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Co-integration and VECM Yi-Nung Yang CYCU, Taiwan May, 2012 不 列 1

Learning objectives Integrated variables Co-integration Vector Error correction model (VECM) Engle-Granger 2-step co-integration test Johansen co-integration test Page 2

Integrated variables Y t is an I(n) ) variable if its n-th difference n Y t ~ I(0) We call Y t is integrated of order n. n An I(0) variable is a stationary variable E.g., Y t is an I(1) variable if its 1 st -difference: Y t ~ I(0) Page 3

Summation of I(n) variables I(m) ) + I(n) ) == I(m) if m n (m,n are integers) Y ~ I(1), X ~ I(0) Let z=y + X, then z ~I(1) I(n) ) + I(n) ) == I(n) Example: Y ~ I(1), X ~ I(1) Let z=y + X, then z ~I(1) The only exception is If Y and X are cointegrated Page 4

Co-integration If Y ~ I(n), X ~ I(n), and n 1 Let z=β 1 Y + β 2 X if z ~I(0), then we call Y and X are cointegrated or There is a co-integration relationship between these two none-stationary variables, Y and X. Page 5

Def: Co-integration There exists a linear combination of the none- stationary variables (integrated of the same orders) that is stationary There could be more than one such linear combination if there are more than 2 variables involved in studies The basic idea of cointegration relates closely to the concept of unit roots Economic implications There exists (long-run) equilibrium relationships between (among) none-stationary variables. Page 6

Co-integrating vectors for example, a Demand of money model: m p = γ 0 + γ 1 y + γ 2 r + e where m: nominal money supply p: price level y: income r: interest rate e: error term Page 7

Co-integrating vectors In equilibrium m p γ 1 y γ 2 r γ 0 = e The above model can be rewritten as x β = e where x = (m, p, y, r, const) β = (1,( -1, -γ 1, -γ 2, -γ 0 ) β is called Co-integrating vector Page 8

VAR of order p and VECM Consider a VAR of order p with a deterministic part given by A 0 (all in matrix form) y t = A 0 +A 1 y t 1 + A 2 y t 2 + + A p y t p + e t we can re-write the above as VECM (it is a long story ) y t =A 0 + y t 1 + 1 y t 1 + + p-1 y t p+1 p+1+ + e t Please refer to Enders (2008) Page 9

VAR of order p and VECM For example, a VAR(1) model y t = A 0 +A 1 y t 1 + e t we can re-write the above as VECM y t =A 0 + y t 1 + e t can be decomposed into =αβ α: : matrix of short-run run adjusting coefficients β: : co-integrating vector (matrix) Page 10

Granger representation theorem Let Y = (y 1, y 2,, y k ), Y~I(n) If all variables in Y are cointegrated Co-integration There exist at least one co-integrating vector to let: βy Y ~ I(n-j), for any j 1. j VECM If and only if the variables in Y are co-integrated, there must exist a VECM Cointegration VECM Page 11

VAR(1) system as an example y t = A 0 +A 1 y t 1 + e t VECM y t =A 0 +αβ y t 1 + e t (if all variables are co-integrated) Page 12

Key concepts before running software All non-stationary variables integrated of the same order may have co-integration relationship Single equation version of cointegration example y t = a + βx t + u t, y t, x t ~ I(1) but u t ~I(0) Co-integrating vector (1, -β, -a) ECM (error correction model) y t = a -α(y t-1 -x t-1 )+β x t + u t =a -α u t-1 +β x t + u t Page 13

Engle-Granger co-integration test Engle-Granger 2-step co-integration test make sure orders of integrated variables are the same. Usually, they are all I(1) variable Step 1. run OLS by regressing y t on x t s save residuals as a variable, e.g., e t Step 2. use ADF test to see whether e t is a I(0) variable: If e t ~ I(0), then the variables are cointegrated; otherwise, there is no cointegration among(between) those variables. Page 14

Plots of Y1 and X1 Page 15

Case 9.1 spurious regression and Engle-Granger co-integration test 1. use gretl to open ex-coint.wf1.run regression as follows y1 t =a 0 +a 1 x1 t Results Page 16

Case 9.1 spurious regression and Engle-Granger co-integration test 2. Conduct ADF tests on Y1 and X1 (you may use or as follows) Strategies (a) show ADF statistics with various lags: no matter how many lags, they all have unit roots (b) use any information criterions to choose the optimal lags and report them in tables Notes: ADF tests should be done with level and 1 st differenced variables Page 17

Case 9.1 spurious regression and Engle-Granger co-integration test 2.(a) show ADF statistics with various lags: no matter how many lags, they all have unit roots Variable ADF-statistic 1% c.v. 5% c.v. Note: c.v. are from Eviews. Page 18

Case 9.1 spurious regression and Engle-Granger co-integration test 2.(b) use any information criterions to choose the optimal lags and report them in tables Suggestions: Sample size <200, use AIC Sample size >300, use BIC or HQC Page 19

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Case 9.1 spurious regression and Engle-Granger co-integration test 2.(b) use any information criterions to choose the optimal lags and report them in tables variable ADF P-value lags AIC Y1 t X1 t Y1 t X1 t -0.351 0.155-12.188-11.816 0.56 0.73 0.00 0.00 0 0 0 0 93.39 148.62 72.17 135.88 Page 22

Case 9.1 spurious regression and Engle-Granger co-integration test 3(a). Save residuals from OLS in step 1 as e1 t, then do ADF test as in step 2(a) or 2(b) on these residuals. Variable ADF-statistic 1% c.v. 5% c.v. Note: c.v. are from Eviews. Page 23

Case 9.1 spurious regression and Engle-Granger co-integration test 3(b). Use gretl: [model]->[time series] ->[Cointegration test] ->[Engle-Granger] Page 24

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9.5 Johansen co-integration test Johansen co-integration test procedure 1. use VAR lags selection to choose p 2. by Johansen s method, testing if variables are co-integrated. 3. make sure rank( ) to see how many cointegrating vectors: i.e., based on trace test or max tests. 4. normalizing integrating vectors and give an interpretations Page 26

Rank and the number of co-integrating vectors The number of Rank of = the number of co-integrating vectors Page 27

trace test or max tests Two often used in determine the number of co- integrating vector (number of Rank( ) Page 28

Normalizing integrating vectors If β 1 Y + β 2 X = e this can be re-written as Y + (β( 2 /β 1 ) X = e/β 1 This is called normalization (on β,, co- integrating vector). Page 29

Case 9.2 Johansen co-integration test 1. plot x, y, z Page 30

Case 9.2 Johansen co-integration test Selection of VAR lags (e.g., max lag=8) Use BIC to select p=1 Page 31

Case 9.2 Johansen co-integration test Fill with p+1 Page 32

Case 9.2 Johansen co-integration test results (I) Page 33

Case 9.2 Johansen co-integration test results (II) Page 34

Case 9.2 Johansen co-integration test There is only 1 Cointegrating vector β=(1, β y, β z, const) =(1.000, -0.984, -1.018, -0.013) This implies a long-run relationship: e t-1 = X t-1-0.984y t-1-1.018z t-1-0.013 Page 35