In a simple linear model with a single explanatory variable,
$\alpha_i = \beta_0 + \beta_1 \delta_i + \epsilon_i$
I find that removing the intercept term improves the fit greatly (value of $R^2$ goes from 0.3 to 0.9). However, the intercept term appears to be statistically significant.
With intercept:
Call: lm(formula = alpha ~ delta, data = cf) Residuals: Min 1Q Median 3Q Max -0.72138 -0.15619 -0.03744 0.14189 0.70305 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.48408 0.05397 8.97 <2e-16 *** delta 0.46112 0.04595 10.04 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2435 on 218 degrees of freedom Multiple R-squared: 0.316, Adjusted R-squared: 0.3129 F-statistic: 100.7 on 1 and 218 DF, p-value: < 2.2e-16
Without intercept:
Call: lm(formula = alpha ~ 0 + delta, data = cf) Residuals: Min 1Q Median 3Q Max -0.92474 -0.15021 0.05114 0.21078 0.85480 Coefficients: Estimate Std. Error t value Pr(>|t|) delta 0.85374 0.01632 52.33 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2842 on 219 degrees of freedom Multiple R-squared: 0.9259, Adjusted R-squared: 0.9256 F-statistic: 2738 on 1 and 219 DF, p-value: < 2.2e-16
How would you interpret these results? Should an intercept term be included in the model or not?