Module M08 Quiz: Time Series — Stationarity, Trends & Distributed Lags
12 questions · Graduate (ECMT5001) · MCQ, calculation, short answer
How to use
Attempt each question before clicking Show Answer. Time-series questions often require careful thought about the direction of causality and stationarity.
Question 1
Lesson L01 · Data Types
Which feature distinguishes time series data from cross-sectional data?
Type: Multiple Choice
- A) Time series data always has more observations
- B) In time series, observations are ordered by time and serial correlation in errors is a realistic concern; randomness of sampling cannot be assumed
- C) Time series data cannot be used for regression
- D) Cross-sectional data has more variables
Show Answer
Answer: B
In cross-sectional data, observations are assumed to be independently drawn from a population (random sampling). In time series, each observation follows the previous one chronologically — errors are likely correlated across time (serial correlation), and the random sampling assumption (MLR.2) is replaced by a stationarity/weak dependence assumption. The ordering also means we must be careful about temporal causality (past causes present, not vice versa).
Question 2
Lesson L01 · Strict vs Weak Exogeneity
Strict exogeneity requires \(E(u_t \mid x_1, x_2, \ldots, x_T) = 0\) for all \(t\). This is violated when:
Type: Multiple Choice
- A) The error has a non-normal distribution
- B) A lagged dependent variable \(y_{t-1}\) appears as a regressor, or when there is feedback from \(y_t\) to future \(x_{t+1}\)
- C) The regressors are correlated with each other
- D) The sample size is small
Show Answer
Answer: B
If \(y_{t-1}\) is a regressor: \(y_{t-1} = \beta_0 + \beta_1 x_{t-1} + u_{t-1}\), so \(y_{t-1}\) depends on \(u_{t-1}\). But the strict exogeneity condition requires \(\text{Cov}(u_{t-1}, x_t) = 0\) for ALL lags/leads — which fails if \(x_t\) depends on \(y_{t-1}\) through feedback. Similarly, if government policy (\(x\)) responds to last period's GDP (which contains \(u_{t-1}\)), strict exogeneity fails. Contemporaneous exogeneity \(E(u_t \mid x_t) = 0\) is weaker and more plausible.
Question 3
Lesson L04 · Stationarity
A weakly stationary time series requires:
Type: Multiple Choice
- A) \(E(y_t) = \mu\) (constant mean), \(\text{Var}(y_t) = \sigma^2\) (constant variance), and \(\text{Cov}(y_t, y_{t-s})\) depends only on the lag \(s\) (not on \(t\))
- B) All observations are normally distributed
- C) No serial correlation: \(\text{Cov}(y_t, y_{t-s}) = 0\) for all \(s > 0\)
- D) Constant mean only
Show Answer
Answer: A
Weak (covariance) stationarity requires three conditions: 1. Constant mean: \(E(y_t) = \mu\) for all \(t\) 2. Constant variance: \(\text{Var}(y_t) = \sigma^2 < \infty\) for all \(t\) 3. Autocovariance depends only on lag: \(\text{Cov}(y_t, y_{t-s}) = \gamma(s)\) — not on when you are in the series, only on how far apart
Option C (no serial correlation) is much stronger — white noise has zero autocovariance, but stationary series can have nonzero autocovariance (they just need it to depend only on \(s\), not \(t\)).
Question 4
Lesson L04 · AR(1) Stationarity
[Calculation] For an AR(1) process \(y_t = \rho y_{t-1} + e_t\) with \(e_t\) i.i.d., stationarity requires \(|\rho| < 1\). If \(\rho = 1\) (random walk), what happens to \(\text{Var}(y_t)\) as \(t \to \infty\)?
Type: Calculation
Show Answer
For a random walk (\(\rho = 1\)): \(y_t = y_0 + \sum_{s=1}^t e_s\)
The variance grows linearly with time — the series wanders without bound and never returns to a fixed mean. This is the defining feature of non-stationarity for a random walk.
Contrast: for \(|\rho| < 1\) (stationary AR(1)): $\(\text{Var}(y_t) = \frac{\sigma_e^2}{1 - \rho^2} \quad \text{(finite and constant)}\)$
Question 5
Lesson L03 · Spurious Regression
A researcher regresses non-stationary GDP on non-stationary CO₂ emissions and gets \(R^2 = 0.91\), \(t = 24.3\) (highly significant). What is the most likely explanation?
Type: Short Answer
Show Answer
Spurious regression. Both GDP and CO₂ emissions have upward trends (they are non-stationary, likely \(I(1)\) — integrated of order 1). When two trending series are regressed on each other, OLS will find a strong statistical relationship even if they are completely unrelated (driven by independent random walks with drift).
The high \(R^2\) and large \(t\)-statistic are artefacts of the shared trend, not a causal link. Standard OLS inference is completely invalid here — \(t\)-statistics don't follow \(t\) distributions with non-stationary variables.
Remedies: - Test for a unit root (ADF test) - If both are \(I(1)\), first-difference both series and re-estimate - Or test for cointegration (if they share a common stochastic trend, the error-correction model is appropriate)
Question 6
Lesson L02 · Static vs FDL Models
A Finite Distributed Lag (FDL) model \(y_t = \alpha + \delta_0 x_t + \delta_1 x_{t-1} + \delta_2 x_{t-2} + u_t\) is preferred over a static model \(y_t = \alpha + \delta_0 x_t + u_t\) when:
Type: Multiple Choice
- A) The sample size is small
- B) The effect of \(x\) on \(y\) takes time to materialise — there are delayed or persistent effects
- C) \(x\) and \(y\) are both stationary
- D) Heteroskedasticity is suspected
Show Answer
Answer: B
The FDL captures dynamic effects: the impact of a change in \(x\) at time \(t\) on \(y\) at times \(t\), \(t+1\), \(t+2\), etc. Classic examples: - Monetary policy: interest rate change affects output with a 12–18 month lag - Advertising: spending in one month affects sales across multiple future months
\(\delta_0\) = impact multiplier (immediate effect) \(\delta_0 + \delta_1 + \delta_2\) = long-run multiplier (total cumulative effect after 2 periods)
The static model (\(\delta_1 = \delta_2 = 0\)) is misspecified if lagged effects exist — OVB results from omitting \(x_{t-1}\) and \(x_{t-2}\).
Question 7
Lesson L02 · Long-Run Multiplier
[Calculation] FDL(2) model: \(y_t = 1 + 0.4x_t + 0.3x_{t-1} + 0.1x_{t-2} + u_t\). Calculate the impact multiplier, the 1-period dynamic multiplier, and the long-run multiplier.
Type: Calculation
Show Answer
-
Impact multiplier \(\delta_0 = \mathbf{0.4}\): a one-unit increase in \(x_t\) immediately raises \(y_t\) by 0.4, holding all else fixed
-
1-period dynamic multiplier \(\delta_0 + \delta_1 = 0.4 + 0.3 = \mathbf{0.7}\): the cumulative effect after one additional period
-
Long-run multiplier \(= \sum_{j=0}^{2} \delta_j = 0.4 + 0.3 + 0.1 = \mathbf{0.8}\): total cumulative effect of a permanent one-unit increase in \(x\) on \(y\) after 2 periods
Interpretation: a permanent $1 increase in advertising spending raises sales by $0.40 immediately, $0.70 within one period, and $0.80 in the long run.
Question 8
Lesson L04 · Detrending
A time-series regression of \(y_t\) on \(x_t\) produces spuriously high \(R^2\) because both series trend upward. What are two remedies?
Type: Short Answer
Show Answer
Remedy 1 — Include a time trend \(t\):
Add \(t\) (a linear time index) as a regressor: \(y_t = \beta_0 + \beta_1 x_t + \beta_2 t + u_t\). This "detrends" both \(y\) and \(x\) within the regression — \(\hat{\beta}_1\) now measures the relationship after removing common linear trend effects. Works when trends are deterministic (constant slope).
Remedy 2 — First-difference:
Estimate \(\Delta y_t = \beta_1 \Delta x_t + \Delta u_t\) where \(\Delta y_t = y_t - y_{t-1}\). Differencing removes a stochastic (random-walk) trend. More appropriate than detrending when the trend is stochastic (unit root present). If \(y\) and \(x\) are both \(I(1)\), \(\Delta y\) and \(\Delta x\) are \(I(0)\) (stationary) and OLS on the differenced model is valid.
Question 9
Lesson L04 · Unit Root Intuition
An AR(1) with \(\rho = 0.95\) is stationary. An AR(1) with \(\rho = 1.00\) is a random walk (non-stationary). In practice, these are hard to distinguish. What test is used?
Type: Multiple Choice
- A) Breusch-Pagan test
- B) Augmented Dickey-Fuller (ADF) test
- C) Durbin-Watson test
- D) White test
Show Answer
Answer: B — Augmented Dickey-Fuller (ADF) test
The ADF tests \(H_0: \rho = 1\) (unit root, non-stationary) vs \(H_1: |\rho| < 1\) (stationary). The test statistic does not follow a standard \(t\)-distribution under \(H_0\) — special critical values (Dickey-Fuller critical values) are needed.
"Augmented" means the regression includes lagged differences \(\Delta y_{t-1}, \ldots\) to absorb serial correlation in errors. The ADF is known for low power — it often fails to reject \(H_0\) even when \(\rho\) is close to but below 1. This is why non-stationarity tests should be interpreted carefully alongside graphical inspection.
Question 10
Lesson L03 · Spurious Regression Detection
After estimating a time-series regression, which diagnostic would most directly reveal if the results are spurious (due to both variables trending)?
Type: Multiple Choice
- A) High \(R^2\) and significant \(t\)-statistics (these are signs of spurious regression, not diagnostics for it)
- B) Plot residuals over time: if the residuals show strong persistence (slow mean reversion), this suggests the regression is spurious — the residuals themselves look like a random walk
- C) Breusch-Pagan test for heteroskedasticity
- D) RESET test for functional form
Show Answer
Answer: B
In a valid cointegrating regression, residuals are stationary (mean-reverting). In a spurious regression, the residuals look like a random walk — they drift far from zero and take a long time to return. Visual inspection of the residual plot, together with a unit root test on the residuals (Engle-Granger cointegration test), is the standard diagnostic.
High \(R^2\) and significant \(t\)-statistics are actually warning signs of spurious regression, not confirmations of a genuine relationship — the counterintuitive opposite of cross-sectional data.
Question 11
Lesson L01 · Weak Dependence
OLS is consistent in time series under weak exogeneity and weak dependence (rather than the i.i.d. assumption). What does weak dependence require?
Type: Short Answer
Show Answer
Weak dependence requires that the correlation between \(y_t\) and \(y_{t-s}\) decays to zero as the lag \(s \to \infty\): \(\gamma(s) \to 0\) as \(s \to \infty\).
This ensures that distant observations are "nearly independent," allowing a Law of Large Numbers and Central Limit Theorem to apply to time-series data. It is satisfied by: - Stationary ARMA processes with \(|\rho| < 1\) - Most stable economic time series with short-lived shocks
It is violated by random walks (\(\rho = 1\)) — shocks never die out, so observations arbitrarily far apart remain strongly correlated. This is why non-stationary processes require special treatment (differencing, cointegration).
Question 12
Lesson L02 · FDL Application
[Short Answer] A government spending shock of \(\$1\) billion occurs at time \(t\). Using the FDL model \(\Delta\text{GDP}_t = 0.6\Delta G_t + 0.3\Delta G_{t-1} + 0.1\Delta G_{t-2}\), what is the total multiplier effect on GDP over 3 periods?
Type: Short Answer
Show Answer
The Keynesian fiscal multiplier through a distributed-lag model:
- Period \(t\): \(\Delta\text{GDP}_t = 0.6 \times 1 = \mathbf{\$0.6B}\)
- Period \(t+1\): \(\Delta\text{GDP}_{t+1} = 0.3 \times 1 = \mathbf{\$0.3B}\)
- Period \(t+2\): \(\Delta\text{GDP}_{t+2} = 0.1 \times 1 = \mathbf{\$0.1B}\)
Total (long-run multiplier) = 0.6 + 0.3 + 0.1 = $1.0B
This is a fiscal multiplier of 1.0 — each dollar of government spending raises GDP by $1 in total over 3 periods (the effect is fully transmitted but with delays). Note this is below the simple Keynesian multiplier of \(k = 1/(1-c)\) because we are not modelling the full income-expenditure dynamics here — just the direct statistical relationship from the FDL.