Thirlwall

Applied EconometricsSecond editionDimitrios Asteriou and Stephen G. Hall

Applied EconometricsMULTIPLE REGRESSION1. The Multiple Regression Model 2. The OLS Method of Estimation3. The R2 and the Adjusted R24. Hypothesis Testing5. How to Estimate a Simple Regression in EViews

Applied Econometrics2Learning ObjectivesDerive mathematically the regression coefficients of a multiple regression model.Understand the difference between the R2 and the adjusted R2 for a multiple regression model. Appreciate the importance of the various selection criteria for the best regression model.Conduct hypothesis testing and test linear restrictions, omitted and redundant variables as well as the overall significance of the explanatory variables.

Applied Econometrics3Learning Objectives (2)Obtain the output of a multiple regression estimation using econometric software.Interpret and discuss the results of a multiple regression estimation output.

Applied Econometrics4Multiple Regression Derivation of the OLSThe three variables case (explain on board)The k-variables case (explain on board)Requires matrix algebra and it is quite complicatedLuckily Eviews, Mfit and Stata give results very quickly and efficiently (always correct calculations)

Applied Econometrics5Assumptions of the Multiple Regression Model

Applied Econometrics6R2 and adjusted R2R2 measures goodness of fit as in Simple RegressionHowever, it cannot be used for comparing two different equations containing differentnumbers of explanatory variables. When adding more explanatory variables R2, will always be increased. Therefore we need a different measure (the adjusted R2)Applied Econometrics7R2 and adjusted R2R2 = ESS/TSS = 1 RSS/TSS

Adj R2=

Similar to R2 but adjusts for degrees of freedom

Applied Econometrics8Criteria for Model SelectionAkaike Information Criterion (AIC)

Finite Prediction Error (FPE)

Schwarz Bayesian Criterion (SBC)

Hannan and Quin Criterion (HQC)

Applied Econometrics9Multiple Regression in EViewsStep 1 Open EViews.Step 2 Click File/New/Workfile in order to create a new file or File/Open to open an existing file.Step 3 Enter the dataStep 4 Type in the EViews command line:ls y c x2 x3 . . . xk (press enter)Applied Econometrics10Multiple Regression in EViews

Applied Econometrics11Hypothesis TestingTesting Individual Coefficients (t-tests)Testing for Linear Restrictions (Wald Test)Cobb Douglas Production FunctionTesting for the Overall Significance (F-test)Testing for Omitted Variables (Wald Test)Testing for Redundant Variables (Wald Test)Explain all the tests on boardApplied Econometrics12Hypothesis TestingTesting Individual Coefficients (t-tests)

Same as before (simple regression)Applied Econometrics13Hypothesis TestingTesting for Linear Restrictions (Wald Test)Cobb Douglas Production Function (+=1)

Applied Econometrics14Hypothesis TestingTesting for Linear Restrictions (Wald Test)Impose more than one restriction

Applied Econometrics15Hypothesis TestingTesting for the Overall Significance (F-test)

Applied Econometrics16Testing Multiple Hypotheses: The F-testWe used the t-test to test single hypotheses, i.e. hypotheses involving only one coefficient. But what if we want to test more than one coefficient simultaneously?

We do this using the F-test. The F-test involves estimating 2 regressions.

The unrestricted regression is the one in which the coefficients are freely determined by the data, as we have done before.

The restricted regression is the one in which the coefficients are restricted, i.e. the restrictions are imposed on some s.

Applied EconometricsThe F-test: Restricted and Unrestricted Regressions

ExampleThe general regression isyt = 1 + 2x2t + 3x3t + 4x4t + ut(1)

We want to test the restriction that 3+4 = 1 (we have some hypothesis from theory which suggests that this would be an interesting hypothesis to study). The unrestricted regression is (1) above, but what is the restricted regression?yt = 1 + 2x2t + 3x3t + 4x4t + uts.t. 3+4 = 1

We substitute the restriction (3+4 = 1) into the regression so that it is automatically imposed on the data. 3+4 = 1 4 = 1- 3

Applied EconometricsThe F-test: Forming the Restricted Regression

yt = 1 + 2x2t + 3x3t + (1-3)x4t + ut yt = 1 + 2x2t + 3x3t + x4t - 3x4t + ut

Gather terms in s together and rearrange(yt - x4t) = 1 + 2x2t + 3(x3t - x4t) + ut

This is the restricted regression. We actually estimate it by creating two new variables, call them, say, Pt and Qt.Pt = yt - x4tQt = x3t - x4t soPt = 1 + 2x2t + 3Qt + ut is the restricted regression we actually estimate.Applied EconometricsCalculating the F-Test Statistic

The test statistic is given by

where URSS= RSS from unrestricted regression RRSS= RSS from restricted regression m = number of restrictions T= number of observations k= number of regressors in unrestricted regression including a constant in the unrestricted regression (or the total number of parameters to be estimated).

Applied EconometricsThe F-DistributionThe test statistic follows the F-distribution, which has 2 d.f. parameters.

The value of the degrees of freedom parameters are m and (T-k) respectively (the order of the d.f. parameters is important).

The appropriate critical value will be in column m, row (T-k).

The F-distribution has only positive values and is not symmetrical. We therefore only reject the null if the test statistic > critical F-value.

Applied EconometricsDetermining the Number of Restrictions in an F-testExamples :H0: hypothesisNo. of restrictions, m1 + 2 = 212 = 1 and 3 = -122 = 0, 3 = 0 and 4 = 03If the model is yt = 1 + 2x2t + 3x3t + 4x4t + ut,then the null hypothesisH0: 2 = 0, and 3 = 0 and 4 = 0 is tested by the regression F-statistic. It tests the null hypothesis that all of the coefficients except the intercept coefficient are zero.Note the form of the alternative hypothesis for all tests when more than one restriction is involved: H1: 2 0, or 3 0 or 4 0Applied EconometricsWhat we Cannot Test with Either an F or a t-test

We cannot test using this framework hypotheses which are not linear or which are multiplicative, e.g.H0: 2 3 = 2 or H0: 2 2 = 1 cannot be tested.Applied Econometrics