ECS4863 Advanced Econometrics Prep:
Master Time Series and Forecasting Practice
Questions & Detailed Explanations
Subject: ECS4863 Econometrics - Time Series Analysis and Dynamic Modeling
Question 1: In the context of a stationary $AR(1)$ process defined by $y_t = \phi y_{t-1} +
\epsilon_t$, what is the necessary and sufficient condition for the process to be covariance-
stationary, and what is the implication for the impulse response function?
A) $|\phi| < 1$; the impulse response dies out exponentially.
B) $|\phi| \leq 1$; the impulse response persists indefinitely.
C) $\phi = 1$; the impulse response is constant.
D) $|\phi| > 1$; the impulse response grows without bound.
Correct Answer: A) $|\phi| < 1$; the impulse response dies out exponentially.
Explanation: For a first-order autoregressive process to be covariance-stationary, the root of the
characteristic equation must lie outside the unit circle, which implies $|\phi| < 1$. When this
condition holds, the effect of a shock to $\epsilon_t$ at time $t$ on $y_{t+k}$ is given by
$\phi^k$, which converges to zero as $k \to \infty$, characterizing exponential decay.
Question 2: Consider a $VAR(p)$ model. If the roots of the characteristic polynomial $\det(I_n -
\Phi_1 z - ... - \Phi_p z^p) = 0$ lie outside the unit circle, what can be concluded about the
system?
A) The system contains a deterministic trend.
B) The system is non-stationary and requires cointegration analysis.
C) The system is stable and covariance-stationary.
D) The system exhibits explosive behavior.
Correct Answer: C) The system is stable and covariance-stationary.
Explanation: In a Vector Autoregression, stability (and thus stationarity) is ensured if the
eigenvalues of the companion matrix are less than one in modulus, which is equivalent to the
characteristic roots lying outside the unit circle. If they were inside, the process would be
explosive; if on the unit circle, the process would contain unit roots.
,Question 3: When testing for cointegration using the Johansen procedure, what is the
fundamental difference between the Trace Statistic and the Maximum Eigenvalue Statistic?
A) The Trace statistic tests the null hypothesis of $r$ cointegrating vectors against the alternative
of $n$ vectors, while the Max-Eigen tests $r$ against $r+1$.
B) The Trace statistic is always more powerful than the Max-Eigen statistic.
C) The Max-Eigen statistic accounts for structural breaks, while Trace does not.
D) There is no difference; they are mathematically identical.
Correct Answer: A) The Trace statistic tests the null hypothesis of $r$ cointegrating vectors
against the alternative of $n$ vectors, while the Max-Eigen tests $r$ against $r+1$.
Explanation: The Trace test evaluates the null $H_0: rank(\Pi) \leq r$ against $H_A: rank(\Pi) =
n$. The Maximum Eigenvalue test evaluates $H_0: rank(\Pi) = r$ against $H_A: rank(\Pi) =
r+1$. The Max-Eigen test provides a more specific alternative hypothesis.
Question 4: In the presence of a structural break in the intercept of an $I(1)$ series, how does a
standard Augmented Dickey-Fuller (ADF) test perform?
A) The test becomes more powerful, making it easier to reject the unit root null.
B) The test suffers from size distortion, biasing the result toward failing to reject the unit root
null.
C) The test remains unbiased as long as the sample size is large.
D) The test correctly identifies the trend without needing modification.
Correct Answer: B) The test suffers from size distortion, biasing the result toward failing to
reject the unit root null.
Explanation: Perron (1989) demonstrated that standard unit root tests have very low power
against stationary alternatives if the series contains a structural break. The test tends to mistake
the break for non-stationarity, leading to a failure to reject the null hypothesis of a unit root even
when the series is stationary around a broken trend.
Question 5: A researcher estimates an $ARDL(p, q)$ model. Why is the ARDL approach
generally preferred over the Engle-Granger two-step procedure for cointegration?
A) It requires all variables to be integrated of order $I(1)$.
B) It allows for cointegration testing even if variables are a mix of $I(0)$ and $I(1)$.
,C) It does not require a long-run equilibrium relationship.
D) It is immune to endogeneity issues without using instrumental variables.
Correct Answer: B) It allows for cointegration testing even if variables are a mix of $I(0)$
and $I(1)$.
Explanation: The ARDL (Autoregressive Distributed Lag) bounds testing approach is uniquely
robust because it does not require pre-testing for unit roots. It remains valid regardless of
whether the regressors are purely $I(0)$, purely $I(1)$, or mutually cointegrated.
Question 6: In a GARCH(1,1) model defined by $\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2
+ \beta \sigma_{t-1}^2$, what is the condition for the variance process to be stationary?
A) $\omega > 0$
B) $\alpha + \beta = 1$
C) $\alpha + \beta < 1$
D) $\alpha + \beta > 1$
Correct Answer: C) $\alpha + \beta < 1$
Explanation: For the conditional variance $\sigma_t^2$ to be covariance-stationary, the sum of
the ARCH ($\alpha$) and GARCH ($\beta$) parameters must be less than unity. If $\alpha +
\beta = 1$, the model is an Integrated GARCH (IGARCH) model, where shocks to volatility are
persistent.
Question 7: How does the "Long-Run Multiplier" in a dynamic model relate to the estimated
coefficients of an $ARDL(1, 1)$ model, $y_t = \alpha + \beta_1 y_{t-1} + \delta_0 x_t +
\delta_1 x_{t-1} + \epsilon_t$?
A) $(\delta_0 + \delta_1) / (1 - \beta_1)$
B) $\delta_0 / (1 - \beta_1)$
C) $(\delta_0 + \delta_1) / \beta_1$
D) $\delta_1 / \beta_1$
Correct Answer: A) $(\delta_0 + \delta_1) / (1 - \beta_1)$
Explanation: In the steady state where $y_t = y_{t-1} = y^$ and $x_t = x_{t-1} = x^*$, the
equation becomes $y^* = \alpha + \beta_1 y^* + \delta_0 x^* + \delta_1 x^*$. Rearranging gives
, $y^*(1 - \beta_1) = \alpha + (\delta_0 + \delta_1)x^*$. The long-run derivative $\partial y^* /
\partial x^*$ is $(\delta_0 + \delta_1) / (1 - \beta_1)$.*
Question 8: Which assumption of the Classical Linear Regression Model is most severely
violated in a regression using levels of two independent $I(1)$ series?
A) Homoskedasticity.
B) No perfect multicollinearity.
C) Independence of regressors from the error term (in the context of spurious regression).
D) Normality of errors.
Correct Answer: C) Independence of regressors from the error term (in the context of
spurious regression).
Explanation: Granger and Newbold (1974) showed that regressing one independent random
walk on another leads to "spurious regression." The $t$-statistics will be highly significant and
$R^2$ will be high, but the relationship is a mathematical artifact of the trends, not a causal
link, effectively violating the assumption of a well-behaved error term structure.
Question 9: What does the "Granger Causality" test actually detect?
A) True physical causality.
B) Whether the past values of variable $X$ provide statistically significant information about the
future of variable $Y$, given the past of $Y$.
C) Whether variable $X$ is a prerequisite for variable $Y$.
D) Whether the correlation between $X$ and $Y$ is positive.
Correct Answer: B) Whether the past values of variable $X$ provide statistically significant
information about the future of variable $Y$, given the past of $Y$.
Explanation: Granger causality is based on predictive precedence. It tests whether $X_t$
"Granger-causes" $Y_t$ if the forecast of $Y_{t+1}$ based on both past $Y$ and past $X$ is
significantly better than the forecast based on past $Y$ alone. It does not imply economic or
structural causality.
Question 10: In a VAR model, what is the purpose of an Orthogonalized Impulse Response
Function (OIRF)?
A) To identify the causal structure by imposing a recursive (Cholesky) ordering on the variables.
Master Time Series and Forecasting Practice
Questions & Detailed Explanations
Subject: ECS4863 Econometrics - Time Series Analysis and Dynamic Modeling
Question 1: In the context of a stationary $AR(1)$ process defined by $y_t = \phi y_{t-1} +
\epsilon_t$, what is the necessary and sufficient condition for the process to be covariance-
stationary, and what is the implication for the impulse response function?
A) $|\phi| < 1$; the impulse response dies out exponentially.
B) $|\phi| \leq 1$; the impulse response persists indefinitely.
C) $\phi = 1$; the impulse response is constant.
D) $|\phi| > 1$; the impulse response grows without bound.
Correct Answer: A) $|\phi| < 1$; the impulse response dies out exponentially.
Explanation: For a first-order autoregressive process to be covariance-stationary, the root of the
characteristic equation must lie outside the unit circle, which implies $|\phi| < 1$. When this
condition holds, the effect of a shock to $\epsilon_t$ at time $t$ on $y_{t+k}$ is given by
$\phi^k$, which converges to zero as $k \to \infty$, characterizing exponential decay.
Question 2: Consider a $VAR(p)$ model. If the roots of the characteristic polynomial $\det(I_n -
\Phi_1 z - ... - \Phi_p z^p) = 0$ lie outside the unit circle, what can be concluded about the
system?
A) The system contains a deterministic trend.
B) The system is non-stationary and requires cointegration analysis.
C) The system is stable and covariance-stationary.
D) The system exhibits explosive behavior.
Correct Answer: C) The system is stable and covariance-stationary.
Explanation: In a Vector Autoregression, stability (and thus stationarity) is ensured if the
eigenvalues of the companion matrix are less than one in modulus, which is equivalent to the
characteristic roots lying outside the unit circle. If they were inside, the process would be
explosive; if on the unit circle, the process would contain unit roots.
,Question 3: When testing for cointegration using the Johansen procedure, what is the
fundamental difference between the Trace Statistic and the Maximum Eigenvalue Statistic?
A) The Trace statistic tests the null hypothesis of $r$ cointegrating vectors against the alternative
of $n$ vectors, while the Max-Eigen tests $r$ against $r+1$.
B) The Trace statistic is always more powerful than the Max-Eigen statistic.
C) The Max-Eigen statistic accounts for structural breaks, while Trace does not.
D) There is no difference; they are mathematically identical.
Correct Answer: A) The Trace statistic tests the null hypothesis of $r$ cointegrating vectors
against the alternative of $n$ vectors, while the Max-Eigen tests $r$ against $r+1$.
Explanation: The Trace test evaluates the null $H_0: rank(\Pi) \leq r$ against $H_A: rank(\Pi) =
n$. The Maximum Eigenvalue test evaluates $H_0: rank(\Pi) = r$ against $H_A: rank(\Pi) =
r+1$. The Max-Eigen test provides a more specific alternative hypothesis.
Question 4: In the presence of a structural break in the intercept of an $I(1)$ series, how does a
standard Augmented Dickey-Fuller (ADF) test perform?
A) The test becomes more powerful, making it easier to reject the unit root null.
B) The test suffers from size distortion, biasing the result toward failing to reject the unit root
null.
C) The test remains unbiased as long as the sample size is large.
D) The test correctly identifies the trend without needing modification.
Correct Answer: B) The test suffers from size distortion, biasing the result toward failing to
reject the unit root null.
Explanation: Perron (1989) demonstrated that standard unit root tests have very low power
against stationary alternatives if the series contains a structural break. The test tends to mistake
the break for non-stationarity, leading to a failure to reject the null hypothesis of a unit root even
when the series is stationary around a broken trend.
Question 5: A researcher estimates an $ARDL(p, q)$ model. Why is the ARDL approach
generally preferred over the Engle-Granger two-step procedure for cointegration?
A) It requires all variables to be integrated of order $I(1)$.
B) It allows for cointegration testing even if variables are a mix of $I(0)$ and $I(1)$.
,C) It does not require a long-run equilibrium relationship.
D) It is immune to endogeneity issues without using instrumental variables.
Correct Answer: B) It allows for cointegration testing even if variables are a mix of $I(0)$
and $I(1)$.
Explanation: The ARDL (Autoregressive Distributed Lag) bounds testing approach is uniquely
robust because it does not require pre-testing for unit roots. It remains valid regardless of
whether the regressors are purely $I(0)$, purely $I(1)$, or mutually cointegrated.
Question 6: In a GARCH(1,1) model defined by $\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2
+ \beta \sigma_{t-1}^2$, what is the condition for the variance process to be stationary?
A) $\omega > 0$
B) $\alpha + \beta = 1$
C) $\alpha + \beta < 1$
D) $\alpha + \beta > 1$
Correct Answer: C) $\alpha + \beta < 1$
Explanation: For the conditional variance $\sigma_t^2$ to be covariance-stationary, the sum of
the ARCH ($\alpha$) and GARCH ($\beta$) parameters must be less than unity. If $\alpha +
\beta = 1$, the model is an Integrated GARCH (IGARCH) model, where shocks to volatility are
persistent.
Question 7: How does the "Long-Run Multiplier" in a dynamic model relate to the estimated
coefficients of an $ARDL(1, 1)$ model, $y_t = \alpha + \beta_1 y_{t-1} + \delta_0 x_t +
\delta_1 x_{t-1} + \epsilon_t$?
A) $(\delta_0 + \delta_1) / (1 - \beta_1)$
B) $\delta_0 / (1 - \beta_1)$
C) $(\delta_0 + \delta_1) / \beta_1$
D) $\delta_1 / \beta_1$
Correct Answer: A) $(\delta_0 + \delta_1) / (1 - \beta_1)$
Explanation: In the steady state where $y_t = y_{t-1} = y^$ and $x_t = x_{t-1} = x^*$, the
equation becomes $y^* = \alpha + \beta_1 y^* + \delta_0 x^* + \delta_1 x^*$. Rearranging gives
, $y^*(1 - \beta_1) = \alpha + (\delta_0 + \delta_1)x^*$. The long-run derivative $\partial y^* /
\partial x^*$ is $(\delta_0 + \delta_1) / (1 - \beta_1)$.*
Question 8: Which assumption of the Classical Linear Regression Model is most severely
violated in a regression using levels of two independent $I(1)$ series?
A) Homoskedasticity.
B) No perfect multicollinearity.
C) Independence of regressors from the error term (in the context of spurious regression).
D) Normality of errors.
Correct Answer: C) Independence of regressors from the error term (in the context of
spurious regression).
Explanation: Granger and Newbold (1974) showed that regressing one independent random
walk on another leads to "spurious regression." The $t$-statistics will be highly significant and
$R^2$ will be high, but the relationship is a mathematical artifact of the trends, not a causal
link, effectively violating the assumption of a well-behaved error term structure.
Question 9: What does the "Granger Causality" test actually detect?
A) True physical causality.
B) Whether the past values of variable $X$ provide statistically significant information about the
future of variable $Y$, given the past of $Y$.
C) Whether variable $X$ is a prerequisite for variable $Y$.
D) Whether the correlation between $X$ and $Y$ is positive.
Correct Answer: B) Whether the past values of variable $X$ provide statistically significant
information about the future of variable $Y$, given the past of $Y$.
Explanation: Granger causality is based on predictive precedence. It tests whether $X_t$
"Granger-causes" $Y_t$ if the forecast of $Y_{t+1}$ based on both past $Y$ and past $X$ is
significantly better than the forecast based on past $Y$ alone. It does not imply economic or
structural causality.
Question 10: In a VAR model, what is the purpose of an Orthogonalized Impulse Response
Function (OIRF)?
A) To identify the causal structure by imposing a recursive (Cholesky) ordering on the variables.
