What tests do I need to perform for VECM and VAR to be considered robust? I know LM test for residual autocorrelation is mandatory, but what about Jarque-Bera test? Is that necessary?And what should I do if my regressions don't pass that test?
Solved – Jarque-Bera test mandatory for VECM and VAR
vector-autoregressionvector-error-correction-model
Related Solutions
So you have three nonstationary series and one stationary series. Let us call them $x_1$, $x_2$, $x_3$, and $x_4$, respectively. Suppose the nonstationarity of $x_1$, $x_2$, $x_3$ is of a unit-root kind (rather than of some other kind); that is, each of $x_1$, $x_2$, $x_3$ is integrated of order one, I(1). You can determine the order of integration using, for example, the augmented Dickey-Fuller test (ADF test).
Test each pair of the nonstationary series ($x_1$ and $x_2$; $x_1$ and $x_3$; $x_2$ and $x_3$) for cointegration using the Johansen or the Engle-Granger test.
Then test all three series ($x_1$, $x_2$, $x_3$) for cointegration using the Johansen test.
Depending on the results of the tests, you may find yourself in one of the following situations:
(A) No cointegration
(B) Two of the variables (say, $x_1$ and $x_2$) are cointegrated while the third variable (say, $x_3$) is not
(C) The three variables ($x_1$, $x_2$, $x_3$) are cointegrated
In general, you want the following:
- Models for cointegrated variables should have an error-correction representation; otherwise the model would be misspecified (cointegration goes hand-in-hand with the error correction representation).
- Models for stationary dependent variables should not have nonstationary explanatory variables (except perhaps for stationary combinations of cointegrated nonstationary variables); otherwise the linear combination of the regressors would diverge from the regressand.
- Models for nonstationary dependent variables should have at least one nonstationary explanatory variable; otherwise the regressand would diverge from the linear combination of the regressors. Mind nonstandard distributions of estimators for the integrated variables.
Based on these principles, you may do the following:
If (A) then first-difference each of the three variables ($x_1$, $x_2$, $x_3$), and use them together with the stationary variable $x_4$ to build a VAR model.
If (B) then build a model where
- $\Delta x_1$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $\Delta x_2$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $\Delta x_3$ depends on lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $x_4$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$.
If (C) then build a model where
- $\Delta x_1$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $\Delta x_2$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $\Delta x_3$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
- $x_4$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$.
These are pretty general models with lots of regressors. You may find it beneficial to exclude some variables from some equations or use penalization to avoid overfitting.
Here is an example of where three positive and one negative loading on the error correction term makes intuitive sense.
Consider a four-variable cointegrated system $(x_t, y_t, z_t, w_t)$ with $(x_t, y_t, z_t)$ being the three underlying stochastic trends and $w_t := x_t + y_t + z_t + \varepsilon_t$ where $\varepsilon_t$ is a stationary process.
Define the error correction term as $ect_t := w_t - x_t - y_t - z_t (=\varepsilon_t)$. This is obviously stationary as $\varepsilon_t$ is stationary.
Then it is natural to expect that the error correction term will have positive loadings in the equations for $\Delta x_t, \Delta y_t, \Delta z_t$ and a negative one in the equation for $\Delta w_t$, because:
- If $x_t$ deviates from the long run equilibrium by getting "too high", $ect_t$ will become negative, and then the positive loading on $ect_t$ will drag $x_{t+1}$ down, so back to equilibrium. The same holds for $y_t$ and $z_t$.
- If $w_t$ deviates from the long run equilibrium by getting "too high", $ect_t$ will become positive, and then the negative loading on $ect_t$ will drag $w_{t+1}$ down, so back to equilibrium.
- And the reverse for the cases of variables getting "too low".
Best Answer
Jarque-Bera is not mandatory for either VAR or VECM. Not passing a test for normality (or, more precisely a test for symmetry and no excess kurtosis) at least asymptotically has no implications on the validity of either tests or estimators in VECMs.
It is true that, e.g., Johansen derives MLEs under the assumption of a normal likelihood (i.e. normal errors), but then derives the large-sample distribution of his tests/estimators under much broader moment conditions. In the Palgrave handbook of econometrics, Johansen writes: