10.4 Time series analysis

Time series analysis is a powerful tool in economics, allowing us to study patterns and trends in data collected over time. It's crucial for forecasting, understanding economic behavior, and making informed policy decisions.

From basic concepts like stationarity to advanced models like ARIMA and GARCH, time series techniques offer a range of methods to analyze economic data. These tools help economists uncover relationships between variables, predict future values, and assess the impact of economic policies over time.

Fundamentals of time series

• Analyzes data points collected over time to identify patterns, trends, and make predictions
• Crucial in economic forecasting, financial modeling, and policy analysis
• Enables economists to understand long-term economic behavior and short-term fluctuations

Components of time series

• Trend represents long-term movement in the data (upward, downward, or horizontal)
• Seasonal variations occur at fixed intervals (quarterly sales patterns)
• Cyclical fluctuations show longer-term oscillations (business cycles)
• Irregular components capture random, unpredictable variations in the data
• Decomposition methods separate these components for analysis and forecasting

Stationarity vs non-stationarity

• Stationary time series maintains constant statistical properties over time
• Mean, variance, and autocorrelation structure remain unchanged
• Non-stationary series exhibit trends, seasonality, or changing variances
• Differencing transforms non-stationary series into stationary ones
• Unit root tests (Augmented Dickey-Fuller) determine stationarity

Autocorrelation and partial autocorrelation

• Autocorrelation measures correlation between a series and its lagged values
• Partial autocorrelation removes intermediate effects between lags
• Autocorrelation function (ACF) plots show overall dependency structure
• Partial autocorrelation function (PACF) plots identify direct relationships
• Used to identify appropriate models and lag structures for time series data

Time series models

• Provide mathematical frameworks to describe and forecast time-dependent data
• Essential for understanding economic dynamics and making informed decisions
• Range from simple to complex, accommodating various data characteristics

Autoregressive (AR) models

• Express current value as a linear combination of past values
• Order p determines how many past observations are considered
• AR(1) model uses only one lagged term: $Y_t = c + \phi Y_{t-1} + \epsilon_t$
• Higher-order AR models incorporate more lags for complex patterns
• Useful for modeling series with persistent effects (GDP growth)

Moving average (MA) models

• Represent current value as a function of past forecast errors
• Order q specifies the number of lagged error terms
• MA(1) model uses one lagged error: $Y_t = \mu + \epsilon_t + \theta \epsilon_{t-1}$
• Capture short-term fluctuations and random shocks in the data
• Effective for modeling series with temporary effects (stock returns)

ARMA and ARIMA models

• ARMA combines autoregressive and moving average components
• ARIMA adds integrated component for non-stationary series
• ARIMA(p,d,q) where p = AR order, d = differencing, q = MA order
• Flexible framework accommodates various time series behaviors
• Box-Jenkins methodology guides model selection and estimation

Seasonal ARIMA models

• Extend ARIMA to account for seasonal patterns in data
• Denoted as SARIMA(p,d,q)(P,D,Q)m, where m = seasonal period
• Capture both regular and seasonal components simultaneously
• Useful for economic data with yearly patterns (retail sales)
• Allow for forecasting that respects seasonal fluctuations

Forecasting techniques

• Utilize historical data to predict future values in time series
• Critical for economic planning, policy-making, and business strategy
• Combine statistical methods with domain expertise for accurate projections

Trend analysis and decomposition

• Isolates underlying trend from seasonal and cyclical components
• Methods include moving averages and regression-based techniques
• Additive decomposition: $Y_t = T_t + S_t + C_t + I_t$
• Multiplicative decomposition: $Y_t = T_t \times S_t \times C_t \times I_t$
• Helps identify long-term patterns and seasonal adjustments in economic data

Exponential smoothing methods

• Assign exponentially decreasing weights to older observations
• Simple exponential smoothing for series without trend or seasonality
• Holt's method incorporates trend component
• Holt-Winters method accounts for trend and seasonality
• State space models provide a unified framework for exponential smoothing

Box-Jenkins methodology

• Systematic approach to identify, estimate, and diagnose ARIMA models
• Steps include model identification, parameter estimation, and diagnostic checking
• Uses ACF and PACF plots to determine appropriate model orders
• Maximum likelihood estimation for parameter values
• Residual analysis ensures model adequacy and forecast accuracy

Econometric applications

• Apply time series techniques to analyze economic relationships and causality
• Crucial for understanding complex economic systems and policy impacts
• Integrate economic theory with statistical methods for robust analysis

Granger causality

• Tests whether one time series helps predict another
• Null hypothesis: X does not Granger-cause Y
• Involves regressing Y on lagged values of Y and X
• F-test determines statistical significance of X's predictive power
• Widely used in macroeconomics to study relationships between variables (inflation and unemployment)

Vector autoregression (VAR)

• Extends univariate autoregression to multivariate time series
• Each variable depends on its own lags and lags of other variables
• Captures dynamic interactions between economic variables
• Impulse response functions show effects of shocks on system
• Forecast error variance decomposition analyzes contributions to variability

Cointegration analysis

• Studies long-run equilibrium relationships between non-stationary series
• Two series are cointegrated if their linear combination is stationary
• Engle-Granger two-step method tests for cointegration
• Johansen test handles multiple cointegrating relationships
• Error correction models incorporate short-run dynamics and long-run equilibrium

Time series regression

• Extends classical regression to account for time-dependent structures
• Addresses issues of autocorrelation and non-stationarity in economic data
• Crucial for analyzing dynamic relationships between economic variables

Dynamic regression models

• Incorporate lagged dependent and independent variables
• Account for persistence and delayed effects in economic relationships
• General form: $Y_t = \beta_0 + \beta_1 X_t + \beta_2 X_{t-1} + \gamma Y_{t-1} + \epsilon_t$
• Allow for both immediate and long-run impacts of explanatory variables
• Useful for modeling policy effects and economic adjustments over time

Distributed lag models

• Express dependent variable as function of current and lagged independent variables
• Finite distributed lag (FDL) models specify a fixed number of lags
• Infinite distributed lag models allow for indefinite lag effects
• Almon lag structure imposes polynomial restrictions on lag coefficients
• Koyck transformation converts infinite lag to manageable form

Error correction models

• Combine long-run equilibrium relationship with short-run dynamics
• Based on cointegration between non-stationary variables
• General form: $\Delta Y_t = \alpha(\beta Y_{t-1} - X_{t-1}) + \gamma \Delta X_t + \epsilon_t$
• α represents speed of adjustment to equilibrium
• Widely used in macroeconomics to model relationships (consumption and income)

Advanced topics

• Explore sophisticated techniques for complex time series behaviors
• Address specific challenges in economic and financial data analysis
• Provide powerful tools for modeling volatility, cyclical patterns, and state dynamics

ARCH and GARCH models

• Autoregressive Conditional Heteroskedasticity models time-varying volatility
• ARCH(q) specifies volatility as function of past squared errors
• Generalized ARCH (GARCH) includes lagged conditional variances
• GARCH(p,q): $\sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^p \beta_j \sigma_{t-j}^2$
• Essential for modeling financial time series with volatility clustering

Spectral analysis

• Decomposes time series into frequency components
• Fourier transform converts time domain to frequency domain
• Periodogram estimates power spectral density
• Identifies cyclical patterns and hidden periodicities in data
• Useful for analyzing business cycles and long-term economic fluctuations

State space models

• Represent time series as function of unobserved state variables
• Consist of measurement equation and state transition equation
• Kalman filter estimates unobserved states from observed data
• Flexible framework for modeling trend, seasonality, and structural changes
• Applications include economic forecasting and signal processing

Diagnostic tools

• Ensure model adequacy and reliability of time series analysis
• Critical for validating assumptions and improving model performance
• Guide model selection and refinement process in economic applications

Residual analysis

• Examines model errors to check for remaining patterns or autocorrelation
• Ljung-Box test assesses overall randomness of residuals
• ACF and PACF of residuals should show no significant correlations
• Q-Q plots check for normality of residuals
• Heteroskedasticity tests ensure constant variance of errors

Information criteria

• Guide model selection by balancing goodness-of-fit and complexity
• Akaike Information Criterion (AIC): $AIC = 2k - 2\ln(\hat{L})$
• Bayesian Information Criterion (BIC): $BIC = k\ln(n) - 2\ln(\hat{L})$
• Lower values indicate better models
• Help prevent overfitting by penalizing excessive parameters

Forecast evaluation metrics

• Assess accuracy and reliability of time series predictions
• Mean Absolute Error (MAE) measures average magnitude of errors
• Root Mean Squared Error (RMSE) penalizes larger errors more heavily
• Mean Absolute Percentage Error (MAPE) provides scale-independent measure
• Theil's U statistic compares forecast performance to naive models
• Out-of-sample testing evaluates model performance on unseen data

Software and implementation

• Facilitate practical application of time series techniques in economics
• Enable efficient data manipulation, analysis, and visualization
• Support reproducible research and collaboration in economic studies

Time series packages

• R offers forecast, tseries, and zoo packages for comprehensive analysis
• Python provides statsmodels and pandas for time series functionality
• MATLAB includes Econometrics Toolbox for advanced econometric modeling
• EViews specializes in econometric time series analysis and forecasting
• Gretl offers open-source platform for econometric time series work

Data visualization techniques

• Time plots display series evolution over time
• Seasonal plots highlight recurring patterns
• Lag plots reveal autocorrelation structure
• ACF and PACF plots guide model identification
• Interactive dashboards (Tableau, Power BI) for dynamic exploration of economic data

Case studies in economics

• GDP forecasting using ARIMA models to project economic growth
• Inflation dynamics analysis with VAR models to inform monetary policy
• Exchange rate prediction using GARCH models for currency risk management
• Stock market volatility modeling with ARCH techniques for financial risk assessment
• Consumer spending patterns analysis with seasonal decomposition for retail strategy