MS-DS Master of Data science Master of Data Science: Statistical Inference & Regression Models 1

Question 1

In simple linear regression, which condition does the Gauss-Markov theorem require for the OLS estimator to be BLUE (Best Linear Unbiased Estimator)?

Accepted Answer

The error terms must have constant variance and be uncorrelated

Answer

The Gauss-Markov theorem guarantees OLS is BLUE under the conditions of linearity, exogeneity, homoscedasticity (constant error variance), and no autocorrelation. Normality of errors is NOT required for BLUE status—it is only needed for exact inference (t-tests, F-tests) in finite samples.

Question 2

A researcher fits a multiple regression model and finds the F-statistic is highly significant (p < 0.001), yet none of the individual t-tests for the coefficients are significant. What is the most likely explanation?

Accepted Answer

Severe multicollinearity is inflating the standard errors of individual coefficients

Answer

When the overall F-test is significant but individual t-tests are not, severe multicollinearity is the classic culprit. Highly correlated predictors inflate standard errors, making it difficult to isolate each variable's unique contribution, even though collectively they explain significant variance.

Question 3

What does the p-value of 0.03 in a hypothesis test formally represent?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme than the one obtained, given that H₀ is true

Answer

A p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming the null hypothesis is true. It does NOT give the probability that H₀ is true or false—that would require a Bayesian framework.

Question 4

In a regression context, what is the consequence of omitting a variable that is correlated with both the outcome and an included predictor?

Accepted Answer

Biased and inconsistent coefficient estimates for the included predictors

Answer

Omitting a confounding variable causes omitted variable bias (OVB). Because the omitted variable correlates with an included predictor, the OLS estimator absorbs part of the omitted variable's effect into the included predictor's coefficient, making it biased and inconsistent.

Question 5

Which transformation is most appropriate to stabilize variance (correct heteroscedasticity) when the residual spread increases proportionally with the fitted values?

Accepted Answer

Log transformation of the response variable

Answer

When variance grows proportionally with the mean (a common pattern in count or ratio data), a log transformation of the response variable compresses large values more than small ones, stabilizing variance and often linearizing a multiplicative relationship.

MS-DS Master of Data science Practice Test

MS-DS Master of Data science Practice Test

MS-DS Master of Data science Master of Data Science: Statistical Inference & Regression Models 1