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### Chapter 9

Dynamic Models, Autocorrelation and Forecasting

Prepared by Vera Tabakova, East Carolina University

Chapter 9: Dynamic Models, Autocorrelation and Forecasting 9.2 Lags in the Error Term: Autocorrelation 9.3 Estimating an AR(1) Error Model 9.4 Testing for Autocorrelation 9.5 An Introduction to Forecasting: Autoregressive Models 9.6 Finite Distributed Lags 9.7 Autoregressive Distributed Lag Models

- 9.1 Introduction

Principles of Econometrics, 3rd Edition

9.1 Introduction

Principles of Econometrics, 3rd Edition

9.1 Introduction

Figure 9.2(a) Time Series of a Stationary Variable

Principles of Econometrics, 3rd Edition

9.1 Introduction

Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’

Principles of Econometrics, 3rd Edition

9.1 Introduction

Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’

Principles of Econometrics, 3rd Edition

9.2 Lags in the Error Term: Autocorrelation

9.2.1 Area Response Model for Sugar Cane

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Figure 9.3 Least Squares Residuals Plotted Against Time

Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors

Principles of Econometrics, 3rd Edition

9.3 Estimating an AR(1) Error Model

The existence of AR(1) errors implies:

- The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance.
- The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading.

Principles of Econometrics, 3rd Edition

9.3 Estimating an AR(1) Error Model

Sugar cane example

The two sets of standard errors, along with the estimated equation are:

The 95% confidence intervals for β2 are:

Principles of Econometrics, 3rd Edition

9.3.2 Nonlinear Least Squares Estimation

Principles of Econometrics, 3rd Edition

9.3.2 Nonlinear Least Squares Estimation

Principles of Econometrics, 3rd Edition

9.3.2a Generalized Least Squares Estimation

It can be shown that nonlinear least squares estimation of (9.24) is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure. Details are provided in Appendix 9A.

Principles of Econometrics, 3rd Edition

9.3.3 Estimating a More General Model

Principles of Econometrics, 3rd Edition

9.4.1 Residual Correlogram

Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example

Principles of Econometrics, 3rd Edition

9.4.1 Residual Correlogram

Principles of Econometrics, 3rd Edition

9.4.1 Residual Correlogram

Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example

Principles of Econometrics, 3rd Edition

9.4.2 A Lagrange Multiplier Test

Principles of Econometrics, 3rd Edition

9.4.2 A Lagrange Multiplier Test

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.5 An Introduction to Forecasting: Autoregressive Models

Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags

Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags

Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models

Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model

Principles of Econometrics, 3rd Edition

Keywords

- autocorrelation
- autoregressive distributed lag models
- autoregressive error
- autoregressive model
- correlogram
- delay multiplier
- distributed lag weight
- dynamic models
- finite distributed lag
- forecast error
- forecasting
- HAC standard errors
- impact multiplier
- infinite distributed lag
- interim multiplier
- lag length
- lagged dependent variable
- LM test
- nonlinear least squares
- sample autocorrelation function
- standard error of forecast error
- total multiplier
- form of LM test

Principles of Econometrics, 3rd Edition

Chapter 9 Appendices

- Appendix 9A Generalized Least Squares Estimation
- Appendix 9B The Durbin Watson Test
- Appendix 9C Deriving ARDL Lag Weights
- Appendix 9D Forecasting: Exponential Smoothing

Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation

Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation

Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation

Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation

Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test

Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test

Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test

Principles of Econometrics, 3rd Edition

Appendix 9B 9B.1 The Durbin-Watson Bounds Test

The Durbin-Watson bounds test.

- if the test is inconclusive.

Principles of Econometrics, 3rd Edition

Appendix 9C Deriving ARDL Lag Weights

Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag

Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag

Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag

Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag

Principles of Econometrics, 3rd Edition

Appendix 9C 9C.2 Lag Weights for More General ARDL Models

Principles of Econometrics, 3rd Edition

Appendix 9D Forecasting: Exponential Smoothing

Principles of Econometrics, 3rd Edition

Appendix 9D Forecasting: Exponential Smoothing

Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α

Principles of Econometrics, 3rd Edition

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