Scenarios, Uncertainty and Conditional Forecasts of the World Population

Current official population forecasts differ little from those that Whelpton made 50 years ago either in the cohort–component methodology used or in the arguments used to motivate the assumptions. However, Whelpton produced some of the most erroneous forecasts of this century. This suggests that current forecasters should ensure that they give users an assessment of the uncertainty of their forecasts. We show how simple statistical methods can be combined with expert judgment to arrive at an overall predictive distribution for the future population. We apply the methods to a world population forecast that was made in 1994. Accepting that point forecast, we find that the probability is only about 2% that the world population in the year 2030 will be less than the low scenario of 8317 million. The probability that the world population will exceed the high scenario of 10 736 million is about 13%. Similarly, the probability is only about 51% that the high–low interval of a recent United Nations (UN) forecast will contain the true population in the year 2025. Even if we consider the UN high–low intervals as conditional on the possible future policies of its member states, they appear to have a relatively small probability of encompassing the future population.     

Stochastic population forecasts for the United States: beyond high,medium, and low

"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."     

On future household structure

Summary.  We develop a method for computing probabilistic household forecasts which quantifies uncertainty in the future number of households of various types in a country. A probabilistic household forecast helps policy makers, planners and other forecast users in the fields of housing, energy, social security etc. in taking appropriate decisions, because some household variables are more uncertain than others. Deterministic forecasts traditionally do not quantify uncertainty. We apply the method to data from Norway. We find that predictions of future numbers of married couples, cohabiting couples and one-person households are more certain than those of lone parents and other private households. Our method builds on an existing method for computing probabilistic population forecasts, combining such a forecast with a random breakdown of the population according to household position (single, cohabiting, living with a spouse, living alone etc.). In this application, uncertainty in the total numbers of households of different types derives primarily from random shares, rather than uncertain future population size. A similar method could be applied to obtain probabilistic forecasts for other divisions of the population, such as household size, health or disability status, region of residence and labour market status.     

北京市人口老龄化系数预测

本文利用生命表理论、联合国平均预期寿命增长模型及年龄移算法,预测北京市老年人口峰值期在2037年。根据北京市1978-2010年户籍人口数据,建立预测值满足老年人口峰值期的向量自回归模型。在设定市际净迁移人数及低、中、高三种总和生育率方案下,利用模型预测北京市2011-2050年的户籍总人数、老年人数以及老龄化系数。预测结果表明：在低、中、高三种方案下,总人数分别在2023年、2024年和2025年达到峰值,峰值分别为1377万人、1389万人和1397万人,老年人数均在2037年达到峰值242万人,在预测期内老龄化系数一直呈上升趋势,至2050年依次为21.25%、20.25%和19.82%。最后,针对预测结果提出了应对北京市人口老龄化的建议。     

Uncertain population forecasting

"Errors in population forecasts arise from errors in the jump-off population and errors in the predictions of future vital rates. The propagation of these errors through the linear (Leslie) growth model is studied, and prediction intervals for future population are developed. For U.S. national forecasts, the prediction intervals are compared with the U.S. Census Bureau's high-low intervals." In order to assess the accuracy of the predictions of vital rates, the authors "derive the predictions from a parametric statistical model and estimate the extent of model misspecification and errors in parameter estimates. Subjective, expert opinion, so important in real forecasting, is incorporated with the technique of mixed estimation. A robust regression model is used to assess the effects of model misspecification."     

When and how much do average forecasts improve predictive accuracy

A composite forecast combines two or more individual forecasts into a single estimate by way of a number of different averaging schemes. The easiest way to combine forecasts is through using a simple average. In this paper, the authors show that in many instances the simple average of individual forecasts approximates the optimal combining scheme. Results are expressed in terms of the probability that a composite forecast will improve upon an individual forecast.     

Functional time series approach for forecasting very short-term electricity demand

This empirical paper presents a number of functional modelling and forecasting methods for predicting very short-term (such as minute-by-minute) electricity demand. The proposed functional methods slice a seasonal univariate time series (TS) into a TS of curves; reduce the dimensionality of curves by applying functional principal component analysis before using a univariate TS forecasting method and regression techniques. As data points in the daily electricity demand are sequentially observed, a forecast updating method can greatly improve the accuracy of point forecasts. Moreover, we present a non-parametric bootstrap approach to construct and update prediction intervals, and compare the point and interval forecast accuracy with some naive benchmark methods. The proposed methods are illustrated by the half-hourly electricity demand from Monday to Sunday in South Australia.     

Stochastic population models for the analysis of the effects of demographic processes on social security systems

The aim of this paper is to describe several methods for quantifying the amount of uncertainty inherent in population forecasts used to assess the impact of demographic processes on social security systems. Each method is briefly outlined, and its advantages and disadvantages are discussed. The primary emphasis is on stochastic population models, and the geographic focus is on the Federal Republic of Germany.     

Estimation Bias in the First-Order Autoregressive Model and Its Impact on Predictions and Prediction Intervals

The least squares estimate of the autoregressive coefficient in the AR(1) model is known to be biased towards zero, especially for parameters close to the stationarity boundary. Several methods for correcting the autoregressive parameter estimate for the bias have been suggested. Using simulations, we study the bias and the mean square error of the least squares estimate and the bias-corrections proposed by Kendall and Quenouille.

We also study the mean square forecast error and the coverage of the 95% prediction interval when using the biased least squares estimate or one of its bias-corrected versions. We find that the estimation bias matters little for point forecasts, but that it affects the coverage of the prediction intervals. Prediction intervals for forecasts more than one step ahead, when calculated with the biased least squares estimate, are too narrow.     

Exact joint forecast regions for vector autoregressive models

Assume that a k-element vector time series follows a vector autoregressive (VAR) model. Obtaining simultaneous forecasts of the k elements of the vector time series is an important problem. Based on the Bonferroni inequality, Lutkepohl (1991) derived the procedures which construct the conservative joint forecast regions for the VAR model. In this paper, we propose to use an exact method which provides shorter prediction intervals than does the Bonferroni method. Three illustrative examples are given for comparison of the various VAR forecasting procedures.     

Improved multivariate prediction regions for Markov process models

This paper concerns the specification of multivariate prediction regions which may be useful in time series applications whenever we aim at considering not just one single forecast but a group of consecutive forecasts. We review a general result on improved multivariate prediction and we use it in order to calculate conditional prediction intervals for Markov process models so that the associated coverage probability turns out to be close to the target value. This improved solution is asymptotically superior to the estimative one, which is simpler but it may lead to unreliable predictive conclusions. An application to general autoregressive models is presented, focusing in particular on AR and ARCH models.     

LOCAL LINEAR FORECASTS USING CUBIC SMOOTHING SPLINES

This paper shows how cubic smoothing splines fitted to univariate time series data can be used to obtain local linear forecasts. The approach is based on a stochastic state‐space model which allows the use of likelihoods for estimating the smoothing parameter, and which enables easy construction of prediction intervals. The paper shows that the model is a special case of an ARIMA(0, 2, 2) model; it provides a simple upper bound for the smoothing parameter to ensure an invertible model; and it demonstrates that the spline model is not a special case of Holt's local linear trend method. The paper compares the spline forecasts with Holt's forecasts and those obtained from the full ARIMA(0, 2, 2) model, showing that the restricted parameter space does not impair forecast performance. The advantage of this approach over a full ARIMA(0, 2, 2) model is that it gives a smooth trend estimate as well as a linear forecast function.     

Robust bootstrap densities for dynamic conditional correlations: implications for portfolio selection and Value-at-Risk

ABSTRACT

Many financial decisions such as portfolio allocation, risk management, option pricing and hedge strategies are based on the forecast of the conditional variances, covariances and correlations of financial returns. Although the decisions depend on the forecasts covariance matrix little is known about effects of outliers on the uncertainty associated with these forecasts. In this paper we analyse these effects on the context of dynamic conditional correlation models when the uncertainty is measured using bootstrap methods. We also propose a bootstrap procedure to obtain forecast densities for return, volatilities, conditional correlation and Value-at-Risk that is robust to outliers. The results are illustrated with simulated and real data.     

Predictive Accuracy Gain From Disaggregate Sampling in ARIMA Models

We compare the forecast accuracy of autoregressive integrated moving average (ARIMA) models based on data observed with high and low frequency, respectively. We discuss how, for instance, a quarterly model can be used to predict one quarter ahead even if only annual data are available, and we compare the variance of the prediction error in this case with the variance if quarterly observations were indeed available. Results on the expected information gain are presented for a number of ARIMA models including models that describe the seasonally adjusted gross national product (GNP) series in the Netherlands. Disaggregation from annual to quarterly GNP data has reduced the variance of short-run forecast errors considerably, but further disaggregation from quarterly to monthly data is found to hardly improve the accuracy of monthly forecasts.     

The relationship between the length of the base period and population forecast errors

"The base period of a population forecast is the time period from which historical data are collected for the purpose of forecasting future population values. The length of the base period is one of the fundamental decisions made in preparing population forecasts, yet very few studies have investigated the effects of this decision on population forecast errors. In this article the relationship between the length of the base period and population forecast errors is analyzed, using three simple forecasting techniques and data from 1900 to 1980 for states in the United States. It is found that increasing the length of the base period up to 10 years improves forecast accuracy, but that further increases generally have little additional effect. The only exception to this finding is long-range forecasts of rapidly growing states, in which a longer base period substantially improves forecast accuracy for two of the forecasting techniques."     

Assessing Macro Uncertainty in Real-Time When Data Are Subject To Revision

Model-based estimates of future uncertainty are generally based on the in-sample fit of the model, as when Box–Jenkins prediction intervals are calculated. However, this approach will generate biased uncertainty estimates in real time when there are data revisions. A simple remedy is suggested, and used to generate more accurate prediction intervals for 25 macroeconomic variables, in line with the theory. A simulation study based on an empirically estimated model of data revisions for U.S. output growth is used to investigate small-sample properties.     

Improved autoregressive forecasts in the presence of non-normal errors

This paper is concerned with obtaining more accurate point forecasts in the presence of non-normal errors. Specifically, we apply the residual augmented least-squares (RALS) estimator to autoregressive models to utilize the additional moment restrictions embodied in non-normal errors. Monte Carlo experiments are performed to compare our RALS forecasts to forecasts based on the ordinary least-squares estimator and the least absolute deviations (LAD) estimator. We find that the RALS approach provides superior forecasts when the data are skewed. Compared to the LAD forecast, the RALS forecast has smaller mean squared prediction errors in the baseline case with normal errors.     

Dynamic financial distress prediction based on Kalman filtering

In models for predicting financial distress, ranging from traditional statistical models to artificial intelligence models, scholars have primarily paid attention to improving predictive accuracy as well as the progressivism and intellectualization of the prognostic methods. However, the extant models use static or short-term data rather than time-series data to draw inferences on future financial distress. If financial distress occurs at the end of a progressive process, then omitting time series of historical financial ratios from the analysis ignores the cumulative effect of previous financial ratios on the current consequences. This study incorporated the cumulative characteristics of financial distress by using the characteristics of a state space model that is able to perform long-term forecasts to dynamically predict an enterprise's financial distress. Kalman filtering is used to estimate the model parameters. Thus, the model constructed in this paper is a dynamic financial prediction model that has the benefit of forecasting over the long term. Additionally, current data are used to forecast the future annual financial position and to judge whether the establishment will be in financial distress.     

Some Problems in Forecasting Population for Government Planning Purposes

Some governments rely on centralized, official sets of population forecasts for planning capital facilities. But the nature of population forecasting, as well as the milieu of government forecasting in general, can lead to the creation of extrapolative forecasts not well suited to long-range planning. This report discusses these matters, and suggests that custom-made forecasts and the use of forecast guidelines and a review process stressing forecast assumption justification may be a more realistic basis for planning individual facilities than general-purpose, official forecasts.     

Error models for official mortality forecasts

"The Office of the Actuary, U.S. Social Security Administration, produces alternative forecasts of mortality to reflect uncertainty about the future.... In this article we identify the components and assumptions of the official forecasts and approximate them by stochastic parametric models. We estimate parameters of the models from past data, derive statistical intervals for the forecasts, and compare them with the official high-low intervals. We use the models to evaluate the forecasts rather than to develop different predictions of the future. Analysis of data from 1972 to 1985 shows that the official intervals for mortality forecasts for males or females aged 45-70 have approximately a 95% chance of including the true mortality rate in any year. For other ages the chances are much less than 95%."