基于AR(8)模型的消费数据IRF的95%自助法置信区间构建问询

Hey there! Let's walk through exactly how to build those 95% bootstrap confidence intervals for your AR(8) impulse response function (IRF), and integrate this with your existing code. I'll break this down step by step so it's easy to follow.

For univariate AR models like your AR(8), the standard approach is residual bootstrapping. Here's why:

• We first estimate the original model to get coefficients and residuals (the unobserved error terms from the model).
• Since AR models assume residuals are iid (independent and identically distributed), we resample these residuals with replacement to generate new simulated time series.
• We then re-estimate the AR model on each simulated series and compute its IRF—this gives us a distribution of IRFs we can use to calculate confidence intervals.

Below is a modified, integrated version of your code that handles the bootstrap process, computes bootstrapped IRFs, and calculates the 95% CI. I've added detailed comments to explain each step:

% --- Configuration ---
L = 8;                  % AR(8) lag order
T = 20;                 % Number of periods for IRF (adjust to your needs)
B = 1000;               % Number of bootstrap replications (1000+ is ideal for stability)

% --- Step 1: Estimate Original AR(8) Model & IRF ---
clear;
% Load and demean data
X = xlsread('data.xls');
mean_u = mean(X);
X_demean = X - mean_u*ones(length(X),1);

% Construct lag matrix for AR estimation
LM = lagmatrix(X_demean, 1:L);
X_used = X_demean(L+1:end);  % Trim first L observations (no lags available)
LM_used = LM(L+1:end,:);

% Estimate AR coefficients
beta_original = LM_used \ X_used;

% Calculate original residuals (mean-zero adjusted)
e_original = X_used - LM_used * beta_original;
e_original = e_original - mean(e_original);  % Ensure residuals have mean 0

% Compute original IRF
F_original = companion(beta_original);
sigma_e = std(e_original);
irf_original = zeros(L, T);
irf_original(1,1) = sigma_e;  % Initial impulse to the first lag
for t = 2:T
    irf_original(:,t) = F_original * irf_original(:,t-1);
end
irf_original = irf_original(1,:);  % Keep only the response of the variable itself (univariate AR)

% --- Step 2: Bootstrap Loop ---
irf_boot = zeros(B, T);  % Store IRF for each bootstrap replication

rng(1);  % Set seed for reproducibility
for i = 1:B
    % 1. Resample residuals with replacement
    idx = randi(length(e_original), length(e_original), 1);
    e_boot = e_original(idx);
    
    % 2. Generate a bootstrap time series using original coefficients
    X_boot = zeros(length(X_demean), 1);
    X_boot(1:L) = X_demean(1:L);  % Initialize with original first L observations
    for t = L+1:length(X_demean)
        X_boot(t) = LM_used(t-L,:) * beta_original + e_boot(t-L);
    end
    
    % 3. Re-estimate AR(8) on the bootstrap sample
    LM_boot = lagmatrix(X_boot, 1:L);
    X_boot_used = X_boot(L+1:end);
    LM_boot_used = LM_boot(L+1:end,:);
    beta_boot = LM_boot_used \ X_boot_used;
    
    % 4. Compute IRF for this bootstrap replication
    F_boot = companion(beta_boot);
    irf_temp = zeros(L, T);
    irf_temp(1,1) = sigma_e;
    for t = 2:T
        irf_temp(:,t) = F_boot * irf_temp(:,t-1);
    end
    irf_boot(i,:) = irf_temp(1,:);  % Store the variable's response
end

% --- Step 3: Calculate 95% Confidence Intervals ---
ci_lower = quantile(irf_boot, 0.025, 1);  % 2.5th percentile
ci_upper = quantile(irf_boot, 0.975, 1);  % 97.5th percentile

% --- Step 4: Plot IRF with Confidence Intervals ---
figure;
plot(1:T, irf_original, 'b-', 'LineWidth', 2);
hold on;
plot(1:T, ci_lower, 'r--', 'LineWidth', 1.5);
plot(1:T, ci_upper, 'r--', 'LineWidth', 1.5);
fill([1:T, T:-1:1], [ci_lower, fliplr(ci_upper)], 'r', 'FaceAlpha', 0.1);  % Shaded CI

xlabel('Period');
ylabel('Impulse Response');
title('AR(8) Impulse Response Function with 95% Bootstrap Confidence Intervals');
legend('Original IRF', '95% CI Bounds');
grid on;

• Bootstrap replications (B): Use at least 1000 replications for stable confidence intervals. If computation time is an issue, 500 can work, but 1000 is more reliable.
• Residual handling: We adjust residuals to have mean zero to match the AR model's assumption of mean-zero errors.
• Initialization of bootstrap series: Using the original first L observations ensures the simulated series starts in a stationary state (critical for AR models).

内容的提问来源于stack exchange，提问作者I.B