Cointegration and Error-Correction

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Chapter 9 Cointegration and Error-Correction In this chapter we will estimate structural VAR models that include nonstationary variables. This exploits the possibility that there could be a linear combination of integrated variables that is stationary. Then, such variables are said to be cointegrated. The key idea behind cointegrated variables is that any equilibrium relationship among a set of nonstationary variables implies that their stochastic trends must be linked. Since the trends of cointegreated variables are linked, the dynamic paths of such variables must bear some relation to the current deviation from the equilibrium relationship. 9.1 Linear Combination of Integrated Variables Following Enders (2010), consider the following example of money demand: where m t = demand for money p t = price level y t = real income r t = interest rate e t = stationary disturbance term β i = parameters to be estimated m t = β 0 + β 1 p t + β 2 y t + β 3 r t + e t (9.1) and all variables but the interest rate are expressed in logarithms. Economic theory predicts that β 1 = 1, β 2 > 0, and β 3 < 0. Moreover, it also predicts that the{e t } sequence is stationary. The problem for the researcher is that real income, demand for money, price level, and interest rate can all be be characterized as nonstationary I(1) variables. This is a problem because Equation 9.1 predicts that the linear combination of these nonstationary variables is stationary. By solving for 105

106 9 Cointegration and Error-Correction the error: e t = m t β 0 β 1 p t β 2 y t β 3 r t (9.2) we can see that because{e t } must be stationary, the linear combination of integrated variables on the right-hand side must also be stationary. Hence, the time paths of the nonstationary variables{m t },{p t },{y t }, and {r t } should be linked. The concept of cointegration was introduced by Engle and Granger (1987). The long-run equilibrium is defined to be: β 1 x 1t + β 2 x 2t + +β n x nt = 0. (9.3) Using matrix notation, the long-run equilibrium can be written as βx t = 0. Then, deviations from the long-run equilibrium e t (called the equilibrium error) are such that e t = βx t (9.4) The equilibrium error must be stationary for the equilibrium to be meaningful. The following definition of cointegration was provided by Engle and Granger (1987): The component of the vector x t =(x 1t,x 2t,...,x nt ) are said to be cointegrated of order d, b, denoted by x t CI(d,b) if 1. All components of x t are integrated of order d. 2. There exists a vector β = (β 1,β 2,...,β n ) such that the linear combination βx t = β 1 x 1t + β 2 x 2t + β n x nt is integrated of order(d b) where b>0. The vector β is called the cointegrating vector. In the example of Equation 9.1, money demand, price level, real income, and interest rate are all I(1) and the linear combination m t β 0 β 1 p t β 2 y t β 3 r t = e t is stationary. Hence, the variables are cointegrated of order (1,1). The vector x t is (m t,1, p t,y t,r t ) and the cointegrating vector β is(1, β 0, β 1, β 2, β 3 ). Some important point about this definition are: 1. Cointegration typically refers to a linear combination of nonstationary variables. Moreover, the cointegrating vector is not unique. If (β 1,β 1,...,β n ) is a cointegrating vector, then for any nonzero value of λ, (λβ 1,λβ 1,...,λβ n ) is also a cointegrating vector. 2. The definition refers to variables that are integrated of the same order; however, this does not mean that all integrated variables are cointegrated. In addition, if two variables are integrated of different orders, they cannot be cointegrated. 1 3. If x t has n nonstationary components, there may be as many as n 1 linearly independent cointegrating vectors. The number of cointegrating vectors is called the cointegrating rank of x t. Consider the following example. If monetary authorities follow the simple feedback rule that says that they decrease the money supply when y t was high and increase the money supply when y t was low. Hence, we have the following representation of this feedback rule: 1 Multicointegration is the term used to refer to relationships among groups of variables that are integrated of different orders.

9.2 Cointegration and Common Trends 107 m t = γ 0 γ 1 (y t + p t )+e 1t (9.5) = γ 0 γ 1 y t γ 1 p t + e 1t with {e 1t } being a stationary error in the money supply feedback rule. Given the money demand function in Equation 9.1, there are actually two cointegrating vectors: [ ] 1 β0 β β = 1 β 2 β 3 1 γ 0 γ 1 γ 1 0 This means that there are two combinations given by βx t that are stationary. Hence, the cointegrating rank of x t is 2. 9.2 Cointegration and Common Trends Following Enders (2010), consider the case in which the vector x t contains only two variables, x t =(y t,z t ). If we write each of these variables as a random walk plus a random component we have: y t = µ yt + e yt (9.6) z t = µ zt + e zt where µ it is a random walk process representing the stochastic trend in variable i, and e it is the stationary component of variable i. If {y t } and {z t } are cointegrated of order (1,1), there must be nonzero values of β 1 and β 2 for which the linear combination β 1 y t + β 2 z t is stationary. Let s consider the sum β 1 y t + β 2 z t = β 1 (µ yt + e yt )+β 2 (µ zt + e zt ) (9.7) = (β 1 µ yt + β 2 µ zt )+(β 2 e yt + β 2 e zt ). Because the second term is stationary, we need the term(β 1 µ yt +β 1 e yt ) to disappear in order to have β 1 y t +β 2 z t stationary. That is, the necessary and sufficient condition for{y t } and {z t } to be CI(1,1) is: β 1 µ yt + β 2 µ zt = 0 (9.8) Since we ruled out the case in which both β 1 and β 2 are zero, Equation 9.8 will hold for all t if and only if: µ yt = β 2 β 1 µ zt (9.9) That is, the only way to achieve equality equality is for the stochastic trends to be identical up to a scalar. Hence, we say that up to the scalar β 2 /β 1, two I(1)

108 9 Cointegration and Error-Correction stochastic processes {y t } and {z t } must have the same stochastic trend if they are cointegrated of order (1,1). 9.3 Cointegration and Error Correction The key idea behind cointegrated variables is that their time paths are influenced by the extent of any deviation from long-run equilibrium. To illustrate the idea behind error-correction models, consider the relationship between long-term an short-term interest rates. In an error-correction model, the short-term dynamics of the variables in the system are influenced by the deviations from the equilibrium. If we assume that both interest rates are I(1), then the model is r St = +α S (r Lt 1 βr St 1 )+ε St α S > 0 (9.10) r Lt = α L (r Lt 1 βr St 1 )+ε Lt α L > 0 (9.11) where ε St and ε Lt are white-noise disturbance terms which may be correlated, r Lt and r St are long- and short-term interest rates, and α S, α L, and β are parameters. The terms in parentheses are deviations from long-run equilibrium, where long-run equilibrium is reached when r Lt = βr St. If deviations are positive(r Lt 1 βr St 1 )> 0, the short-term interest rate would rise and the long-term rate would fall. The key things noticing from Equations 9.10 and 9.10 are: 1. These equations show the relatinship between error-correcting models and cointegrated variables. 2. r St and r Lt are stationary, so the left hand sides are I(0). 3. The right-hand side must be I(0) too. 4. Because ε St and ε Lt are stationary, it must be that the linear combination r Lt 1 βr St 1 is also stationary. 5. The two rates are cointegrated with vector(1, β). 6. The key point is that the two variables need to be cointegrated of order CI(1,1). 7. α S and α L are interpreted as the speed-of-adjustment parameters. The larger the values, the larger the responses from previous period deviations from long-run equilibrium. A more general formulation of these two equations is: r St = a 10 + α S (r Lt 1 βr St 1 ) (9.12) + p i=1 a 11 (i) r St i + p i=1 a 12 (i) r Lt i + ε St α S > 0 r Lt = a 10 α L (r Lt 1 βr St 1 ) (9.13) + p i=1 a 21 (i) r St i + p i=1 a 22 (i) r Lt i + ε Lt α L > 0

9.4 Estimation in Stata 109 11.2 11.4 11.6 11.8 12 12.2 1990m1 1995m1 2000m1 2005m1 time ln of house prices in houston ln of house prices in dallas Fig. 9.1 Logarithm of housing prices in Houston and Dallas where the previous results hold because the additional variables are all stationary. 2 Notice the similarity to the VAR models we say before. This two-variable errorcorrection model is simply a bivariate VAR in first differences augmented by the error correction terms α S (r Lt 1 βr St 1 ) and α L (r Lt 1 βr St 1 ). Estimating r Lt 1 and r St 1 as a VAR in first differences is inappropriate if they have an errorcorrection representation. The omission of the term (r Lt 1 βr St 1 ) entails a misspecification error. 9.4 Estimation in Stata This estimation example follows the Stata manual. Consider the monthly data of the logarithm of average selling prices of houses in Houston and Dallas (houston and dallas, respectively). The series are from January 1990 through December 2003, for a total of 168 observations. The graph of these two variables is shown in Figure 9.1. The plot shows that both variables are trending and potential I(1) processes. 2 For a generalization to the n-variable model, see Hamilton (1994) or Enders (2010).

110 9 Cointegration and Error-Correction 9.4.1 Selection of the Number of Lags Before we test for cointegration or fit the cointegrating vector-error correction model, we must specify how many lags to include. Nielsen (2001) note that the methods used to determine the lag order for a VAR model can also be used for I(1) variables. Hence, we use the same command we employed in the previous chapter: use http://www.stata-press.com/data/r11/txhprice varsoc dallas houston Selection-order criteria Sample: 1990m5-2003m12 Number of obs = 164 +---------------------------------------------------------------------------+ lag LL LR df p FPE AIC HQIC SBIC ----+---------------------------------------------------------------------- 0 299.525.000091-3.62835-3.61301-3.59055 1 577.483 555.92 4 0.000 3.2e-06-6.9693-6.92326-6.85589 2 590.978 26.991* 4 0.000 2.9e-06* -7.0851* -7.00837* -6.89608* 3 593.437 4.918 4 0.296 2.9e-06-7.06631-6.95888-6.80168 4 596.364 5.8532 4 0.210 3.0e-06-7.05322-6.9151-6.71299 +---------------------------------------------------------------------------+ Endogenous: dallas houston Exogenous: _cons As indicated by the star (*) in the output, the HannanQuinn information criterion (HQIC), the Schwarz Bayesian information criterion (SBIC), and the Akaike Information criterion, all selected the model with two lags. 9.4.2 Testing for Cointegration The command vecrank implements the cointegration tests based on the Johansen s method (see Enders, 2010, p.401). In the log likelihood of the unconstrained model that includes the cointegrating equations is significantly different from the log likelihood of the constrained model that does not include the cointegrating equation, we reject the null hypothesis of no cointegration. We usevecrank to determine the number of cointegrating equations: vecrank dallas houston Johansen tests for cointegration Trend: constant Number of obs = 166 Sample: 1990m3-2003m12 Lags = 2 ------------------------------------------------------------------------------- 5% maximum trace critical rank parms LL eigenvalue statistic value 0 6 576.26444. 46.8252 15.41 1 9 599.58781 0.24498 0.1785* 3.76 2 10 599.67706 0.00107 ------------------------------------------------------------------------------- The body of the test presents test statistics and their critical values of the null hypothesis of no cointegration (line 1) and one of fewer cointegrating equations (line 2). In this output we strongly reject the null hypothesis of no cointegration and fail

9.4 Estimation in Stata 111 to reject the null hypothesis of at most one cointegrating equation. Hence, there is one cointegrating equation in the bivariate model. 9.4.3 Fitting the Vector Error-Correction Model The command to estimate the parameters in the vector error-correction model is vec. After determining that there is a single cointegrating equation between dallas and houston, we now want to estimate the parameters of the bivariate cointegrating vector error-correction model using vecrank dallas houston Vector error-correction model Sample: 1990m3-2003m12 No. of obs = 166 AIC = -7.115516 Log likelihood = 599.5878 HQIC = -7.04703 Det(Sigma_ml) = 2.50e-06 SBIC = -6.946794 Equation Parms RMSE R-sq chi2 P>chi2 ---------------------------------------------------------------- D_dallas 4.038546 0.1692 32.98959 0.0000 D_houston 4.045348 0.3737 96.66399 0.0000 ---------------------------------------------------------------- ------------------------------------------------------------------------------ Coef. Std. Err. z P>z [95% Conf. Interval] -------------+---------------------------------------------------------------- D_dallas _ce1 L1. -.3038799.0908504-3.34 0.001 -.4819434 -.1258165 dallas LD. -.1647304.0879356-1.87 0.061 -.337081.0076202 houston LD. -.0998368.0650838-1.53 0.125 -.2273988.0277251 _cons.0056128.0030341 1.85 0.064 -.0003339.0115595 -------------+---------------------------------------------------------------- D_houston _ce1 L1..5027143.1068838 4.70 0.000.2932258.7122028 dallas LD. -.0619653.1034547-0.60 0.549 -.2647327.1408022 houston LD. -.3328437.07657-4.35 0.000 -.4829181 -.1827693 _cons.0033928.0035695 0.95 0.342 -.0036034.010389 ------------------------------------------------------------------------------ Cointegrating equations Equation Parms chi2 P>chi2 ------------------------------------------- _ce1 1 1640.088 0.0000 ------------------------------------------- Identification: beta is exactly identified Johansen normalization restriction imposed ------------------------------------------------------------------------------ beta Coef. Std. Err. z P>z [95% Conf. Interval] -------------+---------------------------------------------------------------- _ce1 dallas 1..... houston -.8675936.0214231-40.50 0.000 -.9095821 -.825605

112 9 Cointegration and Error-Correction _cons -1.688897..... ------------------------------------------------------------------------------ The header contains information about the sample, the fit of each equation, and overall model fit statistics. The first table contains the estimates of the short-run parameters. The second table contains the estimated parameters of the cointegrating vector. Using the notation in Equation 9.12 and 9.14, the estimates can be presented as: dallas t = 0.00561 0.3039(dallas t 1 0.867houston t 1 ) 0.1647 dallas t 1 0.0998 houston t 1 + e 1t houston t = 0.00339+0.5027(dallas t 1 0.867houston t 1 ) 0.0619 dallas t 1 0.3328 houston t 1 + e 2t The variable houston is statistically significant in the cointegrating equation. There is also a relatively fast adjustment toward long-run equilibrium. The estimates show that when the average housing price in Dallas is too high, it quickly falls back towards the Houston level (-0.3039). On the other hand, based on the estimate 0.5027, when the average housing price in Dallas is too high, the average price in Houston quickly adjusts towards the Dallas level. 9.5 Supporting.do files Code to obtain Figure 9.1: use http://www.stata-press.com/data/r11/txhprice rename t time twoway line houston dallas time, m(o) c(l) scheme(sj)

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