题目2493 · 机器学习
Projection Error Is Orthogonal to the Fitted Subspace 23
Why is y - X beta_hat orthogonal to every fitted vector Xv?
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中文题目题目2482 · 机器学习
Projection Interpretation of the Fitted Vector 12
In one sentence, what geometric object is X beta_hat in OLS?
打开 →题目1531 · 数学
Residual-Space Projection Rank 1
P is a projection on R^8 with rank 3. What is rank(I - P)?
打开 →题目631 · 概率
Terminal-Variable Projection 1
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{X_1+X_2+X_3 >= 2} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目632 · 概率
Terminal-Variable Projection 2
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{X_1+X_2+X_3+X_4 = 0} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目633 · 概率
Terminal-Variable Projection 3
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = 1{max(X_1,X_2,X_3) = 1} and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目634 · 概率
Terminal-Variable Projection 4
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = X_1+X_2+X_3+X_4 and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目635 · 概率
Terminal-Variable Projection 5
Let X_1, X_2, X_3, X_4 be iid symmetric ±1 variables with natural filtration F_n. Define Y = (X_1+X_2+X_3)^2 and M_n = E[Y | F_n]. Is (M_n) a martingale?
打开 →题目1752 · 统计
Why OLS Hedging Is a Projection Problem
Why is the OLS hedge ratio often described as projecting a cash-book return stream onto the span of hedge instruments?
打开 →模块2.2.2 · 数学与统计能力 · 统计推断
回归与广义线性模型
statistical-inference · regression · linear-regression · ordinary-least-squares · normal-equations · design-matrix · hat-matrix · projection
打开 →课程回归与广义线性模型 · 统计推断
线性回归:最小二乘法与几何视角
周二开盘前 30 分钟,你在一家百亿规模的私募(private fund)接手了今早的因子配置(factor allocation)任务。手头是沪深300 成份股过去 60 个交易日的日收益,以及 4 个候选风格因子——规模、价值、动量、低波——在同期的横截面暴露。你的 PM 只问一句:「把这批个股的今日预期收益,拟合成这 4 个因子的线性组合,残差还剩多少...
打开 →题目2076 · 数理金融
Affine Replication Check in an Incomplete Market 6
The stock is 100 today and ends at 120, 100, or 80 next period. Consider a claim paying 20, 10, and 0 in those three states. Can it be replicated exactly using only the stock and cash? If yes, give the hedge. If not, identify the replication obstruction.
打开 →题目3346 · 数学
Closest Point to the Origin Above a Budget Line
Minimize $x^2+y^2$ subject to $x+y\ge 1$. Find $(x^*,y^*)$ and the optimal KKT multiplier for the inequality $g(x,y)=1-x-y\le 0$.
打开 →题目2091 · 数理金融
Incomplete-Market Pricing Intuition 21
Why does a unique no-arbitrage price disappear as soon as the trinomial market has more states than traded securities?
打开 →题目2092 · 数理金融
Incomplete-Market Pricing Intuition 22
Why does the superhedge naturally sit at the top of an incomplete-market price interval?
打开 →题目2093 · 数理金融
Incomplete-Market Pricing Intuition 23
Why can a minimum-variance hedge still fail to pin down a unique no-arbitrage price?
打开 →题目2094 · 数理金融
Incomplete-Market Pricing Intuition 24
Why can one extra state-contingent quote complete the market in a trinomial model even if the stock and bond alone cannot?
打开 →题目2095 · 数理金融
Incomplete-Market Pricing Intuition 25
Why do indifference prices depend on risk aversion while no-arbitrage intervals do not?
打开 →题目3345 · 数学
Negative Target, Mildly Active Floor
Minimize $(x--2)^2$ subject to $x\ge -1$. Using the KKT form $g(x)=-1-x\le 0$, find the optimizer $x^*$ and the optimal multiplier $\lambda$.
打开 →题目3269 · 数学
One-PC Reconstruction of a Two-Asset Move
Let $\Sigma=egin{pmatrix}9&-3\-3&9\end{pmatrix}$, whose first principal direction is along $(1,-1)$. For the observed move $x=(2,-1)$, what are the rank-1 reconstruction using only the first principal component and the residual?
打开 →题目2071 · 数理金融
Price a Trinomial Claim After Market Completion 1
A one-period stock is 100 today and ends at 120, 100, or 80. The risk-free rate is 0. A quoted up-state digital that pays 1 only in the up state trades at 0.2, which completes the market. What unique no-arbitrage price does this imply for the claim paying 5, 1, and 0 in the up, m
打开 →题目3341 · 数学
Projected Optimum Above the Floor
Minimize $(x-2)^2$ subject to $x\ge 0$. Using the KKT form $g(x)=0-x\le 0$, find the optimizer $x^*$ and the optimal multiplier $\lambda$.
打开 →题目4251 · 机器学习
Rank-1 PCA Reconstruction 1
A centered two-feature dataset has covariance matrix [[1.8, 2.4], [2.4, 8.2]]. What is the second principal-component direction and its variance?
打开 →题目2086 · 数理金融
Recover the Completing Digital Quote 16
A one-period trinomial stock ends at 120, 100, or 80 with zero interest. An up-state digital paying 1 only in the up state completes the market and trades at an unknown price q. A claim paying 5, 1, and 0 in the three states is observed to trade at 1.8. What q is implied?
打开 →题目2081 · 数理金融
Worst Shortfall of a Simple Hedge 11
A non-traded payoff pays 4, 1, and 6 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash -8 and Delta = 0.1 shares of stock. What is the worst-case shortfall of that hedge across the three states?
打开 →题目2082 · 数理金融
Worst Shortfall of a Simple Hedge 12
A non-traded payoff pays 3, 5, and 1 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash 7 and Delta = -0.05 shares of stock. What is the worst-case shortfall of that hedge across the three states?
打开 →课程回归与广义线性模型 · 统计推断
回归诊断、多重共线性与正则化
上海某私募的多因子研究员把过去 250 个交易日的沪深300 成分股横截面回归刚跑完——12 个风格因子,公式 达到 0.41,看着挺漂亮。可是把 5 月那一周的极端行情样本剔掉再跑一次,某个动量因子的系数从 公式 翻成 公式;再换一种风险因子的口径,价值因子又从显著变成不显著。模型「拟合得很好」却一推就倒——这正是前两课没有触及的现实: 普通最小二乘...
打开 →课程回归与广义线性模型 · 统计推断
广义线性模型
上海某量化私募的两位研究员同一天上午被同一类工具卡住:小赵在搭一个「明日是否跑赢沪深300」的择时信号,标签是二元的 0/1;小李在 50ETF 期权做市数据上估「下一分钟到单笔数」,响应是非负整数 公式。模块前三课的普通最小二乘(ordinary least squares, OLS)对这两个任务都派不上用场——OLS 默认响应在正态分布(Gaussian...
打开 →课程线性代数核心 · 线性代数与微积分
正交性、投影与最小二乘
北京某私募的量化研究员手头有 1,500 个交易日的沪深300 ETF（300ETF）收益序列，外加 12 个候选因子——动量、价值、低波、三个流动性代理、六个宏观贝塔。她想要的是这 12 个因子在 L2 意义下最接近 ETF 收益的线性组合。1,500 个方程对 12 个未知数，这是高度超定的方程组，根本不存在精确解，她只能挑出 最佳近似 。给出 ...
打开 →课程回归与广义线性模型 · 统计推断
线性回归的统计推断
上海某私募的量化研究员在沪深300（CSI 300）成分股的三年日频收益里跑了一支六因子模型，回归表打出来：动量项系数 0.18、t 统计量 3.2，整体显著性 F 统计量 18.4。组合经理盯着她问：「这几个数字，到底说明因子真的有 alpha，还是只是回归噪音被你刚好捞到了？」她手里的工具不能回答这个问题——上一课的 公式 是点估计，没有不确定性。本节要...
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