Calculation Result
| Item | Value | Meaning |
|---|
Advanced Calculator
Example Data Table
| Period | Y | X1 | X2 | Use |
|---|---|---|---|---|
| 1 | 101 | 50 | 22 | Base observation |
| 2 | 103 | 51 | 23 | Model fitting |
| 3 | 106 | 53 | 24 | Residual estimate |
| 4 | 108 | 54 | 25 | Spread building |
Formula Used
This calculator estimates a cointegrating vector through an ordinary least squares relation. It treats one column as the dependent series and the selected columns as explanatory series.
Cointegrating vector = [1, -β1, -β2, ...]
Spread = Yt - α - β1X1,t - β2X2,t - ...
The residual spread is then checked with summary diagnostics. A lower residual standard deviation and a reasonable mean-reverting half-life can support a stronger relationship.
How to Use This Calculator
- Paste your time series data into the data box.
- Use commas between values and rows for each period.
- Enter the dependent column name or number.
- Enter the independent column names or numbers.
- Choose whether the model should include an intercept.
- Select a transform when your data needs log scaling.
- Press the calculate button to view the vector.
- Download the result as a CSV or PDF file.
About Cointegrating Vector Analysis
What the Vector Means
A cointegrating vector shows a stable long-run balance between non-stationary series. Each coefficient works like a weight. These weights combine the series into one spread. A useful spread should move around a steady average. It should not drift forever. Traders often call these weights hedge ratios. Analysts use them in macro models too. The goal is not simple correlation. The goal is long-run equilibrium.
Why Residuals Matter
The residual is the distance from the estimated balance. A small and stable residual is helpful. It suggests the variables share a common trend. This calculator builds the residual after fitting the selected columns. It reports spread mean, spread deviation, latest spread, and z-score. These values help you inspect possible mean reversion. They do not replace formal testing. They give a fast practical review.
Using Advanced Options
Use the intercept option when the spread has a constant offset. Remove it only when theory requires a zero intercept. Use log transform for price-like series with growth behavior. Use standardization when variables have very different scales. Choose dependent normalization when you want a traditional vector. Choose unit length when you want comparable weights. Keep enough observations for stable estimates. Very small samples can mislead.
Practical Interpretation
The vector should be read with context. A coefficient near negative two means one unit of that variable receives that spread weight. The intercept shifts the spread level. The half-life estimates how quickly deviations decay. A shorter half-life suggests faster adjustment. A very long half-life suggests weak mean reversion. Always review plots and domain logic. Good models need statistical and practical support.
Frequently Asked Questions
What is a cointegrating vector?
A cointegrating vector is a set of weights that combines multiple time series into a spread. The spread should be more stable than the original series.
Does this calculator perform a full Johansen test?
No. It estimates a practical vector through regression and residual diagnostics. Use specialist software for complete Johansen rank testing and critical values.
Which column should be dependent?
Choose the series you want normalized to one. For pairs trading, this is often the target asset or the asset being explained.
Should I include an intercept?
Use an intercept when the long-run spread may have a constant offset. Remove it only when theory requires the spread to pass through zero.
When should I use log transform?
Use log transform for positive price, index, or economic level data. It can make proportional changes easier to model and interpret.
What does half-life mean?
Half-life estimates how many periods a residual shock may take to decay by half. Lower values suggest faster mean reversion.
What is the residual z-score?
The residual z-score compares the latest spread to its recent average. It helps show whether the current spread is unusually high or low.
Can I download the calculation?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a cleaner report that can be saved or shared.