cross sectional data and simple regression

Question 1

what does cross sectional data mean ?

Answer 1

Interested in the variables (Y,X) for example the relationship between demand D and price P

Question 2

What is a population ?

Answer 2

A group of data, e.g. various individuals, firms, supermarkets and countries, we observe this simultaneously for N subjects drawn from this population

Question 3

What are we traditionally interested in

Answer 3

The single variable Y for example as a return of a financial asset

Question 4

Can data be regarded as a random sample

Answer 4

NO, it is important to account for trends, seasonality, temporal and serial dependence

Question 5

So how is cross-sectional data indexed?

Answer 5

Yt, t=1,…,T

Question 6

Explain simple regression

Answer 6

We are interested in a dependent (LHS, Explained, Response) variable Y, which is supposed to depend on an explanatory (RHS, Independent, Control, Predictor) variable X

Question 7

Give an example of a simple regression

Answer 7

Demand is a response variable and price is the predictor variable. Another one is wage is the response variable and years of education is the predictor variable

Question 8

What do we assume dealing with Cross-Sectional data means our observation pairs are?

Answer 8

(Y1, X1),…,(Yn, Xn)

These are stochastically independent and follow the same probability distribution drawn from the same population

Question 9

What is the conditional mean?

Answer 9

The avg behaviour of a dependent variable Y can be summarised with its mean E(y)

Question 10

What can we do if the explanatory variable X entails info about about Y ?

Answer 10

we can use it to refine the definition of the mean, we can express the avg behaviour of Y given X as the conditional mean of Y given X, E(Y/X)

Question 11

Examples of the conditional mean

Answer 11

If wage is the response Y and years of education is the predictor X, then we are interested in the avg wage of a person with X years of education, E(Y/X)

Question 12

If the outcome of throwing a dice is
E(Y) = (1+2+3+4+5+6)/6

Consider the explanatory variable:

X = 1 if Y is even
x = 0 if Y is odd

Calculate the conditional mean

Answer 12

E(Y/X) = 1: (2+4+6)/3 = 4

E(Y/X) = 0: (1+3+5)/3 = 3

we can summerise this as:
E(Y/X) = 4x+3(1-x)=3+x

This means that it is a function of x,
meaning E(Y/X) = g(x)

Question 13

What is the alternative way of giving 2 defining properties to the conditional mean

Answer 13

1. It is a function of X: E(Y/X) = g(X)

2 E(Y/X∈A) = E(E(Y/X/X∈A) for all sets of A ⊆ R

What this equation essentially says that the expected values of the conditional mean is equal to the unconditional expected values

In this course, we often assume that E(Y/X) is a linear function in X: E(Y/X) = β0 + β1X

Question 14

What are some properties of the conditional mean?

Answer 14

• If X and Y are independent than E(Y/X) = E(Y)
• If Y = f(x) then E(f(X)/(X)) = f(X)
• if E(f(X)/(X)) = f(X)E(Y/X)

Law of total/iterated expectation: E(E(Y/X)) = E(Y)

Linearity:
E(Y1 + Y2|X ) = E(Y1|X ) + E(Y2|X ) , and E(aY |X ) = aE(Y |X )
I If Y ≥ 0 then: E(Y |X ) ≥ 0