5-1 Chapter 5 Univariate time series modelling and forecasting
5-1 Chapter 5 Univariate time series modelling and forecasting
5-2 1 introduction ·单变量时间序列模型 只利用变量的过去信息和可能的误差项的当前和过去值来建模和预测 的一类模型(设定]。 与结构模型不同;通常不依赖于经济和金融理论 用于描述被观测数据的经验性相关特征 ARIMA(Auto Regressive Integrated Moving Average)ie- 类重要的时间序列模型 Box-enkins 1976 ·当结构模型不适用时,时间序列模型却很有用 如引起因变量变化的因素中包含不可观测因素,解释变量等观测频率 较低。结构模型常常不适用于进行预测 ·本章主要解决两个问题 个给定参数的时间序列模型,其变动特征是什么? 给定一组具有确定性特征的数据,描述它们的合适模型是什么?
5-2 1 introduction • 单变量时间序列模型 – 只利用变量的过去信息和可能的误差项的当前和过去值来建模和预测 的一类模型(设定)。 – 与结构模型不同；通常不依赖于经济和金融理论 – 用于描述被观测数据的经验性相关特征 • ARIMA（AutoRegressive Integrated Moving Average）是一 类重要的时间序列模型 – Box-Jenkins 1976 • 当结构模型不适用时，时间序列模型却很有用 – 如引起因变量变化的因素中包含不可观测因素，解释变量等观测频率 较低。结构模型常常不适用于进行预测 • 本章主要解决两个问题 – 一个给定参数的时间序列模型，其变动特征是什么？ – 给定一组具有确定性特征的数据，描述它们的合适模型是什么？
5-3 2 Some Notation and concepts a Strictly Stationary Process A strictly stationary process is one where Pmv1≤b,…,yn≤bhn}=P{y+m≤b,…,yn+m≤bn} For any t1,t2,…,tn∈Z,anym∈Z,n=1,2, A Weakly stationary process If a series satisfies the next three equations, it is said to be weakly or covariance stationary 1.E0y)=p,t=1,2,…, 2.E(v2-)(v2-)=a2<0 3.E(y21-A)(y2-A)=y2-Vt,z2
5-3 • A Strictly Stationary Process A strictly stationary process is one where • For any t1 ,t2 ,…, tn∈ Z, any m ∈ Z, n=1,2,… • A Weakly Stationary Process If a series satisfies the next three equations, it is said to be weakly or covariance stationary 1. E(yt ) = , t = 1,2,..., 2. 3. t 1 , t 2 2 Some Notation and Concepts P{yt b ,..., yt n bn } P{yt m b ,..., yt n m bn } 1 1 = 1+ 1 + E y y t t t t ( )( ) 1 2 2 1 − − = − E y y t t ( − )( − ) = 2
5-4 Some Notation and Concepts So if the process is covariance stationary, all the variances are the same and all the covariances depend on the difference between t, and t. The moments E(y-E(y)(y+s-E(y+s)=ys,s=0,1,2, are known as the covariance function The covariances, y are known as autocovariances However, the value of the autocovariances depend on the units of measurement of It is thus more convenient to use the autocorrelations which are the autocovariances normalised by dividing by the variance. ,s=0,1,2,… If we plot t against S=0, 1, 2,. then we obtain the autocorrelation function(acf or correlogram
5-4 • So if the process is covariance stationary, all the variances are the same and all the covariances depend on the difference between t 1 and t 2 . The moments , s = 0,1,2, ... are known as the covariance function. • The covariances, s , are known as autocovariances. • However, the value of the autocovariances depend on the units of measurement of yt . • It is thus more convenient to use the autocorrelations which are the autocovariances normalised by dividing by the variance: , s = 0,1,2, ... If we plot s against s=0,1,2,... then we obtain the autocorrelation function (acf) or correlogram. Some Notation and Concepts s s = 0 E y E y y E y t t t s t s s ( − ( ))( + − ( + )) =
5-5 A White noise process a white noise process is one with no discernible structure. E(y7)= Var(t otherwise Thus the autocorrelation function will be zero apart from a single peak of l ats=0. 如果假设服从标准正态分布,则。~ approximately n0,/n) We can use this to do significance tests for the autocorrelation coefficients by constructing a confidence interval a 95% confidence interval would be given by +196x If the sample autocorrelation coefficient, te, falls outside this region for any value of s, then we reject the null hypothesis that the true value of the coefficient at lag s is zero
5-5 • A white noise process is one with no discernible structure. • Thus the autocorrelation function will be zero apart from a single peak of 1 at s = 0. • 如果假设yt服从标准正态分布,则 approximately N(0,1/T) • We can use this to do significance tests for the autocorrelation coefficients by constructing a confidence interval. • a 95% confidence interval would be given by . • If the sample autocorrelation coefficient, , falls outside this region for any value of s, then we reject the null hypothesis that the true value of the coefficient at lag s is zero. A White Noise Process E y Var y if t r otherwise t t t r ( ) ( ) = = = = − 2 2 0 s T 1 1.96 s ˆ