Assessing the Order of Integration (with Time Series Data)

[Sandy Flannery]

Hi all,

I am working with time series data that contains 3 variables. How would I go about assessing these three variables cointegration and ultimately testing their cointegration? What Stata commands can accomplish this goal?

Best,

Sandy

[Andrew Musau]

For economic variables, we assume that they are either integrated of order 0 or integrated of order 1 (i.e., \(I0\) or \(I1\), respectively). We do not consider any other cases and therefore, a unit root test is used to determine the order of integration, most prominently, the Dickey Fuller test.

help dfuller

The intuition is simple. Consider the autoregressive equation:

$$Y_{t}= \zeta Y_{t-1}+ \epsilon_{t}$$

where \(\epsilon_{t}\) is a classical error term. If you subtract \(Y_{t-1}\) from both sides of the equation, you have

$$Y_{t}- Y_{t-1} = (\zeta-1)Y_{t-1}+ \epsilon_{t}$$


Defining \(\Delta Y_{t}= Y_{t}-Y_{t-1}\), you have the simplest form of the Dickey-Fuller test:

$$\Delta Y_{t} = \beta_{1} Y_{t-1}+ \epsilon_{t}$$

where \(\beta_{1}= \zeta-1\). Alternative Dickey-Fuller tests additionally include a constant and/or a constant and a trend term (see options of the dfuller command). Thus:


$$\text{H}_{0}:\beta_{1}=0\text{ (unit root)} $$
$$\text{H}_{A}:\beta_{1}<0\text{ (stationary)} $$

To test for cointegration, you estimate the intended time-series regression and use the Dickey-Fuller test to test whether the ensuing residuals have a unit root. If you are able to reject the null hypothesis of a unit root in the residuals, then you can conclude that the variables are cointegrated and your OLS estimates are not spurious.