Multiple Regression Hypothesis Testing PDF

Hypothesis testing in Multiple Regression Session 7 UKZN INSPIRING  Intro- methodology – econometrics & Data considerations DESCRIPTIVE STATISTICS Univariate data Mechanics of...

Hypothesis testing in Multiple Regression Session 7 UKZN INSPIRING  Intro- methodology – econometrics & Data considerations DESCRIPTIVE STATISTICS Univariate data Mechanics of analysis OLS : two variables Bivariate data Hypothesis analysis Testing & Regression Results Quan LINEAR OLS:Multiple t. REGESSION regression MODEL: estimation Econ BASICS Hypothesis. Testing: Multiple ECONOMETRI regression CS Functional forms Multicollinearit y REGRESSION Autocorrelati ANALYSIS on IN PRACTICE Heteroscedastici ty Model specification  Learning Objectives In this session we are going to focus on some additional aspects of hypothesis testing, such as 1. Hypotheses testing of Individual coefficients in a multiple regression model. 2. The fit of the model as a whole 3. Joint hypotheses testing - the joint significance two or more variable in a model  Note: We can test hypotheses about a multiple regression model using similar methods to the two-variable model. UKZN INSPIRING  u is normally distributed with mean 0 and variance σ 2 b is normally j distributed with mean βj and variance σbj2 What does this mean?... UKZN INSPIRING  b1 and b2 (our sample estimators) are functions of Y (the value of the dependent variable in the population), which is a function of u (the stochastic error terms in the population). We cannot say what the distribution of b1 or b2 is, or derive a confidence interval for them, without knowing what u’s distribution is. Since we cannot observe u, but we make an assumption that the error terms (u) are normally distributed i.e. How do we illustrate this distribution?... UKZN INSPIRING  * Note that the * The diagram diagram illustrates illustrates the standard deviations, assumed distribution σ, and not variances of the population σ2 from the mean error, not the sample error. 0 UKZN INSPIRING  Since b1 , b2 … bj (sample estimators) are linear functions of u (population error), this means that they are themselves normally distributed. For example, if we have a population of 50 million and we drew 10,000 samples from the population, each sample consisting of 500 observations (n=500), then we would calculate 10,000 different values for b2 We would expect the distribution of these 10,000 different values for b2 to be normally distributed as below: However, in order to use the normal The mean value of all the distribution, we must know the true estimates would σ2. Since its unknown, but we estimate be the true it using its estimator the expression above population parameter β2. follows the t distribution, not the standard normal distribution. β2 Graphically, the t-distribution looks similar to the normal distribution Using STATA results to test hypotheses UKZN INSPIRING Method 1: Confidence Intervals UKZN INSPIRING Method 2: Comparing t-values UKZN INSPIRING Method 3: p-values Can only test if βj is equal to zero. UKZN INSPIRING  ‘Manual’ hypothesis tests  Test whether a 1 year increase in educational attainment raises the wages of individuals by more than R500/month. H0: educ ≤ 0.5 *note wages are in thousands of Rands H1: educ > 0.5  We cannot use the regression output (statistical tests) to do the test (why not?). UKZN INSPIRING  In this case, H0: educ ≤ 0.5 H1: educ > 0.5 beduc   educ 0.673  0.5 t  0.840 ˆ educ 0.206 At a 5% significance level for a one-tail test, with df = 98 (use the value for df = 120), tc = 1.658. Since the calculated t falls within the acceptance region, we cannot reject the null hypothesis. UKZN INSPIRING  Testing Model Fitness: F-test Critical Value: P-value: Reject H0 if F-Stat > Fα, k, n-k-1 Reject H0 if p- value < α UKZN INSPIRING UKZN INSPIRING Example: Calculating the F statistic For k - 1 = 3 and n - k = 98 = 0.1878/(3) (1-0.1878)/ = 7.55 (98) UKZN INSPIRING At a 5% level of significance, the critical F-value (with 3 df in the numerator and 98 df in the denominator) from an F-table is 2.68  Testing Model Fitness: F-test At a 5% level of significance, the critical F-value (with 3 df in the numerator and 98 df in the denominator) from an F-table is 2.68 (round off the degrees of freedom). The calculated statistic is greater than the critical value (7.55 > 2.68), therefore we must reject the null hypothesis. We conclude that the model as a whole offers a significant degree of explanation. UKZN INSPIRING  Example: Using the p-value To test whether the degree of explanation provided by the model (given by R2 = 18.78%) is significantly different from zero using the STATA results, we can simply use the p-value Therefore we must reject the null hypothesis at all significance levels (equal to or) above 0.01%. Thus the model does have a significant degree of explanatory power at such levels (e.g. 1%, 5%, etc.) UKZN INSPIRING  Individual/Group Activity 1 Question Two In the following regression: income = monthly income in thousands of Rands educ = education in years jobexp = job experience in years 1. Test whether the degree of explanation provided by the model is significantly different from zero, at the 5 % level of significance. UKZN INSPIRING Joint Hypothesis Testing / Restricted Least Squares UKZN INSPIRING  Testing for joint significance Consider the model: UKZN INSPIRING Example: Test whether exp and IQ belong in the model, at a 5% significance level. We estimate the restricted regression, with the following results (standard errors in brackets): Critical value from F table: 3.07 Since F < Fc, we cannot reject H0 Therefore exp and IQ are jointly insignificant   Do the price of beef and consumers’ income belong in the Example model? (i.e. do they have any effect on consumption?) H0: βpb = βpcy = 0 Unrestricted model: H1: at least one of βpb , βpcy ≠ 0 cc = –73.08 – 1.95pc + 1.15pb + 0.1pcy n = 40, R2 = 0.936 Restricted model: cc = 41.97 +1.30pc n = 40, R2 = 0.725 Test statistic: = Critical values, with df = 2; 36 are: Fcrit = 3.23 (5%) and Fcrit = 5.18 (1%)  Fcalc > Fcrit therefore reject H0 at both significance levels.  Therefore, the price of beef and consumers’ income do belong in the model. Testing a relationship between coefficients 1. Obtain R2 from each regression. 2. Calculate f- statistic (m=1). 3. Compare to critical value  Individual/Group Activity 2 1. Test whether education and job experience have the same effect on income, at the 5 % level of significance. When the restricted regression is estimated, the stata output is as follows: Multiple Linear Regression UKZN INSPIRING