Economics 430
Lecture 10 Cointegration
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Spurious Regressions
In reality, rw1 and rw2 are completely unrelated.
Random Walk 1 (rw1): Random Walk 2 (rw2):
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Spurious Regressions
• Suppose we did not know that rw1 and rw2 were unrelated, so we fit a regression model.
• A simple regression of series one (rw1) on series two (rw2) yields:
2 rw+=17.818 0.842r=w , R 0.70
1t 2t (t) (40.837)
These results are completely meaningless, or spurious.
• The apparent significance of the relationship is false
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Spurious Regressions
• When nonstationary time series are used in a regression model, the results may spuriously indicate a significant relationship when there is none
– In these cases the least squares estimator and least squares predictor do not have their usual properties, and t-statistics are not reliable.
– Since many macroeconomic time series are nonstationary, it is particularly important to take care when estimating regressions with macroeconomic variables.
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Unit Root Tests for Stationarity
• Consider an AR(1) process: y = 𝜙𝜙y + e t t-1 t
• When 𝜙𝜙 =1, we get a random walk, therefore
H :𝜙𝜙=1(thisisknownas‘unitroottestfor 0
we can test for stationarity by testing the null
stationarity’)
• We can re-arrange the AR(1) equation as:
Δy =(𝜙𝜙-1)y +e,anddefineγ=𝜙𝜙-1 t t-1t
H :𝜙𝜙=1isequivalenttoH :γ=0 00
(The null is that the series is nonstationary)
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Unit Root Tests for Stationarity
• There are many tests for determining whether a series is stationary or nonstationary
– The most popular is the Dickey–Fuller (DF) test • There are three popular cases:
– DF Test 1 (No Constant and No Trend): Δyt = γyt-1 + et
– DF Test 2 (With Constant but No Trend): Δyt = α+ γyt-1 + et
– DF Test 3 (With Constant and With Trend): Δyt = α+ γyt-1 +λt+ et
Notice that for DF 1, when γ =1, we get a random walk
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DF Test 1
DF Test 1
No Constant and
No Trend
DF Test 2
DF Test 2
With Constant but
No Trend
Constant=0
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DF Test 3
DF Test 3
With Constant and
With Trend
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Unit Root Tests for Stationarity
• To test the hypothesis in all three cases, we simply estimate the test equation by least squares and examine the t-statistic for the hypothesis that γ = 0.
– Unfortunately this t-statistic no longer has the t-distribution.
– Instead, we use the statistic often called a τ (tau) statistic.
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Unit Root Tests for Stationarity
• To carry out a one-tail test of significance, if τc is the critical value obtained from the DF Table, we reject the null hypothesis of nonstationarity if τ ≤ τc
– If τ > τc then we do not reject the null hypothesis that the series is nonstationary
• An important extension of the Dickey–Fuller test allows for the possibility that the error term is autocorrelated. These tests are referred to as augmented Dickey–Fuller (ADF) tests.
– When γ = 0, in addition to saying that the series is nonstationary, we also say the series has a unit root.
– In practice, we always use the augmented Dickey–Fuller test.
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The Dickey-Fuller Testing Procedure
• Plotthetimeseriesofthevariableandselecta suitable Dickey-Fuller test based on a visual inspection of the plot
– If the series appears to be wandering or fluctuating around a sample average of zero, use DF Test 1.
– If the series appears to be wandering or fluctuating around a sample average which is nonzero, use DF Test 2.
– If the series appears to be wandering or fluctuating around a linear trend, use DF Test 3.
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Cointegration
• Whenwecanfindalinearcombinationofintegrated variables that is stationary, we say that these variables are cointegrated.
• Ifytandxtarecointegrated,weexpectthemtoshare similar stochastic trends.
• Cointegrationisthestatisticalnotionthat corresponds to the economic notion of equilibrium.
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Cointegration
• Forexample,assumeytandxtarenonstationaryI(1) variables, then et = yt – β1 – β2 xt is a stationary I(0) process.
• We can test whether yt and xt are cointegrated by using a Dickey-Fuller test (for stationarity) on the LS residuals (since we cannot observe et).
• Basically,iftheresidualsarestationary,thenytandxt are cointegrated.
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Cointegration
• Def: Cointegration Relation yt= β1 – β2 xt +et
• Def: Disequilibrium Error et measures how far the system
yt, xt is from the equilibrium path.
• Equilibrium: et=0
• Disequilibrium: et >0 or et<0
• For Stationary data we can use models that capture short-term features.
• For Non-Stationary data we can use models that capture long-term features.
• Q: How can we integrate short- and long-term dynamics in a time series?
• A: The Error Correction Model (based on cointegration)
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Cointegration
• The test for stationarity of the residuals is based on the test equation:
∆e垐= γe v+ t t−1 t
– The regression has no constant term because the mean of the regression residuals is zero.
– We are basing this test upon estimated values of the residuals.
– The rhs of the equation can include higher order lags if needed to eliminate any dynamics in vt.
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Cointegration
• There are three sets of critical values
– Which set we use depends on whether the
residuals are derived from:
Equation1:eˆ=y b−x ttt
Equation2: eˆ=y− bx− b tt2t1
ˆ Equation3:eˆ=y b−x b− −tδ
tt2t1
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Cointegration
• As an example, consider the two interest rate series (both are non stationary I(1)):
– The federal funds rate (Ft)
– The three-year bond rate (Bt)
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垐?
Cointegration
• Consider the estimated model: ˆ2
B =+1.140 0.914F=, R 0.881 tt
(t) (6.548) (29.421)
• The unit root test for stationarity in the
estimated residuals is:
∆e =− 0.225e + 0.25∆4 e
t t−1 t−1
(tau) (−4.196)
• Use Equation 2 since:
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Cointegration
• The null and alternative hypotheses in the test for cointegration are:
• Similar to the one-tail unit root tests, we reject the null hypothesis of no cointegration if τ ≤ τc, and we do not reject the null hypothesis that the series are not cointegrated if τ > τc.
Result: Reject H0Federal Funds and Bond Rates are cointegrated!
Implication: It means that when the Fed implements monetary policy by changing the federal funds rate, the bond rate will also change thereby ensuring that the effects of monetary policy are transmitted to the rest of the economy.
H0 : the series are not cointegrated ⇔ residuals are nonstationary H1 : the series are cointegrated ⇔ residuals are stationary
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Error Correction Equation
• Consider a general model that contains lags of yt and xt.
• Namely, the autoregressive distributed lag model (ARDL), except the variables are
nonstationary:
where vt are the residuals.
y=+δ θy+ δ+x δx+ v t 1t−10t1t−1t
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Error Correction Equation
• If yt and xt are cointegrated, it means that there is a long-run relationship between them
– To derive this exact relationship, we set yt =yt-1=y,xt =xt-1=xandvt =0
– Imposing this concept in the ARDL, we obtain: y = β1 + β2 x, where β1 = δ/(1-θ1) and
β2 = (δ0+δ1)/(1-θ1)
Cointegrating (or long-run) relationship between x and y
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Error Correction Equation
• Add the term -yt-1 to both sides of the equation: y − y = +δ ( θ − 1 ) y + δ x + δ x + v
t t−1 1 t−1 0t 1t−1 t • Add the term – δ0xt-1+ δ0xt-1:
∆yt =δ+(θ1−1)yt−1+δ0(xt−xt−1)+(δ0+δ1)xt−1+vt – Manipulating this we get:
( ) δ (δ0+δ1) ∆y=θ1− +y + x+δx∆+v
t 1()t−1()t−10tt θ1 −1 θ1 −1
∆y =− α y − β− β x + δ∆ x+ v ()
t t−112t−10tt
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Error Correction Equation
• The expression ∆y =− α y − β− β x + δ∆ x+ v is ()
t t−112t−10tt called an error correction equation
• This is a very popular model because:
– It allows for an underlying or fundamental link between
variables (the long-run relationship).
– It allows for short-run adjustments (i.e. changes) between variables, including adjustments to achieve the cointegrating relationship.
– It also shows we can work with I(1) variables (yt-1,xt-1) and I(0) variables (Δyt-1,Δxt-1) in the same equation .
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Error Correction Equation
• For the bond and federal funds rates example, we have:
ˆ
∆−B = 0.142−B 1−.429 0.777+F 0∆.84−2 F 0∆.327 F
()
t t−1 t−1 t t−1
(t ) (2.857) (9.387) (3.855) – The estimated residuals are
e =B−1.429−0.777F ˆ()
t −1 t −1 t −1
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Error Correction Equation
• The result from applying the ADF test for stationarity is:
∆e =− 0.169e + 0.18∆0 e
t t −1 t −1
(t ) (−3.929)
• Comparing the calculated value (-3.929) with the critical value, we reject the null hypothesis and conclude that (B, F) are cointegrated.
垐?
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