Quantitative Analysis
Quantitative Analysis supplies the estimation toolkit used in every later domain: probability, descriptive statistics, sampling distributions, hypothesis testing, regression, simulation, and volatility modeling. Questions are computational but short - you typically apply one formula correctly rather than derive it.
Core rules
- Bayes' theorem updates a prior with a likelihood: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ)].
- OLS requires linearity in parameters, no perfect collinearity, zero conditional mean of errors, homoskedasticity, and no autocorrelation for classic standard errors. R² always rises with added regressors; adjusted R² penalizes them.
- The CLT makes the sample mean approximately normal for large n regardless of the population; the LLN says the sample mean converges to the population mean.
- EWMA: σ²t = λσ²(t−1) + (1−λ)r²(t−1). GARCH(1,1) adds a long-run term: σ²t = ω + αr²(t−1) + βσ²(t−1), with persistence α + β; EWMA is the special case ω = 0, α + β = 1.
Worked example
A default-screening model flags firms as high risk. The base default rate is 2%; 90% of true defaulters are flagged, and 5% of non-defaulters are also flagged. P(flag) = 0.9(0.02) + 0.05(0.98) = 0.018 + 0.049 = 0.067. P(default | flag) = 0.018/0.067 = 26.9%. The flag lifts the probability from 2% to about 27%, yet most flagged firms still do not default - the base rate dominates.
Traps
- Reversing conditional probabilities (base-rate neglect).
- Treating a high R² as evidence of causation or of correct specification.
- Mixing the t-test for a single coefficient with the F-test for joint significance.
- Believing GARCH persistence above 1 is stable - it is not, and forecasts explode.