Lista1seriestemporaisgabarito

ANÁLISE DE SÉRIES TEMORDAS\nLISTA 1\nGAJARDO\n(1)\ny(t) = g(t) + 4ξt\nCov(ξt, ξn) = 0 para t ≠ n.\n= Σ(ι + nξt) , ξt ∈ [0, θ], t = 1, 2, 3,...\nE(g(t)) = E(g(1) + ... + g(t)) = E(g(1)) + ... + E(g(t)) = 0\nVar(g) = Var(g(1) + ... + g(t)) = Var(g) + ... + Var(g(t)) = t σ²\nComo Var(g(t)) = t mais é constante, o processo é estacionário\n(b)\nY(t, n) = Cov(y(t), y(n)) = Cov(g(t), g(n)) + Cov(ξ(t), ξ(n))\n= Cov(ξ(t), ξ(n)) + ... + Cov(ξ(t), ξ(n)) = t², t < n\n(Cov(ξ(t), ξ(n)) = 0; t < n)\n\nCombin, Y(t, n) = t² min(ν, t)\n(c)\n(c) Ψ > 0\nY(t, n) = Cov(ω(t), ω(n)) = Cov(ω(t) − ω(n) + ω(n), ω(n))\n= Cov(ω(t), ω(n)) − Cov(ω(t), ω(n)) + Cov(ω(n), ω(n))\n= Var(ω(n)) = D\n(d)\n(e) ǫ < ν\nY(t, n) = Cov(ω(t), ω(n)) = Cov(ω(t), ω(n)) = Cov(ω(1), ω(n)) − ω(t) + ω(n)\n= Cov(ω(t), ω(n) - ω(t)) = Var(ω(n)) = t\n doyg o, Y(t, n) = min(t, δ)\nP(t, λ) = Y(t, n) \n ____________\nV(t,t)(δ, n)\n min(t, δ)\n\n3)\ny(t) = p1ξt + ςt, ǫ = ℕ ∪ [0, 1), Cov(ξt, ξn) = 0 para t ≠ n\nE(y(t)) = E(p1 + p2 + ς1) = p1 + p2 + ς(t) + θ\n= Cov(ξ1, ς1) + Cov(ξ1, ω(ς)))\n= 0, t ≠ n\n= f(x) + g(y) + Cov(ω(n), ω(n))\n= ... = Cov(ξ, ρ) = 0 para t ≠ n\n(d)\nS(p1, p2) = Σ∞I=t(y(t) − p1 − p2(t))² \nqueremos minimizar S(p1, p2)\n\n∂S(p1, p2)\n∂p1 = −2Σ(t=y...(y(t) − p1 − p(t) = 0 (1)\n\n∂S(p1, p2)\n∂p1 = −2 Σ(t, u)\n(y(t) − p1 − p(t)t = 0 (2) D(t, 1) = Σξyt − mg1 − p1ξt = 0\n(a) => p0 = ΣEyt − (p1)i1ξt = ḡ − p1t\n\nD(t, 2) = Σξ2t − p1ξ2t − ρ1ξt = 0\n = Σyt − [ΣEξ(t) − ρ(t)ξt + p1ξ2t − 0\n=) ΣEyti − mg1 + p1m 2 − β1ξ2 = 0\n= p1 = ΣEyt = mg1|\n\nCov(p1t, p1t) = ∑\n(iñdt = E[y(t)] = (...\n\nCorretora, p̄1 = Σgα(t + t) = Sy + x1 − βt − ρ̄1(t)\nΣ(ξt + ║)\n\nE(p̄1) = E[g − ρ̄1(t)] = E[g] − ρ̄1(t) = E(ΣE(m)\n\n= 1/ΣE(y) = 1/Σ/Σ(p1 + p(t)) = p1ρ0 + p1αξt /μ\n\x1 + ρ1 + ρ1 = 0\n... Var(y0) = Var(y) - E[f(t)]2 Var(y0) - 2 E[Cov(y0, y1)]\n = \u03C32 m + f2 u2 S\u20AC - 2t Cov(y0, y1)\n = u2 m + f2 u2 S\u20AC - 2t Cov(y0, y1)\n\nCov(y0, y1) = 1 \n\t\t Cov( \u2211\u03BCk, \u2211\u03B8k(t\u2212l)) = 1/m\u03BC \u2211\u2211(t-l)2[Cov(yk,yj)]\n = 1/m S\u20AC \n = 1/m S\u20AC t2 \n\nCurr., Var(y0) = u2(1/m + t2/S\u20AC). \n\nlog(y0) = \u03B8\u0302 \u223C N(0, \u03C32/S\u20AC) + \u03B8\u0303 \u223C N(\u03B2, u(1/m + t2/S\u20AC))\n\n\n4)\n yt = \u03BC cos(2\u03C0mt) + n new( \u2202mt) \nE(yt) = \u03C9(2\u03C0)t E(\u03BC) + n\u03C9(2\u03C0E(y)) = 0\nVar(yt) = \u03C9(2\u03C0)Var(yt) + min(2\u03B2\u03C0mt)Var(Var(y)) = 1 \nY(t) = Cov(y,...)\n = Cov( \u03BC Cos(2\u03C0mt) + n new(2\u03C0mt), ...)\n= \u03C9(2\u03C0mt) = \u03C9(2\u03C0wH) Y(h) = \u03B8(h) = R(h) u = 1/12\nh=0 1\nh=-1 0,366\nh=-2 0,200\nh=-3 0\nh=-4 -0,950\nh=-5 -0,986\nh=-6 -1\nh=-7 -0,836\n\nA fun\u00E7\u00E3o de autoind\u00FAncia apresenta componentes que respeitam regularidade\n\nyk = \u03A6[ \u03A6yt-2 + \u03B8t-1 + \u03B8t-2 + ...] \ne \u03E4 \nE(yt) = E(yt)\n= 0\nE(y) = \u03C6u2 + u3 \sum_{0}^{h}\u03A6t+ \u03B8t... \n \nVar(y) = Var(Et)\nVar(y) = Var(y) + \u03A6[u]+ \u03A6t-2...\n = \u03B8m Cov(\u2212,\u03B8t) + \u03B8(u)+Cov(\u2212)\n =\n2 / {u\u03B8t... + \u03B8n+4}\n= Var + E(y) \u03C6=0.5\u02C6 \n \u03C6=-0.5 \n \n\n6) (a) yt = E + \u03B1t + \u03B8t-1 + \u03B8t+1, \u03C6 t\u2208[0,1)\n Y(0) = \u03BA + \u03B8t = Cov(yt,y(t-n))\n = Cov(y(y... + \u03B8t-1) + \u02C6(y+1) + \u03B1t-1 + \u03B1t...) + ....\n= R(x + y) + C ...\n=... \n= \n\nh=0 \nY(0) = \u03C0/2 + 0\u03B0 = \u03C6(2)\n h=1 \nY(1)= \u03B0 + \u03B2 = \u03C3//(1 + 1 + 0)\n h=3 \nY(1) = 0 p(h) = {\n h = 0\n 0, (1 + 0), h = 1\n 1, 1 + 0, h = 2\n 0, cero constante\n}\n\nE(Y) = E(h) = E(0) + E(4) + E(1) + E(4) + ... > 0\nVal(y|h) = y^2 (1 + 0^2) + constante.\n\ncomo p(h) depende, siendo de h, recto o creciente, decreciente.