Ordinary least squares

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the given dataset and those predicted by the linear function of the independent variable.
Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation.
The OLS estimator is consistent when the regressors are exogenous, and—by the Gauss–Markov theorem—optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors are normally distributed, OLS is the maximum likelihood estimator.
Linear model.
Suppose the data consists of formula_1 observations formula_2. Each observation formula_3 includes a scalar response formula_4 and a column vector formula_5 of formula_6 parameters (regressors), i.e., formula_7. In a linear regression model, the response variable, formula_4, is a linear function of the regressors:
or in vector form,
where formula_5, as introduced previously, is a column vector of the formula_3-th observation of all the explanatory variables; formula_13 is a formula_14 vector of unknown parameters; and the scalar formula_15 represents unobserved random variables (errors) of the formula_3-th observation. formula_15 accounts for the influences upon the responses formula_4 from sources other than the explanators formula_5. This model can also be written in matrix notation as
where formula_21 and formula_22 are formula_23 vectors of the response variables and the errors of the formula_24 observations, and formula_25 is an formula_26 matrix of regressors, also sometimes called the design matrix, whose row formula_3 is formula_28 and contains the formula_3-th observations on all the explanatory variables.
As a rule, the constant term is always included in the set of regressors formula_25, say, by taking formula_31 for all formula_32. The coefficient formula_33 corresponding to this regressor is called the "intercept".
Regressors do not have to be independent: there can be any desired relationship between the regressors (so long as it is not a linear relationship). For instance, we might suspect the response depends linearly both on a value and its square; in which case we would include one regressor whose value is just the square of another regressor. In that case, the model would be "quadratic" in the second regressor, but none-the-less is still considered a "linear" model because the model "is" still linear in the parameters (formula_13).
Matrix/vector formulation.
Consider an overdetermined system
of formula_24 linear equations in formula_6 unknown coefficients, formula_38, with formula_39. (Note: for a linear model as above, not all elements in formula_40 contains information on the data points. The first column is populated with ones, formula_41. Only the other columns contain actual data. So here formula_6 is equal to the number of regressors plus one.) This can be written in matrix form as
where
Such a system usually has no exact solution, so the goal is instead to find the coefficients formula_13 which fit the equations "best", in the sense of solving the quadratic minimization problem
where the objective function formula_47 is given by
A justification for choosing this criterion is given in Properties below. This minimization problem has a unique solution, provided that the formula_6 columns of the matrix formula_40 are linearly independent, given by solving the normal equations
The matrix formula_52 is known as the normal matrix and the matrix formula_53 is known as the moment matrix of regressand by regressors. Finally, formula_54 is the coefficient vector of the least-squares hyperplane, expressed as
Estimation.
Suppose "b" is a "candidate" value for the parameter vector "β". The quantity , called the residual for the "i"-th observation, measures the vertical distance between the data point and the hyperplane , and thus assesses the degree of fit between the actual data and the model. The sum of squared residuals (SSR) (also called the error sum of squares (ESS) or residual sum of squares (RSS)) is a measure of the overall model fit:
where "T" denotes the matrix transpose, and the rows of "X", denoting the values of all the independent variables associated with a particular value of the dependent variable, are "Xi = xi"T. The value of "b" which minimizes this sum is called the OLS estimator for "β". The function "S"("b") is quadratic in "b" with positive-definite Hessian, and therefore this function possesses a unique global minimum at formula_57, which can be given by the explicit formula:[proof]
The product "N"="X"T "X" is a normal matrix and its inverse, "Q"="N"–1, is the "cofactor matrix" of "β", closely related to its covariance matrix, "C""β".
The matrix ("X"T "X")–1 "X"T="Q" "X"T is called the Moore–Penrose pseudoinverse matrix of X. This formulation highlights the point that estimation can be carried out if, and only if, there is no perfect multicollinearity between the explanatory variables (which would cause the normal matrix to have no inverse).
After we have estimated "β", the fitted values (or predicted values) from the regression will be 
where "P" = "X"("X"T"X")−1"X"T is the projection matrix onto the space "V" spanned by the columns of "X". This matrix "P" is also sometimes called the hat matrix because it "puts a hat" onto the variable "y". Another matrix, closely related to "P" is the "annihilator" matrix ; this is a projection matrix onto the space orthogonal to "V". Both matrices "P" and "M" are symmetric and idempotent (meaning that and ), and relate to the data matrix "X" via identities and . Matrix "M" creates the residuals from the regression:
Using these residuals we can estimate the value of "σ" 2 using the reduced chi-squared statistic:
The denominator, "n"−"p", is the statistical degrees of freedom. The first quantity, "s"2, is the OLS estimate for "σ"2, whereas the second, formula_62, is the MLE estimate for "σ"2. The two estimators are quite similar in large samples; the first estimator is always unbiased, while the second estimator is biased but has a smaller mean squared error. In practice "s"2 is used more often, since it is more convenient for the hypothesis testing. The square root of "s"2 is called the regression standard error, standard error of the regression, or standard error of the equation.
It is common to assess the goodness-of-fit of the OLS regression by comparing how much the initial variation in the sample can be reduced by regressing onto "X". The coefficient of determination "R"2 is defined as a ratio of "explained" variance to the "total" variance of the dependent variable "y", in the cases where the regression sum of squares equals the sum of squares of residuals:
where TSS is the total sum of squares for the dependent variable, formula_64, and formula_65 is an "n"×"n" matrix of ones. (formula_66 is a centering matrix which is equivalent to regression on a constant; it simply subtracts the mean from a variable.) In order for "R"2 to be meaningful, the matrix "X" of data on regressors must contain a column vector of ones to represent the constant whose coefficient is the regression intercept. In that case, "R"2 will always be a number between 0 and 1, with values close to 1 indicating a good degree of fit.
The variance in the prediction of the independent variable as a function of the dependent variable is given in the article Polynomial least squares.
Simple linear regression model.
If the data matrix "X" contains only two variables, a constant and a scalar regressor "xi", then this is called the "simple regression model". This case is often considered in the beginner statistics classes, as it provides much simpler formulas even suitable for manual calculation. The parameters are commonly denoted as :
The least squares estimates in this case are given by simple formulas
Introducing formula_72 and a matrix "K" with the assumption that a matrix formula_73 is non-singular and "K"T "X" = 0 (cf. Orthogonal projections), the residual vector should satisfy the following equation:
The equation and solution of linear least squares are thus described as follows:
Another way of looking at it is to consider the regression line to be a weighted average of the lines passing through the combination of any two points in the dataset. Although this way of calculation is more computationally expensive, it provides a better intuition on OLS.
Maximum likelihood.
The OLS estimator is identical to the maximum likelihood estimator (MLE) under the normality assumption for the error terms.[proof] This normality assumption has historical importance, as it provided the basis for the early work in linear regression analysis by Yule and Pearson. From the properties of MLE, we can infer that the OLS estimator is asymptotically efficient (in the sense of attaining the Cramér–Rao bound for variance) if the normality assumption is satisfied.
Generalized method of moments.
In iid case the OLS estimator can also be viewed as a GMM estimator arising from the moment conditions
These moment conditions state that the regressors should be uncorrelated with the errors. Since "xi" is a "p"-vector, the number of moment conditions is equal to the dimension of the parameter vector "β", and thus the system is exactly identified. This is the so-called classical GMM case, when the estimator does not depend on the choice of the weighting matrix.
Note that the original strict exogeneity assumption implies a far richer set of moment conditions than stated above. In particular, this assumption implies that for any vector-function "ƒ", the moment condition will hold. However it can be shown using the Gauss–Markov theorem that the optimal choice of function "ƒ" is to take , which results in the moment equation posted above.
Properties.
Assumptions.
There are several different frameworks in which the linear regression model can be cast in order to make the OLS technique applicable. Each of these settings produces the same formulas and same results. The only difference is the interpretation and the assumptions which have to be imposed in order for the method to give meaningful results. The choice of the applicable framework depends mostly on the nature of data in hand, and on the inference task which has to be performed.
One of the lines of difference in interpretation is whether to treat the regressors as random variables, or as predefined constants. In the first case (random design) the regressors "xi" are random and sampled together with the "yi"'s from some population, as in an observational study. This approach allows for more natural study of the asymptotic properties of the estimators. In the other interpretation (fixed design), the regressors "X" are treated as known constants set by a design, and "y" is sampled conditionally on the values of "X" as in an experiment. For practical purposes, this distinction is often unimportant, since estimation and inference is carried out while conditioning on "X". All results stated in this article are within the random design framework.
Classical linear regression model.
The classical model focuses on the "finite sample" estimation and inference, meaning that the number of observations "n" is fixed. This contrasts with the other approaches, which study the asymptotic behavior of OLS, and in which the number of observations is allowed to grow to infinity.
Independent and identically distributed (iid).
In some applications, especially with cross-sectional data, an additional assumption is imposed — that all observations are independent and identically distributed. This means that all observations are taken from a random sample which makes all the assumptions listed earlier simpler and easier to interpret. Also this framework allows one to state asymptotic results (as the sample size ), which are understood as a theoretical possibility of fetching new independent observations from the data generating process. The list of assumptions in this case is:
Finite sample properties.
First of all, under the "strict exogeneity" assumption the OLS estimators formula_82 and "s"2 are unbiased, meaning that their expected values coincide with the true values of the parameters:[proof]
If the strict exogeneity does not hold (as is the case with many time series models, where exogeneity is assumed only with respect to the past shocks but not the future ones), then these estimators will be biased in finite samples.
The "variance-covariance matrix" (or simply "covariance matrix") of formula_82 is equal to
In particular, the standard error of each coefficient formula_86 is equal to square root of the "j"-th diagonal element of this matrix. The estimate of this standard error is obtained by replacing the unknown quantity "σ"2 with its estimate "s"2. Thus,
It can also be easily shown that the estimator formula_82 is uncorrelated with the residuals from the model:
The Gauss–Markov theorem states that under the "spherical errors" assumption (that is, the errors should be uncorrelated and homoscedastic) the estimator formula_82 is efficient in the class of linear unbiased estimators. This is called the best linear unbiased estimator (BLUE). Efficiency should be understood as if we were to find some other estimator formula_91 which would be linear in "y" and unbiased, then 
in the sense that this is a nonnegative-definite matrix. This theorem establishes optimality only in the class of linear unbiased estimators, which is quite restrictive. Depending on the distribution of the error terms "ε", other, non-linear estimators may provide better results than OLS.
Assuming normality.
The properties listed so far are all valid regardless of the underlying distribution of the error terms. However, if you are willing to assume that the "normality assumption" holds (that is, that ), then additional properties of the OLS estimators can be stated.
The estimator formula_82 is normally distributed, with mean and variance as given before:
where "Q" is the cofactor matrix. This estimator reaches the Cramér–Rao bound for the model, and thus is optimal in the class of all unbiased estimators. Note that unlike the Gauss–Markov theorem, this result establishes optimality among both linear and non-linear estimators, but only in the case of normally distributed error terms.
The estimator "s"2 will be proportional to the chi-squared distribution:
The variance of this estimator is equal to , which does not attain the Cramér–Rao bound of . However it was shown that there are no unbiased estimators of "σ"2 with variance smaller than that of the estimator "s"2. If we are willing to allow biased estimators, and consider the class of estimators that are proportional to the sum of squared residuals (SSR) of the model, then the best (in the sense of the mean squared error) estimator in this class will be , which even beats the Cramér–Rao bound in case when there is only one regressor ().
Moreover, the estimators formula_82 and "s"2 are independent, the fact which comes in useful when constructing the t- and F-tests for the regression.
Influential observations.
As was mentioned before, the estimator formula_97 is linear in "y", meaning that it represents a linear combination of the dependent variables "yi". The weights in this linear combination are functions of the regressors "X", and generally are unequal. The observations with high weights are called influential because they have a more pronounced effect on the value of the estimator.
To analyze which observations are influential we remove a specific "j"-th observation and consider how much the estimated quantities are going to change (similarly to the jackknife method). It can be shown that the change in the OLS estimator for "β" will be equal to 
where is the "j"-th diagonal element of the hat matrix "P", and "xj" is the vector of regressors corresponding to the "j"-th observation. Similarly, the change in the predicted value for "j"-th observation resulting from omitting that observation from the dataset will be equal to 
From the properties of the hat matrix, , and they sum up to "p", so that on average . These quantities "hj" are called the leverages, and observations with high "hj" are called leverage points. Usually the observations with high leverage ought to be scrutinized more carefully, in case they are erroneous, or outliers, or in some other way atypical of the rest of the dataset.
Partitioned regression.
Sometimes the variables and corresponding parameters in the regression can be logically split into two groups, so that the regression takes form
where "X"1 and "X"2 have dimensions "n"×"p"1, "n"×"p"2, and "β"1, "β"2 are "p"1×1 and "p"2×1 vectors, with .
The Frisch–Waugh–Lovell theorem states that in this regression the residuals formula_101 and the OLS estimate formula_102 will be numerically identical to the residuals and the OLS estimate for "β"2 in the following regression:
where "M"1 is the annihilator matrix for regressors "X"1.
The theorem can be used to establish a number of theoretical results. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the de-meaned variables but without the constant term.
Constrained estimation.
Suppose it is known that the coefficients in the regression satisfy a system of linear equations
where "Q" is a "p"×"q" matrix of full rank, and "c" is a "q"×1 vector of known constants, where . In this case least squares estimation is equivalent to minimizing the sum of squared residuals of the model subject to the constraint "A". The constrained least squares (CLS) estimator can be given by an explicit formula:
This expression for the constrained estimator is valid as long as the matrix "XTX" is invertible. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails, "β" will not be identifiable. However it may happen that adding the restriction "A" makes "β" identifiable, in which case one would like to find the formula for the estimator. The estimator is equal to 
where "R" is a "p"×("p" − "q") matrix such that the matrix is non-singular, and . Such a matrix can always be found, although generally it is not unique. The second formula coincides with the first in case when "XTX" is invertible.
Large sample properties.
The least squares estimators are point estimates of the linear regression model parameters "β". However, generally we also want to know how close those estimates might be to the true values of parameters. In other words, we want to construct the interval estimates.
Since we haven't made any assumption about the distribution of error term "εi", it is impossible to infer the distribution of the estimators formula_97 and formula_108. Nevertheless, we can apply the central limit theorem to derive their "asymptotic" properties as sample size "n" goes to infinity. While the sample size is necessarily finite, it is customary to assume that "n" is "large enough" so that the true distribution of the OLS estimator is close to its asymptotic limit.
We can show that under the model assumptions, the least squares estimator for "β" is consistent (that is formula_97 converges in probability to "β") and asymptotically normal:[proof]
where formula_111
Intervals.
Using this asymptotic distribution, approximate two-sided confidence intervals for the "j"-th component of the vector formula_112 can be constructed as
where "q" denotes the quantile function of standard normal distribution, and [·]"jj" is the "j"-th diagonal element of a matrix.
Similarly, the least squares estimator for "σ"2 is also consistent and asymptotically normal (provided that the fourth moment of "εi" exists) with limiting distribution
These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose formula_115 is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The mean response is the quantity formula_116, whereas the predicted response is formula_117. Clearly the predicted response is a random variable, its distribution can be derived from that of formula_112:
which allows construct confidence intervals for mean response formula_120 to be constructed:
Hypothesis testing.
Two hypothesis tests are particularly widely used. First, one wants to know if the estimated regression equation is any better than simply predicting that all values of the response variable equal its sample mean (if not, it is said to have no explanatory power). The null hypothesis of no explanatory value of the estimated regression is tested using an F-test. If the calculated F-value is found to be large enough to exceed its critical value for the pre-chosen level of significance, the null hypothesis is rejected and the alternative hypothesis, that the regression has explanatory power, is accepted. Otherwise, the null hypothesis of no explanatory power is accepted.
Second, for each explanatory variable of interest, one wants to know whether its estimated coefficient differs significantly from zero—that is, whether this particular explanatory variable in fact has explanatory power in predicting the response variable. Here the null hypothesis is that the true coefficient is zero. This hypothesis is tested by computing the coefficient's t-statistic, as the ratio of the coefficient estimate to its standard error. If the t-statistic is larger than a predetermined value, the null hypothesis is rejected and the variable is found to have explanatory power, with its coefficient significantly different from zero. Otherwise, the null hypothesis of a zero value of the true coefficient is accepted.
In addition, the Chow test is used to test whether two subsamples both have the same underlying true coefficient values. The sum of squared residuals of regressions on each of the subsets and on the combined data set are compared by computing an F-statistic; if this exceeds a critical value, the null hypothesis of no difference between the two subsets is rejected; otherwise, it is accepted.
Example with real data.
The following data set gives average heights and weights for American women aged 30–39 (source: "The World Almanac and Book of Facts, 1975").
When only one dependent variable is being modeled, a scatterplot will suggest the form and strength of the relationship between the dependent variable and regressors. It might also reveal outliers, heteroscedasticity, and other aspects of the data that may complicate the interpretation of a fitted regression model. The scatterplot suggests that the relationship is strong and can be approximated as a quadratic function. OLS can handle non-linear relationships by introducing the regressor HEIGHT2. The regression model then becomes a multiple linear model:
The output from most popular statistical packages will look similar to this:
In this table:
Ordinary least squares analysis often includes the use of diagnostic plots designed to detect departures of the data from the assumed form of the model. These are some of the common diagnostic plots:
An important consideration when carrying out statistical inference using regression models is how the data were sampled. In this example, the data are averages rather than measurements on individual women. The fit of the model is very good, but this does not imply that the weight of an individual woman can be predicted with high accuracy based only on her height.
Sensitivity to rounding.
This example also demonstrates that coefficients determined by these calculations are sensitive to how the data is prepared. The heights were originally given rounded to the nearest inch and have been converted and rounded to the nearest centimetre. Since the conversion factor is one inch to 2.54 cm this is "not" an exact conversion. The original inches can be recovered by Round(x/0.0254) and then re-converted to metric without rounding. If this is done the results become:
Using either of these equations to predict the weight of a 5' 6" (1.6764 m) woman gives similar values: 62.94 kg with rounding vs. 62.98 kg without rounding. Thus a seemingly small variation in the data has a real effect on the coefficients but a small effect on the results of the equation.
While this may look innocuous in the middle of the data range it could become significant at the extremes or in the case where the fitted model is used to project outside the data range (extrapolation).
This highlights a common error: this example is an abuse of OLS which inherently requires that the errors in the independent variable (in this case height) are zero or at least negligible. The initial rounding to nearest inch plus any actual measurement errors constitute a finite and non-negligible error. As a result, the fitted parameters are not the best estimates they are presumed to be. Though not totally spurious the error in the estimation will depend upon relative size of the "x" and "y" errors.
Another example with less real data.
Problem statement.
We can use the least square mechanism to figure out the equation of a two body orbit in polar base co-ordinates. The equation typically used is formula_129 where formula_130 is the radius of how far the object is from one of the bodies. In the equation the parameters formula_6 and formula_132 are used to determine the path of the orbit. We have measured the following data. 
We need to find the least-squares approximation of formula_132 and formula_6 for the given data.
Solution.
First we need to represent e and p in a linear form. So we are going to rewrite the equation formula_130 as formula_136. Now we can use this form to represent our observational data as:
formula_137 where formula_138 is formula_139 and formula_140 is formula_141 and formula_142 is constructed by the first column being the coefficient of formula_139 and the second column being the coefficient of formula_141 and formula_145 is the values for the respective formula_146 so formula_147 and formula_148
On solving we get formula_149
so "formula_150 and formula_151"