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Econometrics

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

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  1. Econometrics Lecture Notes Hayashi, Chapter 6f Large Sample Theory

  2. 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?

  3. Law of Large Number • Let {yt} be covariance stationary with mean m and autovariances {gj}. Then

  4. 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.

  5. 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

  6. 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.

  7. 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 < 

  8. 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}.

  9. 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-jms 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|) < 

  10. 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

  11. Multivariate Sample Mean • The sample mean of a vector process {yt}: • if each diagonal element of Gj goes to zero as j  

  12. 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’)

  13. Multivariate Sample Mean • If {yt} is vector MA() with absolutely summable coefficients and {et} is vector independent white noise,

  14. 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,…)

  15. 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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