Free CAS MAS-I (Modern Actuarial Statistics I) Extended Linear Models Practice Questions

For an AR(1) model $Z_t = c + \phi Z_{t-1} + a_t$ with $\phi = 0.85$ and $\hat{\sigma}_a^2 = 13.645$:

The two-step ahead forecast error variance is:
$\text{Var}(e_t(2)) = \sigma_a^2(1 + \phi^2) = 13.645(1 + 0.85^2) = 13.645(1 + 0.7225) = 13.645(1.7225) = 23.504$

Forecast standard error:
$S(B) = \sqrt{23.504} = 4.848$

The answer is at least 4.0, but less than 7.0.

Choice A is incorrect because 4.848 > 4.0.
Choice C is incorrect because 4.848 < 7.0.
Choice E is incorrect because 4.848 < 10.0.
Choice B is incorrect because 4.848 < 13.0.

Choice D is correct.

The residual plot shows a clear funnel (megaphone) pattern: residuals near fitted value 2 are tightly clustered within $\pm 0.3$, but by fitted value 10 they spread to $\pm 2.5$. This increasing spread is the hallmark of heteroscedasticity — the variance of the response is not constant but grows with the mean. For a Poisson GLM, this suggests overdispersion (the actual variance exceeds what the Poisson model assumes). A quasi-Poisson or negative binomial model would be more appropriate.

Choice A is incorrect because the funnel pattern is a clear departure from a good fit, which would show uniform spread.
Choice B is incorrect because nonlinearity would appear as a curved pattern in the residuals (e.g., a U-shape), not a fan shape.
Choice C is incorrect because while a few residuals are large, the pattern is systematic across all fitted values — it is not a few isolated outliers.
Choice E is incorrect because there is no wave-like periodic pattern visible; the spread simply increases monotonically.

Let us evaluate each statement:

A: TRUE. Standardizing predictors before PCA is recommended because PCA is sensitive to the scale of variables.

D: FALSE. PCR does NOT perform feature selection in the traditional sense. It selects principal components (linear combinations of all original features), but it does not select or exclude individual features. All original variables contribute to each principal component. Feature selection methods like LASSO actually zero out coefficients.

C: TRUE. This is a known limitation/assumption of PCR -- it assumes that the directions of maximum variance in the predictors are the directions most associated with the response.

(B): TRUE. PCR can reduce overfitting by using fewer principal components than original features, effectively reducing dimensionality.

E: TRUE. By definition, the first principal component is the direction of maximum variance in the data.

Choice D is incorrect because this statement is true.
Choice
Choice A is correct because PCR does not perform feature selection -- it uses linear combinations of all features, not individual feature selection.
Choice C is incorrect because this statement is true -- it is a known assumption of PCR.
Choice B is incorrect because this statement is true.
Choice E is incorrect because this statement is true.