Testing For Autocorrelated Errors

Question 1

Non-rigorous way of detecting autocorrelated errors

Answer 1

Analyse the plots.

E.g if spikey likely to be negative autocorrel, if smooth then pos, random then p=0 likely!)

Question 2

Formal method to test for autocorrelation

Answer 2

Durbin Watson test (DW) test

Question 3

What does the DW test test for

Answer 3

First order autocorrelation, (what we have learnt i.e
error term written as

εt = pεt-₁ + ut

Question 4

What is null hypothesis for DW

Answer 4

H₀: P=0 (means CLRA satisfied, so no auto!

So if we reject means autocorrelated, so error is a function of its past.

Question 5

Test statistic formula for DW

(Most software compute for u)

Answer 5

Σ(εhatt-Ehatt-₁)²
/
ΣεHat²

~ Look up in table n, k to find dl and du
(n is observations, k is amount of explanatory variables)

Question 6

We get 2 critical values left and right side where our hypothesis of non-autocorrelation p=0 can be rejected.

(Value will only be between 0-4)

Question 7

If DW lies between…

Where is positive, where is negative, where is no correlation, where is inconclusive

Answer 7

<dL = reject p=0, positive autocorrelation
>4-dL =reject p=0, negative autocorrelation

In between dU and 4-dU = no autocorrel

Betwwen dL and dU = inconclusive
Between 4-dU and 4-dL = inconclusive

Question 8

Note;

around 2 = likely no autocorrelation (cannot reject null)

Since it is in between dU and 4 - dU where no autocorrelation!

Question 9

When is this DW test invalid (2)

Answer 9

For things other than first order regression! I.e if not in form εt = pεt-₁ + ut

If dependent value is lagged! Yt-f

Question 10

What test overcomes faults of DW test (of only being able to test first order autocorrelation)

Answer 10

Breusch-Godfrey test (BG)

(Allows to test for pth order autocorrelation, not just first order)

Question 11

Now we have a new test regression, what does this look like?

Answer 11

Ehatt = β₀ + βhat₁X₁+…βhatkXkt +

Question 12

What is the null and alternate in the BG test.

Answer 12

H₀=P₁=P₂=…=Pp=0 (no autocorrelation)
H₁: at least one autocorrelation coefficient is not 0 (autocorrelation)

Question 13

Test statistic for BG

Answer 13

Lagrance multiplier (LM)

LM = nR² ~ X²p

Sample size n x R² (goodness of fit)

p degrees of freedom using the X² distribution table
E.g if want 2nd order autocorrel p=2, so can still do 1st order just like the DW test, just set p=1!)

Question 14

So if no autocorrelation: We can use OLS fine (given other assumptions hold as well)

If autocorrelated erros exist (which we test for via DW or BG): it means OLS estimates incorrect, so what 2 methods can we use?

Answer 14

Generalised least squares: Removes autocorrelation from the errors, so it is then OK to use OLS

Cochrane-Orcutt - extension of GLS

(So DW and BG determine whether autocorrelation exists, GLS and CO are for actually continuing with the estimation!)

Question 15

GLS process.

Start with:
Yt=β₀ +β₁Xt+εt

And we have used BG or DW and established autocorrelation so errors can be written as
εt= pεt-₁ +ut

Answer 15

Lag the model by one period.
Yt-₁ =β₀ + β₁Xt-₁ + εt-₁

Multiply by p
pYt-₁ =pβ₀ + pβ₁Xt-₁ + pεt-₁

Subtract this from original (Yt - pYt-₁)
β₀* + β₁Xt* + ut

Where
Yt* = Yt - pYt-₁
Xt* = Xt- pXt-₁ are quasi differences.
Here also, error term is uncorrelated since it is ut (uncorrelated as mentioned earlier)

Question 16

why is GLS often infeasible

Answer 16

in order to know the quasi differences, we need the value of p, which is unknown in most situations.

so we need another method that overcomes this

Question 17

how do we overcome this GLS problem (of having to know p)

Answer 17

Cochrane-Orcutt procedure

Question 18

Cochrane Orcutt steps

Same model we used in OLS
Yt=β₀ +β₁Xt+εt

Answer 18

Estimate parameters of model
Yt= Bhat₀ + Bhat₁Xt + εhatt

Estimate (by OLS) p from εt=pεt-₁ + ut
so get hat version of each to find p hat

Use phat to transoform variables into their quasi differences
Yt=Yt- phatYt-₁ and
Xt = phatXt- pXt-₁

Estimate using OLS on transformed model to find second estimate of p (phathat) from AR1 (εt=εt−₁+ ut)

Create quasi differences for Y** and X** now using phathat.

Repeat till parameters converge (change by small amount - hence why called iterative)

Question 19

How is the Prais-Winsten technique used along with the Cochrane Orcutt

Answer 19

to allow for the loss of observations