ECS4863
Assignment 1
DUE: 12 MAY 2026
, ECS4863
Advanced Econometrics: Time Series Analysis
, QUESTION 1 (24 Marks)
This question covers core theoretical concepts in time series and introductory econometrics.
1.1 Omitted Variable Bias: Positive and Negative Bias (4 marks)
Omitted variable bias (OVB) arises when a relevant explanatory variable is excluded from a
regression model. Because the omitted variable's effect is absorbed into the error term, the error
term becomes correlated with the included regressors, violating the Ordinary Least Squares
(OLS) assumption of zero conditional mean: E(u|X) = 0. As a result, the estimated coefficients
are biased and inconsistent.
The direction of the bias is formally expressed as:
₁ − β̃₁= β̃ ₂· δ̃₁
Bias = β̃
where β ₂is the true effect of the omitted variable on the dependent variable, and δ̃.. ₁ is the
coefficient from regressing the omitted variable on the included regressor.
Positive bias: Occurs when the omitted variable is positively correlated with both the dependent
variable and the included regressor. For example, if ability is omitted from a wage-education
regression, and ability is positively correlated with education and wages, the coefficient on
education is overestimated (biased upward).
Negative bias: Occurs when the correlation between the omitted variable and the included
regressor is in the opposite direction to the omitted variable's effect on the dependent variable.
For example, if work experience is omitted and it is negatively correlated with education (older
workers have less education but more experience), the coefficient on education is biased
downward.
1.2 Weakly Dependent Time Series (8 marks)
A weakly dependent time series is one in which the statistical dependence (correlation) between
observations at two time points diminishes and eventually vanishes as the time gap between
those observations increases. Formally, a stationary time series {x ₜ} is weakly dependent if:
Corr(xₜ , x ₜ₊ₕ) → 0 as h → ∞
This means observations that are far apart in time become nearly uncorrelated, approaching
independence as the lag h increases.
Assignment 1
DUE: 12 MAY 2026
, ECS4863
Advanced Econometrics: Time Series Analysis
, QUESTION 1 (24 Marks)
This question covers core theoretical concepts in time series and introductory econometrics.
1.1 Omitted Variable Bias: Positive and Negative Bias (4 marks)
Omitted variable bias (OVB) arises when a relevant explanatory variable is excluded from a
regression model. Because the omitted variable's effect is absorbed into the error term, the error
term becomes correlated with the included regressors, violating the Ordinary Least Squares
(OLS) assumption of zero conditional mean: E(u|X) = 0. As a result, the estimated coefficients
are biased and inconsistent.
The direction of the bias is formally expressed as:
₁ − β̃₁= β̃ ₂· δ̃₁
Bias = β̃
where β ₂is the true effect of the omitted variable on the dependent variable, and δ̃.. ₁ is the
coefficient from regressing the omitted variable on the included regressor.
Positive bias: Occurs when the omitted variable is positively correlated with both the dependent
variable and the included regressor. For example, if ability is omitted from a wage-education
regression, and ability is positively correlated with education and wages, the coefficient on
education is overestimated (biased upward).
Negative bias: Occurs when the correlation between the omitted variable and the included
regressor is in the opposite direction to the omitted variable's effect on the dependent variable.
For example, if work experience is omitted and it is negatively correlated with education (older
workers have less education but more experience), the coefficient on education is biased
downward.
1.2 Weakly Dependent Time Series (8 marks)
A weakly dependent time series is one in which the statistical dependence (correlation) between
observations at two time points diminishes and eventually vanishes as the time gap between
those observations increases. Formally, a stationary time series {x ₜ} is weakly dependent if:
Corr(xₜ , x ₜ₊ₕ) → 0 as h → ∞
This means observations that are far apart in time become nearly uncorrelated, approaching
independence as the lag h increases.
