Negative prediction values from linear regression in R

Question

So I made a linear regression in R Studio to predict the price of a car based on the year of fabrication. The data set is called "audi" and my linear regression looks like this:

library(tidyverse)
library(modelr)
...
model_price_Year <- lm(data = audi, price ~ year)
summary(model_price_Year)

The result of the summary is this:

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -6.437e+06  8.503e+04  -75.71   <2e-16 
year         3.203e+03  4.215e+01   75.98   <2e-16 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9437 on 10666 degrees of freedom
Multiple R-squared:  0.3512,    Adjusted R-squared:  0.3511 
F-statistic:  5772 on 1 and 10666 DF,  p-value: < 2.2e-16

Then, I made a grid and i added predictions for 100 values of the year. It looks like this:

grid_year <- audi %>%
  data_grid(year = seq_range(year, 100)) %>%
  add_predictions(model_price_Year, "price")

And after that, if i want to see results, they look like this:

  year   price
   <dbl>   <dbl>
 1 1997  -41481.
 2 1997. -40737.
 3 1997. -39993.
 4 1998. -39249.
 5 1998. -38505.
 6 1998. -37761.
 7 1998. -37017.
 8 1999. -36273.
 9 1999. -35529.
10 1999. -34785.

They are all negative, and becuase we are talking about the price, it doesnt really make sense. Why are they negative? How do I interpret this?

Answer 1

You have a linear fit that does not predict well for cars older than ten years.

This is because most data points are for cars younger than 10 years old and these will dominate the fitting. If you would force the fit line to have no negative values from 1997 to 2020, then you would get a fit that has a larger error for the younger cars.

How do I interpret this?

The linear model is false, and does not fit well.

Answer 2

You didn't constrain the output. Without such a constraint, you allow for any real number to be predicted, including numbers that are ridiculous. For instance, logistic regressions constrain the probability predictions to $[0,1]$ by applying a link function that compresses the real line to the unit interval.

An alternative might be to fit a log-linked gamma generalized linear model or a log-linked Gaussian generalized linear model.

A negative price might make sense, but the interpretation would be that, instead of selling the car, you pay someone to take it off your hands. I could imagine this for someone who has a useless item that is expensive to store, so you hire someone to take it away.

It also might be that the year is not a linear predictor of price. I would guess that prices don't just increase by a fixed amount every year. If the price of a car goes up $X\%$ each year (nonlinear), that's different than the price going up $\$Y$ every year (linear).

Answer 3

[UPDATED] A typical times series interpretation doesn't apply in this case. It's more like a panel data, just to point out. There are more than one value per year. I suspect there are many models of cars with same year of fabrication, among other types of heterogeneity. I'll assume you want to say in the OLS model.

So, the negative values can be caused, for example by the use of logarithm in prices (A very common technique), however if prices are adjusted by inflation, a price of 0.8 could turn to a negative value ($log(0.8) = -0.22$) Another cause is a very short time-space of fabrication year (i.e. 1997 to 2022).

If your model is $Price = a + YearFab$, we should remember that the intercept ($a$) is the Price when the Year of Fabrication is 0. Obviously that's not possible. Using your results, the year when prices start being negative is 2009 to the past ($year_{p=0} = -a/b$) However, that last interpretation is useless and misleading.

This is the graph before recoding:

Thus, you may need to recode the year from 1 (first year of fabrication) to T (last year), and then run the regression again, to obtain new coefficients. For example if 1997 is the first year, then it code will be 1, and so on. Please present your results. If the problem persists, perhaps the relation is not trend related to year or you may have huge outliers, among other things.

Basically, the problem is the data set of the model, the results suggest that the independent variable starts at 0 and goes to the end (2022) given a total of 2022 possible units, which is false. Your data starts at 1997 (or similar year) and end in 2022 with a total of 25 units (around).

After the recoding, the intercept will be interpreted as the price when the year series starts, "year 0", that is 1997 (presumably). And any increase will be related to that year.

The marginal interpretation doesn't change, that is, if the car is newer then the price will increase in b units. ( $dy/dx = b$). Forecasts are easy too, if you keep the code for external validation data.