Mastering Econometrics II: OLS, AR(1) Residuals & ML Estimation in MATLAB (Spring 2025)

Introduction: Why This Econometrics Tutorial Matters Now

In Spring 2025, econometrics students are diving deep into time series analysis. With the U.S. economy showing mixed signals – from AI-driven productivity booms to lingering inflation concerns – understanding how to model consumption and income growth is more relevant than ever. This tutorial walks you through the key steps of your Econometrics II homework 9, focusing on MATLAB implementation of OLS, AR(1) residual correction, and Maximum Likelihood estimation. We'll connect each method to real-world data trends, like how analysts at the Federal Reserve use similar techniques to forecast consumer spending.

Step 1: OLS Regression of Consumption Growth on Income Growth

First, load your U.S. real per capita consumption growth and income growth data into MATLAB. Assume your variables are c_growth and y_growth. Run OLS with a constant:

X = [ones(length(c_growth),1), y_growth];
[b_ols, ~, res_ols] = regress(c_growth, X);
se_ols = sqrt(diag(res_ols'*res_ols/(length(c_growth)-2)*inv(X'*X)));
tstat_ols = b_ols ./ se_ols;

This gives you the coefficient estimates and t-statistics. The t-stat for income growth tells you if it significantly predicts consumption – a key insight for policymakers. In our spring 2025 context, a high t-stat would suggest that recent income gains (from tech sector growth) are driving consumer spending, while a low one might indicate that consumers are saving more due to uncertainty.

Step 2: AR(1) Model for Residuals

After OLS, check for autocorrelation. Estimate an AR(1) model on the residuals:

res_ols = res_ols(2:end); % drop first observation for lag
rho = (res_ols(1:end-1)' * res_ols(2:end)) / (res_ols(1:end-1)' * res_ols(1:end-1));

This rho is the autocorrelation coefficient. If it's significant (e.g., >0.3), you need to correct for serial correlation – a common issue in consumption data due to habit formation. Think of it like streaks in video game battles: if a player wins several rounds in a row, past performance predicts future outcomes. Similarly, consumption growth often follows patterns.

Step 3: Cochrane-Orcutt and Prais-Winsten Estimation

Now apply Cochrane-Orcutt (CO) and Prais-Winsten (PW) to get better estimates. For CO, transform the variables using rho from step 2, then re-estimate OLS on the transformed data. For PW, include the first observation to avoid losing data. Here's a simplified CO implementation:

% Cochrane-Orcutt iterative
rho_old = 0; tol = 1e-6;
while abs(rho - rho_old) > tol
    rho_old = rho;
    y_star = c_growth(2:end) - rho * c_growth(1:end-1);
    X_star = [ones(length(y_star),1), y_growth(2:end) - rho * y_growth(1:end-1)];
    b_co = regress(y_star, X_star);
    res_co = y_star - X_star * b_co;
    rho = (res_co(1:end-1)' * res_co(2:end)) / (res_co(1:end-1)' * res_co(1:end-1));
end

For Prais-Winsten, adjust the first observation by multiplying by sqrt(1-rho^2). Compare the t-stats: PW often gives more efficient estimates, similar to how smoothing algorithms in AI models (like those used in stock prediction apps) reduce noise.

Step 4: Maximum Likelihood Estimation

Now estimate the same consumption-income model by Maximum Likelihood (ML). This method finds parameters that make the observed data most probable. In MATLAB, use fminunc or the Econometrics Toolbox's arima function. Here's a custom ML approach:

% Define log-likelihood function
negloglik = @(theta) -sum(log(normpdf(c_growth - [ones(length(c_growth),1), y_growth] * theta(1:2), 0, theta(3))));
theta0 = [b_ols; std(res_ols)];
theta_ml = fminunc(negloglik, theta0);
b_ml = theta_ml(1:2);
sigma_ml = theta_ml(3);
% Compute t-stats using Hessian (numerical)
H = hessian(negloglik, theta_ml);
se_ml = sqrt(diag(inv(H)));
tstat_ml = b_ml ./ se_ml(1:2);

ML estimates are asymptotically efficient, meaning they have the smallest possible variance. This is crucial in high-stakes financial modeling – like when hedge funds use ML to predict market moves based on consumer data trends in 2025.

Comparison of Estimates

After running all methods, compare the coefficients and t-stats. Typically, OLS and ML give similar point estimates if errors are normal. Cochrane-Orcutt and Prais-Winsten correct for autocorrelation, often yielding different t-stats. For your homework, create a table in MATLAB:

results = table(b_ols, b_co, b_pw, b_ml, 'VariableNames', {'OLS','CO','PW','ML'});
disp(results);

In our spring 2025 example, you might find that PW gives a higher t-stat for income growth, suggesting that correcting for autocorrelation strengthens the consumption-income link – a finding that aligns with recent research on 'sticky' consumption patterns.

Practical Tips for Your Homework

• Use autocorr(res_ols) to visualize residual autocorrelation.
• For PW, remember to transform the first observation: y1 = sqrt(1-rho^2)*c_growth(1).
• Check convergence of ML by trying different starting values.
• Relate your results to current events: In 2025, the U.S. economy is experiencing a tech-driven income surge, but consumption growth is lagging due to high savings rates. Your econometric models can quantify this relationship.

Conclusion

By completing this tutorial, you've mastered OLS, AR(1) correction, and ML estimation in MATLAB. These techniques are not just for homework – they're used by economists at the IMF, World Bank, and top investment firms. As you submit your Econometrics II homework 9, remember that the skills you're building are directly applicable to real-world data analysis. Good luck!