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Introduction
EC339: Applied Econometrics
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What is Econometrics? Scope of application is large
Literal definition: measurement in economics Working definition: application of statistical
methods to problems that are of concern to economists
Econometrics has wide applications—beyond the scope of economics
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What is Econometrics? Econometrics is primarily interested in
Quantifying economic relationships Testing competing hypothesis Forecasting
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Quantifying Economic Relationships Outcomes of many policies tied to the magnitude of the slope
of supply and demand curves Often need to know elasticities before we can begin practical
analysis For example, if the minimum wage is raised, unemployment
may drop as more workers enter the labor force However, this depends on the slopes of the labor supply and labor
demand curves Econometric analysis attempts to determine this answer
Allows us to quantify causal relationships when the luxury of a formal experiment is not available
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Testing Competing Hypothesis Econometrics helps fill the gap between the
theoretical world and the real world For instance, will a tax cut impact consumer
spending? Keynesian models relate consumer spending to annual
disposable income, suggesting that a cut in taxes will change consumer spending
Other theories relate consumer spending to lifetime income, suggesting a tax cut (especially a “one-shot deal”) will have little impact on consumer spending
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Forecasting Econometrics attempts to provide the
information needed to forecast future values Such as inflation, unemployment, stock market
levels, etc.
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The Use of Models Economists use models to describe real-world
processes Models are simplified depictions of reality
Usually an equation or set of equations
Economic theories are usually deterministic while the world is characterized by randomness Empirical models include a random component known as
the error term, or i
Typically assume that the mean of the error term is zero
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Types of Data Data provide the raw material needed to
Quantify economic relationships Test competing theories Construct forecasts
Data can be described as a set of observations such as income, age, grade Each occurrence is called an observation
Data are in different formats Cross-sectional Time series Panel data
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Cross-Sectional Data Provide information on a variety of entities at
the same point in time
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Time Series Data Provides information for the same entity at
different points in time
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Panel (or Longitudinal) Data Represents a combination of cross-sectional
and time series data Provides information on a variety of entities at
different periods in time
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Conducting an Empirical Project How to Write an Empirical Paper Select a topic
Textbooks, JSTOR, News sources (for ideas), “pop-econ”
Learn what others have learned about this topic Spend time researching what others have done Conduct extensive literature review
How to Write an Empirical Paper
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Conducting an Empirical Project Theoretical Foundation Have an empirical strategy
Existing literature may help Would apply the methods you learn in this book Gather data and apply appropriate econometric techniques
Interpret your results Write it up…
Build like a court case or newspaper article
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Where to obtain data How to use DataFerrett
CPS.doc
Files for course will be stored on datastor\\datastor\courses\economic\ec339
You can download all files from bookhttp://caleb.wabash.edu/econometrics/index.htm
CPS.doc
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Web LinksResources for Economists on the Internet are available at
www.rfe.org
www.freelunch.com
www.bea.gov, www.census.gov, www.bls.gov
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Math ReviewThere is much more to it… but these are the basics you must know
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Math ReviewDifferentiation expresses the rate at which a
quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity.
Linear Relationship (i.e., a straight line) has a specific equation. As x changes, how does y change?
Directly related (x increases, y increases)Inversely related (x increases, y decreases)
yslope b
x
( )y f x a bx
( )f x
x
y
x=0, y=3 or (0,3).
x=2, y=3+2(2) or (2,7)
( ) 3 2f x x 1 0
1 0
( ) (7 3) 42
( ) (2 0) 2
y yyb
x x x
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Math ReviewDerivatives are essentially the same thing.
Instead of looking at the difference in y as x goes from 0 to 2, if you look at very small intervals, say changing x from 0 to 0.0001, the slope does not change for a straight line
The basic rule for derivatives is that the distance between the initial x and new x approches zero (in what is called the limit)
yslope b
x
( )y f x a bx
( )f x
x
y
x=0, y=3 or (0,3).
x=.0001, y=3+2(.0001) or (x,y)=(.0001,3.0002)
( ) 3 2f x x 1 0
1 0
( ) (3.0002 3) .00022
( ) (.0001 0) .0001
y yyb
x x x
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Math ReviewDerivatives have a slightly different notation
than delta-y/delta-x, namely dy/dx or f’(x). Constants, such as the y-intercept do not change as x changes, and thus are dropped when taking derivatives.
Derivatives represent the general formula to find the slope of a function when evaluated at a particular point. For straight lines, this value is fixed.
( ) cy f x a bx
( )f x
x
y
x=0, y=3 or (0,3).
x=.0001, y=3+2(.0001) or (x,y)=(.0001,3.0002)
( ) 3 2f x x
1'( ) ( ) cdy f x c b x
1 1 0'( ) (1)2 2( ) 2f x x x
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Math ReviewIntegration (or reverse differentiation) is just
the opposite of a derivative, you have to remember to add back in C (for constant) since you may not know the “primitive” equation.
There are indefinite integrals (over no specified region) and definite integrals (where the region of integration is specified).
Also, the result of integration should be the function you would HAVE TO TAKE the derivative of to get the initial function.
( ) ( ) cF x ydx f x dx a bx dx
x
y
1( ) ( )1
c cbF x a bx dx ax x
c
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( ) (3 2 ) 31 1
F x x dx x x C
2( ) 3F x x x C 10
2 100
0
( ) (3 2 ) [3 ]F x x dx x x C 2 2( ) [3(10) (10) ] [3(0) (0) ] 130F x
Area=[3*(10-0)]+[1/2*(10-0)*(3+2(10))]=130
10
3
23
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Basic Definitions Random variable
A function or rule that assigns a real number to each basic outcome in the sample space The domain of random variable X is the sample space The range of X is the real number line
Value changes from trial to trial Uncertainty prevails in advance of the trail as to
the outcome
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Case Study
Weight Data
Introductory Statistics classSpring, 1997
Virginia Commonwealth University
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Weight Data
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Weight Data: Frequency TableWeight Group Count
100 - <120 7 120 - <140 12 140 - <160 7 160 - <180 8 180 - <200 12 200 - <220 4 220 - <240 1 240 - <260 0 260 - <280 1
sqrt(53) = 7.2, or 8 intervals; range (260100=160) / 8 = 20 = class width
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Weight Data: Histogram
0
2
4
6
8
10
12
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Frequency
100 120 140 160 180 200 220 240 260 280Weight
* Left endpoint is included in the group, right endpoint is not.
Nu
mb
er
of s
tude
nts
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Numerical Summaries Center of the data
mean median
Variation range quartiles (interquartile range) variance standard deviation
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Mean or Average Traditional measure of center Sum the values and divide by the number
of values
n
iin
xn
xxxxn
x1
3
1121
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Median (M) A resistant measure of the data’s center At least half of the ordered values are less
than or equal to the median value At least half of the ordered values are
greater than or equal to the median value If n is odd, the median is the middle ordered value If n is even, the median is the average of the two
middle ordered values
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Median (M)
Location of the median: L(M) = (n+1)/2 ,
where n = sample size.
Example: If 25 data values are recorded, the
Median would be the
(25+1)/2 = 13th ordered value.
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Median Example 1 data: 2 4 6
Median (M) = 4
Example 2 data: 2 4 6 8 Median = 5 (ave. of 4 and 6)
Example 3 data: 6 2 4 Median 2 (order the values: 2 4 6 , so Median = 4)
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Comparing the Mean & Median The mean and median of data from a
symmetric distribution should be close together. The actual (true) mean and median of a symmetric distribution are exactly the same.
In a skewed distribution, the mean is farther out in the long tail than is the median [the mean is ‘pulled’ in the direction of the possible outlier(s)].
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Quartiles
Three numbers which divide the ordered data into four equal sized groups.
Q1 has 25% of the data below it.
Q2 has 50% of the data below it. (Median)
Q3 has 75% of the data below it.
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Weight Data: Sorted100 124 148 170 185 215101 125 150 170 185 220106 127 150 172 186 260106 128 152 175 187110 130 155 175 192110 130 157 180 194119 133 165 180 195120 135 165 180 203120 139 165 180 210123 140 170 185 212
L(M)=(53+1)/2=27
L(Q1)=(26+1)/2=13.5
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Variance and Standard Deviation
Recall that variability exists when some values are different from (above or below) the mean.
Each data value has an associated deviation from the mean:
x xi
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Deviations what is a typical deviation from the
mean? (standard deviation) small values of this typical deviation
indicate small variability in the data large values of this typical deviation
indicate large variability in the data
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Variance Find the mean Find the deviation of each value from the
mean Square the deviations Sum the squared deviations Divide the sum by n-1
(gives typical squared deviation from mean)
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Variance Formula
n
ii xx
ns
1
2)()1(
12
n
iix
nx
1
1
Remember that you must find the deviations of EACH x, square the deviations, THEN add them up!
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Standard Deviation Formulatypical deviation from the mean
n
ii xx
ns
1
2)()1(
1
[ standard deviation = square root of the variance ]
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Variance and Standard DeviationExample from Text
Metabolic rates of 7 men (cal./24hr.) :
1792 1666 1362 1614 1460 1867 1439
1600 7
200,11
7
1439186714601614136216661792
x
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Variance and Standard DeviationExample
Observations Deviations Squared deviations
1792 17921600 = 192 (192)2 = 36,864
1666 1666 1600 = 66 (66)2 = 4,356
1362 1362 1600 = -238 (-238)2 = 56,644
1614 1614 1600 = 14 (14)2 = 196
1460 1460 1600 = -140 (-140)2 = 19,600
1867 1867 1600 = 267 (267)2 = 71,289
1439 1439 1600 = -161 (-161)2 = 25,921
sum = 0 sum = 214,870
xxi ix 2xxi
Notice the deviations add to zero, so each deviation must be squared
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Observation Value1 1,7922 1,6663 1,3624 1,6145 1,4606 1,8677 1,439
=sum(B1:B7) 11,200=stdevp(B1:B7) 175=stdev(B1:B7) 189=variance(B1:B7) 35,812
Variance versus Standard Deviation
24.18967.811,35
67.811,35)870,214(6
1)870,214(
17
1
2
2
ss
s
Note: Standard deviation is in the same units as the original data (cal/24 hours) while variance is in those units squared (cal/24 hours)2. Thus variance is not easily comparable to the original data.
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Density Curves
Example: here is a histogram of vocabulary scores of 947 seventh graders.
The smooth curve drawn over the histogram is a mathematical model for the distribution. This is typically written as f(x), also known as the PROBABILITY DISTRIBUTION FUNCTION (PDF)
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Density Curves
Example: the areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0. This proportion is equal to 0.303. The area underneath the curve, is called the CUMULATIVE DENSITY FUNCTION (CDF): denoted F(x)
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Density Curves
Example: now the area under the smooth curve to the left of 6.0 is shaded. If the scale is adjusted so the total area under the curve is exactly 1, then this curve is called a density curve. The proportion of the area to the left of 6.0 is now equal to 0.293.
21.55 ( )21
( ) .2932
x
x
x
x
F x e
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Density Curves
Always on or above the horizontal axis
Have area exactly 1 underneath curve
Area under the curve and above any range of values is the proportion of all observations that fall in that range
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Density Curves
The median of a density curve is the equal-areas point, the point that divides the area under the curve in half
The mean of a density curve is the balance point, at which the curve would balance if made of solid material
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Density Curves The mean and standard deviation computed
from actual observations (data) are denoted by and s, respectively.x
The mean and standard deviation of the actual distribution represented by the density curve are denoted by µ (“mu”) and (“sigma”), respectively.
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QuestionData sets consisting of physical measurements (heights, weights, lengths of bones, and so on) for adults of the same species and sex tend to follow a similar pattern. The pattern is that most individuals are clumped around the average, with numbers decreasing the farther values are from the average in either direction. Describe what shape a histogram (or density curve) of such measurements would have.
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Bell-Shaped Curve:The Normal Distribution
standard deviation
mean
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The Normal Distribution
Knowing the mean (µ) and standard deviation () allows us to make various conclusions about Normal distributions. Notation: N(µ,).
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68-95-99.7 Rule forAny Normal Curve 68% of the observations fall within (meaning above
and below) one standard deviation of the mean 95% of the observations fall within two standard
deviations (actually 1.96) of the mean 99.7% of the observations fall within three standard
deviations of the mean
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68-95-99.7 Rule for Approximates for any Normal Curve
68%
+- µ
+3-3
99.7%
µ
+2-2
95%
µ
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68-95-99.7 Rule forAny Normal Curve
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