ECS3706
Assignment 2
(ANSWERS)
Semester 2
2023
ADMIN
[COMPANY NAME]
, QUESTION A1 (15 marks) (a) One of the most challenging concepts to
master in this module is distinguishing between the stochastic error term and
the residual. List three differences between the stochastic error term and the
residual (3) (b) Explain in detail how Ordinary Least Squares (OLS) works in
estimating the coefficients of a linear regression model. (3) (c) Demonstrate
your understanding of the goodness of fit by referring to the total, explained,
and residual sum of squares. (6) (d) Explain the Gauss Markov Theorem and
discuss the importance of this theorem in econometrics. (3) [15]
(a) Three Differences between the Stochastic Error Term and the Residual:
1. Nature:
Stochastic Error Term: The stochastic error term (ε) represents the unobservable
random variation or noise inherent in any statistical model. It is a theoretical
concept that cannot be directly observed or measured in practice.
Residual: The residual (e) is the difference between the observed data points and
the predicted values obtained from a regression model. It is a calculated value
and represents the unexplained variation in the data.
2. Calculation:
Stochastic Error Term: It is assumed to follow certain statistical properties, such
as having a mean of zero (E[ε] = 0) and constant variance (Var[ε] = σ^2). Its
specific values are not estimated or calculated in regression analysis.
Residual: Residuals are calculated as e = observed value - predicted value for
each data point. They are computed for each observation in the dataset and can
be positive or negative.
3. Role:
Stochastic Error Term: It is a theoretical construct used in the mathematical
formulation of regression models. It represents the random deviations from the
true relationship between variables and is crucial for understanding the
probabilistic nature of regression.
Residual: Residuals play a practical role in model assessment and diagnostics.
They help assess how well the regression model fits the data by measuring the
discrepancies between observed and predicted values.
(b) Explanation of Ordinary Least Squares (OLS):
Ordinary Least Squares is a method used to estimate the coefficients of a linear
regression model. It works by minimizing the sum of squared residuals, making the
model fit the data as closely as possible. Here's a detailed explanation:
1. Linear Regression Model: Start with a linear regression model in the form:
�=�0+�1�1+�2�2+...+����+�Y=β0+β1X1+β2X2+...+βkXk+ε
Y is the dependent variable.
X₁, X₂, ..., Xₖ are the independent variables.
β₀, β₁, β₂, ..., βₖ are the coefficients to be estimated.
ε is the stochastic error term.
Assignment 2
(ANSWERS)
Semester 2
2023
ADMIN
[COMPANY NAME]
, QUESTION A1 (15 marks) (a) One of the most challenging concepts to
master in this module is distinguishing between the stochastic error term and
the residual. List three differences between the stochastic error term and the
residual (3) (b) Explain in detail how Ordinary Least Squares (OLS) works in
estimating the coefficients of a linear regression model. (3) (c) Demonstrate
your understanding of the goodness of fit by referring to the total, explained,
and residual sum of squares. (6) (d) Explain the Gauss Markov Theorem and
discuss the importance of this theorem in econometrics. (3) [15]
(a) Three Differences between the Stochastic Error Term and the Residual:
1. Nature:
Stochastic Error Term: The stochastic error term (ε) represents the unobservable
random variation or noise inherent in any statistical model. It is a theoretical
concept that cannot be directly observed or measured in practice.
Residual: The residual (e) is the difference between the observed data points and
the predicted values obtained from a regression model. It is a calculated value
and represents the unexplained variation in the data.
2. Calculation:
Stochastic Error Term: It is assumed to follow certain statistical properties, such
as having a mean of zero (E[ε] = 0) and constant variance (Var[ε] = σ^2). Its
specific values are not estimated or calculated in regression analysis.
Residual: Residuals are calculated as e = observed value - predicted value for
each data point. They are computed for each observation in the dataset and can
be positive or negative.
3. Role:
Stochastic Error Term: It is a theoretical construct used in the mathematical
formulation of regression models. It represents the random deviations from the
true relationship between variables and is crucial for understanding the
probabilistic nature of regression.
Residual: Residuals play a practical role in model assessment and diagnostics.
They help assess how well the regression model fits the data by measuring the
discrepancies between observed and predicted values.
(b) Explanation of Ordinary Least Squares (OLS):
Ordinary Least Squares is a method used to estimate the coefficients of a linear
regression model. It works by minimizing the sum of squared residuals, making the
model fit the data as closely as possible. Here's a detailed explanation:
1. Linear Regression Model: Start with a linear regression model in the form:
�=�0+�1�1+�2�2+...+����+�Y=β0+β1X1+β2X2+...+βkXk+ε
Y is the dependent variable.
X₁, X₂, ..., Xₖ are the independent variables.
β₀, β₁, β₂, ..., βₖ are the coefficients to be estimated.
ε is the stochastic error term.
