Economics, with Computers

"People use statistics as the drunken man uses lamp posts - for support rather than illumination" Andrew Lang (or not)

"Econometrics is seens a vehicle for fundamental innovations in scientific method, above all, in the development of operative forecasting procedures in non-experimental situations"", see Wold (1969) Econometrics as Pioneering in Nonexperimental Model Building

Historically, econometric models were designed for macroeconomic data

Nowadays, more and more econometric models designed for microeconomic data

See Jensen (2007) The Digital Provide: Information, Market Performance, and Welfare in the South Indian Fisheries Sector

A supervised model for \(y\in R~\) given \(\boldsymbol{x}\in R^k~\) is a mapping \(m:R^k\rightarrow R~\). A dataset is a collection of observations \((\boldsymbol{x}_i,y_i)~\), where \(i=1,\cdots,n\).

In Econometrics, given some probability space \((\Omega,A,P)\), assume that \(y_i\) are realization of i.i.d. variables \(Y_i\) (given \(\boldsymbol{X}_i=\boldsymbol{x}_i\)) with distribution \(F_i\). Then solve \[ \widehat{m}(\cdot)=\underset{m(\cdot)\in F}{\text{argmax}}\left\lbrace \log\mathcal{L}(m(\boldsymbol{x});\boldsymbol{y}) \right\rbrace = \underset{m(\cdot)\in F}{\text{argmax}}\left\lbrace \sum_{i=1}^n \log f(y_i; m(\boldsymbol{x}_i)) \right\rbrace \] where \(\log\mathcal{L}\) denotes the log-likelihood.

In Machine Learning, given some dataset \((\boldsymbol{x}_i,y_i)\), solve \[ \widehat{m}(\cdot)=\underset{m(\cdot)\in F}{\text{argmin}}\left\lbrace \sum_{i=1}^n \ell(y_i,m(\boldsymbol{x}_i)) \right\rbrace \] for some loss functions \(\ell(\cdot,\cdot)\). See Varian (2015) or Kaldero & Taurasi (2015)

Consider an optimal smoothing problem: optimal \(h\) for \(\widehat{m}_h(\cdot)\)

Standard criteria, minimize the mean integrated squared error \(\widehat{m}_{\color{red}{h}}(\cdot)\), \[ mise(\color{red}{h})=\mathbb{E}\left[\int\left( m(\boldsymbol{x})-\widehat{m}_{\color{red}{h}}(\boldsymbol{x}) \right)^2d\boldsymbol{x}\right] \]

In Econometrics, massive use of asymptotic statistical properties, i.e. CLT, \[ \sqrt{\color{blue}{n}}(\overline{X} - \mathbb{E}[X])\ \overset{L}\rightarrow N(0,\;\text{Var}[X]) \] and Taylor expansion (delta method).

For some kernel based smoother, as \(\color{blue}{n}\rightarrow\infty\) \[ mise(\color{red}{h}) \sim \frac{\color{red}{h}^4}{4}\left(\int x^2K(\boldsymbol{x})d\boldsymbol{x} \right)^2\int\left(m''(\boldsymbol{x})+2m'(\boldsymbol{x})\frac{f'(\boldsymbol{x})}{f(\boldsymbol{x})}\right)d\boldsymbol{x} \\ +\frac{1}{\color{blue}{n}\color{red}{h}}\sigma^2\int K^2(\boldsymbol{x})d\boldsymbol{x} \int\frac{d\boldsymbol{x}}{f(\boldsymbol{x})} \] Thus, standard rule of thumb, \(\color{red}{h^\star \sim \color{blue}{n}^{-\frac{1}{5}}}\) (up to some multiplicative constant).

In Machine Learning, use bootstrap / cross validation.

E.g. for leave-on-out cross validation define \[ \widehat{mise}({\color{red}{h}})=\frac{1}{n}\sum [y_i - \widehat{m}_{\color{red}{h},(i)}(\boldsymbol{x}_i)]^2 \] then set \(\color{red}{h^\star}=\text{argmin}\lbrace\widehat{mise}({\color{red}{h}})\rbrace\)

Machine learning methods are about algorithms, more than asymptotic statistical properties.

Big data, scalability issues but good news to test microeconomic models, e.g. oil price, prixdescarburants.info and data.gouv.fr

Small data,

• sampling techniques (e.g. selection bias)

• aggregation techniques (e.g. ecological fallacy)

E.g. Simpson's Paradox, Blyth (1972), or Charpentier (2015)

For healthy people

while for sick people