Cointegration in Theory and Practice. A Tribute to Clive Granger. ASSA Meetings January 5, 2010

Transcription

1 Coinegraion in heory and Pracice A ribue o Clive Granger ASSA Meeings January 5, 00 James H. Sock Deparmen of Economics, Harvard Universiy and he NBER /4/009

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3 Coinegraion: he Hisorical Seing Granger and Newbold (Journal of Economerics, 974) I is very common o see repored in applied economeric lieraure ime series regression equaions wih an apparenly high degree of fi, as measured by he coefficien of muliple correlaion R or he correced coefficien R, bu wih an exremely low value for he Durbin-Wason saisic. We find i very curious ha whereas virually every exbook on economeric mehodology conains explici warnings of he dangers of auocorrelaed errors, his phenomenon crops up so frequenly in wellrespeced work (p. ) /4/009 3

4 Dickey-Fuller (JASA, 979): Y = ρy + ε he hypohesis ha ρ = is of some ineres in applicaions because i corresponds o he hypohesis ha i is appropriae o ransform he ime series by differencing. Currenly, praciioners may decide o difference a ime series on he basis of visual inspecion of he auocorrelaion funcion (p. 47) /4/009 4

5 Davidson, Hendry, Srba and Yeo (Economics Journal, 978) Δ 4 c = 0.49Δ 4 y 0.7ΔΔ 4 y.06(c -4 y 4 ) + 0.0D (4) (.04) (.05) (.0) (.004) cf. Hall (978) /4/009 5

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8 Coinegraion: Economeric heory riumphs and Disappoinmens riangular model coinegraed (,) wih n=, r=: Δx = v y = θx + u Main aims of iniial economeric heory:.superconsisency of OLS esimaor ˆ θ.use of esimaed ECM erm z ˆ ˆ = y θ x as a regressor wihou he generaed regressor problem 3.esing for coinegraion (e.g. EG-ADF es on OLS residual) 4.Efficien (Gaussian) esimaion of θ 5.Inference for θ /4/009 8

9 Simple exposiional model: Δx = v y = θx + u, Assume: (v, u ) ~ (0,Σ), serially uncorrelaed, and [.] = [.] = v u B B v u (.), B is BM(Σ) (.) /4/009 9

10 Superconsisency and disribuion ( ˆ = θ θ ) = ECM as a regressor = x u x σ + B db uv v u w = βz + ζ = βz ˆ + [β(z z ˆ ) + ζ ] = βz + ζ = βz ˆ + [( ˆ θ θ)βx + ζ ] Efficien esimaion and inference MLE: f(y,x θ,γ) = f(y X,θ,γ )f(x γ ) so y = θx + γ (L)Δx + u ( ˆ = θ θ ) = = x u x + o p () v B B db B v v u σ ~ u N(0, ) df Bv B v /4/009 0

11 Which of hese resuls are robus o changes in assumpions abou long run properies? x = αx + v, α = + c/ (local o uniy model) y = θx + u, (v, u ) saisfy same assumpions; c is unknown Superconsisency and disribuion xu ( ˆ σ = θ θ ) = x = ECM as a regressor + JdB J uv v u w = βz + ζ = βz ˆ + [β(z z ˆ ) + ζ ] = βz + ζ = βz ˆ + [( ˆ θ θ)βx + ζ ] v, dj v = cj v + dw v /4/009

12 Efficien esimaion and inference MLE: f(y,x θ,γ) = f(y X,θ,γ )f(x γ ) so y = θx + γ (L)(x αx ) + u = θx + γ (L)Δx + [γ(l)( α)x + u ] = θx + γ (L)Δx + [ γ(l)cx + u ] so (Ellio (998)) ( ˆ θ θ ) = = ( γ () + ) x cx u = = γ()c + = = x u x x γ()c + + o p () JdB v J v u σ ~ u N( γ () c, ) df Jv J v /4/009

13 Some recen work on his problem Jansson and Moreira (006) he OLS and MLE disribuions are sensiive o oher models of long-run behavior e.g. fracional inegraion essenially have nuisance parameers ha are no esimable. See Müller and Wason (008) esing has he same issues /4/009 3

14 Coinegraion: Empirical Legacy wo examples () Consumpion/income and consumpion/income/wealh DHSY (978). Leau and Ludvigson (004) /4/009 4

15 ln(c/y) ln(a/y) 950q 960q 970q 980q 990q 000q 00q ime ln(c/y) ln(a/y) /4/009 5

16 cay q 960q 970q 980q 990q 000q 00q ime /4/009 6

17 cay q 00q3 005q 007q3 00q ime /4/009 7

18 () Housing values and median income /4/009 8

19 housing price - income EC erm 970q 980q 990q 000q ime /4/009 9

20 housing price - income EC erm 970q 980q 990q 000q 00q ime /4/009 0

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