# Why can't I calculate the $R^2$ in some regression models if I use the method of maximum likelihood estimation?

I've modeled two regression models the first is a multiple linear regression (OLS) $$Y=\beta_0+\beta_1X_1+\cdots+\beta_nX_n+e$$ and I can get its $$R^2$$. The second model is a spatial autoregressive model (SAR) $$Y=\rho W + \beta_0 + \beta_1 X_1 + \cdots+\beta_nX_n+e$$ where $$W$$ is the contiguity matrix and $$\rho$$ is an unknown parameter. This model is estimated by the method of maximum likelihood but I cannot calculate its $$R^2$$ and rather I have to use the $$R^2$$ Nalgerkerke. I've found this "There is no direct equivalent to the OLS R-squared, these models are fitted by maximum likelihood." from http://r-sig-geo.2731867.n2.nabble.com/How-to-calculate-squared-R-of-spatial-autoregressive-models-td5762576.html but I'd like to know why I cannot calculate $$R^2$$ for this model if the formula is just $$R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\overline{y})^2}$$

• "There is no direct equivalent" does not mean "you cannot calculate it." The former needs to be interpreted as a warning about the applicability and interpretation of $R^2.$
– whuber
Jun 28, 2019 at 18:13
• Alright but why? I mean I can calculate it for both but is the interpretation different or no equivalent? Do you know some paper or book where I could read more about this? Thanks for your answer. Jun 28, 2019 at 18:42
• If you have a likelihood, then you can compute deviance. For Gaussian likelihood, deviance is the r-squared. Compare both formulas to understand if and why they differ. Aug 15, 2021 at 12:39
• Fit the model and find its predictions. What is $\sum_i\Big[ (y_i - \hat{y_i})(\hat{y_i} - \bar{y}) \Big]$? If that sum is not zero (or some tiny number that close enough to zero for arithmetic on a computer), then $R^2$ loses its usual "proportion of variance explained" interpretation.
– Dave
Dec 16, 2021 at 21:19

First, the definition of $$R^2$$ originates from the decomposition formula, i.e. $$S_T=S_R+S_e,$$ where $$S_T=\sum\limits_{i=1}^n(y_i-\bar y)^2$$, $$S_R=\sum\limits_{i=1}^n(\hat y_i-\bar y)^2$$, $$S_e=\sum\limits_{i=1}^n(y_i-\hat y_i)^2$$. Assume that the matrix form of the multiple linear regression is $$Y_{n\times 1}=X_{n\times (p+1)}\beta_{(p+1)\times 1}+\varepsilon_{n\times 1},$$ the proof of the above decomposition formula uses the normal equations of the OLS estimator $$\hat\beta_{OLS}$$ $$X^T(Y-X\hat\beta_{OLS})=0_{(p+1)\times 1}.$$ Second, the MLE estimator $$\hat\beta_{MLE}$$ of $$\beta$$ doesn't satisfy the normal equations unless $$\varepsilon_{n\times 1}$$ follows $$N(0_{n\times 1},I_n)$$.