Parameter Stability & Structural Change Flashcards
What CLRA is rejected now?
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)
How to work with non-constant parameters
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
Using this, when does a structural break/change exist?
If β¹j ≠ β²j
What does this mean for the pooled original equation (standard multi regression expression)
If β¹j ≠ β²j
OLS estimator βj is biased! It is a weighted average of unbiased estimators βhat¹j and βhat²j
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!
It will be biased.
Why is structural change problematic (2)
Violates CLRA1 of constant parameters
Can lead to misleading parameter estimates (by using the full pooled regression PAGE 5 VISUALISED!)
How and where do structural breaks arise (2)
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
2 formal tests for TESTING structural change
1 informal test to identify potential breaks
Chow
Predictive failure
Recursive least squares
Chow test - same set up i.e splitting up into 2 categories again etc.
What is the hypothesis
H₀:β¹₀ = β²₀ , β¹₁=β²₁ ,…, β¹k=β²k (No structural break)
H₁:β¹j ≠ β²j for at least 1 j (Structural break)
Test statistic for Chow test
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!
What is the RSSr and RSSu obtained from?
RSSr is from the pooled regression
RSSu = RSS1 (n₁) + RSS2 (n-n₁)
Add together to get RSSu.
Then calculate F statistic
Chow test with dummy variables
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.
What does our hypothesis’ become
H0 :δ₀ =δ₁ =…=δk =0
H1 :δj≉0 for any j
Test statistic for Chow test with dummy variables
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)
Drawbacks of Chow test (4)
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
Predictive failure test
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)
Predictive failure test process
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
F test
RSSr-RSSu/q
/
RSSu/(n-(k+1))
~Fq,n-(k+1)
Steps of predictive failure test
Run full regression on whole sample (n₁ and n-n₁) to get RSSr
Get RSSu by just running regression on n₁
Find F statistic
Predictive failure test using dummy variables
Add dummt variables to the original model by +ΣyjDj + εi
Dj= 1 if i=n₁+j
Hypothesis for predictive failure test with dummy variables
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!)
Same F test
RSSr-RSSu / n₂
/
RSSu / n₁ - (k+1)
Both tests require us to know where the break is.
How can we find break?
Seeing differences between 2 groups in cross-sectional data
break after a major event in time-series data
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
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!
Recursive least squares steps
- Fit model on smallest possible subsample (first k + 1 observations) to obtain βˆj
- Extend sample by one observation and fit model again to obtain another set of βˆj
- Repeat until entire sample used till we have a sequence of sets of βˆj (this is the “recursive” bit)
- Plot, and visual inspection will tell you where the break might be
