Understanding the Johansen trace test can be a bit tricky. Breaking it down step-by-step can help clarify how the eigenvalues and the test statistic are derived. First, we will revisit the Vector Autoregressive (VAR) model and its Vector Error Correction Model (VECM) representation:
VAR and VECM Models
For an $m$-dimensional VAR($p$) model:
\begin{equation}
Y_t = \Pi_1 Y_{t-1} + \Pi_2 Y_{t-1} + \dots + \Pi_p Y_{t-p} + \varepsilon_t
\end{equation}
The corresponding VECM is:
\begin{equation}
\Delta Y_t = \Pi Y_{t-1} + \sum_{j=1}^{p-1} \Gamma_j \Delta Y_{t-j} + \varepsilon_t
\end{equation}
where
\begin{equation}
\Pi = \Pi_1 + \Pi_2 + \dots + \Pi_p - I_m
\end{equation}
Here, $\Pi$ is important because it contains information about the long-term relationships between the variables.
Canonical Correlations and Eigenvalues
The eigenvalues used in the Johansen test are indeed derived from canonical correlations. Let’s clarify why these eigenvalues, which are important for the trace test, lie between 0 and 1:
1. Canonical Correlations :
Canonical correlations measure the strength of the linear relationship between two sets of variables. In this context, the two sets are the differenced series $\Delta Y_t$ and the lagged levels $Y_{t-1}$.
When computing these canonical correlations, we derive values that lie between -1 and 1.
2. Squared Canonical Correlations :
The eigenvalues $\hat{\lambda}_j$ used in the Johansen test are the squares of these canonical correlations, hence they lie between 0 and 1.
Why Eigenvalues Lie Between 0 and 1
You correctly identified that the eigenvalues $\hat{\lambda}_1, \hat{\lambda}_2, \ldots, \hat{\lambda}_m$ are not the direct eigenvalues of $\Pi$. They are related to the canonical correlations between $\Delta Y_t$ and $Y_{t-1}$.
Connection to $\Pi$
The matrix $\Pi$ plays an important role in determining the cointegrating relationships:
- Rank of $\Pi$ :
The rank of $\Pi$ indicates the number of cointegrating relationships. The eigenvalues (squared canonical correlations) of the matrix formed in the canonical correlation analysis help us determine this rank.
Johansen Trace Test Statistic
The Johansen trace test statistic is:
$$
\begin{equation}
LR(r) = -\frac{n-p}{2} \sum_{j=r+1}^m \log(1 - \hat{\lambda}_j)
\end{equation}
$$
where $\hat{\lambda}_j$ are the squared canonical correlations. These values are ordered as $1 \geq \hat{\lambda}_1 \geq \hat{\lambda}_2 \geq \dots \geq \hat{\lambda}_m \geq 0$.
Why the Test Works
The test statistic involves the sum of the log transformations of $1 - \hat{\lambda}_j$. This ensures that the contribution of each canonical correlation to the test statistic is appropriately weighted. The test compares the null hypothesis of $r$ cointegrating vectors against the alternative of $r+1$ or more.
Summary
To summarise, the eigenvalues $\hat{\lambda}_j$ used in the Johansen trace test lie between 0 and 1 because they are derived from the squared canonical correlations between $\Delta Y_t$ and $Y_{t-1}$. This is different from the eigenvalues of the companion matrix of the VAR process, which must lie inside the unit circle to ensure stationarity. The trace test statistic uses these eigenvalues to determine the rank of $\Pi$, thus identifying the number of cointegrating relationships.
Addressing the first comment to this answer:
Thank you! That makes a lot more sense. I still need to read up on how canonical correlation analysis works to understand, but I can see that you will can get $m$ values from performing CCA on $\Delta Y$ and $Y_{t-1}$. Is there any direct interpretation of the canonical correlations as eigenvalues? For example, are they the eigenvalues of the variance-covariance matrix of $\Delta Y_t$ and $Y_{t-1}$
Canonical correlations do have a direct interpretation in terms of eigenvalues, but they are not simply the eigenvalues of the variance-covariance matrix of $\Delta Y_t$ and $Y_{t-1}$. Rather,they are the eigenvalues of a related matrix product involving the covariance and cross-covariance matrices of $\Delta Y_t$ and $Y_{t-1}$. CCA finds linear combinations of the variables in $\Delta Y_t$ and $Y_{t-1}$ that are maximally correlated.
Let $\Sigma_{\Delta Y_t, \Delta Y_t}$ be the covariance matrix of $\Delta Y_t$, $\Sigma_{Y_{t-1}, Y_{t-1}}$ be the covariance matrix of $Y_{t-1}$, and $\Sigma_{\Delta Y_t, Y_{t-1}}$ be the cross-covariance matrix between $\Delta Y_t$ and $Y_{t-1}$.
The squared canonical correlations, $\lambda_j$, are the eigenvalues of the matrix product:
$$
\large \Sigma_{\scriptscriptstyle \Delta Y_t, Y_{t-1}} \, \Sigma_{\scriptscriptstyle Y_{t-1}, Y_{t-1}}^{-1} \, \Sigma_{\scriptscriptstyle Y_{t-1}, \Delta Y_t} \, \Sigma_{\scriptscriptstyle \Delta Y_t, \Delta Y_t}^{-1}
$$
These eigenvalues represent the squared canonical correlations between the optimal linear combinations of $\Delta Y_t$ and $Y_{t-1}$.
