ESTIMATING A VECM in STATA IN STATA type: webuse urates vec missouri indiana illinois, trend(rconstant) rank(2) lags(2) and we get the output on the following page. This is a VECM a vector error correction model. It is linked in to Johansen’s methodology related to cointegration. Consider a VAR with p lags.

where π is to be interpreted as in Johansen lectures.

not identiified. We will not go into the nature of those restrictions. β is a matrix containing the cointegrating vectors (if more than one, otherwise just a vector). Hence βyt-1 can be regarded as an error correction element. α is then a speed of adjustment vector. For example with 3 variables and two cointegrating vectorsand ignoring v and setting π= αβ then we can rewrite (8) as’: Δy1t = α11(β11y1t-1 +β12y2t-1 β13y3t-1) + α12(β21y1t-1 +β22y2t-1 β23y3t-1) +γ11Δy1t-1 + γ12Δy2t-1 + γ13Δy3t-1 Δy2t = α21(β11y1t-1 +β12y2t-1 β13y3t-1) + α22(β21y1t-1 +β22y2t-1 β23y3t-1) +γ11Δy1t-1 + γ12Δy2t-1 + γ13Δy3t-1 Δy3t = α31(β11y1t-1 +β12y2t-1 β13y3t-1) + α32(β21y1t-1 +β22y2t-1 β23y3t-1) +γ11Δy1t-1 + γ12Δy2t-1 + γ13Δy3t-1

This implies that all three variables adjust to the two error correction terms, but at different speeds relating to the α’s. OK now lets look at the output. There are three variables and rank(2) specified to estimate two equilbrium relationships (the optimum number should have been tested for earlier). 310 observations for each variable. Each of the three equations has its own R2. The two speeds of adjustment in the first equation are: α11= -0.065 and 0.044 The first equilbrium relationship (see end of output) is Missouri -078Illinois -0.405 The second is Indiana-1.294Illinois +2.76 I believe the variables relate to unemployment. This suggests that the labour market in Illinois is linked to both Missouri and to Indiana, but there is no direct link between the labour markets in Missouri and Indiana. This makes sense Illinois shares borders with both the other states, but there is no border between Missouri and Indiana. Oddly enough both error correction terms are significant for Missouri but only one each for the other two states. Given

what I have just said I would have expected them both to be significant for Illinois not Missouri. But this is only a very preliminary analysis. The lag depth says 2. I believe this means that t-2 is the farthest back the data goes which allows us to include e.g. Δy2t-1 in the short run adjustment factors, but not Δy2t-2 for that we would lag depth of 3. The short run adjustment parameters are e.g. γ11 =0.301 with a t statistic of 5.16. This is significant but not all the short run parameters are.

_cons 2.760894 .6408888 4.31 0.000 1.504775 4.017012 illinois -1.294305 .0898285 -14.41 0.000 -1.470366 -1.118245 indiana 1 . . . . . missouri 5.55e-17 . . . . ._ce2 _cons -.4050136 .4773932 -0.85 0.396 -1.340687 .53066 illinois -.7797688 .0669126 -11.65 0.000 -.910915 -.6486225 indiana (omitted) missouri 1 . . . . ._ce1 beta Coef. Std. Err. z P>|z| [95% Conf. Interval] Johansen normalization restrictions imposed

Identification: beta is exactly identified

_ce2 1 207.6084 0.0000_ce1 1 135.805 0.0000 Equation Parms chi2 P>chi2

Cointegrating equations

LD. .1950483 .057376 3.40 0.001 .0825935 .3075032 illinois LD. .1201587 .0635256 1.89 0.059 -.0043492 .2446665 indiana LD. .1398911 .0705116 1.98 0.047 .0016909 .2780914 missouri L1. .0512544 .0129247 3.97 0.000 .0259225 .0765864 _ce2 L1. .0242664 .0236293 1.03 0.304 -.0220462 .0705791 _ce1 D_illinois LD. .1006186 .0559413 1.80 0.072 -.0090244 .2102616 illinois LD. .1861303 .0619372 3.01 0.003 .0647357 .307525 indiana LD. .0864837 .0687485 1.26 0.208 -.0482609 .2212284 missouri L1. .0385532 .0126015 3.06 0.002 .0138546 .0632517 _ce2 L1. -.0229019 .0230385 -0.99 0.320 -.0680565 .0222527 _ce1 D_indiana LD. .1225543 .0475326 2.58 0.010 .0293922 .2157164 illinois LD. .0171618 .0526271 0.33 0.744 -.0859855 .1203091 indiana LD. .3013632 .0584147 5.16 0.000 .1868726 .4158539 missouri L1. .0437556 .0107073 4.09 0.000 .0227695 .0647416 _ce2 L1. -.0647633 .0195755 -3.31 0.001 -.1031305 -.026396 _ce1 D_missouri Coef. Std. Err. z P>|z| [95% Conf. Interval]

D_illinois 5 .194787 0.2197 85.57777 0.0000D_indiana 5 .189917 0.1499 53.58711 0.0000D_missouri 5 .161369 0.2407 96.37098 0.0000 Equation Parms RMSE R-sq chi2 P>chi2

Det(Sigma_ml) = .0000287 SBIC = -1.592109Log likelihood = 301.2743 HQIC = -1.729574 AIC = -1.821125Sample: 1978m3 - 2003m12 No. of obs = 310

Vector error-correction model