⏳Intro to Time Series Unit 15 – Time Series in Finance and Economics

Key Concepts and Definitions

• Time series data consists of observations collected sequentially over time at regular intervals (hourly, daily, monthly)
• Stationarity assumes the statistical properties of a time series remain constant over time
  • Constant mean and variance
  • Autocovariance depends only on the lag between observations
• Autocorrelation measures the linear relationship between a time series and its lagged values
• Partial autocorrelation measures the correlation between a time series and its lagged values after removing the effect of intermediate lags
• White noise is a sequence of uncorrelated random variables with zero mean and constant variance
• Unit root tests (Dickey-Fuller, Phillips-Perron) determine if a time series is stationary or non-stationary
• Cointegration occurs when two or more non-stationary time series have a linear combination that is stationary

Time Series Components and Patterns

• Trend represents the long-term increase or decrease in the level of a time series
  • Can be linear, exponential, or polynomial
• Seasonality refers to regular, predictable fluctuations within a fixed period (year, quarter, month)
  • Seasonal patterns can be additive or multiplicative
• Cyclical component captures medium to long-term oscillations around the trend, typically related to business cycles
• Irregular component represents random, unpredictable fluctuations not captured by other components
• Decomposition methods (additive, multiplicative) separate a time series into its components for analysis and modeling
• Autocorrelation function (ACF) and partial autocorrelation function (PACF) help identify patterns and lags in a time series
• Cross-correlation measures the relationship between two time series at different lags

Statistical Tools for Time Series Analysis

• Moving averages smooth out short-term fluctuations and highlight long-term trends
  • Simple moving average (SMA) assigns equal weights to all observations
  • Exponential moving average (EMA) assigns higher weights to recent observations
• Differencing removes trend and seasonality by computing the difference between consecutive observations
  • First-order differencing: $\Delta y_t = y_t - y_{t-1}$
  • Seasonal differencing: $\Delta_s y_t = y_t - y_{t-s}$, where $s$ is the seasonal period
• Autoregressive (AR) models express a time series as a linear combination of its past values
  • AR(1) model: $y_t = c + \phi_1 y_{t-1} + \varepsilon_t$
• Moving average (MA) models express a time series as a linear combination of past forecast errors
  • MA(1) model: $y_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1}$
• Autoregressive moving average (ARMA) models combine AR and MA components
  • ARMA(1,1) model: $y_t = c + \phi_1 y_{t-1} + \varepsilon_t + \theta_1 \varepsilon_{t-1}$
• Autoregressive integrated moving average (ARIMA) models extend ARMA to handle non-stationary time series through differencing
  • ARIMA(p,d,q) model: $\Delta^d y_t = c + \sum_{i=1}^p \phi_i \Delta^d y_{t-i} + \varepsilon_t + \sum_{j=1}^q \theta_j \varepsilon_{t-j}$

Forecasting Methods and Models

• Naive forecasting assumes the next observation will be equal to the most recent observation
• Drift method accounts for the average change between consecutive observations in the forecast
• Exponential smoothing models (simple, Holt's, Holt-Winters) assign exponentially decreasing weights to past observations
  • Simple exponential smoothing: $\hat{y}_{t+1} = \alpha y_t + (1-\alpha) \hat{y}_t$
  • Holt's linear trend method: $\hat{y}_{t+h} = l_t + hb_t$
  • Holt-Winters' seasonal method: $\hat{y}_{t+h} = (l_t + hb_t)s_{t-m+h_m^+}$
• ARIMA models are widely used for short-term forecasting and can capture complex patterns
• Vector autoregressive (VAR) models extend univariate autoregressive models to multivariate time series
• Error correction models (ECM) incorporate cointegration relationships for long-run equilibrium and short-run dynamics
• Forecast accuracy measures (MAE, MAPE, RMSE) evaluate the performance of forecasting models

Applications in Finance and Economics

• Stock market forecasting uses time series models (ARIMA, GARCH) to predict future stock prices and volatility
• Yield curve modeling analyzes the relationship between bond yields and maturities using techniques like principal component analysis (PCA)
• Macroeconomic forecasting employs time series models to predict key indicators (GDP, inflation, unemployment)
  • Leading indicators (stock market indices, consumer confidence) provide early signals of economic trends
  • Coincident indicators (industrial production, retail sales) move in tandem with the overall economy
  • Lagging indicators (unemployment rate, average duration of unemployment) confirm long-term trends
• Risk management uses time series models (Value at Risk, Expected Shortfall) to quantify and manage financial risks
• Pairs trading identifies co-moving assets and exploits temporary deviations from their long-run relationship
• Event studies analyze the impact of specific events (earnings announcements, mergers) on asset prices using time series data

Data Handling and Visualization

• Data preprocessing involves handling missing values, outliers, and transformations (log, power)
  • Interpolation fills in missing values based on surrounding observations
  • Outlier detection methods (Z-score, Tukey's fences) identify and treat extreme values
• Resampling changes the frequency of a time series through aggregation (upsampling) or interpolation (downsampling)
• Rolling statistics (mean, variance) compute summary statistics over a moving window of fixed size
• Time series plots display observations over time, revealing patterns and trends
  • Line plots connect consecutive observations with lines
  • Scatter plots show individual observations as points
• Seasonal subseries plots group observations by seasonal periods (months, quarters) to highlight seasonal patterns
• Autocorrelation and partial autocorrelation plots visualize the correlation structure of a time series
• Heatmaps and correlation matrices display the relationships between multiple time series

Common Challenges and Pitfalls

• Spurious regression occurs when two unrelated time series appear to have a significant relationship due to common trends
  • Leads to misleading conclusions and invalid inferences
• Overfitting happens when a model is too complex and fits noise rather than the underlying pattern
  • Results in poor out-of-sample performance and unreliable forecasts
• Structural breaks and regime shifts can cause sudden changes in the behavior of a time series
  • Chow test and CUSUM test help detect structural breaks
• Heteroscedasticity refers to non-constant variance in the errors of a time series model
  • ARCH and GARCH models capture time-varying volatility
• Multicollinearity arises when predictor variables in a time series model are highly correlated
  • Variance inflation factor (VIF) measures the severity of multicollinearity
• Ignoring seasonality can lead to biased estimates and inaccurate forecasts
  • Seasonal adjustment methods (X-11, SEATS) remove seasonal effects from time series data
• Misspecification of the model order (p, q) in ARIMA models can result in suboptimal forecasts
  • Information criteria (AIC, BIC) help select the appropriate model order

Real-World Case Studies

• Forecasting electricity demand using ARIMA and regression models with weather variables
  • Helps utility companies plan production and avoid shortages or surpluses
• Predicting exchange rates using VAR models and macroeconomic fundamentals (interest rates, inflation)
  • Informs currency hedging strategies and international investment decisions
• Analyzing the impact of oil price shocks on stock market returns using VAR and impulse response functions
  • Helps investors understand the sensitivity of different sectors to energy prices
• Modeling and forecasting volatility in financial markets using GARCH models
  • Crucial for risk management, option pricing, and portfolio optimization
• Detecting and forecasting business cycle turning points using leading economic indicators and Markov switching models
  • Assists policymakers in implementing timely monetary and fiscal measures
• Forecasting sales and demand for products using exponential smoothing and ARIMA models
  • Enables businesses to optimize inventory management and production planning
• Analyzing the transmission of monetary policy shocks using structural VAR models
  • Provides insights into the effectiveness of central bank actions on the economy
• Modeling the spread of infectious diseases using time series SIR (Susceptible-Infected-Recovered) models
  • Helps public health officials plan interventions and allocate resources during outbreaks