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Econometrics. Lecture Notes Hayashi, Chapter 6f Large Sample Theory. Sample Mean. Given a serially correlated process {y t }, what is the asymptotic properties of the sample mean? What are the restrictions on covariance stationary processes for the consistency of the sample mean?.

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**Econometrics**Lecture Notes Hayashi, Chapter 6f Large Sample Theory**Sample Mean**• Given a serially correlated process {yt}, what is the asymptotic properties of the sample mean? • What are the restrictions on covariance stationary processes for the consistency of the sample mean?**Law of Large Number**• Let {yt} be covariance stationary with mean m and autovariances {gj}. Then**Law of Large Number**• The long-run variance is • In terms of autocovariance-generating function, the long-run variance is gy(z) at z=1 or 2p times the spectrum at frequency zero.**Central Limit Theorem**• CLT for MA()Let yt = m + j=0,…yjet-j where {et} is independent white noise and j=0,…|yj|< (ergodic stationarity). Then**Central Limit Theorem**• Gordin’s condition for ergodic stationary process {yt}: • E(yt2) < (strict stationarity) • E(yt|yt-j,yt-j-1,…) ms 0 as j E(yt) = 0Conditional expectations (forecasts) will approach unconditional expectation, as less and less information becomes available.**Central Limit Theorem**• Gordin’s condition (continued): • Let It = (yt,yt-1,yt-2,…), and writertj = E(yt|It-j) - E(yt|It-j-1). Define yt = j=0,… rtj (telescoping sum). Then j=0,…[E(rtj2)]1/2 < **Central Limit Theorem**• Gordin’s condition explained: • The revision of expectation about yt as the information set increases from It-j-1 to It-j, yt-(rt0+rt1+…+rtj-1 ) ms 0 as j . • The telescoping sum indicates how the shocks represented by (rt0, rt1 , …) influence the current value of yt. • The shocks occurred a long time ago do not have disproportionately large influence. This condition restricts the extent of serial correlation in {yt}.**Central Limit Theorem**• Gordin’s condition (example): • yt = f yt-1+et, |f|<1, {et} independent white noise with s2 = Var(et). • E(yt2) < • E(yt|yt-j,yt-j-1,…) = fjyt-jms 0 as j • rtj = fjyt-j–fj+1yt-j-1 = fj(yt-j– fyt-j-1) = fjet-j • yt = j=0,… rtj (telescoping sum) is MA() • j=0,…[E(rtj2)]1/2 = j=0,…|f|js = s/(1- |f|) < **Central Limit Theorem**• CLT for zero-mean ergodic stationary process: Suppose {yt} is stationary and ergodic and suppose Gordin’s condition is satisfied. Then E(yt) = 0, the autocovariance {gj} are absolutely summable, and**Multivariate Sample Mean**• The sample mean of a vector process {yt}: • if each diagonal element of Gj goes to zero as j **Multivariate Sample Mean**• (long-run covariance matrix of {yt}) equals j=-,…Gj if {Gj} is summable. • The long-run covariance matrix of {yt} can be written as Gy(1) = 2psy(0) = j=-,…Gj = G0 +j=1,…(Gj + Gj’)**Multivariate Sample Mean**• If {yt} is vector MA() with absolutely summable coefficients and {et} is vector independent white noise,**Multivariate Sample Mean**• Gordin’s condition on ergodic stationary process: • E(ytyt’) exists and is finite • E(yt|yt-j,yt-j-1,…) ms0 as j • j=0,…[E(rtj’rtj)]1/2 is finite, wherertj = E(yt|yt-j,yt-j-1,…) - E(yt|yt-j-1,yt-j-2,…)**Multivariate Sample Mean**• Suppose Gordin’s condition holds for vector ergodic stationary process {yt}. Then E(yt) = 0, {Gj} is absolutely summable, and

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