Econometric analysis Sample Selection参考.ppt

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Interpreting a Time Series Time domain: A “process” y(t) = ax(t) + by(t-1) + … Regression like approach/interpretation Frequency domain: A sum of terms y(t) = Contribution of different frequencies to the observed series. (“High frequency data and financial econometrics – “frequency” is used slightly differently here.) For example,… In parts… Studying the Frequency Domain Cannot identify the number of terms Cannot identify frequencies from the time series Deconstructing the variance, autocovariances and autocorrelations Contributions at different frequencies Apparent large weights at different frequencies Using Fourier transforms of the data Does this provide “new” information about the series? Autocorrelation in Regression Yt = b’xt + εt Cov(εt, εt-1) ≠ 0 Ex. RealConst = a + bRealIncome + εt U.S. Data, quarterly, 1950-2000 Autocorrelation How does it arise? What does it mean? Modeling approaches Classical – direct: “Corrective” Estimation that accounts for autocorrelation Inference in the presence of autocorrelation Contemporary – structural Model the source Incorporate the time series aspect in the model Stationary Time Series zt = b1yt-1 + b2yt-2 + … + bPyt-P + et Autocovariance: γk = Cov[yt,yt-k] Autocorrelation: ?k = γk / γ0 Stationary series: γk depends only on k, not on t Weak stationarity: E[yt] is not a function of t, E[yt * yt-s] is not a function of t or s, only of |t-s| Strong stationarity: The joint distribution of [yt,yt-1,…,yt-s] for any window of length s periods, is not a function of t or s. A condition for weak stationarity: The smallest root of the characteristic polynomial: 1 - b1z1 - b2z2 - … - bPzP = 0, is greater than one. The unit circle Complex roots Example: yt = ?yt-1 + ee, 1 - ?z = 0 has root z = 1/ ?, | z | 1 = | ? | 1. Stationary vs. Nonstationary Series The Lag Operator Lxt = xt-1 L2 xt = xt-2 LPxt + LQxt = xt-P + xt-Q