ECS4863
Advanced econometrics assignment 01
Question 1
1.1. Omitted variable bias occurs when a variable that belongs in a regression model is
excluded, causing the estimated coefficients on the included variables to be distorted
because they absorb the effect of the missing variable. For bias to arise, the omitted
variable must both affect the dependent variable and be correlated with at least one
included regressor. The direction of the bias depends on the signs of these two
relationships. Positive bias occurs when the omitted variable has the same directional
relationship with both the dependent variable and the included regressor — for example,
omitting ability from a wage-education regression inflates the education coefficient
because ability is positively correlated with both. Negative bias arises when one of those
relationships reverses sign, pulling the estimate below its true value.
1.2. A weakly dependent time series is one in which the statistical dependence between
observations diminishes as the time gap between them increases, eventually becoming
negligible as that gap approaches infinity. Formally, a stationary sequence {xₜ} is weakly
dependent if the correlation between xₜ and xₜ₊ₕ converges to zero sufficiently fast as h →
∞. This property is important because it serves as the time series counterpart to the
random sampling assumption used in cross-sectional analysis, allowing the Law of Large
Numbers and the Central Limit Theorem to remain applicable even when observations
are not independent.
The practical significance of weak dependence is that it places a restriction on how much
"memory" a series can have. A stable AR(1) process with |ρ| < 1 is a standard example
— while adjacent observations are correlated, that correlation decays geometrically over
time. By contrast, a unit root process (ρ = 1) is not weakly dependent because its
correlations do not decay, which undermines standard inference. Weak dependence
therefore defines the class of time series for which OLS-based estimation and hypothesis
testing remain asymptotically valid without requiring the stronger assumption of strict
exogeneity.
1.3. Heteroscedasticity refers to the condition in which the variance of the error term in a
regression model is not constant across observations, but instead varies with the level of
one or more regressors or across time. This violates one of the classical OLS assumptions.
While OLS coefficient estimates remain unbiased and consistent in the presence of
heteroscedasticity, they are no longer efficient — meaning they do not have the minimum
, variance among linear unbiased estimators. More critically, the standard errors produced
by OLS are incorrect, which renders t-statistics, F-statistics, and confidence intervals
unreliable, leading to invalid hypothesis tests. The standard remedy is to use
heteroscedasticity-robust standard errors, which correct the inference problem without
changing the point estimates.
1.4. A covariance stationary process is a time series in which the mean and variance are
constant over time, and the covariance between any two observations depends only on
the length of the time gap between them, not on where in time those observations fall.
This stability of the first and second moments is a foundational requirement for much of
time series analysis. Sequential exogeneity, on the other hand, is the assumption that the
error term at time t is uncorrelated with the current and all past values of the regressors
— formally, E(uₜ | xₜ, xₜ₋₁, …, x₁) = 0. It is weaker than strict exogeneity, which
additionally requires the error to be uncorrelated with future values of the regressors.
1.5. Relaxing the strict exogeneity assumption is important in time series analysis because
many realistic economic models involve dynamics that strict exogeneity cannot
accommodate. Strict exogeneity requires that the error term be uncorrelated with
regressors across all time periods — past, present, and future — which rules out any
feedback from the outcome variable back to future regressors. In practice, this is
frequently violated. For instance, including a lagged dependent variable as a regressor
automatically violates strict exogeneity because past errors influence past values of the
dependent variable, which then appear directly as regressors. By relaxing to sequential
exogeneity, which only requires errors to be uncorrelated with current and past
regressors, a much broader and more realistic class of dynamic models can be estimated
consistently, provided the series is also weakly dependent.
__________________________________________________________________________
Question 2
2(a)
Advanced econometrics assignment 01
Question 1
1.1. Omitted variable bias occurs when a variable that belongs in a regression model is
excluded, causing the estimated coefficients on the included variables to be distorted
because they absorb the effect of the missing variable. For bias to arise, the omitted
variable must both affect the dependent variable and be correlated with at least one
included regressor. The direction of the bias depends on the signs of these two
relationships. Positive bias occurs when the omitted variable has the same directional
relationship with both the dependent variable and the included regressor — for example,
omitting ability from a wage-education regression inflates the education coefficient
because ability is positively correlated with both. Negative bias arises when one of those
relationships reverses sign, pulling the estimate below its true value.
1.2. A weakly dependent time series is one in which the statistical dependence between
observations diminishes as the time gap between them increases, eventually becoming
negligible as that gap approaches infinity. Formally, a stationary sequence {xₜ} is weakly
dependent if the correlation between xₜ and xₜ₊ₕ converges to zero sufficiently fast as h →
∞. This property is important because it serves as the time series counterpart to the
random sampling assumption used in cross-sectional analysis, allowing the Law of Large
Numbers and the Central Limit Theorem to remain applicable even when observations
are not independent.
The practical significance of weak dependence is that it places a restriction on how much
"memory" a series can have. A stable AR(1) process with |ρ| < 1 is a standard example
— while adjacent observations are correlated, that correlation decays geometrically over
time. By contrast, a unit root process (ρ = 1) is not weakly dependent because its
correlations do not decay, which undermines standard inference. Weak dependence
therefore defines the class of time series for which OLS-based estimation and hypothesis
testing remain asymptotically valid without requiring the stronger assumption of strict
exogeneity.
1.3. Heteroscedasticity refers to the condition in which the variance of the error term in a
regression model is not constant across observations, but instead varies with the level of
one or more regressors or across time. This violates one of the classical OLS assumptions.
While OLS coefficient estimates remain unbiased and consistent in the presence of
heteroscedasticity, they are no longer efficient — meaning they do not have the minimum
, variance among linear unbiased estimators. More critically, the standard errors produced
by OLS are incorrect, which renders t-statistics, F-statistics, and confidence intervals
unreliable, leading to invalid hypothesis tests. The standard remedy is to use
heteroscedasticity-robust standard errors, which correct the inference problem without
changing the point estimates.
1.4. A covariance stationary process is a time series in which the mean and variance are
constant over time, and the covariance between any two observations depends only on
the length of the time gap between them, not on where in time those observations fall.
This stability of the first and second moments is a foundational requirement for much of
time series analysis. Sequential exogeneity, on the other hand, is the assumption that the
error term at time t is uncorrelated with the current and all past values of the regressors
— formally, E(uₜ | xₜ, xₜ₋₁, …, x₁) = 0. It is weaker than strict exogeneity, which
additionally requires the error to be uncorrelated with future values of the regressors.
1.5. Relaxing the strict exogeneity assumption is important in time series analysis because
many realistic economic models involve dynamics that strict exogeneity cannot
accommodate. Strict exogeneity requires that the error term be uncorrelated with
regressors across all time periods — past, present, and future — which rules out any
feedback from the outcome variable back to future regressors. In practice, this is
frequently violated. For instance, including a lagged dependent variable as a regressor
automatically violates strict exogeneity because past errors influence past values of the
dependent variable, which then appear directly as regressors. By relaxing to sequential
exogeneity, which only requires errors to be uncorrelated with current and past
regressors, a much broader and more realistic class of dynamic models can be estimated
consistently, provided the series is also weakly dependent.
__________________________________________________________________________
Question 2
2(a)
