FAME
Cointegration
 Most economics variables have trend, either
deterministic or stochastic and can lead to spurious
regressions
 Model looks good in terms of summary stats but
difficult to evaluate
 We have talked about taking differences of SP to make
them stationary
 From the differenced series can use ARIMA models to
evaluate SP or with stationary variables can carry out
regression analysis
 One serious problem using first differences is that can
not recover long-run properties of the data
 First differences can measure short run properties
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 But, a rather new concept called Cointegration looks at
the levels of each series not differences.
 Say we have two series 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 in level form
 So both not stationary
 (draw Fig 1)
 Both 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 have trend but not sure ~I(1) or ~I(2) in
addition two series drift apart overtime
 This implies that the difference between the series is
not stable.
 (draw Fig 2)
 In Fig 2 variables drifting together over time
 Each series could be ~I(1) and perhaps the difference
𝑦𝑡 − 𝑥𝑡 is stationary!
 The idea that differences between two SP in levels is
stationary defines the concept of cointegration
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 For two SP processes 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 are cointegrated of
order (d, b) where 𝑑 ≥ 𝑏 ≥ 0
 If a) 𝑥𝑡 , 𝑦𝑡 ~ 𝐶𝐼(𝑑, 𝑏) 𝑖𝑓 𝑥𝑡 , 𝑦𝑡 ~𝐼(𝑑)
 b) there exists a linear combination 𝛼1 𝑥𝑡 +
𝛼2 𝑦𝑡 ~ 𝐼(𝑑 − 𝑏)
 The vector 𝛼1 , 𝛼2 called cointegrating vector
 Now, what is interesting for econometrics is when
(d – b) = 0 because this says that there exists a
cointegrating vector that makes the linear combination
between 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 stationary.
 Moreover it defines the long run relationship between
𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡
 E.g. say 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 both ~I(1) and long run relationship
𝑦𝑡∗ = 𝛽𝑥𝑡
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 If the cointegrating vector is
[𝛽, −1] ~ 𝐶𝐼(𝑑, 𝑏) 𝑜𝑟 ~𝐶𝐼(1, 1)
 Then deviations of 𝑦𝑡 − 𝑦𝑡∗ from Long Run path are
~I(0)
 And [𝛽, −1] defines the long run relationship
 Now, 𝑦𝑡 − 𝑦𝑡∗ = 𝑦𝑡 − 𝛽𝑥𝑡 = 𝜇𝑡 is called
error correction mechanism ECM.
 Think of it as forcing the model to maintain the long run
relationship between 𝑦𝑡 𝑎𝑛𝑑 𝑥𝑡 .
 Lets extend basic regression model and write the model in
stationary variables
 Δ𝑦𝑡 = 𝛽1 Δ𝑥𝑡 + 𝛽2 𝐸𝐶𝑀𝑡−1 + 𝜀𝑡
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 Each argument is ~I(0), so reasonable model which includes
both short run and long run properties.
 𝛽1 defines short run relationship and 𝛽2 speed of adjustment
to deviations from the long run trend.
 Testing for cointegration
 Engle and Granger first method for testing
 Idea is that if cointegrated then error terms should be
stationary, so from the long run model test the errors for
stationary properties using D-F statistic
 But a bit of a problem, how to write the model
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 𝑦𝑡 = 𝑏01+ 𝑏11 𝑥𝑡 + 𝜀𝑡1 𝑜𝑟
 𝑥𝑡 = 𝑏02 + 𝑏12 𝑦𝑡 + 𝜀𝑡2
 large samples it does not matter get same result but in small
samples get different results.
 Problem just gets worse with more than two variables.
 Johansen test is a procedure that allows multiple variables
and multiple cointegrating vectors and can test
 The test relies on the relationship between rank of a matrix
and its characteristic roots.
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 Technical procedure but really a multivariate generalization
of the DF test.
 Enders provides a useful description (page 386)
 In the single variable case
 𝑦𝑡 = 𝑎1 𝑦𝑡−1 + 𝜀𝑡 𝑎𝑛𝑑 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑦𝑡−1
 Δ𝑦𝑡 = (𝑎1 − 1)𝑦𝑡−1 + 𝜀𝑡 𝑖𝑓 𝑎1 − 1 = 0 unit root --nonstationary but if 𝑎1 − 1 < 0 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦
 Now, if N variables 𝑋𝑡 = 𝐴𝑖 𝑋𝑡−1 + 𝜀𝑡 ,
 𝑋 𝑖𝑠 (𝑥1𝑡 , 𝑥2𝑡 , ⋯ 𝑥𝑁𝑡 )′ 𝑎𝑛𝑑 𝜀 = (𝜀1𝑡 , 𝜀2𝑡 , ⋯ 𝜀𝑁𝑡 )′
𝐴1 𝑖𝑠 𝑛 ∗ 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥
 Then Δ𝑋𝑡 = −(𝐼 − 𝐴1 )𝑋𝑡−1 + 𝜀𝑡 = (𝐴1 − 1)𝑋𝑡−1 + 𝜀𝑡
 = Π𝑋𝑡−1 + 𝜀𝑡
 The rank of (A1 -1) = number of cointegrating vectors
 If rank = 0 all Δ𝑋 are unit root, no cointegration
 If rank = N full rank all variables are stationary
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 Could add a drift term for trend
 The model can be generalized to allow for higher – order
auto-regressive process
 Number of distinct cointegrating vectors = number of
significant characteristic roots of Π .
 Two tests for determining significant roots
 Trace test
 𝜆𝑡𝑟𝑎𝑐𝑒 = −𝑇 ∑𝑛𝑖=𝑟+1 𝑙𝑛 (1 − 𝜆̂𝑖 )
 𝜆̂𝑡 estimated values of characteristic roots
 𝑇 number of observations
 Ho number of cointegrating vectors ≤ 𝑟
 Ha : Ho not true


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 Max test
 𝜆𝑚𝑎𝑥 = −𝑇𝑙𝑛(1 − 𝜆̂𝑟+1 )
 Ho number of cointegrating vectors = 𝑟
 Ha : number of cointegrating vectors =r+1
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