CFA Level II Quantitative Methods

CFA Level II Quantitative Methods Volume 1 Review Notes - Main Concepts, Definitions, and Formulas This PDF is a concise, original study guide based on the uploaded CFA Program 2026 Level II Volume 1 Quantitative Methods material. It is structured by learning module and focuses on review-ready concepts and formulas. Section Topics covered LM1 Multiple regression basics and assumptions LM2 Regression fit, hypothesis tests, and forecasting LM3 Model misspecification and assumption violations LM4 Influence analysis, dummy variables, and logistic regression LM5 Time-series analysis LM6 Machine learning LM7 Big data projects Appendix Formula reference and exam reminders CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 1 - Basics of Multiple Regression and Underlying Assumptions • Multiple linear regression models one dependent variable using two or more independent variables. • Investment uses include factor exposure analysis, distress prediction, country-risk analysis, forecasting, and testing theories. • A regression model must be specified before estimation: choose the dependent variable, relevant independent variables, functional form, and objective. • A slope coefficient in multiple regression is a partial effect: it measures the change in the dependent variable for a one-unit change in one independent variable, holding the others constant. • If the dependent variable is binary or categorical rather than continuous, logistic regression is usually more appropriate than linear regression. Multiple linear regression model Y_i = b_0 + b_1 X_1i + b_2 X_2i + ... + b_k X_ki + e_i Component Meaning Y_i Dependent variable for observation i. b_0 Intercept. Expected value of Y when all X variables are zero. b_j Slope coefficient for independent variable X_j. It is a partial effect. X_ji Value of independent variable j for observation i. e_i Residual/error term for observation i. k Number of independent variables. n Number of observations. n must exceed k for estimation. Use: Estimate how several explanatory variables jointly relate to a dependent variable. Estimated/predicted regression value Y_hat_i = b_hat_0 + b_hat_1 X_1i + ... + b_hat_k X_ki Component Meaning Y_hat_i Predicted value of the dependent variable for observation i. b_hat_j Estimated coefficient from the sample data. X_ji Observed value of independent variable j. Use: Use the estimated coefficients to calculate fitted or forecast values. Core regression assumptions Assumption Meaning Warning sign Linearity Relationship between Y and the X variables is linear in parameters. Scatterplot suggests curvature or missing nonlinear term. Homoskedasticity Residual variance is constant across observations. Residuals fan out or cluster as predicted values change. Independence of errors Residuals are not correlated across observations. Pattern over time or serial correlation in residuals. Normality Residuals are normally distributed. Q-Q plot deviates materially from diagonal line. Independent variables not perfectly collinear No exact linear relationship among the X variables. High correlations or exact redundancy among regressors. Diagnostic plots Page 2 CFA Level II Quantitative Methods - Volume 1 Review Notes • Scatterplot of Y versus each X: checks whether a linear relation is plausible and whether outliers exist. • Residuals versus predicted values: checks homoskedasticity, nonlinearity, and outliers. • Residuals versus each X: checks for omitted patterns and extreme values in the independent variables. • Normal Q-Q plot of residuals: checks whether residuals are approximately normal; tail deviations suggest non-normality/fat tails. Page 3 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 2 - Evaluating Regression Model Fit and Interpreting Model Results • Goodness-of-fit measures describe how well the independent variables explain variation in the dependent variable. • In multiple regression, adjusted R-squared is preferred to raw R-squared because raw R-squared never decreases when variables are added. • AIC and BIC compare models with the same dependent variable; lower is better. AIC is often preferred for prediction, while BIC penalizes complexity more heavily. • Use t-tests for individual coefficients and F-tests for joint hypotheses or overall model significance. • Forecasting requires including the intercept and all variables in the estimated model, even if some are not individually significant. Total, explained, and unexplained variation SST = SSR + SSE Component Meaning SST Sum of squares total: total variation in Y around its mean. SSR Sum of squares regression: variation explained by the model. SSE Sum of squares error: unexplained variation/residual variation. Use: ANOVA decomposes the variation of Y into explained and unexplained parts. R-squared R^2 = SSR / SST = 1 - (SSE / SST) Component Meaning R^2 Fraction of total variation in Y explained by the regression. SSR Explained sum of squares. SST Total sum of squares. SSE Error sum of squares. Use: Higher R-squared means more in-sample variation is explained, but it can rise just because more variables are added. Adjusted R-squared Adj R^2 = 1 - [ ((n - 1) / (n - k - 1)) * (1 - R^2) ] Component Meaning Adj R^2 Goodness-of-fit measure adjusted for model size. n Number of observations. k Number of independent variables. R^2 Regular R-squared. Use: Use to compare multiple regression models while penalizing unnecessary variables. It increases if a newly added variable has |t-stat| greater than 1.0. Page 4 CFA Level II Quantitative Methods - Volume 1 Review Notes Akaike information criterion (AIC) AIC = n * ln(SSE / n) + 2 * (k + 1) Component Meaning AIC Model selection statistic; lower is better. n Number of observations. SSE Sum of squared residuals/errors. k Number of independent variables. k+1 Number of estimated coefficients including the intercept. Use: Often useful when the model objective is prediction. Bayesian information criterion (BIC) BIC = n * ln(SSE / n) + ln(n) * (k + 1) Component Meaning BIC Model selection statistic; lower is better. ln(n) Natural log of sample size; creates a stronger penalty for complexity than AIC. Other terms Same as AIC. Use: Often preferred when the objective is parsimonious goodness of fit. t-test for one coefficient t = (b_hat_j - B_j) / SE(b_hat_j) Component Meaning b_hat_j Estimated slope coefficient. B_j Hypothesized value under the null, usually 0. SE(b_hat_j) Standard error of the estimated slope coefficient. df Degrees of freedom = n - k - 1. Use: Tests whether one variable has a statistically significant coefficient, holding other variables constant. Joint F-test for nested models F = [ (SSE_R - SSE_U) / q ] / [ SSE_U / (n - k - 1) ] Component Meaning SSE_R SSE from the restricted model, with q variables excluded or constrained. SSE_U SSE from the unrestricted model, with the full set of variables. q Number of restrictions or excluded variables. k Number of independent variables in the unrestricted model. n-k-1 Denominator degrees of freedom. Use: Tests whether excluded variables jointly add explanatory power. Reject H0 if calculated F exceeds the critical F-value. Page 5 CFA Level II Quantitative Methods - Volume 1 Review Notes Overall regression F-test F = MSR / MSE Component Meaning MSR Mean square regression = SSR / k. MSE Mean square error = SSE / (n - k - 1). H0 All slope coefficients equal zero. Ha At least one slope coefficient is not zero. Use: Tests whether the full regression has explanatory power. Forecast using multiple regression Y_hat_f = b_hat_0 + b_hat_1 X_1f + b_hat_2 X_2f + ... + b_hat_k X_kf Component Meaning Y_hat_f Forecasted value of the dependent variable. X_jf Assumed or forecasted value of independent variable j. b_hat_j Estimated slope coefficient for variable j. b_hat_0 Estimated intercept. Use: Forecast intervals widen when the independent variables are themselves forecasts because model error and sampling error both matter. Page 6 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 3 - Model Misspecification • Model misspecification occurs when the regression model does not match the true relationship in the data. • Common specification errors include omitted variables, wrong functional form, inappropriate scaling, using the wrong variable form, and inappropriate pooling of data. • Misspecification can create biased coefficients, invalid inference, unreliable forecasts, and misleading economic interpretations. • Always combine statistical tests with economic reasoning. A statistically significant model can still be economically nonsensical. Common specification problems Problem Typical issue Possible fix Omitted variable A relevant variable is excluded and correlated with included variables. Add the missing variable or a reasonable proxy. Wrong functional form True relation is nonlinear, but a linear model is imposed. Add nonlinear terms, logs, or transformations. Wrong variable form Level, change, percentage change, or log form is inappropriate. Use economically justified transformations. Inappropriate scaling Variables are not scaled consistently across companies or time. Scale by size, market value, assets, sales, etc. Inappropriate pooling Different groups/time periods are combined despite different relationships. Use dummy variables, interaction terms, or separate models. Heteroskedasticity • Heteroskedasticity means the variance of residuals is not constant across observations. • Unconditional heteroskedasticity is usually not a major problem; conditional heteroskedasticity is related to the level of independent variables and affects inference. • OLS coefficients may remain consistent, but standard errors are biased, so t-tests and F-tests can be unreliable. • Detection: residual plots, Breusch-Pagan test, White test. Correction: robust standard errors or respecify the model. Serial correlation • Serial correlation means residuals are correlated across time. It is common in time-series regressions. • Positive serial correlation often makes standard errors too small, causing t-statistics to look too significant. • Detection: residual autocorrelations, Durbin-Watson statistic for AR(1) error patterns, Breusch-Godfrey in broader settings. • Correction: use robust standard errors, add lagged variables, or respecify the time-series model. Durbin-Watson statistic DW = sum_{t=2 to T} (e_t - e_{t-1})^2 / sum_{t=1 to T} e_t^2 Component Meaning DW Test statistic for first-order serial correlation in residuals. e_t Regression residual at time t. T Number of time-series observations. Use: DW near 2 suggests no first-order serial correlation. DW below 2 suggests positive serial correlation; above 2 suggests negative serial correlation. Multicollinearity • Multicollinearity means independent variables are highly correlated with one another or form near-linear combinations. Page 7 CFA Level II Quantitative Methods - Volume 1 Review Notes • It does not necessarily bias coefficients, but it inflates standard errors and lowers individual t-statistics. • The overall regression may be significant even when individual coefficients are not. • Detection: high pairwise correlations, high variance inflation factors, unstable coefficient signs/magnitudes. • Correction: drop redundant variables, use a different proxy, combine variables, or increase sample size. Variance inflation factor (VIF) VIF_j = 1 / (1 - R_j^2) Component Meaning VIF_j Variance inflation factor for independent variable X_j. R_j^2 R-squared from regressing X_j on the other independent variables. Use: VIF = 1 indicates no relation with other regressors. VIF above 5 deserves investigation; above 10 is often serious. Page 8 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 4 - Extensions of Multiple Regression Influence analysis • An outlier is an observation with an unusual value relative to the rest of the sample. • A high-leverage observation has unusual values for independent variables and can pull the fitted regression line. • An influential observation materially changes regression coefficients or fitted values if removed. • Influential points are not automatically errors; investigate whether they are data mistakes, special events, or economically valid observations. Predicted residual for identifying unusual residuals Standardized residual = e_i / estimated standard deviation of residuals Component Meaning e_i Residual for observation i. Standardized residual Residual measured in standard-deviation units. Use: Large absolute standardized residuals suggest potential outliers. Dummy variables and interactions • A dummy variable equals 1 if an observation belongs to a category and 0 otherwise. • With m categories, include m - 1 dummy variables to avoid perfect multicollinearity; the omitted category is the base case. • An intercept dummy shifts the intercept for a group. A slope interaction changes the slope for a group. • Interaction terms are useful when the effect of one variable depends on the level or category of another variable. Regression with one dummy variable Y_i = b_0 + b_1 X_i + b_2 D_i + e_i Component Meaning D_i Dummy variable equal to 1 for the group of interest and 0 otherwise. b_0 Intercept for the base group where D_i = 0. b_2 Difference in intercept between the D_i = 1 group and the base group. Use: Use when the average level of Y differs by category. Regression with dummy interaction term Y_i = b_0 + b_1 X_i + b_2 D_i + b_3 (D_i * X_i) + e_i Component Meaning b_1 Slope of X for the base group. b_3 Difference in slope for the D_i = 1 group. b_1 + b_3 Slope of X when D_i = 1. Use: Use when the relation between X and Y differs by category. Qualitative dependent variables and logistic regression • When the dependent variable is binary, a linear probability model can predict probabilities outside the 0 to 1 range and often has heteroskedastic errors. • Logistic regression models the probability of a binary outcome and keeps predicted probabilities between 0 and 1. • Logistic coefficients are interpreted through log-odds or changes in probability, not as direct changes in Y like OLS coefficients. Page 9 CFA Level II Quantitative Methods - Volume 1 Review Notes • Classification output is assessed using confusion matrices, accuracy, precision, recall, F1 score, ROC curves, and AUC. Logistic regression probability p_i = 1 / (1 + exp[-(b_0 + b_1 X_1i + ... + b_k X_ki)]) Component Meaning p_i Predicted probability that observation i belongs to class 1. exp Exponential function. b_j Logistic regression coefficients. X_ji Independent variables for observation i. Use: Use for binary dependent variables, such as default/non-default or target/non-target. Log odds form of logistic regression ln[p_i / (1 - p_i)] = b_0 + b_1 X_1i + ... + b_k X_ki Component Meaning p_i / (1 - p_i) Odds of class 1 versus class 0. ln Natural logarithm. b_j Change in log-odds for a one-unit change in X_j, holding other variables constant. Use: A positive coefficient increases the odds of class 1; a negative coefficient decreases the odds. Page 10 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 5 - Time-Series Analysis • A time series is a set of observations indexed in time order. Time-series modeling requires attention to trend, seasonality, serial correlation, changing variance, and nonstationarity. • A covariance-stationary time series has constant mean, constant finite variance, and autocovariances that depend only on lag, not on calendar time. • Forecasts should be evaluated out of sample when possible. A model that fits the past well can still forecast poorly. Trend models Linear trend model y_t = b_0 + b_1 * t + e_t Component Meaning y_t Value of the time series at time t. t Time index, such as 1, 2, 3, ... b_0 Intercept or starting level. b_1 Change in y per time period. e_t Error term at time t. Use: Use when the series changes by approximately the same absolute amount each period. Log-linear trend model ln(y_t) = b_0 + b_1 * t + e_t Component Meaning ln(y_t) Natural log of the time-series value. b_1 Approximate continuously compounded growth rate. Growth rate Approximate growth = exp(b_1) - 1. Use: Use when the series grows at an approximately constant percentage rate. Autoregressive models AR(1) model x_t = b_0 + b_1 x_{t-1} + e_t Component Meaning x_t Current value of the time series. x_{t-1} One-period lagged value. b_0 Intercept. b_1 Autoregressive coefficient. e_t Random error. Use: Use when the current value depends on its own past value. Page 11 CFA Level II Quantitative Methods - Volume 1 Review Notes Mean-reverting level for AR(1) Mean-reverting level = b_0 / (1 - b_1) Component Meaning b_0 Intercept in the AR(1) model. b_1 Coefficient on the lagged variable. Condition For covariance stationarity, |b_1| must be less than 1. Use: If the series is stationary, forecasts tend to move toward this long-run level. One-step-ahead AR(1) forecast x_hat_{t+1} = b_0 + b_1 x_t Component Meaning x_hat_{t+1} Forecast for next period. x_t Current observed value. b_0, b_1 Estimated AR(1) coefficients. Use: Apply the AR equation using the latest observed value. Two-step-ahead AR(1) forecast x_hat_{t+2} = b_0 + b_1 x_hat_{t+1} Component Meaning x_hat_{t+2} Forecast two periods ahead. x_hat_{t+1} One-period-ahead forecast used recursively. Use: Multi-period forecasts use previous forecasts as inputs. Random walks and unit roots • A random walk is nonstationary: shocks have permanent effects and the variance increases with the forecast horizon. • A random walk with drift has a predictable average change but remains nonstationary. • Unit root tests assess whether the autoregressive coefficient is equal to 1. If a unit root exists, first differencing can transform the series into a form suitable for analysis. • Do not use trend or AR models designed for stationary series on a nonstationary series without appropriate transformation. Random walk x_t = x_{t-1} + e_t Component Meaning x_t Current value. x_{t-1} Previous value. e_t Random shock. Use: Best forecast of the next value is often the current value. Page 12 CFA Level II Quantitative Methods - Volume 1 Review Notes Random walk with drift x_t = b_0 + x_{t-1} + e_t Component Meaning b_0 Drift term: average change per period. Other terms Same as random walk. Use: Forecast changes by the drift amount each period. First difference Delta x_t = x_t - x_{t-1} Component Meaning Delta x_t Change in the series from t-1 to t. x_t Current level. x_{t-1} Previous level. Use: Often used to transform a unit-root series into a stationary series. Moving-average, seasonality, ARMA, and ARCH n-period moving average MA_t = (x_t + x_{t-1} + ... + x_{t-n+1}) / n Component Meaning MA_t Moving average at time t. n Number of periods included. x_t Most recent observation. Use: Smooths short-term fluctuations by averaging recent observations. ARCH(1) variance model sigma_t^2 = a_0 + a_1 * e_{t-1}^2 Component Meaning sigma_t^2 Conditional variance at time t. a_0 Base variance term. a_1 Sensitivity of current variance to last period squared residual. e_{t-1}^2 Previous period squared residual/shock. Use: Models volatility clustering, where large shocks tend to be followed by high volatility. Root mean squared error (RMSE) RMSE = sqrt[ (1 / T) * sum_{t=1 to T} (actual_t - forecast_t)^2 ] Component Meaning RMSE Forecast error metric; lower is better. T Number of forecast periods. actual_t Observed value at time t. forecast_t Predicted value at time t. Use: Useful for comparing out-of-sample forecast accuracy. Page 13 CFA Level II Quantitative Methods - Volume 1 Review Notes Regressions with more than one time series • Regressing nonstationary time series on each other can produce spurious relationships unless the series are cointegrated. • Cointegration means two nonstationary series have a stable long-term relationship; the residual from their relationship is stationary. • If variables are cointegrated, an error-correction model can capture both short-run dynamics and long-run equilibrium adjustment. Page 14 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 6 - Machine Learning • Machine learning uses algorithms to identify patterns and generate predictions from data, often with less emphasis on explicitly specified economic relationships. • Supervised learning uses labeled data with a target variable; unsupervised learning searches for structure without a target variable. • Training error measures in-sample fit. Validation and test sets are used to evaluate generalization to unseen data. • Overfitting occurs when a model learns noise rather than signal; underfitting occurs when a model is too simple to capture the true pattern. • In finance, ML is useful for prediction, classification, signal extraction, clustering, risk detection, text analysis, and alternative data projects. Supervised algorithms Algorithm Main idea Typical use Penalized regression Adds a penalty to reduce coefficient size and control overfitting. High-dimensional prediction and feature selection. Support vector machine Finds a separating boundary with the largest margin. Classification with complex boundaries. K-nearest neighbor Classifies/predicts based on nearby observations. Simple nonlinear classification/regression. CART Splits data into decision-tree regions. Interpretable classification/regression. Random forest Averages many randomized decision trees. Improved predictive accuracy and lower overfitting. Neural networks Layers of connected nodes model complex nonlinear patterns. Large datasets, images, text, speech, complex interactions. Penalized regression Ridge regression objective Minimize: SSE + lambda * sum_{j=1 to k} b_j^2 Component Meaning SSE Sum of squared prediction errors. lambda Penalty strength chosen by tuning/cross-validation. b_j^2 Squared coefficient for variable j. Use: Shrinks coefficients toward zero but usually does not set them exactly to zero. LASSO objective Minimize: SSE + lambda * sum_{j=1 to k} |b_j| Component Meaning |b_j| Absolute value of coefficient j. lambda Penalty strength. SSE Sum of squared prediction errors. Use: Can shrink some coefficients exactly to zero, so it can perform feature selection. Page 15 CFA Level II Quantitative Methods - Volume 1 Review Notes Elastic net objective Minimize: SSE + lambda1 * sum |b_j| + lambda2 * sum b_j^2 Component Meaning lambda1 LASSO-style penalty strength. lambda2 Ridge-style penalty strength. b_j Coefficient for feature j. Use: Combines LASSO feature selection with ridge shrinkage. Unsupervised algorithms • Principal components analysis reduces many correlated variables into fewer uncorrelated components that explain maximum variance. • K-means clustering assigns observations to k clusters by minimizing within-cluster distances. • Hierarchical clustering builds nested groups and is often visualized with a dendrogram. • Clustering can identify groups based on data characteristics rather than conventional sectors, countries, or labels. PCA linear component PC_1 = w_1 X_1 + w_2 X_2 + ... + w_k X_k Component Meaning PC_1 First principal component. w_j Weight/loadings on variable X_j. X_j Original standardized variable j. Use: The first component captures the largest possible variance; later components are uncorrelated with earlier components. K-means objective Minimize: sum over clusters c sum over points i in c distance(x_i, center_c)^2 Component Meaning x_i Observation i. center_c Centroid of cluster c. distance Usually Euclidean distance. k Number of clusters chosen by the analyst. Use: Assigns observations to clusters with similar features. Evaluation and overfitting controls • Use cross-validation, holdout test sets, regularization, pruning, early stopping, and simpler models to reduce overfitting. • Bias-variance trade-off: simple models may have high bias/underfit; complex models may have high variance/overfit. • Classification performance should be evaluated using metrics appropriate to the cost of false positives and false negatives. Page 16 CFA Level II Quantitative Methods - Volume 1 Review Notes Accuracy Accuracy = (TP + TN) / (TP + TN + FP + FN) Component Meaning TP True positives: predicted positive and actually positive. TN True negatives: predicted negative and actually negative. FP False positives: predicted positive but actually negative. FN False negatives: predicted negative but actually positive. Use: Overall share of correct classifications. Can be misleading when classes are imbalanced. Precision Precision = TP / (TP + FP) Component Meaning Precision Among predicted positives, the share that are actually positive. TP, FP True positives and false positives. Use: Important when false positives are costly. Recall Recall = TP / (TP + FN) Component Meaning Recall Among actual positives, the share correctly detected. TP, FN True positives and false negatives. Use: Important when missing a positive case is costly. F1 score F1 = 2 * (Precision * Recall) / (Precision + Recall) Component Meaning Precision Quality of positive predictions. Recall Ability to capture actual positives. Use: Balances precision and recall in one metric. Page 17 CFA Level II Quantitative Methods - Volume 1 Review Notes Learning Module 7 - Big Data Projects • Big data projects require clear business objectives, data sourcing, data preparation, data exploration, model training, performance evaluation, and communication of results. • Investment data may be structured, semi-structured, or unstructured. Text data require special preparation before modeling. • Data quality issues include incompleteness, invalidity, inconsistency, non-uniformity, and duplication. • The project team should combine domain knowledge, quantitative modeling, technology, and clear governance. Big data characteristics Characteristic Meaning in investment context Volume Large amount of observations, features, documents, or transactions. Velocity Data arrive or need to be processed quickly, sometimes in real time. Variety Data come in many forms: numeric tables, text, images, audio, web data. Veracity Data reliability, noise, bias, errors, or fake/low-quality observations. Project workflow Stage Purpose Business understanding Define the investment question, success metric, constraints, and user of the model. Data collection/curation Identify sources, permissions, coverage, reliability, and relevance. Preparation/wrangling Clean, validate, transform, aggregate, filter, and convert data. Data exploration Use EDA, visualization, feature selection, and feature engineering. Model training Choose method, train model, tune parameters, and control overfitting. Evaluation Assess out-of-sample performance, robustness, errors, and model limitations. Interpretation/results Translate statistical output into investment decisions and communicate clearly. Structured data preparation • Extraction: create a variable or feature from existing fields. • Aggregation: combine multiple observations into a summary measure. • Filtration: remove irrelevant observations or variables. • Selection: keep chosen features for analysis or model training. • Conversion: change data types, scales, units, or categories into usable model inputs. Min-max normalization Scaled value = (x - x_min) / (x_max - x_min) Component Meaning x Original value. x_min Minimum value of that feature in the sample. x_max Maximum value of that feature in the sample. Use: Transforms values to a 0-to-1 range. Page 18 CFA Level II Quantitative Methods - Volume 1 Review Notes Standardization (z-score) z = (x - mean_x) / standard_deviation_x Component Meaning z Standardized value. x Original value. mean_x Average of the feature. standard_deviation_x Standard deviation of the feature. Use: Transforms a feature to mean 0 and standard deviation 1. Text data preparation • Text cleansing commonly removes irrelevant HTML tags, unnecessary punctuation, extra white space, and many numbers; useful symbols may need to be preserved or annotated. • Tokenization splits text into tokens, usually words or word-like units. • Normalization may include lowercasing, stop-word removal, stemming, and lemmatization. • A bag-of-words represents text as a set of tokens without preserving full grammar or word order. • N-grams preserve limited word order by creating sequences of n tokens. • A document term matrix (DTM) stores documents as rows, tokens as columns, and token counts or weights in the cells. Term frequency - inverse document frequency (TF-IDF) TF-IDF(term, document) = TF(term, document) * IDF(term) Component Meaning TF Term frequency: how often a term appears in a document or sentence. IDF Inverse document frequency: downweights terms that appear in many documents. Term Token/word being measured. Document Text unit such as article, filing, paragraph, or sentence. Use: Highlights terms that are frequent in one document but not common across all documents. Performance evaluation in big data projects • Regression problems often use RMSE or similar forecast-error measures. • Classification problems use a confusion matrix and metrics such as accuracy, precision, recall, and F1 score. • ROC curves compare true positive rates and false positive rates across thresholds; higher AUC generally indicates better classification ability. • Tuning balances bias and variance; performance should be judged primarily on validation/test data, not training data. Page 19 CFA Level II Quantitative Methods - Volume 1 Review Notes Quick Exam Review Checklist Regression and time-series reminders • For coefficient interpretation in multiple regression, always add: holding all other independent variables constant. • Adjusted R-squared, AIC, and BIC are model comparison tools; raw R-squared can reward overfitting. • Reject a joint F-test only if the calculated F-statistic exceeds the critical F-value. • Heteroskedasticity and serial correlation mainly threaten standard errors and inference; multicollinearity mainly inflates standard errors and destabilizes individual coefficients. • For time series, check stationarity before trusting AR models. Unit roots imply nonstationarity and may require differencing. • Use out-of-sample RMSE to compare forecasts when possible. Machine learning and big data reminders • Supervised learning has a target label; unsupervised learning does not. • Overfitting means the model captures noise. Use regularization, pruning, cross-validation, or simpler models. • Precision answers: of the predicted positives, how many were correct? Recall answers: of the actual positives, how many were captured? • Feature selection removes weak or noisy features; feature engineering creates or transforms features to improve model usefulness. • Data veracity is often the key concern with alternative data, web data, and social media data. Formula Reference Concept Formula Multiple regression Y_i = b_0 + b_1 X_1i + ... + b_k X_ki + e_i R-squared R^2 = SSR / SST = 1 - SSE / SST Adjusted R-squared Adj R^2 = 1 - [((n - 1) / (n - k - 1)) * (1 - R^2)] AIC AIC = n * ln(SSE / n) + 2 * (k + 1) BIC BIC = n * ln(SSE / n) + ln(n) * (k + 1) t-statistic t = (b_hat_j - B_j) / SE(b_hat_j) Joint F-test F = [ (SSE_R - SSE_U) / q ] / [ SSE_U / (n - k - 1) ] Overall F-test F = MSR / MSE VIF VIF_j = 1 / (1 - R_j^2) Durbin-Watson DW = sum(e_t - e_{t-1})^2 / sum(e_t^2) AR(1) x_t = b_0 + b_1 x_{t-1} + e_t AR(1) mean Mean = b_0 / (1 - b_1) Random walk x_t = x_{t-1} + e_t First difference Delta x_t = x_t - x_{t-1} RMSE RMSE = sqrt[(1/T) * sum(actual_t - forecast_t)^2] Logistic probability p = 1 / (1 + exp[-(b_0 + b_1 X_1 + ... + b_k X_k)]) Accuracy (TP + TN) / (TP + TN + FP + FN) Precision TP / (TP + FP) Recall TP / (TP + FN) F1 score 2 * precision * recall / (precision + recall) Page 20 CFA Level II Quantitative Methods - Volume 1 Review Notes Min-max scaling (x - x_min) / (x_max - x_min) Z-score (x - mean) / standard deviation End of notes. 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