how to calculate the residual sum of squares
residual sum of squares rss vs residual standard error rse the residual standard error rse is another statistical term used to describe the difference in standard deviations of observed values versus predicted values as shown by points in a regression analysis it is a goodness of fit measure that can be used to analyze how well a set of data points fit with the actual model rse is computed by dividing the rss by the number of observations in the sample less 2 and then taking the square root rse rss n 2 1 2minimizing rss for optimal fitin the realm of regression analysis minimizing the residual sum of squares is crucial for achieving the best possible fit of a model to the data among the different techniques to make this happen one of the most fundamental and widely used approaches is least squares regression least squares regression is a method that aims to find the line or curve that minimizes the sum of the squared differences these differences will be between the observed values and the values predicted by the model in essence the least squares regression seeks to strike a balance where the model captures the underlying trend of the data while still minimizing the discrepancies between what s been observed and what s been predicted the process of minimizing rss through least squares regression involves iteratively adjusting the parameters of the model this is usually done until the optimal fit is achieved for a simple linear regression model this typically entails finding the slope and intercept of the line that best fits the data in more complex scenarios the process becomes more intricate but has many of the same principles limitations of rssrss has some limitations to it first rss gives equal weight to all residuals this means that outliers can disproportionately influence the rss meaning that estimated coefficients may be negatively skewed another downside is that rss relies on several assumptions if any assumption such as linearity independence of errors or homoscedasticity are violated rss may lead to biased estimates and incorrect inferences while rss is useful for evaluating the fit of a single model comparing the fit across multiple models using rss alone can be tough this is because rss depends on the number of parameters in the model it isn t really meant to compare models with a different number of parameters last while rss is easy to compute and interpret it provides limited insight into the underlying structure of the data in cases where understanding the relationship between predictors and the response variable is important there may be better metrics to use in some ways rss can act somewhat like a black box where the relationships aren t entirely known only the end value is of most importance special considerationsfinancial markets have increasingly become more quantitatively driven as such in search of an edge many investors are using advanced statistical techniques to aid in their decisions big data machine learning and artificial intelligence applications further necessitate the use of statistical properties to guide contemporary investment strategies the residual sum of squares or rss statistics is one of many statistical properties enjoying a renaissance statistical models are used by investors and portfolio managers to track an investment s price and use that data to predict future movements the study called regression analysis might involve analyzing the relationship in price movements between a commodity and the stocks of companies engaged in producing the commodity finding the residual sum of squares by hand can be difficult and time consuming because it involves a lot of subtracting squaring and summing the calculations can be prone to errors for this reason you may decide to use software such as excel to do the calculations any model might have variances between the predicted values and actual results although the variances might be explained by the regression analysis the rss represents the variances or errors that are not explained since a sufficiently complex regression function can be made to closely fit virtually any data set further study is necessary to determine whether the regression function is in fact useful in explaining the variance of the dataset typically however a smaller or lower value for the rss is ideal in any model since it means there s less variation in the data set in other words the lower the sum of squared residuals the better the regression model is at explaining the data example of the rssfor a simple but lengthy demonstration of the rss calculation consider the well known correlation between a country s consumer spending and its gdp the following chart reflects the published values of consumer spending and gross domestic product for the 27 states of the european union note that this information may have slightly changed since it has been published but the example of residual sum of squares remains valid 12consumer spending and gdp have a strong positive correlation and it is possible to predict a country s gdp based on consumer spending cs using the formula for a best fit line this relationship can be approximated as the units for both gdp and consumer spending are in millions of u s dollars this formula is highly accurate for most purposes but it is not perfect due to the individual variations in each country s economy the following chart compares the projected gdp of each country based on the formula above and the actual gdp as recorded by the world bank the column on the right indicates the residual squares the squared difference between each projected value and its actual value the numbers appear large but their sum is actually lower than the rss for any other possible trendline if a different line had a lower rss for these data points that line would be the best fit line 