Module 5

Time-Series Analysis

Overview

This module develops time-series methods for analyzing how financial variables evolve over time. The central questions are whether past returns predict future returns, whether interest rates tend to revert to long-run equilibria, and whether return volatility is constant or changes systematically over time. These questions are not merely theoretical: autocorrelation tests inform trading strategy design, mean-reversion models underpin fixed-income analysis, and volatility models are essential inputs for option pricing, portfolio risk management, and the measurement of the variance risk premium.

A recurring theme is the importance of inference under non-standard conditions. Because financial time series exhibit heteroskedasticity and autocorrelation in their residuals, standard OLS standard errors are invalid. Throughout the module we use Newey-West heteroskedasticity-and-autocorrelation-consistent (HAC) standard errors to draw reliable conclusions.

Topics

Autocorrelation in Stock Returns

  • The efficient market hypothesis in its weak form: past prices should not predict future returns
  • Autoregressive AR(1) and AR(k) models for testing whether lagged returns predict current returns
  • The problem of autocorrelated residuals and why standard OLS standard errors fail
  • Newey-West HAC standard errors and lag selection using \(T^{1/4}\)
  • Application to Nvidia (NVDA) daily returns over 2015–2025

Predicting Short-Rate Changes

  • The Vasicek model and the implication that the short rate follows a mean-reverting process
  • The spurious regression problem: why regressing persistent levels on each other inflates \(t\)-statistics
  • The augmented Dickey-Fuller (ADF) test for unit roots and its application to the level and first difference of the risk-free rate
  • Estimating the mean-reversion regression \(\Delta r^f_{t+1} = \alpha + \beta r^f_t + e_{t+1}\) with Newey-West standard errors on the full Fama-French sample from 1926
  • Extending the model with the lagged term spread and lagged rate change; interpretation through the expectations hypothesis
  • A regression comparison table across three specifications, showing that the term spread absorbs the level effect

Time-Varying Volatility

  • The constant-volatility model and its limitations: real returns exhibit volatility clustering
  • Rolling window estimation of conditional volatility and its sensitivity to window length
  • Annualizing daily variance estimates using the 252 trading days convention
  • Stylized facts of financial volatility: clustering, asymmetry, and fat tails

The GARCH Model

  • ARCH and GARCH(1,1): conditional variance as a weighted combination of the long-run variance, the last squared return, and the lagged conditional variance
  • Maximum likelihood estimation of GARCH parameters; comparison with the OLS objective
  • Estimating GARCH models with the arch library; recovering the long-run annualized standard deviation from \(\omega\), \(\alpha\), and \(\beta\)
  • Multi-step GARCH variance forecasts and the 21-day cumulated sum needed to approximate the VIX horizon
  • The variance risk premium; predictive regressions at 1-, 3-, and 6-month horizons with Newey-West standard errors