Classical Assumptions 1.Regression is linear, correctly specified, and has additive error term 2.E(ε i )=0 3.Correlation between X ki and ε i is 0 for all k. 4.ε t is uncorrelated with ε t+1 for all t. 5.Var(ε i )=σ 2 [No Heteroskedasticity] 6.No perfect multicollinearity and sometimes: 7.ε i ~N(0, σ 2 )
Sampling Distribution of is assumed to be normally distributed because the stochastic error is assumed to be normally distributed (assumption 7) Usually, we take a sample of size N from a population to produce a single estimator of β, which we call. But what if we took a different sample? We should get a different result for
Sampling Distribution of β
In OLS, is unbiased, so E( )=β OLS estimators also have the smallest variance possible at any sample size (efficiency) Finally, OLS estimators are consistent. As N increases, variance shrinks. As N->∞, β->
Hypothesis Testing Most times, we only take one sample, so we only get one estimate of How do we know if is meaningful I we can only observe one value in the distribution?
Example Suppose we are interested in whether school size has an effect on student performance. Specifically, do students at small schools do better? We estimate the following equation: math10 i = β 0 +β 1 enroll i +β 2 staff i +β 3 totcomp i +ε i
Example math10 i = β 0 +β 1 enroll i +β 2 staff i +β 3 totcomp i +ε i Where: math10 = % of students passing the 10 th grade math portion of the Michigan Educational Assessment Program (MEAP) test enroll = school size staff = number of staff/1000 students (to control for how much attention students get) totcomp = average annual teaching compensation (to control for teacher quality)
Hypothesis Testing We need to develop a null and alternative hypothesis before running the regression. Null Hypothesis (H 0 ) Usually, you want to reject the null hypothesis Most common null hypothesis: “there is no effect of X on Y” or “ β 1 =0” Alternative Hypothesis (H A or H 1 ) Usually, what you are trying to prove
Hypothesis Testing In our example, we would pick H 0 :β 1 ≥0 “there is no negative effect of school size on student performance” H A :β 1 <0 “There is a negative effect of school size on student performance” Test this using meap93.gdt
Example 2 Consider the wage equation log(wage i )=β 0 +β 1 educ i +β 2 exer i +β 3 tenure i +ε i The null hypothesis H 0 : β 2 =0 says: once education and tenure have been accounted for, the number of years in the workforce has no effect on hourly wage If β 2 >0, prior work experience contributes to productivity, and to wage.
Alternative Hypothesis Usually, we want to reject the null hypothesis. We form an alternative hypothesis – values we don’t expect. One-sided Alternatives We expect there to be a sign on a particular variable based on our economic model e.g. H A : β K >0.
Hypothesis Testing log(wage i )=β 0 +β 1 educ i +β 2 exer i +β 3 tenure i +ε i In our example, we might set our hypotheses as H 0 :β 2 ≤0 H A :β 2 >0 We believe that the effect of experience on wages is positive, holding education and tenure fixed.
Hypothesis Testing log(wage i )=β 0 +β 1 educ i +β 2 exer i +β 3 tenure i +ε i What should the null and alternative hypotheses for the other coefficients be? H 0 :β 1 ≤0 H A :β 1 >0 H 0 :β 3 ≤0 H A :β 3 >0
Two sided alternatives Y i =β 0 +β 1 X 1i +…+β k X ki +ε i H 0 :β 1 =0 H A :β 1 ≠0 Under the alternative, X 1i has a significant effect on the dependent variable without specifying if it’s positive or negative You should use this if you don’t know what sign β k has (not well defined by theory) Or…sometimes it is better to use because it prevents us from forming our hypothesis after looking at the results
Other Hypotheses Although H 0 :β k =0 is the most common null hypothesis, sometimes, we want to test whether or not β k is equal to some other constant – usually 1 or -1. Example: Suppose we want to look at the effect of college enrollment on crime. log(crime i )=β 0 +β 1 log(enroll i )+ε i This is a constant elasticity model, where β 1 is the elasticity of crime with respect to enrollment.
Other hypotheses log(crime i )=β 0 +β 1 log(enroll i )+ε i We could test, H 0 :β 1 =0 & H A :β 1 ≠0 But more interesting would be to test if β 1 =1 If β 1 >1, then a 1% increase in enrollment leads to a greater than 1% increase in crime, so crime is a bigger problem at large campuses Set up our hypotheses as follows H 0 :β 1 =1 H A :β 1 ≠1
t-test Y i =β 0 +β 1 X 1i +…+β k X ki +ε i t-statistic: = estimated regression coefficient of the k th variable = The border value (usually zero) implied by the null hypothesis = The estimated standard error of the coefficient on the k th variable
t-test For example, suppose our hypotheses were: H 0 :β 1 =0 H A :β 1 >0 Then, suppose that we estimate that =6, and that =2 We would calculate t as
How does the t-test work? β1β1 Distribution of if null is true Suppose we found a value of way out here It’s not very likely that the null hypothesis is true…
t-test How does this look for our example? =6 and =
t-test We want to know, if H 0 really is true (i.e. β 1 really is 0), how likely is it that we could have observed a value of 6? Not very. We can probably say that H 0 is not true. But we need a rule to decide.
Hypothesis Testing How do we decide when to reject the null? Choose a level of significance Rule of thumb: 5% level of significance This means that we will rule out H 0 if we would have expected a value of at least as extreme as 6 less than 5% of the time. Instead of trying to figure out this probability using the sampling distribution, we transform the distribution to the t-distribution The t-distribution is almost the same as the standard normal distribution.
t-test In our example, t=6-0/2 = 3 Suppose our sample size was 23 We need to compare our t-statistic to the critical t-value, which distinguishes the acceptance region from the rejection region. Look at inside cover of book We want the t-value for = 20 degrees of freedom. For a one sided test with 5% significance, this is t c =1.725 Decision Rule: Reject H 0 if |t k |>t c, and has the sign implied by H A, otherwise do not reject. Here, we reject the null in favor of the alternative, suggesting that X 1 is significant
Choosing a Level of Significance Rule of thumb – Significance level = 5% If significance level is too low, we risk what is called a type II error, where we reject the null hypothesis when it is actually true. If we reject H 0 at the 5% level, we say that the coefficient is “statistically significant at the 5% level” Sometimes researchers use asterisks * means significant at 10% ** means significant at 5% *** means significant at 1%
Confidence Intervals Confidence Interval - The range that contains the population value a specified percent of the time. The two-sided t-critical value at a specific significance level gives the (1-sig level) confidence interval. So, the 5% significance level is equivalent to the 95% CI.
Confidence Intervals For our example, the t-critical value was So the 95% CI= 6 ± 2*2.086 = 6±4.172 Or to We could say that with 95% confidence, the true value of β is between and Notice that 0 is not in this range. We can reject H 0
P-value Alternative to t-test If the true population value was really 0, what is the probability we would have observed a value as extreme as 6? If p is small, reject the null. This is calculated automatically by most econometrics software Reject the null if p is less than the significance level
Example Student performance and school size using data.
F-test (Appendix Ch. 5) What if you want to test a hypothesis that involves multiple coefficients? For example: Suppose we run this regression (data7-2.gdt): wage i = β 0 +β 1 educ i +β 2 exper i +β 3 clerical i +β 4 maint i +β 5 crafts i +ε i clerical, maint, and crafts are job type “dummies” We want to test whether job type matters We would need to test whether β 3, β 4, and β 5 are “jointly significant. H 0 :β 3 =β 4 =β 5 =0 H A : The null hypothesis is not true.
F-test Steps 1. Run full regression, get RSS 2. Run constrained regression (without job type variables), get RSS M RSS = RSS from step 1 RSS M = RSS from step 2 M = # of excluded coeffs N = # observations K = # of coefficients in overall equation
F-stat Calculate F-stat, and compare it to the critical value of F (from F-table) Degrees of freedom numerator = M Degrees of freedom denominator = N-K-1 If F>F crit reject null hypothesis The variables are jointly significant if you can reject the null.
F-test In Gretl Run the model Select test>omit variables Gives F-stat and related p-value