3
$\begingroup$

Let us say that I have two economic time series $Y_t$ and $X_t$ (both $I(1)$) and that I have good theoretical reasons to expect that the values of $X$ influence $Y$ with a one-period lag. Then, I want to test for cointegration, to see if a long-run relationship like $Y_t = \beta_0 + \beta_1 X_{t-1} $ holds.

When performing the Augmented Dickey-Fuller test, should I test $Y_t$ and $X_t$ or $Y_t$ and $X_{t-1}$? By logic, I would say that using the latter is the right thing to do, but is it legitimate from an econometric point of view? In other words, should I test for

$Y_t = \beta_0 + \beta_1 X_{t-1} $

or

$Y_t = \beta_0 + \beta_1 X_{t} $ (and just select the right lag structure for the short-term coefficients)?

$\endgroup$
1
$\begingroup$

Suppose you have two $I(1)$ series, $y_{1,t}$ and $y_{2,t}$. They can be decomposed into \begin{aligned} y_{1,t} &= x_{1,t} + s_{1,t}, \\ y_{2,t} &= x_{2,t} + s_{2,t}; \\ \end{aligned} where $x_{1,t}$ and $x_{2,t}$ are stochastic trends while $s_{1,t}$ and $s_{2,t}$ are stationary components.

If $y_{1,t}$ and $y_{2,t}$ are cointegrated, then $x_{1,t} \equiv c x_{2,t}$ for some $c$. Without loss of generality, let us assume $c=1$. That means $y_{1,t}$ and $y_{2,t}$ share the same stochastic trend $x_{t}:=x_{1,t}=x_{2,t}$ and we can write \begin{aligned} y_{1,t} &= x_{t} + s_{1,t}, \\ y_{2,t} &= x_{t} + s_{2,t}. \\ \end{aligned}

If you take $y_{1,t}$ and $y_{2,t-h}$ for some $h>0$, will they be cointegrated? \begin{aligned} y_{1,t} - y_{2,t} &= ( x_t + s_{1,t} ) - ( x_{t-h} + s_{2,t-h} ) \\ &= ( x_t - x_{t-h} ) + ( s_{1,t} - s_{2,t-h} ) \end{aligned} which is a sum of two stationary series, because a difference at lag $h$ of $x_t$ is stationary and a difference of two stationary series $s_{1,t}$ and $s_{2,t-h}$ is stationary. Thus we have found a stationary combination of $y_{1,t}$ and $y_{2,t-h}$, which means they are cointegrated. In a similar way you could also show that if $y_{1,t}$ and $y_{2,t-h}$ are cointegrated, then $y_{1,t}$ and $y_{2,t}$ are cointegrated, too.

Thus in theory you can test for cointegration either between $y_{1,t}$ and $y_{2,t}$ or $y_{1,t}$ and $y_{2,t-h}$ and the answer should be the same. Empirically the answer may differ, but hopefully you have a large enough sample so that it does not differ in your case.

Which pair to test is then a matter of taste from the statistical point of view. So probably you want to test the pair that makes the most subject-matter sense.

P.S. This holds for the case of $y_{1,t}$ and $y_{2,t}$ being $I(1)$. If $y_{1,t}$ and $y_{2,t}$ are $I(d)$ with $d>1$, the answer will be different, but similar reasoning as above applies.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.