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A Note on a Matrix Inversion in Multiple Exponential Smoothing

多重指数平滑中的矩阵求逆札记

Summary In discussing the equivalence of certain polynomial predictors, Godolphin and Harrison (1975) note a matrix U employed in Multiple Exponential Smoothing whose inverse has not been obtained explicitly. The inverse of a related matrix p is obtained here and the relationship between the two matrices U and p reveals why no inverse has been obtained in general for U.

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Linear Prediction by Autoregressive Model Fitting in the Time Domain

通过时域自回归模型拟合进行线性预测

Let $\{x_t\}$ be a purely nondeterministic stationary process satisfying all the assumptions made by Berk (1974), and $\{y_t\}$ be another purely nondeterministic stationary process. Assume that $y_t$ is independent of $x_t$ but has exactly the same statistical properties as that of $x_t$. Consider the linear prediction of future values of $y_t$ on the basis of past values, using prediction constants estimated from a realisation of $T$ observations of $x_t$ by least-squares…

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A General Computer for Econometric Models--Shazam

计量经济模型的通用计算程序:SHAZAM

to prepare and allow a large number of options. The program can be run in batch mode or interactively at a computer terminal. Computer core storage is dynamically allocated so that large problems are only limited by the size of the machine. SHAZAM is designed to grow so that new algorithms and procedures can easily be added by any programmer familiar with the internal structure of the program. Features of SHAZAM include ordinary least squares, two-stage least squares,…

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The Estimation of the Prediction Error Variance

预测误差方差的估计

Spectral methods are used to construct an estimate of the variance of the prediction error for a normal, stationary process. The estimate obtained is shown to be strongly consistent and asymptotically normally distributed. Some aspects of the computations with respect to the fast Fourier transform are considered. The latter half of the article consists of a number of simulations, based on both generated and real data, which illustrate the results obtained. The relation…

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Analysis of Autoregressive-Moving Average Models: Estimation and Prediction

自回归移动平均模型分析:估计与预测

The exact likelihood function from a set of observations generated by univariate stationary normal autoregressive–moving average models is derived. Convenient methods are developed to evaluate the exact likelihood function, specifically the inverse and determinant of the T × T covariance matrix of the observations, ΣT and the quadratic form, Z′ΣT-1Z, where Z is a T × 1 vector of observations. Computations of the least squares and the maximum likelihood estimates of the…

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Asymptotic Properties of the Wiener–Kolmogorov Predictor. Ii

Wiener–Kolmogorov 预测器的渐近性质(二)

Summary The asymptotic properties of the autoregressive estimates a (T)(u) obtained using the fles method (Bhansali, 1974) suggest that, in practice, a decision on truncating the a (T)(u) at u = p, say, will need to be taken. This truncated sequence of the auto-regressive estimates is used to form alternative estimates of the linear predictor, the autoregressive spectral density function and the innovation variance. Some of the asymptotic properties of these estimates are…

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Sequential Procedures for Detecting Parameter Changes in a Time-Series Model

检测时间序列模型参数变化的序贯程序

Procedures are proposed for monitoring forecast errors in order to detect changes in a time-series model. These procedures are based on likelihood ratio statistics which consist of cumulative sums. An extension of Page's method is presented which tests for changes in the parameter values of autoregressive integrated moving average (arima) models. The distributional properties of the statistics are approximated under the assumption that the series follows an integrated…

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Prediction Functions and Mean-Estimation Functions for a Time Series

时间序列的预测函数与均值估计函数

Let $T \subseteqq I$ be sets of real numbers. Let $\{Y(t): t \in I\}$ be a real time series whose mean is an unknown element of a known class of functions on $I$ and whose covariance kernel $k$ is assumed known. For each fixed $s \in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t \in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given a set of observations $\{Y(t) = g(t): t \in T\}$, then the function…

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Autoregressive Integrated Moving Average (ARIMA) Models for Birth Forecasting

用于出生人数预测的自回归单整移动平均模型

Autoregressive integrated moving average (ARIMA) models are developed for birth time series, and their relationship with classical models for population growth is investigated. Parsimonious versions for the ARIMA models are obtained which retain the most important pieces of information including the length of generation of the population. The technique is applied to birth time series data for Norway.

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Analysis of autoregressive-moving average models: Estimation and prediction

自回归移动平均模型分析:估计与预测

The exact likelihood function from a set of observations generated by univariate stationary normal autoregressive–moving average models is derived. Convenient methods are developed to evaluate the exact likelihood function, specifically the inverse and determinant of the T × T covariance matrix of the observations, ΣT and the quadratic form, Z′ΣT-1Z, where Z is a T × 1 vector of observations. Computations of the least squares and the maximum likelihood estimates of the…

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Current- and Constant-Dollar Input-Output Forecasts for the U.S. Economy

美国经济的现价与不变价投入产出预测

Utilizing recently deflated and reconciled U.S. Department of Commerce input-output data from 1947, 1958, 1961, 1963, and 1966, this paper presents an empirical test of the accuracy of input-output forecasts in current dollars and in constant dollars. While the results vary considerably among industries, in general constant-dollar forecasts are more accurate in the long run while in the short run the results are mixed. The implications of these findings for input-output…

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Elements of Spatial Structure: A Quantitative Approach.

空间结构要素:一种定量方法

1. Introduction Part I. Static Aspects of Regional Structure: 2. Regions as combinatorial structures 3. Regions as ordered series 4. Regions as surfaces Part II. Dynamic Aspects of Regional Structure: 5. Spatial comparison of time series: a framework 6. Spatial comparison of time series: I. Contagious processes 7. Spatial comparison of time series: II. Unemployment in South-West England Part III. Autocorrelation and Forecasting: 8. Spatial autocorrelation 9. The analysis of…

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An Econometric Analysis of Wilderness Area Use

荒野地区游憩使用量的计量经济分析

Although outdoor recreational use fluctuates daily, this time-varying feature has received little attention by economists and statisticians. In this paper, wilderness recreational use relationships are estimated, based on daily observations over the length of a given season, by (i) traditional econometric techniques and via (ii) time series analysis procedures promoted by Box and Jenkins [3]. In applying both sets of estimated relationships for forecasting purposes, the…

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An Econometric Analysis of Wilderness Area Use

荒野地区游憩使用的计量经济分析

Although outdoor recreational use fluctuates daily, this time-varying feature has received little attention by economists and statisticians. In this paper, wilderness recreational use relationships are estimated, based on daily observations over the length of a given season, by (i) traditional econometric techniques and via (ii) time series analysis procedures promoted by Box and Jenkins [3]. In applying both sets of estimated relationships for forecasting purposes, the…

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Rational Expectations and Distributed Lag Expectations Proxies

理性预期与分布滞后预期代理

exogenous variables which actually generate the variable to be predicted. Empirical researchers, however, have employed a modified (and weaker) form of Muth's hypothesis. Expectations are defined as rational if they fully incorporate all of the information contained in a specified information set that is available at the time that the forecast is made.3 When using distributed lag proxies, researchers implicitly assume that the information set used to generate forecasts…

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Autoregressive and Nonautoregressive Elements in Cross-Section Forecasts of Inflation

通胀横截面预测中的自回归与非自回归成分

Using cross-section data collected from a panel of institutional investors in 1969, 1970, 1972, this paper focuses on how knowledgeable individuals formulate forecasts of future rates of price inflation, We estimate return-to-normality and error-learning forecasting models and inquire whether such equations can be interpreted simply as reduced forms of an autoregressive forecasting structure. Assuming that respondents' expectations were formed rationally in the sense of…

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An Experimental Study of Expectation Formation

预期形成的实验研究

This paper reports on an experimental study of expectation formation-and revision in a time series context. In an adaptive expectations framework, it is shown that the speed of adjustment seems to fall in turning point periods. Expectations are considered as probability density functions, and a scoring system is devised and employed that gives subjects an incentive to report a measure of the dispersion of these functions. This measure, which is inversely related to the…

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Effect of temporal aggregation on the dynamic relationship of two time series variables

时间聚合对两个时间序列变量动态关系的影响

This paper considers the effect of temporal aggregation on the dynamic relationships between two discrete time series variables. Given the model in terms of some basic time unit, the corresponding model for the aggregates is obtained. Examples are given showing that temporal aggregation can lead to a substantial loss in parameter estimation while the loss in prediction efficiency is much less severe.

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Statistics for Management and Economics.

管理与经济学统计

What is Statistics? Describing Sets of Measurements Probability Random Variables and Probability Distributions Three Useful Discrete Probability Distributions The Normal and Other Continuous Probability Distributions Sampling and Sampling Distributions Large sample Estimation Large Sample Tests of Hypotheses Inferences from Small Samples The Analysis of Variance Linear Regression and Correlation Multiple Linear Regression Elements of Time Series Analysis Forecasting Models…

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A direct basic form for predictors of autoregressive integrated moving average processes

ARIMA过程预测量的一种直接基本形式

A direct method of formulating predictors of autoregressive integrated moving average processes is given which has certain advantages over the three basic forms proposed by Box & Jenkins (1970). Some ways of making use of this representation are discussed.

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A Direct Basic Form for Predictors of Autoregressive Integrated Moving Average Processes

ARIMA过程预测器的一种直接基本形式

A direct method of formulating predictors of autoregressive integrated moving average processes is given which has certain advantages over the three basic forms proposed by Box & Jenkins (1970). Some ways of making use of this representation are discussed.

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