Tài liệu về hồi quy với chuỗi thời gian

Nội dung

Section 1: Introduction to Time Series Stationarity

• • What is stationarity?

  • Why stationarity is crucial for time series modeling and forecasting

  • Strict stationarity vs. weak stationarity

  • Trend stationarity and difference stationarity

    1. 1. 1. • Consequences of non-stationarity

  • Spurious regression and misleading statistical inference

  • Implications for model estimation, interpretation, and forecasting

• Real-world examples of stationary and non-stationary series in economics, finance, and environmental data

Section 2: Theoretical Foundations of Stationarity

• Mathematical Representation of Stationarity

  • Definition of a stationary process: mean, variance, and autocovariance

  • Autoregressive Moving Average (ARMA) models and stationarity

• Random Walk and Unit Root Processes

  • Definition and characteristics of random walks

  • Understanding unit root processes and their implications for non-stationarity

• Trend and Seasonal Components

  • Decomposing time series into trend, seasonal, and cyclical components

  • Understanding deterministic and stochastic trends

Section 3: Visual and Descriptive Methods for Detecting Stationarity

• Time Series Plots

  • Plotting time series data to visually inspect for trends, seasonality, and stationarity

  • Identifying patterns that indicate non-stationarity

• Summary Statistics and Autocorrelation Function (ACF)

  • Examining mean and variance over time

  • Using ACF and Partial ACF (PACF) to detect stationarity and identify appropriate models

• Rolling Statistics

  • Calculating rolling mean and variance for visual inspection

  • Using rolling correlation to assess changes in relationships over time

Section 4: Statistical Tests for Stationarity

• Unit Root Tests

  • Augmented Dickey-Fuller (ADF) Test

    • Explanation of the ADF test and null hypothesis

    • Step-by-step implementation and interpretation

  • Phillips-Perron (PP) Test

    • Difference from ADF and situations where PP is preferred

    • Implementation and interpretation

      • • • The DF–GLS test

        • Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

    • Testing for stationarity as the null hypothesis

    • Complementary use with ADF and PP tests

• The Leybourne–McCabe test

• Additional Stationarity Tests

  • Elliott-Rothenberg-Stock (ERS) Test: Testing for unit roots with higher power

  • Ng-Perron Test: Modified tests for unit roots with improved size and power properties

    • Testing for unit roots with structural breaks 

  • Zivot-Andrews Test: Testing for unit roots with a structural break in the series

  • Perron Test (Perron 1989, Perron 1997): Tests for a unit root in the presence of a known (exogenous) structural break.

  •  Lumsdaine-Papell Test: Extends the Zivot-Andrews test by allowing for two endogenous structural breaks in the series.

  • Lee-Strazicich Test (Minimum LM Unit Root Test): Tests for a unit root with structural breaks, addressing issues related to earlier tests such as Perron and Zivot-Andrews. The test allows for endogenous structural breaks.

  •  Clemente-Montañés-Reyes Test: Tests for a unit root with two structural breaks, allowing for both additive and innovative outliers in the data.

  • Narayan-Popp Test: Tests for a unit root with two structural breaks in both the intercept and the slope (trend) of the series.

  • Carrion-i-Silvestre Test: Tests for a unit root with multiple structural breaks (up to five breaks). (link)

  • Perron-Rodriguez Test:  A variant of Perron’s test that allows for structural breaks with known or exogenous break dates.

  •  Bai-Perron Multiple Structural Break Test: Tests for multiple structural breaks in time series data and identifies the number of breaks and their locations.

  • Kapetanios Unit Root Test with a Break: Tests for a unit root allowing for a structural break with an exponential smooth transition.

  • Ng-Perron Modified Unit Root Test (with Structural Breaks): Extends the Ng-Perron test to handle structural breaks. 

Section 5: Dealing with Non-Stationarity

• Transformations for Stationarity

  • Log transformation, differencing, and detrending

  • Seasonal differencing for seasonal time series

• Testing and Applying Differences

  • First and second differencing to achieve stationarity

  • Determining the order of differencing using ADF test and ACF plots

• Structural Breaks and Non-Stationarity

  • Detecting structural breaks with Chow test and Bai-Perron test

  • Accounting for breaks in stationarity testing

Section 6: Practical Applications and Case Studies

• Case Study 1: GDP Growth and Stationarity

  • Analyzing GDP growth rates for stationarity and trend components

• Case Study 2: Financial Time Series

  • Testing for stationarity in stock prices and returns

• Case Study 3: Macroeconomic Indicators

  • Analyzing inflation rates and interest rates for unit roots

Section 7: Implementation in Statistical Software

• Implementation in R

  • Using the tseries and forecast packages for stationarity tests and model estimation

  • Example code for ADF, PP, KPSS, and other tests

• Implementation in Stata

  • Using the dfuller, pp, and kpss commands for stationarity testing

  • Example workflows for model building with stationary series

• Implementation in Python

  • Using statsmodels and arch libraries for stationarity testing and ARMA modeling

  • Practical examples and code snippets

• Other Software Tools

  • Implementing stationarity tests and models in EViews, MATLAB, and SAS

Section 8: Common Pitfalls and Best Practices

• Misinterpretation of Stationarity Tests

  • Understanding the limitations of ADF and PP tests

  • Proper use of stationarity tests in model selection

• Over-Differencing and Data Transformation

  • Avoiding over-differencing and loss of valuable information

  • Choosing appropriate transformations based on data characteristics

• Best Practices in Time Series Analysis

  • Guidelines for achieving and verifying stationarity

  • Integrating stationarity checks into the time series modeling workflow

Section 9: Summary and Conclusions

• Recap of Key Concepts

  • Summary of stationarity, its importance, and testing methods

• Practical Guidelines for Researchers

  • Best practices for testing and achieving stationarity in time series data

• Future Directions

  • Emerging techniques in stationarity testing and handling non-stationarity

Sách

•  Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press (Chapters 1–5, 7, and 8)

• Hall, P., & Heyde, C. C. (2014). Martingale limit theory and its application. Academic press. (Chapter 3).

• Brockwell, P. J., & Davis, R. A. (1991). Time series: theory and methods. Springer science & business media. (Chapters 1, 3, and Section 5.7)

Các bài giảng tham khảo

•  Boston College - Unit root tests  (link)

• MIT - Unit Roots (link)

Nội dung

Section 1: Introduction to Exponential Smoothing

• • Key principles behind exponential smoothing

  • Why exponential smoothing is popular for time series forecasting

• Types of Data Suitable for Exponential Smoothing

  • Short-term vs. long-term forecasting

  • Time series characteristics: trend, seasonality, and noise

Section 2: Simple Exponential Smoothing (SES)

• The SES Model

  • Formula and structure

  • Explanation of the smoothing parameter 

• Assumptions of SES

  • Stationarity and no trend or seasonality

  • Homoscedasticity in time series data

• Estimation of Smoothing Parameter α\alphaα

  • Methods for selecting α\alphaα: Optimization techniques, trial and error

  • Impact of different values of α\alphaα on the forecasts

• Practical Applications of SES

  • Case study: Forecasting monthly sales

  • Applications in finance, economics, and business

Section 3: Holt’s Linear Trend Model (Double Exponential Smoothing)

• The Holt’s Model

  • Accounting for trends in the data and formula for the trend component

  • Explanation of smoothing parameters

• Estimating the Trend Component

  • Methods for determining the trend smoothing parameter β\betaβ

  • Impact of varying β\betaβ on trend forecasts

• Forecasting with Holt’s Model

  • Making multi-step forecasts using the Holt’s model

  • Practical examples in economic forecasting

• Applications of Holt’s Model

  • Case study: Forecasting GDP growth with a linear trend

Section 4: Holt-Winters Seasonal Smoothing Model (Triple Exponential Smoothing)

• The Holt-Winters Model

  • Accounting for both trend and seasonality

  • Additive model formula and multiplicative model formula

  • Explanation of smoothing parameters

• Seasonality in Time Series

  • Dealing with additive and multiplicative seasonality

  • Identifying seasonal cycles in time series data

• Choosing Between Additive and Multiplicative Models

  • Guidelines for selecting the right model based on data characteristics

  • Practical differences between the two approaches

• Forecasting with the Holt-Winters Model

  • Multi-step forecasts for seasonal data

  • Case study: Forecasting retail sales with seasonal patterns

• Applications of the Holt-Winters Model

  • Seasonal demand forecasting in supply chain management

  • Time series forecasting in tourism and climate data

Section 5: Advanced Exponential Smoothing Methods

• Damped Trend Exponential Smoothing

  • Handling situations where the trend may level off over time and formula for damped trend smoothing

  • Explanation of the damping parameter

  • Case study: Forecasting long-term energy consumption

Section 6: Model Selection and Evaluation

• Model Selection Criteria

  • Choosing the appropriate exponential smoothing model

  • Criteria for model selection: AIC, BIC, RMSE, MAPE

• Evaluating Forecast Accuracy

  • Calculating forecast errors: MAE, MSE, and RMSE

  • Cross-validation techniques for evaluating out-of-sample forecasts

• Dealing with Model Overfitting

  • Dangers of overfitting in time series models

  • Methods for preventing overfitting

Section 7: Practical Applications of Exponential Smoothing Methods

• Case Study 1: Forecasting Macroeconomic Indicators

  • Using exponential smoothing methods to forecast inflation rates

• Case Study 2: Inventory Management and Demand Forecasting

  • Practical application in forecasting demand in supply chain management

• Case Study 3: Financial Market Forecasting

  • Forecasting stock prices using exponential smoothing techniques

• Case Study 4: Forecasting Tourism Demand

  • Applying Holt-Winters model to forecast seasonal tourism trends

Section 8: Limitations and Extensions of Exponential Smoothing

• Limitations of Exponential Smoothing

  • Limitations in capturing long-term patterns, non-linear trends, and structural breaks

  • Challenges in forecasting highly volatile time series data

• Combining Exponential Smoothing with Other Models

  • Hybrid models: Combining exponential smoothing with ARIMA or machine learning models

  • Benefits of combining methods for improved forecast accuracy

Section 9: Implementation in Statistical Software

• Implementation in R

  • Using forecast and smooth packages for exponential smoothing

  • Example code for SES, Holt’s, and Holt-Winters models

• Implementation in Python

  • Using statsmodels and prophet libraries for exponential smoothing methods

  • Practical examples and code snippets

• Implementation in Stata

  • Using Stata’s sts and tssmooth commands for exponential smoothing

  • Practical application and forecast interpretation

• Other Software Tools

  • Implementing exponential smoothing methods in Excel, MATLAB, and SAS

Section 10: Common Pitfalls and Best Practices

• 10.1 Over-smoothing and Under-smoothing

  • Understanding the effects of selecting inappropriate smoothing parameters

  • Guidelines for proper parameter selection

• 10.2 Misinterpretation of Trend and Seasonality

  • Avoiding common errors when interpreting trend and seasonal components

• 10.3 Best Practices in Time Series Forecasting

  • Guidelines for improving the accuracy and reliability of forecasts using exponential smoothing

Section 11: Summary and Conclusions

• Recap of Key Concepts

  • Summary of exponential smoothing methods and their applications

• Choosing the Right Exponential Smoothing Model

  • Practical advice on selecting the appropriate model based on data characteristics

• Future Trends and Directions

  • Emerging trends in exponential smoothing and time series forecasting

Sách

•  Abraham, B., & Ledolter, J. (2009). Statistical methods for forecasting. John Wiley & Sons.

• Becketti, S. (2013). Introduction to time series using Stata (Vol. 4905, pp. 176-182). College Station, TX: Stata Press.

• Chatfield, C. (2000). Time-series forecasting. Chapman and Hall/CRC.

• Montgomery, D. C., Johnson, L. A., & Gardiner, J. S. (1990). Forecasting and time series analysis.  (link)

Bài giảng tham khảo

•   Exponential Smoothing (link)

• Berkeley: Exponential Smoothing With Trend and Seasonality (link)

Bài báo

•  Holt, C. C. (2004). Forecasting seasonals and trends by exponentially weighted moving averages. International journal of forecasting, 20(1), 5-10.

• Chatfield, C., & Yar, M. (1988). Holt-Winters forecasting: some practical issues. Journal of the Royal Statistical Society Series D: The Statistician, 37(2), 129-140.

Nội dung

Section 1: Introduction to Time Series Decomposition

• • Definition and purpose of time series decomposition

  • Why decomposing time series is important for analysis and forecasting

    1. 1. 1. 1. • Components of a Time Series

                  • Trend: Long-term direction in the data

                  • Seasonality: Regular patterns that repeat over a fixed period

                  • Cyclical: Fluctuations over irregular periods, distinct from seasonality

                  • Irregular component: Random, unsystematic fluctuations

Section 2: Additive and Multiplicative Decomposition Models

• Additive Model

  • Assumes components are added together

  • Suitable for time series where seasonal variations remain constant over time

  • Examples of additive decomposition

• Multiplicative Model

  • Assumes components are multiplied

  • Suitable for time series where seasonal variation increases or decreases with trend

  • Examples of multiplicative decomposition

• Choosing Between Additive and Multiplicative Models

  • Guidelines for determining the appropriate model based on data characteristics

Section 3: Classical Decomposition Methods

• Moving Average Method for Trend Extraction

  • Using moving averages to estimate the trend component

  • Selection of the window size for moving averages

  • Advantages and limitations of moving average methods

• Seasonal Decomposition Using Moving Averages (SDMA)

  • Step-by-step guide to decomposing time series into trend, seasonality, and residuals

  • Estimating the seasonal index using centered moving averages

  • Case study: Decomposing retail sales data using classical decomposition

Section 4: STL (Seasonal-Trend Decomposition Using LOESS)

• Overview of STL Decomposition

  • What is STL decomposition, and how it differs from classical decomposition

  • Flexible handling of non-linear trends and seasonal patterns

• The LOESS Smoothing Technique

  • Introduction to LOESS (locally estimated scatterplot smoothing) for trend and seasonal estimation

  • How STL handles seasonality with varying lengths and non-constant trends

• Implementation and Interpretation of STL Decomposition

  • Practical example: Decomposing time series with multiple seasonal patterns

  • Benefits of STL over traditional decomposition methods

Section 5: X-12-ARIMA/X-13-ARIMA-SEATS Decomposition

• Introduction to X-12-ARIMA/X-13-ARIMA-SEATS

  • Overview of ARIMA-based decomposition models developed by the U.S. Census Bureau

  • How these models extend classical decomposition with advanced forecasting features

• Steps in X-12-ARIMA Decomposition

  • Pre-adjustments for outliers, calendar effects, and trading days

  • ARIMA model fitting for trend and seasonal components

  • Residual diagnostics and seasonal adjustments

• Applications of X-12-ARIMA/X-13-ARIMA-SEATS

  • Case study: Forecasting macroeconomic indicators using X-12-ARIMA

  • Example: Seasonal adjustment in national income accounting

Section 6: Decomposition Based on Exponential Smoothing (ETS Decomposition)

• ETS Models for Time Series Decomposition

  • Introduction to Exponential Smoothing State Space (ETS) models

  • How ETS models estimate trend, seasonality, and error components

• Comparison with Classical Decomposition Methods

  • When to use ETS decomposition over moving averages or STL

• Case Study

  • Example: Decomposing electricity consumption data using ETS models

Section 7: Advanced Time Series Decomposition Techniques

• Wavelet Decomposition

  • Introduction to wavelet analysis and its applications in time series decomposition

  • How wavelet decomposition handles non-stationary and non-linear time series data

• Empirical Mode Decomposition (EMD)

  • Overview of EMD and its use for adaptive time series decomposition

  • Application to time series with complex, non-stationary characteristics

  • Examples of EMD in financial and environmental data

Section 8: Handling Irregular Components and Structural Breaks

• Identifying and Dealing with Irregular Components

  • Methods to isolate and analyze the residual (irregular) component

  • Techniques for smoothing irregular components

• Detecting and Addressing Structural Breaks

  • Using statistical tests for structural breaks in time series (e.g., Chow Test, Bai-Perron Test)

  • Impact of structural breaks on trend and seasonality estimation

  • Strategies for adjusting decomposition models to account for breaks

Section 9: Practical Applications and Case Studies

• Case Study 1: Macroeconomic Data Decomposition

  • Decomposing GDP data into trend, seasonal, and irregular components

• Case Study 2: Decomposing Stock Market Prices

  • Time series decomposition of stock market data using multiplicative models

• Case Study 3: Climate Data Decomposition

  • Seasonal-trend decomposition of temperature data with STL and LOESS smoothing

• Case Study 4: Retail Sales Forecasting

  • Using decomposition to analyze and forecast seasonal retail sales patterns

Section 10: Model Selection and Evaluation

• Model Selection Criteria

  • Guidelines for selecting the appropriate decomposition method

  • Comparing models using AIC, BIC, and cross-validation

• Evaluating Decomposition Accuracy

  • Metrics for assessing the quality of decomposition (e.g., RMSE, MAE)

  • Techniques for evaluating the accuracy of the trend, seasonal, and residual components

• Dealing with Overfitting

  • Avoiding overfitting in trend and seasonal estimation

  • Best practices for handling noisy and irregular data

Section 11: Implementation in Statistical Software

• Implementation in R

  • Using the decompose(), stl(), and forecast packages for time series decomposition

  • Practical examples and code snippets for additive and multiplicative decomposition

• Implementation in Python

  • Using statsmodels and Prophet libraries for classical and STL decomposition

  • Example applications of decomposition techniques in Python

• Implementation in Stata

  • Decomposing time series data using Stata’s built-in commands (tsfilter and stsmooth)

  • Case study: Forecasting and adjusting for seasonality using Stata

• Other Software Tools

  • Implementation of decomposition techniques in EViews, MATLAB, and SAS

Section 12: Common Pitfalls and Best Practices

• Misinterpretation of Components

  • Avoiding common errors in interpreting trend, seasonal, and residual components

  • Recognizing over-differencing and underfitting in decomposition

• Seasonality and Trend Confusion

  • Best practices for correctly separating trend and seasonal effects in complex data

• Best Practices in Time Series Decomposition

  • Guidelines for selecting appropriate models and ensuring accurate decomposition

Section 13: Summary and Conclusions

• Recap of Key Concepts

  • Summary of time series decomposition methods and their applications

• Choosing the Right Decomposition Method

  • Practical guidelines for selecting decomposition techniques based on the type of data and the forecasting objective

• Future Directions

  • Emerging trends in time series decomposition, including machine learning applications

Sách

•  Fuller, W. A. (2009). Introduction to statistical time series. John Wiley & Sons.

• Harvey, A. C. (1990). The econometric analysis of time series. Mit Press.

• William, W., & Wei, S. (2006). Time series analysis: univariate and multivariate methods. USA, Pearson Addison Wesley, Segunda edicion. Cap, 10, 212-235.

• Baum, C. F., & Hurn, S. (2021). Environmental econometrics using Stata. College Station, TX: Stata Press. (datasets: link)

Bài giảng tham khảo

• Decomposition Models (link)

• Decomposition of Time Series  (link)

Bài báo

•  Beveridge, S., & Nelson, C. R. (1981). A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. Journal of Monetary economics, 7(2), 151-174.

Nội dung

Section 1: Introduction to Spectral Analysis

• • Comparison of time domain vs. frequency domain analysis

  • Why and when spectral analysis is useful in econometrics

  • Applications of spectral analysis in economics and finance

    1. 1. 1. • Key Concepts in Spectral Analysis

               1. • Periodicity and cycles in time series data

                  • Definition of frequency, period, and wavelength

                  • The importance of identifying cyclical patterns

Section 2: Theoretical Foundations of Spectral Analysis

• Fourier Transform and Time Series

  • Introduction to Fourier transforms and their role in spectral analysis

  • The Fourier series decomposition: representing a time series as a sum of sinusoids

  • Intuition behind the use of sine and cosine waves to capture periodicity

• Power Spectrum and Spectral Density

  • Definition of the power spectrum: measuring how variance is distributed across frequencies

  • Spectral density function: f(λ)f(\lambda)f(λ), the strength of different frequency components in the time series

  • The relationship between autocovariance and the spectral density

• Understanding Cycles and Frequencies

  • Interpreting low-frequency (long-term trends) vs. high-frequency (short-term fluctuations)

  • Relationship between period (T) and frequency (λ): T=2πλT = \frac{2\pi}{\lambda}T=λ2π​

  • Identifying business cycles, seasonal effects, and irregular cyclical behavior

Section 3: The Periodogram

• Definition of the Periodogram

  • Constructing the periodogram: decomposing the time series into its frequency components

  • Formula and estimation

  • Interpretation of the periodogram: identifying peaks and corresponding frequencies

• Limitations of the Periodogram

  • Bias and variance trade-off in periodogram estimates

  • Smoothing techniques to improve interpretation (e.g., Daniell smoothing)

• Example: Periodogram of a Simulated Time Series

  • Step-by-step construction and interpretation of the periodogram for synthetic data

Section 4: Spectral Density Estimation

• Nonparametric Spectral Density Estimation

  • Understanding kernel smoothing and its application to spectral density estimation

  • Applying the Daniell, Parzen, and Bartlett kernels to smooth the periodogram

• Parametric Spectral Density Estimation

  • Estimating the spectral density function using ARMA models

  • Maximum likelihood estimation of the spectral density for parametric models

• Example: Spectral Density Estimation for Financial Time Series

  • Practical example using real-world data (e.g., stock prices or GDP data)

  • Comparing parametric and nonparametric spectral estimates

Section 5: Coherence and Cross-Spectral Analysis

• Introduction to Coherence

  • Measuring the relationship between two time series in the frequency domain

  • Definition and interpretation of the coherence function

• Cross-Spectral Density

  • Understanding the cross-spectrum

  • Phase and gain: interpreting the lead-lag relationship in frequency domain

• Practical Application of Cross-Spectral Analysis

  • Example: Analyzing the relationship between GDP growth and inflation cycles using cross-spectral analysis

Section 6: The Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)

• The Discrete Fourier Transform

  • Introduction to the DFT for finite time series data

  • Formula and properties of the DFT: Xk=∑t=1Nxte−i2πkt/NX_k = \sum_{t=1}^{N} x_t e^{-i 2 \pi k t / N}Xk​=∑t=1N​xt​e−i2πkt/N

  • Understanding the inverse DFT to reconstruct the time series

• The Fast Fourier Transform (FFT) Algorithm

  • Importance of FFT for computational efficiency in large datasets

  • Practical implementation of FFT in spectral analysis

• Example: Applying FFT to Economic Data

  • Example using the FFT algorithm to analyze frequency components in real-world data

Section 7: Practical Applications of Spectral Analysis

• Business Cycle Analysis

  • Identifying economic cycles and periodicity using spectral methods

  • Case study: Decomposing GDP data to extract business cycles

• Seasonality in Economic Data

  • Using spectral analysis to detect and quantify seasonal patterns in time series

  • Case study: Seasonal effects in retail sales or energy consumption

• Volatility and Frequency Analysis in Financial Markets

  • Application of spectral analysis in identifying short-term and long-term market volatility cycles

  • Case study: Analyzing stock market returns or exchange rate data

Section 8: Spectral Analysis with Non-Stationary Data

• Limitations of Spectral Analysis for Non-Stationary Time Series

  • Challenges in applying spectral methods to non-stationary data

  • Addressing non-stationarity with differencing and detrending

• Wavelet Analysis as an Alternative

  • Introduction to wavelet analysis for non-stationary time series

  • Benefits of wavelets for capturing both time and frequency information

• Example: Wavelet Decomposition of Exchange Rate Volatility

  • Applying wavelet analysis to a non-stationary economic time series

Section 9: Software Implementation of Spectral Analysis

• Implementation in R

  • Using the spectrum() and fft() functions for spectral analysis in R

  • Practical examples of periodogram and spectral density estimation in R

• Implementation in Python

  • Using scipy and matplotlib for FFT and spectral analysis in Python

  • Example code for conducting spectral analysis on economic and financial data

• Implementation in Stata

  • Using Stata’s tsfilter command for spectral density estimation

  • Practical application of spectral techniques in Stata for time series analysis

• Other Software Tools

  • Implementing spectral analysis in MATLAB, EViews, and SAS

Section 10: Common Pitfalls and Best Practices in Spectral Analysis

• Overfitting in Spectral Analysis

  • Avoiding over-interpretation of spurious peaks in the periodogram

  • Best practices for selecting smoothing parameters

• Handling Noise in Spectral Density Estimation

  • Techniques for filtering noise and identifying true periodic components

• Ensuring Stationarity Before Spectral Analysis

  • Guidelines for differencing and detrending non-stationary time series data

Section 11: Summary and Conclusions

• Recap of Key Concepts

  • Review of the fundamental principles of spectral analysis

  • Importance of spectral methods in time series econometrics

• When to Use Spectral Analysis

  • Guidelines for choosing spectral analysis over traditional time series methods

  • Situations where spectral analysis provides unique insights

• Future Directions in Spectral Analysis

  • Advances in wavelet analysis and multiscale methods for economic data

Bài giảng tham khảo

• Spectral Analysis (link)

Sách

• Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press (Chapter 6).

• Brockwell, P. J., & Davis, R. A. (1991). Time series: theory and methods. Springer science & business media. (Chapters 4, and 10)

• Brillinger, D. R. (2001). Time series: data analysis and theory. Society for Industrial and Applied Mathematics. (Chapters. 3-5)

Bài báo

• Baxter, M., & King, R. G. (1995). Approximate band-pass filters for economic time series. NBER Working Paper, 5022. 

• Berk, K. N. (1974). Consistent autoregressive spectral estimates. The Annals of Statistics, 489-502.

• Hodrick, R. J., & Prescott, E. C. (1997). Postwar US business cycles: an empirical investigation. Journal of Money, credit, and Banking, 1-16.

• Christiano, L. J., & Fitzgerald, T. J. (2003). The band pass filter. International economic review, 44(2), 435-465.

Nội dung: 

Section 1: Introduction to ARIMA Models

• • Importance of ARIMA models in econometrics

  • Use cases for ARIMA in economics, finance, and other fields

    1. 1. • Components of ARIMA Models

            1. • Autoregressive (AR) component

               • Moving Average (MA) component

               • ARMA(p,q) model

               • ARIMA(p, d, q) model specification

Section 2: Understanding the Box-Jenkins Methodology

• Overview of the Box-Jenkins Approach

  • A systematic process for identifying, estimating, and diagnosing ARIMA models

  • Focus on iterative steps for building accurate models

• The Three Stages of the Box-Jenkins Method

  • Identification: Selecting the appropriate ARIMA model

  • Estimation: Fitting the selected model to the data

  • Diagnostic Checking: Validating the model and refining it if necessary

• The Box-Jenkins Model-Building Process

  • Iterative process of improving models based on diagnostic checks

Section 4: Autoregressive (AR) and Moving Average (MA) Components

• Autoregressive (AR) Models

  • Definition and explanation of the AR component

  • Partial autocorrelation function (PACF) for identifying the order of AR (p)

  • Example: Building an AR(1) model

• Moving Average (MA) Models

  • Definition and explanation of the MA component

  • Autocorrelation function (ACF) for identifying the order of MA (q)

  • Example: Building an MA(1) model

• Combined ARMA Models

  • ARMA(p, q) model specification for stationary series

  • Practical examples of ARMA(2, 2) models and interpretation

Section 5: Identification of ARIMA Models

• Analyzing the ACF and PACF

  • Interpreting ACF and PACF plots to identify ARIMA model orders

  • Guidelines for selecting AR and MA terms based on ACF/PACF behavior

• Model Identification Strategy

  • Using time series plots and correlograms to identify trends and seasonality

  • Step-by-step process for identifying ARIMA(p, d, q)

• Case Study: Identifying an ARIMA Model for Real-World Data

  • Practical example: Identifying the ARIMA order for a macroeconomic time series (e.g., GDP growth)

Section 6: Estimation of ARIMA Models

• Maximum Likelihood Estimation for ARIMA Models

  • Explanation of how ARIMA models are estimated using maximum likelihood

  • Importance of parameter estimation for accurate model fitting

• Example of Estimating ARIMA Parameters

  • Practical example of estimating ARIMA parameters using maximum likelihood in software

• Challenges in Estimation

  • Addressing issues such as non-convergence and multicollinearity

  • Solutions for improving the estimation process

Section 7: Diagnostic Checking of ARIMA Models

• Residual Analysis

  • Assessing model adequacy by analyzing residuals

  • Autocorrelation of residuals: The Ljung-Box test

• Model Validation Techniques

  • Ensuring residuals are white noise (no autocorrelation or pattern)

  • Checking for overfitting and underfitting

Section 10: Forecasting with ARIMA Models

• Generating Forecasts with ARIMA

  • Steps for generating short-term and long-term forecasts using ARIMA

  • Constructing forecast intervals and understanding forecast accuracy

• Example of Forecasting with ARIMA

  • Forecasting economic variables such as inflation or unemployment rates using ARIMA models

• Evaluating Forecast Performance

  • Measures of forecast accuracy: Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE)

  • Cross-validation techniques for improving forecast accuracy

Section 10: Practical Applications of ARIMA and Box-Jenkins Method

• Case Study 1: Forecasting Stock Prices

  • Applying ARIMA models to forecast daily or weekly stock prices

• Case Study 2: Macroeconomic Forecasting

  • Forecasting GDP growth using ARIMA and SARIMA models

• Case Study 3: Energy Consumption Forecasting

  • Modeling and forecasting energy consumption with ARIMA models

• Case Study 4: Retail Sales Forecasting

  • Using ARIMA and SARIMA for short-term retail sales forecasting

Section 11: Implementation of ARIMA in Statistical Software

• Implementation in R

  • Using auto.arima() and forecast package for ARIMA modeling

  • Example code and case study for ARIMA in R

• Implementation in Python

  • Using statsmodels library for ARIMA modeling in Python

  • Practical examples of ARIMA in Python for financial data

• Implementation in Stata

  • Using arima command in Stata for ARIMA model estimation

  • Example of fitting ARIMA models to economic data in Stata

• Implementation in EViews and MATLAB

  • Implementing ARIMA models using EViews and MATLAB with practical examples

Section 12: Limitations and Extensions of ARIMA Models

• Limitations of ARIMA Models

  • Issues with non-linearity, structural breaks, and volatility in ARIMA models

  • Handling time series with complex seasonal and cyclical components

• Combining ARIMA with Other Methods

  • Hybrid models: Combining ARIMA with GARCH for volatility forecasting

  • Machine learning extensions of ARIMA (e.g., ARIMA with neural networks)

•  ARIMAX (ARIMA with Exogenous Variables)

  • 1. 1. • ARIMA(p, d, q) with additional terms for the exogenous variables.

          • Forecasting when the time series is influenced by external factors, such as forecasting sales based on marketing expenditure or weather conditions.

• Long Memory Processes

  • Introduction to ARFIMA models for time series with long memory

  • Practical applications of ARFIMA in econometrics

Section 13: Summary and Conclusions

• Recap of ARIMA and Box-Jenkins Method

  • Review of the model identification, estimation, and diagnostic process

  • Practical insights for using ARIMA in econometric analysis

• Guidelines for Applying ARIMA

  • Best practices for applying ARIMA models to real-world time series data

• Future Directions

  • Emerging trends in time series modeling, including deep learning and non-linear time series analysis

Sách

• Enders, W. (2008). Applied econometric time series. John Wiley & Sons (Section 2.1-2.11) 

• Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice. OTexts.(Chapter 8) 

• Shumway, R. H., Stoffer, D. S., & Stoffer, D. S. (2000). Time series analysis and its applications (Vol. 3). New York: springer. (Chapter 3)

• Enders, W. (2008). Applied econometric time series. John Wiley & Sons. (pages 78-87)

• Diebold, F. X. (2001). Elements of forecasting. (Chapter 7)

• Granger, C. W. J. (2014). Forecasting in business and economics. Academic Press. (pages 65-75).

• Becketti, S. (2013). Introduction to time series using Stata (Vol. 4905, pp. 176-182). College Station, TX: Stata Press.

Bài giảng tham khảo

•  MIT- Autoregressive Integrated Moving Average Models (link)

• ARIMA – Identification, Estimation & Seasonalities (link)

• ARIMA – Estimation & Diagnostic Testing (link)

• Berkeley - Introduction to Time Series Analysis (link)

Bài báo

•  Holan, S. H., Lund, R., & Davis, G. (2010). The ARMA alphabet soup: A tour of ARMA model variants.

Nội dung

Section 1: Introduction to Volatility Modeling

• • Importance of volatility modeling in financial and economic data

  • Time-varying volatility and its economic significance

  • Why models like ARIMA fail to capture volatility clustering and heteroskedasticity

  • The need for models that handle conditional variance (heteroskedasticity)

Section 2: ARCH Models

• The ARCH Process

  • Definition and concept of Autoregressive Conditional Heteroskedasticity (ARCH)

  • ARCH(q) model: Modeling time-varying variance with past squared residuals

• Properties of ARCH Models

  • Conditional mean and conditional variance

  • Volatility clustering: Understanding periods of high and low volatility

  • Stationarity conditions for ARCH models

• Identification of ARCH Models

  • Analyzing autocorrelation in squared residuals to detect heteroskedasticity

  • Use of Lagrange Multiplier (LM) tests for ARCH effects

• Example: Fitting an ARCH(1) Model to Financial Returns

  • Practical example of estimating and interpreting an ARCH model for stock returns

Section 3: The GARCH Model

• Generalizing ARCH: The GARCH Model

  • Introduction to the Generalized ARCH (GARCH) model

  • GARCH(p, q) model specification

  • How GARCH allows for more parsimonious models than ARCH

• Properties of GARCH Models

  • Conditional variance as a weighted sum of past squared residuals and past variances

  • Interpretation of parameters α\alphaα and β\betaβ

  • Stationarity conditions: Ensuring the sum of α\alphaα and β\betaβ is less than one

• Estimating GARCH Models

  • Maximum likelihood estimation (MLE) of GARCH parameters

  • Practical example: Estimating a GARCH(1,1) model for daily returns

• Forecasting with GARCH Models

  • Short-term and long-term volatility forecasting with GARCH models

  • Example: Volatility forecast for financial market data

Section 4: Model Selection and Diagnostics

• Model Selection Criteria for ARCH/GARCH Models

  • Information criteria: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)

  • Selecting appropriate lag orders (p and q) for GARCH models

• Diagnostic Checking

  • Residual diagnostics for ARCH/GARCH models

  • Ljung-Box test for autocorrelation in standardized residuals

  • Testing for remaining ARCH effects in residuals

• Practical Example: Refining an ARCH/GARCH Model Based on Diagnostics

  • Iterative process for improving model fit based on diagnostic checks

Section 5: Extensions of the GARCH Model

• EGARCH (Exponential GARCH)

  • Modeling asymmetries in volatility: Allowing for different impacts of positive and negative shocks

  • Formula and interpretation of the EGARCH model

  • Example: Fitting an EGARCH model to stock returns with leverage effects

• TGARCH (Threshold GARCH)

  • Threshold GARCH for capturing different volatility regimes based on the size or sign of past shocks

  • Practical application of TGARCH models in financial data

• GARCH-M (GARCH in Mean)

  • Incorporating volatility into the mean equation: Modeling risk-return trade-offs

  • GARCH-M model for risk-premium estimation in asset returns

  • Example: Applying GARCH-M to estimate the risk premium in stock returns

• IGARCH (Integrated GARCH)

  • Long-memory GARCH models: IGARCH for modeling persistent volatility

  • Use cases and estimation of IGARCH models in high-frequency financial data

Section 6: Volatility Forecasting with ARCH/GARCH Models

• Forecasting Conditional Variance

  • Step-by-step guide to forecasting volatility using GARCH models

  • Forecast intervals and confidence bounds for volatility predictions

• Practical Application: Volatility Forecasting for Option Pricing

  • Example: Using GARCH models to forecast volatility for option pricing (Black-Scholes model)

• Comparing ARCH/GARCH Forecasts with Other Volatility Forecasting Models

  • Comparison of GARCH forecasts with realized volatility, historical volatility, and implied volatility

Section 7: Practical Applications of ARCH/GARCH Models

• Case Study 1: Stock Market Volatility

  • Modeling and forecasting stock return volatility using GARCH models

• Case Study 2: Foreign Exchange Rate Volatility

  • Applying GARCH models to analyze exchange rate volatility

• Case Study 3: Commodity Prices

  • Using ARCH/GARCH models to model the volatility of commodity prices (e.g., oil, gold)

• Case Study 4: Interest Rate Volatility

  • Modeling volatility in short-term interest rates with GARCH models

Section 8: Implementation in Statistical Software

• Implementation in R

  • Using the rugarch package for estimating ARCH/GARCH models

  • Example code for fitting GARCH models and volatility forecasting

• Implementation in Python

  • Using arch and statsmodels libraries for ARCH/GARCH estimation

  • Example of modeling financial time series with GARCH in Python

• Implementation in Stata

  • Using arch command in Stata for GARCH model estimation

  • Practical examples and interpretation of results

• Other Software Tools

  • Implementing ARCH/GARCH models in EViews, MATLAB, and SAS

Section 9: Common Pitfalls and Best Practices in ARCH/GARCH Modeling

• Misidentification of Lag Orders

  • Guidelines for selecting appropriate lags in ARCH and GARCH models

• Overfitting and Underfitting in GARCH Models

  • Balancing model complexity and fit: Avoiding overfitting in ARCH/GARCH models

• Handling Non-Normality in Residuals

  • Dealing with non-Gaussian innovations: Student’s t-distribution or generalized error distribution (GED)

  • Practical guidelines for robust estimation in the presence of heavy tails or outliers

• Best Practices for ARCH/GARCH Models in Financial Econometrics

  • Recommendations for applying ARCH/GARCH models to real-world data

Section 10: Summary and Conclusions

• Recap of ARCH/GARCH Models

  • Summary of key concepts, including ARCH, GARCH, and their extensions

• Practical Guidelines for Using ARCH/GARCH Models

  • Best practices for applying these models in empirical research

• Future Directions in Volatility Modeling

  • Emerging techniques and models for volatility analysis, such as multivariate GARCH, high-frequency volatility models, and machine learning approaches

Bài giảng tham khảo

• ARCH/GARCH Models (link)

• Financial time series, ARCH and GARCH (link)

•  Modeling Daily Returns with the GARCH Model (link)

Sách

• Wei, W. W. (2006). Time Series Analysis: Univariate and Multivariate Methods (Chapter 15)

• Enders, W. (2008). Applied econometric time series. John Wiley & Sons  (Chapter 3)

• Shumway, R. H., Stoffer, D. S., & Stoffer, D. S. (2000). Time series analysis and its applications (Vol. 3). New York: springer. (Chapter 5.3)

Bài báo

• Diebold, F. X. (2004). The nobel memorial prize for Robert F. Engle. (link)

Nội dung

Section 1: Introduction to Causality in Time Series

• • The concept of causality and correlation in econometric models

  • Causality in time series: Predictive relationships between variables

  • Why correlation does not imply causation

  • The importance of distinguishing between correlation and causality in time series

  • Introduction to Granger’s concept of causality

  • How Granger causality relates to prediction: X "Granger causes" Y if past values of X improve the prediction of Y

Section 2: Granger Causality Test

• Granger’s Definition of Causality

  • Definition and formalization of Granger causality

  • Granger causality as a hypothesis test

• Assumptions of the Granger Causality Test

  • Stationarity of the time series

  • Linearity assumption and implications for Granger causality

• Granger Causality in the Context of Vector Autoregressive (VAR) Models

  • Introduction to VAR models for multiple time series

  • Role of VAR in testing for Granger causality

Section 3: Granger Causality in Bivariate Models

• Testing for Granger Causality in a Bivariate Time Series Model

  • Basic setup for Granger causality in a two-variable (bivariate) system

  • Model specification: Testing whether past values of X help predict Y

  • Example: Testing whether GDP growth "Granger causes" investment

• The Granger Causality Test Procedure

  • Steps in conducting the Granger causality test

  • Estimating VAR models and choosing optimal lags using AIC or BIC

  • Hypothesis testing and interpreting the F-statistic

Section 4: Granger Causality in Multivariate Models

• Extending Granger Causality to Multivariate Systems

  • Multivariate Granger causality tests: Testing causality between multiple variables

  • Granger causality in systems of more than two variables

  • Example: Testing causality between GDP, inflation, and interest rates

• Joint Tests for Granger Causality

  • Testing whether multiple variables jointly Granger cause another variable

  • Wald tests for joint significance in multivariate models

• Example: Granger Causality in a Multivariate System

  • Practical example of testing Granger causality between macroeconomic variables

Section 5: Interpreting Granger Causality Results

• Understanding the Results of the Granger Causality Test

  • Interpreting the test statistics and p-values

  • Causality in one direction vs. bidirectional causality (feedback)

• Limitations of the Granger Causality Test

  • Potential pitfalls: Spurious causality, omitted variables, and model misspecification

  • Non-linearity and other limitations of the Granger causality test in real-world applications

• Example: Granger Causality Between Exchange Rates and Interest Rates

  • Case study demonstrating how to interpret results in a financial context

Section 6: Non-Stationary Time Series and Granger Causality

• Granger Causality in Non-Stationary Time Series

  • The issue of non-stationarity in time series data

  • Differencing and detrending to achieve stationarity before testing

• Granger Causality with Cointegrated Variables

  • Introduction to cointegration and its implications for Granger causality

  • Using Vector Error Correction Models (VECM) to test Granger causality in cointegrated systems

• Example: Granger Causality in a Cointegrated System

  • Case study: Testing Granger causality between consumption and income in the presence of cointegration

Section 7: Variants and Extensions of Granger Causality Tests

• oda-Yamamoto Causality Test

  • Overview of the Toda-Yamamoto approach: Testing causality without the need for differencing

  • Example: Applying Toda-Yamamoto test to real-world data

• Nonlinear Granger Causality

  • Introduction to nonlinear Granger causality tests

  • Capturing nonlinear relationships in time series data

  • Example: Testing for nonlinear Granger causality in stock returns and volatility

• Frequency-Domain Granger Causality

  • Testing for Granger causality in different frequency ranges

  • Example: Using spectral Granger causality to analyze business cycle synchronization between countries

Section 8: Practical Applications of Granger Causality Tests

• Case Study 1: Granger Causality Between Monetary Policy and Inflation

  • Testing whether central bank interest rates "Granger cause" inflation rates

• Case Study 2: Granger Causality Between Energy Consumption and Economic Growth

  • Analyzing the relationship between energy use and GDP growth

• Case Study 3: Financial Market Causality

  • Testing causality between stock market returns and economic indicators such as industrial production or consumer sentiment

Section 9: Granger Causality in Statistical Software

• Implementation in R

  • Using the VAR and causality functions in the vars package for Granger causality testing in R

  • Practical example of running Granger causality tests in R

• Implementation in Python

  • Using statsmodels and grangercausalitytests in Python for time series Granger causality tests

  • Practical example with Python code for testing causality in financial data

• Implementation in Stata

  • Using Stata’s vargranger command for testing Granger causality

  • Step-by-step guide to implementing Granger causality tests in Stata

• Other Software Tools

  • Implementing Granger causality tests in EViews, MATLAB, and SAS

Section 10: Common Pitfalls and Best Practices

• Avoiding Spurious Granger Causality

  • Ensuring stationarity and model specification are correct to avoid false causality results

• Lag Length Selection

  • Importance of choosing the correct lag length for Granger causality tests

  • Methods for selecting optimal lags: AIC, BIC, and cross-validation

• Handling Structural Breaks

  • Accounting for structural breaks in the data before testing for Granger causality

• 10.4 Best Practices for Granger Causality Tests

  • Practical guidelines for conducting reliable and valid Granger causality tests

Section 11: Summary and Conclusions

• Recap of Granger Causality Concepts

  • Review of the key concepts and methods introduced in the chapter

• Guidelines for Applying Granger Causality in Empirical Research

  • Practical recommendations for researchers using Granger causality in time series analysis

• Future Directions in Granger Causality Testing

  • Emerging techniques and future developments in causality testing, including machine learning-based approaches

Sách

• Lütkepohl, H. (2013). Introduction to multiple time series analysis. Springer Science & Business Media. (link) (link)

• Granger, C. W. (2001). Essays in econometrics: collected papers of Clive WJ Granger (Vol. 32). Cambridge University Press. 

Bài báo

• Shojaie, A., & Fox, E. B. (2022). Granger causality: A review and recent advances. Annual Review of Statistics and Its Application, 9(1), 289-319.

• White, H., & Pettenuzzo, D. (2014). Granger causality, exogeneity, cointegration, and economic policy analysis. Journal of Econometrics, 178, 316-330.

• 
  

Nội dung

Section 1: Introduction to VAR Models

• • The need for modeling multiple interrelated time series

  • Examples of multivariate time series in economics, finance, and macroeconomic data

  • Definition and basic structure of VAR models

  • Why VAR models are useful for capturing dynamic relationships among multiple variables

  • Example: VAR for GDP, interest rates, and inflation

Section 2: VAR Models

• 2.1 The General VAR(p) Model

  • Mathematical formulation

  • Explanation of matrices for endogenous variables (Y), lagged terms (A), and error terms (ε)

  • Example: Two-variable VAR(1) model

• Stationarity and Stability Conditions for VAR Models

  • Stationarity in multivariate time series: Why it is important

  • Testing for stationarity using unit root tests in VAR models

  • Stability conditions for VAR models: Determinants of the companion matrix

• Autocorrelation and Lag Structure

  • Explaining the role of lags in capturing temporal relationships

  • Choosing the lag length: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and likelihood ratio tests

Section 3: Estimating VAR Models

• Ordinary Least Squares (OLS) Estimation for VAR

  • Step-by-step OLS estimation for each equation in a VAR system

  • Estimation under stationarity: Consistency and efficiency of OLS

• Maximum Likelihood Estimation (MLE) for VAR

  • Using MLE for more efficient parameter estimation in VAR models

  • Comparison between OLS and MLE estimation in VAR systems

• Example: Estimating a VAR Model for GDP and Inflation

  • Step-by-step guide for estimating a bivariate VAR model using real economic data

  • Interpreting parameter estimates and checking residuals

Section 4: Model Selection and Diagnostic Checking

• Choosing the Optimal Lag Length

  • Criteria for selecting the lag length: AIC, BIC, and likelihood ratio test

  • Example: Determining the lag length for a three-variable VAR model

• Diagnostic Checking of VAR Models

  • Residual analysis: Checking for autocorrelation, heteroskedasticity, and normality

  • Ljung-Box test for serial correlation in VAR residuals

  • Testing for Granger causality in VAR models (Granger causality block exogeneity tests)

• Model Specification Tests

  • Wald test for parameter restrictions

  • Testing for structural breaks using Chow tests or recursive residuals

Section 5: Impulse Response Functions (IRFs)

• Understanding Impulse Response Functions

  • Definition and purpose of IRFs in VAR models

  • How IRFs trace out the response of endogenous variables to shocks

• Estimating and Interpreting Impulse Response Functions

  • Step-by-step calculation of IRFs using Cholesky decomposition

  • Interpretation of IRFs: Short-run vs. long-run effects of shocks

  • Example: Analyzing the impact of an interest rate shock on GDP and inflation using IRFs

• Confidence Intervals for IRFs

  • Bootstrapping methods for estimating confidence intervals for IRFs

  • Example: Estimating and interpreting confidence intervals for impulse responses

Section 6: Variance Decomposition

• Concept of Variance Decomposition

  • Decomposing the forecast error variance of each variable in the VAR system

  • How variance decomposition helps in understanding the contribution of shocks to each variable

• Estimating Variance Decomposition in VAR

  • Practical example of estimating variance decomposition for a two-variable VAR model

  • Interpreting the results: Relative importance of shocks in explaining the variation in each variable

• Example: Variance Decomposition for Macro Variables

  • Application: Analyzing the variance decomposition of inflation and output growth

Section 7: Extensions and Advanced Topics in VAR Models

• Structural VAR (SVAR) Models

  • Difference between VAR and SVAR models

  • Imposing economic theory-based restrictions on VAR models to identify structural shocks

    1. 1. 1. • Identifying Restrictions in SVAR (next section)

• Vector Error Correction Model (VECM)

  • Introduction to VECM for cointegrated time series 

  • Continue at next section

• Factor-Augmented VAR (FAVAR)

  • Extending VAR models to include factor analysis

  • Application of FAVAR models in macroeconomic policy analysis

• Bayesian VAR (BVAR)

  • Introduction to Bayesian estimation methods for VAR models

  • Incorporating prior information into VAR estimation

  • Example: Forecasting using BVAR models in macroeconomics

• Time-Varying Parameter VAR (TVP-VAR)

  • Modeling changes in relationships over time with TVP-VAR

  • Application in financial markets and macroeconomic forecasting

Section 10: Practical Applications of VAR Models

• Case Study 1: VAR Model for Macroeconomic Policy Analysis

  • Analyzing the effects of fiscal and monetary policy on output and inflation

• Case Study 2: VAR in Financial Markets

  • Using VAR models to study the relationship between stock returns and interest rates

• Case Study 3: International Economics

  • Analyzing the relationship between exchange rates and international trade using a VAR model

Section 11: Implementation of VAR Models in Statistical Software

• Implementation in R

  • Using the vars package to estimate and analyze VAR models

  • Practical example: Estimating a VAR model in R for economic forecasting

• Implementation in Python

  • Using the statsmodels library to estimate VAR models in Python

  • Example: Building and interpreting a VAR model in Python

• Implementation in Stata

  • Using Stata’s var command for VAR estimation and analysis

  • Step-by-step guide for estimating VAR models in Stata

• Other Software Tools

  • Implementing VAR models in EViews, MATLAB, and SAS

Section 12: Common Pitfalls and Best Practices in VAR Modeling

• Overfitting and Underfitting in VAR Models

  • Avoiding common mistakes in selecting lag lengths and model complexity

• Spurious Causality and Model Specification Errors

  • Ensuring stationarity and correct model specification to avoid spurious results

• Best Practices for VAR Model Estimation and Interpretation

  • Guidelines for accurate estimation and interpretation of VAR models in empirical research

Section 13: Summary and Conclusions

• Recap of Key Concepts in VAR Modeling

  • Summary of the key points discussed in the chapter, including estimation, interpretation, and forecasting with VAR models

• Guidelines for Applying VAR Models in Empirical Research

  • Practical advice for using VAR models in time series econometrics

• Future Directions in VAR Modeling

  • Emerging trends and future research directions in VAR models, including machine learning approaches

Sách

• Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press (Chapters 10, and 11)

• Lütkepohl, H. (2005). New introduction to multiple time series analysis. Springer Science & Business Media. (Chapters 2, and 3) 

• Watson, M. “Vector Autoregressions and Cointegration.” Chapter 47 in Handbook of Econometrics. Vol. 4. North Holland, 1999. ISBN: 9780444887665.

• Wei, W. W. (2006). Time Series Analysis: Univariate and Multivariate Methods  (Chapter 16 and 17)

• Enders, W. (2008). Applied econometric time series. John Wiley & Sons  (Section 5.5-5.13) 

• Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: principles and practice. OTexts. (Chapter 11.2) 

• Shumway, R. H., Stoffer, D. S., & Stoffer, D. S. (2000). Time series analysis and its applications (Vol. 3). New York: springer. (Chapter 5.6) 

• Pesaran, M. H. (2015). Time series and panel data econometrics. Oxford University Press. (Chapter 24)

Bài báo

•  Stock, J. H., & Watson, M. W. (2001). Vector autoregressions. Journal of Economic perspectives, 15(4), 101-115.

• Wright, J. H. (2000). Confidence intervals for univariate impulse responses with a near unit root. Journal of Business & Economic Statistics, 18(3), 368-373.

• Kilian, L. (1998). Small-sample confidence intervals for impulse response functions. Review of economics and statistics, 80(2), 218-230.

• Mikusheva, A. (2012). One‐dimensional inference in autoregressive models with the potential presence of a unit root. Econometrica, 80(1), 173-212.

• Lippi, M., & Reichlin, L. (1994). VAR analysis, nonfundamental representations, Blaschke matrices. Journal of Econometrics, 63(1), 307-325.

• Rudebusch, G. D. (1998). Do measures of monetary policy in a VAR make sense?. International economic review, 907-931.

Nội dung

Section 1: Introduction to Structural VAR Models

• • Difference between standard VAR and Structural VAR (SVAR) models

  • Why we need SVARs to identify economic shocks

  • Examples of economic shocks: monetary policy, supply shocks, demand shocks

    1. 1. 1. 1. • Key Concepts of SVAR Models

                  • Identifying structural relationships between variables

                  • Role of contemporaneous and lagged relationships in SVARs

                  • Example: Identifying monetary policy shocks in a system of GDP, inflation, and interest rates

Section 2: SVAR Models

• Mathematical Representation of SVAR Models

  • Basic SVAR model

  • Structural shocks vs. reduced-form residuals

  • Difference between structural parameters and reduced-form VAR coefficients

• Identifying Restrictions in SVARs

  • The importance of identifying restrictions to solve for structural shocks

  • Short-run vs. long-run restrictions

  • Economic theory and policy insights as a basis for identification

Section 3: Identification Strategies in SVAR Models

• Short-Run Identification Using Cholesky Decomposition

  • Introduction to Cholesky decomposition as an identification method

  • Imposing recursive structure: Ordering variables based on economic theory

  • Example: Identifying monetary policy shocks in a Cholesky-identified SVAR

• Long-Run Identification Using Blanchard-Quah Decomposition

  • Identifying permanent vs. transitory shocks using long-run restrictions

  • Blanchard-Quah (1989) model for distinguishing supply and demand shocks

  • Example: Identifying supply and demand shocks in output and unemployment

• Sign Restrictions for Identifying Structural Shocks

  • Overview of sign restrictions as an alternative identification strategy

  • Imposing economically meaningful signs on impulse responses

  • Example: Identifying shocks in financial markets using sign restrictions

Section 4: Estimation of Structural VAR Models

• Estimating Reduced-Form VAR Models

  • Estimating the reduced-form VAR before applying structural identification

  • OLS estimation of reduced-form VARs

• Estimating Structural Parameters

  • Solving for structural parameters using the identified system of equations

  • Estimating contemporaneous impact matrix (A0A_0A0​)

• Maximum Likelihood Estimation (MLE) for SVAR Models

  • MLE for estimating structural parameters in SVAR models

  • Example: Estimating a monetary SVAR using MLE

Section 5: Impulse Response Functions (IRFs) in SVAR Models

• Impulse Response Functions in SVARs

  • Definition and purpose of IRFs in SVAR models

  • How IRFs show the dynamic response of variables to structural shocks

• Estimating IRFs in SVARs

  • Calculating IRFs after identifying structural shocks

  • Example: Analyzing the response of inflation and output to a monetary policy shock

• Interpreting IRFs in SVAR Models

  • Short-run vs. long-run effects of shocks

  • Interpreting IRFs in policy analysis: What the responses tell us about economic mechanisms

• Confidence Intervals for IRFs

  • Bootstrapping methods for calculating confidence intervals for SVAR IRFs

  • Example: Estimating confidence intervals for an IRF after a fiscal policy shock

Section 6: Variance Decomposition in SVAR Models

• Forecast Error Variance Decomposition (FEVD) in SVARs

  • Understanding the contribution of each shock to the forecast error variance of variables

  • Short-term and long-term variance decomposition

• Estimating and Interpreting Variance Decomposition

  • Calculating variance decomposition from SVAR models

  • Example: Analyzing how much of GDP growth volatility is explained by monetary policy shocks vs. supply shocks

Section 7: Practical Applications of SVAR Models

• SVAR in Macroeconomic Policy Analysis

  • Using SVARs to analyze the effects of monetary policy shocks on inflation and output

  • Case study: Identifying and interpreting monetary policy shocks using an SVAR model

• SVAR for Fiscal Policy Analysis

  • Analyzing the impact of fiscal shocks (e.g., government spending or tax changes) on economic activity

  • Example: Identifying fiscal policy shocks and their effects on GDP and unemployment

• SVAR in International Economics

  • Applying SVAR models to analyze the effects of external shocks (e.g., oil price shocks or exchange rate shocks) on the domestic economy

  • Case study: Identifying external shocks in a small open economy using an SVAR model

• SVAR in Financial Markets

  • Analyzing the impact of financial shocks (e.g., stock market volatility or interest rate changes) on the real economy

  • Example: Identifying and interpreting financial market shocks in a SVAR system

Section 8: Extensions of Structural VAR Models

• Time-Varying Parameter SVAR (TVP-SVAR)

  • Modeling changes in structural relationships over time

  • Application: Analyzing how the effects of monetary policy shocks evolve over time

• SVAR with Heteroskedasticity (SVAR-GARCH)

  • Modeling time-varying volatility in SVAR models using GARCH

  • Example: Identifying structural shocks in a SVAR-GARCH model for financial markets

• Bayesian Structural VAR (B-SVAR)

  • Introducing Bayesian methods to SVAR estimation

  • Incorporating prior information into the identification process

  • Example: Bayesian estimation of structural shocks in a macroeconomic SVAR model

Section 9: Criticisms and Limitations of SVAR Models

• Criticisms of Structural VAR Models

  • Common criticisms: Identification uncertainty, overreliance on theoretical assumptions

  • Sensitivity to the choice of identification strategy: Short-run vs. long-run restrictions

• Limitations in Real-World Applications

  • Limitations in capturing nonlinear dynamics and structural breaks

  • Handling model misspecification and overfitting in SVARs

• Addressing Criticisms: Robustness Checks and Sensitivity Analysis

  • Conducting sensitivity analysis on identification schemes

  • Example: Checking robustness of SVAR results with different lag lengths and variable orderings

Section 10: Implementation of SVAR Models in Statistical Software

• Implementation in R

  • Using the vars and svars packages to estimate SVAR models

  • Practical example: Estimating and interpreting an SVAR model in R

• Implementation in Python

  • Using statsmodels and linearmodels libraries for SVAR estimation in Python

  • Example: Building and estimating an SVAR in Python for financial data

• Implementation in Stata

  • Using Stata’s svar command for structural VAR estimation

  • Step-by-step guide for identifying and estimating SVAR models in Stata

• Other Software Tools

  • Implementing SVAR models in EViews, MATLAB, and SAS

Section 11: Common Pitfalls and Best Practices in SVAR Modeling

• Misinterpretation of Structural Shocks

  • Avoiding incorrect interpretations of structural shocks and impulse responses

• Ensuring Proper Identification

  • Best practices for choosing identification strategies: Short-run vs. long-run restrictions

• Checking for Robustness in SVAR Models

  • Sensitivity testing of identification assumptions and model stability

• 1Guidelines for Applying SVAR Models in Empirical Research

  • Practical tips for applying SVAR models in macroeconomic and financial analysis

Section 12: Summary and Conclusions

• Recap of Key Concepts in SVAR Models

  • Summary of the structural identification process, impulse responses, and variance decomposition in SVAR models

• Practical Guidelines for Applying SVAR Models

  • Best practices for using SVARs in policy analysis and forecasting

• Future Directions in Structural VAR Modeling

  • Emerging trends in SVAR research, including machine learning approaches and non-linear SVAR models

Sách

• Wei, W. W. (2006). Time Series Analysis: Univariate and Multivariate Methods  (Chapter 16 and 17)

Bài báo

• Sims, C. A. (1980). Macroeconomics and reality. Econometrica: journal of the Econometric Society, 1-48.

• Blanchard, O. J., & Quah, D. (1988). The dynamic effects of aggregate demand and supply disturbances.

• Blanchard, O. J. (1989). A traditional interpretation of macroeconomic fluctuations. The American Economic Review, 1146-1164.

• King, R. G., Plosser, C. I., Stock, J. H., & Watson, M. W. (1987). Stochastic trends and economic fluctuations.

• Cooley, T. F., & LeRoy, S. F. (1985). Atheoretical macroeconometrics: A critique. Journal of Monetary Economics, 16(3), 283-308.

• Braun, P. A., & Mittnik, S. (1993). Misspecifications in vector autoregressions and their effects on impulse responses and variance decompositions. Journal of econometrics, 59(3), 319-341.

• Cooley, T. F., & Dwyer, M. (1998). Business cycle analysis without much theory A look at structural VARs. Journal of econometrics, 83(1-2), 57-88.

• Wright, J. H. (2012). What does monetary policy do to long‐term interest rates at the zero lower bound?. The Economic Journal, 122(564), F447-F466.

• Moon, H. R., Schorfheide, F., Granziera, E., & Lee, M. (2011). Inference for VARs identified with sign restrictions (No. w17140). National Bureau of Economic Research.

• Amir‐Ahmadi, P., & Drautzburg, T. (2021). Identification and inference with ranking restrictions. Quantitative Economics, 12(1), 1-39.

• Chari, V. V., Kehoe, P. J., & McGrattan, E. R. (2005). A critique of structural VARs using business cycle theory. Federal Reserve Bank of Minneapolis.

• Christiano, L. J., Eichenbaum, M., Vigfusson, R., Kehoe, P. J., & Watson, M. W. (2006). Assessing structural VARs [with comments and discussion]. NBER macroeconomics annual, 21, 1-105. 

• Erceg, C. J., Guerrieri, L., & Gust, C. (2005). Can long-run restrictions identify technology shocks?. Journal of the European Economic Association, 3(6), 1237-1278.

• Faust, J., & Leeper, E. (1997). Do long run restrictions really identify anything. Journal of Business and Economic Statistics, 15, 345-353.

• Uhlig, H. (2017). Shocks, sign restrictions, and identification. Advances in economics and econometrics, 2, 95.

Nội dung

Section 1: Introduction to Multivariate Volatility Modeling

• • Importance of modeling volatility in multivariate time series

  • Examples of multivariate data: Asset returns, exchange rates, interest rates

• Limitations of Univariate GARCH Models

  • Why modeling only individual series is not sufficient for multivariate systems

  • The need for capturing time-varying covariances and correlations between variables

• Introduction to Multivariate GARCH Models

  • The basic idea of extending GARCH to multiple time series

  • Application in portfolio risk management, asset pricing, and financial stability analysis

Section 2: Multivariate GARCH Models

• Structure of Multivariate GARCH Models

  • Definition and structure of multivariate GARCH models

  • Multivariate conditional variance-covariance matrix: HtH_tHt​

• Conditional Variance and Covariance Dynamics

  • Time-varying volatility and correlation structures in multivariate systems

  • Relationship between conditional covariances and correlations

• Assumptions and Properties of Multivariate GARCH

  • Stationarity conditions for multivariate GARCH models

  • Positive definiteness of the covariance matrix HtH_tHt​

Section 3: Types of Multivariate GARCH Models

• Diagonal VECH Model

  • Definition and structure of the Diagonal VECH model

  • Modeling each element of the conditional variance-covariance matrix

  • Practical issues: Large number of parameters and computational challenges

• BEKK Model (Baba, Engle, Kraft, Kroner)

  • Definition and structure of the BEKK model

  • Advantages of the BEKK model: Reduced parameter space and guaranteed positive definiteness

  • Example: Estimating a BEKK model for a system of stock returns

• Constant Conditional Correlation (CCC) Model

  • Structure and assumptions of the CCC model

  • Assumption of constant conditional correlations across time

  • Application of the CCC model in portfolio optimization

• Dynamic Conditional Correlation (DCC) Model

  • Overview of the DCC model: Time-varying conditional correlations

  • Two-step estimation: Estimating conditional variances and then conditional correlations

  • Example: Using the DCC model to estimate time-varying correlations between asset returns

• Factor Multivariate GARCH Models

  • Structure of factor models: Reducing dimensionality using latent factors

  • Application of factor models in large multivariate systems (e.g., bond yields, stock returns)

Section 4: Estimation of Multivariate GARCH Models

• Maximum Likelihood Estimation (MLE) for Multivariate GARCH

  • The likelihood function for multivariate GARCH models

  • Iterative estimation procedures for maximizing the log-likelihood

• Two-Step Estimation for DCC Models

  • Step 1: Estimating univariate GARCH models for individual series

  • Step 2: Estimating the dynamic conditional correlations

  • Example: Step-by-step estimation of a DCC-GARCH model

• Estimation Challenges and Computational Considerations

  • Dealing with high-dimensional systems and computational constraints

  • Strategies for reducing parameter complexity (e.g., using parsimonious models like BEKK)

Section 5: Model Selection and Diagnostic Checking

• Model Selection Criteria for Multivariate GARCH Models

  • Information criteria (AIC, BIC) for selecting the best model specification

  • Trade-offs between model complexity and goodness of fit

• Diagnostic Checking of Multivariate GARCH Models

  • Residual analysis: Checking for remaining ARCH effects

  • Multivariate Ljung-Box test for autocorrelation in standardized residuals

• Example: Selecting and Diagnosing a Multivariate GARCH Model for Exchange Rates

  • Step-by-step example of model selection and diagnostics for an exchange rate dataset

Section 6: Applications of Multivariate GARCH Models

• Portfolio Risk Management

  • Estimating dynamic covariances for portfolio risk measurement

  • Application of multivariate GARCH in Value-at-Risk (VaR) calculations

  • Example: Using a DCC-GARCH model to estimate the VaR of a stock portfolio

• Asset Pricing and Financial Market Volatility

  • Modeling volatility and correlations in asset returns for pricing derivatives

  • Case study: Estimating time-varying correlations in equity and bond markets

• Macro-Financial Linkages

  • Using multivariate GARCH to analyze volatility spillovers between financial markets and the real economy

  • Example: Modeling volatility spillovers between stock markets and macroeconomic variables

• Financial Stability and Systemic Risk

  • Application of multivariate GARCH models in stress testing and systemic risk analysis

  • Case study: Identifying systemic risk using a BEKK model for global banking systems

Section 7: Forecasting with Multivariate GARCH Models

• Volatility Forecasting Using Multivariate GARCH

  • Step-by-step guide to generating volatility forecasts from multivariate GARCH models

  • Short-term vs. long-term forecasts of variances and covariances

• Forecasting Portfolio Volatility and Risk

  • Using multivariate GARCH forecasts for portfolio risk management and optimization

  • Example: Forecasting portfolio volatility using a DCC-GARCH model

• Forecast Accuracy and Evaluation

  • Evaluating the accuracy of volatility forecasts: Mean Squared Error (MSE), Mean Absolute Error (MAE)

  • Backtesting volatility forecasts in risk management applications

Section 8: Extensions of Multivariate GARCH Models

• Asymmetric Multivariate GARCH Models

  • Modeling asymmetries in volatility: Allowing for different impacts of positive and negative shocks

  • Example: Estimating an asymmetric BEKK model for stock returns

• Multivariate GARCH with Exogenous Variables (GARCH-X)

  • Extending multivariate GARCH models to include exogenous variables (e.g., macroeconomic indicators)

  • Example: Incorporating macroeconomic variables into a multivariate GARCH model for bond yields

• High-Dimensional Multivariate GARCH Models

  • Factor GARCH and other approaches for handling large datasets with many variables

  • Application of dimensionality reduction techniques in large-scale financial systems

• Copula-Based Multivariate GARCH Models

  • Introduction to copulas for modeling complex dependency structures in multivariate systems

  • Example: Using copulas to model non-linear correlations in financial markets

Section 9: Implementation of Multivariate GARCH Models in Statistical Software

• Implementation in R

  • Using the rugarch and rmgarch packages for multivariate GARCH models in R

  • Example: Estimating a DCC-GARCH model in R for financial data

• Implementation in Python

  • Using the arch and statsmodels libraries to estimate multivariate GARCH models in Python

  • Example: Implementing a BEKK-GARCH model in Python for a stock portfolio

• Implementation in Stata

  • Using Stata’s mgarch command for estimating multivariate GARCH models

  • Step-by-step guide to estimating a CCC-GARCH model in Stata

• Other Software Tools

  • Implementing multivariate GARCH models in MATLAB, EViews, and SAS

Section 10: Common Pitfalls and Best Practices in Multivariate GARCH Modeling

• Overfitting and Underfitting in Multivariate GARCH Models

  • Avoiding overparameterization and model complexity issues

  • Choosing the right model structure: BEKK, DCC, CCC, or Factor models

• Ensuring Positive Definiteness

  • Handling challenges related to maintaining the positive definiteness of the covariance matrix

• Best Practices for Multivariate GARCH Modeling

  • Practical tips for estimating, diagnosing, and interpreting multivariate GARCH models in empirical research

Section 11: Summary and Conclusions

• Recap of Key Concepts in Multivariate GARCH Models

  • Summary of the main types of multivariate GARCH models and their applications

• Guidelines for Applying Multivariate GARCH Models in Empirical Research

  • Practical advice for using multivariate GARCH models in risk management, asset pricing, and volatility forecasting

• Future Directions in Multivariate Volatility Modeling

  • Emerging trends and future research directions in multivariate GARCH models, including high-frequency data and machine learning approaches

Sách

• Boffelli, S., & Urga, G. (2016). Financial econometrics using Stata (pp. 3-5). College Station, United States: Stata Press.

• Baum, C. F., & Hurn, S. (2021). Environmental econometrics using Stata. College Station, TX: Stata Press.

Bài báo

•  Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. The review of economics and statistics, 498-505.

• Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. Journal of political Economy, 96(1), 116-131.

Nội dung

Section 1: Introduction to Cointegration

• • Concept of long-run equilibrium between non-stationary variables

  • Difference between spurious regression and cointegration

  • Relevance to long-term relationships in time series

  • Implications for forecasting and policy analysis

• • Importance of confirming unit roots before testing for cointegration

Section 2: The Concept of Cointegration

• • Formal definition of cointegration

  • Long-run relationships versus short-run dynamics

  • Link between cointegration and error correction models

  • Error correction term as an adjustment mechanism

Section 3: Engle-Granger Two-Step Method

• The Engle-Granger Approach to Testing for Cointegration

  • Step-by-step process: Estimation of long-run equation and unit root test on residuals

  • Interpretation of results: Testing residuals for stationarity

• Strengths and Limitations of the Engle-Granger Approach

  • Discussion on the drawbacks of the two-step method

  • Issues with single-equation models

Section 4: Johansen's Approach to Cointegration

• Vector Autoregressive (VAR) Model as a Basis for Johansen’s Test

  • Introduction to VAR models and their role in cointegration analysis

  • Estimating VAR models and identifying lag length

• The Johansen Cointegration Test

  • Likelihood ratio tests for cointegration: Trace test and Max-Eigenvalue test

  • Step-by-step implementation of the Johansen test

  • Choosing the rank of the cointegration matrix

• Economic Interpretation of Johansen’s Test Results

  • How to interpret the cointegration rank

  • Economic and policy implications of cointegrated systems

Section 5: Vector Error Correction Models (VECM)

• The Role of VECM in Modeling Cointegrated Variables

  • Concept of short-run dynamics and long-run equilibrium adjustment

  • VECM as an extension of the VAR model for cointegrated variables

• Specifying a Vector Error Correction Model

  • The error correction term and its role in adjustment

  • Estimation procedure for VECM

  • Identifying lag lengths and interpreting coefficients

• Interpreting VECM Results

  • Adjustment coefficients (speed of adjustment) and their economic meaning

  • Short-run and long-run causal relationships

  • Impulse response functions and variance decompositions in VECM

Section 6: Diagnostics and Model Selection in VECM

• Residual Diagnostics in Cointegration Models

  • Testing for autocorrelation, heteroscedasticity, and normality in residuals

  • Implications of diagnostic failures

• Model Selection Criteria

  • Akaike Information Criterion (AIC), Schwarz Bayesian Criterion (SBC), and likelihood-based criteria

  • Lag selection in the VAR and VECM context

Section 7: Practical Applications of Cointegration and VECM

• Empirical Example: Long-run Relationship Between GDP and Investment

  • Step-by-step empirical example of cointegration analysis and VECM estimation

• Financial Applications: Interest Rates and Exchange Rates

  • Example of cointegration in financial time series

  • Practical use in risk management and forecasting

• Macroeconomic Applications: Money Demand and Inflation

  • Application of VECM in policy analysis and economic forecasting

Section 8: Limitations and Extensions of VECM and Cointegration Analysis

• Challenges with Cointegration Testing and VECM Estimation: 

  • Small sample issues and sensitivity to lag length

• Non-linear Cointegration and Threshold Error Correction Models

  • Introduction to more advanced models for dealing with non-linearity

• Structural Breaks and Cointegration

  • Testing for cointegration in the presence of structural breaks

Section 9: Software Implementation of Cointegration and VECM

• Using Stata for Cointegration and VECM Analysis

  • Commands and procedures for testing cointegration and estimating VECM in Stata

• Implementing Cointegration and VECM in R

  • Overview of relevant packages and step-by-step guide for estimation

• Other Software Options (EViews, MATLAB)

  • Using alternative software tools for cointegration and VECM analysis

Section 10: Conclusion and Further Reading

• Summary of Key Concepts

  • Review of the importance of cointegration and VECM in econometrics

• Suggested Further Reading and Key Papers

  • Foundational papers by Engle and Granger, Johansen, and others

  • Recommended textbooks and articles for deeper understanding

Sách

• Enders, W. (2008). Applied econometric time series. John Wiley & Sons  (Section 6.1-6.6)

• Becketti, S. (2013). Introduction to time series using Stata (Vol. 4905, pp. 176-182). College Station, TX: Stata Press.

• Hamilton, J. D. (2020). Time series analysis. Princeton university press.

• Maddala, G. S. (1998). Unit roots, cointegration, and structural change. Cambridge university press. (link)

Bài giảng tham khảo

•  Vector error correction model, VECM - Cointegrated VAR (link)

• Vector Error Correction (VEC) Models (link)

• n Introduction to Vector Error Correction Models (VECMs) (link)

Bài báo

•  Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of economic dynamics and control, 12(2-3), 231-254.

• 
  

Nội dung

Section 1: Introduction to Distributed Lag Models

• • Definition and importance of lag structures in time series models

  • The economic intuition behind lagged responses in economic variables (e.g., policy effects, price adjustments)

  • Static models vs. dynamic models with lags

  • When and why to use lag structures in regression models

• 
  

Section 2: The Basic Form of ARDL Models

• Types of Distributed Lag Models

  • Finite distributed lag (FDL) models

  • Infinite distributed lag models

• Structure of ARDL Models

  • Mathematical representation of ARDL models: AR(p) + DL(q)

  • Explanation of the autoregressive (AR) and distributed lag (DL) components

  • Interpretation of coefficients on lagged variables

  • How lagged variables capture delayed responses in time series data

• Economic Interpretation of ARDL Models

  • Real-world applications: Monetary policy, fiscal policy, and consumption functions

    1. 1. 1. • Short-run vs. long-run multipliers in ARDL models

             • Interpreting the long-run impact of variables through ARDL estimation

• Estimating ARDL Models: General-to-Specific Approach

  • Estimation strategy for ARDL models using ordinary least squares (OLS)

  • Conditions for consistent and unbiased estimation

Section 3: Model Specification and Estimation

• Specifying an ARDL Model

  • Steps for setting up an ARDL model: Identification of dependent and independent variables

  • The role of lagged dependent and independent variables in capturing dynamics

  • Criteria for selecting lag length: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Hannan-Quinn (HQ)

  • Practical examples of lag length selection in ARDL models

• Estimation of ARDL Models Using OLS

  • Practical approach to estimating ARDL models using OLS

  • Dealing with endogeneity in ARDL estimation

• Dealing with Multicollinearity and Autocorrelation

  • Diagnosing multicollinearity and autocorrelation in ARDL models

  • Solutions for correcting autocorrelation (e.g., Newey-West standard errors)

Section 4: Short-run and Long-run Dynamics in ARDL Models

• Short-run Coefficients

  • Understanding short-run dynamics and their significance in ARDL models

  • Interpretation of short-run coefficients and their relation to economic theory

• Long-run Coefficients and the Long-run Equilibrium

  • How to derive and interpret long-run coefficients in ARDL models

  • Calculating the long-run equilibrium relationships between variables

• Adjustment Mechanism: Error Correction Representation of ARDL Models

  • How ARDL models incorporate error correction terms (ECT)

  • Interpretation of the speed of adjustment parameter and its role in reverting to the long-run equilibrium

Section 5: Bounds Testing for Cointegration in ARDL Models

• Introduction to Bounds Testing

  • Explanation of the bounds testing approach developed by Pesaran, Shin, and Smith (2001)

  • Why bounds testing is used in ARDL models for cointegration

• Steps in Conducting a Bounds Test

  • Estimation of an unrestricted error correction model (UECM)

  • Testing for the existence of a long-run relationship using F-statistics

• Interpreting Bounds Test Results

  • Decision rules based on critical values (lower bound and upper bound)

  • How to conclude cointegration from the bounds test

Section 6: Diagnostics and Model Stability

• Diagnostic Tests for ARDL Models

  • Residual diagnostics: Testing for normality, serial correlation, and heteroscedasticity

  • Stability tests: Cumulative sum of recursive residuals (CUSUM) and CUSUM of squares

• Model Misspecification

  • Dealing with omitted variables, wrong lag length, and multicollinearity

  • Practical ways to handle model misspecification in ARDL models

Section 7: Extensions and Variants of ARDL Models

• Panel ARDL Models

  • Explanation of ARDL models for panel data

  • Differences between time series ARDL and panel ARDL models

• Nonlinear ARDL (NARDL) Models

  • Introduction to nonlinear ARDL models and asymmetric adjustment

  • Practical applications of NARDL in economics and finance

• Structural Breaks and ARDL Models

  • Incorporating structural breaks in ARDL models using dummy variables or break tests

  • Practical implications of structural breaks for long-run relationships

Section 8: Practical Applications of ARDL Models

• Empirical Example 1: The Phillips Curve

  • ARDL analysis of inflation and unemployment relationship over time

  • Estimation of short-run and long-run effects of inflationary expectations

• Empirical Example 2: Demand for Money

  • Application of ARDL to analyze the long-run demand for money function

  • Interpretation of short-run and long-run coefficients in the money demand context

• Empirical Example 3: Energy Consumption and Economic Growth

  • Using ARDL to examine the long-run and short-run relationship between energy consumption and economic growth

  • Policy implications derived from the results

Section 9: Software Implementation of ARDL Models

• Using Stata for ARDL Estimation

  • Commands and procedures for specifying and estimating ARDL models in Stata

• Implementing ARDL Models in EViews

  • Step-by-step guide to estimating ARDL models using EViews

• ARDL Estimation in R and MATLAB

  • Overview of relevant packages and commands for ARDL in R and MATLAB

Section 10: Conclusion and Further Reading

• Summary of Key Concepts

  • Recap of ARDL models' role in capturing short-run and long-run dynamics

  • Importance of bounds testing for cointegration in mixed-order time series

• Further Reading

  • Foundational papers and textbooks on ARDL models

  • Suggested reading for advanced applications of ARDL

Sách

•  Enders, W. (2008). Applied econometric time series. John Wiley & Sons  (Section 6.1-6.6)

• Becketti, S. (2013). Introduction to time series using Stata (Vol. 4905, pp. 176-182). College Station, TX: Stata Press.

• Hamilton, J. D. (2020). Time series analysis. Princeton university press.

• Maddala, G. S. (1998). Unit roots, cointegration, and structural change. Cambridge university press. (link)

Bài báo

•  Kuma, B., & Gata, G. (2023). Factors affecting food price inflation in Ethiopia: An autoregressive distributed lag approach. Journal of Agriculture and Food Research, 12, 100548. 

• Hurley, D. T., & Papanikolaou, N. (2021). Autoregressive distributed lag (ARDL) analysis of US-China commodity trade dynamics. The Quarterly Review of Economics and Finance, 81, 454-467.

Nội dung

Section 1: Introduction to State Space Models

• • Definition and key concepts of state space models

  • The importance of state space models in time series analysis

  • Comparison with other time series models (ARIMA, VAR, etc.)

  • Common applications in economics and finance: Modeling unobserved components, dynamic factor models, and structural time series models

Section 2: Mathematical Representation of State Space Models

• The General Form of State Space Models

  • State equation: First-order Markov process for the unobserved state

  • Observation equation: Linear function of the state variable

  • Noise processes: Disturbance terms and their properties

• Time-Varying Parameters and State Space Models

  • How state space models accommodate time-varying coefficients

  • Applications in modeling macroeconomic variables (e.g., time-varying inflation or interest rates)

• Linear vs. Nonlinear State Space Models

  • Linear state space models (e.g., for ARMA processes)

  • Nonlinear models: Introduction to extended state space models

Section 3: The Kalman Filter and Smoother

• The Kalman Filter Algorithm

  • Purpose of the Kalman filter: Real-time updating of state estimates

  • Step-by-step derivation of the Kalman filter equations: Prediction and updating

  • Kalman filter as a recursive estimator

• Properties of the Kalman Filter

  • Optimality of the Kalman filter under Gaussian assumptions

  • Dealing with non-stationary data and non-Gaussian noise

• Kalman Smoother

  • Introduction to smoothing: Estimating past states given future information

  • Forward and backward recursions in the Kalman smoother

• Implementation of the Kalman Filter in Econometrics

  • Common applications of the Kalman filter in macroeconomics and finance

  • Estimation of potential output, inflation trends, and financial market volatility

Section 4: Estimation of State Space Models

• Maximum Likelihood Estimation (MLE)

  • How MLE is used to estimate the parameters of state space models

  • Likelihood function for state space models

  • Numerical optimization for MLE: Expectation-Maximization (EM) algorithm

• Bayesian Estimation

  • Bayesian inference and state space models

  • Using Markov Chain Monte Carlo (MCMC) for parameter estimation

• Dealing with Missing Data

  • Handling missing observations in state space models

  • Kalman filter’s ability to handle missing data and measurement error

Section 5: Dynamic Linear Models (DLM) as State Space Models

• Definition of Dynamic Linear Models

  • Overview of DLMs as a special case of state space models

  • Specification of DLMs: Time-varying regression coefficients

• Estimation of DLMs Using Kalman Filter

  • Implementing Kalman filtering and smoothing in DLMs

  • Forecasting with DLMs

• Applications of DLMs in Macroeconomic and Financial Models

  • Modeling the dynamic behavior of GDP, inflation, and interest rates

  • Time-varying betas in asset pricing models

Section 6: Unobserved Components Models (UCM)

• Definition and Structure of UCMs

  • UCMs as a type of state space model: Breaking down time series into trend, cycle, and seasonal components

• Estimation of UCMs

  • Applying the Kalman filter and smoother for UCM estimation

  • Trend-cycle decomposition: Estimating potential GDP or output gaps

• Applications of UCMs in Economic Analysis

  • Analyzing unobserved economic factors like potential growth, inflation gaps, and labor market slack

  • UCMs in business cycle analysis

Section 7: Time-Varying Parameter Models

• Time-Varying Coefficients in State Space Models

  • Introduction to time-varying parameter (TVP) models

  • Examples of TVP models in macroeconomics: Modeling inflation with time-varying coefficients

• Estimation of TVP Models Using Kalman Filter

  • Recursive updating of time-varying parameters

  • Applications: Monetary policy rules, interest rate models, and financial risk modeling

• Comparing TVP Models with Fixed-Parameter Models

  • Advantages of time-varying parameters in capturing economic shocks and regime changes

Section 8: Structural Time Series Models (STSM)

• Definition and Structure of Structural Time Series Models

  • How structural models decompose time series into components (trend, cycle, seasonal)

• Estimating Structural Models Using State Space Representation

  • Specifying and estimating the structural model through the Kalman filter

• Applications of Structural Models

  • Estimating the natural rate of unemployment (NAIRU), potential output, and structural inflation

Section 9: Nonlinear State Space Models and Particle Filtering

• Introduction to Nonlinear State Space Models

  • Nonlinear extensions of state space models: Modeling regimes and breaks

• The Particle Filter Algorithm

  • Overview of particle filtering for nonlinear/non-Gaussian state space models

  • Step-by-step explanation of particle filtering

• Applications of Nonlinear State Space Models

  • Nonlinear models in finance: Asset pricing with regime shifts

  • Nonlinear dynamic factor models in macroeconomics

Section 10: Model Diagnostics and Validation

• Model Diagnostics for State Space Models

  • Residual analysis: Autocorrelation, heteroscedasticity, and normality tests

  • Assessing model fit: Likelihood-based criteria (AIC, BIC)

• Testing for Structural Breaks and Regime Changes

  • Detecting breaks in the state space: Time-varying parameters and regime-switching models

• Stability and Robustness of State Space Models

  • Assessing parameter stability and robustness in time series analysis

Section 11: Forecasting Using State Space Models

• Forecasting with State Space Models

  • How state space models generate dynamic forecasts for unobserved states

• Forecasting Performance Evaluation

  • Techniques for evaluating forecasting accuracy: Mean Squared Error (MSE), Root Mean Squared Error (RMSE)

• Practical Forecasting Applications

  • Forecasting inflation, GDP, and other macroeconomic variables using state space models

Section 12: Software Implementation of State Space Models

• Using Stata for State Space Models

  • Commands and procedures for estimating state space models in Stata

• Implementing State Space Models in R

  • Overview of R packages (e.g., KFAS, dlm) and step-by-step guide

• State Space Models in MATLAB and EViews

  • Implementing state space estimation and Kalman filter in MATLAB and EViews

Section 13: Practical Applications and Case Studies

• Estimating Potential Output and the Output Gap

  • Case study: Using UCM to estimate potential GDP and the output gap in macroeconomic policy

• Dynamic Factor Models in Financial Markets

  • Case study: Estimating latent factors in stock markets using state space models

• Estimating Time-Varying Betas in Asset Pricing Models

  • Practical example: Time-varying beta estimation in the CAPM framework

Section 14: Conclusion and Further Reading

• Summary of Key Concepts

  • Recap of the main features and applications of state space models

  • Importance of the Kalman filter and its role in econometric modeling

• Further Reading and Key Papers

  • Seminal papers and books on state space models, Kalman filtering, and unobserved components

  • Suggested further reading for advanced topics in nonlinear state space modeling

Sách

•  Wei, W. W. (2006). Time Series Analysis: Univariate and Multivariate Methods (Chapter 18)

• Brockwell, P. J., & Davis, R. A. (1991). Time series: theory and methods. Springer science & business media.

• Hamilton, J. D. (2020). Time series analysis. Princeton university press.

• Lütkepohl, H. (2005). New introduction to multiple time series analysis. Springer Science & Business Media. (link)

Bài giảng tham khảo

•  State Space Models (link)

Bài báo

Nội dung

Section 1: Introduction to Structural Breaks

• • What is a structural break in time series?

  • Why structural breaks matter in econometrics: Economic crises, policy changes, and regime shifts

  • Real-world examples: Oil price shocks, financial crises, exchange rate regime shifts, and policy interventions

    1. 1. 1. • Consequences of Ignoring Structural Breaks

               • Bias and inconsistency in parameter estimates

               • Spurious relationships and model misspecification

               • Impact on forecasting accuracy

Section 2: Theoretical Foundations of Structural Breaks

• 2.2 Types of Structural Breaks

  • Single break vs. multiple breaks

  • Break in mean (level shift) and break in variance (volatility shift)

  • Gradual vs. abrupt breaks

  • Breaks in trend vs. breaks in intercept

• Visual Inspection of Structural Breaks

  • Plotting time series data: Identifying potential breaks visually

  • Understanding the limitations of visual inspection

• The Chow Test

  • Overview of the Chow test for a single known break

  • Step-by-step procedure: Estimating the model with and without the break

  • Strengths and limitations of the Chow test (e.g., assumption of a known break date)

• Tests for Multiple Structural Breaks

  • Issues with testing for multiple breaks

  • Overview of tests for multiple breaks (Bai-Perron methodology)

• The Quandt-Andrews Test

  • Testing for unknown breakpoints within a time series

  • Procedure for calculating the Sup-Wald statistic

Section 4: Formal Tests for Structural Breaks

• Bai-Perron Multiple Structural Break Test

  • Introduction to the Bai-Perron method for multiple breaks

  • Estimating the number and location of breaks

  • Breakpoint estimation and confidence intervals

  • Practical applications and interpretation of results

• The Zivot-Andrews Test for Unit Roots with Structural Breaks

  • Testing for unit roots in the presence of structural breaks

  • Zivot-Andrews test procedure: Testing for a single break in the series

  • Interpretation of the test results: Null and alternative hypotheses

• The Perron Test for Unit Roots with Structural Breaks

  • Comparison with the Zivot-Andrews test

  • Perron’s approach to endogenous breaks: Breaks in trend or intercept

• Tests for Structural Breaks in Cointegrated Systems

  • Gregory-Hansen test for cointegration with regime shifts

  • Johansen cointegration tests with structural breaks

Section 5: Estimation Methods in Models with Structural Breaks

• Estimating Structural Breaks in Regression Models

  • Including break dummies in regression models

  • Estimating pre- and post-break relationships

  • Accounting for endogeneity and heteroscedasticity in break models

• Threshold Models and Regime-Switching Models

  • Introduction to threshold autoregressive (TAR) models

  • Estimation of regime-switching models: Markov switching models and smooth transition regression models

• Dynamic Models with Time-Varying Parameters

  • Structural breaks in ARDL and VAR models

  • Time-varying coefficient models as an alternative to discrete breaks

Section 6: Implications of Structural Breaks for Forecasting

• Impact of Structural Breaks on Forecast Accuracy

  • How structural breaks affect forecasting models

  • The cost of ignoring breaks: Poor predictive performance

• Incorporating Breaks into Forecasting Models

  • Accounting for breaks in ARIMA and VAR forecasts

  • Estimating models with regime-switching for improved forecasts

• Forecasting in the Presence of Breaks: Empirical Examples

  • Forecasting GDP during periods of economic shocks

  • Practical examples from financial markets (e.g., stock prices and volatility)

Section 7: Structural Breaks in Financial Time Series

• tructural Breaks and Volatility Shifts in Financial Markets

  • Breaks in volatility: GARCH models and structural breaks

  • Estimating structural breaks in financial returns

• Structural Breaks in Asset Pricing Models

  • How breaks influence CAPM, Fama-French models, and factor models

  • Time-varying betas and risk premiums

• Applications in Risk Management

  • Accounting for structural breaks in risk assessment

  • Practical examples in portfolio management and option pricing

Section 8: Structural Breaks in Macroeconomic Time Series

• Structural Breaks in Economic Growth

  • Identifying breaks in GDP and economic cycles

  • Structural breaks during periods of economic crisis or reforms

• Breaks in Monetary and Fiscal Policy

  • Analyzing changes in central bank behavior (e.g., policy rate regimes)

  • Fiscal policy shifts: Changes in government spending and taxation

• International Applications: Breaks in Trade and Exchange Rates

  • Structural breaks in exchange rate regimes (fixed vs. floating)

  • Trade liberalization and its impact on time series data

Section 9: Structural Breaks and Nonlinear Time Series Models

• Introduction to Nonlinear Models with Structural Breaks

  • Nonlinear models as an extension to models with discrete breaks

• Markov-Switching Models

  • Regime shifts in time series with unobserved switching

  • Estimation and interpretation of Markov switching models

• Smooth Transition Regression Models (STR)

  • Modeling gradual shifts with STR models

  • Application in macroeconomic and financial time series

Section 10: Practical Implementation of Structural Break Tests

•  Implementing Structural Break Tests in Stata

  • Commands and procedures for performing Chow test, Bai-Perron test, Zivot-Andrews test in Stata

• Using R for Structural Break Analysis

  • R packages and functions for detecting and estimating structural breaks

• Structural Breaks in EViews and MATLAB

  • Step-by-step guide to performing structural break tests in EViews and MATLAB

Section 11: Case Studies and Empirical Applications

• Case Study 1: Structural Breaks in Exchange Rate Volatility

  • Practical example using exchange rate data

  • Identifying multiple breaks during financial crises

• Case Study 2: Structural Breaks in Inflation Dynamics

  • Using the Bai-Perron test to detect breaks in inflation trends

  • Implications for monetary policy and inflation targeting

• Case Study 3: Structural Breaks in Stock Market Returns

  • Identifying regime shifts in stock price data

  • Application of regime-switching models for asset pricing

Section 12: Challenges and Limitations in Structural Break Analysis

• Small Sample Issues in Structural Break Tests

  • Power and size problems in small samples

  • Remedies for small sample bias in structural break tests

• Multiple Breaks and Model Complexity

  • Overfitting and model selection issues with multiple breaks

  • Bayesian methods and model averaging for structural break models

• Dealing with Gradual Structural Change

  • Identifying gradual shifts vs. abrupt breaks

  • Estimating models with gradual structural change (e.g., STR models)

Section 13: Conclusion and Further Reading

• Summary of Key Concepts

  • Recap of structural break theory and methods in time series analysis

  • Importance of considering breaks in econometric models

• Suggested Further Reading

  • Key papers and foundational research on structural breaks

  • Recommended textbooks and articles for deeper exploration

Sách

• Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press (Chapter 22).

Bài báo

•  Andrews, D. W. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica: Journal of the Econometric Society, 821-856.

• Hansen, B. E. (2001). The new econometrics of structural change: Dating breaks in US labor productivity. Journal of Economic perspectives, 15(4), 117-128.

• Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica: journal of the Econometric Society, 1361-1401.

• Andrews, D. W., & Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica: Journal of the Econometric Society, 1383-1414.

• Bai, J. (1997). Estimating multiple breaks one at a time. Econometric theory, 13(3), 315-352.

• Bai, J. (1997). Estimation of a change point in multiple regression models. Review of Economics and Statistics, 79(4), 551-563.

• Bai, J., & Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 47-78.

• Bai, J., Lumsdaine, R. L., & Stock, J. H. (1998). Testing for and dating common breaks in multivariate time series. The Review of Economic Studies, 65(3), 395-432.

• Zivot, E., & Andrews, D. W. K. (2002). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of business & economic statistics, 20(1), 25-44.

Nội dung

Section 1: Introduction to Thresholds in Time Series

• • 1. 1. 1. • Definition and Concept of Threshold Autoregressive (TAR) Models

  • The concept of thresholds in time series.

  • The Role of Thresholds in Time Series Models

  • Economic intuition behind the threshold: Interpreting the threshold variable.

  • Relationship between regimes and different economic conditions.

  • Economic applications: Business cycles, exchange rate regimes, stock market bubbles, and interest rate changes.

Section 2:  TAR Models

• General Form of Threshold Autoregressive Models

  • Mathematical representation of TAR models.

  • The notion of regime switching based on the threshold variable.

• Single-Threshold vs. Multiple-Threshold Models

  • One threshold vs. multiple thresholds: TAR vs. SETAR models.

  • The flexibility of multiple-regime models in capturing complex dynamics.

Section 3: Stationarity and Stability of TAR Models

• Stationarity in Threshold Autoregressive Models

  • Conditions for stationarity in nonlinear models.

  • How thresholds affect the stationarity properties of the series.

• Stability Conditions for TAR Models

  • Stability analysis for regime-switching models.

  • The role of autoregressive coefficients in each regime for ensuring model stability.

Section 4: Model Specification and Identification

• Specifying the TAR Model

  • Steps to specify a TAR model: Choosing the threshold variable and lag structure.

  • Key choices in TAR models: Threshold variable, number of regimes, and lag orders.

• Identifying the Threshold and Regime-Switching Mechanism

  • How to determine the threshold level(s).

  • Identifying breakpoints and regime shifts.

• Testing for Threshold Effects

  • Introduction to statistical tests for threshold nonlinearity.

  • The Sup-LR test for the presence of threshold effects.

  • Alternative tests for threshold effects (e.g., Hansen’s test).

Section 5: Estimation of TAR Models

• Estimation via Conditional Least Squares (CLS)

  • Step-by-step procedure for estimating TAR models using CLS.

  • Interpretation of regime-specific parameter estimates.

• Maximum Likelihood Estimation (MLE)

  • Introduction to MLE for TAR models.

  • Estimation challenges: Numerical optimization and convergence.

• Estimation of the Threshold Parameter

  • Grid search methods for estimating the threshold.

  • Confidence intervals and inference for the estimated threshold.

• Estimating Multiple Thresholds in SETAR Models

  • Extension to models with more than one threshold (SETAR).

  • Identifying and estimating multiple regimes.

Section 6: Diagnostics and Model Selection

• Residual Diagnostics for TAR Models

  • Checking for serial correlation and heteroscedasticity in the residuals.

  • Graphical diagnostic tools for TAR models.

• Model Selection Criteria for TAR Models

  • Choosing the number of regimes: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and likelihood-ratio tests.

  • Trade-offs in model complexity: Overfitting vs. underfitting in nonlinear models.

• Testing for the Number of Thresholds

  • Sequential testing approach for determining the appropriate number of regimes.

  • Comparison of models with different threshold structures.

Section 7: Forecasting with TAR Models

• Forecasting in a Threshold Context

  • How regime-switching affects forecasting in TAR models.

  • Practical strategies for forecasting across regimes.

• Forecasting Performance Evaluation

  • Comparing forecast accuracy of TAR models vs. linear models.

  • Out-of-sample forecasting performance: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and other criteria.

• Empirical Example of Forecasting

  • Forecasting macroeconomic indicators using TAR models.

  • Practical applications in exchange rate forecasting and stock market prediction.

Section 8: Applications of TAR Models in Economics and Finance

• TAR Models in Macroeconomic Analysis

  • Application of TAR models to study business cycle dynamics.

  • Modeling asymmetric responses in monetary policy or fiscal shocks.

• TAR Models in Financial Markets

  • Using TAR models to capture nonlinearities in stock returns and volatility.

  • Threshold effects in bond yields and interest rates.

• TAR Models in International Economics

  • Applications to exchange rate dynamics and trade balances.

  • Thresholds in capital flows and foreign direct investment.

Section 9: Structural Breaks and Threshold Effects

• Structural Breaks vs. Threshold Effects

  • Differences between structural breaks and threshold nonlinearity.

  • Testing for structural breaks within a TAR framework.

• Combining TAR Models with Structural Breaks

  • Estimating TAR models in the presence of structural breaks.

  • Practical challenges in separating breaks from nonlinear dynamics.

Section 10:  Self-Exciting Threshold Autoregressive (SETAR) Models

• Definition

• Key Features:

• Applications: Business cycle analysis, stock market dynamics, exchange rate modeling.

Section 11:  Smooth Transition Autoregressive (STAR) Models

• • • • • • Definition

          • Key Features

            • Logistic STAR (LSTAR)

            • Exponential STAR (ESTAR)

          • Applications: Modeling gradual adjustments in economic variables, such as inflation or interest rates.

Section 12: Threshold Vector Autoregressive (TVAR) Models

• Definition

• Key Features:

  • Thresholds can affect multiple variables simultaneously.

  • Vector autoregressive processes within each regime.

  • Useful for capturing regime-dependent relationships in multivariate systems.

• Applications

Section 13: Threshold Error Correction Models (TECM)

• • • • • • Definition

          • Key Features:

            • Nonlinear error correction mechanism.

            • Threshold-based adjustments towards equilibrium.

            • Common in cointegrated time series with threshold effects.

          • Applications

Section 14: Markov-Switching Autoregressive (MSAR) Models

• Definition

• Key Features:

  • Regimes switch based on probabilities rather than a threshold.

  • Suitable for capturing hidden states and regime shifts driven by unobserved factors.

• Applications

Section 15: . Threshold Generalized Autoregressive Conditional Heteroscedasticity (T-GARCH) Models

• Definition

• Key Features

  • Conditional variance depends on threshold levels.

  • Allows for different volatility regimes depending on the threshold variable.

• Applications

Section 16: Regime-Switching Smooth Transition Autoregressive (RS-STAR) Models

• Definition

• Key Features:

  • Smooth transitions between regimes with probabilistic switching.

  • Captures complex dynamics with both abrupt and smooth regime changes.

• Applications

Section 17: Exogenous Threshold Autoregressive (XTAR) Models

• Definition

• Key Features:

  • Threshold depends on an external variable (e.g., interest rates, inflation, policy changes).

  • Captures the influence of external shocks or policies on the dynamics of the system.

• Applications

Section 18: . Threshold Moving Average (TMA) Models

• Definition: An extension of TAR models to include moving average components, allowing past forecast errors to affect the current regime.

• Key Features:

  • Incorporates past innovations (errors) in regime determination.

  • Useful for modeling systems with nonlinear dynamics in the residuals or shocks.

• Applications

Section 19: Nonlinear ARDL (NARDL) Models

• Definition

• Key Features:

  • Asymmetric dynamics in response to positive and negative changes in the threshold variable.

  • Can capture both short-term and long-term nonlinear effects.

• Applications

Section 20: Piecewise Linear Threshold Models

• Definition

• Key Features:

  • The model switches between different linear regimes.

  • Captures changes in behavior at different thresholds, but each regime is linear within itself.

• Applications

Section 21: Threshold Cointegration Models

• Definition

• Key Features:

  • Nonlinear adjustment towards equilibrium depending on the magnitude of the deviation.

  • Useful for studying asymmetric error correction in cointegrated systems.

• Applications

Section 22: . Time-Varying Threshold Models

• Definition

• Key Features:

  • The threshold is not constant and can depend on external or temporal factors.

  • Allows for more flexibility in capturing evolving nonlinear dynamics.

• Applications

Section 23: Software Implementation of TAR Models

• Implementing TAR Models in Stata

  • Commands and procedures for estimating TAR models in Stata.

• Using R for TAR Model Estimation

  • Overview of relevant packages (e.g., TAR package) and implementation steps in R.

• TAR Models in EViews and MATLAB

  • Step-by-step guide to estimating TAR models using EViews and MATLAB.

Section 24: Conclusion and Further Reading

• Summary of Key Concepts

  • Recap of the importance of TAR models in capturing nonlinear time series dynamics.

  • Review of key estimation techniques and empirical applications.

• Suggested Further Reading

  • Foundational papers and advanced readings on TAR and other nonlinear models.

  • Recommended textbooks and articles for deeper exploration.

Sách

• Baum, C. F., & Hurn, S. (2021). Environmental econometrics using Stata. College Station, TX: Stata Press.

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