Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator

Mary Wangui Wanjohi; Josephine Njeri Ngure; Martin Mutweri Kithinji

doi:doi:10.11648/j.ajtas.20261504.13

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Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator

Mary Wangui Wanjohi^*

, Josephine Njeri Ngure, Martin Mutweri Kithinji

Published in American Journal of Theoretical and Applied Statistics (Volume 15, Issue 4)

Received: 1 June 2026 Accepted: 10 June 2026 Published: 28 July 2026

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Abstract

Exchange rate volatility poses significant challenges for economic planning, risk management, and policy formulation in emerging economies such as Kenya. This study models the volatility of the United States Dollar/ Kenyan Shillings (USD/KES) exchange rate using the nonparametric Nadaraya-Watson kernel regression estimator. Daily buying rates from the Central Bank of Kenya spanning January 2, 2003, to December 29, 2023 (4,764 observations) were used. The Nadaraya-Watson estimator smooths the data using kernel functions and bandwidth parameters, enabling it to adapt to localized features in the volatility pattern that conventional parametric models may overlook. The conditional mean and conditional variance functions were estimated using a Gaussian kernel with cross-validated fixed bandwidths. The optimal bandwidth for the conditional mean was 0.02750928 and for the conditional variance was 0.1349632, with a bandwidth ratio of 4.905869, indicating that the volatility function requires smoother estimation. The conditional variance function exhibited a distinct U-shape, showing higher volatility following extreme positive or negative lagged returns. The model achieved a Root Mean Squared Error (RMSE) of 1.919 for variance, capturing major volatility episodes including the 2008-2009 financial crisis, the 2011 currency crisis, and the 2016 reserves depletion shock. The study concludes that the Nadaraya-Watson kernel regression estimator successfully captures nonlinear volatility dynamics without imposing rigid parametric assumptions, making it suitable for emerging market currencies like the Kenyan shilling.

Published in	American Journal of Theoretical and Applied Statistics (Volume 15, Issue 4)
DOI	10.11648/j.ajtas.20261504.13
Page(s)	141-148
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Nadaraya-watson Estimator, Kernel Regression, Exchange Rate Volatility, USD/KES, Nonparametric Estimation, Bandwidth Selection, Volatility Clustering

References

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[2]	Nadaraya, E. A. (1964). On estimating regression. Theory of Probability & Its Applications, 9(1), 141-142. https://doi.org/10.1137/1109020
[3]	Watson, G. S. (1964). Smooth regression analysis. Sankhya: The Indian Journal of Statistics, Series A}, 26(4), 359-372.
[4]	Rahma, I., & Setiawan, I. (2022). Comparison of the nonparametric regression Nadaraya-Watson estimator kernel function and local polynomial regression in predicting USD against IDR. Tadulako Science and Technology Journal, 2(2), 10-16.
[5]	H"ardle, W. (1990). Applied Nonparametric Regression. Cambridge University Press. https://doi.org/10.1017/CCOL0521382483
[6]	Ngure, J. N., & Waititu, A. G. (2021). Consistency of an estimator for change point in volatility of financial returns. Journal of Mathematics Research, 13(1), 56-66. https://doi.org/10.5539/jmr.v13n1p56
[7]	Ngure, J. N., Waititu, A. G., & Mundia, S. M. (2023). Detection and estimation of change point in volatility function of foreign exchange rate returns. International Journal of Data Science and Analysis, 9(1), 1-7. https://doi.org/10.11648/j.ijdsa.20230901.11ext
[8]	Silverman, B. W. (1986). Density estimation for statistics and data analysis. Routledge. https://doi.org/10.1201/9781315140919
[9]	H"ardle, W., Huet, S., Mammen, E., & Sperlich, S. (2004). Bootstrap inference in semiparametric generalized additive models. Econometric Theory, 20(2), 265-300. https://doi.org/10.1017/S026646660420202X
[10]	Wand, M. P., & Jones, M. C. (1994). Kernel smoothing. CRC Press. https://doi.org/10.10170S026646660420202X
[11]	Tsay, R. S. (2005). Analysis of financial time series. John Wiley & Sons. https://doi.org/10.1002/0471264105
[12]	Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327. https://doi.org/10.1016/0304-4076(86)90063-1
[13]	Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007. https://doi.org/10.2307/1912773
[14]	Yuan, K. H., Bentler, P. M., & Zhang, W. (2005). The effect of skewness and kurtosis on mean and covariance structure analysis: The univariate case and its multivariate implication. Sociological Methods & Research, 34(2), 240-258. https://doi.org/10.1177/0049124105280200
[15]	Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1), 246-256. https://doi.org/10.1016/j.jeconom.2010.03.034
[16]	Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH (1, 1)? Journal of Applied Econometrics, 20(7), 873-889. https://doi.org/10.1002/jae.800

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APA Style

Wanjohi, M. W., Ngure, J. N., Kithinji, M. M. (2026). Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. American Journal of Theoretical and Applied Statistics, 15(4), 141-148. https://doi.org/10.11648/j.ajtas.20261504.13

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ACS Style

Wanjohi, M. W.; Ngure, J. N.; Kithinji, M. M. Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. Am. J. Theor. Appl. Stat. 2026, 15(4), 141-148. doi: 10.11648/j.ajtas.20261504.13

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AMA Style

Wanjohi MW, Ngure JN, Kithinji MM. Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. Am J Theor Appl Stat. 2026;15(4):141-148. doi: 10.11648/j.ajtas.20261504.13

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@article{10.11648/j.ajtas.20261504.13,
  author = {Mary Wangui Wanjohi and Josephine Njeri Ngure and Martin Mutweri Kithinji},
  title = {Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {4},
  pages = {141-148},
  doi = {10.11648/j.ajtas.20261504.13},
  url = {https://doi.org/10.11648/j.ajtas.20261504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261504.13},
  abstract = {Exchange rate volatility poses significant challenges for economic planning, risk management, and policy formulation in emerging economies such as Kenya. This study models the volatility of the United States Dollar/ Kenyan Shillings (USD/KES) exchange rate using the nonparametric Nadaraya-Watson kernel regression estimator. Daily buying rates from the Central Bank of Kenya spanning January 2, 2003, to December 29, 2023 (4,764 observations) were used. The Nadaraya-Watson estimator smooths the data using kernel functions and bandwidth parameters, enabling it to adapt to localized features in the volatility pattern that conventional parametric models may overlook. The conditional mean and conditional variance functions were estimated using a Gaussian kernel with cross-validated fixed bandwidths. The optimal bandwidth for the conditional mean was 0.02750928 and for the conditional variance was 0.1349632, with a bandwidth ratio of 4.905869, indicating that the volatility function requires smoother estimation. The conditional variance function exhibited a distinct U-shape, showing higher volatility following extreme positive or negative lagged returns. The model achieved a Root Mean Squared Error (RMSE) of 1.919 for variance, capturing major volatility episodes including the 2008-2009 financial crisis, the 2011 currency crisis, and the 2016 reserves depletion shock. The study concludes that the Nadaraya-Watson kernel regression estimator successfully captures nonlinear volatility dynamics without imposing rigid parametric assumptions, making it suitable for emerging market currencies like the Kenyan shilling.},
 year = {2026}
}

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TY  - JOUR
T1  - Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator
AU  - Mary Wangui Wanjohi
AU  - Josephine Njeri Ngure
AU  - Martin Mutweri Kithinji
Y1  - 2026/07/28
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261504.13
DO  - 10.11648/j.ajtas.20261504.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 141
EP  - 148
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261504.13
AB  - Exchange rate volatility poses significant challenges for economic planning, risk management, and policy formulation in emerging economies such as Kenya. This study models the volatility of the United States Dollar/ Kenyan Shillings (USD/KES) exchange rate using the nonparametric Nadaraya-Watson kernel regression estimator. Daily buying rates from the Central Bank of Kenya spanning January 2, 2003, to December 29, 2023 (4,764 observations) were used. The Nadaraya-Watson estimator smooths the data using kernel functions and bandwidth parameters, enabling it to adapt to localized features in the volatility pattern that conventional parametric models may overlook. The conditional mean and conditional variance functions were estimated using a Gaussian kernel with cross-validated fixed bandwidths. The optimal bandwidth for the conditional mean was 0.02750928 and for the conditional variance was 0.1349632, with a bandwidth ratio of 4.905869, indicating that the volatility function requires smoother estimation. The conditional variance function exhibited a distinct U-shape, showing higher volatility following extreme positive or negative lagged returns. The model achieved a Root Mean Squared Error (RMSE) of 1.919 for variance, capturing major volatility episodes including the 2008-2009 financial crisis, the 2011 currency crisis, and the 2016 reserves depletion shock. The study concludes that the Nadaraya-Watson kernel regression estimator successfully captures nonlinear volatility dynamics without imposing rigid parametric assumptions, making it suitable for emerging market currencies like the Kenyan shilling.
VL  - 15
IS  - 4
ER  -

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Author Information

Mary Wangui Wanjohi

Department of Pure and Applied Sciences, Kirinyaga University, Kerugoya, Kenya

Contact Email

http://orcid.org/0009-0003-6592-2727
Josephine Njeri Ngure

Department of Pure and Applied Sciences, Kirinyaga University, Kerugoya, Kenya
Martin Mutweri Kithinji

Department of Pure and Applied Sciences, Kirinyaga University, Kerugoya, Kenya

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APA Style

Wanjohi, M. W., Ngure, J. N., Kithinji, M. M. (2026). Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. American Journal of Theoretical and Applied Statistics, 15(4), 141-148. https://doi.org/10.11648/j.ajtas.20261504.13

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ACS Style

Wanjohi, M. W.; Ngure, J. N.; Kithinji, M. M. Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. Am. J. Theor. Appl. Stat. 2026, 15(4), 141-148. doi: 10.11648/j.ajtas.20261504.13

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AMA Style

Wanjohi MW, Ngure JN, Kithinji MM. Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator. Am J Theor Appl Stat. 2026;15(4):141-148. doi: 10.11648/j.ajtas.20261504.13

Copy | Download

@article{10.11648/j.ajtas.20261504.13,
  author = {Mary Wangui Wanjohi and Josephine Njeri Ngure and Martin Mutweri Kithinji},
  title = {Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {4},
  pages = {141-148},
  doi = {10.11648/j.ajtas.20261504.13},
  url = {https://doi.org/10.11648/j.ajtas.20261504.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261504.13},
  abstract = {Exchange rate volatility poses significant challenges for economic planning, risk management, and policy formulation in emerging economies such as Kenya. This study models the volatility of the United States Dollar/ Kenyan Shillings (USD/KES) exchange rate using the nonparametric Nadaraya-Watson kernel regression estimator. Daily buying rates from the Central Bank of Kenya spanning January 2, 2003, to December 29, 2023 (4,764 observations) were used. The Nadaraya-Watson estimator smooths the data using kernel functions and bandwidth parameters, enabling it to adapt to localized features in the volatility pattern that conventional parametric models may overlook. The conditional mean and conditional variance functions were estimated using a Gaussian kernel with cross-validated fixed bandwidths. The optimal bandwidth for the conditional mean was 0.02750928 and for the conditional variance was 0.1349632, with a bandwidth ratio of 4.905869, indicating that the volatility function requires smoother estimation. The conditional variance function exhibited a distinct U-shape, showing higher volatility following extreme positive or negative lagged returns. The model achieved a Root Mean Squared Error (RMSE) of 1.919 for variance, capturing major volatility episodes including the 2008-2009 financial crisis, the 2011 currency crisis, and the 2016 reserves depletion shock. The study concludes that the Nadaraya-Watson kernel regression estimator successfully captures nonlinear volatility dynamics without imposing rigid parametric assumptions, making it suitable for emerging market currencies like the Kenyan shilling.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Modelling the Volatility of the USD/KES Exchange Rate Using the Nadaraya-watson Kernel Regression Estimator
AU  - Mary Wangui Wanjohi
AU  - Josephine Njeri Ngure
AU  - Martin Mutweri Kithinji
Y1  - 2026/07/28
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261504.13
DO  - 10.11648/j.ajtas.20261504.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 141
EP  - 148
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261504.13
AB  - Exchange rate volatility poses significant challenges for economic planning, risk management, and policy formulation in emerging economies such as Kenya. This study models the volatility of the United States Dollar/ Kenyan Shillings (USD/KES) exchange rate using the nonparametric Nadaraya-Watson kernel regression estimator. Daily buying rates from the Central Bank of Kenya spanning January 2, 2003, to December 29, 2023 (4,764 observations) were used. The Nadaraya-Watson estimator smooths the data using kernel functions and bandwidth parameters, enabling it to adapt to localized features in the volatility pattern that conventional parametric models may overlook. The conditional mean and conditional variance functions were estimated using a Gaussian kernel with cross-validated fixed bandwidths. The optimal bandwidth for the conditional mean was 0.02750928 and for the conditional variance was 0.1349632, with a bandwidth ratio of 4.905869, indicating that the volatility function requires smoother estimation. The conditional variance function exhibited a distinct U-shape, showing higher volatility following extreme positive or negative lagged returns. The model achieved a Root Mean Squared Error (RMSE) of 1.919 for variance, capturing major volatility episodes including the 2008-2009 financial crisis, the 2011 currency crisis, and the 2016 reserves depletion shock. The study concludes that the Nadaraya-Watson kernel regression estimator successfully captures nonlinear volatility dynamics without imposing rigid parametric assumptions, making it suitable for emerging market currencies like the Kenyan shilling.
VL  - 15
IS  - 4
ER  -

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