Solved – How to interpret a low coefficient yet statistically significant with a high R-squared

interpretationr-squaredregressionregression coefficients

Consider the data below. (Dependent Variable: Human Development Index (HDI))

           Coefficient   Std. Error   t-ratio   p-value
const        -0.260523      0.129316     2.0146    0.05693  *
LITRATE       0.0031374    0.000965695   3.2488    0.00384  ***
LIFEXP        0.0051052    0.00268771    1.8995    0.07133  *
GINI          0.00598775   0.00175813    3.4057    0.00266  ***

R-squared           0.826974
Adjusted R-squared  0.802256
S.D. dependent var  0.053285
S.E. of regression  0.023695
Sum squared resid   0.011791
Durbin-Watson       0.618921

How do you interpret these results? Given the independent variables are significant yet it result to a small coefficient, but the R-squared is high?

Best Answer

$R^2$ is easily misinterpreted (see this excellent answer: is-r2-useful-or-dangerous). Note that $R^2$ is a function of several things: the marginal variance of $Y$, the residual variance, and both the slope of the relationships between your $X$s & $Y$ and how spread out your $X$s are. Consider the figure below:

enter image description here

The underlying data generating process is the same: the data were generated with the same slope and intercept, and with the same residual variance. Only the amount by which the $X$ values are spread out differs. However, the $R^2$ changes as we move from left to right. The three observed values are: $0.02$, $0.10$, and $0.24$. If I were to spread the values far enough apart, I could get the $R^2$ to asymptotically approach $1.0$.

Thus, how small the coefficients are is not sufficient to determine how large $R^2$ can be.

Further notes: @Martyn's candidate solution is popularly held, however, it won't necessarily answer your question, unfortunately. You will simply lose information about how far your $X$ values are spread out in terms of the original units. In addition, I don't know about the nature of the Human Development Index, but if @DJE is correct, the whole issue would be misconceived.

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