2. Parameter stability and structural change

What does CLRA1 mean?

What does β_j interpreted as?

What are non constant parameters

We have n observations and that for the first set we have n₁ observations and the second set has n -n₁ observations and the β are not the same

What does it mean if we have a structural break/change?

β_j¹ =/ β_j² - 1 and 2 indicate subsets

• the parameter is not constant

• OLS estimater β_jhat will be biased

• It will be be some unbiased estimators of β_j²hat and β_j² hat - overestimatye for one and underestimate for the other

What can a sructural break look like?

What does the Simpsons paradox look like

Why can structural change be problematic?

• breaks assumption of constant parameters

• May lead to misleading parameter estimates

How does structural change problems arise in cross-sectional data?

Cross-sectional data, relationships between variables may vary across different subgroups

• determinants of wage may be different for men and women

• determinants of life satisfaction may differ by age

How does structural change problems arise in time-series data?

Time-series data, relationships between variables may vary across different subperiods

• relationships between macroeconomic aggregates change due to shocks

• Determinants of aggregate excess mortality change due to Covid

What are the tests for structural change?

formal test:

• Chow Test

• Predictive Failure test

Informal diagnostic to idnetify potential structural breaks

• recursive Least squares

What are we testing in the Chow test?

The null hypothesis is that the model 1 is correct: Y_i = β₀ + β₁X_1i + β₂X_2i + … + β_kX_ki + e_i

Alternative hypothesis is that any parameter is non-constant

• H₀: β₀¹ = β₀², β₁¹ = β₁², …. , β_k¹ = β_k²

• H₁: β_j¹ =/ β_j² for atleast one j

How is the Chow test an F-Test

What are the 5 easy steps to do the F stat of a Chow test?

1. Run the pooled, restricted regression using the whole sample (1,…, n) and obtain RSS_R

2. Run regression in group 1 using observations 1,…n₁ obtain RSS₁

3. Run regression in group 2 using observations n₁+1,…,n and obtain RSS₂

4. Calculated RSS_u = RSS₁ + RSS₂

5. Calculate the F-stat and compare it to the relevant critical value of the F-distribution table - bigger values of F are rejected - if RSS_R is much bigger than RSS_U → more residual variation in RSS_R than RSS_U

What are the steps for a chow test

1. need critical components

2. need hypothesis

3. write F stat/ test stat

4. Need critical value

What are the drawbacks of the Chow Test

1. to calculate RSS_u = RSS₁ + RSS₂ we require:

   • e_i¹ and e_i² are homoscedastic and independently distributed

2. The test does not tell us which parameter is unstable

   • only that any of them may be

3. Procedure requires us to know where structural break occurs

   • bc/ we need to specify where to split the sample - in practice we don’t know this exactly

4. May not have enough data to estimate both models seperately

   • esp if one sub-sample is relatively small

What’s different between the predictive failure test and the chow test

• instead of estimating models for 2 samples and detecting differences We estimate the model for one sample and check whether it can accurately predict outcomes in the other sample

How does predictive failure test show there is a structural break?

If model cannot predict outcomes in the second sample → then structural break has occured

What are the hypothesis for a predictive failure test?

Null: the same model can be fitted to the second sample

H₀: Y_i = β₀ + β₁X_1i + β₂X_2i + … + β_kX_ki + e_i i = n₁ + 1, …. n₁ + n₂

If we impose the null on our unrestricted model what do we get

Y_i = β₀ + β₁X_1i + β₂X_2i + … + β_kX_ki + e_i i = 1,2,..n₁+n₂

How is the predictive failure test an F-test of multiple regressions

What are the steps for a predictive failure test

1. Run the full, restricted regression using the whole sample (inc all observations 1, .., n₁, n₁ + 1, …, n₁ + n₂) and obtain RSS_R

2. Run regression in the first subgroup using only observations 1,…,n₁ and obtain RSS_U

3. Calculate the F-stat and compare it to the relevant critical value from the F-distribution table

How can we use the predictive failure test using dummy variables?

How can we know when where a break is?

• differences btw/ groups in cross-sectional data

• break after a major event in time-series

• Data-driven way to identify potential breaks - recursive least squares

What is the procedure for Recursive Least squares

1. Fit model on smallest possible subsample (first k+1 observations)

   • Obtain β_jhat

2. Extend sample by 1 observation and fit model again

   • Obtain another set for β_jhat

3. Repeat adding one observation at a time, until entire sample used

   • Obtaining a sequence of sets for β_jhat

4. Plot the sequence of values for each parameter

   • visual inspection will tell you where the break might be

What could it look like if there is a break using reclusive least squares