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Marietta College Spring 2011 Econ 420: Applied Regression Analysis Dr. Jacqueline Khorassani Week 6

Tuesday, February 15 Exam 2: Tuesday, March 22 Exam 3: Monday, April 25, 12- 2:30PM

Note In this equation when your replace with =Z , you are not changing the meaning of the coefficient

Note In this equation when you replace with (β1 log X1i ), your are changing the meaning of the coefficient

ASST 8: Due Thursday in class • # 5 , Page 111

Assumption 5 • The error term has a constant variance • What if it is violated? • Problem of Heteroskedasticity • Example: Consumptioni = β0 + β1 (Income)i+…+ єi • Suppose we use cross sectional data on 70 individuals to estimate the above model. • Graph • People with low levels of income will probably spend most of their income. (The variance of the error is small) • People with high levels of income may spend anywhere between 10% to 99% of their income. (The variance of the error is high.)

Assumption 6 Two or more independent variables are not perfectly and linearly correlated with each other. • If violated  Perfect Multicollinearity • Example • Consumption = f (inflation, real interest rate, nominal interest rate, ….) • Problem • real interest = nominal interest – inflation • The 3 independent variables are perfectly and linearly correlated with each other. OLS can not capture the effect of one variable in isolation. • It will give you an error message

Let’s look at Q 3, Page 111 • What is the answer?

Assumption 7 (Not Necessary) • The error term is normally distributed • What is a normal distribution? • Symmetric, continuous, bell shaped • Can be characterized by its mean and variance • Must know if it is violated • If violated, some statistical tests are not applicable • But, as the size of sample goes up  the distribution becomes more normal

Recap • Suppose the population of students at Marietta College = 1400 • The model is Yi = β0 + β1X1i + β2X2i + єi • Y = GPA • X1 = hours of study • X2 = IQ score • We don’t see the true βs • We choose a sample of 50 students and estimate β^s • Are our β^s the same as true βs? • No • What if we chose another sample of 50 observations? • We will get different β^s

The sampling distribution of the estimated coefficients • Displays the values of all possible β^s that we can get if we select an infinite number of samples from the population to estimate our equation using a given procedure. • If the error term is normally distributed  the estimated coefficients are normally distributed too

So the distribution of β^s will be just like the Z distribution below. ………. ……. ……. ……..

Unbiased Estimator • Is a method of estimation which results in β^s that belong to distributions whose means are equal to the true βs . . . .

Best (most efficient) Estimator • Is a method of estimation whose β^s belong to distributions with the lowest possible variances. . . . Β^

Consistent Estimator Is a method of estimation that results in β^s that get closer and closer to the true βs as the sample size is increased. . . . Β^ β

The Gauss-Markov Theorem • Given assumptions 1 through 6, the OLS estimator is BLUE (Best Linear Unbiased Estimator)

Asst 9 (in teams of 2-3) • Are all unbiased estimators efficient? Draw a graph and explain. • Are all unbiased estimators consistent? Draw a graph and explain. • Are all consistent estimators unbiased? Draw a graph and explain

Thursday, February 17 • Exam 2: Tuesday, March 22 • Exam 3: Monday, April 25, 12- 2:30PM

Collect ASST 8 • # 5 , Page 111

Asst 10: Due Tuesday in class • # 9, Page 114 • #11, Page 116

Please • Bring your laptops to class on Tuesday

Return and discuss Asst 9 • Are all unbiased estimators efficient? Draw a graph and explain. • Are all unbiased estimators consistent? Draw a graph and explain. • Are all consistent estimators unbiased? Draw a graph and explain

Answers • Are all unbiased estimators efficient? • Graph of two unbiased estimator; one is not efficient. • No; If several unbiased estimators are compared, only the estimator with the lowest variance is considered efficient.

Answers 2. Are all unbiased estimators consistent? • Graph of unbiased estimator with 2 sample sizes • Yes; an estimator is consistent if its estimates approach the true value when sample size becomes very large. • Since an unbiased estimator has a sampling distribution centered on the true value, a large sample size will give estimates that approach the true value. All unbiased estimators are consistent.

Answers 3. Are all consistent estimators unbiased? • Graph of consistent estimator that is not unbiased • No. Consistency is a weaker condition than unbiasedness. • If an estimator is unbiased, that is better than if it is only consistent. • However, if an estimator is biased but it’s consistent, that is still better than nothing.

Chapter 5: Hypothesis Testing • Yi = β0 + β1X1i + β2X2i + β3X3i + єi • Recall that we don’t see the true line • So we don’t know the true intercept or the slope coefficients. • We collect a sample data. • We use OLS to estimate the coefficients. • Hypothesis testing refers to using sample information to draw conclusions about the true population coefficients .

Three Steps of Hypothesis Testing Step One • Set the null and alternative hypotheses about the true coefficients. • Alternative hypothesis is consistent with our common sense or theory. • It is what we expect to be true. • It is what we expect to fail to reject. • Null hypothesis is what we expect to not be true. • It is what we expect to reject.

Two Sided Hypotheses • Suppose, all you want to show is that something affects something else. But you don’t want to show the direction of the relationship • In this case, you will set a two sided hypothesis • Example • All you want to show is that age affects a person’s weight. • In this case you would set up a two sided test.

Suppose β3 is the true coefficient of age, then the two sided hypothesis looks like this • H0: β3 =0 • HA: β3 ≠0

You expect to • Reject the null hypothesis H0 in favor of alternative hypothesis, HA • And if you do, • You have found empirical evidence that age affects weight. • However, you are not making any statements with regards to the nature of the relationship between age and weight.

One sided hypotheses • Suppose you want to show that something has a positive (or negative) effect on something else. In this case, you will set a one sided hypothesis • Example • you want to show that calorie intake affects the weight in a positive way • In this case you would set up a one sided test

Suppose β3 is the true coefficient of calorie intake, then the one sided hypothesis looks like this H0: β3 ≤0 HA: β3 >0

You expect to • Reject the null hypothesis H0 in favor of alternative hypothesis, HA • And if you do, • You have found empirical evidence that calorie intake has a positive effect on weight. • Notes • A one sided test is stronger than a two sided test. • You must test a stronger hypothesis when possible.

Asst 11 (teams of 2) • Parts “b” and “c”, Question 3, Page 150

Type I Error • Refers to rejecting a true null hypothesis • Example H0: β3 ≤0 HA: β3 >0 • If you reject a true H0, you will conclude that the higher the calorie intake the higher the weight; while in reality, calorie intake does not matter.

Type II Error • Refers to failing to reject a false null hypothesis H0: β3 ≤0 HA: β3 >0 • If you fail to reject a false H0, you will conclude that calorie intake does not matter; while in reality, it does.

Type I/Type II Errors • Type I error • No error • No error • Type II error

Which type of error is more serious? • Type I error: We conclude that the higher calorie in take, the height the weight. (While in reality there is no correlation between the two.) So we put a lot of effort into watching our calorie intake while we should not. • Type II error: We conclude that calorie intake does not affect the weight, while it actually does. So, we do not watch our diet. (no effort) • When testing hypotheses we try to minimize the type I error

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