Parameter Stability & Structural Change

Question 1

What CLRA is rejected now?

Answer 1

CLRA 1 - model is expressed as… where βs are unknown parameters (constants) e.g β₁=0.9 £1, increase in income increases consumption by 0.9

Now we assume NON-constant parameters! (Hence why called we need parameter stability)

Question 2

How to work with non-constant parameters

Answer 2

Assume we have n observations, and we split it into 2 categories: n₁ and n-n₁. (Similar to GQ for heteroscedasticity where we split it into low and high group)

δYi/δXji = β¹j
δYi/δXji = β²j

Question 3

Using this, when does a structural break/change exist?

Answer 3

If β¹j ≠ β²j

Question 4

What does this mean for the pooled original equation (standard multi regression expression)

Answer 4

If β¹j ≠ β²j

OLS estimator βj is biased! It is a weighted average of unbiased estimators βhat¹j and βhat²j

Question 5

Structural break visualisation pg 4

Page 5 shows if we predicted OLS using full sample (not splitting n up into 2)

Page 6 shows OLS using 2 separate regressions for each subgroup of n, which captures the 2 true relationships better!

Answer 5

It will be biased.

Question 6

Why is structural change problematic (2)

Answer 6

Violates CLRA1 of constant parameters

Can lead to misleading parameter estimates (by using the full pooled regression PAGE 5 VISUALISED!)

Question 7

How and where do structural breaks arise (2)

Answer 7

Cross-sectional - relationships with variables may vary across subgroups (e.g determinants of wages different by gender)

Time series - relationships with variables may vary over time

Question 8

2 formal tests for TESTING structural change

1 informal test to identify potential breaks

Answer 8

Chow
Predictive failure

Recursive least squares

Question 9

Chow test - same set up i.e splitting up into 2 categories again etc.

What is the hypothesis

Answer 9

H₀:β¹₀ = β²₀ , β¹₁=β²₁ ,…, β¹k=β²k (No structural break)

H₁:β¹j ≠ β²j for at least 1 j (Structural break)

Question 10

Test statistic for Chow test

Answer 10

F test

RSSr-RSSu / Q
/
RSSu/ (n-2(k+1))

~Q, n-2(k+1)

RSSr is from the pooled eq (restricted model)
RSSu is from the split equations (unrestricted model)
So different from standard F test!

Question 11

What is the RSSr and RSSu obtained from?

Answer 11

RSSr is from the pooled regression

RSSu = RSS1 (n₁) + RSS2 (n-n₁)
Add together to get RSSu.

Then calculate F statistic

Question 12

Chow test with dummy variables

Answer 12

Add dummy vriables to the original model

Yi = γ₀ + γ₁X₁i + … + γkXki + δ₀Di + δ₁X₁iDi + … + δkXkiDi +εi

(So original + the same for every X interacted with D.

Question 13

What does our hypothesis’ become

Answer 13

H0 :δ₀ =δ₁ =…=δk =0

H1 :δj≉0 for any j

Question 14

Test statistic for Chow test with dummy variables

Answer 14

The same, but our

RSSr is from the restricted model (original standard multi regression expression)

RSSu is from the unrestrcited model (the original model with the dummy variables added)

Question 15

Drawbacks of Chow test (4)

Answer 15

For RSSu = RSS₁ +RSS₂ , we require errors of both regressions to be homescedastic (the same), and independently distributed.

Doesn’t tell us which specific parameter(s) are unstable

Need to know where structural break occurs to split the sample (i.e for n₁ and n-n₁) (RECURSIVE LEAST SQUARES!)

May not have enough data to estimate both models separately especially if one sample is small

Question 16

Predictive failure test

Answer 16

Estimates the model for one sample (unlike Chow) , and see whether it can predict outcomes in the other sample.

If model cannot predict outcomes in other sample accurately, a structural break has probably occured.

(Hence the name)

Question 17

Predictive failure test process

Answer 17

Start with 2 samples with n₁ and n₂ observations as usual.

Null hypothesis is that the first model n₁ can be fitted to the second sample

Question 18

F test

Answer 18

RSSr-RSSu/q
/
RSSu/(n-(k+1))

~Fq,n-(k+1)

Question 19

Steps of predictive failure test

Answer 19

Run full regression on whole sample (n₁ and n-n₁) to get RSSr

Get RSSu by just running regression on n₁

Find F statistic

Question 20

Predictive failure test using dummy variables

Answer 20

Add dummt variables to the original model by +ΣyjDj + εi

Dj= 1 if i=n₁+j

Question 21

Hypothesis for predictive failure test with dummy variables

Answer 21

H₀ :γ₁ =γ₂ =…=γn₂ =0 (no prediction error, so no structural break)

H₁ =yj≉0 for any j (prediction error, n₁ doesn’t predict outcomes of n-n₁ accurately, so structural break!)

Question 22

Same F test

Answer 22

RSSr-RSSu / n₂
/
RSSu / n₁ - (k+1)

Question 23

Both tests require us to know where the break is.

How can we find break?

Answer 23

Seeing differences between 2 groups in cross-sectional data

break after a major event in time-series data

Question 24

So Chow and Predictive failure test both require us to know where the break is. (They both split n into subsamples)

if we dont know, how do we identify potential breaks

Answer 24

Recursive least squares (INFORMAL), so use to indicate potential breaks, then apply the break to split the sample using Chow/predictive and see if hypothesis holds or not!

Question 25

Recursive least squares steps

Answer 25

1. Fit model on smallest possible subsample (first k + 1 observations) to obtain βˆj
2. Extend sample by one observation and fit model again to obtain another set of βˆj
3. Repeat until entire sample used till we have a sequence of sets of βˆj (this is the “recursive” bit)
4. Plot, and visual inspection will tell you where the break might be