程序代写 ECON7350 Cointegration - I

ECON7350 Cointegration – I

The University of Queensland

Applied Econometrics for Macro and Finance

Dynamic Relationships Between I(1) Processes
What are the implications of working with I(1) processes?
In general, linear combinations of I(1) processes yield another I(1) process; however, in
some cases a linear combination can result in I(0).
Cointegration: I(1) processes are related such that there exists a linear combination
that yields a I(0) process.
Spurious regression: Infer a significant relationship between I(1) processes when in theory one does not exist.

Applied Econometrics for Macro and Finance

Cointegration
Consider the following example:
yt = ξt + νy,t, xt = β1ξt +νx,t,
ξt = ξt−1 + ηt, where the errors νy,t, νx,t and ηt are given by
νy,t =−βε1,t, νx,t =−β1ε2,t, ηt =βε1,t +ε2,t,
and {ε1,t}, {ε2,t} are both (uncorrelated) white noise processes.
This implies that {νy,t}, {νx,t} and {ηt} are also white noise processes.

Applied Econometrics for Macro and Finance

Cointegration
Observe that
∆yt =∆ξt +∆νy,t =ηt +νy,t −νy,t−1
= ε2,t + βε1,t−1 ≡ uy,t,
∆xt = β1∆ξt +∆νx,t = β1ηt +νx,t −νx,t−1, = ε1,t + β1ε2,t−1 ≡ ux,t,
where uy,t and ux,t are uncorrelated.
When examined individually, yt = yt−1 + uy,t and xt = xt−1 + ux,t are random walks.
But the linear combination
yt −βxt =νy,t −βνx,t =−βε1,t +ε2,t
is also I(0).

Applied Econometrics for Macro and Finance

Cointegration and Common Stochastic Trends
In this example, yt and xt are cointegrated.
While each is process is a RW, they in fact wander together.
There exists a long-run stable relationship between yt and xt, such that they never wander too far away from each other.
The key component that ties together yt and xt is the common stochastic trend ξt. Both yt and xt embody stationary movements around the non-stationary ξt.
In economics, many theories suggest that observed economic data are driven by common stochastic trends—variables evolve together over time and there exist long-run equilibria.

Applied Econometrics for Macro and Finance

Common Stochastic Trend Example
Expectation Theory of the Term Structure of Interest Rates suggests rates are I(1), but spreads are I(0).
55 60 65 70 75 80 85 90 95 00 05 10 15
GS3 GS5 SPREAD

Applied Econometrics for Macro and Finance

Definition of Cointegration
Let wt = (w1,t, . . . , wn,t)′ be a n × 1 vector with variables wi,t integrated of order d.
Definition
The components of wt are cointegrated of order d, b, denoted CI(d, b), iff
1 all components of wt are I(d), and
2 there exists a n×1 vector β ̸= 0 such that
zt =β′wt ∼I(d−b), b>0.
The vector β is called the cointegrating vector.

Applied Econometrics for Macro and Finance

Cointegration Rank
For any given n × 1 vector wt, there may exist 0 ≤ r ≤ n − 1 linearly independent cointegrating vectors β1, . . . , βr. We say that wt has cointegration rank r.
Supposewi,t ∼I(1)fori=1,…,nandletB=(β1,…,βr)bean×rmatrixof cointegrating vectors. Then
zt = B′wt is a r × 1 vector with elements zj,t ∼ I(0).
If wt has cointegration rank r, then there are exactly n − r common stochastic trends driving the variables in wt.

Applied Econometrics for Macro and Finance

Real Business Cycles Example
King RG, Plosser CI, Stock JH, and Watson MW, “Stochastic trends and economic fluctuations”, The American Economic Review, 81: 819-840 (1991).
General class of RBC models can be characterized by
Yt = λtKtθN1−θ, t
ln λt = μλ + ln λt−1 + ξt,
where Yt is output, Kt is capital stock, Nt is labour input, and λt is total factor
productivity, which is a RW with drift.
The resource constraint implies Yt = Ct + It, where Ct is consumption and It is investment.

Applied Econometrics for Macro and Finance

Real Business Cycles Example
ln λt/θ is a common stochastic trend with growth rate (μλ + ξt)/θ.
Slight extension of the Neoclassical Growth Model yields that the great ratios Ct and
It are stationary stochastic processes.
Therefore, theory implies that ln Ct, ln It, and ln Yt are all I(1) but the quantities lnCt − lnYt and lnIt − lnYt are I(0).
That is, there exist two cointegrating vectors such that
′ 􏰁1 0 −1􏰂lnCt 􏰁lnCt −lnYt􏰂
Bwt = 0 1 −1 lnIt= lnI −lnY ∼I(0).
KPSW use this to compute impulse responses of economic variables to productivity shocks, identified as innovations in the common stochastic trend. Such IRFs are features of the business cycle.

Applied Econometrics for Macro and Finance

Testing for Cointegration
When I(1) variables cointegrate, a linear combination will be I(0); otherwise all linear combinations will be I(1).
Basic idea to test for cointegration: given a vector wt, estimate the regression wn,t =β0 +β1w1,t +···+βn−1wn−1,t +εt,
and test the residual
ε􏰐 = w − β − β w − · · · − β w
t n,t 0 1 1,t n−1 n−1,t
for a unit root.
Can use the Cointegrating ADF or Cointegrating Regression DW test for this: rejecting
H implies ε􏰐 ∼ I(0), and therefore, evidence of cointegration. 0t

Applied Econometrics for Macro and Finance

Income Consumption Example
ct: (log) consumption; yt: (log) income;
at: (log) wealth;
and arrange them into a 3 × 1 vector wt = (ct, yt, at)′.
Extension of the Permanent Income Hypothesis states that for β = (1, −1, β)′,
zt = β′wt = ct − yt − βat is stationary. To test this, regress
ct − yt = β0 + β1at + εt,
􏰐􏰐 andtestforstationarityofε􏰐 =c −y −β −β a.
Use, for example, Cointegrating ADF with (in this case) two variables—Table C of Enders.

Applied Econometrics for Macro and Finance

Cointegration and ARDLs
Cointegration may also be tested with an ARDL. In general, an ARDL(p, l) may contain the following cases.
Case yt SS Relationship Cointegration 1 I(0)YN
4 I(0)NN 5 I(1)NN 6 I(2)NN 7 I(1)NN 8 I(1)NN
Only cases 1-2 result in an equilibrium relationship. Only cases 1-4 admit an ECM representation.
Case 8 is an ARDL in stationary ∆yt and ∆xt.
a(1)=0 b(1)=0 xt
N N I(0) N N I(1) N Y I(0) N Y I(1) Y N I(0) Y N I(1) Y Y I(0) Y Y I(1)

Applied Econometrics for Macro and Finance

Estimating Cointegrating Relations
We are often interested in the cointegrating relation itself, not just that it exists.
All examples so far assumed cointegrating vectors are either fully or partially known.
In the permanent income hypothesis example,
1 B = −1.
In the RBC example, we can generalize slightly to obtain
1 0 B=0 1.

Applied Econometrics for Macro and Finance

Estimating Cointegrating Relations
In the permanent income hypothesis example, there is one cointegrating vector, and therefore, one cointegrating relation.
In the RBC example, there are two cointegrating vectors, and therefore, two cointegrating relations.
In both cases, the cointegrating vectors are unique and we can estimate unknown components from data.
PIH: regressing ct − yt on at yields a consistent OLS estimator of β.
RBC: regressing ln Ct on ln Yt yields a consistent OLS estimator of β1 and
regressing ln It on ln Yt yields a consistent OLS estimator of β2.
This works because B in both examples contains sufficient restrictions: 1s and 0s in pre-determined places.

Applied Econometrics for Macro and Finance

Cointegration Space
In general, if zt = B′wt is I(0), then for any invertible K, B􏰜 = BK is another n × r matrix that yields 􏰜zt = B􏰜′wt = K′zt being I(0).
Multiplying B by K yields linear combinations of the vectors in B. The r vectors formed by these linear combinations are also valid cointegrating vectors.
Implication: unrestricted B cannot be estimated from data because only the space spanned by (true) cointegrating vectors β1, . . . , βr (cointegration space) is identified.
For identification, restrictions on B are necessary:
Impose 0 and 1 restrictions based on theoretical considerations (as in examples), or Impose the restriction B′B = Ir, i.e., identify the basis of the cointegration space.

Applied Econometrics for Macro and Finance

Properties of OLS for Cointegrated Variables
If yt and xt are cointegrated, then OLS estimator β0,β1 in yt = β0 + β1xt + εt
will be superconsistent. Normally,
√ T(β􏰐−β)−→N(0,V).
That is, β􏰐 converges at rate T. For cointegrated time-series,
T (β􏰐 − β) −→ N (0, V). That is, β􏰐 converges at rate T .
When cointegrating relationships exist, OLS estimates reliably even in small samples; when they do not exist, OLS leads to spurious relationships.

Applied Econometrics for Macro and Finance

Spurious Regression
Let xt = xt−1 + εx,t and yt = yt−1 + εy,t, εx,t and εy,t are independent white noise. Since xt and yt are independent random walks, there is no relationship between them.
However, regressing yt = β0 + β1xt + εt typically yields large, significant β1.
The sampling distribution of β1−β1 under H : β = 0 is not Student-t; it is more 0 1
se(β1) 􏰒􏰀􏰀β􏰀􏰀 􏰓
= Pr(reject H0 |H0 is true) > α.
A t-test will tend to reject H0 incorrectly (i.e. α is not the correct significance level).
dispersed so that
􏰀se(β1)􏰀 > tα,T−k 􏰀􏰐􏰀

Applied Econometrics for Macro and Finance

Spurious Regression
In fact, as T −→ ∞, the estimator β1 diverges.
OLS is not consistent, and there is no well-defined sampling distribution.
t-statistics based on OLS increase with sample size—more data exacerbates the problems with t-tests.
Significant t-tests occur because even pure random walks tend to resemble trends; when two I(1) variables wander in the same direction for a while, we might mistake this for cointegration.
But in yt = β0 + β1xt + εt, the residual εt is I(1), so we would expect low Durbin-Watson statistics.

Applied Econometrics for Macro and Finance

Estimating Cointegrating Relations In Practice
Practical signs of a spurious regression:
Significant t-statistics; high R2 values; low Durbin-Watson statistics.
However, a regression of yt on xt is also an ARDL(0, 0).
If we use the familiar approach to construct an adequate set of ARDL models, then
there is no explicit need to worry about spurious regressions!
The ARDL(0, 0) would be excluded from such a set based on considerations other
than R2 and t-tests.
A cointegrating relation in an ARDL model is simply the long-run equilibrium, as
represented by the error correction term.
It is a cointegrating relation if and only if a(1) ̸= 0, b(1) ̸= 0 and xt is I(1).

Applied Econometrics for Macro and Finance

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