diff --git a/nbs/docs/models/ARCH.ipynb b/nbs/docs/models/ARCH.ipynb index 2d61a2fe3..789113d4c 100644 --- a/nbs/docs/models/ARCH.ipynb +++ b/nbs/docs/models/ARCH.ipynb @@ -673,7 +673,7 @@ "source": [ "### Return Series\n", "\n", - "Since the 1970s, the financial industry has been very prosperous with advancement of computer and Internet technology. Trade of financial products (including various derivatives) generates a huge amount of data which form financial time series. For finance, the return on a financial product is most interesting, and so our attention focuses on the return series. If {Pt } is the closing price at time t for a certain financial product, then the return on this product is\n", + "Since the 1970s, the financial industry has been very prosperous with advancement of computer and Internet technology. Trade of financial products (including various derivatives) generates a huge amount of data which form financial time series. For finance, the return on a financial product is most interesting, and so our attention focuses on the return series. If \\{Pt } is the closing price at time t for a certain financial product, then the return on this product is\n", "\n", "$$X_t = \\frac{(P_t − P_{t−1})}{P_{t−1}} ≈ log(P_t ) − log(P_{t−1}).$$\n", "\n", @@ -12906,7 +12906,7 @@ 0.11683902533544767, 0.18038662130227442, 1.1396760668737118, - 8.429606751246858e-06, + 0.000008429606751246858, 0.9301650273653356, 0.16603494698002158, 0.02552244687162859, @@ -13023,7 +13023,7 @@ 0.333507961464413, 1.5341682132027463, 0.535647628131223, - 8.047187854454163e-06, + 0.000008047187854454163, 0.05159803214177818, 0.07600424912094962, 0.05061172231156831, @@ -13163,7 +13163,7 @@ 2.7279986776819762, 2.7160487251081724, 0.21768883267417796, - 6.77720693099992e-06, + 0.00000677720693099992, 2.0892382083596366, 1.5511137261921708, 0.19711705562956672, @@ -13207,7 +13207,7 @@ 0.09514048933694128, 0.005799502263676435, 0.26977917189300543, - 2.2905689893454364e-05, + 0.000022905689893454364, 0.032835101620127545, 0.035071983113154785, 0.02720364389284937, @@ -13299,7 +13299,7 @@ 0.2733247895916533, 0.03816868105362628, 0.05644730956949203, - 1.7126891073291785e-05, + 0.000017126891073291785, 0.17648748135786346, 0.08890408218010715, 0.00021428585381320998, @@ -13310,7 +13310,7 @@ 0.0034536384504556326, 1.0219740565421616, 0.14230208866334162, - 3.4607522944888557e-06, + 0.0000034607522944888557, 0.0008948379096681811, 1.1918442949862227, 0.4224720090152985, @@ -13334,7 +13334,7 @@ 0.379505710351215, 0.04804736114077823, 0.018926161782214495, - 7.079826049627675e-05, + 0.00007079826049627675, 0.2256097560481151, 0.14422736286851592, 0.03029031934041486, @@ -13469,7 +13469,7 @@ 0.016163575341260606, 0.0033884362508481143, 0.16717428315038643, - 1.4022086968800128e-05, + 0.000014022086968800128, 0.010511348063911363, 0.012782595072851783, 0.04677744379099467, @@ -13517,7 +13517,7 @@ 0.5337190473107196, 0.03514066158205324, 0.21841633296069093, - 2.799770311426146e-05, + 0.00002799770311426146, 0.0035743299293066215, 0.2886525495692123, 0.0002361118071194596, @@ -13567,7 +13567,7 @@ 0.09277613223137564, 0.004196366830944074, 0.04937508180038409, - 5.208049534825782e-05, + 0.00005208049534825782, 0.16688400504491227, 0.014510991850446027, 0.13727838266375458, @@ -13710,7 +13710,7 @@ 0.006911690943276298, 0.3278850494791402, 0.7287238222344438, - 3.161982797127493e-05, + 0.00003161982797127493, 1.7903942515655977, 0.033747867911173186, 1.0887262880763404, @@ -13821,7 +13821,7 @@ 0.017032753347700658, 0.10819350754652532, 0.07635754962107597, - 4.720076651680163e-07, + 4.720076651680163e-7, 0.13257647932461017, 0.0015736305321550584, 0.005062602892380193, @@ -13876,7 +13876,7 @@ 0.0003997208956144086, 3.6430248324576744, 4.315374983468133, - 7.465081431494341e-05, + 0.00007465081431494341, 2.3702802404040986, 2.4875933720473613, 4.238751501049985, @@ -13947,14 +13947,14 @@ 0.12884377091778848, 0.45350527664342116, 1.3383250835766, - 3.047038069794671e-06, + 0.000003047038069794671, 0.046155230627765706, 0.04345704527385812, 0.21496506713259997, 0.010971728644473754, 0.3681449089203887, 0.12095591183388035, - 1.4533571009694057e-05, + 0.000014533571009694057, 0.43683167649436533, 0.003961054815055465, 0.002594457624136788, @@ -14057,7 +14057,7 @@ 1.1755067590686328, 1.692552536701068, 0.008292017842472079, - 8.837910181512196e-05, + 0.00008837910181512196, 0.0010388011063882155, 0.52268289783404, 0.08289521198197039, @@ -14065,9 +14065,9 @@ 0.09831904474102832, 0.06665444896152177, 0.0011743773216012492, - 3.989948937286202e-06, + 0.000003989948937286202, 0.23966670389611294, - 9.396568391014351e-05, + 0.00009396568391014351, 0.7083530264126577, 0.379280036868932, 0.05899634912283012, @@ -14125,7 +14125,7 @@ 0.012032641276976222, 0.08457382662710584, 0.73534494099483, - 5.26813972937628e-05, + 0.0000526813972937628, 0.5109083285028833, 0.0011242101166246262, 0.0018688345123409381, @@ -14134,7 +14134,7 @@ 0.0075020470243941095, 0.00038200004553110315, 0.26298092501994424, - 1.1549731265675906e-05, + 0.000011549731265675906, 0.33417910089435754, 0.0867904320408933, 0.7020433764156192, @@ -14609,7 +14609,7 @@ 0.6766417031004891, 0.0030353502921928455, 0.52166759677021, - 1.1534388436087901e-06, + 0.0000011534388436087901, 0.1494000082288102, 0.06768454071199759, 0.11456736834367175, @@ -14960,13 +14960,13 @@ 0.06211911053832683, 0.12810408947533514, 0.009926192984207037, - 1.7100299484364275e-05, + 0.000017100299484364275, 0.17097265655211852, 1.7589894800913939, 0.042823064448460374, 0.10930672666497082, 0.007314017675982816, - 7.100057638853501e-05, + 0.00007100057638853501, 0.3543283102780241, 0.00815750397918031, 0.00725186857060821, @@ -14994,7 +14994,7 @@ 0.5356191175456438, 0.7669523057389738, 1.7027866477021951, - 2.756746937148935e-06, + 0.000002756746937148935, 0.3731529145352457, 0.9711025783145061, 2.1124949696707556, diff --git a/nbs/src/core/models.ipynb b/nbs/src/core/models.ipynb index 5684d66a3..64926e297 100644 --- a/nbs/src/core/models.ipynb +++ b/nbs/src/core/models.ipynb @@ -10428,7 +10428,7 @@ " \n", " $$\\sigma_t^2 = w + \\sum_{i=1}^p a_i y_{t-i}^2 + \\sum_{j=1}^q b_j \\sigma_{t-j}^2$$. \n", " \n", - " Here {$v_t$} is a sequence of iid random variables with zero mean and unit variance. \n", + " Here \\{$v_t$} is a sequence of iid random variables with zero mean and unit variance. \n", " The coefficients $w$, $a_i$, $i=1,...,p$, and $b_j$, $j=1,...,q$ must satisfy the following conditions: \n", " \n", " 1. $w > 0$ and $a_i, b_j \\geq 0$ for all $i$ and $j$. \n",