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Econometrics Week 3 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Further issues in using OLS with time series data. Chapter 11 (pp. 347 – 370). For the next week Serial correlation and heteroskedasticity in time series regressions. Chapter 12 (pp. 376 – 404). 2 / 23 Today’s Lecture We have learned that OLS has exactly same desirable finite sample properties in the time-series case as in the cross-sectional case. But we need somewhat altered set of assumptions. Today, we will study the large sample properties, which are much more problematic in time-series than in cross-sectional case. I will introduce the key concepts needed to apply the usual large sample approximations in regression analysis with time series. Trend analysis - crucial in economic analysis, most of the time, you will deal with these data. 3 / 23 Stationary and Nonstationary Time Series A stationary process is important for time series analysis. PDF of stationary process is stable over time. If we shift the sequence of the collection of random variables in any time, joint probability distribution must remain unchanged. Stationarity The stochastic process {xt : t = 1, 2, . . .} is stationary if for every collection of time indices 1 ≤ t1 < t2 < . . . < tm , the joint probability distribution of (xt1 , xt2 , . . . , xtm ) is the same as the joint probability distribution of (xt1 +h , xt2 +h , . . . , xtm +h ) for all integers h ≥ 1. Definition may seem little abstract, but meaning is pretty straightforward ⇒ see example on the next page. Time series with trend are nonstationary ⇒ its mean changes in time. 4 / 23 STATIONARY NONSTATIONARY Time Series 1 4 Time Series 2 120 110 2 yt xt 100 0 90 -2 -4 80 0 200 400 600 800 1000 70 0 200 400 600 t 800 1000 t ⇓ ⇓ Histograms Histograms 35 20 30 25 15 20 10 15 10 5 5 0 0 -3 -2 -1 Hx100, ... ,x300L 0 1 Hx600, ... ,x800L 2 80 85 90 Hy100, ... ,y300L 95 100 105 110 Hx600, ... ,x800L 5 / 23 Covariance Stationary Process Covariance Stationary Process A stochastic process {xt : t = 1, 2, . . .} with finite second moment [E(x2t ) < ∞] is covariance stationary if: E(xt ) is constant. V ar(xt ) is constant. for any t, h ≥ 1, Cov(xt , xt+h ) depends only on h and not on t. Covariance stationarity focuses only on the first two moments of a stochastic process and the covariance between xt and xt+h . If a stationary process has a finite second moment, it must be covariance stationary, BUT NOT VICE VERSA. ⇒ strict stationarity is stronger than covariance stationarity. 6 / 23 Weakly Dependent Time Series A very different concept from stationarity. Weakly Dependent Process A stationary time series process {xt : t = 1, 2, . . .} is said to be weakly dependent if xt and xt+h are “almost independent” as h increases. Very vague definition, as there are many cases of weak dependence ⇒ Not covered by this course. If for a covariance stationary process, Corr(xt , xt+h ) → 0 as h → ∞, we will say this process is weakly dependent. In other words, as the variables get farther apart in time, the correlation between them becomes smaller and smaller. Technically, we assume that the correlation converges to zero fast enough. 7 / 23 Weakly Dependent Time Series cont. ⇒ Asymptotically uncorrelated sequences will characterize weak dependence. Why we need it? ⇒ It replaces assumption of random sampling in implying that law of large numbers (LLN) and central limit theorem (CLT) holds. The simplest example of Weakly Dependent Time Series Independent, identically distributed sequence. A sequence that is independent is trivially weakly dependent. 8 / 23 Moving Average – MA(1) – Process More interesting example is Moving Average Process of order one – MA(1). xt = t + α1 t−1 , t = 1, 2, . . . where {t : t = 0, 1, . . .} is an i.i.d. sequence with zero mean and variance σ2 . Is MA(1) weakly dependent? YES! ⇒ its consecutive terms xt+1 and xt are correlated, with Corr(xt , xt+1 ) = α1 /(1 + α12 ), but xt+2 and xt are independent. Is MA(1) stationary? YES! (Ex.1): Interactive Example in Mathematica. (NOTE: To make this link work, you need to have Lecture3.cdf at the same directory and Free CDF player installed on your computer.) 9 / 23 Autoregressive – AR(1) – Process The process {yt } is called Autoregressive process of order one – AR(1), when it follows: yt = ρyt−1 + t , t = 1, 2, . . . with starting point y0 at time t = 0. where {t : t = 1, 2, . . .} is an i.i.d. sequence with zero mean and variance σ2 This process is weakly dependent and stationary in case that |ρ < 1|. (Ex.2): Interactive Example in Mathematica. (NOTE: To make this link work, you need to have Lecture3.cdf at the same directory and Free CDF player installed on your computer.) 10 / 23 Autoregressive – AR(1) – Process cont. To show that AR(1) is asymptotically uncorrelated, we will assume that it is covariance stationary. Then, E(yt ) = E(yt−1 ) ⇐ E(yt ) = 0. V ar(yt ) = ρ2 V ar(yt−1 ) + V ar(t ), and under covariance stationarity, we must have: σy2 = ρ2 σy2 + σ2 . For ρ2 < 1, we can easily solve for σy2 : σy2 = σ2 . 1−ρ2 11 / 23 Autoregressive – AR(1) – Process cont. The covariance between yt and yt+h for h ≥ 1 can be found by repeated substitution: yt+h = ρyt+h−1 + t+h = ρ(ρyt+h−2 + t+h−1 ) + t+h ... yt+h = ρh yt + ρh−1 t+1 + . . . + ρt+h−1 + t+h As E(yt ) = 0, we can multiply it by yt , take expectations and obtain Cov(yt , yt+h ). Cov(yt , yt+h ) = E(yt , yt+h ) = ρh E(yt2 )+ρh−1 E(yt t+1 )+. . .+E(yt t+h ) = ρh E(yt2 ) = ρh σy2 . As t+j is uncorrelated with yt . Finally, Corr(yt , yt+h ) = Cov(yt ,yt+h ) σy σy = ρh So ρ is correlation between any adjacent 2 terms, and for large h, correlation gets very low (ρh → 0 as h → ∞.) 12 / 23 Trends Revisited A trending series can not be stationary. A trending series can be weakly dependent. Trend-stationarity A series that is stationary about its time trend, as well as weakly dependent, is called trend-stationary process. As long as a trend is included in the regression, everything is OK. 13 / 23 Asymptotic Properties of OLS Following assumptions will alter the assumptions for unbiasedness we introduced last lecture (Ass. 1, Ass. 2 and Ass. 3). Ass. 1A: Linearity and Weak Dependence In a linear model: yt = β0 + β1 xt1 + . . . + βk xtk + ut , we assume that {(xt , yt ) : t = 1, 2, . . .} is weakly dependent. Thus law of large numbers and central limit theorem can be applied to sample averages. Ass2: Zero Conditional Mean For each t, E(ut |xt ) = 0 Ass. 2A: is much weaker than original one ⇐ no restriction on the relationship of ut and X. 14 / 23 Asymptotic Properties of OLS cont. Ass. 3A: No Perfect Collinearity No independent variable is constant or a perfect linear combination of the others. remains the same Theorem 1: Consistency of OLS Under Ass. (1A), (2A) and (3A), the OLS estimators are consistent: plim βˆj = βj , j = 0, 1, . . . , k. The theorem is similar to the one we used in cross-sectional data, but we have omitted random sampling assumption (thanks to the Ass 2.). 15 / 23 Large-Sample Inference We have weaker Ass. 4 and Ass. 5 than in CLM counterpart: Ass. 4A: Homoskedasticity For all t, V ar(ut |xt ) = σ 2 . Ass. 5A: No Serial Correlation For all t 6= s, E(ut , us |xt ) = 0 Theorem 2: Asymptotic Normality of OLS Under the Ass. (1A–5A) for time series, the OLS estimators are asymptotically normally distributed, and the usual OLS standard errors, t statistics, F statistics and LM statistics are asymptotically valid. 16 / 23 Large-Sample Inference Why is the Theorem 2 so important? Even if the CLM assumptions do not hold, OLS may still be consistent, and the usual inferences are valid! Of course, only if Ass. (1A – 5A) are met !!! In other words: Provided the time series are weakly dependent, usual OLS inference is valid user assumptions weaker than CLM assumptions. Under Ass. 1A – 5A, we can show that the OLS estimators are asymptotically efficient (as in the cross-sectional case). Finally, when an trend-stationary variable satisfy the Ass. 1A – 5A, and we add time trend to the regression, usual inference is valid. 17 / 23 Highly Persistent Time Series in Regression Analysis Highly persistent = strongly dependent. Unfortunately, many economic time series have strong dependence. Ass. on weak dependence violated. This is no problem, if the CLM assumptions (1 – 6) from previous Lecture holds. But, usual inference is susceptible to violation. 18 / 23 Random Walk A simple example of highly persistent series often found in economics is AR(1) with ρ = 1: yt = yt−1 + t , t = 1, 2, . . . where {t : t = 1, 2, . . .} is an i.i.d. with zero mean and variance σ2 . Expected value of {yt } do not depend on t, as yt = t + t−1 + . . . + 1 + y0 : E(yt ) = E(t ) + E(t−1 ) + . . . + E(1 ) + E(y0 ) = E(y0 ) If y0 = 0, then E(yt ) = 0 for all t. V ar(yt ) = V ar(t ) + V ar(t−1 ) + . . . + V ar(1 ) + V ar(y0 ) = σ2 t. Process is not stationary, as variance increases in time. 19 / 23 Random Walk Thus RW is highly persistent, as value of yt today influences all the future values of yt+h : E(yt+h |yt ) = yt for all h ≥ 1. Do you remember an AR(1) with ρ < 1?: E(yt+h |yt ) = ρh yt for all h ≥ 1 ⇒ with h → ∞, E(yt+h |yt ) = 0. (Ex.3): Interactive Example in Mathematica. (NOTE: To make this link work, you need to have Lecture3.cdf at the same directory and Free CDF player installed on your computer.) A random walk is a special case of an unit root process. 20 / 23 Random Walk It is important to distinguish between highly persistent time series and trending time series. Interest rate, inflation rate, unemployment rate can be highly persistent, but they do not have trend. Finally, we can have highly persistent time series with trend: random walk with trend: yt = α + yt−1 + t , t = 1, 2, . . . where α is a drift term (see interactive Ex. 3). E(yt+h |yt ) = αh + y0 . 21 / 23 Transforming Highly Persistent Time Series Highly persistent time series can lead to very misleading results if CLM are violated. (do you remember spurious regression problem?) We need to transform it into a weekly dependent process. Weakly dependent process will be referred to as integrated of order zero, I(0). Random Walk is integrated of order one, I(1). It’s first difference is integrated of order zero, I(0): ∆yt = yt − yt−1 = t , t = 1, 2, . . . Therefore {∆yt } is i.i.d. sequence and we can use it for analysis. (Ex.3.3): Interactive Example in Mathematica. 22 / 23 Thank you Thank you very much for your attention! 23 / 23