From Uncorrelated to Correlated Random Variables: The Gauss+ Case

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1. Introduction

Let’s say we start with two standard normal random variables, $ X $ and $ Y $, which are uncorrelated. You can easily observe this lack of correlation in a scatter plot of $ X $ versus $ Y $, which would show no particular structure.

But what if your task requires two correlated standard normal variables?

2. Constructing Correlated Variables

Here’s the trick:

We can choose $ X $ as our first variable and construct a new variable $ Z $, which:

  • Also follows a standard normal distribution, and
  • Has a specified correlation $ \rho $ with $ X $.

The variable $ Z $ is created as a linear combination of $ X $ and $ Y $:

$$ Z = \rho X + \sqrt{1 – \rho^2} \, Y $$

Using basic statistics, you can verify that:

  • $ Z $ remains standard normal, and
  • $ \text{Corr}(X, Z) = \rho $ — exactly the correlation we want.

3. Visual Example

Let’s take a specific example:

If $ \rho = +0.75 $, then:

$$ Z = 0.75 X + \sqrt{1 – 0.75^2} \, Y $$

A scatter plot of $ X $ versus $ Z $ would show a positively tilted ellipse — giving us visual confirmation of the $ +0.75 $ correlation.

If we repeat this for $ \rho = -0.75 $, then:

$$ Z = -0.75 X + \sqrt{1 – (-0.75)^2} \, Y $$

The scatter plot now shows an ellipse tilted in the negative direction, confirming the $ -0.75 $ correlation.

4. Why Does This Matter?

This method of constructing correlated random variables is not just a mathematical trick — it extends directly to more complex models.

For example, in the Gauss+ model, this same technique is used to correlate the innovations (or shocks) to different factors: The shocks to the medium-term factor are correlated to the shocks to the long-term factor using exactly this approach.

This is why understanding how to go from uncorrelated to correlated variables is so useful in quantitative modeling and risk management.

5. Watch the Video

For a full explanation with visual examples, watch the video below:

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