Here’s a pretty cool and useless way to understand how to obtain the $\beta$ in linear regression. Let’s begin with (real valued) Hilbert space, $\mathcal H$ which comes with inner product $\lang \cdot, \cdot\rang$ that are symmetric, linear and positive definite. (Might as well think classical $x, y\in \R^m$ and $\lang x, y\rang = x^Ty = \sum_{i \in [m]}x_iy_i$ in standard linear algebra) Now suppose we are given $y\in \mathcal H$ and $X := [x_1, x_2, \ldots, x_N]\in \mathcal H^N$. The way to “project” $y$ onto $x$ is ...

Today The University of Chicago Symphony Orchestra performs our 2025 Halloween Concert. During the the William (Guillaume) Tell Overture by Gioachino Rossini, there’s a lovely slow passage in the middle of the piece: This pastorale section in G major and in an A-B-A-Coda form, signifying the calm after the storm, begins with a Ranz des vaches or “Call to the Cows”, featuring the cor anglais (English horn). The English horn then plays in alternating phrases with the flute, culminating in a duet with the triangle accompanying them in the background. ...

College classrooms can sometimes become awkwardly dead silence. This happens often when I was in Shanghai. The cringe culture can explain it a little. But I think all of us would be better off if we can shed that away. Here’s a new podcast report by Ariana Grande, snippet: The world had a difficult time with earnestness… We should be able to be happy and not be corny or cringe or classified to be something like that. Being passionate/happy/excited about something, loving people. What else do you want to me be, sitting here numb? ...

The game Counter-Strike kinda nutured a market: the weapons and gloves that players use in battle are tradable aesthetic assets known as skins. (i) some skins are rare (eg. getting them isn’t simple: players can either open loot boxes each costing about 2.5usd, with less than a 0.3% chance of lotteried on a rare item), and (ii) tradable (one can purchase them on the Steam Market, Valve’s global exchange where millions of players list and trade items in real time). ...

Some research doesn’t really start with a question, but a survey form waiting to be filled. The pattern is predictable: pick a “socially important” topic — say, “Do people feel more ethical after recycling a paper cup?” Then split participants into two groups. One group watches a short clip of someone recycling. The other sees the same cup tossed in the trash. Everyone rates their feelings on a 1-to-7 scale. And just like that, we’ve produced “evidence.” ...

The Wall-Street Journal has this wonderful video The Money-Making Secrets Behind Hotel Design: “From vanishing minibars to disappearing closets, hotel rooms are shrinking. With the rise of Airbnb and hotel occupancy rates plateauing, operators are on the hunt for the most profitable design to maximize profits. Take Marriott’s Moxy brand, for example — its rooms are less than half the average size yet can generate up to 20% more revenue than its peers.” ...

A pause on my econometric homework, here’s what seems to have happened:

The music industry and the academia has something in common: composers/academics are basically, content creators. Looking into a bit music history helps understand the general dynamic of both worlds. From the book The Orchestra And Orchestral Music (W. J. Henderson, 1899): The older composers are like ancient history; one must have sufficient information to know what to accept and what to reject in order to read them with advantage. (From chapter The Strings, where the author is trying to make a point on which composer to look to to study the best string orchestration) I think there is something profound in how the author view (and try to let the readers understand his way of viewing) the development of woodwinds and harmonies through time: ...

Stephen Dubner is in town today for the 20th anniversary of Freakonomics, at an open conversation event hosted by Chicago Humanities. It’s even better than a live podcast. Slay. Stephen called the podcast industry in general as one of Steve Jobs’ greatest inventions. By introducing a simple but open content-upload API, that once “stupid purple app” slashed the cost of journalism to nearly zero — making great ideas freely available to anyone, anywhere. ...

Class note from Price Theory course. Consider an individual looks to buy $m$ goods. Let there be nice assumptions: Continuous: that consumption can be expressed as $x\in \R^m$. (Strictly?) concave utilities. Linear budget: $p\cdot x \le \texttt{Budget}$. Back of (a big) envelope Let $\mathbf x^H(p, u), \mathbf x^M(p, I)$ be the Hicksian and Marshallian demand functions. If we focus on one single good $i\in [m]$, and with a tiny abuse of notation, we can write $$ \begin{align*} & x_i^{H(\bar U, p_{-i})}(p_i) :=x_i^H(p_i, p_{-i}, \bar U)\cr & x_i^{M(I, p_{-i})}(p_i) :=x_i^M(p_i, p_{-i}, I) \end{align*} $$ When economists scribble a downward sloping demand curve on a 2D plane, we are basically holding $(\bar U, p_{-i})$ (or $(I, p_{-i})$) fixed and drawing the Hicksian demand curve $D^H:\lbrace (x_i^H(p_i), p_i)\rbrace$ or Marshallian demand curve: $D^M:\lbrace(x_i^M(p_i), p_i)\rbrace$. ...