4. Multiple Regression Analysis: Estimation -Most econometric regressions are motivated by a question -ie: Do Canadian Heritage commercials have a positive impact on Canadian identity? -Once a regression has been run, hypothesis tests work to both refine the regression and answer the question -To do this, we assume that the error is normally distributed -Hypothesis tests also assume no statistical issues in the regression

4. Multiple Regression Analysis: Inference 4.1 Sampling Distributions of the OLS Estimators 4.2 Testing Hypotheses about a Single Population Parameter: The t test 4.3 Confidence Intervals 4.4 Testing Hypothesis about a Single Linear Combination of the Parameters 4.5 Testing Multiple Linear Restrictions: The F test 4.6 Reporting Regression Results

4.1 Sampling Distributions of OLS -In chapter 3, we formed assumptions that make OLS unbiased and covered the issue of omitted variable bias -In chapter 3 we also obtained estimates for OLS variance and showed it was smallest of all linear unbiased estimators -Expected value and variance are just the first two moments of B j hat, its distribution can still have any shape

4.1 Sampling Distributions of OLS -From our OLS estimate formulas, the sample distributions of OLS estimators depends on the underlying distribution of the errors -In order to conduct hypothesis tests, we assume that the error is normally distributed in the population -This is the NORMALITY ASSUMPTION:

Assumption MLR. 6 (Normality) The population error u is independent of the explanatory variables x 1, x 2,…,x k and is normally distributed with zero mean and variance σ 2 :

Assumption MLR. 6 Notes MLR. 6 is much stronger than any of our previous assumptions as it implies: MLR. 4: E(u|X)=E(u)=0 MLR. 5: Var(u|X)=Var(u)=σ 2 Assumptions MLR. 1 through MRL. 6 are the CLASSICAL LINEAR MODEL (CLM) ASSUMPTIONS used to produce the CLASSICAL LINEAR MODEL -CLM assumptions are all the Gauss-Markov assumptions plus a normally distributed error

4.1 CLM Assumptions Under the CLM assumptions, the efficiency of OLS’s estimators is enhanced -OLS estimators are now the MINIMUM VARIANCE UNBIASED ESTIMATORS -the “linear” requirement has been dropped and OLS is now BUE -the population assumptions of CLM can be summarised as: -conditional on x, y has a normal distribution with mean linear in X and a constant variance

4.1 CLM Assumptions The normal distribution of errors assumption is driven by the following: 1)u is the sum of many unobserved factors that affect y 2)By the CENTRAL LIMIT THEOREM (CLT), u has an approximately normal distribution (see appendix C)

4.1 Normality Assumption Problems This normality assumption has difficulties: 1)Factors affecting u can have widely different distributions -this assumption becomes worse depending on the number of factors in u and how different their distributions are 2) The assumption assumes that u factors affect y in SEPARATE, additive fashions -if u factors affect y in a complicated fashion, CLT doesn’t apply

4.1 Normality Assumption Problems In general, the normality of u is an empirical (not theoretical) matter -if empirical evidence shows that a distribution is NOT normal, we can practically ask if it is CLOSE to normal -often applying a transformation (such as logs) can make a non-normal distribution normal -Consequences of nonnormality are covered in Chapter 5

4.1 Nonnormality -In some cases, MLR. 6 is clearly false -Take the regression: -since dental visits have only a few values for most people, our dependent variable is far from normal -we will see that nonnormality is not a difficulty in large samples -for now, we simply assume normality -error normality extends to the OLS estimators:

Theorem 4.1 (Normal Sampling Distributions) Under the CLM assumptions MLR.1 through MLR. 6, conditional on the sample values of the independent variable, Where Var(B j hat) was given in Chapter 3 [equation 3.51]. Therefore,

Theorem 4.1 Proof Where r j hat and SSR j come from the regression of x j on all other x’s -Therefore w is non random -B j hat is therefore a linear combination of the error terms (u) -MLR. 6 and MLR. 2 make the errors independent, normally distributed random variables

Theorem 4.1 Proof Any linear combination of independent normal random variables is itself normally distributed (Appendix B) This proves the first equation in the proof -The second equation comes from the fact that standardizing a normal random variable (by subtracting its mean and dividing by its standard deviation) gives us a standard normal random variable (statistical theory)

4.1 Normality Theorem 4.1 allows us to do simple hypothesis tests by assigning a normal distribution to OLS estimators Furthermore, since any linear combination of OLS estimators has a normal distribution, and any subset of OLS estimators has a joint normal distribution, more complicated tests can be done

4.2 Single Hypothesis Tests: t tests -this section covers testing hypotheses about single parameters from the population regression function -Consider the population model: -we assume this model satisfies CLM assumptions -we know that OLS’s estimate of B j is unbiased -the true B j is unknown, and in order to hypothesis about B j ’s true value, we use statistical inference and the following theorem:

Theorem 4.2 (t Distribution for the Standardized Estimators) Under the CLM assumptions MLR.1 through MLR. 6, where k+1 is the number of unknown parameters in the population model (k slope parameters and the intercept B 0 ).

Theorem 4.2 Notes Theorem 4.2 differs from theorem 4.1: -4.1 deals with a normal distribution and standard deviation -4.2 deals with a t distribution and standard error -replacing σ with σhat causes this -see section B.5 for more details

4.2 Null Hypothesis -In order to perform a hypothesis test, we first need a NULL HYPOTHESIS of the form: -which examines the idea that x j has no partial effect on y -For example, given the regression and null hypothesis: -We examine the idea that roses don’t make no impression (good or bad) in a bouquet

4.2 Hypothesis Tests -Hypothesis tests are easy, the hard part is calculating the needed values in the regression -our T STATISTIC or S RATIO is calculated as: -therefore, given an OLS estimate for B j and its standard error, we can calculate a t statistic -note that our t stat will have the same sign as B j hat -note also that larger B j hats cause larger t stats

4.2 Hypothesis Tests -Note that B j hat will never EXACTLY equal zero -instead we ask: How far is B j hat from zero? -a B j hat far from zero provides evidence that B j isn’t zero -but the sampling error (standard deviation) must also be taken into account -Hence the t statistic -t measures how many estimated standard deviations B j hat is from zero

4.2 Hypothesis Tests -values of t significantly far from zero cause the null hypothesis to be rejected -to determine how far t must be from zero, we select a SIGNIFICANCE LEVEL – a probability of rejecting H 0 when it is true -we know that the sample distribution of t is t n-k-1, which is key -note that we are testing the population parameters (B j ) not the estimates (B j hat)