【正文】 ? It is still of concern in the context of simultaneous systems whether the CLRM assumptions are supported by the data. ? If the disturbances in the structural equations are autocorrelated, the 2SLS estimator is not even consistent. ? The standard error estimates also need to be modified pared with their OLS counterparts, but once this has been done, we can use the usual t and Ftests to test hypotheses about the structural form coefficients. Estimation of Systems Using TwoStage Least Squares 169。 Chris Brooks 2021, 陳磊 2021 661 Example: Say equations (14)(16) are required. Stage 1: ? Estimate the reduced form equations (17)(19) individually by OLS and obtain the fitted values, . Stage 2: ? Replace the RHS endogenous variables with their stage 1 estimated values: (24)(26) ? Now and will not be correlated with u1, will not be correlated with u2 , and will not be correlated with u3 . Estimation of Systems Using TwoStage Least Squares ?, ? , ?Y Y Y1 2 3Y Y Y X X uY Y X uY Y u1 0 1 2 3 3 4 1 5 2 12 0 1 3 2 1 23 0 1 2 3? ? ? ? ? ?? ? ? ?? ? ?? ? ? ? ?? ? ?? ?? ????Y2?Y3?Y3?Y2169。 Chris Brooks 2021, 陳磊 2021 658 ? Equation 23: Contains both Y1 and Y2。 前定變量的通常定義：包括外生變量和滯后的內(nèi)生變量 169。 Example of the order condition 169。 What Determines whether an Equation is Identified or not? 169。 More than one set of structural coefficients could be obtained from the reduced form. What Determines whether an Equation is Identified or not? 169。 we cannot get the structural coefficients from the reduced form estimates 2. An equation is exactly identified what we wanted were the original parameters in the structural equations ?, ?, ?, ?, ?, ?. 3 Avoiding Simultaneous Equations Bias P T S? ? ? ?? ? ? ?10 11 12 1Q T S? ? ? ?? ? ? ?20 21 22 2169。(?1111??????????????169。(39。 )? ??? ?? ???11E ( ?)? ??uXXXuXXXXXXXuXXXX39。? ? ?X X X y1 y X u? ??E E E X X X uX X E X u( ? ) ( ) (( 39。 Chris Brooks 2021, 陳磊 2021 644 ? Multiplying (7) through by ??, (9) ? (8) and (9) are the reduced form equations for P and Q. Obtaining the Reduced Form ? ?? ?? ? ? ?? ?? ?Q S u Q T v? ? ? ? ? ? ?( ) ( ) ( )? ? ?? ?? ?? ?? ? ?? ? ? ? ? ? ?Q T S u vQ T S u v? ?? ? ? ? ? ? ???? ??? ? ??? ? ??? ? ? ?? ?169。 Chris Brooks 2021, 陳磊 2021 640 Chapter 6 Multivariate models 169。 從而有 tqittiit uLuuLy )(1?? ??? ??qq LLLL ???? ????? ?2211)(0)( ?z?tittiitt uyyLayLayL ???? ????11 )()(??????1ittiit uyLcy課件 530 ? By bining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model: where and or with 6 ARMA Processes ? ? ? ?( ) . . .L L L Lp p? ? ? ? ?1 1 2 2qq LLLL ???? ????? . ..1)( 221tt uLyL )()( ??? ??tqtqttptpttt uuuuyyyy ?????????? ?????? ??????? . ... .. 22112211stuuEuEuE sttt ???? ,0)(。 ? The condition for stationarity of a general AR(p) model is that the roots of 特征方程 all lie outside the unit circle. ? Example 1: Is yt = yt1 + ut stationary? The characteristic root is 1, so it is a unit root process (so nonstationary) ? Example 2: p241 ? A stationary AR(p) model is required for it to have an MA(?) representation. The Stationary Condition for an AR Model 1 01 2 2? ? ? ? ?? ? ?z z zp p. . .課件 517 ? States that any stationary series can be deposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part, which will be an MA(?). ? For the AR(p) model, , ignoring the intercept, the Wold deposition is where, 可以證明 ， 算子多項(xiàng)式 R(L)的集合與代數(shù)多項(xiàng)式 R(z)的集合是同結(jié)構(gòu)的 ， 因此可以對算子 L做加 、 減 、 乘和比率運(yùn)算 。 – 與結(jié)構(gòu)模型不同；通常不依賴于經(jīng)濟(jì)和金融理論 – 用于描述被觀測數(shù)據(jù)的經(jīng)驗(yàn)性相關(guān)特征 ? ARIMA（ AutoRegressive Integrated Moving Average）是一類重要的時(shí)間序列模型 – BoxJenkins 1976 ? 當(dāng)結(jié)構(gòu)模型不適用時(shí)，時(shí)間序列模型卻很有用 – 如引起因變量變化的因素中包含不可觀測因素，解釋變量等觀測頻率較低。結(jié)構(gòu)模型常常不適用于進(jìn)行預(yù)測 ? 本章主要解決兩個(gè)問題 – 一個(gè)給定參數(shù)的時(shí)間序列模型，其變動(dòng)特征是什么？ – 給定一組具有確定性特征的數(shù)據(jù)，描述它們的合適模型是什么？ 課件 53 ? 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. ? t1 , t2 2 Some Notation and Concepts P y b y b P y b y bt t n t m t m nn n{ , . . . , } { , . . . , }1 11 1? ? ? ? ?? ?E y yt t t t( )( )1 2 2 1? ? ? ?? ? ?E y yt t( )( )? ? ? ? ?? ? ? 2課件 54 ? So if the process is covariance stationary, all the variances are the same and all the covariances depend on the difference between t1 and t2. 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?0E y E y y E yt t t s t s s( ( ))( ( ))? ? ?? ? ?課件 55 ? A white noise process is one with no discernible structure. ? Thus the autocorrelation function