Lecture 1

Question 1

What are the equations of the expectation of X in a discrete and continuous case

Answer 1

Question 2

For a single draw of random variable Y, the best constant predictor C in the sense of minimising is

Answer 2

Question 3

what is MSE

Answer 3

Mean Squared Error
1. means Y is our random variable
2. we are then taking c away from Y, no matter Y value
3. We square the result
4. We then Add the Sum of scenarios

uy is inbiased predictor of Y
MSE is E [(Y-uy)^2] = the var

Question 4

What is a conditional Moment

Answer 4

the joint probability distribution of two random variables of X and Y

Question 5

Linear Regress

Answer 5

Assumes that the regression function or Conditonal Expectation Function (CEF) given by

Question 6

Linear Regress- what happens if Y and X are independent

Answer 6

look at the bottom part

Question 7

Linear Regress- what if both Y and X are jointly normally distributed?

Answer 7

Linear model holds with:

Question 8

Linear Regress– what if true CEF is not linear,

Answer 8

Called a projection

Question 9

What is the best predictor h(x) of Y in the sense of minimising for a single draw of a pair of random variables

Answer 9

MSE = E [(Y-h(x))^2]
where the conditional mean h(x) = E [Y|x]

uy|x is an unbiased predictor ie. E [ Y- E [ Y-X]]
MSE is E [(Y- E [Y|X)^2] = E [ Var of y | X ]
Linear projection E [ Y|X] is the best predictor among the class of all linear functions

Question 10

Types of Data

Answer 10

Cross section - one or more variables at a single point of time
Time Series- data collected over a period of time

Question 11

Random Sampling

Answer 11

Assume it can be obtained from the underlying pop
- each person has the same chance of getting picked
- may describe random smapling of size T as a collection of iid (Independent and identically distrusted)

Question 12

Model a Time- Series

Answer 12

we often capture a temporal relationship
- infinite stochastic process is a sequence

Question 13

Goal of modeling a time series

Answer 13

1. describe prob behaviour of the underlying stochastic process that is believe to have generated the data
2. use the data to estimate moments of the process

Question 14

Modeling time-series 2 = how do we esitmate the moments of the process

Answer 14

Make Assumptions, regarding the joint behavior of the random variables
Preserve:
- identical distribution assumption from cross sectional analysis
-allow for dependencies between variables close together in time

Question 15

White Noise are independent stand normally distrubited with

Answer 15

E [Yt] = 0 all of time
Var [ Yt] = 1 for all of time

Question 16

White Noise Conditional Expectation of 2 adjacent pairs Yt, Yt+1?

Answer 16

E [Y|Y-1] = E [Yt] =0
should be zero as Yt given Yt-1 is equal to the unconditional expectation , E [Yt]. Normally distrbuted with mean zero

Question 17

Auto regssion - to introduce serial correlation, let’s construct the following
1. draw T = 100 observations from ut ∼ N(0, 1)
2. set Y0 = u0
3. generate the remaining Yt ’s as follow
Yt = 0.9 · Yt−1 + σut (t = 1, 2, . . . , 99) (1)
with σ = √(1 − 0.9^2)

Answer 17

Because Y1 is a linear combination of two independent N(0, 1) variables with σ^2 y = 0.9^2 + σ^2 = 1, it follows that

Yt ∼ N(0, 1) ∀t

However, they are not independent:

C[Yt , Yt+1] = C[Yt , 0.9 · Yt + σut+1]
= 0.9 C[Yt , Yt ]+σ C[Yt , ut+1]
= Var[Yt ]=1 =0
= 0.9 · 1 + σ · 0
= 0.9

{z }

Can also do this for any 2 obs which are 2 periods apart:
C[Yt , Yt+2] = 0.9^2

Question 18

What do we call the covaraince with its self, even though they are at diffrent points of time

Answer 18

AutoCovariance

Question 19

Stationarity conditions

Answer 19

E (Yt ) is constant for all t

▶ 0 < Var (Yt ) < ∞,
Var (Yt ) is constant and finite for all t

▶ 0 ⩽ |C (Yt , Yt±s ) | < ∞,
C (Yt , Yt±s ) is constant for all t and any s,
C (Yt , Yt±s ) = f (s, Θ

the 3 moments dont change over time

Question 20

What does Statistical independence imply

Answer 20

Mean independence
Uncorrelatedness
Hence Mean independence means uncorrelatedness

Question 21

Time series process is stationary if:

Answer 21

1. uncodntional mean E [yt] is not a function of time- constant
2. the unconditional Var (yt) is constant
3. auto covariance C (yt, yt+s) is nit a function of time and only dpends on the lag s- t is constant for any s