The VAR Model We observe a vector time series $Z_1, Z_2, \ldots, Z_T \in \mathbb{R}^m$ following a VAR($\ell$): $$ Z_{t+1} = \tilde {\mathbb N} + D \begin{pmatrix} Z_t \\ Z_{t-1} \\ \vdots \\ Z_{t-\ell+1} \end{pmatrix} + F\,W_{t+1}, \qquad W_{t+1} \sim \mathcal{N}(0, I_k). \tag{1} $$Known: the signals $Z_1, \ldots, Z_T$. Unknown: the coefficient matrices $\tilde {\mathbb N}$, $D$, and the covariance $FF'$. Key idea. Treat $(\tilde {\mathbb N}, D)$ as hidden states that never change, so the problem becomes an Hidden Markdov Model (HMM) with trivial state dynamics $\beta_{t+1} = \beta_t$ and a linear observation equation. We then update beliefs about $\beta$ recursively as data arrives. ...

以史为鉴, 可以知兴替 Using history as a mirror, one can predict a dynasty’s future. Some operations research archeology: Kantorovich and Zalgaller (1951): the 0-th column generation algorithm Eduardo Uchoa, Ruslan Sadykov. Mathematical Programming, 2026. This article probes the origins of the Column Generation technique. It begins with Kantorovich’s classic 1939 work, correcting widespread misconceptions about his contributions to the Cutting Stock Problem. It then brings to light Kantorovich and Zalgaller’s lesser-known 1951 book, which is revealed to contain a complete Column Generation algorithm. The article also places these contributions in the context of the turbulent USSR’s political and ideological environment, essential for a deeper understanding of their significance. ...

Professor Shleifer come over to Booth to talk in our Monday macro/intermational economic workshop. He’s so popular that we have to move to the largest classroom and it is filled. The admin bought enough sandwich for everyone though The Psychology of Macroeconomic Expectations Bordalo, Gennaioli, Lopez de Silanes, Schroeder, Shleifer, van Rooij (2026) There’s a version online in Booth’s website: https://www.chicagobooth.edu/-/media/project/chicago-booth/faculty-and-insights/research-workshops/shleifer.pdf Behavioral economics and general economic theory have left an open question in modeling: whether non-domain-specific (NDS) experiences — a health crisis, a divorce, financial hardship — could causally shift macro beliefs through a psychological, non-informational channel. This paper did the job: ...

Contemporary program: Joyce Light and Dark Villa-Lobos Assobio a Jato (“Jet Whistle”) for flute and cello Shaw Boris Kerner for cello + flower pots Chen Yi Qi for flute, cello, percussion, piano Dehnhard WAKE UP! for piccolo and alarm clock Gosfield Daughters of the Industrial Revolution Farrenc Piano Trio No. 4 in E Minor, Op. 45 What a program. When UChicago hands the reins to the artists themselves (according to Cynthia Yeh in the post-concert panel) — letting artists choose what they actually want to play — the results speak for themselves. No institutional hedging, no safe overtures. What we got instead was a concert of proper, fearless, genuinely good contemporary music. And the ensemble really CSO’s chamber music dream team: ...

You can shuffle allocation, but competition is still there: 重磅：广州将不分重点班！小学和初中新生或“一键分班” Source: 羊城晚报 (Yangcheng Evening News), by 蒋隽, April 15, 2026. Link Breaking News: Starting September 2026, Guangzhou will ban tracked classes (重点班) in all Grade 1 and Grade 7 cohorts. Students and teachers will be randomly assigned via a centralized “one-click” system. The core mechanism is DOUBLE RANDOMIZATION! “9月新学年入学的一年级和初一新生将在教育部门组织及监管下’一键分班’，同时老师也将随机匹配” “Grade 1 and Grade 7 students entering in September will be ‘one-click assigned’ under the organization and supervision of the education bureau; teachers will also be randomly matched.” ...

Erdodic’s Wikipedia definition is very general (and confusing) — “time averages equal ensemble averages”? Here’s a simplified version for finite-state markov chain: Let $\{X_k\}_{k \geq 0}$ be a Markov chain on state $S$. Let $\pi\in \Delta(S)$. Let Let transition operatro be $P$, so $\pi^{t + 1} = P \pi^t$. Define stationary distribution $\bar \pi$ as $X_0 \sim \bar \pi$ where $P\bar\pi = \bar\pi$, the stationary process satisfies $$ (X_0, X_1, \dots) \overset{d}{=} (X_1, X_2, \dots) $$ Definition. The stationary process $\{X_k\}$ is ergodic if for every bounded function $f: S \to \mathbb{R}$ the following convergence hold a.s.: ...

Evgeny Kissin returned to Chicago Symphony Orchestra, this time with Andrey Boreyko, opened a short run of concerts built around a thoughtfully balanced program: Rimsky-Korsakov: Russian Easter Overture, Op. 36 Mozart: Piano Concerto No. 12 in A Major, K. 414 Rimsky-Korsakov: Suite from The Tale of Tsar Saltan, Op. 57 Scriabin: Piano Concerto in F-sharp Minor, Op. 20 Rating: ⭐️⭐️⭐️⭐️⭐️. The program is repeated in Chicago the upcoming weekends then next week in Boston. Kissin played Mozart K414 with breathtakingly excitement in elegance. The opening Allegro carried a faint trace of caution, as though he were testing the acoustic and the air of the hall and feeling his way into the space. Even in that slight reserve his articulation remained crystalline. And by the Andante he had fully arrived: the phrasing deepened, the tone softened into something inward and luminous, the music seemed to breathe with so much intimacy. The final Allegretto is buoyant, danced lightly—quick yet unhurried, delicate without fragility. ...

Side Note of Professor Luciano Pomatto from Caltech’s Theory Seminar Talk at UChicago. Kudos to Zizhe for the summary. Ben said it was a great model as it was also well received in the internal student’s discussion. Disclaimer: All intellectual and copy rights belong to the author of the paper. If you like the economic model, make sure to check out Professor Pomatto’s webpage for the OG paper. All mistakes are mine. Setup A decision maker acquires information, choose action, and output. ...

Some sidenotes from price theory iii: Definitions For distributions (pdf) $f_1(\cdot), f_2(\cdot)$: Monotone Likelihood Ratio (MLR): let ratio $$ \phi(l) = \frac{f_1(l)}{f_2(l)}. $$ $f_1$ dominate $f_2$ (in MLR sense) iff. $\phi(l)$ is weakly increasing in $l$. Intuition: Pointwise relative density shifts up Hazard Rate Dominance (HRD): define the hazard rate (aka conditional failure rate): $$ h_{f_i}(l) = \frac{f_i(l)}{1 - F_i(l)} \quad i =1, 2. $$ $f_1$ dominate $f_2$ (in HRD sense) iff. $h_{f_1}(l)\le h_{f_2}(l), \forall l$. ...

It’s very intellectually stimulating to read Professor Stiglitz’s book, or if you’re as lazy as me, playing his YouTube archive recording of the book launch at UChicago. Professor Stiglitz’s most provocative point is almost saying the invisible hand doesn’t exist. Making this claim at the University of Chicago is dangerous… But basically, Kenneth Arrow (also a Chicago veteran) proved the conditions under which Adam Smith’s invisible hand claim holds — and showed how narrow those conditions are. Stiglitz’s argument is that then economists cherry-picked Arrow’s conclusion and ignored his caveats. They took a theorem with heavy asterisks and preached it as gospel. ...