Research objective and question

In this publication, I address a precise operational question:

Did Brexit strengthen or weaken the correlation between the FTSE 100 and the major equity indices of France, Switzerland, Germany, the United States, and Japan?

This is not only a descriptive question. It is a portfolio-strategy question: when correlation changes over time, the quality of international diversification changes as well. The central idea of this work is that a political shock such as Brexit does not generate one linear effect, but a sequence of different regimes.

What I did, operationally

I used daily Bloomberg data for:

  • FTSE 100 (dependent variable)
  • CAC 40, SMI, DAX 40, S&P 500, NIKKEI 225 (candidate regressors)

Sample horizon:

  • market data: 02/01/2014 - 31/05/2024
  • final rolling series: 01/05/2014 - 31/05/2024

Estimation structure:

  • rolling window length: 86 observations
  • total rolling regressions: 2632

This architecture was chosen for one specific reason: avoiding a single full-sample "average" regression that hides structural change, and observing instead how the FTSE-external-market relationship evolves day by day.

Why not a single static model

A static 10-year model would be too rigid relative to events such as:

  • Brexit referendum (23/06/2016)
  • official UK exit from the EU (31/01/2020)
  • pandemic shock and the post-2023 phase

For this reason, the methodology was built around:

  1. rolling coefficient estimation
  2. continuous comparison between reduced and full models
  3. daily best-model selection through F-testing
  4. dynamic analysis of the selected model's R2R^2

Key econometric steps

Below I keep only the central formulas of the empirical exercise, i.e., those required to run the estimation and make model-selection decisions.

1) Rolling regression with time-varying parameters

yt(n)=Xt(n)βt(n)+ϵt(n),t=n,…,Ty_t(n) = X_t(n)\beta_t(n) + \epsilon_t(n), \qquad t=n,\dots,T

2) OLS estimation in each window

β^t(n)=[Xt(n)′Xt(n)]−1Xt(n)′yt(n)\hat{\beta}_t(n)=\left[X_t(n)'X_t(n)\right]^{-1}X_t(n)'y_t(n) σ^t2(n)=1n−kϵ^t(n)′ϵ^t(n)\hat{\sigma}_t^2(n)=\frac{1}{n-k}\hat{\epsilon}_t(n)'\hat{\epsilon}_t(n)

3) Full model and baseline model

Full model:

FTSEt=β0+β1CACt+β2SWISSt+β3DAXt+β4S&Pt+β5NIKKEIt+utFTSE_t = \beta_0 + \beta_1 CAC_t + \beta_2 SWISS_t + \beta_3 DAX_t + \beta_4 S\&P_t + \beta_5 NIKKEI_t + u_t

Baseline model (from the preliminary forward phase):

FTSEt=α0+α1CACt+α2SWISSt+etFTSE_t = \alpha_0 + \alpha_1 CAC_t + \alpha_2 SWISS_t + e_t

4) Unrestricted vs restricted model comparison

For each window I computed:

RNV2=(β^1,β^2,β^3,β^4,β^5)(SY,X1,SY,X2,SY,X3,SY,X4,SY,X5)TSY^2R^2_{NV}=\frac{(\hat{\beta}_1,\hat{\beta}_2,\hat{\beta}_3,\hat{\beta}_4,\hat{\beta}_5)(S_{Y,X_1},S_{Y,X_2},S_{Y,X_3},S_{Y,X_4},S_{Y,X_5})^T}{S_{\hat{Y}}^2} RV2=(α^1,α^2)(SY,X1,SY,X2)TSY^2R^2_{V}=\frac{(\hat{\alpha}_1,\hat{\alpha}_2)(S_{Y,X_1},S_{Y,X_2})^T}{S_{\hat{Y}}^2}

with joint test:

H0:β3=β4=β5=0H_0:\beta_3=\beta_4=\beta_5=0

5) F-statistic used in model selection

F=(RNV2−RV2q)(1−RNV2Tτ−kNV−1)F=\frac{\left(\frac{R^2_{NV}-R^2_{V}}{q}\right)}{\left(\frac{1-R^2_{NV}}{T_{\tau}-k_{NV}-1}\right)}

I then defined a decision dummy:

Dj=1(p-valuej<0.05)D_j = \mathbf{1}(p\text{-value}_j < 0.05)

and the average rejection rate over all 2632 windows:

12632∑j=12632Dj\frac{1}{2632}\sum_{j=1}^{2632}D_j

In the final results this frequency is approximately 19.41%.

6) Turning-point detection

To distinguish strengthening/weakening regimes I compared 10-day and 60-day moving averages:

t=R2‾10−R2‾60SE(R2‾10−R2‾60)t=\frac{\overline{R^2}_{10}-\overline{R^2}_{60}}{SE(\overline{R^2}_{10}-\overline{R^2}_{60})}

Using one-sided thresholds -2.39 and +2.39 (58 d.f., α=0.01\alpha=0.01).

Selection logic implemented (detailed)

The "modified stepwise backward" procedure does not remove variables once in a static way; it does so day by day inside each rolling window.

Order of candidate models compared:

  1. CAC + SWISS
  2. CAC + SWISS + S&P
  3. CAC + SWISS + DAX
  4. CAC + SWISS + NIKKEI
  5. CAC + SWISS + S&P + DAX
  6. CAC + SWISS + S&P + NIKKEI
  7. CAC + SWISS + DAX + NIKKEI
  8. CAC + SWISS + S&P + DAX + NIKKEI

Applied decision rule:

  • if the reduced model is not rejected against the full model, keep the reduced model
  • if multiple models are admissible within the same block, keep the one with the best R2R^2
  • store that selected model's R2R^2 as the daily value of the final series

This is the core contribution of the exercise: the analysis is not estimating one "average" correlation, but a selected and time-updated correlation structure based on statistical testing.

Main empirical results

Selection frequencies over 2632 days:

Selected modelCountRelative frequency
CAC + SWISS21210.8059
CAC + SWISS + S&P2810.1068
CAC + SWISS + DAX1260.0479
CAC + SWISS + NIKKEI510.0194
CAC + SWISS + S&P + DAX400.0152
CAC + SWISS + S&P + NIKKEI90.0034
CAC + SWISS + DAX + NIKKEI40.0015
Full model (5 regressors)00.0000

Structural summary:

  • 2-variable models: 80.59%
  • 3-variable models: 17.40%
  • 4-variable models: 2.01%
  • full model: 0%

Interpretation: FTSE dependence remains mainly Eurocentric (CAC-SWISS), while additional regressors enter only episodically and in a regime-dependent way.

Rolling R2 and regime reading

Rolling R2 and models selected each day (data from 01/05/2014 to 31/05/2024)

Rolling R2 and selected models per day (01/05/2014 - 31/05/2024)

Sample references used in the chart:

  • Q1 = 0.6029
  • Q3 = 0.7788
  • observed R2R^2 range: approximately 0.30 - 0.90

Argument-based phase reading:

  • Post referendum (2016-2018): persistent downward path and progressive decorrelation.
  • 2018-early 2020: return to interquartile range and stabilization.
  • 2020-2021: correlation peak during systemic global shock.
  • 09/2023-05/2024: new weakening phase with recent lows.

The results chapter identifies 38 turning points, consistent with non-linear dynamics and multiple regime changes.

What this work demonstrates, in substance

The strongest conclusion is not simply "Brexit increases" or "Brexit decreases" correlation in absolute terms. The strongest conclusion is both methodological and economic:

  • the FTSE-external-market relationship is time-varying
  • Brexit acts as a shock that initiates a first decorrelation phase
  • subsequent global shocks (COVID, geopolitical context) temporarily re-synchronize correlations
  • in the last sample segment, a renewed weakening of co-movement emerges

Conclusion

This analysis answers the research question with a non-simplistic conclusion: the Brexit effect is mixed and dynamic. The rolling procedure with F-test-based model selection is what makes the result robust, because it translates historical change into a statistically traceable day-by-day sequence.