A researcher regresses U.S. GDP levels on Danish butter production over 1950–2010, finding R² = 0.94 and a t-statistic of 18 on the slope. What is the most likely explanation?
AButter production genuinely drives U.S. GDP through a food-economy linkage
BThe regression is spurious — both series are I(1) and trend upward, and OLS mistakes their shared stochastic trend for a causal relationship
CThe result is valid because R² = 0.94 always indicates a meaningful statistical relationship
DThe t-statistic of 18 is so extreme that spurious regression can be ruled out by any significance threshold
This is the classic spurious regression problem. Both GDP and butter production are I(1) — their levels wander upward due to stochastic trends. OLS interprets their shared upward drift as a relationship, producing high R² and inflated t-statistics. Crucially, standard t-test critical values are invalid for I(1) variables — even a t-statistic of 18 provides no evidence of a true relationship. Testing for stationarity before running regressions is not optional.
Question 2 Multiple Choice
You apply the ADF test to monthly CPI levels and fail to reject the null hypothesis at the 5% level. What is the most accurate conclusion?
AThe series definitely has a unit root and is non-stationary
BThe series is probably stationary, since failing to reject usually means the null is true
CYou cannot reject the presence of a unit root, but this does not prove non-stationarity — the test may have insufficient power
DThe series should be differenced twice, since failing the ADF test implies it is I(2)
The ADF test has a specific directionality: the null hypothesis IS unit root (non-stationarity). Failing to reject means you lack sufficient evidence to conclude stationarity — not that you've proven non-stationarity. ADF tests are known to have low power, especially with short samples or near-unit-root processes (ρ close to but less than 1). The correct conclusion is 'we cannot reject a unit root,' not 'this series has a unit root.' Combine ADF results with economic reasoning and visual inspection.
Question 3 True / False
A time series following a linear deterministic trend y_t = a + bt + ε_t is non-stationary and is expected to be first-differenced to achieve stationarity.
TTrue
FFalse
Answer: False
A deterministic trend can be removed by detrending — regressing y_t on time t and keeping the residuals — which yields a stationary series. First-differencing also removes the trend, but over-differences a trend-stationary series, introducing a unit root in the MA component and distorting the model. The key distinction: a stochastic trend (unit root) requires differencing; a deterministic trend requires detrending. Treating a deterministic trend as a unit root is a specification error.
Question 4 True / False
In a random walk y_t = y_{t-1} + ε_t, a positive shock to ε_t permanently raises the level of the series, with no tendency to revert toward a long-run mean.
TTrue
FFalse
Answer: True
This is the defining feature distinguishing I(1) series from stationary ones. In a stationary AR(1) with |ρ| < 1, shocks decay geometrically and the series returns to its mean. In a random walk (ρ = 1), the series simply starts from a new, permanently higher level after a positive shock and wanders from there — there is no mean to return to. Variance grows without bound (Var(y_t) = σ²t), directly violating constant-variance stationarity.
Question 5 Short Answer
Explain why regressing two unrelated I(1) series on each other typically produces a high R² and significant t-statistics, even though no true relationship exists.
Think about your answer, then reveal below.
Model answer: Both I(1) series have stochastic trends causing their levels to drift over the sample period. OLS minimizes squared residuals, and the best way to fit one drifting series with another is to find a slope that makes them track each other's long-run drift. The resulting fit appears strong (high R²) not because of causation, but because both series move in similar directions over time. Additionally, standard OLS critical values assume well-behaved (stationary) residuals; residuals from regressing two unrelated I(1) series are themselves non-stationary, so conventional t-test distributions don't apply and t-statistics can be extreme under the null of no relationship.
The diagnostic is to test whether regression residuals are stationary (cointegration, which would indicate a genuine long-run relationship) or non-stationary (spurious). High R² and significant coefficients alone cannot distinguish the two cases — only residual stationarity tests can.