Structural breaks

Quantitative Risk Management in Python

Jamsheed Shorish

Computational Economist

Risk and distribution

Risk management toolkit
- Risk mitigation: MPT
- Risk measurement: VaR, CVaR
Risk: dispersion, volatility
- Variance (standard deviation) as risk definition
Connection between risk and distribution of risk factors as random variables

Stationarity

Assumption: distribution is same over time
Unchanging distribution = stationary
Global financial crisis period efficient frontier
- Not stationary
Estimation techniques require stationarity
- Historical: unknown stationary distribution from past data
- Parametric: assumed stationary distribution class
- Monte Carlo: assumed stationary distribution for random draws

Structural breaks

Non-stationary => perhaps distribution changes over time
Assume specific points in time for change
- Break up data into sub-periods
- Within each sub-period, assume stationarity
Structural break(s): point(s) of change
- Change in 'trend' of average and/or volatility of data

Example: China's population growth

Examine period 1950 - 2019
Trend is roughly linear...

Graph of China's population over time, 1950-2019

Example: China's population growth

Examine period 1950 - 2019
Trend is roughly linear...
...but seems to slow down from around 1990
Possible structural break near 1990.
Implies distribution of net population (births - deaths) changed
Possible reasons: government policy, standard of living, etc.

Graph of China's population over time with trend arrows, 1950-2019

The Chow test

Previous example: visual evidence for structural break
Quantification: statistical measure
Chow Test:
- Test for existence of structural break given linear model
- Null hypothesis: no break
- Requires three OLS regressions
  - Regression for entire period
  - Two regressions, before and after break
- Collect sum-of-squared residuals
- Test statistic is distributed according to "F" distribution

The Chow test in Python

Hypothesis: structural break in 1990 for China population
Assume linear "factor model": $$\log(\text{Population}_t) = \alpha + \beta * \text{Year}_t + u_t$$
OLS regression using statsmodels's OLS object over full period 1950 - 2019
- Retrieve sum-of-squared residual res.ssr

import statsmodels.api as sm
res = sm.OLS(log_pop, year).fit()

print('SSR 1950-2019: ', res.ssr)

SSR 1950-2019: 0.29240576138055463

The Chow test in Python

Break 1950 - 2019 into 1950 - 1989 and 1990 - 2019 sub-periods
Perform OLS regressions on each sub-period
- Retrieve res_before.ssr and res_after.ssr

pop_before = log_pop.loc['1950':'1989']; year_before = year.loc['1950':'1989'];
pop_after  = log_pop.loc['1990':'2019']; year_after =  year.loc['1990':'2019'];

res_before = sm.OLS(pop_before, year_before).fit()
res_after  = sm.OLS(pop_after, year_after).fit()

print('SSR 1950-1989: ', res_before.ssr)
print('SSR 1990-2019: ', res_after.ssr)

SSR 1950-1989: 0.011741113017411783
SSR 1990-2019: 0.0013717593339608077

The Chow test in Python

Compute the F-distributed Chow test statistic
- Compute the numerator
  - k = 2 degrees of freedom = 2 OLS coefficients $\alpha$, $\beta$
- Compute the denominator
  - 66 degrees of freedom = total number of data points (70) - 2*k

numerator = (ssr_total - (ssr_before + ssr_after)) / 2

denominator = (ssr_before + ssr_after) / 66

chow_test = numerator / denominator
print("Chow test statistic: ", chow_test, "; Critical value, 99.9%: ", 7.7)

Chow test statistic: 702.8715822890057; Critical value, 99.9%: 7.7

Let's practice!

Quantitative Risk Management in Python

Preparing Video For Download...