, QUESTION A1 (15 marks)
(a) Brief explanations (2 marks each)
i. Stochastic error term
The stochastic error term;call it uᵢ;is the catch-all component tacked onto a regression
equation. It scoops up every bit of variation in your dependent variable Y that your chosen X
variables simply cannot explain. Think of it as the "everything else" basket. The specific
value it takes for any given observation is determined by chance. No model is perfect; we
always leave something out, measure something wrong, or pick the wrong functional form.
The error term is where all that unavoidable imperfection lives.
ii. A regression analysis
Regression analysis is a statistical tool that tries to pin down how one variable (the
dependent variable, Y) moves when another variable (or several others, the Xs) changes.
It's not about proving causation;it's about quantifying association. You feed it data, and it
spits out an equation that summarises the average relationship. The goal? To get the most
accurate estimates of the coe icients, whatever data problems you run into along the way.
iii. The total sum of the squares (SST)
The total sum of squares measures the total variation in the dependent variable Y around
its sample mean. It's the sum of squared deviations of each Yᵢ from Ȳ:
SST = Σ(Yᵢ − Ȳ)²
Think of it as the total amount of "movement" in Y that you're trying to explain. Regression
analysis then splits this total into two parts: the part your model explains (ESS) and the part
it doesn't (RSS). SST = ESS + RSS.
iv. The degrees of freedom
Degrees of freedom are about how much "wiggle room" your data has left after you've
estimated parameters. In a regression with *k* independent variables
and *n* observations, the residual degrees of freedom are n − k − 1. Why the minus one?
Because you've already used up one degree of freedom estimating the intercept. For the
total sum of squares, you lose one degree of freedom because you estimated the mean Ȳ.
In essence: the more parameters you estimate, the fewer degrees of freedom you have left.
(a) Brief explanations (2 marks each)
i. Stochastic error term
The stochastic error term;call it uᵢ;is the catch-all component tacked onto a regression
equation. It scoops up every bit of variation in your dependent variable Y that your chosen X
variables simply cannot explain. Think of it as the "everything else" basket. The specific
value it takes for any given observation is determined by chance. No model is perfect; we
always leave something out, measure something wrong, or pick the wrong functional form.
The error term is where all that unavoidable imperfection lives.
ii. A regression analysis
Regression analysis is a statistical tool that tries to pin down how one variable (the
dependent variable, Y) moves when another variable (or several others, the Xs) changes.
It's not about proving causation;it's about quantifying association. You feed it data, and it
spits out an equation that summarises the average relationship. The goal? To get the most
accurate estimates of the coe icients, whatever data problems you run into along the way.
iii. The total sum of the squares (SST)
The total sum of squares measures the total variation in the dependent variable Y around
its sample mean. It's the sum of squared deviations of each Yᵢ from Ȳ:
SST = Σ(Yᵢ − Ȳ)²
Think of it as the total amount of "movement" in Y that you're trying to explain. Regression
analysis then splits this total into two parts: the part your model explains (ESS) and the part
it doesn't (RSS). SST = ESS + RSS.
iv. The degrees of freedom
Degrees of freedom are about how much "wiggle room" your data has left after you've
estimated parameters. In a regression with *k* independent variables
and *n* observations, the residual degrees of freedom are n − k − 1. Why the minus one?
Because you've already used up one degree of freedom estimating the intercept. For the
total sum of squares, you lose one degree of freedom because you estimated the mean Ȳ.
In essence: the more parameters you estimate, the fewer degrees of freedom you have left.
