| DF regression |
Δyt=γyt−1+et, γ=ρ−1 (13A.2) |
| DF limits (no constant) |
T(ρ^−1)⇒∫W221(W(1)2−1); τ^⇒(∫W2)1/221(W(1)2−1); P(ρ^<1)→0.6827 (13A.4) |
| DF 5% critical values |
none −1.94; constant −2.86; constant+trend −3.41 |
| ADF reparametrisation |
γ=∑ϕi−1, δj=−∑i>jϕi (13A.6) |
| Spurious regression |
β^, R2 non-degenerate; t=Op(T) (13A.8) |
| VECM |
Δyt=αβ′yt−1+∑ΓiΔyt−i+εt (13A.9–10) |
| Granger representation, VAR(1) |
β′yt=(I+β′α)β′yt−1+β′εt; C=β⊥(α⊥′β⊥)−1α⊥′ (13A.11–13) |
| Engle–Granger 5% (pair, constant) |
−3.34 (vs DF −2.86) |
| Johansen |
∣λS11−S10S00−1S01∣=0; trace −T∑i>rln(1−λ^i); max-eig −Tln(1−λ^r+1) (13A.18–21) |
| OU exact AR(1) |
b=e−κΔ, a=θ(1−b), σe2=σ2(1−b2)/(2κ); half-life ln2/κ (13A.23) |
| OU MLE |
κ^=−lnb^/Δ, θ^=a^/(1−b^), σ^=σ^e2κ^/(1−b^2) (13A.24) |
| Precision of κ^ |
rel. s.e. ≈2/(κS); rel. bias ≈4/(κS), S = span (13A.25) |
| OU passage time |
Exτℓ=κπ∫zzℓerfcx(−u)du (13A.27) |
| Bertram cycle |
ET=κπ[erfi(zm)−erfi(za)]; rate (m−a−c)/ET (13A.26, 13A.28) |
| Optimal symmetric band |
(2u−c^)eu2=πerfi(u); u∗≈(3c^/4)1/3; zero-cost rate σκ/(2π) (13A.29) |
| TLS slope |
2Sxy(Syy−Sxx)+(Syy−Sxx)2+4Sxy2; β^y∣xβ^x∣y=ρ^2 (13A.30) |
| Kalman filter |
Pt∣t−1=Pt−1∣t−1+Q; Ft=xt′Pt∣t−1xt+R; Kt=Pt∣t−1xt/Ft; bt∣t=bt∣t−1+Ktet (13A.32–34) |
| Kalman steady state |
Pˉ=21(Q+Q2+4QR/x2); EWMA weight λ=Kˉx (13A.36) |
| Chow test |
F=(SSR1+SSR2)/(n−2k)(SSRR−SSR1−SSR2)/k∼F(k,n−2k) (13A.37) |
| Unknown date |
use supF (Andrews 1993); Chow at a data-chosen date rejects ~37% at nominal 5% (T=200, k=2) |
| CUSUM |
Wr=σ^−1∑wt; bounds ±0.948[n−k+2(r−k)/n−k] (13A.38–39) |