When is R squared negative? [duplicate]

Have you forgotten to include an intercept in your regression? I'm not familiar with SPSS code, but on page 21 of Hayashi's Econometrics:

If the regressors do not include a constant but (as some regression software packages do) you nevertheless calculate $R^2$ by the formula

$R^2=1-\frac{\sum_{i=1}^{n}e_i^2}{\sum_{i=1}^{n}(y_i-\bar{y})^2}$

then the $R^2$ can be negative. This is because, without the benefit of an intercept, the regression could do worse than the sample mean in terms of tracking the dependent variable (i.e., the numerator could be greater than the denominator).

I'd check and make sure that SPSS is including an intercept in your regression.

This can happen if you have a time series that is N.i.i.d. and you construct an inappropriate ARIMA model of the form(0,1,0) which is a first difference random walk model with no drift then the variance (sum of squares - SSE ) of the residuals will be larger than the variance (sum of squares SSO) of the original series. Thus the equation 1-SSE/SSO will yield a negative number as SSE execeedS SSO . We have seen this when users simply fit an assumed model or use inadequate procedures to identify/form an appropriate ARIMA structure. The larger message IS that a model can distort (much like a pair of bad glasses ) your vision. Without having access to your data I would otherwise have a problem in explaining your faulty results. Have you brought this to the attention of IBM ?

The idea of an assumed model being counter-productive has been echoed by Harvey Motulsky. Great post Harvey !

For those from the Machine Learning field:
A negative R squared ($R^2$) means that the model is predicting worse than a dummy model that simply uses the mean of the target values ($\bar{y}$) as the prediction for all instances.

Mathematically:

$R^2 = 1 - \frac{MSE(y, \hat{y})}{MSE(y, \bar{y})}$

Where:

• y = the true target values
• ŷ (y_pred) = the predicted values from the model
• ȳ (y_mean) = the mean of the true target values
• MSE(y, y_pred) = mean squared error of the model
• MSE(y, y_mean) = mean squared error of a dummy model that always predicts the mean

If the model is bad enough that MSE(y, y_pred) is greater than MSE(y, y_mean), the R² score becomes negative.

Here's an example in Python:

from sklearn.metrics import r2_score, mean_squared_error import numpy as np # True target values y = np.array([3, 5, 7, 9, 11]) # Poor model predictions y_pred = np.array([10, 10, 10, 10, 10]) # Dummy model predictions (mean of y) y_mean = np.full_like(y, y.mean()) # R squared calculation r2 = r2_score(y, y_pred) r2_using_mean = r2_score(y, y_mean) # Output results print("True values (y):", y) print("Model predictions (y_pred):", y_pred) print("Mean of y (ȳ):", y_mean[0]) print("R² score:", r2) print("R² score using mean as pred:", r2_using_mean) 

Which prints:

True values (y): [ 3 5 7 9 11] Model poor predictions (y_pred): [10 10 10 10 10] Mean of y (ȳ): 7 R² score: -1.125 R² score using mean as pred: 0.0 

Also here's a plot:

import matplotlib.pyplot as plt plt.plot(y, label="True values (y)", marker='o') plt.plot(y_pred, label=f"Model predictions (y_pred) - R2: {r2}", linestyle='--', marker='x') plt.plot(y_mean, label=f"Dummy predictions (y_mean) - R2: {r2_using_mean}", linestyle=':', marker='s') plt.title("True vs Model vs Mean Predictions") plt.xlabel("Sample Index") plt.ylabel("Value") plt.legend() plt.grid(True) plt.tight_layout() plt.show()