interaction and dummy term interpretation WITH partial derivatives using inflection point

Question

I have the following model for returns as a function of firm_profitability, sentiment, and additional predictors (included in the Gamma * Z term), with error term e.

positive_sentiment is a dummy variable representing sentiment > 0, so that the association between returns and sentiment can differ between positive and negative sentiment values.

returns = -0.23 + 0.91 firm_profitability + 
          0.32 sentiment  +
          0.52 (positive_sentiment * sentiment) +
          0.66 (sentiment * firm_profitability) +
          Gamma * Z + 
          e

Can someone please help me interpret the equation using the partial derivative with respect to sentiment? Can we interpret it as 1 unit increase in negative sentiment increases returns by 0.32, and when we interact sentiment with the positive sentiment dummy, the total impact of positive sentiment on returns is 0.84 (= 0.32 + 0.52)?

I also would like to find the inflection point where the effect of firm profitability changes direction based on the effect of sentiment interaction, as I see the negative and positive sentiments affect returns differently.

Answer 1

Can we interpret [the coefficient for sentiment] as 1 unit increase in negative sentiment increases returns by 0.32?

Only if firm_profitability = 0, as you have an interaction between sentiment and firm_profitability that isn't restricted to positive values of sentiment. Within negative values of sentiment, a 1-unit change in sentiment means a change of 0.32+0.66*firm_profitability in returns.

when we interact sentiment with the positive sentiment dummy, the total impact of positive sentiment on returns is 0.84 (= 0.32 + 0.52)?

Again, only if firm_profitability = 0 as that otherwise would still ignore the interaction term 0.66 (sentiment * firm_profitability). You need to take the firm_profitability into account. When a predictor is involved in an interaction, is has no single association with outcome unless you specify the values of all its interacting predictors.

I also would like to find the inflection point where the effect of firm profitability changes direction based on the effect of sentiment interaction.

I take this to mean that you are looking for the level of sentiment at which the association between returns and increasing firm_profitability changes from negative to positive. The first derivative of returns with respect to firm_profitability is a function of sentiment: 0.91 + 0.66*sentiment. This equals 0 when sentiment = -0.91/0.66, approximately -1.38. Note that this is below the value of 0 at which you allowed for a change in the association between returns and sentiment.

In response to comment

You can ask a related question: what's the level of firm_profitability where the association between sentiment and returns changes sign? In your model, that depends on whether sentiment is negative or positive.

For positive sentiment, the first derivative of returns with respect to sentiment is: 0.32 + 0.52 + 0.66*firm_profitability. That equals 0 (sign of association switches) at approximately firm_profitability = -1.27.

For negative sentiment, the first derivative of returns with respect to sentiment is: 0.32 + 0.66*firm_profitability. That equals 0 at approximately firm_profitability = -0.49.

Whether those predictions make sense depends on your understanding of the subject matter.

Specifying a sharp change in one part of the model at sentiment = 0 while having the interaction between sentiment and firm_profitability independent of the sign of sentiment might be leading to predictions that don't represent reality well. You might consider a model that handles sentiment more smoothly, for example with regression splines, instead.