Cointegration: A First Look

Transcription

1 Cointegration: A First Look Professor: Alan G. Isaac Last modified: 2006 Apr 01 Contents 1 Deterministic and Stochastic Trends GDP A Difficulty Error Correction 8 3 Cointegration ECMs VECMs An Alternative Representation Application Cointegration Deterministic and Stochastic Trends Time trend decomposition is a common practice. One traditional practice is to decompose output into two components: a non-stationary growth trend and a stationary cyclical 1

2 page 2 component. 1 There is no general agreement on the appropriate way to detrend data. 2 The traditional practice is close to Hooker s (1901) suggestion to focus on deviations from trend. For example, output is often regressed on (a polynomial in) time; residuals are interpreted as the cyclical component. Nelson and Plosser (JME,1982) argue that this is a misspecification, that secular movement in output is better modeled as the result of a random walk. Implications: fluctuations in output are largely due to permanent real shocks, and a large fraction of output fluctuations cannot be explained by money induced transitory fluctuations. Consider the simple example discussed by Nelson and Plosser (JME,1982). If y t = α + βt + u t (1) where u t is a stationary shock (the cyclical component). Then, although we may exploit any autocorrelation in u t for SR forecasting, our LR forecasts are just α + βt: current and past events don t affect our LR expectations. Such a process is called trend stationary (TS) since it is stationary once we remove a time trend. In contrast, if (1 L)y t = β + u t (2) then y t = y 0 + βt + t u j (3) so that current and past shocks are relevant even for our LR forecasts. Note that although just removing the trend doesn t render y t stationary, the series is stationary after differencing. Such data generating processes (DGPs) are called integrated or difference stationary (DS). 3 Note that in contrast to the bounded variance of the forecast error for a TS process, the 1 We will call a random variable stationary when its mean and auto-covariance are independent of time. 2 See the discussion in Stock and Watson, JEP A DGP with a spectrum that is finite but non-zero at all frequencies is said to be integrated of order zero or I(0). A stationary DGP is an example of an I(0) process. If a series needs to be differenced d times to become I(0), we say it is integrated of order d or I(d). Thus a difference stationary process is I(1). i=1

3 page 3 forecast error of a DS process increases without bound (since we must allow for a new permanent shock every period). In our simple example, if u t is white noise then the t period ahead forecast has forecast error variance σu 2 for the TS process but tσu 2 for the DS process. 1.1 GDP In the trend stationary case, we sometimes refer to β as the natural rate of growth of GNP. If output follows a TS process, then the natural rate of growth will give us a good prediction of future GNP. As a result we can refer to α + βt as natural real GNP. In contrast, if output follows a DS process, then predictions of future output that are based on the natural rate of growth deteriorate as the forecast horizon increases. There is no natural level of GNP. Advocates of the DS characterization of GNP, the real business cycle (RBC) theorists, emphasize supply side rather than demand side shocks to the level of output. Technology shocks tend to be emphasized as positive permanent shocks, but any effect on factor supplies should also be included (e.g., demographic shocks, changes in preferences, regulatory shocks, tax shocks). Comment: RBC theorist often motivate autocorrelation in u t (persistence of the shock) by including the time to build new capital and the consumption smoothing (permanent income) behavior of consumers. Comment: the RBC framework was motivated partly by the observation of procyclical real wages, but it is also compatible with countercyclical wage behavior (given, say, a labor supply shock.) 1.2 A Difficulty Problem: we can get the wrong econometric answers if we treat a DS process as if it were TS. 4 For example, Shiller (JPE,1979) rejects bond market efficiency under the TS assumption 4 The intuition for this turns on the convergence of the moments of an integrated variable to random variables, in contrast to stationary variables whose moments converge to constants that are the population counterparts. See Pagan and Wickens p and Also, see the example constructed by Stock and

4 page 4 for SR interest rates but not under the DS assumption. This suggests that we should not use the TS assumption without testing it. Consider a special case of our simple example: u t iid(0, σ 2 ). Both the TS and the DS case can be written y t = α + βt + u t /(1 φl) (4) where φ is zero or one. Multiplying by (1 φl), y t = [α(1 φ) + φβ] + φy t 1 + β(1 φ)t + u t (5) So we can run the regression y t = µ + ρy t 1 + γt + u t (6) and test the null hypothesis ρ = 1, γ = 0 in order to test the null of φ = 1, i.e., that y t is DS. Problem: the t-ratios are not t-distributed under the null: standard t-tests are biased toward rejecting the null. Solution: Fuller (1976) and Evans and Savin (Econometrica 1981,1984) provide tabulations of the t-ratio distribution under the null. Also see Dickey and Fuller (Econometrica 1981) for likelihood ratio statistics and for results on higher order AR representations. Comment: Nelson and Plosser (1982) suggest some natural but informal additional diagnostics based on sample autocorrelations. Comment: For the special case we have considered, it may be more natural to just examine the Durbin-Watson (DW) statistic from the regression of y on a constant (y t = c + u t ). Sargan and Bhargava (Econometrica,1983) tabulate the appropriate critical values for this test, which is invariant to the presence of a trend in the true model and is the uniformly most powerful invariant test of a simple random walk null against a stationary AR(1) alternative. However the Sargan and Bhargava DW test is only applicable in this special case. Watson J.Econ.Persp.,1988.

5 page 5 Comment: Caution! Inclusion of an intercept or trend, actual drift in the series, and autocorrelation in u t all of these changes affect the distribution of ˆρ. Comment: Autocorrelation in u t is most often handled by the augmented Dickey-Fuller (ADF) test: estimate by OLS y t = µ (1 ρ)y t 1 + p θ i y t i + e t (7) i=1 The original Dickey-Fuller tables tabulate results for the test statistic T (ˆρ 1). Unfortunately, procedures for determining the appropriate number of lags remain ad hoc. Note that the presence of a unit root in a series implies that, with probability one, economically absurd values for many real variables must eventually be realized. Further, as indicated by Perron (1989), unit root tests are sensitive to missing regressors: mean shifts do to missing regressors may be interpreted as unit roots. A big puzzle in this context is the blithe reliance on unit root tests where acceptance of the null hypothesis of the existence of a unit root is based on low Type I error. As emphasized by R.A. Fisher, failure to reject should not be equated with acceptance. (Nelson and Plosser do acknowledge this.) Current practice becomes all the more puzzling given the universal acknowledgement of the low power of the tests against the more believable autoregressive alternatives. The real reason for using a DS null is probably because it would natural to test a simple null that the process is TS by looking for a unit root in moving average component of the innovation to the differenced series. (E.g., from the simple example, estimate y t = β + u t φu t 1 and test the null of φ = 1.) Unfortunately, testing for a unit MA root is known to be problematic. 5 It may also be worth noting that current unit roots tests rule out fractional 5 Plosser and Schwert,1977,J.Econometrics; but see the exact MLE estimation suggested by Campbell and Mankiw, QJE 102(4), Nov.1987.

6 page 6 differencing a priori (see F. Sowell, Econometrica 58(2), March 1990, p ). Despite these many difficulties, while some economists such as Cochrane (JPE,1988) continue to dispute the Nelson and Plosser story, many economists have accepted it. Econometric considerations arise when regressors are DS. Suppose y t = x t β + z t γ + u t (8) where z t is stationary (and can include a constant) but x t is DS. If x t includes a drift (so that E x t 0) then standard t-ratios for β and γ should be checked against a standardized normal table. 6 However, if a trend is included in the regression then the Dickey-Fuller tables remain relevant for β. Related to this last point, if x t is without drift then the t-ratios for β lose their general usefulness: significance testing for integrated regressors is a problem. Further problems arise if there is more than one DS regressor with drift, unless they are detrended (see Park and Phillips,1988, in Econometric Theory vol 4). An exception to the problem: if it is possible to rewrite the regression equation so that the coefficients of interest are on mean zero stationary variables the F -statistic and t-statistic have the usual asymptotic distributions (Stock and West, JME,1988). Note that Stock and West claim that this result does not depend on being able to rewrite the entire regression in this fashion! Caution! This blessing is a curse in the presence of cointegration: then, in the absence of strict exogeneity, the stationary regressor will typically be correlated with the error term and the parameter estimate will be inconsistent. (Stock and Watson 1988,p.166). That is, simultaneous equations bias (etc.) is relevant once again see below. Granger and Newbold (J.Econometrics,1974) joined Yule (J.Roy.Stat.Soc., 1926) in warning that regressions involving trending or integrated variables can yield spurious (misleadingly strong) correlations, and they urged differencing as a propadeutic. (For example, 6 Roughly: see West, Econometrica 56(6),1988; of course this relies on the asymptotic distribution. West s results apply for univariate x t, i.e., a single DS regressor. He also shows (p.1408) that the results extend the the case of several DS variables if those variables cointegrate. The results also extend to the IV case.

7 page 7 regression of a DS variable on another independent DS variable yields parameter estimates that converge to random variables.)

8 page 8 2 Error Correction Suppose we relate two difference stationary variables as follows: y t = β x t + ε t (9) where ε t is white noise. Noting that y t = t y s + y 0 (10) i=1 is an identity, we can substitute for y t to get t y t = (β x t + ε s ) + y 0 i=1 = y 0 + β(x t x 0 ) + = (y 0 βx 0 ) + βx t + t i=1 ε s t i=1 ε s (11) There is no long-run equilibrium relationship between y and x. Davidson et al. and Hendry and Mizon (EJ,1978) worried about this implied loss of information about any LR steady state solution. Error correction models (ECMs) were introduced basically in order to recover this information. Suppose for example that we relate two difference stationary variables as follows: y t = β x t γ(y t 1 αx t 1 ) + ε t (12) The new term is called the error correction term. The basic ECM derives from classical control theory and has the form y t = β y t + γ(y t 1 y t 1 ) + δs t 1 + e t (13)

9 page 9 where y is the control variable, y t is the target, and S t 1 = δ (y t j y t j ) (14) Note that y = y in the long run steady state. In practice, S t 1 is generally dropped from the model and the target is expressed as a linear function of exogenous variables (yt = x t β). As Nickell (Oxford Bulletin of Economics and Statistics,1985) notes, the minimization of discounted costs of deviation from target and of adjustment introduces lagged values of y, where the number of lags depends on the data generating process for y. 3 Cointegration This sections draws on N. Mark section 2.6. Consider I(1) processes: q t and f t. We say these two series are cointegrated if they have a stationary linear combination. Otherwise they are not cointegrated. We will model q t and f t as a sum of nonstationary and stationary components. q t = ξ 1 t + z 1 t (15) f t = ξ 2 t + z 2 t (16) The components are two stationary contributions, z 1 t and z 2 t, and two random walks: ξ 1 t = ξ 1 t 1 + u 1 t (17) ξ 2 t = ξ 2 t 1 + u 2 t (18) (Here u 1 t and u 2 t are independent white noise processes.)

10 page 10 Now consider possible linear combinations of our two variables: q t βf t = (ξ 1 t βξ 2 t ) + (z 1 t βz 2 t ) (19) This can only be stationary if ξ 1 t βξ 2 t is stationary. That is, we need q t and f t to share a common stochastic trend. 3.1 ECMs Interest in ECMs has surged recently as a result of developments in the unit root literature. In particular, the Granger Representation Theorem (Granger, UCSC disc.paper 83-13,1983) established an equivalence ECMs and what is called cointegration: ECMs imply cointegration, and driftless cointegrated variables have an ECM representation. 7 Cointegration exists when a linear combination of DS variables is stationary. Recall our example y t = x t β + z t γ + u t where z t is stationary but x t is DS. In this case y t and x t are cointegrated, since (1 β)(y t x t ) is stationary. We call (1 β) the cointegrating vector. The good (and interesting) news is that since the variance in x t will dominate the variance in the stationary variables z t, it is acceptable to simply regress y t on x t in order to get a consistent estimate of β. In fact, the resulting estimate converges to β very fast a result known as the super-consistency of the cointegrating vector. 8 It may also be viewed as good news that this super-consistency is preserved even if you regress x t on y t : considerations of endogeneity and of errors in variables become relatively unimportant. (I believe this is a major source of the favor 7 More precisely, if the components of x t are I(1) and αx t is I(0), then there are finite order lag polynomials A(L) and d(l) such that A(L)(1 L)x t = γαx t + d(l)ɛ t 8 In a multiple regression, this super-consistency requires that no cointegrating relationship exists among the regressors. See Hendry (1986, Ox.Bul.Ec.Stat.,p.208).

11 page 11 cointegration receives in the profession: it supports single equation techniques by minimizing the importance of simultaneous equations bias. See Stock and Watson s JEP, 1988 p.168 comment on Haavelmo s problem.) The bad news is that, despite this super-consistency, small sample bias can be very large (see Banerjee et al., Oxford Bulletin of Economics and Statistics,1986). In order to get a feel for the link between cointegration and ECMs, recall our example y t = x t β + z t γ + u t Subtracting y t 1 from each side yields, with y t = x t β, y t = x t β + z t γ y t 1 + u t = x t β + z t γ (y t 1 x t 1 β) + u t = y t + (y t 1 y t 1 ) + z t γ + u t Granger and Engle (1987,p.263) provide a more interesting example. 9 The usefulness of this transformation is clear: the regression now involvles only I(0) variables and so our usual regression theory applies. Note that the close links between ECMs and cointegration has led some economists to mistakenly interpret a cointegrating vector as characterizing long run economic equilibrium. As noted by Hall and Henry (1988), The term equilibrium has many meanings in economics and its use in the cointegration literature is rather different from most definitions of equilibrium. Within the cointegration literature all that is meant by equilibrium is that it is an observed relationship which has, on average, been maintained by a set of variables for a long period. It implies none of the usual theoretical implications of market clearing or full 9 We can also write y t = x t β + z t γ (y t 1 x t 1 β z t 1 γ) + u t of course; but z t is already I(0) by assumption. Our example is close to that in Phillips 1988 Cowles Foundation Discussion Paper #893, as presented by Pagan and Wickens, except that we did not presume that yt should be dumped into the error term just because it is I(0).

12 page 12 employment and neither does it imply that the system is at rest. See Pagan and Wickens (EJ,1989,p.1003), Swamy (J.Econometrics,1989), or even Granger (Ox.Bul.Ec.Stat., 1986, p.216) for for discussion of this error. 3.2 VECMs This section draws on N. Mark ch.2 Let y t = (q t, f t ). Suppose we have the following DGP: y t = A 1 y t 1 + A 2 y t 2 + u t (20) Subtract y t 1 from both sides: y t = (A 1 I)y t 1 + A 2 y t 2 + u t (21) We can rewrite this as y t = (A 1 + A 2 I)y t 1 A 2 y t 1 + u t (22) or y t = Ry t 1 A 2 y t 1 + u t (23) This is just the vector analog of our Dickey-Fuller discussion. If y t I(1), then y t is stationary. One way to achive this is R = 0, by direct analogy to our Dickey-Fuller discussion. In this case y t follows a standard vector autoregression. But with two variables, we can also potentially find linear combinations of q t and f t that are stationary. For example, if q t βf t is stationary, then this becomes the error correction

13 page 13 term, and we can write y t = r [ ] 11 1 β y t 1 A 2 y t 1 + u t r 21 = r 11 βr 11 y t 1 A 2 y t 1 + u t r 21 βr 21 (24) Note that the matrix is singular. Letting ζ t = (1 R = r 11 βr 11 (25) r 21 βr 21 β)y t be the error correction term, we now have y t = r 11 ζ t 1 A 2 y t 1 + u t (26) r 21 Equation (26) is called the vector error correction representation of y t. Note that is looks like a VAR in first differences, except for the addtional error correction term. The presence of the error correction term suggests that a simple VAR in differences would be misspecified: it wd have an omitted variable. 3.3 An Alternative Representation Suppose the cointegrating vector is given by theory. Then we can form the error correction term without estimation. Multiply (26) by the cointegrating vector to get ζ t = (r 11 βr 21 )ζ t 1 (1 β)a 2 y t 1 + (1 β)u t (27)

14 page 14 which implies ζ t = (1 + r 11 βr 21 )ζ t 1 (1 β)a 2 y t 1 + (1 β)u t (28) This relationship between stationary variables looks like it could be included in the system (26). But since we produced this as a linear combination of the equations in (26), this clearly adds no new information to that system. But we can use it to replace either one of the equations in (26). For example, we can use f t = (q t ζ t )/β to substitute everywhere for f t in (26) and (28) to get a two equation system in q t and ζ t. ζ t ζ t 1 ζ t 2 q t = Â1 q t 1 + Â2 q t u t (29) 1 β Or, we can use q t = (βf t + ζ t ) to substitute everywhere for q t in (26) and (28) to get a two equation system in ζ t and f t. ζ t = Ã1 ζ t 1 + Ã2 ζ t 2 f t f t 1 f t β u t (30) 0 1 So, why would we bother to do this? It allows us to use the simpler VAR estimation framework, with all its standard results, rather than relying on the more complicated VECM estimation framework. And in particular, the forecasting algebra for the VAR representation is simple Application Nelson Mark presents a reestimation of the MacDonald and Taylor (1993) application of these observations to the monetary approach model. (This in turn derives from earlier work by? on the present value model.)

15 page 15 Recall our core implication of the monetary model: s deviates from m to the extent that there is expected depreciation. s t = m t + λ(s e t+1 s t ) (31) Mark argues as follows: s is nonstationary, but s e t+1 s t is stationary, therefore s and m must be cointegrated. s t m t = λ(s e t+1 s t ) (32) What is more, the cointegrating vector is known to be (1 1). Defining ζ t = s t m t, our earlier work suggests the VAR m t = ζ t maxlag j=1 A j m t 1 + v t (33) ζ t 1 This can be estimated as a VAR, and the cross equation restrictions implied by the rational expectations hypothesis can be tested. Mark estimates the model using US and German data from to , and he uses a Wald test of the cross equation restrictions. Note that he imposes values for λ: he does not estimate these! But for robustness he tries several different values. His conclusion: the restrictions are rejected for reasonable values of λ. 3.4 Cointegration Testing for cointegration just involves testing that the residuals from a cointegrating regression do not contain a unit root. Thus when cointegration tests are applied to a pre-specified model, they are just another specification test, testing the econometric model s assumption of a stationary error term. 10 One approach is to just run the DF test on the residuals, but 10 In this light, it is easy to understand Pagan and Wickens suggestion (p.1009) that there seems little to be gained in running the cointegrating regression rather than running the entire prespecified regression from the beginning. (The stationary variables coefficients will be consistently estimated and their test

16 page 16 Figure 1: Theoretical and Actual Spreads

17 page 17 the bias in the estimate of the cointegrating vector leads too often to the rejection of the null of no cointegration. The most popular test is the cointegrating regression Durbin-Watson test (CRDW), which relies on the Durbin-Watson statistic, but only the bounds of the distribution are available (as opposed to its use in testing individual series for integration). Engle and Granger (Econometrica,1987) suggest that a better procedure is to apply the ADF test to the residuals of the cointegrating regression. Engle and Yoo (J.Econometrics,1987) suggest changes in the critical values of the ADF in order to account for the dependence of these residuals on estimated parameters. These unit root tests of course retain the weaknesses noted above, making it perhaps surprising that so many researchers find cointegration. It seems to me, however, that this is an easily explained artifact of the autocorrelation in the time series data. High autocorrelation makes it difficult to reject the null that a series contains a unit root, but the same autocorrelation in multiple series makes it relatively easy to get lower autocorrelation in the residuals of a cointegrating regression. References Chow, Gregory C. (1987). Money and Price Level Determination in China. Journal of Comparative Economics 11, Fuller, Wayne A. (1976). Introduction to Statistical Time Series (1st ed.). Wiley-Interscience. MacDonald, Ronald and Mark P. Taylor (1993). The Monetary Approach to the Exchange Rate: Rational Expectations, Long-Run Equilibrium, and Forecasting. International Monetary Fund Staff Papers 40, Nelson, Charles R. and Charles I. Plosser (1982). Trends and Random Walks in Macroecostatistics will have the usual large sample properties: Stock and Watson,JEP 1988,p.167.) On the other hand, cointegrating regressions have been viewed by some as an aid to specification search, preliminary to the formulation of an ECM. Since cointegrating vectors are not unique in fact, both sign and magnitude are completely up for grabs this approach appears to have obvious dangers.

18 page 18 nomic Time Series: Some Evidence and Implications. Journal of Monetary Economics 10, Engle, Robert F. and Granger, C.W.J., 1987, Cointegration and Error Correction: Representation, Estimation, and Testing, Econometrica 55(2), Hooker, R.H., 1901, Correlation of the Marriage Rate with Trade, Journal of the Royal Statistical Society 64, p , cited by Hendry, 1986, Ox.Bul.Ec.Stat. 48(3). Perron, Pierre, 1989, The Great Crash, the Oil Price Shock and the Unit Root Hypothesis, Econometrica 57, Stock, James H., and Watson, Mark W., Variable Trends in Economic Time Series, JEP 2(3), Summer 1988,