EC339: Applied Econometrics

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EC339 Applied Econometrics

• Introduction

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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 applicationsbeyond 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

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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 book
• http//caleb.wabash.edu/econometrics/index.htm

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Web Links

• Resources 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 Review

• There is much more to it but these are the
  basics you must know

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Math Review

• Differentiation 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 ? 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)

y
x
x0, y3 or (0,3). x2, y32(2) or (2,7)
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Math Review

• Derivatives 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)

y
x
x0, y3 or (0,3). x.0001, y32(.0001) or
(x,y)(.0001,3.0002)
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Math Review

• Derivatives 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.

y
x
x0, y3 or (0,3). x.0001, y32(.0001) or
(x,y)(.0001,3.0002)
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Math Review

• Integration (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.

y
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3
x
10
Area3(10-0)1/2(10-0)(32(10))130
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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 Table
sqrt(53) 7.2, or 8 intervals range
(260?100160) / 8 20 class width
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Weight Data Histogram
Number of students
Weight Left endpoint is included in the group,
right endpoint is not.
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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

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Median (M)

• A resistant measure of the datas 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) (n1)/2 ,where n
  sample size.
• Example If 25 data values are recorded, the
  Median would be the (251)/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 Sorted
L(M)(531)/227
L(Q1)(261)/213.5
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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

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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
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
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

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Variance and Standard DeviationExample
Observations Deviations Squared deviations

1792 1792?1600 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
Notice the deviations add to zero, so each
deviation must be squared
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Variance versus Standard Deviation
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.

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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.

• 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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Question
Data 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 CurveThe Normal Distribution
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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
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68-95-99.7 Rule forAny Normal Curve
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