 2015

Haldrup, N. — Fri, 01 Jan 2016 10:20:20 +0100

Common long-range dependence in a panel of hourly Nord Pool electricity prices and loads

Ergemen, Y. E., Haldrup, N., Rodríguez-Caballero, C. V. — Fri, 01 Jan 2016 10:20:20 +0100

Equilibrium electricity spot prices and loads are often determined simultaneously in a day-ahead auction market for each hour of the subsequent day. Hence daily observations of hourly prices take the form of a periodic panel rather than a time series of hourly observations. We consider novel panel data approaches to analyse the time series and the cross-sectional dependence of hourly Nord Pool electricity spot prices and loads for the period 2000–2013. Hourly electricity prices and load data are characterized by strong serial long-range dependence in the time series dimension in addition to strong seasonal periodicity, and along the cross-sectional dimension, i.e. the hours of the day, there is a strong dependence which necessarily has to be accounted for in order to avoid spurious inference when focusing on the time series dependence alone. The long-range dependence is modelled in terms of a fractionally integrated panel data model and it is shown that both prices and loads consist of common factors with long memory and with loadings that vary considerably during the day. Due to the competitiveness of the Nordic power market the aggregate supply curve approximates well the marginal costs of the underlying production technology and because the demand is more volatile than the supply, equilibrium prices and loads are argued to identify the periodic power supply curve. The estimated supply elasticities are estimated from fractionally co-integrated relations and range between 0.5 and 1.17 with the largest elasticities being estimated during morning and evening peak hours.

A generalized exponential time series regression model for electricity prices

Haldrup, N., Knapik, O., Proietti, T. — Mon, 21 Mar 2016 10:20:20 +0100

We consider the issue of modeling and forecasting daily electricity spot prices on the Nord Pool Elspot power market. We propose a method that can handle seasonal and non-seasonal persistence by modelling the price series as a generalized exponential process. As the presence of spikes can distort the estimation of the dynamic structure of the series we consider an iterative estimation strategy which, conditional on a set of parameter estimates, clears the spikes using a data cleaning algorithm, and reestimates the parameters using the cleaned data so as to robustify the estimates. Conditional on the estimated model, the best linear predictor is constructed. Our modeling approach provides good fit within sample and outperforms competing benchmark predictors in terms of forecasting accuracy. We also find that building separate models for each hour of the day and averaging the forecasts is a better strategy than forecasting the daily average directly.

Long Memory, Fractional Integration, and Cross-Sectional Aggregation

Haldrup, N., Vera-Valdés, E. — Mon, 14 Dec 2015 10:20:20 +0100

It is commonly argued that observed long memory in time series variables can result from cross-sectional aggregation of dynamic heterogeneous micro units. For instance, Granger (1980) demonstrated that aggregation of AR(1) processes with a Beta distributed AR coefficient can exhibit long memory under certain conditions and that the aggregated series will have an autocorrelation function that exhibits hyperbolic decay. In this paper, we further analyze this phenomenon. We demonstrate that the aggregation argument leading to long memory is consistent with a wide range of definitions of long memory. In a simulation study we seek to quantify Granger's result and find that indeed both the time series and cross-sectional dimensions have to be rather significant to reflect the theoretical asymptotic results. Long memory can result even for moderate T,N dimensions but can vary considerably from the theoretical degree of memory. Also, Granger's result is most precise in samples with a relatively high degree of memory. Finally, we show that even though the aggregated process will behave as generalized fractional process and thus converge to a fractional Brownian motion asymptotically, the fractionally differenced series does not behave according to an ARMA process. In particular, despite the autocorrelation function is summable and hence the fractionally differenced process satisfy the conditions for being I(0), it still exhibits hyperbolic decay. This may have consequences for the validity of ARFIMA time series modeling of long memory processes when the source of memory is due to aggregation.

CREATES Annual Report 2014

Haldrup, N. — Thu, 01 Jan 2015 10:20:20 +0100

Deterministic and stochastic trends in the Lee-Carter mortality model

Callot, L., Haldrup, N., Kallestrup-Lamb, M. — Thu, 01 Jan 2015 10:20:20 +0100

The Lee and Carter (1992) model assumes that the deterministic and stochastic time series dynamics load with identical weights when describing the development of age-specific mortality rates. Effectively this means that the main characteristics of the model simplify to a random walk model with age-specific drift components. But restricting the adjustment mechanism of the
stochastic and linear trend components to be identical may be too strong a simplification. In fact, the presence of a stochastic trend component may itself result from a bias induced by properly fitting the linear trend that characterizes mortality data. We find empirical evidence that this feature of the Lee–Carter model overly restricts the system dynamics and we suggest to separate the deterministic and stochastic time series components at the benefit of improved fit and forecasting performance. In fact, we find that the classical Lee–Carter model will otherwise overestimate the reduction of mortality for the younger age groups and will underestimate the reduction of mortality for the older age groups. In practice, our recommendation means that the Lee–Carter model instead of a one-factor model should be formulated as a two- (or several) factor
model where one factor is deterministic and the other factors are stochastic. This feature generalizes to the range of models that extend the Lee–Carter model in various directions.

Space-time modeling of electricity spot prices

Abate, G. D., Haldrup, N. — Fri, 08 May 2015 10:20:20 +0200

In this paper we derive a space-time model for electricity spot prices. A general spatial Durbin model that incorporates the temporal as well as spatial lags of spot prices is presented. Joint modeling of space-time effects is necessarily important when prices and loads are determined in a network of power exchange areas. We use data from the Nord Pool electricity power exchange area bidding markets. Different spatial weight matrices are considered to capture the structure of the spatial dependence process across different bidding markets and statistical tests show significant spatial dependence in the spot price dynamics. Estimation of the spatial Durbin model show that the spatial lag variable is as important as the temporal lag variable in describing the spot price dynamics. We use the partial derivatives impact approach to decompose the price impacts into direct and indirect effects and we show that price effects transmit to neighboring markets and decline with distance. In order to examine the evolution of the spatial correlation over time, a time varying parameters spot price spatial Durbin model is estimated using recursive estimation. It is found that the spatial correlation within the Nord Pool grid has been increasing over time which we interpret as evidence for an increasing degree of market integration.

CREATES Annual Report 2013

Haldrup, N. — Wed, 01 Jan 2014 10:20:20 +0100

Deterministic and stochastic trends in the Lee-Carter mortality model

Callot, L., Haldrup, N., Kallestrup-Lamb, M. — Thu, 20 Nov 2014 10:20:20 +0100

The Lee and Carter (1992) model assumes that the deterministic and stochastic time series dynamics loads with identical weights when describing the development of age specific mortality rates. Effectively this means that the main characteristics of the model simplifies to a random walk model with age specific drift components. But restricting the adjustment mechanism of the stochastic and linear trend components to be identical may be a too strong simplification. In fact, the presence of a stochastic trend component may itself result from a bias induced by properly fitting the linear trend that characterizes mortality data. We find empirical evidence that this feature of the Lee-Carter model overly restricts the system dynamics and we suggest to separate the deterministic and stochastic time series components at the benefit of improved fit and forecasting performance. In fact, we find that the classical Lee-Carter model will otherwise over estimate the reduction of mortality for the younger age groups and will under estimate the reduction of mortality for the older age groups. In practice, our recommendation means that the Lee-Carter model instead of a one-factor model should be formulated as a two (or several)-factor model where one factor is deterministic and the other factors are stochastic. This feature generalizes to the range of models that extend the Lee-Carter model in various directions.

Discriminating between fractional integration and spurious long memory

Haldrup, N., Kruse, R. — Mon, 30 Jun 2014 10:20:20 +0200

Fractionally integrated processes have become a standard class of models to describe the long memory features of economic and financial time series data. However, it has been demonstrated in numerous studies that structural break processes and non-linear features can often be confused as being long memory. The question naturally arises whether it is possible empirically to determine the source of long memory as being genuinely long memory in the form of a fractionally integrated process or whether the long range dependence is of a di¤erent nature. In this paper we suggest a testing procedure that helps discriminating between such processes. The idea is based on the feature that nonlinear transformations of stationary fractionally integrated Gaussian processes decrease the order of memory in a speci…c way which is determined by the Hermite rank of the transformation. In principle, a non-linear transformation of the series can make the series short memory I(0). We suggest using the Wald test of Shimotsu (2007) to test the null hypothesis that a vector time series of properly transformed variables is I(0). Our testing procedure is designed such that even non-stationary fractionally integrated processes are permitted under the null hypothesis. The test is shown to have good size and to be robust against certain types of deviations from Gaussianity. The test is also shown to be consistent against a broad class of processes that are non-fractional but still exhibit (spurious) long memory. In particular, the test is shown to have excellent power against a class of stationary and non-stationary random level shift models as well as Markov switching GARCH processes where the break and transition probabilities are allowed to be time varying.

Essays on Non-linear Time Series Econometrics

Haldrup, N., Meitz, M., Saikkonen, P. — Wed, 01 Jan 2014 10:20:20 +0100

CREATES Annual Report 2012

Haldrup, N. — Tue, 01 Jan 2013 10:20:20 +0100

CREATES Annual Report 2011

Haldrup, N. — Sun, 01 Jan 2012 10:20:20 +0100

Unit Roots, Non-linearities, and Structural Breaks

Haldrup, N., Kruse, R., Teräsvirta, T., Varneskov, R. T. — Tue, 01 Jan 2013 10:20:20 +0100

Unit roots, nonlinearities and structural breaks

Haldrup, N., Kruse, R., Teräsvirta, T., Varneskov, R. T. — Tue, 24 Apr 2012 10:20:20 +0200

One of the most influential research fields in econometrics over the past decades concerns unit root testing in economic time series. In macro-economics much of the interest in the area originate from the fact that when unit roots are present, then shocks to the time series processes have a persistent effect with resulting policy implications. From a statistical perspective on the other hand, the presence of unit roots has dramatic implications for econometric model building, estimation, and inference in order to avoid the so-called spurious regression problem. The present paper provides a selective review of contributions to the field of unit root testing over the past three decades. We discuss the nature of stochastic and deterministic trend processes, including break processes, that are likely to affect unit root inference. A range of the most popular unit root tests are presented and their modifications to situations with breaks are discussed. We also review some results on unit root testing within the framework of non-linear processes.

Detection of Additive Outliers in Seasonal Time Series

Haldrup, N., Montanés, A., Sansó, A. — Sat, 01 Jan 2011 10:20:20 +0100

Periodicity, Non-stationarity, and Forecasting of Economic and Financial Time Series: Editors' Introduction

Bollerslev, T., Christensen, B. J., Haldrup, N., Lunde, A. — Sat, 01 Jan 2011 10:20:20 +0100

Separation in Cointegrated Systems

Haldrup, N. — Fri, 01 Jan 2010 10:20:20 +0100

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 10:20:20 +0100

Detection of additive outliers in seasonal time series

Haldrup, N., Montañés, A., Sansó, A. — Thu, 01 Jan 2009 10:20:20 +0100

The detection and location of additive outliers in integrated variables
has attracted much attention recently because such outliers tend to affect
unit root inference among other things. Most of these procedures have
been developed for non-seasonal processes. However, the presence of seasonality
in the form of seasonally varying means and variances affect the
properties of outlier detection procedures, and hence appropriate adjustments
of existing methods are needed for seasonal data. In this paper we
suggest modifications of tests proposed by Shin et al. (1996) and Perron
and Rodriguez (2003) to deal with data sampled at a seasonal frequency
and the size and power properties are discussed. We also show that the
presence of periodic heteroscedasticity will inflate the size of the tests and
hence will tend to identify an excessive number of outliers. A modified
Perron-Rodriguez test which allows periodically varying variances is suggested
and it is shown to have excellent properties in terms of both power
and size

Sequential Versus Simultaneous Market Delineation

Haldrup, N., Møllgaard, P., Nielsen, C. K. — Tue, 01 Jan 2008 10:20:20 +0100

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

Haldrup, N., Nielsen, F., Nielsen, M. Ø. — Mon, 01 Jan 2007 10:20:20 +0100

A regime dependent VAR model is suggested that allows long memory (fractional
integration) in each of the regime states as well as the possibility of fractional cointegra-
tion. The model is relevant in describing the price dynamics of electricity prices where the
transmission of power is subject to occasional congestion periods. For a system of bilat-
eral prices non-congestion means that electricity prices are identical whereas congestion
makes prices depart. Hence, the joint price dynamics implies switching between essen-
tially a univariate price process under non-congestion and a bivariate price process under
congestion. At the same time it is an empirical regularity that electricity prices tend
to show a high degree of fractional integration, and thus that prices may be fractionally
cointegrated. An empirical analysis using Nord Pool data shows that even though the
prices strongly co-move under non-congestion, the prices are not, in general, fractional
cointegrated in the congestion state.

Improving power and size in unit root testing

Haldrup, N., Jansson, M. — Sun, 01 Jan 2006 10:20:20 +0100

A regime switching long memory model for electricity prices

Haldrup, N., Nielsen, M. Ø. — Sun, 01 Jan 2006 10:20:20 +0100

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

Haldrup, N., Nielsen, M. Ø. — Sun, 01 Jan 2006 10:20:20 +0100

A note on the Vogelsang test for additive outliers

Haldrup, N., Sanso, A. — Tue, 01 Jan 2008 10:20:20 +0100

Labour Market Implications of EU Product Market Integration

Andersen, T. M., Haldrup, N., Sørensen, J. R. — Sat, 01 Jan 2000 10:20:20 +0100

Common Periodic Correlation Features and the Interaction of Stock and Flows in Daily Airport Data

Haldrup, N., Hylleberg, S., Pons, G., Sansó, A. — Mon, 01 Jan 2007 10:20:20 +0100

Udviklingslinier i Økonometrien

Bunzel, H., Christensen, B. J., Haldrup, N., et al. — Sat, 01 Jan 2000 10:20:20 +0100

Near-integration and Deterministic Trends

Haldrup, N., Hylleberg, S. — Wed, 01 Jan 1997 10:20:20 +0100

A Note on: The Distribution of the Least Squares Estimator of a Random Walk with Drift - Some Analytical Evidence

Haldrup, N., Hylleberg, S. — Sun, 01 Jan 1995 10:20:20 +0100

Measurement Errors and Outliers in Seasonal Unit Root Testing

Haldrup, N., Montanes, A., Sanso, A. — Sat, 01 Jan 2005 10:20:20 +0100

Local Power Functions of Tests for Double Unit Roots

Haldrup, N., Lildholt, P. — Sat, 01 Jan 2005 10:20:20 +0100

A Gaussian IV estimator of cointegrating relations

Bårdsen, G., Haldrup, N. — Sun, 01 Jan 2006 10:20:20 +0100

In static single equation cointegration regression modelsthe OLS estimator will have a non-standard distribution unless regressors arestrictly exogenous. In the literature a number of estimators have been suggestedto deal with this problem, especially by the use of semi-nonparametricestimators. Theoretically ideal instruments can be defined to ensure a limitingGaussian distribution of IV estimators, but unfortunately such instruments areunlikely to be found in real data. In the present paper we suggest an IV estimatorwhere the Hodrick-Prescott filtered trends are used as instruments forthe regressors in cointegrating regressions. These instruments are almost idealand simulations show that the IV estimator using such instruments alleviatethe endogeneity problem extremely well in both finite and large samples.

Regression Theory for Nearly Co-Integrated Time Series

Jansson, M., Haldrup, N. — Tue, 01 Jan 2002 10:20:20 +0100

Estimation of fractional integration in the presence of data noise

Haldrup, N., Nielsen, M.Ø. — Wed, 01 Jan 2003 10:20:20 +0100

Long-run forecasting in multi-cointegrated systems

Siliverstovs, B., Haldrup, N., Engsted, T. — Thu, 01 Jan 2004 10:20:20 +0100

Testing for Additive Outliers in Seasonally Integrated Time Series

Haldrup, N., Montanes, A., Sanso, A. — Thu, 01 Jan 2004 10:20:20 +0100

A Regime Switching Long Memory Model for Electricity Prices

Haldrup, N., Nielsen, M.Ø. — Thu, 01 Jan 2004 10:20:20 +0100

Model selection and evaluation in econometrics

Haldrup, N., Hendry, D.F., Dijk, H.K.van — Wed, 01 Jan 2003 10:20:20 +0100

On the Robustness of Unit Root Tests in the Presence of Double Unit Roots

Haldrup, N., Lildholdt, P.M. — Tue, 01 Jan 2002 10:20:20 +0100

Nobel-prisen i Økonomi 2003, Tidsserieøkonometri: Cointegration og ARCH, Clive W.J. Granger og Robert F. Engle

Christiansen, C., Haldrup, N. — Thu, 01 Jan 2004 10:20:20 +0100

Guest editors introduction: Model selection and evaluation in econometrics

Haldrup, N., Hendry, D.F., Dijk, H.K.van — Wed, 01 Jan 2003 10:20:20 +0100

Empirical analysis of price data in the delineation of the relevant geographical market in competition analysis

Haldrup, N. — Wed, 01 Jan 2003 10:20:20 +0100

Estimation af middellevetider for mænd og kvinder i Danmark 2002-2100 baseret på Lee-Carter metoden

Haldrup, N. — Thu, 01 Jan 2004 10:20:20 +0100