Predictor Variable

Study guide: Exam MAS-I

A predictor variable — also called an explanatory or independent variable, a covariate, a feature, or in ratemaking a rating variable — is an input $x_j$ used to explain or predict the response. Each predictor fills one or more columns of the design matrix and receives its own coefficient. Usage of covariate varies: Dobson & Barnett reserve it for a quantitative explanatory variable and call a qualitative one a factor.

$g(\mu_i) = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_p x_{ip}$

$\mu_i = E[Y_i]$ and $g$ is the link function (the identity in Linear Regression). $\beta_j$ is the effect of $x_j$ holding the other predictors fixed; with a log link, $e^{\beta_j}$ is a multiplicative relativity
Types (Data Types): a continuous predictor gets one coefficient, or a transformation when its effect is curved; a nominal categorical predictor with $k$ levels gets $k - 1$ indicator columns; an ordinal one can be treated either way. An Interaction lets one predictor's effect depend on the level of another
Not every input is a free predictor. A control variable is estimated so that it absorbs a real effect (a year trend, say) but is left out of the rates; an offset enters with its coefficient fixed at $1$ ; an identifier is not a predictor at all
Multicollinearity among predictors — vehicle age with vehicle value, building age with construction type — inflates each coefficient's variance by $\text{VIF}_j = 1/(1 - R_j^2)$ , where $R_j^2$ comes from regressing $x_j$ on the other predictors. Fitted values are largely unaffected; the interpretation of each coefficient is not
"Independent" variable does not mean statistically independent of the other predictors. In insurance data they almost never are, which is exactly why a multivariate model is needed instead of one-way analyses
Adding a predictor always lowers training error but can raise test error. Choosing them is variable selection, judged by significance tests, AIC, BIC and cross-validation

Assigning Roles in a Frequency Data Set A personal auto frequency file contains: claim count, earned car-years, accident year, territory ( $12$ levels), driver age, number of prior claims, and policy number. The rating plan will use territory, driver age and prior claims. State each field's role in a Poisson GLM with a log link.

Answer

Claim count — the response

Earned car-years — an offset, entered as $\ln(\text{car-years})$ with coefficient fixed at $1$ , so the model fits a frequency per car-year

Accident year — a control variable: estimated so that trend is not absorbed into the rating variables, then left out of the rates

Territory — a categorical predictor with $11$ indicator columns against a base territory

Driver age — continuous, but its effect on frequency is U-shaped, so it is banded or transformed rather than given a single slope

Prior claims — a count used as a predictor; usually capped (e.g. $0$ , $1$ , $2+$ ) and treated as ordinal

Policy number — an identifier, used to join tables; including it as a predictor would be meaningless

How Collinearity Hides a Real Effect In a severity model, regressing vehicle value on the other predictors gives $R_j^2 = 0.84$ . Had the predictors been uncorrelated, the vehicle-value coefficient of $0.15$ would have had a standard error of $0.04$ . Find the VIF, the actual standard error, and the $t$ -statistic before and after the inflation.
Answer
 $\begin{align*} \text{VIF} &= \frac{1}{1 - 0.84} \\ &= 6.25 \\ \text{SE} &= 0.04\sqrt{6.25} \\ &= 0.10 \end{align*}$ 
The $t$ -statistic falls from $0.15/0.04 = 3.75$ to $0.15/0.10 = 1.50$ . The effect may be entirely real, but the data cannot separate it from the correlated predictors, so vehicle value looks insignificant. The low $t$ -statistic does not show that the effect is absent. The two standard remedies are to drop one of the collinear predictors, usually at little cost to the fit since the information it carries is largely redundant, or to combine them into a single predictor.