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1. The Core Philosophy: Why VECM?

When you use a standard Engle-Granger (EG) regression, you are looking at the where — "Where is the equilibrium, and how far has the spread stretched?"

The Vector Error Correction Model (VECM) tells you the how and the when. It takes that stretched rubber band (the spread) and mathematically answers two critical questions:

• Which of the two assets is going to move to fix this spread?
• Exactly how fast is it going to snap back?

Because a spread can be closed by Asset Y dropping, Asset X rising, or both moving toward each other, looking at just the EG spread leaves you blind. The VECM system splits the physics into two panes: the momentum of Asset Y (the Dependent) and the momentum of Asset X (the Anchor).

2. Decoding the Math: Alphas and Half-Lives

At the top of your panes, you will see $\alpha$ (Speed of Adjustment). The VECM predicts today's price change by looking at yesterday's spread size. The math is essentially: Expected Daily Change = Alpha × Yesterday's Spread.

• Why is $0$ stubborn? If $\alpha = 0$, the asset completely ignores the dislocation. It refuses to mean-revert.
• Why is $-1$ instant? If $\alpha = -1$, the asset will move the exact opposite amount of the error, wiping out 100% of the dislocation in a single day (teleporting to equilibrium).
• Why must it be negative? If an asset is priced too high relative to equilibrium, it must have a negative price change to fix it. If $\alpha$ were positive, the asset would explosively trend away from equilibrium (this means it is driven by momentum, not mean-reversion).

The Half-Life: Because human brains are bad at grasping daily fractional velocities like $\alpha = -0.04$, the dashboard provides the Half-Life. This translates the speed into human days: it tells you exactly how many days it will take to close 50% of the gap.

Note on "Full-Life": In quantitative finance, we never use "Full-Life". Because mean-reverting processes are modeled using exponential decay, the curve approaches the zero line asymptotically. Mathematically, it never reaches exactly zero, meaning the "Full-life" is technically infinite. Half-life is the standard, statistically sound metric to gauge capital lock-up time.

3. Interpreting the 3-Pane Dashboard

Pane 3: The Bottom Pane (The Spring)
This is your Error Correction Term (ECT). It represents the raw dislocation or "tension".

• The Zero Line: Represents the historical average equilibrium.
• You are waiting for this purple line to stretch past the $\pm 1\sigma$ or $\pm 2\sigma$ standard deviation bands. When it breaches these lines, the rubber band is under extreme tension.

Panes 1 & 2: The Top & Middle Panes (Prediction vs. Reality)
These panes plot Daily Changes ($\Delta Y$ and $\Delta X$), not absolute prices.

• The Zero Line: Represents exactly 0% growth (the price didn't change from yesterday).
• The Thin Black Line (Left Axis): The actual historical daily price change of the asset. If it's above the zero line, the asset went up that day.
• The Solid Blue Line (Right Axis): The VECM's prediction for the daily price change. We use a "Zero-Aligned Twin-X Axis" here because actual daily volatility (black line) is massive, while macro-driven predictions (blue line) are tiny. The dual-axis magnifies the blue line so you can clearly see the model's structural bias without it looking flat.
• The Red and Green Dots: These overlay the blue line to highlight the model's directional bias for tomorrow:
  • Green Dot: The model expects the asset's price to go UP tomorrow ($>0$).
  • Red Dot: The model expects the asset's price to go DOWN tomorrow ($<0$).
  • Example: If the black line is above zero today, and tomorrow prints a green dot, the model is predicting two consecutive up-days.

4. The Two R² Paradox (Levels vs. Returns)

You may notice an EG plot has a great R² (e.g., 0.24), while the VECM R² is incredibly tiny (e.g., 0.001). This is completely normal and not a bug.

• The Engle-Granger R² predicts absolute Price Levels. Because both assets generally drift upward over a decade, the model easily finds a strong correlation.
• The VECM R² predicts Daily Percentage Returns. Daily returns are notoriously dominated by random walk noise. Predicting even 0.1% of a daily return using a macro spread is considered a structural edge in quant finance.

5. Blue-Sky vs. Slow-Grind Scenarios

The Takeaway: The VECM just saved you from a bad trade. The EG model proved the equilibrium exists, but the VECM warned you that the return to the average spread will be an agonizingly slow, multi-year grind. Do not trade this with short-dated instruments.

6. Tracking Divergence: How to Act

The true edge in this chart comes from the visual gap between the Black Line (reality) and the Blue Line (the model).

7. The T-1 Reality Check & The LOG Warning (Crucial)

Look at the bottom right of your Engle-Granger chart or the date labels on the dots on our VECM chart panes: "Data valid up to: [Date]". This framework operates on daily close prices. Often the last available price is for yesterday’s market close, sometimes there is a delay and we have the last price before yesterday. The green or red dot on the extreme right edge of the blue line is the model telling you exactly what it expects the asset to do on (last available price day + 1), based on where the tension was on the last available price.

In any case this day-by-day prediction is probably just useful for deciding on a good day to start a position. Most people using our tools start it small and build it larger if needed expecting a return to the Engle-Granger predicted spread. The VECM chart for them is mostly for knowing the half-life and seeing the prediction on how it will exponentially decay towards this half-life (presented on the bottom pane in dotted line).

The day-to-day predictions can be helpful even for investors to know whether they can trim their positions or wait or add to their positions or wait. Before selling or buying today in your brokerage, you must check live intraday prices to ensure the mean-reversion move hasn't already happened. However, you must first check the chart title to see if a LOG transformation is being used (e.g., VECM Dynamics: Δ LOG(T-URA)):

• If LOG is NOT used: The Y-axis represents actual prices or standard percentages. A predicted drop can be directly compared to live market action.
• If LOG IS used: Unless stated otherwise, all our logs are calculated in base 2. This means the Y-axis represents the change in the base-2 logarithm, not a simple percentage. A prediction of -0.05 on a base-2 log scale does not equal a -5% drop. You cannot directly compare a $\Delta \text{LOG}$ red dot to a standard percentage drop on your brokerage screen without converting it first (e.g., calculating $2^{\text{prediction}} - 1$).

If the live market has already moved heavily in the correctly scaled predicted direction, the alpha has executed. Do not chase a spread that the market has already closed intraday.