1. Lecture 1 Cameron Kaplan
Econ 488
Lecture 1
Cameron Kaplan

2. What is Econometrics?
Applying quantitative and statistical methods to study economic principles.
Econometrics has evolved as a separate discipline from statistics because it mainly focuses on non-experimental data
Multiple regression is used in both econometrics and statistics, but the interpretation is different

3. What is Econometrics
Economists have devised new techniques to deal with the complexities of economic data and to test predictions of economic theories

4. Uses of Econometrics Description of economic reality.
Testing hypotheses about economic theory.
Forecasting future economic activity

5. Probability Imagine two dice - a red die and a green die.
We define a random variable X to be the sum of the two dice.
e.g. if we roll a 5 on the red die, and a 2 on the green die, X=7.

6. Probability Distribution
What is the probability the red die=2?
1/6
What is the probability the green die=5?
What is the probability red = 2 and green = 5?
1/6*1/6 = 1/36

7. Probability Distribution
What is the probability X = 2?
1/6*1/6 = 1/36
What is the probability X = 5?

8. Probability Distribution
Green
Red 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

9. Probability DistributionX
Freq
Prob 2 3 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

10. Probability DistributionX
Freq
Prob 2 1 3 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

11. Probability DistributionX
Freq
Prob 2 1
1/36 3 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

12. Probability DistributionX
Freq
Prob 2 1
1/36 3 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

13. Probability DistributionX
Freq
Prob 2 1
1/36 3 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

14. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

15. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

16. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

17. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

18. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

19. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

20. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

21. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

22. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

23. Probability DistributionX
Freq
Prob 2 1
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
Green
Red 1 2 3 4 5 6 7 8 9 10 11 12

25. Expected Value of a Random Variable
E(X) = x1*p1+x2*p2+ x3*p3+…+xn*pn
E(X)=
The expected value is also called the population mean, or x

26. Expected Value xi pi 2
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
xi pi
2/36
6/36
12/36
20/36
30/36
42/36
40/36
36/36
22/36
E(X) = 2/36 + 6/ / / / / / / / / /36
E(X) = 252/36
E(x) = 7

27. Population Variance (2 )
2 = E[(X-) 2 ]
2 =

28. Standard Deviation ()
Population Variance xi pi 2
1/36 3
2/36 4
3/36 5
4/36 6
5/36 7
6/36 8 9 10 11 12
xi - -5 -4 -3 -2 -1 1 2 3 4 5
(xi -)2 25 16 9 4 1
(xi -)2 pi
25/36
32/36
27/36
16/36
5/36
2 = 210/36
2  5.83
Standard Deviation ()
 = 2
 = 5.83
  2.41

29. Continuous Random Variables
Imagine a variable that is equally likely to take on any value between 55 and 75.

30. Continuous Random Variables
What is the probability X= 65 (exactly)
Zero!
We need to think about probabiliy in a range.

31. Continuous Random Variables
f(x) = 0.05 for 55X75
f(x) = 0 otherwise
What is the probability X is between 55 and 56?
= 0.05

32. Continuous Probability Density Functions
Probability Distributions can take on many shapes
The area under the curve must sum to one.

33. Continuous Probability Density Functions
What is f(x)?
f(x) = X for 65X75
f(x) = 0 o.w.

34. The Normal Distribution (AKA Gaussian Distribution)

35. Central Limit Theorem
The sum (or mean) of a large number of independent and identically distributed random variables will be distributed approximately normal.

36. Standard Normal Distribution

37. Standardized Normal Variable
z = (x- )/
Pr[-1 < z < 1] =
Pr[-2 < z < 2] =
Pr[-3 < z < 3] =

38. Height Analyzer Go to http://www.shortsupport.org
Click on the “Research” Tab, and select height analyzer

39. Height Analyzer Men: Mean height = 5’8.5” Standard Dev = 2.75”
Women: Mean height = 5’3.5” Standard Dev = 2.5”
What is the probability that a random woman is between 5’1” and 5’3”?

40. Height Analyzer
Convert to inches: ’1” = 61” 5’3” = 63” 5’3.5” = 63.5”
Standardize z1 = ( )/2.5 = z2 = ( )/2.5 = -0.2
Look up both vales on the z table (pg. 621)

41. Area to the left =

42. Area to the left =

43. Shaded area = = 0.262

44. Height Analyzer What percentage of men are taller than 6’4”?
X = 6’4” = 76”
 = 5’8.5” = 68.5”
Z = ( )/2.75 = 2.727
Only area to the right of on standard normal curve is only
Only 0.32% of men are taller than 6’4” (about one in 300)

45. Sampling
This is the most important thing you could have learned from prob/stats.
Population - entire group (e.g. height for the entire US population)
Mean of population = 
Variance = 2

46. Sampling
Sample - The part of the population you observe (e.g. the subjects in the NHANES)
Sampling mean =
Variance = s2
We use the sample to draw conclusions about the population

47. Sampling Distributions
Suppose we want to estimate 
Sample Average =
Suppose we want to know how good of an estimate x-bar is of 
We create the sampling distribution

48. Sampling Distributions
Sampling Distribution - the probability distribution of all of the possible values of a statistic, in this case x-bar.
Due to the central limit theorem, the sampling distribution of x-bar is approximately normal.

49. Estimators X-bar is an estimator of .
Unbiased Estimator - An estimator is unbiased if it’s mean is equal to the population parameter.
so x-bar is an unbiased estimator!

50. Standard Deviation As N increases, the standard deviation shrinks
Also notice that we can’t calculate this unless we know the population parameter, which is almost never true.

51. Sampling Variance
Notice that this is divided by N-1. If we divide by N, the estimator is too low.

52. Standard Error
When the standard deviation of an estimator is estimated from the data it is called the standard error
The standard error of