Why is the intercept negative, and what does my regression show?

Question

I am trying to get my regression right. I want to see, if subs increase how much increase in revenue is seen. The dependent variable is Revenue while the independent variable is subscribers.

Least squares method regression shows the following equation 24.4x - 189,883,443. I can't make sense of it. If subs increase means increase in revenue, then why this?

There is no 0 value and the lowest value for subscribers starts with 10 m while revenue starts with 180 m

How can i make sense of y = 24.4x - 189,883,443. Why is the intercept negative. Does it mean that one sub increase would mean a revenue increase of 24.4?

Answer 1

Why is the intercept negative. Does it mean that one sub increase would mean a revenue increase of 24.4

The negative intercept does not mean "that one sub increase would mean a revenue increase of 24.4". The slope coefficient means something like that (but different to it). The negative intercept tells you where the linear model predicts revenue (y) would be when subs (x) is 0.

Your question appears to be prompted by confusion about the fact that in your fitted model, $E(Y|x=0)\neq 0$, even though logically, you would expect no revenue then.

This situation is not only common, it's to be expected. You normally cannot expect a relationship identified on data over a limited range to be appropriate everywhere, and so you should beware putting any real meaning on the fitted value at 0 unless your $x$ values encompass 0 (at which point you'll likely see that in fact revenue is much closer to what you'd expect there).

The assumption of linearity of relationship between $y$ and $x$ might be reasonable over the range of your x values, but that gives no basis on which to extrapolate outside the range of your data.

We don't really know what's going on in the green part.

Nevertheless you should check that assumption of linearity within the range of your data, via a residual plot.

[Even when you have good reason to fit a linear model to data that extends over the entire range of possible values of $x$ - including 0 - and to expect that $E(Y|x=0)= 0$ it's still pretty common practice to fit an intercept.]

Answer 2

The negative intercept means that, if subscribers were 0, the predicted revenue would be -189,883,443 and that predicted revenue increases by 24.4 for each subscriber.

This is a nonsensical result, so, it is probable that either 1) You did something wrong in Excel and aren't doing what you think you are doing (very easy to believe, Excel is not great for statistics

2) There are violations of the assumptions. Possibly outliers or leverage points.

3) The form of the relationship is not linear.

(There could be other reasons too).

Answer 3

In a regression model, an attempt is made to explain y (dependent variable) with the help of known independent variables.The simple one independent variable regression could be written as y = a + bx. It is quite possible that x alone cannot explain all the variations in y ie., there may be some error in the predictions of y by the selected model. The Intercept represents the average error that we are going to commit if we predict all the possible y values with the help of x values. The second interpretation is when X=0 what value y could take. So, mathematically, the intercept is the y value when x is made equal to zero. The third explanation could be that it is the unexplained variation in y due to X. It is expected that if a model is perfect, the unexplained variation in y should be 0 and thereby the intercept should be zero. In a regression model where the intercept is negative implies that the model is overestimating on an average the y values thereby a negative correction in the predicted values is needed.

Answer 4

Your lowest data value is 10m fand you are wanting to extrapolate the model down to 1, with the assumption that it remains linear across that range. Perhaps if you plot the line, and it's 95% prediction interval, you'll see a very wide range of possible values where the intercept could be at the low end, which could include positive and negative values for the intercept, including 0. I try and look at the prediction interval for a simple linear regression to get a sense of what extrapolation might be valid. Graphpad Prism does this nicely for you, although you could take advantage of free programs like R and quickly and easily follow any of the multiple online step-by-step guides that could plot it for you.

Answer 5

If you're using Excel and your X-axis are dates, and your linear regression equation looks like y = 0.04x - 1900 (slope = 0.04), that is because Excel converts dates into days since Jan. 1, 1900 (some large numbers), so your intercept is huge. I had this situation, but when I instead used period (counting from 1) for the X axis, my linear equation was: y = 1.3x + 13 (slope = 1.3). It made a LOT more sense!

Answer 6

Although this is an old post, I came across it and have an interpretation of the results to share. The expected revenue given that the number of subscribers is zero implies that the business is paying money rather than receiving it. Consider that at the core of this data exist some fundamental economics. Theoretically, negative revenue suggests a business could be issuing returns beyond their receivables, or they could be paying a subscriber (that doesn’t actually exist because the OP is discussing the intercept, which we are interpreting when the number of subscribers equals zero).

Corporations have reported negative revenues in the past, most notably after the global crisis in 2020. The mathematics here make sense because of the economics and accounting. Sometimes these results are attributable to accounting practices, but recall that the data from which we are making our inferences is derived from those same accounting methods. The coefficients and intercept of the OP’s model are a direct result of those inputs.

Each scenario in one’s data exploration and analyses will come with its own unique set of considerations. It’s normal to question our results and I encourage it. For anyone who lands on this page sometime in the future, I hope you find these comments helpful to think about.

Edit: In response to the comments about mixing revenues and profits, beware that these concepts have different meanings and my post may have not made that clear. In fact, the word revenue is perhaps the most misleading on a balance sheet because the numbers strictly depend on how a business decides to structure its books.

Depending on the type of firm, companies may be reporting net revenue, which is the value after returns, discounts, cost of goods sold (COGS), etcetera. Net revenue is also not profit because it does not deduct all expenses, rather only revenue expenses. By strict definitions, if a business has no sales or goods/services being rendered as a result of having no customers we might say revenues should equal zero. However, revenue can come from activities not related to the original business, such as leasing a building or machinery, known as “other revenue”.

The reality is that accounting practices allow for counterintuitive manipulations of the numbers. If working with real-world data, thinking about such nuances can assist in identifying the sources of discrepancies to clarify our otherwise unintuitive results. For corporate projects, someone in the organisation should be able to answer questions or provide insights about your findings, so by all means ask questions and seek input from the source.