Section 5

Question 1

What are autocorrelated errors?

Answer 1

When there is correlation between the errors

Question 2

What makes a time series variable autocorrelated?

Answer 2

When it is correlated with itself at different points in time

Question 3

What is a first-order autoregressive process (AR(1))?

Answer 3

When a variable is a function of itself from the previous period (see notes on P1 example regarding AR(1))

Question 4

Give an example of an AR process? What is alpha?

Answer 4

Xt=αX(t-1)+εt

Where alpha is between -1 and 1 and is a parameter called the Autocorrelation Coefficient

Question 5

Prove the equation for the variance of X in a AR process? What does this proof assume?

Answer 5

See notes

Assumes that Xt is homoskedastic

Question 6

By comparing the covariance between the variable and its first lag, show there’s a non-zero first-order autocorrelation?

Answer 6

See notes

Question 7

How do we know that as we look at autocorrelations further into the past, the correlation with current Xt decreases?

Answer 7

Since absolute value of alpha is less than 1, and that α^j->0 as n->infinity, the correlation with current Xt must decrease as n-> further into the future

Question 8

See

Answer 8

Bits in notes S5 saying to see bits in lectures

Question 9

See and read 4.2 autocorrelated regression errors

Answer 9

now, read whole section

Question 10

Why, and how, can errors be written as an AR process?

Answer 10

Error will now be correlated with itself in different time periods

Written as: ε(t)=ρε(t-1)+u(t)

ρ=AC coefficient

Question 11

Given we want to estimate a MRM, where error process can be written as AR(1), if we ignore AC in errors and estimate beta parameters, what are the consequences? (2) and what is the solution?

Answer 11

• OLS estimators are still unbiased (didn’t use no AC assumption when proving them to be unbiased)
• When errors are AR(1), the equations for the variances of the OLS estimators are wrong. There is a corrected variance equation in notes BUT it still does not make OLS the best estimator since it doesn’t have the smallest variance tf use GLS

Question 12

Informal way and formal way of testing for AC’ed errors?

Answer 12

Plot graphs and then visually compare residuals to see if any correlation

formal way: durbin watson test

Question 13

Briefly describe what the durbin watson test does?

Answer 13

Tests for first order AC (only), assumes the error term is written as ε(t)=ρε(t-1)+u(t)
Then tests the hypotheses:
H0: ρ=0
H1: ρ not equal to 0

Question 14

See

Answer 14

Durbin watson equations, and the number line thing

Question 15

What values can the DW test take?

Answer 15

anywhere between 0 and 4

Question 16

3 drawbacks of the durbin watson test, and a solution to the third problem?

Answer 16

• 2 inconclusive regions in the distribution
• only tests for first order AC
• test is invalid if one regressor in model is Y(t-1) (lagged dependent variable) (solution is to use durbin’s h test)

Question 17

Explain durbin’s h test?

Answer 17

Use when Y(t-1) is present. Hypotheses are same as before. Test is done off normal distribution (see equations)

Question 18

See

Answer 18

h stat note bottom of side 2 page 1

Question 19

What is the idea behind GLS estimation?

Answer 19

Using OLS estimation on a model that is transformed so it has no AC errors

Question 20

Explain the method of GLS estimation?

Answer 20

Given simple RM and error is AC such that ε(t)=ρε(t-1)+u(t):
1) Lag model by one period by multiplying by ρ
2) Subtract this model from the original to give:
Y(t)=α1+β2X(t)+u(t) (see notes)

Here u(t) satisfies all of the classical assumptions tf just use normal OLS estimators on the transformed model (method can also be used on MRM)

Question 21

When is the cochrane iterative procedure used? Explain the cochrane iterative procedure?

Answer 21

Used when ρ is unknown (extension of GLS)
METHOD:
1) estimate parameters of original model
2) using info from residuals, estimate ρ from:
ε(t)=ρε(t-1)+u(t) to give ρ(hat)
3) use ρ(hat) to estimate the quasi-differences (see form in notes!)
4) use OLS to estimate parameters in the transformed model (a) Y(t)=α1+α2X(t)+u(t)
5) use residuals of (a) to estimate ρ again, denoted ρ(hathat)
6) Repeat steps 3-5 until estimates for ρ converge

Question 22

What is artificial autocorrelation?

Answer 22

see notes 5.6