# A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of the FTSE and S&P500 Indexes

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## Abstract

**:**

## 1. Introduction

## 2. Previous Work and Econometric Models

#### 2.1. The Transmission of Shocks across Markets and Economies

#### 2.2. Stock Price Changes and Volatility Changes

#### 2.3. Bounds Tests

#### 2.4. NARDL Approach

#### 2.5. Quantile Regression

## 3. Results of the Analysis

#### 3.1. Preliminary Analysis

#### 3.2. ARDL Analysis

#### 3.3. NARDL Analysis

#### 3.4. Quantile Regression Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Descriptive Statistics for the continuously compounded returns on the S&P500 and FTSE100 indices and analysis of variance tests.

S&P500 | |

Period 1: 2009-04-24–2015-12-31 | Number of Observations1776 |

Mean | 0.00050218 |

Median | 0.00067228 |

Minimum | −0.068958 |

Maximum | 0.046317 |

Standard deviation | 0.010271 |

Period 2: 2016-01-01–2022-03-11 | Number of Observations 1172 |

Mean | 0.00053136 |

Median | 0.00075748 |

Minimum | −0.12765 |

Maximum | 0.089683 |

Standard deviation | 0.012454 |

FTSE100 | |

Period 1: 2009-04-24–2015-12-31 | Number of Observations1776 |

Mean | 0.00021978 |

Median | 0.00056958 |

Minimum | 0.047798 |

Maximum | 0.050323 |

Standard deviation | 0.010834 |

Period 2: 2016-01-01–2022-03-11 | Number of Observations 1172 |

Mean | $7.9112\times {10}^{-5}$ |

Median | 0.00052255 |

Minimum | −0.11512 |

Maximum | 0.086668 |

Standard deviation | 0.010834 |

Test of Equality of Variance between the two periods: Pre and Post-Brexit | |

S&P500 | |

Sample 1, n = 1776. variance = 0.010271 | Sample 2, n = 1172, variance = 0.012454 |

Test Statistic: F(1171, 1176) = 1.21254 | |

Two-tailed p-value = 0.0002681 *** | |

FTSE100 | |

Sample 1, n = 1776. variance = 0.010517 | Sample 2, n = 1172, variance = 0.010834 |

Test statistic: F(1744, 1203) = 1.03014 | |

Two-tailed p-value = Two-tailed p-value = 0.5742 |

Period 1: April 2009 to December 2015 | ||

ADF test with constant | ADF test with constant and trend | |

SP500 | −1.00892 (0.75) | −2.79693 (0.20) |

FTSE | −2.85173 (0.05) | −3.46467 (0.04) |

Period 2: January 2016–March 2021 | ||

ADF test with constant | ADF test with constant and trend | |

SP500 | −0.3379 (0.92) | −3.15656 (0.09) |

FTSE | −2.42549 (0.13) | −2.54632 (0.31) |

Period 1: April 2009 to December 2015 | ||||

Coefficient | Std. Error | t-Ratio | p-Value | |

Test with constant | ||||

Constant | 3548.82 | 29.97 | 118.4 | 0.00 |

S&P500 | 1.60271 | 0.019 | 83.29 | 0.00 |

Test for unit root residuals | −4.31864 | 0.00 | ||

Test with Constant and Trend | ||||

Constant | 3044.99 | 73.52 | 41.42 | 0.00 |

S&P500 | 2.19491 | 0.0814 | 26.97 | 0.00 |

Time | −0.472097 | 0.0631 | −7.48 | 0.00 |

Test for unit root residuals | −4.6544 | 0.003 | ||

Period 2: January 2016–March 2021 | ||||

Coefficient | Std. Error | t-Ratio | p-Value | |

Test with constant | ||||

Constant | 6951.66 | 97.86 | 71.04 | 0.00 |

S&P500 | −0.00243523 | 0.03535 | −0.069 | 0.95 |

Test for unit root residuals | −2.48 | 0.29 | ||

Test with Constant and Trend | ||||

Constant | 3044.99 | 73.52 | 41.42 | 0.00 |

S&P500 | 2.1949 | 0.0814 | 26.97 | 0.00 |

Time | −0.4720 | 0.0631 | −7.48 | 0.00 |

Test for unit root residuals | −4.654 | 0.003 |

Period 1: 2009–2015 | |||

Variables | Coefficients | Std. Error | t Value |

intercept | 87.414927 | 21.589016 | 4.049 *** |

FTSE(−1) | −0.013315 | 0.005094 | −2.614 *** |

S&P500(diff) | 0.424579 | 0.100060 | 4.243 *** |

S&P500(−1) | −0.019356 | 0.020183 | −0.959 |

S&P500(diff, −1) | 0.035495 | 0.020183 | 0.358 |

Trendvar | 0.026740 | 0.013212 | 2.024 ** |

Adjusted R-squared | 0.01734 | ||

F Statistic | 6.845 *** | ||

Breusch-Godfrey LM Test | 0.267 | p-value: 0.605 | |

Shapiro-Wilk Test | 0.981 | p-value: 0 | |

Period 2: 2016–2021 | |||

Variables | Coefficients | Std. Error | t Value |

intercept | 62.674903 | 25.525247 | 2.455 ** |

FTSE(−1) | −0.008025 | 0.003207 | −2.502 ** |

S&P500(diff) | 0.939819 | 0.053684 | 17.506 *** |

S&P500(−1) | −0.001318 | 0.011459 | −0.115 |

S&P500(diff, −1) | 0.266686 | 0.053847 | 4.953 *** |

Trendvar | −0.006951 | 0.014569 | −0.477 |

Adjusted R-squared | 0.1959 | ||

F Statistic | 63.71 *** | ||

Breusch-Godfrey LM Test | 1.034 | p-value: 0.309 | |

Shapiro-Wilk Test | 0.974 | p-value: 0 |

Period 1: 2009–2015 | ||||

OLS, using observations 2009-04-27–2015-12-31 (T = 1658) | ||||

Dependent variable: ld_FTSEAdjusted | ||||

Coefficient | Std. Error | t-Ratio | p-Value | |

const | 0.000144605 | 0.000250097 | 0.5782 | 0.5632 |

ld_GSPCAdjusted | 0.162625 | 0.0241800 | 6.726 | 0.0000 |

Mean dependent var | 0.000229 | S.D. dependent var | 0.010306 | |

Sum squared resid | 0.171305 | S.E. of regression | 0.010171 | |

${R}^{2}$ | 0.026589 | Adjusted ${R}^{2}$ | 0.026001 | |

$F(1,1656)$ | 45.23380 | p-value(F) | $2.40\times {10}^{-11}$ | |

Log-likelihood | 5255.693 | Akaike criterion | −10507.39 | |

Schwarz criterion | −10,496.56 | Hannan–Quinn | −10503.37 | |

$\widehat{\rho}$ | 0.002305 | Durbin–Watson | 1.994813 | |

Period 2: 2016–2021 | ||||

OLS, using observations 2016-01-04–2021-03-11 (T = 1289) | ||||

Dependent variable: ld_FTSEAdjusted | ||||

Coefficient | Std. Error | t-Ratio | p-Value | |

const | −0.000112716 | 0.000276583 | −0.4075 | 0.6837 |

ld_GSPCAdjusted | 0.400293 | 0.0226603 | 17.66 | 0.0000 |

Mean dependent var | 0.000096 | S.D. dependent var | 0.011054 | |

Sum squared resid | 0.126675 | S.E. of regression | 0.009921 | |

${R}^{2}$ | 0.195148 | Adjusted ${R}^{2}$ | 0.194522 | |

$F(1,1287)$ | 312.0508 | P-value(F) | $1.07\times {10}^{-62}$ | |

Log-likelihood | 4118.274 | Akaike criterion | −8232.548 | |

Schwarz criterion | −8222.225 | Hannan–Quinn | −8228.673 | |

$\widehat{\rho}$ | 0.023553 | Durbin–Watson | 1.952382 |

Period 1: 2009–2015 | ||||||

Estimates | Long Run Coefficients | |||||

Coefficients | Estimate | St. Error | t Value | Estimate | St. Error | t Value |

Const | 80.07220 | 30.46668 | 2.628 *** | |||

FTSE.Adjusted_1 | −0.01636 | 0.00625 | −2.617 *** | |||

GSPC.Adjusted_p | 0.85942 | 0.16947 | 5.071 *** | 52.54374 | 23.05575 | 2.2790 ** |

GSPC.Adjusted_p_1 | −0.74983 | 0.23290 | −3.220 *** | −45.84362 | 22.80100 | −2.0106 ** |

GSPC.Adjusted_p_2 | −0.41464 | 0.23269 | −1.782 * | −25.35061 | 17.25858 | −1.4689 |

GSPC.Adjusted_p_3 | 0.26798 | 0.16826 | 1.593 | 16.38378 | 11.89128 | 1.3778 |

GSPC.Adjusted_n | −0.01127 | 0.02030 | −0.555 | −0.68924 | 1.40084 | −0.4920 |

Trend | 0.16325 | 0.08560 | 1.907 * | 9.98092 | 4.51825 | 2.2090 ** |

Adjusted R-Squared | 0.02418 | |||||

F-statistic | 0.02418 *** | |||||

Model Diagnostic tests | ||||||

JB test | 9.808146 *** | |||||

LM test (1 lag ) | 0.3440852 | |||||

ARCH test (4 lag) | 5.117118 *** | |||||

Long-Run Asymmetry test | ||||||

W. Statistic | 210,664.6 *** | |||||

Bounds F Test | <.......I(0)...................I(1)...> | |||||

1% Critical Value | 5.15..............6.36 | |||||

F Value | 6.85 *** | |||||

Period 2: 2016–2021 | ||||||

Estimates | Long Run Coefficients | |||||

Coefficients | Estimate | St. Error | t Value | Estimate | St. Error | t Value |

Const | 160.443173 | 51.074665 | 3.141 *** | |||

FTSE.Adjusted_1 | 0.005375 | 0.027765 | 0.194 | |||

FTSE.Adjusted_2 | −0.095398 | 0.039504 | −2.415 *** | 17.7480 | 86.4987 | 0.2052 |

FTSE.Adjusted_3 | −0.048372 | 0.038373 | −1.261 | 8.9992 | 47.7393 | 0.1885 |

FTSE.Adjusted_4 | 0.115227 | 0.026474 | 4.352 *** | −21.4370 | 111.2749 | −0.1926 |

GSPC.Adjusted_p | 0.794214 | 0.108815 | 7.299*** | −147.7565 | 761.8276 | −0.1940 |

GSPC.Adjusted_p_1 | −0.633554 | 0.153346 | −4.132 *** | 117.8672 | 604.3643 | 0.1950 |

GSPC.Adjusted_p_2 | −0.428697 | 0.141188 | −3.036 *** | 79.7553 | 414.9075 | 0.1922 |

GSPC.Adjusted_p_3 | 0.268331 | 0.101340 | 2.648 *** | −49.9206 | 257.2427 | −0.1941 |

GSPC.Adjusted_n | 1.199090 | 0.089650 | 13.375 *** | −223.0801 | 1152.7959 | −0.1935 |

GSPC.Adjusted_n_1 | −0.864588 | 0.141390 | −6.115 *** | 160.8490 | 825.7917 | 0.1948 |

GSPC.Adjusted_n_2 | −0.326815 | 0.097825 | −3.341 *** | 60.8012 | 320.2803 | 0.1898 |

Trend | 0.061648 | 0.029695 | 2.076 ** | −11.4690 | 59.8123 | −0.1918 |

Adjusted R-Squared | 0.2212 | |||||

F-statistic | 31.42 *** | |||||

Model Diagnostic tests | ||||||

JB test | 0.9761 *** | |||||

LM test (4 lag ) | 2.6369 | |||||

ARCH test (4 lag) | 100.89 *** | |||||

Long-Run Asymmetry test | ||||||

W. Statistic | 210,664.6 *** | |||||

Bounds F Test | <.......I(0)...................I(1)...> | |||||

1% Critical Value | 5.15..............6.36 | |||||

F Value | 31.42 *** |

Period 1: 2009–2015 | ||||

Quantile estimates, using observations 2009-04-27–2015-12-31 (T = 1659) Dependent | ||||

variable: ld_FTSEAdjusted, Asymptotic standard errors assuming IID errors | ||||

Tau | Coefficient | Std. Error | t-Ratio | |

Constant | 0.05 | −0.0162232 | 0.000593580 | −27.3312 *** |

0.25 | −0.00539134 | 0.000277723 | −19.4127 *** | |

0.50 | 0.000344003 | 0.000242072 | 1.42107 | |

0.75 | 0.00584020 | 0.000352873 | 16.5505 *** | |

0.95 | 0.0166363 | 0.000756901 | 21.9795*** | |

ld_GSPCAdjusted | 0.05 | 0.361510 | 0.0573889 | 6.29931 *** |

0.25 | 0.238989 | 0.0268510 | 8.90058 *** | |

0.50 | 0.157727 | 0.0234042 | 6.73927 *** | |

0.75 | 0.134198 | 0.0341167 | 3.93349 *** | |

0.95 | 0.0715642 | 0.0731792 | 0.977930 | |

Period 2: 2016–2021 | ||||

Quantile estimates, using observations 2016-01-04–2021-03-11 (T = 1290) Dependent | ||||

variable: ld_FTSEAdjusted, Asymptotic standard errors assuming IID errors | ||||

Tau | Coefficient | Std. Error | t-Ratio | |

Constant | 0.05 | −0.0157834 | 0.000883658 | −17.8615 *** |

0.25 | −0.00491128 | 0.000305907 | −16.0548 *** | |

0.50 | 0.000178811 | 0.000234350 | 0.763010 | |

0.75 | 0.00492406 | 0.000319208 | 15.4259 *** | |

0.95 | 0.0154164 | 0.00106163 | 14.5215 | |

ld_GSPCAdjusted | 0.05 | 0.479784 | 0.0723975 | 6.62708 *** |

0.25 | 0.335508 | 0.0250627 | 13.3867 *** | |

0.50 | 0.286440 | 0.0192001 | 14.9187 *** | |

0.75 | 0.265986 | 0.0261525 | 10.1706 | |

0.95 | 0.237152 | 0.0869783 | 2.72656 *** |

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**MDPI and ACS Style**

Allen, D.E.; McAleer, M. A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of the FTSE and S&P500 Indexes. *Risks* **2021**, *9*, 195.
https://doi.org/10.3390/risks9110195

**AMA Style**

Allen DE, McAleer M. A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of the FTSE and S&P500 Indexes. *Risks*. 2021; 9(11):195.
https://doi.org/10.3390/risks9110195

**Chicago/Turabian Style**

Allen, David E., and Michael McAleer. 2021. "A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of the FTSE and S&P500 Indexes" *Risks* 9, no. 11: 195.
https://doi.org/10.3390/risks9110195