Regression models can be used for prediction, but prediction is useful only when model performance is assessed carefully. This chapter uses the Milk Data model to explain fitted values, residuals, R², MAE, and RMSE.
In Chapter 10 we estimated:
\[ \widehat{Price}=516.6+417.0\times Volume1000 \]
Can this equation predict milk prices well?
A fitted value is the value predicted by the regression equation:
\[ \widehat{Y} \]
X = sm.add_constant(milk_data["Volume1000"])
y = milk_data["Price"]
model = sm.OLS(y, X).fit()
milk_data["Predicted"] = model.predict(X)
milk_data[["Price", "Predicted"]].head()| Price | Predicted | |
|---|---|---|
| 0 | 490 | 884.955783 |
| 1 | 1960 | 2136.034010 |
| 2 | 1980 | 2044.288273 |
| 3 | 540 | 780.699264 |
| 4 | 490 | 884.955783 |
The residual is:
\[ Residual_i = Actual_i - Predicted_i \]
milk_data["Residual"] = milk_data["Price"] - milk_data["Predicted"]Residuals measure prediction error.
The model produces:
\[ R^2 = 0.274 \]
Package volume explains approximately 27.4% of the variation in milk prices. The remaining 72.6% is explained by other factors not included in this simple model.
model.rsquarednp.float64(0.27387839998291263)Using the actual Milk Data regression:
\[ MAE = 437.6 \]
\[ RMSE = 772.0 \]
MAE means that predictions differ from actual prices by about 438 price units on average. RMSE is larger because it penalizes large prediction errors more strongly.
mae = mean_absolute_error(milk_data["Price"], milk_data["Predicted"])
rmse = root_mean_squared_error(milk_data["Price"], milk_data["Predicted"])
mae, rmse(437.5678085533861, 772.0000524580844)plt.scatter(milk_data["Price"], milk_data["Predicted"])
plt.xlabel("Actual Price")
plt.ylabel("Predicted Price")
plt.title("Actual versus Predicted Prices")
plt.show()
plt.scatter(milk_data["Predicted"], milk_data["Residual"])
plt.axhline(y=0, linestyle="--")
plt.xlabel("Predicted Price")
plt.ylabel("Residual")
plt.title("Residual Plot")
plt.show()
A good residual plot should show random scatter with no systematic pattern.
Prediction and explanation are related but not identical. Prediction focuses on accuracy. Explanation focuses on economic interpretation. A model can predict well without being causal, and a model can be useful for explanation even with modest prediction power.
A high R² does not prove causality and does not guarantee that the model is correctly specified.
Volume is useful but not sufficient. Additional variables are needed to improve prediction and explanation.