In an important paper, Sims, Stock and Watson(1990) analyzed the behavior of hypothesis tests in linear models in the presence of unit roots in the regressors. It had been known that in the presence of unit roots, certain hypothesis tests had "non-standard" asymptotics, that is, you couldn't just use the standard Normal, t, F or chi-squared distributions to evaluate the statistical significance of a result. The best-known example of this is the unit root test:

\begin{equation} {y_t} = \rho {y_{t - 1}} + {\varepsilon _t} \label{eq:misc_ssw_unitroot} \end{equation}

where if \(\rho  = 1\), it is now well-known that the least-squares estimates of \(\rho\) are strongly skewed left and never come close to being Normal. These non-standard distributions generally can only be approximated by simulation methods and are (in most cases) very specific to a problem. By contrast, the standard statistical distributions have very efficient numerical methods for computing their tail probabilities.

On the other hand, it was also well-known that, even in the presence of unit roots, certain hypotheses did have standard asymptotics. In the augmented Dickey-Fuller test:

\begin{equation} \Delta {y_t} = \left( {\rho - 1} \right){y_{t - 1}} + {\zeta _1}\Delta {y_{t - 1}} + \ldots {\zeta _p}\Delta {y_{t - p}} + {\varepsilon _t} \label{eq:misc_ssw_adfregression} \end{equation}

any hypotheses involving only the \(\zeta\) can be tested using standard methods. In particular, in testing whether lag \(p\) is significant, you can use a standard t test. (The asymptotic distribution is Normal, but standard practice is to use the more conservative t distribution in finite samples).

 

For simplicity, one would want as many hypotheses of interest to be analyzed using standard distributions, to avoid the need for simulation or bootstrapping (or use of table lookups based upon someone else's simulations) to evaluate the significance of a result. While the Sims-Stock-Watson paper is highly technical, the result can be stated in a relatively simple way:

Any (linear) hypothesis which can arranged to be on coefficients which apply solely to stationary variables has standard asymptotics.

 

The "can be arranged" is important to this. The hypothesis as originally written doesn't have to be applied to stationary variables, it just has to be equivalent to a hypothesis that is on stationary variables. For instance, the standard \(p+1\) lag AR on the variable \(y\) can be rearranged into what's basically the form \eqref{eq:misc_ssw_adfregression}:

\begin{equation} \begin{array}{l} {y_t} = {\rho _1}{y_{t - 1}} + \ldots + {\rho _{p + 1}}{y_{t - (p + 1)}} + {\varepsilon _t} = \\ ({\rho _1} + \ldots {\rho _{p + 1}}){y_{t - 1}} - ({\rho _2} + \ldots {\rho _{p + 1}})\Delta {y_{t - 1}} - \ldots - {\rho _{p + 1}}\Delta {y_{t - p}} + {\varepsilon _t} \\ \end{array} \label{eq:misc_ssw_adfregressionrearrange} \end{equation}

If \(y\) has (one) unit root and can be differenced to stationarity, then any hypothesis about the \(\rho\) coefficients except one that restricts the sum has standard asymptotics even if you run it just on the (non-stationary) lags of \(y\) directly. In particular, you can test whether any specific coefficient is zero (that is, the standard output t-tests are asymptotically valid), or a joint test on any (proper) subset of the lag coefficients. A joint test that all lag coefficients are zero however would not have standard asymptotics since that would restrict the sum to be zero.

In a multivariate setting (such as a VAR), you can, similarly, test lag length using standard methods whether or not the series have unit roots (cointegrated or not) because you can rearrange the regression as above so all the trailing lags are on first differenced variables (and as above, you just can't test for lags=0). An important type of hypothesis test that runs afoul of the SSW criterion is the Granger causality test. If a series has a unit root, then there is no way to rewrite a set of lag coefficients on it so that all the coefficients are on differences—as in \eqref{eq:misc_ssw_adfregressionrearrange}, there will always be one coefficient that's on a lagged level. Thus, you can't (with standard methods) test whether or not a series with a unit root Granger causes another series (whether or not the series are cointegrated) using the direct method of estimating a VAR and testing the lags.

 

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