Associate of the Society of Actuaries (ASA/FSA) Exam SRM — Statistics for Risk Modeling Syllabus

Why is adjusted $R^2$ preferred over $R^2$ when comparing models with different numbers of predictors?

$R^2$ never decreases when predictors are added, so it favors larger models. Adjusted $R^2$ penalizes added predictors: $$R^2_{adj} = 1 - \frac{\text{RSS}/(n-p-1)}{\text{TSS}/(n-1)},$$ increasing only if the new predictor improves fit beyond chance.

What hypothesis does the F-statistic in multiple regression test?

It tests $H_0: \beta_1 = \beta_2 = \cdots = \beta_p = 0$ (no predictor is related to $Y$) against the alternative that at least one $\beta_j \neq 0$. A large F (small p-value) rejects $H_0$.

How is the residual standard error (RSE) computed in multiple linear regression?

$$\text{RSE} = \sqrt{\frac{\text{RSS}}{n-p-1}}$$ It estimates the standard deviation of $\varepsilon$, in the units of $Y$.

How are interaction terms used in linear regression, and what does the hierarchy principle say?

An interaction term $X_1 X_2$ lets the effect of one predictor depend on the level of another. The hierarchy principle says if an interaction is included, the main effects $X_1$ and $X_2$ should be included too, even if individually insignificant.

List the key assumptions of the linear regression error term.

The errors $\varepsilon_i$ are independent, have constant variance (homoscedasticity), are normally distributed, and have mean zero. Additionally the relationship between predictors and response is linear.

What patterns in a residuals-versus-fitted-values plot indicate non-linearity and heteroscedasticity?

A curved (e.g., U-shaped) trend in residuals indicates non-linearity. A funnel shape (variance growing or shrinking with fitted values) indicates heteroscedasticity (non-constant variance).

What is multicollinearity and how is the variance inflation factor (VIF) used to detect it?

Multicollinearity is high correlation among predictors, which inflates coefficient standard errors. $$\text{VIF}_j = \frac{1}{1-R_j^{2}}$$ where $R_j^2$ is from regressing $X_j$ on the other predictors. VIF $> 5$ (or 10) signals problematic collinearity.

Distinguish outliers, high-leverage points, and influential points.

An outlier has an unusual $Y$ value given its $X$. A high-leverage point has an unusual $X$ value. An influential point substantially changes the fitted model when removed (often high leverage AND outlier); detected via Cook's distance.

Define the three subset-selection approaches: best subset, forward stepwise, and backward stepwise.

Best subset fits all $2^p$ models and picks the best. Forward stepwise starts with no predictors and adds the one that most improves fit at each step. Backward stepwise starts with all predictors and removes the least useful at each step.

Give the formulas for Mallows' $C_p$, AIC, and BIC used in model selection.

$$C_p = \frac{1}{n}(\text{RSS} + 2d\hat{\sigma}^{2}), \quad \text{AIC} = \frac{1}{n\hat{\sigma}^{2}}(\text{RSS} + 2d\hat{\sigma}^{2}), \quad \text{BIC} = \frac{1}{n}(\text{RSS} + \log(n)\, d\hat{\sigma}^{2})$$ where $d$ is the number of predictors. Lower is better; BIC penalizes more heavily.

Write the ridge regression objective and describe its effect on coefficients.

$$\min_{\beta} \sum_{i=1}^{n}\big(y_i - \beta_0 - \sum_j \beta_j x_{ij}\big)^{2} + \lambda \sum_{j=1}^{p}\beta_j^{2}$$ The $\ell_2$ penalty shrinks coefficients toward zero but never exactly to zero, reducing variance.

Write the lasso objective and state how it differs from ridge regression.

$$\min_{\beta} \text{RSS} + \lambda \sum_{j=1}^{p}|\beta_j|$$ The $\ell_1$ penalty can shrink coefficients exactly to zero, performing variable selection, whereas ridge ($\ell_2$) only shrinks them toward zero.

In regularization, what role does the tuning parameter $\lambda$ play, and how is it chosen?

$\lambda \geq 0$ controls penalty strength: $\lambda=0$ gives least squares; as $\lambda \to \infty$ coefficients shrink toward zero (raising bias, lowering variance). It is typically selected by cross-validation to minimize test error. Predictors should be standardized first.

What are the three components of a generalized linear model (GLM)?

(1) A random component: the response distribution from the exponential family; (2) a systematic component: the linear predictor $\eta = X\beta$; (3) a link function $g$ connecting them via $g(\mu) = \eta$, where $\mu = E[Y]$.

What is a link function in a GLM, and what is the canonical link?

The link function $g$ relates the mean response to the linear predictor: $g(\mu) = X\beta$. The canonical link is the one that maps $\mu$ to the natural parameter of the exponential-family distribution (e.g., logit for binomial, log for Poisson).

Why is ordinary least squares replaced by maximum likelihood estimation in GLMs?

GLM responses are typically non-normal with mean-dependent variance, so least squares assumptions fail. Coefficients are instead estimated by maximizing the likelihood, usually via iteratively reweighted least squares (IRLS).

State the logistic regression model and its link function.

$$\log\!\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p$$ The logit link gives $p = \frac{e^{X\beta}}{1+e^{X\beta}}$, modeling a binary response.

How do you interpret a coefficient $\beta_j$ in logistic regression in terms of odds?

A one-unit increase in $x_j$ multiplies the odds $\frac{p}{1-p}$ by $e^{\beta_j}$ (holding others fixed). Equivalently, $\beta_j$ is the change in log-odds per unit increase in $x_j$.

What link and distribution are used for Poisson regression, and what response does it model?

Poisson regression models count data with the log link: $$\log(\mu) = X\beta, \qquad \mu = e^{X\beta} > 0.$$ The response is assumed Poisson with $E[Y]=\operatorname{Var}(Y)=\mu$.

What is overdispersion in a Poisson GLM, and one way to address it?

Overdispersion occurs when $\operatorname{Var}(Y) > E[Y]$, violating the Poisson mean-variance equality. It can be addressed with a quasi-Poisson model (dispersion parameter $\phi$) or a negative binomial regression.

Define the deviance of a GLM and its use.

Deviance is $D = 2(\ell_{\text{saturated}} - \ell_{\text{model}})$, twice the log-likelihood gap between the fitted model and the saturated model. It measures goodness of fit (analogous to RSS); lower deviance means better fit, and deviance differences test nested models.

What is the AIC for a GLM and how is it used for model comparison?

$$\text{AIC} = -2\ell + 2k$$ where $\ell$ is the maximized log-likelihood and $k$ the number of parameters. Lower AIC indicates a better tradeoff of fit and complexity; it is used to compare (possibly non-nested) GLMs.

In a confusion matrix, define sensitivity, specificity, and precision.

Sensitivity (recall) $= \frac{TP}{TP+FN}$ (true positive rate); specificity $= \frac{TN}{TN+FP}$ (true negative rate); precision $= \frac{TP}{TP+FP}$ (positive predictive value).

What does an ROC curve plot, and what does AUC measure?

An ROC curve plots the true positive rate (sensitivity) against the false positive rate ($1-$specificity) across all classification thresholds. AUC (area under the curve) summarizes overall performance; $0.5$ is random guessing and $1.0$ is perfect.

Describe how a regression tree makes predictions and how splits are chosen.

A regression tree partitions the predictor space into regions and predicts the mean response of the region. Splits are chosen by recursive binary splitting to minimize RSS $\sum_{j}\sum_{i\in R_j}(y_i-\hat{y}_{R_j})^{2}$ greedily at each step.

What impurity measures are used to grow classification trees?

The Gini index $G = \sum_{k} \hat{p}_k(1-\hat{p}_k)$ and cross-entropy $D = -\sum_k \hat{p}_k \log \hat{p}_k$; both are small when a node is pure. Classification error rate is also defined but is less sensitive for growing trees.

What is cost-complexity (weakest-link) pruning of a decision tree?

It controls tree size by minimizing $$\sum_{m}\sum_{i\in R_m}(y_i-\hat{y}_{R_m})^{2} + \alpha |T|$$ where $|T|$ is the number of terminal nodes and $\alpha \geq 0$ is a tuning parameter (chosen by CV) that trades fit against tree complexity.

List key advantages and disadvantages of decision trees.

Advantages: easy to interpret/visualize, handle qualitative predictors without dummies, mirror human decisions. Disadvantages: high variance / non-robust, and generally lower predictive accuracy than ensemble methods.

How does bagging reduce variance, and what are out-of-bag (OOB) observations?

Bagging fits trees on many bootstrap samples and averages their predictions (or majority votes), reducing variance. OOB observations are those not in a given bootstrap sample (~$1/3$); averaging predictions over trees where an observation was OOB gives a built-in error estimate.

How do random forests improve on bagging?

At each split, a random forest considers only a random subset of $m$ predictors (often $m \approx \sqrt{p}$ for classification) instead of all $p$. This decorrelates the trees, so averaging them reduces variance more than bagging.

How does boosting build a predictive model, and how does it differ from bagging?

Boosting grows trees sequentially, each fitting the residuals of the current ensemble, with contributions scaled by a shrinkage parameter $\lambda$. Unlike bagging's independent parallel trees, boosting learns slowly and dependently, using small trees and tuning the number of trees to avoid overfitting.

Describe how K-nearest neighbors (KNN) makes a prediction and the effect of K.

KNN finds the $K$ training points nearest to $x_0$ and predicts by majority vote (classification) or average (regression) of their responses. Small $K$ gives a flexible, low-bias/high-variance fit; large $K$ gives a smoother, higher-bias/lower-variance fit.

What is Principal Components Analysis (PCA) and what does the first principal component maximize?

PCA finds orthogonal linear combinations of standardized variables (principal components) capturing successively maximal variance. The first principal component is the direction (loading vector of unit length) along which the projected data have the largest variance.

In PCA, what is the proportion of variance explained (PVE) by the $m$-th component?

$$\text{PVE}_m = \frac{\sum_{i=1}^{n} z_{im}^{2}}{\sum_{j=1}^{p}\sum_{i=1}^{n} x_{ij}^{2}}$$ the variance of the $m$-th component scores divided by total variance. A scree plot of PVE helps choose how many components to keep.

Why should variables be standardized before performing PCA?

PCA is scale-sensitive: variables with larger variances would dominate the components. Standardizing each variable to mean 0 and standard deviation 1 ensures all variables contribute comparably to the analysis.

Describe the K-means clustering algorithm.

Choose $K$; randomly assign points to clusters. Iterate: (1) compute each cluster's centroid (mean), (2) reassign each point to the nearest centroid. Repeat until assignments stop changing. It minimizes within-cluster sum of squares; rerun with multiple random starts since it finds a local optimum.

Compare K-means and hierarchical clustering.

K-means requires pre-specifying $K$ and partitions data flatly. Hierarchical clustering builds a nested tree (dendrogram) without pre-specifying $K$; cutting the dendrogram at a chosen height yields the clusters and reveals structure at multiple scales.

In agglomerative hierarchical clustering, define complete, single, and average linkage.

Linkage measures inter-cluster dissimilarity: complete linkage uses the largest pairwise distance, single linkage the smallest, and average linkage the mean of all pairwise distances between the two clusters' points.

Define a stationary time series (weak stationarity).

A time series is (weakly) stationary if its mean and variance are constant over time and its autocovariance $\operatorname{Cov}(Y_t, Y_{t+h})$ depends only on the lag $h$, not on $t$.

What is the random walk model and is it stationary?

$$Y_t = Y_{t-1} + \varepsilon_t$$ where $\varepsilon_t$ is white noise. It is non-stationary (its variance grows with $t$); differencing once, $Y_t - Y_{t-1} = \varepsilon_t$, produces a stationary white-noise series.

In time series, what are the trend, seasonal, and cyclical/irregular components of decomposition?

Trend is the long-term increase or decrease; seasonal is a regular pattern of fixed, known period (e.g., monthly); the cyclical/irregular (remainder) component is the residual fluctuation after removing trend and seasonality.

What is exponential smoothing and what does its smoothing parameter control?

Exponential smoothing forecasts with weights that decay geometrically into the past: $$\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha)\hat{Y}_t, \quad 0<\alpha<1.$$ Larger $\alpha$ weights recent observations more (responsive); smaller $\alpha$ gives smoother forecasts.