Stochastic Nonsense

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Regression Questions: A Coin Teaser

This is a straightforward question that elucidates whether you understand regression, particularly the ceteris paribus interpretation of multiple regression.

  • let $Y$ be the total value of change in your pocket;
  • let $X_1$ be the total number of coins;
  • let $X_2$ be the total number of pennies, nickels, and dimes.

Now, regress $Y$ on $X_1$ or $Y$ on $X_2$ alone. Both $\beta_1$ and $\beta_2$ would be positive.

If you regress $Y$ on $X_1 + X_2$, what are the signs of $\beta_1$ and $\beta_2$?

Consider holding $X_2$ constant: if $X_1$ increases by 1, ie you turn a penny, nickle, or dime into a quarter, then $Y$ surely increases. Therefore $\beta_1$ is positive.

Now consider holding $X_1$ constant and increasing $X_2$. If the number of pennies, nickles, and dimes increases while the total number of coins stays constant, you’re replacing quarters with a lower valued coin. Thus increasing $X_2$ can decrease $Y$, so it is entirely possible that $\beta_2$ is negative.

Updated 26 August 2015.