Econometrics 2: Bivariate Linear Regression 2.1 - Discussion of the model Flashcards
What’s regression model also known as?
Econometric Model
What does it mean when ‘we looked at regression models & analysis in the context of Keynesian consumption function’?
A regression model that shows how income influences consumption
What does it mean when there’s ‘a regression model that shows how income influences consumption’?
We looked at regression models & analysis in the context of Keynesian consumption function
What does it mean when one only conerns themselves with ‘simple (bivariate) econometric models’?
They are analysing models that involve only two variables
Describe & explain the use of economics theory to indicate the causation in a bivariate relationship
Is it changes in the value of 𝑋 that cause changes in the value of 𝑌 or vice versa. For notational
purposes we usually indicate the explanatory variable (also called the regressor or predictor variable) by
𝑋 and the dependent variable (also called the regressand or predicted variable) by 𝑌. The causation runs
from 𝑋 to 𝑌; hence we believe that changes in economic variable 𝑋 cause movements in economic variable 𝑌. We are also only interested in linear econometric models, which here implies that the equations represent straight line relationships between the variables 𝑋 and 𝑌.
Introduce & explain the equations that economic theory comes up with for linear bivariate models
An economic theory suggesting that the variation in some economic variable 𝑌 depends linearly upon the variability in another economic variable
𝑋, can be written as
𝑌 = 𝛽bottom right0 + 𝛽bottom right1 𝑋 (1)
This is just the equation of a straight line where 𝛽bottom right0 is the intercept on the 𝑌 axis (i.e. the value of
𝑌 when 𝑋 is equal to 0) and 𝛽bottom right1 is the slope of the line, which represents how much variable 𝑌
changes when the values of variable 𝑋 change. This equation should attempt to mimic the
behaviour of the economic system that it is representing. But we know that the economy would
not move in such an exact way as this. There might be unobservable factors that influence 𝑌 that
we simply cannot include in the equation and hence the line simply mimics the general relationship
between the variables. Suppose that 𝛽̂hat bottom right0 = 2 and 𝛽̂hat bottom right1 = 0.75 such that 𝑌̂hat = 2 + 0.75𝑋. If 𝑋 = 17,
then the equation suggests that 𝑌 = 2 + 0.75(17) = 14.75, but obviously not everyone who
earns £17,000 will spend £14,750. To allow for this, we add an error term or random disturbance term,
often denoted 𝜀, i.e.
𝑌 = 𝛽bottom right0 + 𝛽bottom right1𝑋 + 𝜀 (2)
Equation (2) is what we would call a regression model. 𝜀 takes into account error terms - the true dependant value may be above or below where the model says. The error also accounts
for any factors that influence 𝑌 but are not included in the equation. These maybe factors that we
cannot easily measure or observe.The best way to deal with the error term is to treat it as a
random variable, as it accounts for random behaviour that cannot be quantified or easily modelled.
The properties of this random error are crucially important.
Establish when a model is ‘linear’
A model is defined as linear if it is linear in the parameters. This means that the model’s parameters
do not appear as, for example, exponents or products of other parameters. If the model contains
squares or products of variables, we would still refer to this as a linear regression if it is still linear
in parameters. Another way to think about linearity is that the derivative of 𝑌 with respect to
regression coefficients is a function only of known constants and/or the regressor 𝑋, i.e. it is not
a function of any unknown parameters. That is,
𝑑𝑌bottom right𝑖/𝑑𝛽bottom right𝑗 = 𝑓(𝑋𝑖)
where 𝑓(𝑋𝑖) contains no unknown parameters. As an example, 𝑌𝑖 = 𝛽bottom right0 + 𝛽bottom right1𝑋𝑖 + 𝛽bottom right2𝑋𝑖^2 + 𝜀bottom right𝑖 is
linear because 𝑑𝑌𝑖/𝑑𝛽Bottom right1
= 𝑋𝑖 and 𝑑𝑌𝑖/𝑑𝛽bottom right2 = 𝑋𝑖^2. On the other hand, 𝑌𝑖 = 𝛽bottom right0 + 𝛽bottom right1𝑒^𝛽bottom right2𝑋𝑖 + 𝜀bottom right𝑖 is non-linear
because 𝑑𝑌𝑖/𝑑𝛽1 = 𝑒^𝛽bottom right2𝑋𝑖 and 𝑑𝑌𝑖/𝑑𝛽bottom right2 = 𝛽bottom right1𝑋𝑖𝑒^𝛽bottom right2𝑋𝑖 , both of which are functions of unknown factors. Non-
linear regression techniques would be required to estimate such a model.
