Review I: Basics

1
Review I Basics

• Introduction to Econometrics
• Conducting an Econometric Study
• The Summation Operator
• Properties of Random Variables
• Probability Distribution
• Some Useful Results

2
What Is Econometrics?

• Possible answers
• 1. Econometrics is the science of testing
  economic theories.
• 2. Econometrics is the set of tools used for
  forecasting future values of economic variables,
  such as a firms sales, the growth rate of an
  economy, or stock prices.
• 3. Econometrics is the process of fitting
  mathematical models to economic data.
• 4. Econometrics is the art and science of using
  historical data to make quantitative policy
  recommendations in government and business.

3
What Is Econometrics?

• All these answers are correct!!!
• At a broad level, we define,
• Econometrics is the study of the application of
  statistical methods to economic problems
• Econometric methods are widely used in Finance,
  Labour Economics, Microecononmics,
  Macroeconomics, Public Finance, Marketing,
  Industrial Organization, Health, etc.

4
Why study Econometrics?

• Rare in economics (and many other areas without
  labs!) to have experimental data.
• Need to use non-experimental, or observational
  data to make inferences.
• Important to be able to apply economic theory to
  real world data.

5
Why study Econometrics?

• An empirical analysis uses data to test a theory
  or to estimate a relationship.
• A formal economic model can be tested.
• Theory may be ambiguous as to the effect of some
  policy change can use econometrics to evaluate
  the program.

6
Types of Data Cross Sectional

• Cross-sectional data is a random sample.
• Each observation is a new individual, firm, etc.
  with information at a point in time.
• If the data is not a random sample, we have a
  sample-selection problem.

7
Cross Sectional Data Structure

• A cross-sectional data set on wages and other
  individual characteristics

8
(No Transcript)
9
Types of Data Panel (balanced or unbalanced)

• Can pool random cross sections and treat similar
  to a normal cross section. Will just need to
  account for time differences.
• Can follow the same random individual
  observations over time known as panel data or
  longitudinal data.

10
Pooled Cross Sectional Data Structure

• Pooled cross-sections two years of housing
  prices when there was reduction in property taxes
  in 2002

11
Panel Data Structure

• A two year panel data set on City Crime Statistics

12
(No Transcript)
13
Types of Data Time Series

• Time series data has a separate observation for
  each time period e.g. stock prices, inflation
  rate, unemployment rate, etc.
• Since time series data is not a random sample,
  different problems to consider
• Trends and seasonality will be important.
• Trend Stationary vs. Difference Stationary.

14
Time Series Data Structure

• Time series data for important macroeconomic
  variables (1971-2000)

15
(No Transcript)
16
The Question of Causality

• Simply establishing a relationship between
  variables is rarely sufficient.
• Want to the effect to be considered causal.
• If weve truly controlled for enough other
  variables, then the estimated ceteris paribus
  effect can often be considered to be causal.
• Can be difficult to establish causality.

17
Example Measuring the Return to Education

• A model of human capital investment implies
  getting more education should lead to higher
  earnings.
• In the simplest case, this implies an equation
  like
• Earnings b1 b1Education ?

18
Example (cont.)

• The estimate of b1, is the return to education,
  but can it be considered causal?
• While the error term, ?, includes other factors
  affecting earnings, want to control for as much
  as possible.
• Some things are still unobserved, which can be
  problematic.

19
Conducting an Econometric Study

• Step 1 Develop a Research Question (Hypothesis)
• Step 2 Develop an economic model to frame the
  question
• Step 3 Collecting data to estimate the
  parameters of the model
• Step 4 Model Specification and testing
• Step5 Present the findings and interpret the
  results (prediction or forecasting?)

20
Step 1 Develop a Research Question

• Two Important Considerations
• The question has to be feasible
• Must have an objective answer (positive vs.
  normative)
• Answer can be found using econometric methods
• The Question has to be Practical
• Set up an economic model
• Collect relevant data
• Analyze those data within the available time frame

21
Examples

• Does Reducing Class Size Improve Elementary
  School Education?
• Is there a Racial Discrimination in the Market
  for Home Loans?
• How much Do Cigarette Taxes Reduce Smoking?
• What will the Rate of Inflation be Next Year?

22
Examples (cont.)

• Economic Model of Crime
• Job Training and Worker Productivity
• Effects of Fertilizer on Crop Yield
• The Effect of Law Enforcement on City Crime
  Levels
• The Effect of the Minimum Wage on Unemployment

23
Data Source

• Canadian Sources Visit Data Liberation
  Initiative at the University of Manitoba Library
  to see all Canadian sources of Data.
• US and International Sources Browse Web Links in
  your CD-ROM that come with the Text.
• If you are looking for a particular data set for
  your undergraduate research paper, contact Gary
  Strike, Dafoe Library, tel. [phone] for more
  information and assistance.

24
The Summation Notation

• The sum of a large number of terms occurs
  frequently in econometrics. There is an
  abbreviated notation for such sums. The upper
  case Greek letter ? (sigma) is used to indicate a
  summation and the terms are generally indexed by
  subscripts.

25
Examples
26
Properties of the Summation Operator
27
Sample Average
28
Results
29
Properties of Random Variables

• What is a random Variable?
• it takes a single, specific value
• We dont know in advance what value it takes
• We do know all possible values it may take
• We know the probability that it will take any one
  of those possible values
• Expected value (or population mean value) The
  expected value of a discrete random variable X
  is

30
Properties of Expected Value
31
Properties of Expected Value

• The law of large numbers Suppose one repeatedly
  observes different realized values of a random
  variable and calculates the mean of the realized
  values. The mean will tend to be close to the
  expected value the more times one observes the
  random variable, the closer the mean will tend to
  be.
• The expected value of a random variable (say,
  stock price) does not tell us how much it will go
  up or down. The variance provides a measure of
  how far the random variable is likely to be away
  from its mean.

32
Properties of Random Variables

• For a discrete random variable, the variance (?2
  E(X - ?)2 is calculated by
• Since the variance is the average value of the
  squared distance between Xi and ?, it does not
  have an easy interpretation.
• The standard deviation is a very useful measure.
  The standard deviation ? of a random variable is
  equal to the square root of the variance of the
  random variable.

33
Properties of Variance

• 1. Var(constant) 0
• 2. If X and Y are two independent random
  variables, then
• Var(X Y) Var(X) Var (Y) and
• Var(X - Y) Var(X) Var (Y)
• 3. If b is a constant then Var(bX) Var(X)
• 4. If a is a constant then Var(aX) a2Var(X)
• 5. If a and b are constants then Var(aXb)
  a2Var(X)
• 6. If X and Y are two independent random
  variables and a and b are constants then
  Var(aXbY) a2Var(X) b2Var(Y)

34
Covariance

• Covariance For two discrete random variables X
  and Y with E(X) ?x and E(Y) ?y, the
  covariance between X and Y is defined as Cov(XY)
  ?xy E(X - ?x) E(Y - ?y) E(XY) - ?x ?y.
• To computer the covariance, we use the following
  formula

35
Covariance

• In general, the covariance between two random
  variables can be positive or negative. If two
  random variables move in the same direction, then
  the covariance will be positive, if they move in
  the opposite direction the covariance will be
  negative.
• Properties
• 1.If X and Y are independent random variables,
  their covariance is zero. Since E(XY) E(X)E(Y)
• 2. Cov(XX) Var(X)
• 3. Cov(YY) Var(Y)

36
Correlation Coefficient

• The covariance tells the sign but not the
  magnitude about how strongly the variables are
  positively or negatively related. The correlation
  coefficient provides such measure of how strongly
  the variables are related to each other.
• For two random variables X and Y with E(X) ?x
  and E(Y) ?y, the correlation coefficient is
  defined as

37
Correlation Coefficient

• 1. Like the covariance, the correlation
  coefficient can be positive or negative same
  sign as the covariance.
• 2. The correlation coefficient always lies
  between 1 and 1. 1 perfectly negatively
  correlated and 1 perfectly positively
  correlated.
• 3. Variances of correlated variables
• Var(X Y) Var(X) Var(Y) 2Cov(X,Y)
• Var(X - Y) Var(X) Var(Y) 2Cov(X,Y)

38
The Normal Distribution

• The normal family of distributions occurs much
  more often in econometrics than any other
  parametric family.
• One reason for this is that the sum of a large
  number of independent random variables has an
  approximately normal distribution.
• Normal distributions are symmetrical about the
  mean, and the normal probability curve is the
  familiar bell-shaped curve. The mean, median, and
  mode are equal for this family of distributions

39
Shape of the Normal Distribution
40
Normal Distribution

• A normally distributed random variable X with
  mean ?x and variance ?x2 is written as X N(?x,
  ?x2).
• Standard Normal A normally distributed random
  variable Z with mean 0 and variance 1is written
  as Z N(0,1).

41
Normal Distribution

• The PDF of a normally distributed random
  variable X with mean ?x and variance ?x2 is given
  by
• The PDF of a Standard Normal random variable Z is
  given by

42
Properties of N.D.

• 1. The normal distribution curve is symmetrical
  around its mean value.
• 2. The The PDF of the distribution is highest at
  its mean value. That is, the probability of
  obtaining a value of a normally distributed r.v.
  far away from its mean value becomes
  progressively smaller.

43
Properties of N.D. (cont.)

• 3. Approximately 68 of the area under the
  normal curve lies between
• Approximately 95 of the area under the normal
  curve lies between
• Approximately 99.7 of the area under the normal
  curve lies between

44
Properties of N.D. (cont.)

• 4. A normally distributed random variable is
  fully described by its two parameters mean and
  variance.
• 5. A linear combination of two or more normally
  distributed random variables is itself normally
  distributed.
• 6. For a normal distribution, skewness is zero
  and kurtosis is 3.

45
Properties of N.D. (cont.)

• (Note Skewness is (square of the 3rd
  moment)/(cube of the 2nd moment) and kurtosis is
  (fourth moment)/square of the 2nd moment)
• 7. Z-transformation
• X N(?x, ?x2) then (X -?x)/?x. N(0,1)

46
Useful Results

47
Properties of t distribution

• The t distribution like the normal distribution
  is symmetric. It is a little bit wider and
  flatter than the standard normal distribution.
• The mean of the t distribution, like the standard
  normal distribution is zero, but its variance is
  k/(k-2), where k is the d.f. Thus, variance is
  defined for d.f.gt2.

48
?2 Distribution

• If X N(0,1), then X2 is also random and it
  takes the ?2 (Chi-square) distribution with 1
  d.f. That is, X2 ?2(1).
• If N independent random variables X1, X2, , Xk
  are all distributed N(0,1), then the sum of their
  squares are also random and has the ?2
  distribution with k d.f. That is, ?Xi2 ?2(k).

49
Properties of ?2 Distribution

• 1. Unlike the normal distribution, the ?2
  distribution takes only positive values and
  ranges from 0 to ?.
• 2. Unlike the normal distribution, the ?2
  distribution is a skewed distribution, the degree
  of skewness depending on the d.f. For a
  relatively few d.f., the distribution is highly
  skewed to the right, but as the d.f. increases,
  the distribution becomes increasingly symmetrical
  and approaches the normal distribution.

50
Properties of ?2 Distribution

• 3. The expected value of a ?2 r.v. is k and its
  variance is 2k, where k is the d.f.
• 4. If Z1 and Z2 are two independent ?2 variables
  with k1 and k2 d.f., respectively, then their sum
  is also a ?2 variable with d.f. (k1k2).
• 5. If X N(?, ?2) then (X-?)/?2 ?2(1). If
  X1, X2 , , Xk are all N(?, ?2) then
  ?(Xi-?)/?2 ?2(k)

51
F-Distribution

• If Z1 and Z2 are independently distributed ?2
  variables with k1 and k2 d.f., respectively, then
  the variable (Z1/k1)/(Z2/k2) Fk1,k2
• Like the ?2 distribution, the F-distribution is
  also skewed to the right and also ranges between
  0 and ?.
• Like the ?2 distribution, the F-distribution
  approaches the normal distribution as k1 and k2,
  the d.f., become large.
• The square of a t distributed random variable
  with k d.f. has a F-distribution with 1 and k
  d.f. That is, tk2 F1,k.