1 The graphical approach appears to have served the time series analysts satisfactorily, but on the whole econometricians prefer more formal methods and tests for nonstationarity are no exception. TESTS OF NONSTATIONARITY: INTRODUCTION

2 They are often described as tests for unit roots, for reasons related to the theory of difference equations that need not concern us here. TESTS OF NONSTATIONARITY: INTRODUCTION

3 Before embarking on tests of nonstationarity, it is prudent to recognize that we should not have unrealistic expectations concerning what can be achieved. To illustrate the discussion, suppose that we believe a process can be represented by Y t =  2 Y t–1 +  t. TESTS OF NONSTATIONARITY: INTRODUCTION

4 If  2 = 1, the process is nonstationary. If  2 = 0.99, it is stationary. Can we really expect to be able to discriminate between these two possibilities? TESTS OF NONSTATIONARITY: INTRODUCTION

5 Obviously not. We are seldom able to make such fine distinctions in econometrics. We have to live with the fact that we are unlikely ever to be able to discriminate between nonstationarity and highly autocorrelated stationarity. TESTS OF NONSTATIONARITY: INTRODUCTION

6 We will start with the first-order autoregressive model Y t =  1 +  2 Y t–1 +  t +  t, including a time trend as well as an autoregressive process. TESTS OF NONSTATIONARITY: INTRODUCTION

7 For the time being we will assume that the disturbance term is iid and we will emphasize this by writing it as  t rather than u t. We will relax the iid assumption later on. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives 8 We will consider the various special cases that arise from the following possibilities: –1 <  2 < 1 or  2 = 1  = 0 or  ≠ 0 or TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives 9 We will exclude  2 > 1 because the explosive process implied by it is seldom encountered in economics. We will also exclude  2 < –1 because it implies an implausible process. or TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (a) Case (b) Case (c) Case (d) Case (e) TESTS OF NONSTATIONARITY: INTRODUCTION 10 This gives us four basic possibilities. For one of them, it is important to separate the two sub-cases  1 = 0 and  1 ≠ 0. This gives us five cases in all. In the table,  1 = * means  1 is unrestricted. or

General model Alternatives Case (a) 11 or Case (a) is a stationary AR(1) process of the kind studied at length in Chapter 11. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (b) 12 or Case (b) is a random walk. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (c) 13 or Case (c) is a random walk with drift. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (d) 14 or Case (d) is a stationary autoregressive process around a deterministic trend. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (e) 15 or Case (e) implies a model that is doubly nonstationary, being a random walk (with drift if  1 ≠ 0) around a deterministic time trend. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (e) 16 or Case (e) can be excluded because it is implausible. Lagging and substituting once, we obtain the equation shown. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (e) 17 or Lagging and substituting t times, we can express Y t in terms of the initial Y 0, the innovations, and a convex quadratic expression for t. It is not reasonable to suppose that any (ordinary) time series process can be characterized as a convex quadratic function of time. TESTS OF NONSTATIONARITY: INTRODUCTION

General model Alternatives Case (a) Case (b) Case (c) Case (d) TESTS OF NONSTATIONARITY: INTRODUCTION 18 or Our objective is to determine which of cases (a) – (d) provides the best representation of a given process. Two approaches have been advocated.

General model Alternatives Case (a) Case (b) Case (c) Case (d) TESTS OF NONSTATIONARITY: INTRODUCTION 19 or One is to start with the general model at the top of the slide and then conduct a series of tests that might lead to a simplification to one of cases (a) to (d).

General model Alternatives Case (a) Case (b) Case (c) Case (d) TESTS OF NONSTATIONARITY: INTRODUCTION 20 or This has the intellectual appeal of implementing the general-to-specific approach. However, as will be seen, the tests involved often have low power, and this can give rise to ambiguity that cannot be resolved.

General model Alternatives Case (a) Case (b) Case (c) Case (d) TESTS OF NONSTATIONARITY: INTRODUCTION 21 or The other approach is pragmatic. One starts by plotting the data and assessing whether there is evidence of a trend. If there is not, then one limits the investigation to cases (a) and (b). If there is, one considers cases (c) and (d). This is the approach that will be adopted in the next two slideshows.

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 13.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics