1 / 15

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?.

zion
Download Presentation

Econometrics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related