Dynamic Models, Autocorrelation and Forecasting

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Chapter 9

• Dynamic Models, Autocorrelation and Forecasting

Prepared by Vera Tabakova, East Carolina
University
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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

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9.1 Introduction

• Figure 9.1

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9.1 Introduction

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9.1 Introduction

• Figure 9.2(a) Time Series of a Stationary
  Variable

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9.1 Introduction

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

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9.1 Introduction

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

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9.2 Lags in the Error Term Autocorrelation

• 9.2.1 Area Response Model for Sugar Cane

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9.2.2 First-Order Autoregressive Errors

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9.2.2 First-Order Autoregressive Errors

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9.2.2 First-Order Autoregressive Errors

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9.2.2 First-Order Autoregressive Errors
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9.2.2 First-Order Autoregressive Errors

• Figure 9.3 Least Squares Residuals Plotted
  Against Time

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9.2.2 First-Order Autoregressive Errors

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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.

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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 ß2 are

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9.3.2 Nonlinear Least Squares Estimation

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9.3.2 Nonlinear Least Squares Estimation

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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.

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9.3.3 Estimating a More General Model

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9.4 Testing for Autocorrelation

• 9.4.1 Residual Correlogram

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9.4 Testing for Autocorrelation

• 9.4.1 Residual Correlogram

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9.4.1 Residual Correlogram

• Figure 9.4 Correlogram for Least Squares
  Residuals from Sugar Cane Example

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9.4.1 Residual Correlogram

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9.4.1 Residual Correlogram

• Figure 9.5 Correlogram for Nonlinear Least
  Squares Residualsfrom Sugar Cane Example

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9.4.2 A Lagrange Multiplier Test

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9.4.2 A Lagrange Multiplier Test

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.5 An Introduction to Forecasting
Autoregressive Models

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

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.5 An Introduction to Forecasting
Autoregressive Models

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9.6 Finite Distributed Lags

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9.6 Finite Distributed Lags

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9.6 Finite Distributed Lags

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9.7 Autoregressive Distributed Lag Models

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9.7 Autoregressive Distributed Lag Models

• Figure 9.7 Correlogram for Least Squares
  Residuals fromFinite Distributed Lag Model

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9.7 Autoregressive Distributed Lag Models

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9.7 Autoregressive Distributed Lag Models

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

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9.7 Autoregressive Distributed Lag Models

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9.7 Autoregressive Distributed Lag Models

• Figure 9.9 Distributed Lag Weights for
  Autoregressive Distributed Lag Model

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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

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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

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Appendix 9A Generalized Least Squares Estimation
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Appendix 9A Generalized Least Squares Estimation
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Appendix 9A Generalized Least Squares Estimation
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Appendix 9A Generalized Least Squares Estimation
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Appendix 9B The Durbin-Watson Test
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Appendix 9B The Durbin-Watson Test
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Appendix 9B The Durbin-Watson Test
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Appendix 9B The Durbin-Watson Test

• Figure 9A.1

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

• Figure 9A.2

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

• The Durbin-Watson bounds test.
• if the test is inconclusive.

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Appendix 9C Deriving ARDL Lag Weights
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Appendix 9C 9C.1 The Geometric Lag
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Appendix 9C 9C.1 The Geometric Lag
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Appendix 9C 9C.1 The Geometric Lag
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Appendix 9C 9C.1 The Geometric Lag
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Appendix 9C 9C.2 Lag Weights for More General
ARDL Models
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Appendix 9D Forecasting Exponential Smoothing
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Appendix 9D Forecasting Exponential Smoothing

• Figure 9A.3 Exponential Smoothing Forecasts for
  two alternative values of a