Chapter 11 Ch.11 Time Series Data: Further Issues 1. Stationary & Weakly Dependent Time Time Series Data Series 2. Asymptotic Properties of OLSE y = + x + . .+ x + u 3. Using Highly Persistent Time Series in t 0 1 t1 k tk t Regression Analysis 4. Dynamically Complete Models & the 2. Further Issues Absence of Serial Correlation* 5. The Homoskedasticity Assumption for Time Series Models* Econometrics 1 Econometrics 2 11.1 Stationary & Weakly Dependent Covariance Stationary Process Stationary Stochastic Process A stochastic process is covariance stationary if A stochastic process is stationary if for every 1. E(xt) is constant, collection of time indices 1 ≤ t1 < …< tm, the 2. Var(xt) is constant and joint distribution of (xt1, …, xtm) is the same 3. for any t, h ≥ 1, Cov(xt, xt+h) depends only on h as that of (xt1+h, … xtm+h) for h ≥ 1. and not on t. Thus, stationarity implies that the xt’s are Thus, this weaker form of stationarity requires identically distributed and that the nature of any only that the mean and variance are constant correlation between adjacent terms is the same across time, and the covariance just depends on across all periods. the distance across time. Econometrics 3 Econometrics 4 Weakly Dependent Time Series MA(1) Process A stationary time series is weakly dependent A moving average process of order one if xt and xt+h are “almost independent” as h [MA(1)] can be characterized as one where increases. xt = et + 1et-1, t = 1, 2, … (11.1) If for a covariance stationary process Corr(x ,x ) t t+h with et being an i.i.d. sequence with mean 0 → 0 as h → ∞, we’ll say this covariance and variance 2. stationary process is asymptotically uncorrelated e This is a stationary, weakly dependent sequence (almost equivalently, weakly dependent). as variables 1 period apart are correlated, but 2 So, we can apply the Law of Large Numbers periods apart they are not. (LLN) & CLT to sample averages. Econometrics 5 Econometrics 6 Time Series Data 2: Further Issues 1 Chapter 11 AR(1) Process Trends Revisited An autoregressive process of order one A trending series [AR(1)] can be characterized as one where cannot be stationary, since the mean is changing yt = yt-1 + et , t = 1, 2,… (11.2) over time. with et being an i.i.d. sequence with mean 0 can be weakly dependent. and variance 2. e If a series is weakly dependent and is For this process to be weakly dependent, it must stationary about its trend, we will call it a be the case that || < 1. h trend-stationary process. Corr(yt,yt+h)=Cov(yt,yt+h)/(yy)=1 (11.4) which becomes small as h increases. As long as a trend is included, all is well. Econometrics 7 Econometrics 8 11.2 Asymptotic Properties of OLSE Large-Sample Inference Assumptions for Consistency Asymptotic Normality of OLSE 1. Linearity and Weak Dependence (TS.1’) 4. Weaker assumption of homoskedasticity: 2 2. A weaker zero conditional mean Var (ut|xt) = , for each t (TS4’) assumption: E(ut|xt) = 0, for each t (TS.2’) 5. Weaker assumption of no serial correlation: 3. No Perfect Collinearity (TS.3’) E(utus| xt, xs) = 0 for t s (TS5’) Thus, for asymptotic unbiasedness With these assumptions, we have (consistency), we can weaken the asymptotic normality and the usual standard exogeneity assumptions somewhat relative errors, t statistics, F statistics and LM to those for unbiasedness. statistics are valid. Econometrics 9 Econometrics 10 11.3 Highly Persistent Time Series Cont. Random Walks A random walk is an AR(1) model where A random walk is a special case of what’s 1 = 1, meaning the series is not weakly known as a unit root process. dependent. Note that trending and persistence are yt = yt-1 + et t = 1,2,… (11.20) different things – a series can be trending With a random walk, the expected value of yt is but weakly dependent, or a series can be always y0 – it doesn’t depend on t. highly persistent without any trend. 2 Var(yt) = e t, so it increases with t. A random walk with drift is an example of We say a random walk is highly persistent a highly persistent series that is trending. since E(yt+h|yt) = yt for all h ≥ 1. Econometrics 11 Econometrics 12 Time Series Data 2: Further Issues 2 Chapter 11 Transforming Persistent Series Cont. Transforming Persistent Series In order to use a highly persistent series In a random walk case, where {et} is i.i.d. and get meaningful estimates and make with zero mean and variance 2, correct inferences, we want to transform it yt = yt -yt-1 = et t = 2,3,… (11.24) into a weakly dependent process. If {et} is weakly dependent process, then {yt} We refer to a weakly dependent process as is also weakly dependent. being integrated of order zero, or I(0). To decide whether a time series is I(1) or A random walk is integrated of order one, not, statistical tests can be used. [I(1)], meaning a first difference will be I(0). H0: = 1 and H1: < 1 (see ch.18) Econometrics 13 Econometrics 14 Time Series Data 2: Further Issues 3.