Statistics & Econometrics

What makes one estimator better than another, the bias–variance decomposition of MSE, consistency, efficiency, and the Cramér–Rao bound that caps how good any unbiased estimator can be.

The estimator that asks "what parameters make the data I saw most probable", the score equation, Fisher information, the invariance property, and the asymptotic normality that makes MLE the default engine of parametric inference.

Estimation by matching sample moments to their theoretical counterparts, the classical method of moments, its generalization to more moment conditions than parameters (GMM), and why GMM underlies asset-pricing tests.

The Neyman–Pearson framework for deciding between hypotheses under uncertainty, type I and II errors, power, the most-powerful-test lemma, and the likelihood-ratio test that generalizes it.

What a confidence interval actually claims (and what it does not), how to construct one by inverting a test, its exact duality with hypothesis testing, and why the honest interval on a Sharpe ratio is uncomfortably wide.

What a p-value is and the five things it is not, why testing thousands of signals guarantees false discoveries, and the Bonferroni and Benjamini–Hochberg corrections that keep a backtesting pipeline honest.

The workhorse linear estimator, derived in matrix form, with its geometry, the Gauss-Markov optimality result, its sampling distribution, and the assumptions that break in financial data.

The precise statement that OLS is BLUE, Best Linear Unbiased Estimator, with the assumptions it needs, a full proof that any other linear unbiased estimator has larger variance, and exactly what the theorem does and does not promise.

When error variance is not constant, why it leaves OLS unbiased but wrecks its standard errors, the White/robust sandwich covariance that fixes inference, and the tests that detect it.

When regression errors are correlated across time, why it inflates or deflates naive standard errors, the Newey–West HAC estimator, the Durbin–Watson statistic, and the overlapping-returns trap that fools every backtester.

When regressors are highly correlated, how a near-singular X'X inflates coefficient variances, the variance inflation factor that quantifies it, why signs flip and estimates go unstable, and the shrinkage cure.

The deepest OLS failure, a regressor correlated with the error, which biases and un-fixes the estimator no matter the sample size, and the instrumental-variables / 2SLS machinery that recovers a causal effect.

The property that makes a time series learnable, strict versus weak (covariance) stationarity, why every estimator implicitly assumes it, and how transforming prices into returns is really a hunt for a stationary series.

The linear building blocks of time-series forecasting, AR, MA, and ARMA processes, their ACF/PACF signatures, the stationarity and invertibility conditions via the lag polynomial, and why returns are nearly unforecastable by them.

How to model the one robust feature of returns, volatility clustering, from ARCH to GARCH, the persistence and stationarity conditions, forecasting the variance term structure, and the leverage and fat-tail extensions desks actually use.

The random walk and the difference between a stochastic and deterministic trend, the augmented Dickey–Fuller test, its non-standard null distribution, and the spurious-regression trap that manufactures fake relationships between unrelated prices.

When two non-stationary series share a common stochastic trend so a linear combination is stationary, the Engle–Granger and Johansen tests, the error-correction representation, and the econometric foundation of pairs trading.

Deliberately biasing a regression to cut its variance, the ridge and LASSO penalties, the L2-versus-L1 geometry that makes LASSO select and ridge shrink, and why penalization is essential in the low-signal, high-collinearity world of finance.

Estimating the sampling distribution of any statistic by resampling the data itself, the plug-in principle, why it works, the block bootstrap that preserves time-series dependence, and how to build confidence intervals for things with no formula.

Treating parameters as random and updating beliefs with data, priors, posteriors, conjugacy, credible intervals, and the deep result that shrinkage and regularization are Bayesian estimation in disguise.