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Dynamic Models, Autocorrelation and Forecasting. Chapter 9. Prepared by Vera Tabakova, East Carolina University. Chapter 9: Dynamic Models, Autocorrelation and Forecasting. 9.1 Introduction 9.2 Lags in the Error Term: Autocorrelation 9.3 Estimating an AR(1) Error Model

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Chapter 9

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Dynamic Models, Autocorrelation and Forecasting

## Chapter 9

Prepared by Vera Tabakova, East Carolina University

### Chapter 9: Dynamic Models, Autocorrelation and Forecasting

• 9.1 Introduction

• 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

• Principles of Econometrics, 3rd Edition

### 9.1 Introduction

Figure 9.1

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

### 9.4Testing for Autocorrelation

9.4.1Residual Correlogram

Principles of Econometrics, 3rd Edition

### 9.4Testing for Autocorrelation

9.4.1Residual Correlogram

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

### 9.7Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

### 9.7Autoregressive Distributed Lag Models

Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model

Principles of Econometrics, 3rd Edition

### 9.7Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

### 9.7Autoregressive Distributed Lag Models

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

Principles of Econometrics, 3rd Edition

### 9.7Autoregressive Distributed Lag Models

Principles of Econometrics, 3rd Edition

### 9.7Autoregressive 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 The Durbin-Watson Test

Figure 9A.1:

Principles of Econometrics, 3rd Edition

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

Figure 9A.2:

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