Schwartz-Smith model using SSM

I am looking to use the Schwartz-Smith code here, but adapt the Kalman filter function to use SSM. I haven't been able to find anything to guide me, so I am hoping there might be some hints. Thank you

Answer 1

[Parag]

Hi @Todd,

The Schwartz-Smith 2-factor model decomposes the log spot price of a commodity into two components:

• Long-term equilibrium level (non-stationary, modelled as Brownian motion with drift)
• Short-term deviations (mean-reverting component)

The original code uses a custom Kalman filter for parameter estimation and filtering. To adapt this into MATLAB's SSM (State-Space Model) framework, the model needs to be expressed in the standard linear Gaussian state-space form:

SSM Formulation Overview

State Equation:

Let the state vector be:

xt=[χt

ξt]

• χt ​: long-term (non-stationary)
• ξt: short-term (mean-reverting)

The dynamics are:

χt+1=χt+μ+ηt

​ξt+1​​=χt​+μ+ηt​=ϕξt​+ϵt​​

where ηt∼N(0,ση2) and ϵt∼N(0,σϵ2)

Observation Equation:

yt=χt+ξt+εt , εt∼N(0,σε2)

Please refer the MATLAB pseudo-code for the same:

% Define state-space model for Schwartz-Smith 
% Parameters: mu, phi, sigma_eta, sigma_eps, sigma_obs 

A = @(params) [1 0; 0 params(2)];       % phi 
B = @(params) [1; 0];                   % mu only affects chi_t 
C = @(params) [1 1];                    % observation: chi + xi 
D = @(params) eye(2) .* [params(3); params(4)];  % process noise 
E = @(params) params(5);               % observation noise 

% Define ssm object 
ssmModel = ssm(... 
    @(params) deal(A(params), B(params) * params(1), C(params), E(params), D(params) * D(params)')); 

% Initial parameter guess: [mu, phi, sigma_eta, sigma_eps, sigma_obs] 
param0 = [0.1, 0.9, 0.1, 0.1, 0.1]; 

% Load your observed log spot price time series 
% y = logPrices; 

% Estimate parameters 
[estModel, estParams, estSE] = estimate(ssmModel, logPrices, param0); 

Please refer these MATLAB documentations for more details:

• State-Space Models - https://www.mathworks.com/help/control/ug/state-space-models.html
• Estimate State-Space Model - https://www.mathworks.com/help/ident/ref/estimatestatespacemodel.html
• Kalman Filtering - https://www.mathworks.com/help/control/ug/kalman-filtering.html

Hope this is beneficial!