Unit Roots 31/12/2018

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 31/12/2018

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 31/12/2018

Alternative Formulation of the Dickey-Fuller test The alternative representation is obtained by subtracting yt-1 from both sides 31/12/2018

Critical values for Dickey-Fuller test The critical value is much more negative than it would be in a t-distribution 31/12/2018

Example: Unit root test for three-month T-Bill rates The t-statistic is -2.46. As consequence, the null hypothesis of a unit root cannot be rejected 31/12/2018

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 31/12/2018

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: 31/12/2018

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 31/12/2018

Cointegration 31/12/2018

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" 31/12/2018

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) 31/12/2018

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 31/12/2018

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) 31/12/2018

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 31/12/2018

Example: Cointegration between interest rates (cont.) The Dickey-Fuller strongly rejects a unit root in the spread This means the interest rates are cointegrated 31/12/2018

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 31/12/2018

Critical values for cointegration test 31/12/2018

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 31/12/2018

Critical values for cointegration test including time trend 31/12/2018

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 31/12/2018

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 31/12/2018

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 31/12/2018