ECONOMETRIC THY II
ECONOMETRIC THY II ECON 584
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This is page 429 Printer Opaque this 12 Cointegration 121 Introduction The regression theory of Chapter 6 and the VAR models discussed in the previous chapter are appropriate for modeling 0 data7 like asset returns or growth rates of macroeconomic time series Economic theory often im plies equilibrium relationships between the levels of time series variables that are best described as being 1 Similarly7 arbitrage arguments imply that the 1 prices of certain nancial time series are linked This chapter introduces the statistical concept of cointegration that is required to make sense of regression models and VAR models with 1 data The chapter is organized as follows Section 122 gives an overview of the concepts of spurious regression and cointegration7 and introduces the error correction model as a practical tool for utilizing cointegration with nancial time series Section 123 discusses residual based tests for coin tegration Section 124 covers regression based estimation of cointegrating vectors and error correction models In Section 1257 the connection be tween VAR models and cointegration is made7 and Johansen s maximum likelihood methodology for cointegration modeling is outlined Some tech nical details of the Johansen methodology are provided in the appendix to this chapter Excellent textbook treatments of the statistical theory of cointegration are given in Hamilton 19947 Johansen 1995 and Hayashi 2000 Ap plications of cointegration to nance may be found in Campbell7 Lo and 430 12 Cointegration MacKinlay 1997 Mills 1999 Alexander 2001 Cochrane 2001 and Tsay 2001 122 Spurious Regression and Cointegration 1221 Spurious Regression The time series regression model discussed in Chapter 6 required all vari ables to be 0 In this case the usual statistical results for the linear regression model hold If some or all of the variables in the regression are 1 then the usual statistical results may or may not holdl One important case in which the usual statistical results do not hold is spurious regres sion when all the regressors are 1 and not cointegrated The following example illustrates Example 71 An illustration of spurious regression using simulated data Consider two independent and not cointegrated 1 processes 311 and 312 such that 21in 212171 am Where 5239 N GWva 17 i 17 2 Following Granger and Newbold 1974 250 observations for each series are simulated and plotted in Figure 121 using gt setseed458 gt e1 rnorm250 gt e2 rnorm250 gt y1 cumsume1 gt gt gt 2 cumsume2 tsploty1 y2 ltyc13 legend0 15 Cquotylquot quoty2quot ltyc13 The data in the graph resemble stock prices or exchange rates A visual inspection of the data suggests that the levels of the two series are positively related Regressing 311 on 312 reinforces this observation gt sunmary0LSy1 y2 Call 0LSformula yl y2 1 A systematic technical analysis of the linear regression model with 11 and 10 vari ables is given in Sims Stock and Watson 1990 Hamilton 1994 gives a nice summary of these results and Stock and Watson 1989 provides useful intuition and examples 122 Spurious Regression and Cointegration 431 x x 150 200 250 FIGURE 121 TWO simulated independent 11 processes Residuals Min 1Q Median 3Q Max 16360 4352 0 128 4979 10763 Coefficients Value Std Error t value Prgtt Intercept 67445 03943 17 1033 00000 y2 04083 00508 80352 00000 Regression Diagnostics R Squared 02066 Adjusted R Squared 02034 Durbin Watson Stat 00328 Residual standard error 6217 on 248 degrees of freedom F statistic 6456 on 1 and 248 degrees of freedom the p value is 3797e 014 The estimated slope coef cient is 0408 With a large t statistic of 8035 and the regression R2 is moderate at 0201 The only suspicious statistic is the very low Durbin Watson statistic suggesting strong residual auto correlation These statistics are representative of the spurious regression 432 12 Cointegration phenomenon with 1 that are not cointegrated 1f Ayn is regressed on A312 the correct relationship between the two series is revealed gt summary0LSdiffy1quotdiffy2 Call 0LSformula diffy1 diffy2 Residuals Min 1Q Median 3Q Max 36632 07706 00074 06983 27184 Coefficients Value Std Error t value Prgtt Intercept 00565 00669 08447 03991 diffy2 00275 00642 04290 06683 Regression Diagnostics R Squared 0 0007 Adjusted R Squared 00033 Durbin Watson Stat 19356 Residual standard error 1055 on 247 degrees of freedom F statistic 0184 on 1 and 247 degrees of freedom the p value is 06683 Similar results to those above occur if cov51t52t 7E 0 The levels re gression remains spurious no real long run common movement in levels but the differences regression will re ect the non zero contemporaneous correlation between Ayn and Aym Statistical Implications of Spurious Regression Let Y yup ym denote an n X 1 vector of 1 time series that are not cointegrated Using the partition Y y1Y 2t consider the least squares regression of 311 on Y2 giving the tted mode A yn ngt 12 121 Since yh is not cointegrated with Y2 121 is a spurious regression and the true value of 32 is zero The following results about the behavior of 32 in the spurious regression 121 are due to Phillips 1986 a 82 does not converge in probability to zero but instead converges in distribution to a non normal random variable not necessarily centered at zero This is the spurious regression phenomenon 122 Spurious Regression and Cointegration 433 a The usual OLS i statistics for testing that the elements of 32 are zero diverge to too as T A 00 Hence With a large enough sample it Will appear that Y is cointegrated When it is not if the usual asymptotic normal inference is use The usual R2 from the regression converges to unity as T A 00 so that the model Will appear to t well even though it is misspeci ed Regression With 1 data only makes sense When the data are coin tegrated 1222 Coiutegratiou Let Y 311 ym denote an n X 1 vector of 1 time series Y is cointegrated if there exists an n X 1 vector 3 1 n such that IGIYt 5131M 39 39 39 nynl N 0 122 ln words the nonstationary time series in Y are cointegrated if there is a linear combination of them that is stationary or 0 If some elements of 3 are equal to zero then only the subset of the time series in Y With non zero m 39 t is 39 t t The linear 39 39 IY is often motivated by economic theory and referred to as a longeruu equilibrium relationship The intuition is that 1 time series With a long run equilib rium relationship cannot drift too far apart from the equilibrium because economic forces Will act to restore the equilibrium relationship Normalization The cointegration vector 3 in 122 is not unique since for any scalar c the linear combination c Y 3 Y N 0 Hence some normalization assumption is required to uniquely identify A typical normalization is 175m75ny so that the cointegration relationship may be expressed as lYt 211i 2y2t 39 39 39 57317 N 0 2m zyz nym u 123 Where u N 0 In 123 the error term u is often referred to as the disequilibrium error or the cointegraliug residual ln long run equilibrium the disequilibrium error u is zero and the long run equilibrium relationship is 211i 2y2t nynt 434 12 Cointegration Multiple Cointegrating Relationships If the nx 1 vector Y is cointegrated there may be 0 lt 7 lt n linearly inde pendent cointegrating vectors For example let n 3 and suppose there are T 2 Cgintegrating VeCtOFS I31 51175127513 and 132 5217 5227523 Then I31Yt 511y1i 512y2i lgyai N 07 zYt 521211 522y2i zgyai 0 and the 3 X 2 matrix Br31511 n gm 132 21 22 33 forms a basis for the space of cointegrating vectors The linearly indepen dent vectors l and 32 in the cointegrating basis B are not unique unless some normalization assumptions are made Furthermore any linear combi nation of 1 and 32 eg 33 0131 0232 where cl and 02 are constants is also a cointegrating vector Examples of Cointegration and Common Trends in Economics and Finance Cointegration naturally arises in economics and nance ln economics coin tegration is most often associated with economic theories that imply equi librium relationships between time series variables The permanent income model implies cointegration between consumption and income with con sumption being the common trend Money demand models imply cointe gration between money income prices and interest rates Growth theory models imply cointegration between income consumption and investment with productivity being the common trend Purchasing power parity im plies cointegration between the nominal exchange rate and foreign and domestic prices Covered interest rate parity implies cointegration between forward and spot exchange rates The Fisher equation implies cointegration between nominal interest rates and in ation The expectations hypothesis of the term structure implies cointegration between nominal interest rates at different maturities The equilibrium relationships implied by these eco nomic theories are referred to as longimn equilibrium relationships because the economic forces that act in response to deviations from equilibriium may take a long time to restore equilibrium As a result cointegration is modeled using long spans of low frequency time series data measured monthly quarterly or annually In nance cointegration may be a high frequency relationship or a low frequency relationship Cointegration at a high frequency is motivated by arbitrage arguments The Law of One Price implies that identical assets must sell for the same price to avoid arbitrage opportunities This implies cointegration between the prices of the same asset trading on different markets for example Similar arbitrage arguments imply cointegration be tween spot and futures prices and spot and forward prices and bid and 122 Spurious Regression and Cointegration 435 ask prices Here the terminology long run equilibrium relationship is some what misleading because the economic forces acting to eliminate arbitrage opportunities work very quickly Cointegration is appropriately modeled using short spans of high frequency data in seconds minutes hours or days Cointegration at a low frequency is motivated by economic equilib rium theories linking assets prices or expected returns to fundamentals For example the present value model of stock prices states that a stock s price is an expected discounted present value of its expected future dividends or earnings This links the behavior of stock prices at low frequencies to the behavior of dividends or earnings In this case cointegration is modeled using low frequency data and is used to explain the long run behavior of stock prices or expected returns 1223 Cointegration and Common Trends 1f the n X 1 vector time series Y is cointegrated with 0 lt r lt n coin tegrating vectors then there are n 7 r common 1 stochastic trends To illustrate the duality between cointegration and common trends let Y y1y2 N 1 and a 51 52 53 N 0 and suppose that Y is cointegrated with cointegrating vector 3 17 52 This cointegration relationship may be represented as t 211i 52E 51353t 31 t 212i 251362t 31 The common stochastic trend is 2131 513 Notice that the cointegrating relationship annihilates the common stochastic trend t t HY 52 2513 53t 52 2513 52 53t 525 N 0 31 31 1224 Simulating Cointegrated Systems Cointegrated systems may be conveniently simulated using Phillips7 1991 triangular representation For example consider a bivariate cointegrated system for Y y1y2 with cointegrating vector 3 1 i Q A triangular representation has the orm 21h 2y2t u where u N 0 124 212 2121 12 where v N 0 125 436 12 Cointegration The rst equation describes the long run equilibrium relationship with an 0 disequilibrium error u The second equation speci es 312 as the com mon stochastic trend with innovation 3 t y2t 2120 2117 j1 In general the innovations u and 3 may be contemporaneously and serially correlated The time series structure of these innovations characterizes the short run dynamics of the cointegrated system The system 124 125 with 2 1 for example might be used to model the behavior of the logarithm of spot and forward prices spot and futures prices or stock prices and dividends Example 72 Simulated bivariate cointegrated system Consider simulating T 250 observations from the system 124 125 using 3 1 71 u 075u1 5 a N iidN0052 and v N iidN0 052 The S PLUS code is set seed 432 e rmvnorm250 meanrep0 2 sdc O 5 0 5 u arl arima simmodellist ar0 75 innove 1 y2 cumsume2 yl y2 u arl par mfrowc 2 1 tsploty1 y2 ltyc13 mainquotSimulated bivariate cointegrated systemquot subquot1 cointegrating vector 1 common trendquot legend 0 7 legendcquoty1quot quoty2quot ltyc13 tsplot uar1 mainquotCointegrating residualquot VVVVVVVVV Figure 122 shows the simulated data for 311 and 312 along with the cointe grating residual u 311 7 312 Since 311 and 312 share a common stochas tic trend they follow each other closely The impulse response function for u may be used to determine the speed of adjustment to long run equi librium Since u is an AR1 with 35 075 the half life of a shock is ln05 ln075 24 time periods Next consider a trivariate cointegrated system for Y 311312 313 With a trivariate system there may be one or two cointegrating vectors With one cointegrating vector there are two common stochastic trends and with two cointegrating vectors there is one common trend A triangular representation with one cointegrating vector 3 1 i Q 7 3 and two 122 Spurious Regression and Cointegration 437 Simulated bivariale coinlegraled system 1 cmmamanngvecmm cummunlvend Colnlegraling residual FIGURE 122 Simulated bivariate cointegrated system With 8 17 71 stochastic trends is 111 522 53213 u Where u N 0 126 212 2121 11 Where v N 0 127 213 2131 wt Where w N 0 128 The rst equation describes the long run equilibrium and the second and third equatlons specify the common stochastic trends An example of a trivariate cointegrated system With one cointegrating vector is a system of nominal exchange rates7 home country price indices and foreign country price indices A cointegrating vector 3 17 71 71 implies that the real exchange rate is stationary Example 73 Simulated tn39varz39ate cointegmted system with 1 cointegmting vector The S PLUS code for simulating T 250 observation from 126 128 with 3 1705705 u 075mH 5 5 z39z39dN0052 v N z39z39dN0052 and wt z39z39dN0052 is set seed 573 e rmvnorm250 meanrep03 sdc050505 ul arl arima simmodellistar0 75 innove E 1 gt gt gt gt y2 cumsume2 438 12 Cointegration Simulated trivariate cointegraled system 1 cummmalmu vecluv 2 cummun quotends Coinlegraling residual FIGURE 123 Simulated trivariate cointegrated system with one cointegrating vector 8 17 705 705 and two stochastic trends gt y3 cumsume3 gt yl O5y2 O5y3 u1ar1 gt parmfrowc2l gt tsploty1 y2 y3 ltyc134 mainquotSimulated trivariate cointegrated systemquot subquot1 cointegrating vector 2 common trendsquot gt legend0 12 legendcquoty1quotquoty2quotquoty3quot ltyc134 gt tsplotuar1 mainquotCointegrating residualquot Figure 123 illustrates the simulated data Here7 yg and 113 are the two independent common trends and 21h 05312 05313 u is the average of the two trends plus an AR1 residual Finally7 consider a trivariate cointegrated system with two cointegrat ing vectors and one common stochastic trend A triangular representa tion for this system with cointegrating vectors 31 107 13 and 52 0717 523 is 21h 513313t My Where ut N 0 129 212t 5232131 12 where v N 0 1210 ya 113171 wt where w 0 1211 Here the rst two equations describe two long run equilibrium relations and the third equation gives the common stochastic trend An example in 122 Spurious Regression and Cointegration 439 nance of such a system is the term structure of interest rates where yg represents the short rate and 311 and yg represent two different long rates The cointegrating relationships would indicate that the spreads between the long and short rates are stationary Example 74 Simulated trivariate cointegrated system with 2 cointegrating vectors The S PLUS code for simulating T 250 observation from 129 1211 with 31 10 71 2 0 1 71 u 075u1e a N iidN0052 v 075mH 77 77 z39z39dN0052 and w z39z39dN0052 is gt set seed 573 gt e rmvnorm250meanrep03 sdc050505 gt u arl arima simmodellist ar0 75 innove 1 gt v arl arima simmodellist ar0 75 innove2 gt y3 cumsume 3 gt yl y3 uar1 gt y2 y3 var1 gt parmfrowc 2 1 gt tsploty1 y2 y3 ltyc134 mainquotSimulated trivariate cointegrated systemquot subquot2 cointegrating vectors 1 common trendquot gt legend 0 10 legendcquoty1quot quoty2quot quoty3quot ltyc 1 3 4 gt tsplotuar1 var1 ltyc13 mainquotCointegrated residualsquot gt legend 0 1 legendcquotuquot quotvquot ltyc 1 3 1225 Cointegration and Error Correction Models Consider a bivariate 1 vector Y y1y2 and assume that Y is cointegrated with cointegrating vector 3 17 52 SO that Yt yh 7 ngt is 0 In an extremely in uential and important paper Engle and Granger 1987 showed that cointegration implies the existence of an error correction model ECM of the form Ayn C1 041y1t71 2y2t71 1212 Z 1Ay1tij Z 2Ay2tij 5n 7 7 Alix C2 042211i71 2y2t71 1213 Z w lAymj Z ngAthij 52t 7 j that describes the dynamic behavior of 311 and 312 The ECM links the long run equilibrium relationship implied by cointegration with the short run dynamic adjustment mechanism that describes how the variables react 440 12 Cointegration Simulated lrivariale coinlegraled system 2 cummmalmu vecluvs1 cummun quotand Coinlegraled residuals FIGURE 124 Simulated trivatiate cointegrated system with two cointegrating vectors Gl 107 71 g 017 71 and one common trend when they move out of long run equilibrium This ECM makes the concept of cointegration useful for modeling nancial time series Example 75 Bivarz39ate ECM for stock prices and dividends As an example of an ECM7 let st denote the log of stock prices and dt denote the log of dividends and assume that Y sd is 1 If the log dividend price ratio is 0 then the logs of stock prices and dividends are cointegrated with 3 17 71 That is7 the long run equilibrium is dt 5t IL ut where p is the mean of the log dividend price ratio7 and ut is an 0 random variable representing the dynamic behavior of the log dividend price ratio disequilibrium error Suppose the ECM has the form ASt Csasdt71 5t71 LEst Adt Cd addt71 Stil II Sch where cs gt 0 and Cd gt 0 The rst equation relates the growth rate of dividends to the lagged disequilibrium error d1 7 51 7p and the second equation relates the growth rate of stock prices to the lagged disequilibrium as well The reactions of st and dt to the disequilibrium error are captured by the adjustment coe cients as and ad 122 Spurious Regression and Cointegration 441 Consider the special case of 1212 1213 where ad 0 and as 05 The ECM equations become As CS 05d1 7 51 7 p 5 Ad Cd ash so that only 5 responds to the lagged disequilibrium error Notice that EAslY1 Cs 05d1 7 51 7 p and EAdtlY1 Cd Consider three situations 1 d1 751 7p 0 Then EASlY1 cs and EAdlY1 Cd so that cs and Cd represent the growth rates of stock prices and dividends in long run equilibrium 2 d1 7 51 7 p gt 0 Then EAslY1 Cs 05d1 7 51 7p gt cs Here the dividend yield has increased above its long run mean positive disequilibrium error and the ECM predicts that s will grow faster than its long run rate to restore the dividend yield to its long run mean Notice that the magnitude of the adjustment coef cient as 05 controls the speed at which 5 responds to the disequilibrium error 3 d1 7 51 7 p lt 0 Then EAslY1 Cs 05d1 7 51 7p lt cs Here the dividend yield has decreased below its long run mean negative disequilibrium error and the ECM predicts that s will grow more slowly than its long run rate to restore the dividend yield to its long run mean In Case 1 there is no expected adjustment since the model was in long run equilibrium in the previous period In Case 2 the model was above long run equilibrium last period so the expected adjustment in s is down ward toward equilibrium ln Case 3 the model was below long run equi librium last period and so the expected adjustment is upward toward the equilibrium This discussion illustrates why the model is called an error cor rection model When the variables are out of long run equilibrium there are economic forces captured by the adjustment coef cients that push the model back to long run equilibrium The speed of adjustment toward equilibrium is determined by the magnitude of as In the present example as 05 which implies that roughly one half of the disequilibrium error is corrected in one time period If as 1 then the entire disequilibrium is corrected in one period If as 15 then the correction overshoots the long run equilibrium 442 12 Cointegration 123 Residual Based Tests for Cointegration Let the TL X 1 vector Y be 1 Recall7 Y is cointegrated with 0 lt T lt TL cointegrating vectors if there exists an T X TL matrix B such that I3th uh B Yt 5 5 10 IGLYL um Testing for cointegration may be thought of as testing for the existence of long run equilibria among the elements of Y Cointegration tests cover two situations There is at most one cointegrating vector I There are possibly 0 S T lt TL cointegrating vectors The rst case was originally considered by Engle and Granger 1986 and they developed a simple two step residual based testing procedure based on regression techniques The second case was originally considered by Jo hansen 1988 who developed a sophisticated sequential procedure for de termining the existence of cointegration and for determining the number of cointegrating relationships based on maximum likelihood techniques This section explains Engle and Granger s two step procedure Johansen s more general procedure will be discussed later on Engle and Granger s two step procedure for determining if the TL X 1 vector 3 is a cointegrating vector is as follows a Form the cointegrating residual Y u Perform a unit root test on u to determine if it is 0 The null hypothesis in the Engle Granger procedure is no cointegration and the alternative is cointegration There are two cases to consider In the rst case7 the proposed cointegrating vector 3 is pre speci ed not estimated For example7 economic theory may imply speci c values for the elements in 3 such as 3 17 7 The cointegrating residual is then readily con structed using the prespeci ed cointegrating vector In the second case7 the proposed cointegrating vector is estimated from the data and an estimate of the cointegrating residual aiY 12 is formed Tests for cointegration using a pre speci ed cointegrating vector are generally much more powerful than tests employing an estimated vector 1231 Testing foT CointegTatLon When the Comtegmtmg Vectmquot Is PTespeci ed Let Yt denote an TL X 1 vector of 1 time series7 let 3 denote an TL X 1 prespeci ed cointegrating vector and let u Y denote the prespeci ed 123 ResidualBased Tests for Cointegration 443 my USCA exchange vale data 5pm 197a 1977 197a m man m1 m2 m3 m4 mg was my mg mg Ban 1991 992 1993 1994 1995 1996 USCA away meves1 vale dMevermal 197a 1977 197a m man m1 m2 m3 m4 mg was my mg mg Ban 1991 992 1993 1994 1995 1996 FIGURE 125 Log of USCA spot and 30day exchange rates and 30day interest rate differential cointegrating residual The hypotheses to be tested are H0 u Y N 1 no cointegration 1214 H1 u Y N 0 cointegration Any unit root test statistic may be used to evaluate the above hypotheses The most popular choices are the ADF and PP statistics Cointegration is found if the unit root test rejects the no cointegration null It should be kept in mind however that the cointegrating residual may include deterministic terms constant or trend and the unit root tests should account for these terms accordingly See Chapter 4 for details about the application of unit root tests Example 76 Testing for cointegration between spot and forward exchange rates using a known cointegrating vector In international nance the covered interest rate parity arbitrage re lationship states that the difference between the logarithm of spot and forward exchange rates is equal to the difference between nominal domes tic and foreign interest rates It seems reasonable to believe that interest rate spreads are 0 which implies that spot and forward rates are coin tegrated with cointegrating vector 3 l 71 To illustrate consider the log monthly spot 5 and 30 day forward ft exchange rates between the 444 12 Cointegration US and Canada over the period February 1976 through June 1996 taken from the SFinMetrics timeSeries object lexratesdat uscns lexratesdatquotUSCNSquot uscnstitle quotLog of USCA spot exchange ratequot uscnf lexratesdatquotUSCNFquot uscnftitle quotLog of USCA 30 day forward exchange ratequot 11 uscns uscnf colIdsu quotUSCNIDquot u title quotUSCA 30 day interest rate differentialquot VVVVVVV The interest rate differential is constructed using the pre speci ed cointe grating vector 7 as ut st 7 t The spot and forward exchange rates and interest rate differential are illustrated in Figure 125 Visually the spot and forward exchange rates clearly share a common trend and the interest rate differential appears to be I In addition there is no clear de terministic trend behavior in the exchange rates The SFinMetrics func tion unitroot may be used to test the null hypothesis that st and ft are not cointegrated u N 11 The ADF t test based on 11 lags and a con stant in the test regression leads to the rejection at the 5 level of the hypothesis that s and f are not cointegrated With cointegrating vector 171 gt unitrootu trendquotcquot methodquotadfquot lags11 Test for Unit Root Augmented DF Test Null Hypothesis there is a unit root Type of Test t test Test Statistic 2881 P value 004914 Coefficients lag1 lag2 lag3 lag4 lag5 lag6 lag7 0 1464 0 1171 0 0702 0 1008 0 1234 0 1940 0 0128 lag8 lag9 lag10 lag11 constant 0 1235 0 0550 0 2106 0 1382 0 0002 Degrees of freedom 234 total 222 residual Time period from Jan 1977 to Jun 1996 Residual standard error 8595e 4 123 ResidualBased Tests for Cointegration 445 1232 Testing for Comtegmtion When the Comtegmtmg Vector 5 Estimated Let Y denote an n X 1 vector of 1 time series and let 3 denote an n X 1 unknown cointegrating vector The hypotheses to be tested are given in 1214 Since 3 is unknown to use the Engle Granger procedure it must be rst estimated from the data Before 3 can be estimated some normalization assumption must be made to uniquely identify it A common normalization is to specify the rst element in Y as the dependent variable and the rest as the explanatory variables Then Y y1Y 2t where Y2 112 gm is an 7 1 X 1 vector and the cointegrating vector is normalized as 3 1 73 Engle and Granger propose estimating the normalized cointegrating vector 32 by least squares from the regression yu c 13th u 1215 and testing the no cointegration hypothesis 1214 with a unit root test using the estimated cointegrating residual 12 2m 7 6 7 QYz 1216 where E and 82 are the least squares estimates of c and 32 The unit root test regression in this case is without deterministic terms constant or con stant and trend Phillips and Ouliaris 1990 show that ADF and PP unit root tests t tests and normalized bias applied to the estimated cointegrat ing residual 1216 do not have the usual Dickey Fuller distributions under the null hypothesis 1214 of no cointegration lnstead due to the spurious regression phenomenon under the null hypothesis 1214 the distribution of the ADF and PP unit root tests have asymptotic distributions that are functions of Wiener processes that depend on the deterministic terms in the regression 1215 used to estimate 32 and the number of variables n 7 1 in Y2 These distributions are known as the Phillips Ouliaris PO distributions and are described in Phillips and Ouliaris 1990 To further complicate matters Hansen 1992 showed the appropriate PO distribu tions of the ADF and PP unit root tests applied to the residuals 1216 also depend on the trend behavior of y1 and Y2 as follows Case I Y2 and y1 are both 1 without drift The ADF and PP unit root test statistics follow the PO distributions adjusted for a con stant with dimension parameter n 7 1 Case II Y2 is 1 with drift and 311 may or may not be 1 with drift The ADF and PP unit root test statistics follow the PO distributions adjusted for a constant and trend with dimension parameter n 7 If n 7 2 0 then the ADF and PP unit root test statistics follow the DF distributions adjusted for a constant and trend 446 12 Cointegration Case III Y2 is 1 without drift and y is 1 with drift In this case 32 should be estimated from the regression y c it 3th at 1217 The resulting ADF and PP unit root test statistics on the residuals from 1217 follow the PO distributions adjusted for a constant and trend with dimension parameter n 7 1 Computing Quantiles and P values from the PhillipsOuliaris Distributions Using the SFinMetrics Functions pcoint and qcoint The SFinMetrics functions qcoint and pcoint based on the response surface methodology of MacKinnon 1996 may be used to compute quan tiles and p values from the PO distributions For example to compute the 107 5 and 1 quantiles from the PO distribution for the ADF t statistic adjusted for a constant with n 7 1 3 and a sample size T 100 use gt qcointc01005001 n5ample100 n5eries4 trendquotcquot statisticquottquot 1 38945 42095 48274 Notice that the argument n series represents the total number of variables n To adjust the PO distributions for a constant and trend set trendquotctquot To compute the PO distribution for the ADF normalized bias statistic set statisticquotnquot The quantiles from the PO distributions can be very different from the quantiles from the DF distributions especially if n 7 1 is large To illustrate the 10 5 and 1 quantiles from the DF distribution for the ADF t statistic with a sample size T 100 are gt qunitrootc O 1 0 050 01 n sample100 trendquotcquot statisticquottquot 1 2 5824 2 8906 3 4970 The following examples illustrate testing for cointegration using an esti mated cointegrating vector Example 77 Testing for cointegration between spot and forward exchange rates using an estimated cointegrating vector Consider testing for cointegration between spot and forward exchange rates assuming the cointegrating vector is not known using the same data as in the previous example Let Yt 5 ft and normalize the cointegrating vector on s so that 3 1 7 2 The normalized cointegrating coefficient 2 is estimated by least squares from the regression St052ftut giving the estimated cointegrating residual 12 st 7 E 7 32ft The OLS function in SFinMetrics is used to estimate the above regression 123 ResidualBased Tests for Cointegration 447 gt uscnts seriesMerge uscn s uscn f gt olsfit OLS USCNSquotUSCNFdatauscnts gt olsfit Call 0LSformula USCNS quotUSCNF data uscnts Coefficients Intercept USCNF 0 0023 1 0041 Degrees of freedom 245 total 243 residual Time period from Feb 1976 to Jun 1996 Residual standard error 0001444 The estimated value of 2 is 1004 and is almost identical to the value 2 1 implied by covered interest parity The estimated cointegrating residual zit is extracted from the least squres t using gt uhat residualsolsfit Next the no cointegration hypothesis 1214 is tested using the ADF and PP t tests Because the mean of zit is zero the unit root test regressions are estimated Without a constant or trend The ADF t statistic is computed using 11 lags as in the previous example and the PP t statistic is computed using an automatic lag truncation parameter gt adf fit unitroot u hat trendquotncquot methodquot adf quot lags11 gt adftstat adffitsval gt adf tstat lagl 2 721 gt ppfit unitrootuhattrendquotncquotmethodquotppquot gt pptstat ppfitsval gt pptstat lagl 5416 The ADF t statistic is 72721 Whereas the PP t statistic is 75416 Since 5 and f are both 1 Without drift the 10 5 and 1 quantiles from the approrpiate Phillips Ouliaris distribution for the ADF t statistic is gt qcointc010005001nsamplenrowuscnsnseries2 trendquotcquot statisticquottquot 1 3062 3361 3942 448 12 Cointegration The no cointegration null hypothesis is not rejected at the 10 level using the ADF t statistic but is rejected at the 1 level using the PP t statistic The p values for the ADF and PP t statistics are gt pcoint adf tstat n samplenrowuscn s n series2 trendquotcquot statisticquottquot 1 0 1957 gt pcoint pp tstat n samplenrow uscn s n series2 trendquotcquot statisticquottquot 1 0 00003925 124 Regression Based Estimates of Cointegrating Vectors and Error Correction Models 1241 Least Square Estimator Least squares may be used to consistently estimate a normalized cointe grating vector However7 the asymptotic behavior of the least squares es timator is non standard The following results about the behavior of 82 if Y is cointegrated are due to Stock 1987 and Phillips 1991 I T08 732 converges in distribution to a non normal random variable not necessarily centered at zero a The least squares estimate 82 is consistent for 32 and converges to 32 at rate T instead of the usual rate TlZ That is7 32 is super consistent 82 is consistent even if Y2 is correlated With u so that there is no asymptotic simultaneity bias ln general7 the asymptotic distribution of T82 7 32 is asymptoti cally biased and non normal The usual OLS formula for computing avar2 is incorrect and so the usual OLS standard errors are not correct Even though the asymptotic bias goes to zero as T gets large 82 may be substantially biased in small samples The least squres estimator is also not ef cient The above results indicate that the least squares estimator of the coin tegrating vector 32 could be improved upon A simple improvement is suggested by Stock and Watson 1993 124 RegressionBased Estimates and Error Correction Models 449 1242 Stock and Watson s E icient LeadLag Estimator Stock and Watson 1993 provide a very simple method for obtaining an asymptotically ef cient equivalent to maximum likelihood estimator for the normalized cointegrating vector 32 as well as a valid formula for com puting its asymptotic variance Let Y iv1 Y Qt where Y2 312 ym is an n7 1 X 1 vector and let the cointegrating vector be normalized as 3 17 Stock and Watson s ef cient estimation procedure is Augment the cointegrating regression of yh on Y2 with appropriate deterministic terms D with 17 leads and lags of AYQ 0 ylt Y Dt 52th Z w9AY2H W 1218 j Y Dt 32th WPAYner wiAY2t1 w0AY2t wL1AY2t71 39 39 wLpAY2p ut Estimate the augmented regression by least squares The resulting estimator of 2 is called the dynamic OLS estimator and is denoted Q DOLS It will be consistent asymptotically normally distributed and ef cient equivalent to M under certain assumptions see Stock and Watson 1993 symptotically valid standard errors for the individual elements of Q DOLS are given by the OLS standard errors from 1218 multiplied by the ratio A2 12 01L 124 where 6 is the OLS estimate of varu and is any consistent estimate of the long run variance of at using the residuals 12 from 1218 Alternatively the Newey West HAC standard errors may also be use Example 78 DOLS estimation of cointegrating vector using exchange rate data3 Let 5 denote the log of the monthly spot exchange rate between two currencies at time t and let ftC denote the log of the forward exchange rate at time t for delivery of foreign currency at time t k Under rational 2Harnilton 1994 chapter 19 and Hayashi 2000 chapter 10 give nice discussions of the Stock and Watson procedure 3This example is based on Zivot 2000 450 12 Cointegration expectations and risk neutrality ftC is an unbiased predictor of 5HC7 the spot exchange rate at time t k That is k 5tk f 5tk where stile is a white noise error term This is known as the forward rate unbiasedness hypothesis FRUH Assuming that s and ftC are 1 the FRUH implies that SHk and ftC are cointegrated with cointegrat ing vector 3 17 71 To illustrate7 consider again the log monthly spot7 5 and one month forward7 f exchange rates between the US and Canada over the period February 1976 through June 1996 taken from the SFinMetrics timeSeries object lexrates dat The cointegrating vec tor between stil and ft1 is estimated using least squares and Stock and Watson s dynamic OLS estimator computed from 1218 with 21h stil 17 Y2 ft1 and p 3 The data for the DOLS regression equation 1218 are constucted as uscndf diffuscnf colIds uscndf quotDUSCNFquot uscndf lags tslaguscndf 33trimT uscnts seriesMerge uscn s uscn f uscndf lags colIds uscnts 1 quotUSCNSquot quotUSCNFquot quotD USCNF lead3quot 4 quotD USCNF lead2quot quotD USCNF leadlquot quotD USCNF lagOquot 7 quotDUSCNF laglquot quotDUSCNF lag2quot quotDUSCNFlag3quot VVVV V The least squares estimator of the normalized cointegrating coef cient 2 computed using the SFinMetrics function OLS is gt summaryULS tslag USCNS 1 USCNF datauscn ts na rmT Call 0LSformula tslagUSCNS 1 quotUSCNF data uscnts narm T Residuals Min 1Q Median 3Q Max 00541 00072 00006 00097 00343 Coefficients Value Std Error t value Prgt t I Intercept 00048 00025 19614 00510 USCNF 09767 00110 886166 00000 Regression Diagnostics R Squared 0 9709
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