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Intro to Time Series

Stock price analysis is a crucial skill for investors and financial analysts. By breaking down price movements into components like trend, seasonality, and cyclical fluctuations, we can better understand market behavior and make informed decisions.

Time series modeling takes this analysis further, using techniques like ARIMA and GARCH to forecast future prices and volatility. These models, combined with proper evaluation methods, help investors navigate the complex world of stock markets and manage risk effectively.

Stock Price Time Series Analysis

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  • Time series components break down stock price movements into distinct elements
    • Trend captures the long-term upward or downward movement in stock prices (bull or bear market)
    • Seasonality refers to recurring patterns in stock prices at fixed intervals (quarterly earnings reports)
    • Cyclical fluctuations are related to broader economic or business cycles (expansion or recession)
    • Irregularity encompasses unexpected events or noise that impact stock prices (natural disasters, geopolitical events)
  • Visual analysis techniques provide insights into stock price behavior
    • Line plots help identify overall trends and patterns in stock prices over time
    • Candlestick charts visualize open, high, low, and close prices for each trading period (day, week, month)
    • Moving averages smooth out short-term fluctuations to reveal underlying trends (50-day or 200-day moving average)
  • Statistical analysis techniques quantify relationships and properties of stock price time series
    • Autocorrelation measures the relationship between stock price observations at different lags (correlation between today's price and yesterday's price)
    • Partial autocorrelation measures the relationship between observations at different lags, controlling for intermediate lags (correlation between today's price and yesterday's price, given the price two days ago)
    • Augmented Dickey-Fuller (ADF) test assesses the stationarity of the stock price time series (whether the statistical properties remain constant over time)
  • Anomaly detection methods identify unusual or unexpected behavior in stock prices
    • Outlier detection identifies observations that deviate significantly from the norm (stock price spikes or drops)
    • Change point detection detects abrupt shifts in the time series properties (structural breaks or regime changes)

Calculation of stock returns

  • Simple returns calculate the percentage change in stock price over a single period
    • Formula: Rt=PtPt1Pt1R_t = \frac{P_t - P_{t-1}}{P_{t-1}}
    • Interpretation: Positive returns indicate price increases, while negative returns indicate price decreases
  • Log returns provide a continuously compounded measure of stock returns
    • Formula: rt=ln(PtPt1)r_t = \ln(\frac{P_t}{P_{t-1}})
    • Interpretation: Log returns are symmetric and have an additive property for multi-period returns
    • Advantages: Log returns are more suitable for statistical modeling and assume normality
  • Adjusted returns account for dividends, stock splits, and other corporate actions that affect stock prices
    • Formula: Radj,t=Pt+DtPt1Pt1R_{adj, t} = \frac{P_t + D_t - P_{t-1}}{P_{t-1}}
    • Interpretation: Adjusted returns provide a more accurate measure of the total return to investors
    • Dividends (DtD_t) are cash payments made by companies to shareholders (quarterly or annual)

Time Series Modeling and Forecasting

Time series models for forecasting

  • ARIMA (Autoregressive Integrated Moving Average) models capture linear dependencies in stock price time series
    1. Identify the appropriate order of AR (autoregressive), I (differencing), and MA (moving average) terms using ACF (autocorrelation function), PACF (partial autocorrelation function), and information criteria (AIC, BIC)
    2. Estimate the model parameters using maximum likelihood or least squares methods
    3. Diagnose the model by checking residuals for normality, homoscedasticity (constant variance), and independence
  • GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models capture time-varying volatility in stock returns
    1. Identify the appropriate order of ARCH (autoregressive conditional heteroskedasticity) and GARCH terms using information criteria
    2. Estimate the model parameters using maximum likelihood methods
    3. Diagnose the model by checking standardized residuals for normality and independence
    • GARCH models are useful for risk management and option pricing (Value at Risk, implied volatility)

Evaluation of forecasting models

  • Forecast evaluation metrics quantify the accuracy and precision of stock price predictions
    • Mean Absolute Error (MAE) measures the average absolute difference between forecasts and actual values
    • Mean Squared Error (MSE) penalizes large errors more heavily by taking the average squared difference
    • Root Mean Squared Error (RMSE) is the square root of MSE and has the same unit as the stock prices
    • Mean Absolute Percentage Error (MAPE) expresses the average absolute percentage difference between forecasts and actual values
  • Validation techniques assess the robustness and generalizability of forecasting models
    • Train-test split divides the data into separate training and testing sets to evaluate model performance on unseen data (80% training, 20% testing)
    • Cross-validation iteratively splits the data into training and validation sets to assess model performance across different subsets (k-fold cross-validation)
    • Rolling window updates the model with new data and generates forecasts for a fixed horizon (one-step or multi-step ahead forecasts)
  • Benchmark comparisons evaluate the relative performance of forecasting models
    • Compare the chosen model against naive forecasts (random walk) or other relevant models (historical average, ARIMA, GARCH)
    • Assess the statistical significance of performance differences using appropriate tests (Diebold-Mariano test for equal predictive accuracy)


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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.