for their secretarial and research assistance. Documentos de Trabajo. N.º 0417 2004 The Working P... more for their secretarial and research assistance. Documentos de Trabajo. N.º 0417 2004 The Working Paper Series seeks to disseminate original research in economics and finance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the INTERNET at the following website:
Thanks are due to David Findley and William Bell for their helpful comments, and to Nieves Morale... more Thanks are due to David Findley and William Bell for their helpful comments, and to Nieves Morales and Jorge Carrillo for their secretarial and research assistance. 1 Filters used to estimate unobserved components in time series are often designed on a priori grounds, so as to capture the frequencies associated with the component. A limitation of these filters is that they may yield spurious results. The danger can be avoided if the so-called ARIMA-model-based (AMB) procedure is used to derive the filter. However, parsimony of ARIMA models typically implies little resolution in terms of the detection of hidden components. It would be desirable to combine a higher resolution with consistency with the structure of the observed series. We show first that for a large class of a priori designed filters, an AMB interpretation is always possible. Using this result, proper convolution of AMB filters can produce richer decompositions of the series that incorporate a priori
Time series segmentation has many applications in several disciplines as neurology, cardiology, s... more Time series segmentation has many applications in several disciplines as neurology, cardiology, speech, geology and others. Many time series in this fields do not behave as stationary and the usual transformations to linearity cannot be used. This paper describes and evaluates different methods for segmenting non-stationary time series. We propose a modification of the algorithm in Lee et al. (2003) which is designed to searching for a unique change in the parameters of a time series, in order to find more than one change using an iterative procedure. We evaluate the performance of three approaches for segmenting time series: AutoSLEX (Ombao et al., 2002), AutoPARM (Davis et al., 2006) and the iterative cusum method mentioned above and referred as ICM. The evaluation of each methodology consists of two steps. First, we compute how many times each procedure fails in segmenting stationary processes properly. Second, we analyze the effect of different change patterns by counting how ma...
We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first,... more We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first, the well-known criticism of spurious results due to the ad-hoc character of the filter, and the (often ignored yet important) limitation implied by revisions, which produce imprecision in the cycle estimator for recent periods. We show how the integration of some relatively simple ARIMA-model-based (AMB) techniques with HP filtering can produce important improvements in the performance of the cyclical signal. Finally, the complete procedure of applying the HP filter to a "clean" series is presented within a model-based methodology. This AMB methodology displays several nice features. First, it incorporates automatically optimal treatment of end points and provides a cleaner cyclical signal. Second, it provides an internally consistent full decomposition of the series into "trend + cycle + seasonal irregular" components, where the trend plus cycle aggregate into the s...
In this monograph, first, we analyze in detail some of the major limitations of the standard proc... more In this monograph, first, we analyze in detail some of the major limitations of the standard procedure to estimate business cycles with the Hodrick-Prescott (HP) filter. By incorporating time series analysis techniques, it is seen how some intuitive and relatively simple modifications to the filter can improve significantly its performance, in particular in terms of cleanness of the signal, smaller revision, stability of end-period estimators, and detection of turning points. Then, we show how the modified filter can be seen as the exact solution of a well-defined statistical problem, namely, optimal (minimum mean squared error) estimation of components in a standard unobserved-component model, where the observed series is decomposed into a trend, a cycle, a seasonal, and an irregular component.
The paper details an application of programs TRAMO and SEATS to seasonal adjustment and trend-cyc... more The paper details an application of programs TRAMO and SEATS to seasonal adjustment and trend-cycle estimation of the German Retail Trade Turnover series. When adjusting with X12-ARIMA, the Bundesbank identified two problems: heteroscedasticity in the seasonal component, associated with different moving patterns for some of the months, and unstability of the trend-cycle at the end of the series. It is seen how, starting with the fully automatic procedure and adding some simple modifications, TRAMO and SEATS deal properly with both problems, and provide good, stable, and robust results.
The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for busin... more The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for business-cycle estimation at many economic agencies and institutions. We show that the filter can be obtained from MMSE estimation of the components in an unobserved component model, where the original series is decomposed into (long-term) trend, cyclical, seasonal, and (highlytransitory) irregular components. The component models are sensible and combine desirable “ad-hoc” features with series-dependent features that guarantee consistency with the data. The model-based framework provides improvements having to do with the precision of end-point estimation and the stability of the cyclical signal.
In the analysis of time series, it is frequent to classify perturbations as Additive Outliers (AO... more In the analysis of time series, it is frequent to classify perturbations as Additive Outliers (AO), Innovative Outliers (IO), Level Shift (LS) outliers or Transitory Change (TC) outliers. In this paper, a new outlier type, the Seasonal Level Shift (SLS), is introduced in order to complete the usual classification.
The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for busin... more The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for business-cycle estimation at many economic agencies and institutions. We show that the filter can be obtained from MMSE estimation of the components in an unobserved component model, where the original series is decomposed into (long-term) trend, cyclical, seasonal, and (highlytransitory) irregular components. The component models are sensible and combine desirable "ad-hoc" features with series-dependent features that guarantee consistency with the data. The model-based framework provides improvements having to do with the precision of end-point estimation and the stability of the cyclical signal.
In the framework of decomposing a time series into the sum of signal components plus noise as in ... more In the framework of decomposing a time series into the sum of signal components plus noise as in detrending or seasonal adjustment, we analyze the situation in which the unobserved components may be subject to the influence of sudden shifts. The kind of perturbation that such shifts cause on the observed series can be classified as an outlier, when the shift affects the noise component, or as a structural change, when the shift affects one of the signal components. The consequences of ignoring these perturbations are important for model specification, parameter estimation and forecasting. We extend and modify the iterative procedure of Chen and Liu (1993) to allow the location, classification and estimation of outliers and structural changes affecting the unobserved components of a time series.
In the framework of decomposing a time series into the sum of signal components plus noise as in ... more In the framework of decomposing a time series into the sum of signal components plus noise as in detrending or seasonal adjustment, we analyze the situation in which the unobserved components may be subject to the influence of sudden shifts. The kind of perturbation that such shifts cause on the observed series can be classified as an outlier, when the shift affects the noise component, or as a structural change, when the shift affects one of the signal components. The consequences of ignoring these perturbations are important for model specification, parameter estimation and forecasting. We extend and modify the iterative procedure of Chen and Liu (1993) to allow the location, classification and estimation of outliers and structural changes affecting the unobserved components of a time series.
We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first,... more We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first, the well-known criticism of spurious results due to the ad-hoc character of the filter, and the (often ignored yet important) limitation implied by revisions, which produce imprecision in the cycle estimator for recent periods. We show how the integration of some relatively simple ARIMA-model-based (AMB) techniques with HP filtering can produce important improvements in the performance of the cyclical signal.
Present practice in applied time series work, mostly at economic policy or data producing agencie... more Present practice in applied time series work, mostly at economic policy or data producing agencies, relies heavily on using moving average filters to estimate unobserved components (or signals) in time series, such as the seasonally adjusted series, the trend, or the cycle. The purpose of the present paper is to provide an informal introduction to the time series analysis tools and concepts required by the user or analyst to understand the basic methodology behind the application of filters. The paper is aimed at economists, statisticians, and analysts in general, that do applied work in the field, but have not had an advanced course in applied time series analysis. Although the presentation is informal, we hope that careful reading of the paper will provide them with an important tool to understand and improve their work, in an autonomous manner. Emphasis is put on the model-based approach, although much of the material applies to ad-hoc filtering. The basic structure consists of modelling the series as a linear stochastic process, and estimating the components by means of"signal extraction", i.e., by optimal estimation ofwell-defined components.
In this monograph, first, we analyze in detail some of the major limitations of the standard proc... more In this monograph, first, we analyze in detail some of the major limitations of the standard procedure to estimate business cycles with the Hodrick-Prescott (HP) filter. By incorporating time series analysis techniques, it is seen how some intuitive and relatively simple modifications to ...
Time series segmentation has many applications in several disciplines as neurology, cardiology, s... more Time series segmentation has many applications in several disciplines as neurology, cardiology, speech, geology and others. Many time series in this fields do not behave as stationary and the usual transformations to linearity cannot be used. This paper describes and evaluates different methods for segmenting non-stationary time series.
Present practice in applied time series work, mostly at economic policy or data producing agencie... more Present practice in applied time series work, mostly at economic policy or data producing agencies, relies heavily on using moving average lters to estimate unobserved components (or signals) in time series, such as the seasonally adjusted series, the trend, or the cycle. The purpose of the present paper is to provide an informal introduction to the time series analysis tools and concepts required by the user or analyst to understand the basic methodology behind the application of lters. The paper is aimed at economists, statisticians, and analysts in general, that do applied work in the eld, but have n o t h a d a n a d v anced course in applied time series analysis. Although the presentation is informal, we hope that careful reading of the paper will provide them with an important tool to understand and improve their work, in an autonomous manner. Emphasis is put on the model-based approach, although much of the material applies to ad-hoc ltering. The basic structure consists of modelling the series as a linear stochastic process, and estimating the components by means of "signal extraction", i.e., by optimal estimation of well-de ned components.
for their secretarial and research assistance. Documentos de Trabajo. N.º 0417 2004 The Working P... more for their secretarial and research assistance. Documentos de Trabajo. N.º 0417 2004 The Working Paper Series seeks to disseminate original research in economics and finance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the INTERNET at the following website:
Thanks are due to David Findley and William Bell for their helpful comments, and to Nieves Morale... more Thanks are due to David Findley and William Bell for their helpful comments, and to Nieves Morales and Jorge Carrillo for their secretarial and research assistance. 1 Filters used to estimate unobserved components in time series are often designed on a priori grounds, so as to capture the frequencies associated with the component. A limitation of these filters is that they may yield spurious results. The danger can be avoided if the so-called ARIMA-model-based (AMB) procedure is used to derive the filter. However, parsimony of ARIMA models typically implies little resolution in terms of the detection of hidden components. It would be desirable to combine a higher resolution with consistency with the structure of the observed series. We show first that for a large class of a priori designed filters, an AMB interpretation is always possible. Using this result, proper convolution of AMB filters can produce richer decompositions of the series that incorporate a priori
Time series segmentation has many applications in several disciplines as neurology, cardiology, s... more Time series segmentation has many applications in several disciplines as neurology, cardiology, speech, geology and others. Many time series in this fields do not behave as stationary and the usual transformations to linearity cannot be used. This paper describes and evaluates different methods for segmenting non-stationary time series. We propose a modification of the algorithm in Lee et al. (2003) which is designed to searching for a unique change in the parameters of a time series, in order to find more than one change using an iterative procedure. We evaluate the performance of three approaches for segmenting time series: AutoSLEX (Ombao et al., 2002), AutoPARM (Davis et al., 2006) and the iterative cusum method mentioned above and referred as ICM. The evaluation of each methodology consists of two steps. First, we compute how many times each procedure fails in segmenting stationary processes properly. Second, we analyze the effect of different change patterns by counting how ma...
We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first,... more We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first, the well-known criticism of spurious results due to the ad-hoc character of the filter, and the (often ignored yet important) limitation implied by revisions, which produce imprecision in the cycle estimator for recent periods. We show how the integration of some relatively simple ARIMA-model-based (AMB) techniques with HP filtering can produce important improvements in the performance of the cyclical signal. Finally, the complete procedure of applying the HP filter to a "clean" series is presented within a model-based methodology. This AMB methodology displays several nice features. First, it incorporates automatically optimal treatment of end points and provides a cleaner cyclical signal. Second, it provides an internally consistent full decomposition of the series into "trend + cycle + seasonal irregular" components, where the trend plus cycle aggregate into the s...
In this monograph, first, we analyze in detail some of the major limitations of the standard proc... more In this monograph, first, we analyze in detail some of the major limitations of the standard procedure to estimate business cycles with the Hodrick-Prescott (HP) filter. By incorporating time series analysis techniques, it is seen how some intuitive and relatively simple modifications to the filter can improve significantly its performance, in particular in terms of cleanness of the signal, smaller revision, stability of end-period estimators, and detection of turning points. Then, we show how the modified filter can be seen as the exact solution of a well-defined statistical problem, namely, optimal (minimum mean squared error) estimation of components in a standard unobserved-component model, where the observed series is decomposed into a trend, a cycle, a seasonal, and an irregular component.
The paper details an application of programs TRAMO and SEATS to seasonal adjustment and trend-cyc... more The paper details an application of programs TRAMO and SEATS to seasonal adjustment and trend-cycle estimation of the German Retail Trade Turnover series. When adjusting with X12-ARIMA, the Bundesbank identified two problems: heteroscedasticity in the seasonal component, associated with different moving patterns for some of the months, and unstability of the trend-cycle at the end of the series. It is seen how, starting with the fully automatic procedure and adding some simple modifications, TRAMO and SEATS deal properly with both problems, and provide good, stable, and robust results.
The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for busin... more The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for business-cycle estimation at many economic agencies and institutions. We show that the filter can be obtained from MMSE estimation of the components in an unobserved component model, where the original series is decomposed into (long-term) trend, cyclical, seasonal, and (highlytransitory) irregular components. The component models are sensible and combine desirable “ad-hoc” features with series-dependent features that guarantee consistency with the data. The model-based framework provides improvements having to do with the precision of end-point estimation and the stability of the cyclical signal.
In the analysis of time series, it is frequent to classify perturbations as Additive Outliers (AO... more In the analysis of time series, it is frequent to classify perturbations as Additive Outliers (AO), Innovative Outliers (IO), Level Shift (LS) outliers or Transitory Change (TC) outliers. In this paper, a new outlier type, the Seasonal Level Shift (SLS), is introduced in order to complete the usual classification.
The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for busin... more The Hodrick-Prescott filter applied to seasonally adjusted series has become a paradigm for business-cycle estimation at many economic agencies and institutions. We show that the filter can be obtained from MMSE estimation of the components in an unobserved component model, where the original series is decomposed into (long-term) trend, cyclical, seasonal, and (highlytransitory) irregular components. The component models are sensible and combine desirable "ad-hoc" features with series-dependent features that guarantee consistency with the data. The model-based framework provides improvements having to do with the precision of end-point estimation and the stability of the cyclical signal.
In the framework of decomposing a time series into the sum of signal components plus noise as in ... more In the framework of decomposing a time series into the sum of signal components plus noise as in detrending or seasonal adjustment, we analyze the situation in which the unobserved components may be subject to the influence of sudden shifts. The kind of perturbation that such shifts cause on the observed series can be classified as an outlier, when the shift affects the noise component, or as a structural change, when the shift affects one of the signal components. The consequences of ignoring these perturbations are important for model specification, parameter estimation and forecasting. We extend and modify the iterative procedure of Chen and Liu (1993) to allow the location, classification and estimation of outliers and structural changes affecting the unobserved components of a time series.
In the framework of decomposing a time series into the sum of signal components plus noise as in ... more In the framework of decomposing a time series into the sum of signal components plus noise as in detrending or seasonal adjustment, we analyze the situation in which the unobserved components may be subject to the influence of sudden shifts. The kind of perturbation that such shifts cause on the observed series can be classified as an outlier, when the shift affects the noise component, or as a structural change, when the shift affects one of the signal components. The consequences of ignoring these perturbations are important for model specification, parameter estimation and forecasting. We extend and modify the iterative procedure of Chen and Liu (1993) to allow the location, classification and estimation of outliers and structural changes affecting the unobserved components of a time series.
We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first,... more We consider business cycle estimation with Hodrick-Prescott (HP)-type filters. We address, first, the well-known criticism of spurious results due to the ad-hoc character of the filter, and the (often ignored yet important) limitation implied by revisions, which produce imprecision in the cycle estimator for recent periods. We show how the integration of some relatively simple ARIMA-model-based (AMB) techniques with HP filtering can produce important improvements in the performance of the cyclical signal.
Present practice in applied time series work, mostly at economic policy or data producing agencie... more Present practice in applied time series work, mostly at economic policy or data producing agencies, relies heavily on using moving average filters to estimate unobserved components (or signals) in time series, such as the seasonally adjusted series, the trend, or the cycle. The purpose of the present paper is to provide an informal introduction to the time series analysis tools and concepts required by the user or analyst to understand the basic methodology behind the application of filters. The paper is aimed at economists, statisticians, and analysts in general, that do applied work in the field, but have not had an advanced course in applied time series analysis. Although the presentation is informal, we hope that careful reading of the paper will provide them with an important tool to understand and improve their work, in an autonomous manner. Emphasis is put on the model-based approach, although much of the material applies to ad-hoc filtering. The basic structure consists of modelling the series as a linear stochastic process, and estimating the components by means of"signal extraction", i.e., by optimal estimation ofwell-defined components.
In this monograph, first, we analyze in detail some of the major limitations of the standard proc... more In this monograph, first, we analyze in detail some of the major limitations of the standard procedure to estimate business cycles with the Hodrick-Prescott (HP) filter. By incorporating time series analysis techniques, it is seen how some intuitive and relatively simple modifications to ...
Time series segmentation has many applications in several disciplines as neurology, cardiology, s... more Time series segmentation has many applications in several disciplines as neurology, cardiology, speech, geology and others. Many time series in this fields do not behave as stationary and the usual transformations to linearity cannot be used. This paper describes and evaluates different methods for segmenting non-stationary time series.
Present practice in applied time series work, mostly at economic policy or data producing agencie... more Present practice in applied time series work, mostly at economic policy or data producing agencies, relies heavily on using moving average lters to estimate unobserved components (or signals) in time series, such as the seasonally adjusted series, the trend, or the cycle. The purpose of the present paper is to provide an informal introduction to the time series analysis tools and concepts required by the user or analyst to understand the basic methodology behind the application of lters. The paper is aimed at economists, statisticians, and analysts in general, that do applied work in the eld, but have n o t h a d a n a d v anced course in applied time series analysis. Although the presentation is informal, we hope that careful reading of the paper will provide them with an important tool to understand and improve their work, in an autonomous manner. Emphasis is put on the model-based approach, although much of the material applies to ad-hoc ltering. The basic structure consists of modelling the series as a linear stochastic process, and estimating the components by means of "signal extraction", i.e., by optimal estimation of well-de ned components.
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