Understand how to use a hierarchical model.

A pricing analyst has five years of loss ratios for each of 40 territories. Territory means vary, sample sizes vary, and the grand mean is informative. A hierarchical model gives you per-territory estimates that borrow strength from the rest of the portfolio.

Why hierarchy. Observations inside the same group are correlated. OLS treats them as independent and understates standard errors. A hierarchical model puts the correlation in the model where it belongs and yields a posterior-style estimate for each group that pulls noisy small-sample groups toward the grand mean.

Index Level-1 observations $i = 1,\dots,n_j$ within Level-2 group $j = 1,\dots,J$. The simplest hierarchical model is:

$$y_{ij} = \beta_0 + u_j + \varepsilon_{ij}, \quad u_j \sim N(0,\tau^2),\;\varepsilon_{ij}\sim N(0,\sigma^2)$$

The fixed part is $\beta_0$; the random part is $u_j + \varepsilon_{ij}$. Two variance components: between-group $\tau^2$ and within-group $\sigma^2$.

Intraclass correlation. The correlation between any two observations in the same group is:

$$\rho = \frac{\tau^2}{\tau^2 + \sigma^2}$$

Common mistakes

• Reporting $\bar{y}_j$ as the hierarchical estimate. The shrunk estimate is $\hat{\beta}_0 + \hat{u}_j$, not $\bar{y}_j$. Quoting 0.820 for territory C overstates expected loss ratio by about 5 points.
• Using ML log-likelihoods to compare variance structures. Variance-component comparisons require REML log-likelihoods (with identical fixed effects). ML biases $\hat{\tau}^2$ downward and gives misleading LRTs across random structures.
• Treating the Level-2 group identifier as a fixed effect with 200 dummies. That is the no-pooling OLS approach. Group means have no shrinkage, no borrowing of strength, and inflated MSE for thinly observed groups.

Bottom line

• Hierarchical means grouped data with random effects. Level-1 units (claims, years, students) nest inside Level-2 groups (territories, schools, drivers), and random effects capture group-to-group variation.
• Random intercept model: $y_{ij} = \beta_0 + u_j + \varepsilon_{ij}$, with $u_j \sim N(0,\tau^2)$, $\varepsilon_{ij}\sim N(0,\sigma^2)$. ICC $=\tau^2/(\tau^2+\sigma^2)$ is the share of total variance between groups.
• BLUP shrinkage: the predicted group effect $\hat{u}_j = \lambda_j(\bar{y}_j - \hat{\beta}_0)$, with $\lambda_j = \tau^2/(\tau^2 + \sigma^2/n_j)$. Larger $n_j$ shrinks less.
• Shrinkage is toward the fixed-effect intercept, not zero; the group rate is $\hat{\beta}_0 + \hat{u}_j$, not the raw mean $\bar{y}_j$.

Exam shortcut

If the question gives $\tau^2$, $\sigma^2$, $n_j$, and $\bar{y}_j$, compute the BLUP via $\lambda_j(\bar{y}_j - \hat{\beta}_0)$ and add to $\hat{\beta}_0$; the same skeleton answers most random-intercept items. If two groups share $\bar{y}_j$ but differ in $n_j$, the larger group gets the larger $\lambda_j$ and lands farther from the grand mean; rank by $n_j$ without recomputing.

The full lesson (about 2,679 words, 18 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B2

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