,TAM2601 Advanced Prep: Master
Quantitative Methods & Statistical Analysis
Practice Questions
Subject: TAM2601 (Quantitative Methods) - Advanced Statistical Inference and
Regression Analysis
Question 1: In a multiple linear regression model, which of the following scenarios most strongly
suggests a violation of the homoscedasticity assumption, necessitating a remedial transformation
or the use of Weighted Least Squares (WLS)?
A) The Durbin-Watson statistic is significantly lower than 1.5.
B) A Goldfeld-Quandt test yields a p-value of 0.02, while the residuals exhibit a "fan-shaped"
plot against predicted values.
C) The Variance Inflation Factor (VIF) for two independent variables exceeds 10.
D) The Shapiro-Wilk test on the residuals indicates a p-value of 0.35.
Correct Answer: B) A Goldfeld-Quandt test yields a p-value of 0.02, while the residuals
exhibit a "fan-shaped" plot against predicted values.
Explanation: Homoscedasticity assumes that the error variance is constant across all levels of
the independent variables. A "fan-shaped" plot is the classic diagnostic indicator of
heteroscedasticity, which the Goldfeld-Quandt test statistically confirms (at $\alpha=0.05$).
Option A refers to autocorrelation, Option C refers to multicollinearity, and Option D relates to
the normality of residuals.
Question 2: Given a dataset with $n=100$ and a regressor matrix $X$ that is highly ill-
conditioned, which regularization technique is theoretically most effective at stabilizing
coefficient estimates by introducing a shrinkage penalty without performing model selection?
A) LASSO Regression
B) Ridge Regression (Tikhonov Regularization)
C) Stepwise Forward Selection
D) Ordinary Least Squares (OLS)
Correct Answer: B) Ridge Regression (Tikhonov Regularization)
, Explanation: Ridge regression adds an $L_2$ penalty term ($\lambda \sum \beta_j^2$) to the
loss function. This effectively reduces the variance of the coefficient estimates in the presence of
multicollinearity (an ill-conditioned $X$ matrix). LASSO ($L_1$ penalty) is better for feature
selection, while OLS is BLUE only when assumptions are met and the matrix is not ill-
conditioned.
Question 3: In the context of time-series analysis, if a process is described as $AR(1)$, what
condition must hold for the process to be considered covariance-stationary?
A) The intercept term must be exactly zero.
B) The autoregressive parameter $|\phi| < 1$.
C) The variance of the white noise term must be greater than 1.
D) The mean of the series must be time-dependent.
Correct Answer: B) The autoregressive parameter $|\phi| < 1$.
Explanation: For an $AR(1)$ process $Y_t = c + \phi Y_{t-1} + \epsilon_t$, the process is
covariance-stationary only if the roots of the characteristic equation lie outside the unit circle,
which, in the case of $AR(1)$, simplifies to the absolute value of the autoregressive coefficient
being less than 1.
Question 4: Which of the following best characterizes the impact of Type II error ($\beta$) in a
hypothesis testing framework where the null hypothesis involves a critical safety threshold?
A) Increasing the sample size $n$ will simultaneously increase both $\alpha$ and $\beta$.
B) The Power of the test is defined as $1 - \alpha$.
C) A Type II error occurs when we fail to reject a false null hypothesis, effectively missing a
significant effect or danger.
D) Type II error is independent of the effect size.
Correct Answer: C) A Type II error occurs when we fail to reject a false null hypothesis,
effectively missing a significant effect or danger.
Explanation: Type II error ($\beta$) represents a "false negative." In high-stakes testing, failing
to detect an effect when it exists (e.g., failing to identify a hazardous trend) can be more
dangerous than a Type I error. Power is defined as $1-\beta$, not $1-\alpha$.
Question 5: When performing an Analysis of Variance (ANOVA), what does the "Within-
Groups" Mean Square (MSW) represent?
Quantitative Methods & Statistical Analysis
Practice Questions
Subject: TAM2601 (Quantitative Methods) - Advanced Statistical Inference and
Regression Analysis
Question 1: In a multiple linear regression model, which of the following scenarios most strongly
suggests a violation of the homoscedasticity assumption, necessitating a remedial transformation
or the use of Weighted Least Squares (WLS)?
A) The Durbin-Watson statistic is significantly lower than 1.5.
B) A Goldfeld-Quandt test yields a p-value of 0.02, while the residuals exhibit a "fan-shaped"
plot against predicted values.
C) The Variance Inflation Factor (VIF) for two independent variables exceeds 10.
D) The Shapiro-Wilk test on the residuals indicates a p-value of 0.35.
Correct Answer: B) A Goldfeld-Quandt test yields a p-value of 0.02, while the residuals
exhibit a "fan-shaped" plot against predicted values.
Explanation: Homoscedasticity assumes that the error variance is constant across all levels of
the independent variables. A "fan-shaped" plot is the classic diagnostic indicator of
heteroscedasticity, which the Goldfeld-Quandt test statistically confirms (at $\alpha=0.05$).
Option A refers to autocorrelation, Option C refers to multicollinearity, and Option D relates to
the normality of residuals.
Question 2: Given a dataset with $n=100$ and a regressor matrix $X$ that is highly ill-
conditioned, which regularization technique is theoretically most effective at stabilizing
coefficient estimates by introducing a shrinkage penalty without performing model selection?
A) LASSO Regression
B) Ridge Regression (Tikhonov Regularization)
C) Stepwise Forward Selection
D) Ordinary Least Squares (OLS)
Correct Answer: B) Ridge Regression (Tikhonov Regularization)
, Explanation: Ridge regression adds an $L_2$ penalty term ($\lambda \sum \beta_j^2$) to the
loss function. This effectively reduces the variance of the coefficient estimates in the presence of
multicollinearity (an ill-conditioned $X$ matrix). LASSO ($L_1$ penalty) is better for feature
selection, while OLS is BLUE only when assumptions are met and the matrix is not ill-
conditioned.
Question 3: In the context of time-series analysis, if a process is described as $AR(1)$, what
condition must hold for the process to be considered covariance-stationary?
A) The intercept term must be exactly zero.
B) The autoregressive parameter $|\phi| < 1$.
C) The variance of the white noise term must be greater than 1.
D) The mean of the series must be time-dependent.
Correct Answer: B) The autoregressive parameter $|\phi| < 1$.
Explanation: For an $AR(1)$ process $Y_t = c + \phi Y_{t-1} + \epsilon_t$, the process is
covariance-stationary only if the roots of the characteristic equation lie outside the unit circle,
which, in the case of $AR(1)$, simplifies to the absolute value of the autoregressive coefficient
being less than 1.
Question 4: Which of the following best characterizes the impact of Type II error ($\beta$) in a
hypothesis testing framework where the null hypothesis involves a critical safety threshold?
A) Increasing the sample size $n$ will simultaneously increase both $\alpha$ and $\beta$.
B) The Power of the test is defined as $1 - \alpha$.
C) A Type II error occurs when we fail to reject a false null hypothesis, effectively missing a
significant effect or danger.
D) Type II error is independent of the effect size.
Correct Answer: C) A Type II error occurs when we fail to reject a false null hypothesis,
effectively missing a significant effect or danger.
Explanation: Type II error ($\beta$) represents a "false negative." In high-stakes testing, failing
to detect an effect when it exists (e.g., failing to identify a hazardous trend) can be more
dangerous than a Type I error. Power is defined as $1-\beta$, not $1-\alpha$.
Question 5: When performing an Analysis of Variance (ANOVA), what does the "Within-
Groups" Mean Square (MSW) represent?
