Features

• Python implementation of the Dynamic Nelson-Siegel curve (three factors) with Kalman filter

• Python implementation of the Dynamic Nelson-Siegel-Svensson curve (four factors) with Kalman filter

• Forecasting the yield curve is available

• Log-likelihood is available to use optimize.minimize

Fitting the Dynamic-Nelson-Siegel

To fitting the yield curve, we must set (param,Y,lik,frct,ahead,mty,model)

param: initial parameters vector of Dynamic-Nelson-Siegel models obtained by OLS in the two-step approach =

([

The log of lambdas,

the square root of sample covariance diagonal matrix of VAR(1) residuals (H),

the VAR(1) matrix (phi),

the betas averages (mu), and

the cholesky deconposition of the VAR(1) estimated innovations covariance matrix (Q)

]);

Y: pandas data frame;

lik: log-likelihood;

frct: forecasts;

ahead: months (years) ahead forecasts;

mty: maturities in months (years);

model: ‘NS’ for the Dynamic Nelson-Siegel model or ‘S’ for the Dynamic Nelson-Siegel-Svensson model;

from Dynamic_Nelson_Siegel_Svensson_Kalman_Filter import kalman
import numpy as np
import pandas as pd

url = 'https://www.dropbox.com/s/inpnlugzkddp42q/bonds.csv?dl=1' # US Yield Curve 1972 - 2000
df = pd.read_csv(url,sep=';',index_col=0)

frct = False
ahead = 0
lik = True
mty =  np.array([3,6,9,12,15,18,21,24,30,36,48,60,72,84,96,108,120]) # maturities in months
model = 'NS'

param = np.array([-2.798522, # log of 0.0609
0.14170940,0.07289485,0.11492339,0.11120008,0.09055795,0.07672075,0.07222108,0.07076431,0.07012891,0.07267366,0.10624206,0.09029621,0.10374527,0.09801215,0.09122014,0.11794190,0.13354418, # H
0.99010443,0.02496842,-0.002294319, # phi
-0.02812401,0.94256154, 0.028699387, # phi
0.05178493,0.01247332, 0.788078795, # phi
8.345444,-1.572442,0.2029919,  # mu
0.3408764,-0.07882772,-0.21351036, # Q
0.62661018,-0.00425989, # Q
1.08802059]) # Q

kalman(param = param,Y = df, lik = lik, frct = frct, ahead = ahead, mty = mty, model = model)

which gives the following output

-2281.5860793152

Forecasting the Dynamic-Nelson-Siegel

Kalman function also provides forecasts. We set 12 months ahead, and we define param, Y, mty, and model as above.

frst = True
lik = False
ahead = 12

kalman(param = param,Y = df, lik = lik, frct = frct, ahead = ahead, mty = mty, model = model)

which returns (a_tt,a_t,P_tt,P_t,v2,v1,Yf)

a_tt: the a posteriori state estimate,

a_t: the a priori state estimate,

P_tt: the a posteriori estimate covariance matrix,

P_t: the a posteriori estimate covariance matrix,

v2: filtered yields,

v1: filtered erros, and

Yf: forecasts.

Fitting and Forecasting the Dynamic-Nelson-Siegel-Svensson

We can replicate the steps above but using the following initial parameters.

model = 'S'

param = np.array([-2.798522,-3.55908713, # lambdas
0.14170940,0.07289485,0.11492339,0.11120008,0.09055795,0.07672075,0.07222108,0.07076431,0.07012891,0.07267366,0.10624206,0.09029621,0.10374527,0.09801215,0.09122014,0.11794190,0.13354418, # H
0.99010443,0.02496842,-0.002294319,0.0000, # phi
-0.02812401,0.94256154, 0.028699387,0.000, # phi
0.05178493,0.01247332, 0.788078795,0.0000, # phi
0.00000000,0.00000000,0.0000000000,0.6000, # phi
8.345444,-1.572442,0.2029919,2.96696726, # mu
0.3408764,-0.07882772,-0.21351036,0.1, # Q
0.62661018,-0.00425989,0.2, # Q
1.08802059,0.3, # Q
0.4]) # Q

optimize.minimize

from scipy import optimize

optimize.minimize(fun = kalman, x0 = param, args = (df,lik,frct,ahead,mty,model),method = 'L-BFGS-B', bounds = None,options={'disp':True})

which gives the following output for the Dynamic-Nelson-Siegel model

 ...
 CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
 fun: -3184.551029683603
 ...
 success: True
 x: array([-2.55419095,  0.2682528 ,  0.07550792,  0.09029396,  0.10450556,
 0.09915572,  0.08648558,  0.07862346,  0.07209206,  0.07267585,
 0.07909973,  0.10295337,  0.09260472,  0.10041753,  0.1117615 ,
 0.10696698,  0.15069579,  0.17277911,  0.99437232,  0.02861622,
-0.02214779, -0.02886483,  0.93904762,  0.03963317,  0.02540488,
 0.02288304,  0.84151308,  7.99042415, -1.46634099, -0.43379175,
 0.30276472, -0.02340832,  0.0488872 ,  0.61858155,  0.01026079,
 0.89411573])

Credits

Main developer is Werley Cordeiro.

0.1.0 (2021-02-25)

• First release on PyPI.