The Hidden Weight of GMM Consistency Conditions Consider estimating parameter $\theta\in \Theta$ from data $\lbrace w_i\rbrace_{i \in [N]}$ Assume: Parameter space $\Theta\in \R^K$ is compact. The criterion function of GMM $$ > s_N(\theta) = s(\vec w; \theta) > $$ is continuous in $\theta$ $\forall, \vec w$. $s_N(\cdot)$ well behaves: $$ > \sup_{\theta\in \Theta}|s_N(\theta) - s_\infty(\theta) \xrightarrow{p}0. > $$ $s_\infty(\theta)$ has a unique minimum at $\theta_0$. Then $\hat \theta_{GMM} \xrightarrow{p}\theta$. The proof is essentially a topological argument — uniform convergence of continuous functions on a compact set, plus a unique minimum, pins down the limit of the minimizers. It is clean, elegant, and almost suspiciously general. ...

In heterogeneous agent macroeconomics models, the way to compute (solve) a steady state equilibrium of the market is quite interesting. Consider a unit mass of agents, whose utility functions are the same $u:\R \to \R$. In a continuous time world, at each ifinisimmo time point they receive (stochastic) income and capital rents from their historial savings, consume and save more (sometimes saved capital are also called as ‘assets’ that generate returns, but per the no-arbitrage principle, the return of assets and captial rents net capital depreciation should be the same). So it’s being abstracted as a stochastic continuous time markov system, the state is the asset that the agent hold, action is consumption, plus random income shocks. ...

“Don’t try, just be.” — E Pahud comment on playing Schubert. Franz Schubert was born today 229 years ago. He composed more than 1500 works in his short career (He died 1828, aged 31) Courtesy of Wikipedia. I don’t own copyright! Some of my favourites include Die Forelle, and the Impromptus (No. 3 in G Flat Major). Schubert’s foremost contribution is his Lieders (art songs). The Winterreise was composed just one year before his pass away. Listen to Thomas Quasthoff & Daniel Barenboim performing Gute Nacht. ...

Our macroeconomic class introduced and discussed this paper, in the middle of a series of discussion on growth studied under the framework of neoclassical growth model. Standard approaches to structural change (the shift from agriculture to manufacturing to services as economies develop) have relied on two main mechanisms: Supply-side stories: Differential productivity growth across sectors changes relative prices, causing reallocation Demand-side stories (Stone-Geary preferences): As income rises, people shift spending toward “luxury” sectors Both mechanism’s intuition feels right. But the demand-side’s Stone-Geary preferences have income effects that vanish as income grows. It becomes negligible for rich countries. This is empirically problematic—we observe strong income effects on sectoral composition even at high income levels. So, when the theory tells you one thing but the reality tells you another we need a better model to solve the problem: ...

I sat through a 66-minute Bruckner Symphony (No. 4)—twice. Salonen, Trifonov & Beethoven 2026 January Week 4: Thursday Friday Saturday Sunday https://cso.org/performances/25-26/cso-classical/salonen-trifonov-beethoven/ Program: Beethoven Piano Concerto No. 2 + Bruckner Symphony No. 4 (Romantic) I went to both the Thursday and Friday concerts. The Bruckner is full of moments where the entire brass section erupts into relentless fortississimo, practically blasting your head open—and I was seated on the terrace behind the stage. Bless me. ...

Interestingly, when using implicit updating to solve a continuous time system (of a certain structure), it coincide with Newton’s Method Setup Consider a household with state $(a, z) \in \mathcal{A} \times \mathcal{Z}$, where $a$ denotes assets and $z$ follows a Poisson process with intensity matrix $\Lambda$. Hamilton-Jacobi-Bellman Equation: $$ \rho v(a,z) = \max_c \left \lbrace u(c) + \partial_a v(a,z) \cdot s(a,z) + \sum_{z'} \lambda_{zz'} v(a,z') \right\rbrace $$where savings $s(a,z) = ra + wz - c(a,z)$. ...

Wolfgang Amadeus Mozart (Jan 27 1756 - Dec 5 1791) was born today 270 years ago! (But remember, he was still actively publishing in 2024…) Ganz kleine Nachtmusik (KV 648): Some prev posts about mozart: A Little Mozart Wisdom Mozart K.314 Movt. 3 | 2001 Europakonzert Istanbul Series Catalog | Glenn Gould vs. Mozart’s Greatest Concerto Glenn Gould vs. Mozart’s Greatest Concerto | IV. the piano sonata that Glenn recommends (K.333) Glenn Gould vs. Mozart’s Greatest Concerto | III. the piano concerto that Glenn dislikes for being too ‘boring’ (K.491) Mozart K.550, Symphony No.40 Mozart’s Requiem at the CSO Cosi Fan Tutte, by the LA Opera mozart stories

It’s just January, but there’s good news already! There are possibly hundreds of dishes. My favourites is the Pineapple Bun. (This one’s probably larger than my face…) Guangzhou, my hometown, is the origin of Canton Morning Tea (or dimsum, see leadtochina’s Canton Morning Tea for a crisp and very accurate introduction). But lately as the popularization of pre-made food, some restaurants start to microwave frozen food and put them in steaming plates as if they’re fresh made. It’s cheaper, and not everyone can tell the difference. People who love morning tea are scared that the market might get screwed up — so here comes regulation: ...

Meet my new PR Sleepingbot, an intelligent assistant that answers questions about all my historical blogposts. Its high-level architecture is: ┌─────────────────┐ ┌─────────────────┐ ┌─────────────────┐ │ Hugo Blog │────▶│ Railway API │────▶│ Anthropic API │ │ (GitHub Pages) │ │ (server.py) │ │ (Claude) │ └─────────────────┘ └─────────────────┘ └─────────────────┘ 1. Two-Stage (Retrieval-Augmented Generation) RAG Chatbot Stage 1: Send all post titles to Claude → “Which posts are relevant to this question?” Stage 2: Send full content of relevant posts + question → Get answer This is smarter than simple keyword search, and cheaper than sending all posts every time. ...

The Thaïs Méditation was originally an Entr’acte from the opera Thaïs by Jules Massenet. The opera itself is set in Roman-era Egypt and centers on a monk named Athanaël who tries to convert the courtesan Thaïs to Christianity. The Méditation occurs in Act II between two scenes, at a moment when Thaïs is wrestling with her worldly life and the possibility of spiritual change. It was originally for orchestra plus solo violin. The piece has long line melody that pierce through your heart. ...