This Problem Set has two parts. In the first, we consider univariate autoregressive models with a unit root in Exercise #3.1 and we use the augmented Dickey-Fuller test to test the null of a unit root against the alternative of stationarity in Danish macro variables in Exercise #3.2.

In the second part, which we will work with after the fall break, we consider cointegration and error-correction. We introduce two ways of estimating cointegration relations and error-correction: The two-step Engle-Granger approach and the ADL approach. In the theoretical Exercise #3.3, we consider cointegration between two interest rates and we show that the interest rate spread is (theoretically) stationarity so that the interest rates are cointegrated. Exercise #3.4 is an empirical exercise, where we use the Engle-Granger two-step approach to test for cointegration between Danish consumption, income, and wealth. Finally, Exercise #3.5 considers an empirical analysis for the same data but using the ADL approach.

The problem set can be downloaded here:

Exercise #3.1: Stationarity and a Unit Root in an AR(1) Model

In this theoretical exercise, we derive the MA representation for an AR(1) model under stationarity and with a unit root, and we discuss the role of the constant term, initial value, and shocks in the two cases. Second, we consider the Dickey-Fuller unit root test.

Resources:

Solution Guide.

Video and Review Quiz 2-02: The AR(1) Model.
The video from Topic 2 shows how to derive the MA-representation for a stationary AR(1) model.

Video: The Moving Average Representation for an AR(1) Process with a Unit Root.
The video shows the derivation of the MA-representation for a unit root AR(1) process and it discusses the effect of the initial value, deterministic terms, and the shocks.

Video: Properties of an AR(1) Process with a Unit Root.

Video: The Dickey-Fuller (DF) Unit Root Test in an AR(1) Model.

Video: The Augmented Dickey-Fuller (ADF) Unit Root Test in an AR(p) Model.

Exercise #3.2: Unit Roots in Danish Macro-Variables

In this empirical exercise, you are asked to perform the Dickey-Fuller test for a unit root in Danish macro time-series. The online tutorial below guides you through the steps and the video shows you how to estimate the model and perform the Dickey-Fuller test in practice using OxMetrics.

Resources:

TimeSeries.in7 and TimeSeries.bn7.

Online tutorial.

Video: The Augmented Dickey-Fuller Test in OxMetrics.

Exercise #3.3: Unit Roots and Cointegration

This theoretical exercise considers cointegration between two interest rates. You are asked you to derive the MA-representation of the variables and then show that a linear combination of the two interest rates, given by the interest rate spread, is stationary. Given estimation output, you are asked to formally test for cointegration between the interest rates. Finally, you are asked to show that one of the interest rates is error-correcting.

Resources:

Solution Guide.

Video: An Introduction to Cointegration: A Simple Example.

Exercise #3.4: Engle-Granger Analysis for Danish Consumption

This empirical exercise guides you through the steps of an Engle-Granger two-step cointegration analysis for Danish consumption data. The data analyzed is the same as in Exercise #3.5.

Resources:

Tutorial for Exercise #3.4.

Data files: ConsumptionData.in7 and ConsumptionData.bn7.

Algebra file: ConsumptionData.alg.

Video: The Engle-Granger Two-Step Cointegration Analysis.

Video: The Engle-Granger Two-Step Cointegration Analysis in OxMetrics.

Exercise #3.5: ADL Analysis for Danish Consumption

This empirical exercise guides you through the steps of single-equation cointegration analysis based on the ADL/ECM model for Danish consumption data. The data analyzed is the same as in Exercise #3.4.

Resources:

Tutorial for Exercise #3.5.

Data files: ConsumptionData.in7 and ConsumptionData.bn7.

Algebra file: ConsumptionData.alg.

Video: A ADL/ECM Cointegration Analysis.

Video: Single-Equation Cointegration Analysis Based on the ADL-ECM Model in OxMetrics.