Question bank — Assumptions of linear regression

False. Non-constant variance leaves $\hat{\beta }$ unbiased; it only makes OLS inefficient and its standard errors wrong. The line still lands in the right place on average — you just can't trust the confidence intervals. See Weighted Least Squares for the efficient fix.

False. Gauss–Markov gives BLUE from linearity, exogeneity, independence, and equal variance alone. Normality is only needed for exact $t$-tests and $F$-tests — see Hypothesis Testing in Regression.

False. It means $E[Y\mid X]$ is linear in the coefficients. After a $\log X$ or $X^2$ feature, a visually curved cloud can perfectly satisfy linearity.

False. With perfect collinearity, $X^{T}X$ is exactly singular — the inverse does not exist and OLS has no unique solution at all. Near-perfect correlation is what gives huge-but-computable coefficients.

False. $R^2$ only measures fraction of variance explained. You can hit $R^2=0.95$ while residuals show a clean U-shape (linearity broken) or a funnel (heteroscedasticity).

False. Independence is about the errors ${ϵ}_i$ across observations (rows of $X$). Multicollinearity is about predictor columns of $X$ being linearly related. Different axes of the data, different problems.

False in effect. The Central Limit Theorem makes the sampling distribution of $\hat{\beta }$ approximately normal for large $n$, so $t$-tests stay approximately valid even when individual errors aren't normal.

False. Homoscedasticity is precisely the assumption that the error variance $\text{Var}({ϵ}_i\mid X_i)={\sigma }^2$ is constant. It constrains the spread of the noise $ϵ$ at each $X$, not the marginal spread of $Y$ — those are different quantities.

True (roughly). $d\approx 2$ signals no first-order autocorrelation. But $d$ only catches lag-1 correlation — it can miss seasonal or higher-lag dependence, so it's necessary-ish, not sufficient.

Error: wrong diagnosis. A funnel (see left panel of the first figure) is the signature of heteroscedasticity (growing variance). Nonlinearity shows as a curve or U-shape, not a fan.

Error: it's $\frac{1}{1-R_j^2}$. As $R_j^2\to 1$ (predictor $j$ perfectly explained by the others), the denominator $\to 0$ and VIF blows up — that's the whole point.

Error. That spread is the heteroscedasticity, not a data flaw. The fix is a $\log Y$ transform or Weighted Least Squares with weights $w_i=1/{\sigma }_i^2$, not deletion — deleting valid points biases the model.

Error. Robust SEs correct the reported uncertainty, but the point estimates $\hat{\beta }$ are unchanged and still inefficient. Only reweighting (WLS/GLS) recovers efficiency.

Error: wrong tool. Ridge and Lasso Regression treats multicollinearity (correlated columns of $X$). Correlated errors call for GLS, ARIMA, or mixed-effects models.

Error: it's right-skewed. Points rising above the diagonal in the upper-right (right panel of the second figure) mean the right tail is heavier than normal — large positive residuals — i.e. right skew.

Error: backwards. A large condition number (a huge ratio of largest to smallest eigenvalue) signals near-singularity. A small one means the predictors are comfortably independent.

Positive autocorrelation means nearby observations carry redundant information, so the effective sample size is smaller than $n$. OLS's standard-error formula assumes independence and thus divides by too-large an $n$, shrinking SEs and making $p$-values falsely tiny.

OLS minimizes plain squared residuals, giving every point equal weight. The optimal (GLS) weight is $1/{\sigma }_i^2$, which down-weights high-variance points. Equal weighting therefore lets the noisiest observations pull the fit more than they should — see Ordinary Least Squares (OLS) vs Weighted Least Squares.

When two columns of $X$ nearly overlap, the smallest eigenvalue ${\lambda }_{\min }$ of $X^{T}X$ is close to $0$, and $(X^{T}X{)}^{-1}$ has entries $\propto 1/{\lambda }_{\min }$. Tiny perturbations get amplified enormously, so the model can't decide how to split shared information — signs and magnitudes swing wildly.

If the errors enter multiplicatively rather than additively — a model of the form $Y=({\beta }_0+{\beta }_{1}X)\cdot ϵ$ where the noise scales the mean instead of being added to it — then the spread grows with the mean. Taking logs turns that product into a sum ($\log Y=\log ({\beta }_0+{\beta }_{1}X)+\log ϵ$), converting a proportional (fanning) spread into a constant additive one, flattening the funnel. It only helps when the variance genuinely scales with the mean this way.

Prediction needs only that the fitted line is the right expectation, which BLUE guarantees without normality. Inference (confidence intervals, $p$-values) needs the exact shape of $\hat{\beta }$'s sampling distribution to invoke $t$ and $F$ tables — and that exact shape comes from normal errors (or large-$n$ CLT).

Pairwise correlations only catch two-variable overlap. A predictor can be a linear combination of three or more others while every pairwise correlation stays modest — VIF (which regresses each predictor on all the rest) catches this; a heatmap does not.

The CLT concerns the average behavior of many error terms feeding into $\hat{\beta }$; convergence to normal needs enough terms. At $n=15$ there aren't enough, so heavy tails still distort the sampling distribution — hence bootstrap or robust regression for small samples.

Even a correctly-shaped linear model is pulled off-target if the errors correlate with $X$ (e.g. an omitted variable riding in the noise). Only $E[ϵ\mid X]=0$ guarantees the fitted line's slope has no systematic tilt — so $\hat{\beta }$ is unbiased.

Counts have variance tied to their mean (Poisson-like), so variance cannot be constant — a built-in heteroscedasticity. The principled move is a GLM with a Poisson family, not plain OLS.

No CLT help at $n=8$, so $t$-tests are untrustworthy. Use bootstrap resampling or robust regression, and treat any reported $p$-value with heavy skepticism.

By the usual VIF < 5 rule they scrape under the line, but 4.9 is borderline. Interpret coefficients cautiously; if they concern the same underlying quantity, consider combining them (PCA) or a ridge penalty.

With zero residuals you can't assess linearity, homoscedasticity, or normality — the diagnostics are empty. Perfect fit usually signals overfitting or an exact linear dependence, and screams to inspect for hidden collinearity (see Bias-Variance Tradeoff).

$X^{T}X$ is guaranteed singular (rank at most $n<p$), so OLS has no unique solution — a total collinearity failure. You must reduce dimensions or use a penalized method like Lasso.

Yes, approximately. BLUE holds regardless of shape, and $n=300$ lets the CLT normalize $\hat{\beta }$'s sampling distribution, so intervals and tests are close to correct despite the skew.

A constant column is perfectly collinear with the intercept column of $X$, making $X^{T}X$ singular — extreme multicollinearity. Its coefficient is unidentifiable and must be dropped.