Quantile Autoregression in Value-at-Risk forecasting

Hello,

I've been working on implementing a quantile autoregression model in MATLAB for financial return data to foracast return quantiles (Value-at-Risk) and averages of Quantiles at different levels to obtain forecasts for Expected Shortfall. Previously, I used a differential evolutionary algorithm for optimization and a different coding language - I obtained good results buit the runtime was incredibly high. Using MATLAB, I obtain much lower runtime but discrepancies in the results.

Here's a brief overview of my MATLAB implementation:

%get data
data = readtable('sGARCHnorm_sims.csv');
%define some variables:
data_length = length(data{:,1});
window_size = 2000;
oos_size = data_length-window_size;
q1 = 0.025;
q5 = 0.001;
tau = q1;
a = tau;
%% 

%get logret from data:
logret = data(:,1);

% Initialize outputs:
asl_forecast_q1 = zeros(oos_size,1);
asl_forecast_q5 = zeros(oos_size,1);
asl_ES_equal_2q = zeros(oos_size,1);

logret_to_forecast = zeros(oos_size,1);
%%

%Lets roll:
tic;

for k = 1:oos_size

%initparameters for optimization
beta0 = [0.5,0.5,0.5,0.5];

%get insample return of the considered window:
start = k;
stop = window_size - 1 + k;
return_train = table2array(logret(start:stop,:));
%%

%quantile 1:
%insample hist quantile to initiate optimization:
hist_q = quantile(return_train, q1);
CAViaR = repmat(hist_q, height(return_train), 1);
%optimization:
obj_fnc = @(betas) AS_obj(betas, return_train, CAViaR, q1);
[betas_opt, ~] = fmincon(obj_fnc, beta0);
%forecast 1 step ahead:
insample_q1 = AS_insample(betas_opt, return_train, q1);
asl_forecast_q1(k)= betas_opt(1) + betas_opt(2) * insample_q1(end) + betas_opt(3) * max(return_train(end),0) + betas_opt(4) * max(-return_train(end),0);
%%

%q5
%insample hist quantile to initiate optimization:
hist_q = quantile(return_train, q5);
CAViaR = repmat(hist_q, height(return_train), 1);
%optimization
obj_fnc = @(betas) AS_obj(betas, return_train, CAViaR, q5);
[betas_opt, fval] = fmincon(obj_fnc, beta0);
%forecast 1 step ahead:
insample_q5 = AS_insample(betas_opt, return_train, q5);
asl_forecast_q5(k)= betas_opt(1) + betas_opt(2) * insample_q5(end) + betas_opt(3) * max(return_train(end),0) + betas_opt(4) * max(-return_train(end),0);

%%
%equal weighted averages:
asl_ES_equal_2q(k) = 1/2 * asl_forecast_q1(k) + 1/2 *asl_forecast_q5(k);

disp(k);
end

toc;

output = table( asl_forecast_q1,asl_forecast_q5, asl_ES_equal_2q);
writetable( output, "asl_sim.csv");

%quantile regression obj function:
function [res] = AS_obj(betas, return_train, CAViaR, tau)
    beta1 = betas(1);
    beta2 = betas(2);
    beta3 = betas(3);
    beta4 = betas(4);

    for i = 2:length(return_train)
        CAViaR(i) = beta1 + beta2*CAViaR(i-1) + beta3*max(return_train(i-1),0) + beta4*max(-return_train(i-1),0);
    end

    % Objective function
    res = sum((tau - (return_train < CAViaR)) .* (return_train - CAViaR)) / length(return_train);
    if isnan(res) || isinf(res)
        res = 1e+10;
    end
end

%insample loop for optimal betas:
function [CAViaR] = AS_insample(betas, data,tau)

beta1 = betas(1);
beta2 = betas(2);
beta3 = betas(3);
beta4 = betas(4);

% Create the CAViaR vector
var = quantile(data, tau);
CAViaR = repmat(var, length(data), 1);

for i = 2:length(data)
    CAViaR(i) = beta1 + beta2*CAViaR(i-1) + beta3*max(data(i-1),0) + beta4*max(-data(i-1),0) ;
end

end

The input data is a simualted series of returns using a standard GARCH(1,1) and normal distributed innovation terms.

Is there a better way to do quantile autoregression and quantile forecasting in MATLAB that I am missing?

Best,

Pit

[Lea]

Could you explain what you mean by "discrepancies"? Are [betas_opt, fval] from fmincon different from what you expect?

[Pit]

Yes, exactly.

Are there usefull alternatives to fmincon and fminsearch for (conditional) quantile autoregresion?

[Andreas Apostolatos]

Hi Pit,

fmincon involves gradient-based optimization methods and allows for the incorporation of constraints. For such tasks as quantile regression and Value-at-Risk (VaR) analysis the objective function is oftentimes non-smooth and the use of gradient-based methods might not be the best choice.

Consider please using derivative-free optimization methods for your application, such as fminsearch. The Community Toolbox quantreg is leveraging function fminsearch for quantile regression.

I hope this information helps.

Kind regards

Andreas