Chapter 8 Modelling Long-Run Relationships in Finance PDF

Chapter 8 Modelling long-run relationship in finance ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 1  Stationarity and Unit Root Testing Why do we need to test for Non-Stationarity? The stationarity or otherwise of a series can strong...

Chapter 8 Modelling long-run relationship in finance ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 1  Stationarity and Unit Root Testing Why do we need to test for Non-Stationarity? The stationarity or otherwise of a series can strongly influence its behaviour and properties - e.g. persistence of shocks will be infinite for nonstationary series Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated If the variables in the regression model are not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual “t-ratios” will not follow a t- distribution, so we cannot validly undertake hypothesis tests about the regression parameters. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 2  Value of R2 for 1000 Sets of Regressions of a Non-stationary Variable on another Independent Non-stationary Variable ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 3  Value of t-ratio on Slope Coefficient for 1000 Sets of Regressions of a Non-stationary Variable on another Independent Non-stationary Variable ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 4  Two types of Non-Stationarity Various definitions of non-stationarity exist In this chapter, we are really referring to the weak form or covariance stationarity There are two models which have been frequently used to characterise non-stationarity: the random walk model with drift: yt =  + yt-1 + ut (1) and the deterministic trend process: yt =  + t + ut (2) where ut is iid in both cases. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 5  Stochastic Non-Stationarity Note that the model (1) could be generalised to the case where yt is an explosive process: yt =  + yt-1 + ut where  > 1. Typically, the explosive case is ignored and we use  = 1 to characterise the non-stationarity because –  > 1 does not describe many data series in economics and finance. –  > 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will have an increasingly large influence. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 6  Stochastic Non-stationarity: The Impact of Shocks To see this, consider the general case of an AR(1) with no drift: yt = yt-1 + ut (3) Let  take any value for now. We can write: yt-1 = yt-2 + ut-1 yt-2 = yt-3 + ut-2 Substituting into (3) yields: yt = (yt-2 + ut-1) + ut = 2yt-2 + ut-1 + ut Substituting again for yt-2: yt = 2(yt-3 + ut-2) + ut-1 + ut = 3 yt-3 + 2ut-2 + ut-1 + ut T successive substitutions of this type lead to: yt = T y0 + ut-1 + 2ut-2 + 3ut-3 +...+ Tu0 + ut ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 7  The Impact of Shocks for Stationary and Non-stationary Series We have 3 cases: 1. 1. Now given shocks become more influential as time goes on, since if >1, 3>2> etc. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 8  Detrending a Stochastically Non-stationary Series Going back to our 2 characterisations of non-stationarity, the r.w. with drift: yt =  + yt-1 + ut (1) and the trend-stationary process yt =  + t + ut (2) The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is detrending. The first case is known as stochastic non-stationarity. If we let yt = yt - yt-1 and L yt = yt-1 so (1-L) yt = yt - L yt = yt - yt-1 If we take (1) and subtract yt-1 from both sides: yt - yt-1 =  + ut yt =  + ut We say that we have induced stationarity by “differencing once”. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 9  Detrending a Series: Using the Right Method Although trend-stationary and difference-stationary series are both “trending” over time, the correct approach needs to be used in each case. If we first difference the trend-stationary series, it would “remove” the non-stationarity, but at the expense on introducing an MA(1) structure into the errors. Conversely if we try to detrend a series which has stochastic trend, then we will not remove the non-stationarity. We will now concentrate on the stochastic non-stationarity model since deterministic non-stationarity does not adequately describe most series in economics or finance. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 10  Sample Plots for various Stochastic Processes: A White Noise Process 4 3 2 1 0 -1 1 40 79 118 157 196 235 274 313 352 391 430 469 -2 -3 -4 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 11  Sample Plots for various Stochastic Processes: A Random Walk and a Random Walk with Drift 70 60 Random Walk 50 Random Walk with Drift 40 30 20 10 0 1 19 37 55 73 91 109 127 145 163 181 199 217 235 253 271 289 307 325 343 361 379 397 415 433 451 469 487 -10 -20 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 12  Sample Plots for various Stochastic Processes: A Deterministic Trend Process 30 25 20 15 10 5 0 -5 1 40 79 118 157 196 235 274 313 352 391 430 469 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 13  Autoregressive Processes with differing values of  (0, 0.8, 1) 15 Phi=1 10 Phi=0.8 Phi=0 5 0 1 53 105 157 209 261 313 365 417 469 521 573 625 677 729 781 833 885 937 989 -5 -10 -15 -20 ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 14  Definition of Non-Stationarity Consider again the simplest stochastic trend model: yt = yt-1 + ut or yt = ut We can generalise this concept to consider the case where the series contains more than one “unit root”. That is, we would need to apply the first difference operator, , more than once to induce stationarity. Definition If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order d. We write yt I(d). So if yt  I(d) then dyt I(0). An I(0) series is a stationary series An I(1) series contains one unit root, e.g. yt = yt-1 + ut ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 15  Characteristics of I(0), I(1) and I(2) Series An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. I(1) and I(2) series can wander a long way from their mean value and cross this mean value rarely. I(0) series should cross the mean frequently. The majority of economic and financial series contain a single unit root, although some are stationary and consumer prices have been argued to have 2 unit roots. ‘Introductory Econometrics for Finance’ © Chris Brooks 2013 16  How do we test for a unit root? The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that  =1 in: yt = yt-1 + ut against the one-sided alternative 