# Regress log Y on (log(x1) - log(x2))

I am looking at a paper where we have a regression of the following form:

$log(Y_{price})=\mu log(X_{PredictedPrice})+\lambda(log(X_{LaggedPrice})-log(X_{PredictedPrice}))+\xi(log(X_{LaggedPrice})-log(X_{LaggedPredictedPrice})) +\alpha$

I am trying to figure out the interpretation of $\lambda$. For example, does a one percent difference in the lagged price and the predicted price correspond to a (roughly) $\lambda$ percent change in y, holding the previous predicted price constant? Or is there another interpretation I am missing?

Paper Link (apologies if this is behind a paywall): http://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.99.3.1027

• For small changes, log differences are close to percent changes: $\log x_t - \log x_{t-1} \approx \frac{x_t - x_{t-1}}{x_{t-1}}$. Do you have a link to the paper? Maybe some context would help, but my initial reaction is that this is extremely bizarre. It's like regressing someone's height on the forecasted growth rate of their height. Mar 15 '17 at 20:08
• Added- also changed the notation to conform to that of the paper. Mar 15 '17 at 20:21
• @Matthew One way to guess the paper's motivation is to note that $$-\lambda\log(X_\text{lagged price}) - \log(X_\text{predicted price}))=\lambda\log\left(\frac{X_\text{predicted price}}{X_\text{lagged price}}\right)$$ compares the predicted price to what the prediction would be in the ultra-simple model that predicts the price of $X$ won't change. So, by observing how our predicted $X$ price varies from that, we might get (and exploit, via $\lambda$) information about how to adjust our prediction of $Y.$ This assumes there's some relationship among the variables and the prediction methods... .
– whuber
Sep 18 '20 at 21:12
• Although my comment is pure speculation--I haven't attempted to look at the paper--this slight reformulation provides a good hint concerning how $\lambda$ might be interpreted.
– whuber
Sep 18 '20 at 21:13