ECON7350 Cointegration – II
The University of Queensland
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Applied Econometrics for Macro and Finance
Inference on Long-Run Relationships
Many economic variables “behave like” I(1) processes, but theory suggests there exist linear combinations of them that are I(0).
Such linear combinations, or cointegrating relations, represent long-run equlibrium relationships: we say that the variables involved are cointegrated.
Single equation models such as simple regressions and ARDLs can be used to estimate and test for cointegrating relations.
Data can only identify the cointegration space: linearly independent vectors β1, . . . , βr up to linear combinations, but not the individual β1, . . . , βr themselves.
It’s not a big concern when r = 1, but becomes increasingly complicated for r > 1. Cointegrating relations are driven by common stochastic trends.
Applied Econometrics for Macro and Finance
Common Stochastic Trends
Suppose the elements of a n × 1 vector wt are I(1) and cointegrated, such that all elements of the r × 1 vector zt = B′wt are I(0), for some n × r matrix B with linearly independent columns.
In this case, there exists a representation:
wt = Cξt + ζt,
C = (c1, . . . , cn−r) is n × (n − r) with linearly independent columns; ξt is (n − r) × 1 with each element being I(1);
ζt is n × 1 with each element being I(0);
ξt are the common stochastic trends; we are often interested in estimating these also.
Applied Econometrics for Macro and Finance
Identification of Common Stochastic Trends
If we have estimates of the cointegration space, can we estimate the common stochastic trends? Yes, up to linear combinations.
Firstly, if B′C = 0 for any B, then setting B = BK yields B′C = K′B′C = 0. So, estimating the cointegration space is sufficient.
At the same time, if L is any (n − r) × (n − r) invertible matrix, then setting C = CL yields B′C = B′CL = 0.
Therefore, if ξt = L−1ξt,
w t = C ξ t + ζ t = C ξ t + ζ t and ξt also contains n − r common stochastic trends.
Data can only identify the space spanned by common stochastic trends.
Applied Econometrics for Macro and Finance
UK Earnings Equation
Pesaran MH, , J, “ Approaches to the Analysis of Level Relationships”, Journal of Applied Econometrics, 16: 289-326 (2001), estimate an earnings equation for the UK.
In theory, there exists a long-run equilibrium between the following variables:
wt : URt : wedget : uniont : prodt :
real wage;
unemployment rate;
“real product wage” vs “real consumption wage”; level of unionisation;
labour productivity.
Applied Econometrics for Macro and Finance
UK Earnings Equation Long Run Estimates
Pesaran et al. (2001) use an ARDL in wt to obtain:
wt = 2.701 − 0.105URt − 0.943wedget + 1.481uniont + 1.063prodt + vt
(0.242) (0.034) (0.265) (0.311) (0.050)
What is the interpretation?
Let us focus on the relationship between wages and productivity, both of which behave
like I(1) processes.
These estimates suggest that a 1% increase in productivity leads to approximately a
1% increase in wages in the long-run.
However, such a conclusion requires certain assumptions.
Applied Econometrics for Macro and Finance
Assumptions Needed for the Estimated UK Equation
There are five variables involved, which means that there may exist up to four cointegrating relations.
The ARDL based long-run multiplier effects require the following.
1 Exactly one of these cointegrating relations involves the dependent variable wt.
2 No other variable besides wt adjusts to vt.
3 The independent variable prodt behaves like a random walk (Standard Assumption 2.2).
A shock to prodt is immediate and permanent, such that prodt = prodt−1 + δ implies prodt+h = prodt−1 + δ for all h ≥ 0, assuming no other shocks to productivity occur.
Under these assumptions, a simple regression also yields a consistent but less efficient OLS estimator relative to an ARDL specification.
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
1 1.2 1.4 1.6 1.8 2
w = 5 + 1.063 prod + v ttt
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
1 1.2 1.4 1.6 1.8 2
w = 5 + 1.063 prod + v ttt
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
1 1.2 1.4 1.6 1.8 2
prod prod t-1
w = 5 + 1.063 prod + v ttt
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
1 1.2 1.4 1.6 1.8 2
w = 5 + 1.063 prod + v ttt
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
1 1.2 1.4 1.6 1.8 2
w = 5 + 1.063 prod + v ttt
prod = prod
Applied Econometrics for Macro and Finance
Error Correction in the UK Earnings Equation
w = 5 + 1.063 prod + v ttt
t+2 t+1 t t-1
prod = prod = pro t t+1
1 1.2 1.4 1.6 1.8 2
Applied Econometrics for Macro and Finance
Summary of Pesaran et al. (2001) Results
A 1% permanent increase in labour productivity leads to an estimated 1.063%
permanent increase in wages.
It takes approximately three quarters for wages to fully adjust.
This is obtained from the speed of adjustment coefficient α.
An estimate of α is obtained with an ARDL, but not with a simple regression.
The ARDL specification is not sufficient to infer all the possible cointegrating relationships, or even how many there are.
If we could estimate all possible cointegrating relationships, then we could also estimate the common stochastic trends.
Applied Econometrics for Macro and Finance
Artificial Example with r = 1
Suppose we have a DGP where the three variables yt, xt and wt are each I(1), but zt =yt−xt+0.5wt isI(0)withE(zt)=2.85.
If we generate T = 10000 observations from such a DGP (using, say, a computer), what should we expect by running regressions with different combinations of dependent variables?
Experiment results:
yt = 2.8482 + 0.9933xt − 0.4944wt + vy,t,
xt = 8.4154 + 0.9843yt + 0.5015wt + vx,t, wt = −21.2643 − 1.9331yt + 1.9787xt + vw,t.
This is exactly what we expect to get (with z = 2.8482 + v , etc.)! t y,t
Applied Econometrics for Macro and Finance
Impulse Response Functions, r = 1
0 10 20 30 40 50 60 70 80
Applied Econometrics for Macro and Finance
Common Stochastic Trend, r = 1
The reason both the OLS regression and the ARDL model work well is because there is exactly one cointegrating vector β = (1, −1, −0.5)′.
The entire cointegration space is a scalar multiple of β, so it is easy to estimate with single equation models (note that we fix the scale by choosing one of the variables to be the dependent variable).
Besides dynamic responses and LRMs, we can use these estimates to also construct the common stochastic trends.
With n = 3 and r = 1, there are two common stochastic trends since there exist two linearly independent vectors cj = (c1,j,c2,j,c3,j)′, j = 1,2, that satisfy
c1,j − c2,j − 0.5c3,j = 0.
However, c1 and c2 are not uniquely identified from β—only the linear space spanned by these two vectors is; hence, we need to impose a suitable normalisation in practice.
Applied Econometrics for Macro and Finance
Common Stochastic Trends, r = 1
0 50 100 150 200
Applied Econometrics for Macro and Finance
Artificial Example with r = 2
Now, suppose we have a DGP where the three variables yt, xt and wt are each I(1), but there are two possible linear combinations that yield an I(0) variable:
z1,t = yt − xt, z2,t = xt − wt.
Same experiment as before yields regression results:
yt = 0.1396 + 0.5935xt + 0.4064wt + vy,t, suggesting the LRM of xt on yt is about 0.5935 and wt on yt is 0.4064.
Alternatively, using an ARDL specification, we obtain the LRMx = 0.9645 and LRMw = 0.0356.
Both are consistent estimates of the LRMs, but ARDL in this case is much more efficient (even at T = 10000, consistency isn’t very useful for a reg.).
Applied Econometrics for Macro and Finance
Common Stochastic Trend, r = 2
The reason ARDL still works is because with yt as the dependent variable:
1 exactly one of the cointegrating relations is applicable;
2 xt and wt in the experiment do not adjust to z1,t.
Consequently, the subspace of the cointegration space we are estimating is a single vector, so it is identified up to scale (which is fixed by having yt as the dependent variable).
With n = 3 and r = 2, there is only one common stochastic trend since there is only one vector c = (c1, c2, c3)′ that satisfies
c1 −c2 =0, c2 −c3 =0, up to a scalar multiple.
To construct a common stochastic trend, we can set c = c∗(1, 1, 1)′.
Applied Econometrics for Macro and Finance
Common Stochastic Trend, r = 2
4 3 2 1 0 -1 -2 -3 -4
0 50 100 150 200
Applied Econometrics for Macro and Finance
An simple regression involving I(1) variables:
allows to test if r = 0 versus 0 < r < n, but cannot it does not provide information on r itself in the latter case;
provide consistent estimates of the cointegrating relation involving the depedendent variable when 0 < r < n under certain conditions;
yields spurious results when r = 0 and may be inefficient when r > 0.
An ARDL model involving I(1) variables nests the simple regression.
It still cannot helps us draw inference on r, other than to test H0 : r = 0.
It avoids spurious results and is generally more efficient provided the lag lengths are sufficient to yield a white noise residual.
It is still limited to certain conditions holding.
A more general analysis of cointegrating relations requires multivariate models: we will return to these towards the end of the course.
Applied Econometrics for Macro and Finance
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