Proofs involving ordinary least squares

The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition.
Derivation of the normal equations.
Define the formula_1th residual to be
Then the objective formula_3 can be rewritten
Given that "S" is convex, it is minimized when its gradient vector is zero (This follows by definition: if the gradient vector is not zero, there is a direction in which we can move to minimize it further – see maxima and minima.) The elements of the gradient vector are the partial derivatives of "S" with respect to the parameters:
The derivatives are
Substitution of the expressions for the residuals and the derivatives into the gradient equations gives
Thus if formula_8 minimizes "S", we have
Upon rearrangement, we obtain the normal equations:
The normal equations are written in matrix notation as
The solution of the normal equations yields the vector formula_12 of the optimal parameter values.
Derivation directly in terms of matrices.
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize
Here formula_14 has the dimension 1x1 (the number of columns of formula_15), so it is a scalar and equal to its own transpose, hence formula_16
and the quantity to minimize becomes
Differentiating this with respect to formula_18 and equating to zero to satisfy the first-order conditions gives
which is equivalent to the above-given normal equations. A sufficient condition for satisfaction of the second-order conditions for a minimum is that formula_20 have full column rank, in which case formula_21 is positive definite.
Derivation without calculus.
When formula_21 is positive definite, the formula for the minimizing value of formula_23 can be derived without the use of derivatives. The quantity
can be written as
where formula_26 depends only on formula_27 and formula_28, and formula_29 is the inner product defined by
It follows that formula_31 is equal to
and therefore minimized exactly when
Generalization for complex equations.
In general, the coefficients of the matrices formula_34 and formula_35 can be complex. By using a Hermitian transpose instead of a simple transpose, it is possible to find a vector formula_36 which minimizes formula_37, just as for the real matrix case. In order to get the normal equations we follow a similar path as in previous derivations:
where formula_39 stands for Hermitian transpose.
We should now take derivatives of formula_31 with respect to each of the coefficients formula_41, but first we separate real and imaginary parts to deal with the conjugate factors in above expression. For the formula_42 we have
and the derivatives change into
After rewriting formula_31 in the summation form and writing formula_46 explicitly, we can calculate both partial derivatives with result:
which, after adding it together and comparing to zero (minimization condition for formula_36) yields
In matrix form:
Least squares estimator for "β".
Using matrix notation, the sum of squared residuals is given by
Since this is a quadratic expression, the vector which gives the global minimum may be found via matrix calculus by differentiating with respect to the vector formula_52 (using denominator layout) and setting equal to zero:
By assumption matrix "X" has full column rank, and therefore "XTX" is invertible and the least squares estimator for "β" is given by
Unbiasedness and variance of formula_55.
Plug "y" = "Xβ" + "ε" into the formula for formula_55 and then use the law of total expectation:
where E["ε"|"X"] = 0 by assumptions of the model. Since the expected value of formula_58 equals the parameter it estimates, formula_52, it is an unbiased estimator of formula_52.
For the variance, let the covariance matrix of formula_61 be formula_62
(where formula_63 is the identity formula_64 matrix).
Then,
where we used the fact that formula_66 is just an affine transformation of formula_61 by the matrix formula_68.
For a simple linear regression model, where formula_69 (formula_70 is the "y"-intercept and formula_71 is the slope), one obtains
Expected value and biasedness of formula_73.
First we will plug in the expression for "y" into the estimator, and use the fact that "X'M" = "MX" = 0 (matrix "M" projects onto the space orthogonal to "X"):
Now we can recognize "ε"′"Mε" as a 1×1 matrix, such matrix is equal to its own trace. This is useful because by properties of trace operator, tr("AB") = tr("BA"), and we can use this to separate disturbance "ε" from matrix "M" which is a function of regressors "X":
Using the Law of iterated expectation this can be written as
Recall that "M" = "I" − "P" where "P" is the projection onto linear space spanned by columns of matrix "X". By properties of a projection matrix, it has "p" = rank("X") eigenvalues equal to 1, and all other eigenvalues are equal to 0. Trace of a matrix is equal to the sum of its characteristic values, thus tr("P") = "p", and tr("M") = "n" − "p". Therefore,
Since the expected value of formula_73 does not equal the parameter it estimates, formula_79, it is a biased estimator of formula_79. Note in the later section “Maximum likelihood” we show that under the additional assumption that errors are distributed normally, the estimator formula_73 is proportional to a chi-squared distribution with "n" – "p" degrees of freedom, from which the formula for expected value would immediately follow. However the result we have shown in this section is valid regardless of the distribution of the errors, and thus has importance on its own.
Consistency and asymptotic normality of formula_55.
Estimator formula_55 can be written as
We can use the law of large numbers to establish that 
By Slutsky's theorem and continuous mapping theorem these results can be combined to establish consistency of estimator formula_55:
The central limit theorem tells us that
Applying Slutsky's theorem again we'll have
Maximum likelihood approach.
Maximum likelihood estimation is a generic technique for estimating the unknown parameters in a statistical model by constructing a log-likelihood function corresponding to the joint distribution of the data, then maximizing this function over all possible parameter values. In order to apply this method, we have to make an assumption about the distribution of y given X so that the log-likelihood function can be constructed. The connection of maximum likelihood estimation to OLS arises when this distribution is modeled as a multivariate normal.
Specifically, assume that the errors ε have multivariate normal distribution with mean 0 and variance matrix "σ"2"I". Then the distribution of "y" conditionally on "X" is
and the log-likelihood function of the data will be
Differentiating this expression with respect to "β" and "σ"2 we'll find the ML estimates of these parameters:
We can check that this is indeed a maximum by looking at the Hessian matrix of the log-likelihood function.
Finite-sample distribution.
Since we have assumed in this section that the distribution of error terms is known to be normal, it becomes possible to derive the explicit expressions for the distributions of estimators formula_55 and formula_73:
so that by the affine transformation properties of multivariate normal distribution
Similarly the distribution of formula_73 follows from
where formula_100 is the symmetric projection matrix onto subspace orthogonal to "X", and thus "MX" = "X"′"M" = 0. We have argued before that this matrix rank "n" – "p", and thus by properties of chi-squared distribution,
Moreover, the estimators formula_55 and formula_73 turn out to be independent (conditional on "X"), a fact which is fundamental for construction of the classical t- and F-tests. The independence can be easily seen from following: the estimator formula_55 represents coefficients of vector decomposition of formula_105 by the basis of columns of "X", as such formula_55 is a function of "Pε". At the same time, the estimator formula_73 is a norm of vector "Mε" divided by "n", and thus this estimator is a function of "Mε". Now, random variables ("Pε", "Mε") are jointly normal as a linear transformation of "ε", and they are also uncorrelated because "PM" = 0. By properties of multivariate normal distribution, this means that "Pε" and "Mε" are independent, and therefore estimators formula_55 and formula_73 will be independent as well.
Derivation of simple linear regression estimators.
We look for formula_110 and formula_58 that minimize the sum of squared errors (SSE):
To find a minimum take partial derivatives with respect to formula_110 and formula_58
Before taking partial derivative with respect to formula_58, substitute the previous result for formula_117
Now, take the derivative with respect to formula_58:
And finally substitute formula_58 to determine formula_110