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Marietta College
Spring 2011Econ 420: Applied
Regression AnalysisDr. Jacqueline Khorassani
Week 6
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Tuesday, February 15Exam 2: Tuesday, March 22Exam 3: Monday, April 25, 12- 2:30PM
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Note
iiiii XXXY 333
222110
In this equation when your replace
with =Z , you are not changing the meaning of the coefficient
22iX
22iX
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Noteiiii XXY 23
110
In this equation when you replace with (β1 log X1i ), your are changing the meaning of the coefficient
11iX
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ASST 8: Due Thursday in class
• # 5 , Page 111
Assumption 5
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The error term has a constant varianceWhat if it is violated?Problem of HeteroskedasticityExample: Consumptioni = β0 + β1 (Income)i +…+ єi
Suppose we use cross sectional data on 70 individuals to estimate the above model.GraphPeople 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 6Two or more independent variables are not
perfectly and linearly correlated with each other.If violated Perfect MulticollinearityExample
Consumption = f (inflation, real interest rate, nominal interest rate, ….)Problem
real interest = nominal interest – inflationThe 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
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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 + β1 X1i + β2 X2i + 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
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Best (most efficient) Estimator
• Is a method of estimation whose β^s belong to distributions with the lowest possible variances.
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Β^
Consistent EstimatorIs a method of estimation that results in β^s that get closer and closer to the true βs as the sample size is increased.
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βΒ^
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)
1. Are all unbiased estimators efficient? Draw a graph and explain.
2. Are all unbiased estimators consistent? Draw a graph and explain.
3. Are all consistent estimators unbiased? Draw a graph and explain
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Thursday, February 17
• Exam 2: Tuesday, March 22• Exam 3: Monday, April 25, 12-
2:30PM
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Collect ASST 8
• # 5 , Page 111
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Asst 10: Due Tuesday in class
1. # 9, Page 1142. #11, Page 116
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Please
• Bring your laptops to class on Tuesday
Return and discuss Asst 9
1. Are all unbiased estimators efficient? Draw a graph and explain.
2. Are all unbiased estimators consistent? Draw a graph and explain.
3. Are all consistent estimators unbiased? Draw a graph and explain
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Answers
1. 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.
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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.
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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.
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Chapter 5: Hypothesis Testing
Yi = β0 + β1 X1i + β2 X2i + β3 X3i + єi
Recall that we don’t see the true lineSo 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 =0HA: 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 ≤0HA: 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.
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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 ≤0HA: 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 ≤0HA: 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
β3 is zero or negative
β3 is positive
We conclude that β3 is positiveWe conclude that β3 is zero or negative
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