1. Unit Roots
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2. Dickey-Fuller test The test is based on an AR(1) regression
Under the null hypothesis, the process has a unit root. Under the alternative, it is a stable AR(1) process
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3. Dickey-Fuller test
One can use the t-statistic to test the hypothesis, but under the null, it has not got the t-distribution but the Dickey-Fuller distribution
The Dickey-Fuller distribution has to be looked up in tables
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4. Alternative Formulation of the Dickey-Fuller test
The alternative representation is obtained by subtracting yt-1 from both sides
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5. Critical values for Dickey-Fuller test
The critical value is much more negative than it would be in a t-distribution
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6. Example: Unit root test for three-month T-Bill rates
The t-statistic is As consequence, the null hypothesis of a unit root cannot be rejected
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7. Augmented Dickey-Fuller test
Include lagged differences of dependent variable
The augmented Dickey-Fuller test allows for more serial correlation
The critical values and the rejection rule are the same as before
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8. Dickey-Fuller test for time series that have a time trend
Under the alternative hypothesis of no unit root, the process is trend-stationary
Critical values for Dickey-Fuller test with time trend:
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9. Spurious regression
Regressing one I(1)-series on another I(1)-series may lead to extremely high t-statistics even if the series are completely independent
Similarly, the R-squared of such regressions tends to be very high
This means that regression analysis involving time series that have a unit root may generally lead to completely misleading inferences
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10. Cointegration
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11. Cointegration
Fortunately, regressions with I(1)-variables are not always spurious
If there is a stable relationship between time series that, individually, display unit root behaviour, these time series are called "co-integrated"
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12. Example for time-series that are potentially cointegrated
Spread between interest rates as a function of Interest rates of 6-months bill and of 3-months bill
Individually, it cannot be rejected that the interest rates have a unit root
It is unlikely that the spread has a unit root because this would mean the interest rates can move arbitrarily far away from each other with no tendency to come back together (this is implausible as it contradicts arbitrage arguments)
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13. Example for time-series that are potentially cointegrated
If the spread is an I(0) variable, there is a stable relationship between the interest rates:
With mu being the mean spread between 6-months and 3-months interest rate and epsilon being the temporary deviation from stable relationship
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14. General definition of cointegration
Two I(1)-time series are said to be cointegrated if there exists a stable relationship between them in the sense that
with
i.e. the deviation from the stable relationship is I(0)
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15. Test for cointegration if the cointegration parameters are known
Form residuals of the known cointegration relationship
Test whether the residuals have a unit root
If the unit root can be rejected, are cointegrated
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16. Example: Cointegration between interest rates (cont.)
The Dickey-Fuller strongly rejects a unit root in the spread
This means the interest rates are cointegrated
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17. Testing for cointegration if the parameters are unknown
If the potential relationship is unknown, it can be estimated by OLS
After that, one tests whether the regression residuals have a unit root
If the unit root is rejected, this means that are cointegrated
Due to the pre-estimation of parameters, critical values are different
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18. Critical values for cointegration test
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19. The cointegration relationship may include a time trend
If the two series have differential time trends (=drifts in this case), the deviation between them may still be I(0) but with a linear time trend
In this case one should include a time trend in the first stage regression but one has to use different critical values when testing residuals
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20. Critical values for cointegration test including time trend
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21. Example: Cointegration between fertility and tax exemption
DF-tests suggest that fertility and tax exemption have unit roots
Regressing fertility on tax exemption and a time trend and carrying out a cointegration test suggests there is no evidence for cointegration
This means that the regression in levels is probably spurious
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22. Error correction models
One can show that when variables are cointegrated, their short-term dynamics are related in a so-called error correction representation:
Deviations from the long-term relationship directly feed back into the change of the variables
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23. Summary of cointegration methods
All concepts can be generalized to arbitrarily many time series
Cointegration is the leading methodology in empirical macro/finance as it models equilibrium relationships between nonstationary variables
Estimation and inference is complicated and requires extra care
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