Econometrics II Homework 8: Time Series Models & Matlab Simulation Tutorial

Understanding the Random Walk Plus Noise Model in Econometrics

In modern econometrics, many economic variables like wages, stock prices, or GDP exhibit non-stationary behavior. A classic model decomposes log wages into a random walk and independent white noise. This model is widely used in labor economics and finance. For instance, analyzing AI-driven stock trading algorithms often involves similar stochastic processes. Here, we define y_t = y_{t-1} + u_t, where u_t is white noise with variance σ_u^2, and w_t = y_t + e_t, with e_t white noise variance σ_e^2. Since w_t is non-stationary, we focus on Δw_t.

Variance and Autocovariances of Δw_t

First, compute Δw_t = w_t - w_{t-1} = (y_t + e_t) - (y_{t-1} + e_{t-1}) = u_t + e_t - e_{t-1}. The variance is Var(Δw_t) = Var(u_t) + Var(e_t) + Var(e_{t-1}) = σ_u^2 + 2σ_e^2. Autocovariance of order 1: Cov(Δw_t, Δw_{t-1}) = Cov(u_t + e_t - e_{t-1}, u_{t-1} + e_{t-1} - e_{t-2}) = Cov(-e_{t-1}, e_{t-1}) = -σ_e^2. Autocovariance of order 2: Cov(Δw_t, Δw_{t-2}) = 0. These moments are crucial for matching empirical wage data, similar to how AI models calibrate parameters using moment conditions.

Expected Future Income from an AR(1) Process

Consider an agent's wage following y_t = μ + β y_{t-1} + e_t, with β < 1. This is a common model in macroeconomics for income dynamics. Given y_0 = 100, the expected wage at time t is E[y_t | y_0] = μ (1 - β^t)/(1 - β) + β^t y_0. For the discounted sum of expected future income with discount factor δ = 0.9 (i.e., 0.9% per period?), we assume δ = 0.9 as a decimal? The assignment states discount rate 0.9 percent, so δ = 0.009? Actually, typical discount factor is close to 1. We'll use δ = 0.9 as given. Then Σ = Σ_{t=1}^∞ δ^t E[y_t | y_0]. Using geometric series, Σ = [μ/(1-β)] * [δ/(1-δ)] - [μ/(1-β)] * [δβ/(1-δβ)] + y_0 * [δβ/(1-δβ)]. This formula helps in valuing human capital, akin to AI-driven career planning apps.

Matlab Implementation: AR(1) and AR(2) Models for US Income Growth

Using real per capita US data, estimate an AR(1) model: Δy_t = c + φ Δy_{t-1} + ε_t. In Matlab, use regress or arima. Perform a t-test on φ. If φ is insignificantly different from zero, we cannot reject that income growth is white noise. Next, estimate AR(2): Δy_t = c + φ_1 Δy_{t-1} + φ_2 Δy_{t-2} + ε_t. Use an F-test or likelihood ratio test to compare AR(2) vs AR(1). This is similar to testing whether recent AI stock market predictors have additional memory.

Simulation Study: Bias in AR(1) Coefficient Estimates

Simulate an AR(1) process y_t = a y_{t-1} + ε_t with T=40 for a = 0.5, 0.9, 0.99. Repeat 1000 times, estimate a using OLS. Show that as a increases, the estimate ˆa becomes more biased towards zero. This is the well-known Hurwicz bias. In Matlab, use a loop: for each a, generate data, estimate, store. Plot average estimates. This bias is critical when analyzing persistent processes like AI-driven trend forecasting.

By mastering these concepts, you'll be prepared for advanced econometric analysis in finance, AI, and policy evaluation.