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1
Q

What are autocorrelated errors?

A

When there is correlation between the errors

2
Q

What makes a time series variable autocorrelated?

A

When it is correlated with itself at different points in time

3
Q

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

A

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

4
Q

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

A

Xt=αX(t-1)+εt

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

5
Q

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

A

See notes

Assumes that Xt is homoskedastic

6
Q

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

A

See notes

7
Q

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

A

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

8
Q

See

A

Bits in notes S5 saying to see bits in lectures

9
Q

See and read 4.2 autocorrelated regression errors

A

now, read whole section

10
Q

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

A

Error will now be correlated with itself in different time periods

Written as: ε(t)=ρε(t-1)+u(t)

ρ=AC coefficient

11
Q

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?

A
  • 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
12
Q

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

A

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

formal way: durbin watson test

13
Q

Briefly describe what the durbin watson test does?

A

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

14
Q

See

A

Durbin watson equations, and the number line thing

15
Q

What values can the DW test take?

A

anywhere between 0 and 4

16
Q

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

A
  • 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)
17
Q

Explain durbin’s h test?

A

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

18
Q

See

A

h stat note bottom of side 2 page 1

19
Q

What is the idea behind GLS estimation?

A

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

20
Q

Explain the method of GLS estimation?

A

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)

21
Q

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

A

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

22
Q

What is artificial autocorrelation?

A

see notes 5.6