Statistical Methods, Regression, and Tax-Aware Returns

This lesson pairs two toolkits. The first is statistical: methods that find and test the risk drivers behind alternative-investment returns. The second is tax-aware: methods for valuing and measuring private real estate.

PCA and statistical factors. The curriculum sorts asset factors into three categories: macro, dynamic, and statistical. With macro and dynamic factors, you begin from a series you already know (inflation, productivity, value, momentum, size) and then test whether it explains past return premiums. Statistical factors run the other way. You let the return data reveal candidate factors, and only afterward ask what they represent. Principal component analysis (PCA) is the standard tool for this.

PCA extracts components one at a time. The software first finds the vector of loadings (one per asset) that explains the largest possible share of total return variance, which is the first principal component. It then finds a second vector, orthogonal to the first, that explains as much of the remaining variance as possible. It continues until another component fails a user-specified (or default) stopping criterion.

Common mistakes

• Naming PCA components as economic factors before reading the loadings. PCA factors are pure variance-maximizing combinations, not theory-driven factors. The fix is to interpret ex-post, as with the level, slope, and curvature of Treasury returns. Trap value: assuming a component "is" a credit factor before examining which assets load on it.
• Running a "kitchen sink" regression with multicollinear factors. Correlated independents, such as a US and a non-US equity index, make slope estimates highly inaccurate and inflate standard errors, so coefficients look insignificant despite high R-squared.
• Confusing the stated tax rate with the effective tax rate. When depreciation is disallowed or slower than economic decline, the effective rate exceeds the stated rate and the after-tax IRR falls below pre-tax times one minus t. Trap value: assuming the after-tax IRR is always 60% of the pre-tax IRR.

Bottom line

• PCA extracts orthogonal components sequentially and outputs loadings and eigenvalues; $\lambda_j/\sum\lambda$ is a marginal share, and the sum of the retained eigenvalues divided by the total eigenvalue sum gives the cumulative share.
• The Fama-French model regresses excess returns on market, size, and value spreads; fitting realized returns does not prove a priced premium. Outliers, autocorrelation, and heteroskedasticity persist, and multicollinearity adds inaccurate slopes and inflated standard errors, which may be reduced using...
• The second-order partial autocorrelation equals $(\rho_2 - \rho_1^2)/(1 - \rho_1^2)$; the lag-k partial is the final coefficient in a regression on lags 1 through k, and an n-lag regression leaves only T minus n observations.
• Dummy variable, separate regression, and quadratic models test timing (positive $b_{i,diff}$ or $b_{im}$ suggests skill, after checking the intercept and luck); conditional correlation and rolling windows model changing correlation.

Exam shortcut

Before running a regression, commit to an economically justified factor set and check for multicollinearity by examining correlations among the independents. When two return factors are correlated, multicollinearity may be reduced using a return spread, so recheck the resulting correlations. Stepwise selection can trim the variable set, but it can still dredge up a model that generalizes poorly, so check the result against overfitting.

The full lesson (about 7,677 words, 51 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• model types
• credit risk economics
• structural model overview
• merton model
• kmv model
• reduced form models
• empirical credit models
• multifactor asset pricing
• fama french
• empirical mf challenges
• asset strategy complexity
• factor investing
• adaptive markets
• efficiently inefficient
• trend following
• divergence
• fundamental directional
• behavioral finance
• directional factors
• digital asset valuation
• pca statistical factors
• multifactor regression
• partial autocorrelations
• dynamic risk exposure
• changing correlation
• multifactor return approaches
• rv overview
• statistical pairs equities
• pairs commodity spreads
• pairs rates fx
• rv market neutral risks
• depreciation tax shields
• tax deferral gains
• after tax comparisons
• transaction based indices
• appraisal based indices

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