What it means
The simplest version, linear regression, fits one straight line through a scatter of points. The line has two parts: an intercept, which is the estimated value when the input is zero, and a slope, which is how much the output changes for each one unit change in the input.
Multiple regression does the same job with several inputs at once, which matters because business outcomes rarely have a single cause. A model of weekly sales might include price, advertising spend, competitor promotions and the weather, estimating the effect of each while holding the others steady.
Two numbers tell you whether to trust the result. R squared shows the share of the variation the model explains, on a scale of 0 to 1, and the p value attached to each input shows how likely it is that the relationship appeared by chance.
A high R squared sitting alongside implausible coefficients usually means the model has been fitted to noise rather than signal. Finance uses regression in several standard places.
Beta, the sensitivity of a share to the market, is the slope of a regression of the share's returns on index returns, and cost accountants split semi variable costs into fixed and variable parts by regressing total cost on activity volume. The most repeated warning is that correlation is not causation.
Regression only tells you that two series moved together in the past, so the causal story has to come from knowing the business, and a model built on a period that no longer resembles today will mislead with great confidence.
In practice
Real-world examples.
Example
A food manufacturer regresses monthly electricity cost on machine hours and gets an intercept of $18,000 and a slope of $12 per hour. That tells the plant manager the site carries $18,000 of fixed power cost a month, so a plan for 4,000 machine hours should budget $18,000 + (4,000 x $12) = $66,000.
Example
An asset manager regresses five years of monthly returns for a water utility against a broad market index and gets a slope of 0.65. The beta says the share has historically moved about two thirds as much as the market, which supports its use as a defensive holding.
Example
A consumer lender regresses portfolio default rates on unemployment and loan to value ratios across ten years of data. The model suggests each one percentage point rise in unemployment adds roughly 0.4 percentage points to defaults, which feeds directly into its provisioning.
Formula
Calculation
Simple linear regression: y = a + bx
Slope b = sum of (x - mean of x)(y - mean of y) / sum of (x - mean of x) squared
Intercept a = mean of y - (b x mean of x)
A regional distributor records five months of marketing spend and sales, both in thousands of dollars. Spend was 10, 20, 30, 40 and 50, and sales were 120, 160, 190, 240 and 290. Mean spend is 30 and mean sales is 1,000 / 5 = 200.
The spend deviations from the mean are -20, -10, 0, 10 and 20, and the sales deviations are -80, -40, -10, 40 and 90. Multiplying each pair and adding gives 1,600 + 400 + 0 + 400 + 1,800 = 4,200, while the squared spend deviations add to 400 + 100 + 0 + 100 + 400 = 1,000.
The slope is therefore 4,200 / 1,000 = 4.2 and the intercept is 200 - (4.2 x 30) = 74. The fitted line is sales = 74 + 4.2 multiplied by spend, so each extra $1,000 of marketing is associated with $4,200 of sales, and a $35,000 budget predicts 74 + (4.2 x 35) = 221, or $221,000 of sales.Case study
Seen in the real world.
The following is an illustrative and entirely fictional example. Halverstone Foods, an invented chilled desserts business, had run deep price promotions for years on the belief that volume more than paid for the discount. Its commercial team built a regression on three years of weekly data, including shelf price, promotional display, competitor activity and seasonality.
The model estimated that a 1% price cut lifted volume by about 1.8%, which sounded like a win until the finance team put margins into it. At the normal price of $4.00 and a unit cost of $2.40, the product earned $1.60 per unit and sold 100,000 units a week, a contribution of $160,000. A 10% cut took the price to $3.60 and the margin to $1.20, and even with volume up 18% to 118,000 units the contribution fell to $141,600.
The fictional business was therefore giving away $18,400 of contribution every promotional week. It kept the promotion on two lines where the modelled volume response was far stronger and dropped it on the rest, which is exactly the kind of decision regression is good for: not proving what causes what, but sizing an effect that everyone had been guessing at.
Watch out
Common mistakes.
- Treating a strong statistical relationship as proof that one variable causes the other, when both may simply be driven by something the model never included.
- Chasing a high R squared by adding more and more variables, which fits the past beautifully and predicts the future badly.
- Extrapolating far outside the range of the original data, such as using a model built on spend between $10,000 and $50,000 to predict what $500,000 would do.
Questions
People also ask.
What does R squared actually mean?
It is the proportion of the variation in the outcome that the model accounts for, so 0.80 means four fifths of the movement is explained and the remaining fifth is not.
How much data does a regression need?
There is no fixed rule, but a common guideline is at least ten to fifteen observations for every input variable, and more when the data is noisy.
Is regression the same as forecasting?
No, regression estimates relationships in historical data, and forecasting is one use of it, which only works while those relationships continue to hold.
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