One of the assumptions of the residuals is that there is no real pattern to the residuals and if you were to graph a density to them, you would end up with something approaching a uniform distribution. $\endgroup$ ... Find the answer to your question by asking. Ask question ... See similar questions with these tags. ... 12 Expected Value ... More on stats.stackexchange.com

Indeed the absolute response residuals (on the scale of the original dependent variable) can be calculated as @Roland says with More on stackoverflow.com

Here's the answer I wish I'd had given to me when I asked the same question during my introductory statistics classes. There are many reasons, and the two objectives do not give equivalent results. Minimizing the sum of squared residuals is called "ordinary least squares" and is generally the first technique students learn in estimating functions. Minimizing the sum of absolute is generally called "median regression" for reasons I will discuss later, and is a somewhat less popular technique. Wikipedia indicates that the idea of median regression was actually developed first, which is unsurprising as it is indeed more intuitive. The issue is that there isn't a closed form solution (i.e. a simple formula you can plug numbers into) to find the coefficients that minimize the sum of absolute residuals. In contrast, summing squared residuals gives an objective function that is differentiable: differentiating, setting the derivative equal to zero, and then solving gives a formula for the coefficients that is straightforward to compute. (Technically we are using partial derivatives, and the algebra is a lot easier if you have matrices available, but the basic idea is the same as you would learn in an introductory differential calculus class.) Now, that was a big deal when these ideas were first being developed back in the 18th and 19th century, as then "computer" meant someone who had to perform computations by hand. Algorithms for finding the median regression coefficients existed but were harder to implement. Today we recognize that computing these coefficients is a "linear programming" optimization problem, for which many algorithms exist, most notably the Simplex algorithm. So on a modern computer the two methods are basically equally easy to compute. Then there's the question of inference. In a traditional statistics or econometrics course you would spend a lot of time developing the machinery to do things like hypothesis testing (e.g. suppose I gather a random sample and get an estimated slope coefficient of 0.017, which looks small. It is useful to ask the question "If the true population slope coefficient is 0, how likely is it that we could get an estimated slope coefficient of 0.017 or more extreme?". Very similar is the most basic A/B test, which asks "If the true difference between these two groups is 0, how likely is it that I would get an observed difference as big as that observed in the data just due to random chance?"). The fact we can explicitly write out the OLS formula makes it substantially easier to develop the statistical theory for this. There are also some optimality results, such as the Gauss-Markov Theorem, that say that OLS is "best", albeit in a very specific sense under a very restrictive set of assumptions. The statistical theory of median regression has also been figures out, but it requires more advanced math and is somewhat less elegant. So for both OLS and median regression you can compute the coefficients and perform statistical inference. So why do most students learn about OLS and not about median regression? Part of it is path dependence - OLS was developed first, so lots of people learned it and taught it to others, and it's just easier to stick with what people have learned in the past than switch to something else (e.g. you can keep using the same textbooks). But all the reasons that made it simpler to compute and develop inference for also make it easier for students to learn. Okay, but the pedagogy of introductory courses doesn't really matter once you get into the real world and are choosing which method to use. And these days there are pre-programmed algorithms that will do both estimation and inference for you in a single command, so the differences there don't really matter to a practitioner either. If you've got some data and want to estimate the relationship between Y and X, should you use OLS or median regression? You're absolutely right that squaring does something subtle to the residuals. It "skews" them in the sense of disproportionately trying to reduce large residuals, whereas median regression weights both small and large residuals equally. This is why advocates of median regression say that median regression is more "robust to outliers": if there is a weird Y observation (e.g. someone got a decimal point in the wrong location when transcribing data), OLS is going to try really hard to fit that observation. That is, if you imagine having a bunch of X's and Y's that pretty much follow a straight line, and then one really weird observation, the median regression line is going to be closer to the straight line than the OLS line. But the really exciting part happens when you pause and ask what the heck these estimators are actually estimating. One can show theoretically that minimizing squared errors results in a conditional mean, i.e. given a particular value of X, the predicted value of Y at that X is the "average" (in the sense of arithmetic mean or expected value) value of Y. In contrast, minimizing the sum of absolute errors results in the conditional median: for a fixed X, 50% of Y will be above this number, and 50% below. Upon realizing this people also realized that by weighting negative errors and positive errors differently, one can actually extend median regression to quantile regression (e.g. you can estimate the conditional 0.25 quantile: for a fixed X, 25% of Y will be below it and 75% will be above it). So there's a school of thought out there that quantile regression should be used a lot more: it's robust to outliers and by estimating it for several quantiles you can get a fuller picture of how things are going. On the flip side, however, there are some theoretical results that suggest OLS will give you a more precise estimate. OLS estimates are also much more interpretable (if you are confident your model has a causal interpretation, the slope is the average marginal effect of X on Y). In practice the difference between OLS and median regression usually isn't big enough to matter much - certainly not to the extent that advocates for median regression can say "here are 10,000 cases where using median regression would perform way better". Also, since median regression is a more advanced technique to learn, it would be better to compare it to other more advanced techniques. Median regression has all the issues OLS does in terms of needing to specify exactly what variables are in the model, and in what way (e.g. squared, interaction terms, etc.). If your goal is just to have an algorithm that will give you some sort of sensible prediction, many other tools exist that will do a much better job (see, for example, the later chapters in the book you are reading). And if you for some reason actually do need to estimate a conditional quantile and/or do inference, you might look into the "generalized random forest" R package and associated paper by Athey et al. And the sheer length of this comment reveals to me why my professors did not spend time explaining why they used squared residuals instead of absolute values! Edit: Thanks for the gold! More on reddit.com

You can use the absolute value. This is called the L1 norm and is used for robust regression.

You can read more about it here: http://www.johnmyleswhite.com/notebook/2013/03/22/using-norms-to-understand-linear-regression/

Practically, the math is easier in ordinary least squares regression: You want to minimize the squared residuals so you can take the derivative, set it equal to 0 and solve. Easier to compute the derivative of a polynomial than absolute value.

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