Business Statistics. Tommaso Proietti. Linear Regression. DEF - Università di Roma 'Tor Vergata'

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1 Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Linear Regression

2 Specication Let Y be a univariate quantitative response variable. We model Y as follows: Y = f(x) + ε where f(x) is the systematic part (that can be predicted using the inputs X) and ε is a disturbance (error) term, accounting for the all the variation sources dierent from X. Assumptions: Linearity. The regression function is linear in the inputs: f(x) = β 0 + p β j X j Weak exogeneity: E(ε X) = 0, i.e. X and ε are independent in the mean. j=1 Homoscedasticity: E(ε 2 X) = Var(ε X) = σ 2.

3 The inputs set X includes quantititative variables, nonlinear transformations and basis expansions (to be dened later), as well as dummy variables coding the levels of qualitative inputs. Under these assumptions the regression function is interpreted as the conditional mean of Y given X f(x) = E(Y X). This is the optimal predictor of Y under square loss (minimum mean square estimator of Y ).

4 Estimation Set of training data {(y i, x i ), i = 1,..., N}, with x i = (x i1, x i2,..., x ip ). We assume that the training sample is a sample of size N, drawn independently (with replacement) from the underlying population, so that for the i-th individual we can write p y i = β 0 + β j x ij + ε i, j=1 Writing the N equations in matrix form: y = Xβ + ε i = 1,..., N with E(ε X) = 0 and Var(ε X) = σ 2 I. y 1 1 x 11 x x 1k... x 1p y 2 1 x 21 x x 2k... x 2p y =. y i, X = x i1 x i2... x ik... x ip, ε = y N 1 x N1 x N2... x Nk... x Np β = [β 0, β 1,..., β p] ε 1 ε 2. ε i. ε N

5 We aim at estimating the unknown parameters β 0,..., β p, and σ 2. An important estimation method is Ordinary Least Squares (OLS). OLS obtains an estimate of β denoted ˆβ, as the minimizers of the residual sum of squares RSS(β) = (y Xβ) (y Xβ). There is a closed form solution to the above problem. Computing the partial derivatives of RSS(β) w.r.t. each β j and equating to zero yields a p + 1 linear system of equations in p + 1 unknowns, X Xβ = X y which is solved w.r.t. β. The solution is unique provided that the X's are not collinear. In matrix notation the solution is ˆβ = (X X) 1 X y

6 Example: p = 1 (simple linear regression) Nβ 0 + β 1 i x i = i y i β 0 i x i + β 1 i x2 i = i x iy i We solve this system by substitution: solving wrt β 0 the 1st eqn. yields ˆβ 0 = ȳ ˆβ 1 x Replacing into the 2nd and solving wrt ˆβ 1 gives: 1 N i ˆβ 1 = x iy i xȳ Cov(x, y) i = = (x i x)(y i ȳ) x 2 i x 2 Var(x) i (x i x) 2 1 N

7 Predicted values and residuals Predicted (tted) values ŷ = Xˆβ = X(X X) 1 X y = Hy H = X(X X) 1 X is the hat matrix, projecting y orthogonally onto the vector space generated by X. Residuals The OLS residuals are where M = I H. e = y ŷ = (I H)y = My

8 Properties of ŷ and e The tted values and the LS residuals have properties that are simple to show and are good to know. Orthogonality. X e = 0. The residuals are uncorrelated with each of the variable in X. In the model includes the intercept, the residuals have zero mean. ŷ e = 0 RSS(ˆβ) = e e = i e2 i (henceforth RSS) From y = ŷ + e and the previous properties, y 2 = ŷ 2 + e 2 ȳ = N 1 i y = ŷ (average of predicted values)

9 Estimation of σ 2 For estimating σ 2 (the variance of the error term) we could use the variance of the residuals corrected for the number of degrees of freedom: ˆσ 2 i = e2 i N p 1 N p 1 is known as the number of degrees of freedom. ˆσ is the standard error of regression (SER).

10 Goodness of t Dene T SS = i (y i ȳ) 2 (Total sum of squares - Deviance of y i ) ESS = i (ŷ i ȳ) 2 (Explained sum of squares - Deviance of ŷ i ) RSS = i e2 i (Residual sum of squares - Deviance of e i) It is easy to prove that A relative measure of g.o.f. is T SS = ESS + RSS R 2 = ESS T SS = 1 RSS T SS

11 This measures suers from a serious drawback when used for model selection: the inclusion of (possibly irrelevant) additional regressors always produces an increase in R 2. Adjusted R 2 R 2 = 1 RSS/(N p 1) T SS/(N 1) = 1 N 1 N p 1 (1 R2 )

12 Properties of the OLS estimators We are going to look at the statistical properties of the OLS estimators (and other related quantities, such as the tted values and the residuals), hypothesizing that we can draw repeated training samples from the same population, keeping the X's xed and drawing dierent Y 's. A distribution of outcomes will arise. Under the stated assumptions, E(ˆβ X) = β, Var(ˆβ) = σ 2 (X X) 1 Also, E(ˆσ 2 ) = σ 2. Moreover, the OLS estimators are Best Linear Unbiased (Gauss Markov theorem): E( ˆβ j ) = β j, j and Var( ˆβ j ) = MSE( ˆβ j ) is the smallest among the class of all linear unbiased estimators. Example: p = 1 (simple regression model) [ Var( ˆβ 1 0 ) = σ 2 N + x 2 ] i (x i x) 2, Var( ˆβ σ 2 1 ) = i (x i x) 2,

13 If we further assume ɛ i x i NID(0, σ 2 ), y i x i N(β 0 + β 1 x i, σ 2 ) The OLS estimators have a normal distribution: ˆβ N(β, σ 2 (X X) 1 ). ˆβ is the maximum likelihood estimator of the coecients (we will say more about it). Letting s.e.( ˆβ j ) denote the estimated standard error of the OLS estimator, i.e. the j-th element of Var(ˆβ) ˆ = ˆσ 2 (X X) 1, t j = ˆβ j β j s.e.( ˆβ j ) t N p 1, a Student's t r.v. with N p 1 d.o.f. This result is used to test hypotheses on a single coecient and to produce interval estimates.

14 Suppose we wish to test for the signicance of subset of the coecients. The relevant statistic for the null that J coecients are all equal to zero is F = (RSS 0 RSS)/J RSS/(N p 1) F J,N p 1 where RSS 0 is the residual sum of squares of the restricted model (i.e. it has J less explanatory variables), and F J,N K is Fisher's distribution with J and N p 1 d.o.f. As a special case, the test for H 0 : β 1 = β 2 = = β p = 0 (all the regression coecients are zero, except for the intercept) is: F = ESS/p RSS/(N p 1) = R 2 /p (1 R 2 )/(N p 1)

15 Example: clothing data. Regression of total sales (logs) on hours worked and store size (logs). Call: lm(formula = y ~ x1 + x2, data = clothing) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** x < 2e-16 *** x e-12 *** --- Residual standard error: on 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 2 and 397 DF, p-value: < 2.2e-16

16 Figure : Fitted values log(hoursw) log(ssize) log(tsales)

17 Example: house price regression in R lm(formula = price ~ sqft + Age + Pool + Bedrooms + Pool + Firepl Waterfront + DOM) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) sqft < 2e-16 *** Age e-06 *** Pool Bedrooms e-06 *** Fireplace Waterfront e-11 *** DOM Residual standard error: on 1072 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 7 and 1072 DF, p-value: < 2.2e-16

18 Figure : Histogram of residuals Frequency e+05 0e+00 2e+05 4e+05 6e+05 8e+05 e

19 Properties of the OLS residuals The i-th residual e i has zero expectation (under the model's assumptions) and Var(e i ) = σ 2 (1 h i ), where h i = x i(x X) 1 x i is the leverage of observation i (the i-th diagonal element of H). h i is an important diagnostic quantity. It is a measure of remoteness of the inputs for the i-th individual in the space of the inputs. The barplot of h i versus i illustrates the inuence of the i-th observation on the t. It is possible to show that i h i = p + 1 and the average is (p + 1)/N. The standardized residual is dened as r i = e i ˆσ 1 h i and is used in regression diagnostics (normal probability plot, check for homoscedasticity).

20 Figure : Anscombe Quartet: 4 datasets giving the same set tted values

21 Figure : Anscombe Quartet: Leverage plot (h i vs i)

22 Specication issues: omitted variable bias, collinearity, etc. Consider the regression model y = Xβ + γw + ε, where w is a vector of N measurements on an added variable W. We can show that the OLS estimator of γ is and ˆγ = w My w Mw, M = I X(X X) 1 X, Var(ˆγ) = σ2 w Mw. These expressions have a nice interpretation: ˆγ results from Regress Y on X and obtain the residuals e Y X = My; Regress W on X and obtain the residuals e W X = Mw; Regress e Y X on e W X to obtain the LS estimate ˆγ (partial regr. coe.).

23 Multicollinearity Mw are the residuals of the regression of W on X. w Mw is the residual sum of squares from that regression. If X explains most of the variation in W, this is very small, and Var(ˆγ) will be very high. We can conclude that the eect that W exerts on Y is poorly (i.e. very imprecisely) estimated.

24 Omission of relevant variables If W is omitted from the regression, the OLS estimator ˆβ is biased, but also more precise: E(ˆβ) = β + (X X) 1 X wγ, Var(ˆβ) = σ 2 (X X) 1 σ 2 (X M W X) 1 ; M W = I w(w w) 1 w, where σ 2 (X M W X) 1 is the variance of the estimator of β when W is included in the model. If W is irrelevant (true γ = 0) there is only a cost associated to including it in the model. In any case, if W is orthogonal to X the distribution of ˆβ is not aected by W.

25 Prediction of a new sample observation We are interested in predicting Y for a new unit with values of X in the vector x o = (1, x o1,..., x op ). The optimal predictor under square loss is ŷ o = ˆf(x o ) = ˆβ 0 + ˆβ 1 x o1 + + ˆβ p x op = x o ˆβ Under the stated assumptions, the predictor is unbiased, i.e. the prediction error has zero mean, E(Y o ŷ o ) = 0, and Var(Y o ŷ o ) = σ 2 (1 + x o(x X) 1 x o ) Note: the stated assumptions concern the selection of the set of predictors in X and the properties of ε.

26 Predictive accuracy depends on 1. Ability to select all the relevant variable. Failure to do so will result in an inated σ 2 and E(ε X) 0, so that ˆβ is no longer optimal. 2. How removed is the unit prole x o from the rest: x o(x X) 1 x o measures the (Mahalanobis) distance of the values of the X 1,..., X p for the o-th unit in the space of the X's.

27 If the true model is y = Xβ + γw + ɛ, E(ɛ X, w) = 0, Var(ɛ X, w) = σ 2 ɛ I, the above predictor is biased and its mean square error is with E[(Y 0 ˆf(x o )) 2 X, w, x o, w o ] = σ 2 ɛ (1 + x o(x X) 1 x o ) + Bias 2, Bias = (β E(ˆβ)) x o + γw o = (w o w X(X X) 1 x o )γ The term in parenthesis is the error of prediction of w o using the information in x o.