ECONOMETRICS II

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ECONOMETRICS II

• Overview
• Dr. Liam Delaney

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Last Years Class
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The goal of today's lecture

• To get to know each other
• To review the six topics that form the basis of
  the course

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Topics

• 1. Introduction.
• 2. Dummy dependent variable modelling.
• 3. Simultaneous equation modelling.
• 4. Distributed lag models.
• 5. Other Time series models.
• 6. Panel econometrics.

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1. Introduction

• Discussion of basic principles of economic
  modelling.
• Ensure that you have a familiarity with OLS,
  Deviations from the Classical Model etc.
• Ensure that you have a basic familiarity with the
  principles of model construction, specification
  and testing.

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• FACTORS INFLUENCING THIRD-YEAR STUDENTS SUCCESS
  IN ECONOMETRICS
• Abstract
• The paper aims to investigate the factors
  influencing third-year students success in
  econometrics course being taught at Istanbul
  Bilgi University. To accomplish this, a multiple
  regression model is estimated using survey data
  from the 246 students who have taken the
  econometrics course. The model tries to explain
  the success in the midterm examination of this
  course using factors in accordance with the
  relevant literature. However a factor that has
  not been considered by the literature, namely,
  whether or not the student plans to enter the
  job-market upon graduation is included. The
  results of the present study indicate that the
  students who planned to enter the job market upon
  graduation seemed to be more interested in
  mastering the topic of econometrics.
• Source Gevrek, Z., Kahraman, B., and Kirmanoghu,
  H., (2004) Available from the Social Science
  Research Network (www.ssrn.com)

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• The regression model employed in the present
  study is as follows
• MG a0 a1Ai a2Bi a3Ci a4Di a5Ei a6Fi
  a7Gi a8Hi a9Ii a10Ji a11Ki a12Li
  a13Mi a14Ni a15Oi a16Pi Ui
• Dependent variable
• MG Midterm grade Students grade (out of 100) in
  the econometric midterm examination.
• Explanatory variables
• A Gender A value of 1 is assigned if the
  student is a male a value of 0 is assigned if
  the student is a female.
• B Related grade Students grade (out of 100) in
  the prerequisite statistics midterm examination,
  which reflects prior scholastic performance in a
  related course.

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• C and D Job market This factor has two
  specifications
• a C A value of 1 is assigned if the student
  plans to work as an employee upon graduation a
  value of 0 is assigned otherwise.
• b D A value of 1 is assigned if the student
  plans to work as an employee upon graduation, and
  then subsequently start their own business a
  value of 0 is assigned otherwise.
• E and F Financial aid This factor has two
  specifications
• a. E A value of 1 is assigned if the student
  receives financial aid from Istanbul Bilgi
  Universitys Board of Trustees a value of 0 is
  assigned otherwise.
• b. F A value of 1 is assigned if the student
  receives financial aid from the Turkish Higher
  Education Council a value of 0 is assigned
  otherwise.
• G Mothers education level This is a
  categorical variable, made up of four categories.
  
• H Fathers education level This is a
  categorical variable, made up of four categories.
  

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• I Mothers employment status A value of 1 is
  assigned if the father works as an employee a
  value of 0 is assigned otherwise.
• J Fathers employment status A value of 1 is
  assigned if the mother works as an employee a
  value of 0 is assigned otherwise.
• K Regular student-attendance at lectures.
• L Regular student-attendance at tutorials.
• M Students ability to cope with stress.
• N Suitable study environment, free of
  distractions.
• O Students perceived value of the course.
• P Faculty teaching ability Faculty (both
  instructor and teaching assistants) teaching
  ability is accounted for using factor analysis
  based on the students five-point scale
  evaluations

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OLS

• Find the line of best fit
• min the sum of squared errors (SSE)
• OLS is BLUE and Consistent
• Test Restrictions F,t tests

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• Example
• test H0b1 b2 1
• Calculate
• Note the degrees of freedom
• df1 no. of restrictions, df1 1,
• df2N-K-1
• Get Critical Value from tables Fc(df1,df2, sig
  level)
• Reject the Null Hypothesis if FgtFc

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• How do you interpret Dummy independent variables?
  
• Why do you include one less dummy than number of
  categories.
• Goodness of Fit
• R-squared
• Does it matter?
• Omitted Variable Bias
• Does it matter?

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AC Hetero

• Variance of residuals is non-standard
• AC
• Hetero
• OLS is no-longer best
• Usual formulae for Standard errors not correct
• Tests not correct
• Still unbiased and consistent

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• Solution GLS
• model the error explicitly
• Transform data to eliminate AC or Het
• Do OLS on transformed data

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Detecting AC Het

• AC DW statistic
• Estimate the model by OLS
• Calculate the DW test statistic
• critical values from the DW tables
• Warning! region of indecision and two critical
  values
• upper and lower bound from the tables dL and dU

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• Hetero Park Test
• T-Test of
• Experiment with different structures and
  variables
• Residual Plots
• For both AC and Het
• See if there is a pattern

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2. Dummy Dependent Variables

• REVISION Please Revise statistical sections on
  levels of measurement
• So far, you have almost exclusively considered
  continuous data.
• A large amount of econometrics is concerned with
  discrete data (e.g. 0,1 categories) and other
  forms of limited dependent data.
• Very interesting and wide range of topics e.g.
  consumer theory, voting and any other choice
  topics you can think of.
• Limited Dependent Analysis increasingly common in
  an age of vast survey data sets and powerful
  computing capabilities

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Why not OLS??

• Low Fit
• Heteroscedasticity
• Non-normality
• Does not constrain the predicted values of the
  probabilites between 0 and 1.

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Dummy Dependent Variables

• LPM, Probit, Logit
• LPM
• OLS
• (0,1) barrier
• Marginal effects
• Probit, Logit
• Non-linear ML
• (0,1) barrier
• Marginal effects

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• More formally, we say that the probability that
  Y1 (i.e. that an individual drives) is a
  non-linear function, F, of the variables.
• We choose the function to ensure that it has the
  desired shape.
• In the case of Probit we use F, the cumulative
  distribution function of a normal random
  variable.

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• In general, for a sample of N observations, the
  log-likelihood of the sample will be
• Log likelihood is joint probability that observe
  data
• Computer algorithms find MLE

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Some Issues with Logit/Probit

• How do you test for joint significance?
  (Likelihood Ratio Test).
• Remember that the coefficients are not marginal
  effects.

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Likelihood Ratio Test

• Cant use F-test ---- because there are no SSR
• LR test is the equivalent
• Intuition -- see if the restriction changes the
  likelihood significantly
• Test Statistic
• Critical Value c2 with d.f. equal to no. of
  restrictions

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• The marginal probability is affected by b but it
  is a non-linear function and is not equal to b.
• Wrong to say b equal to 0.2 implies 20 increase
  in travel by car for every 1 increase in bus
• This is a consequence of ensuring that marginal
  probability is low when probability is high and
  vice-versa i.e. as in the diagram.
• The marginal probability will have the same sign
  as b. This is often all that we want.
• Often report marginal probability evaluated at
  the means

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3. Simultaneous Equations

• REVISION please revise the assumptions of the
  OLS model
• Supply and demand
• OLS biased and inconsistent
• Confuses two effects
• Formal proof and intuition
• Identification
• Try to separate the effects
• Need exclusion restrictions

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ct ?1 ?2 yt et
yt ct it
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Illustrating the Identification Problem

• Suppose we observe the following data
• Is this a supply curve or a demand curve?
• It looks like a supply curve

.
.
.
.
.
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• It could be a supply curve, i.e data is generated
  by movements of the demand curve along a supply
  curve -- so trace out the supply curve

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• We can identify (trace out) the supply curve only
  because y is in the demand curve equation but not
  in the supply curve
• It is because y is excluded from the supply curve
  that we can be sure that changes in y move the
  demand curve only
• If y was in the supply curve we could not do this
• We cannot identify (trace out) the demand curve,
  because there is no variable in the supply curve
  that is not in the demand curve

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Estimation- 2SLS

• Two stage least squares
• 1. Estimate the reduced form using OLS.
  
• 2. Do OLS on the structural form with the
  actual values replaced by the fitted values
  from the first stage

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• Why this works for the supply equation
• The fitted values from the first stage are by
  definition the part of the variation in p and q
  that is due to changes in income
• Therefore we are sure that the fitted values lie
  along the supply curve --- so we just do OLS on
  these values
• More formally the fitted value of p is
  uncorrelated with e because it is a function
  solely of y which is uncorrelated with e (i.e.
  exogenous)

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• Why does it not work on the demand equation?
• Computer will generate an error at second stage
  estimation of demand equation because effectively
  the income variable will appear twice

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TIME SERIES - GENERAL

• In time series the order of the data matters
• Very detailed and complex areas
• We examined three main areas
• Distributed Lags
• Univariate Time Series
• Cointegration Models

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4. Distributed Lag

• Effect is distributed through time
• Two questions
• How far back?
• Should the coefficients be restricted?
• Models
• Unrestricted Finite DL
• ADL
• PDL
• Geometric lag (AE, PAM)

yt ? ?0 xt ?1 xt-1 ?2 xt-2 et
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The Distributed Lag Effect
Effect at time t1
Effect at time t2
Effect at time t
Economic action at time t
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The Distributed Lag Effect
Effect at time t
Economic action at time t-2
Economic action at time t
Economic action at time t-1
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Arithmetic Lag Structure (impulse response
function)
?i
.
?0 (n1)?
.
?1 n?
.
?2 (n-1)?
linear lag
structure
.
?n ?
0 1 2 . . .
. . n n1
i
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Polynomial Lag Structure
?i
?2
?1
?3
?0
?4
0 1 2 3 4
i
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n the length of the lag p degree of polynomial
where i 1, . . . , n
For example, a quadratic polynomial
?0 ?0 ?1 ?0 ?1 ?2 ?2 ?0
2?1 4?2 ?3 ?0 3?1 9?2 ?4
?0 4?1 16?2
where i 1, . . . , n p 2 and n 4
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n the length of the lag p degree of polynomial
where i 1, . . . , n
For example, a quadratic polynomial
?0 ?0 ?1 ?0 ?1 ?2 ?2 ?0
2?1 4?2 ?3 ?0 3?1 9?2 ?4
?0 4?1 16?2
where i 1, . . . , n p 2 and n 4
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yt ? ?0 xt ?1 xt-1 ?2 xt-2 ?3 xt-3
??4 xt-4 et
yt ? ?0?xt ??0 ?1 ?2?xt-1 (?0
2?1 4?2)xt-2 (?0
3?1 9?2)xt-3 (?0 4?1 16?2)xt-4 et
Step 2 factor out the unknown coefficients
?0, ?1, ?2.
yt ? ?0 xt xt-1 xt-2 xt-3 xt-4
?1 xt xt-1 2xt-2 3xt-3 4xt-4
?2 xt xt-1 4xt-2 9xt-3 16xt-4 et
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yt ? ?0 xt xt-1 xt-2 xt-3 xt-4
?1 xt xt-1 2xt-2 3xt-3 4xt-4
?2 xt xt-1 4xt-2 9xt-3 16xt-4 et
Step 3 Define zt0 , zt1 and zt2 for ?0 , ?1
, and ?2.
z t0 xt xt-1 xt-2 xt-3 xt-4
z t1 xt xt-1 2xt-2 3xt-3 4xt- 4
z t2 xt xt-1 4xt-2 9xt-3 16xt- 4
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Do OLS on
yt ? ?0 z t0 ?1 z t1 ?2 z t2 et
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Geometric Lag Structure(impulse response
function)
?i
geometrically declining weights
Figure 15.5
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5. Time Series

• Univariate time series.
• AR verus MA processes.
• Unit roots and integration.
• NB testing for unit roots in AR (1).
• Multivariate time series.
• Spurious as opposed to cointegrated
  relationships.
• NB testing for cointegration.

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Key Concepts

• Stationarity and non-stationarity.
• Autoregressive and moving average processes.
• Unit roots.
• Dickey fuller test.
• Cointegration and spurious regressions.
• Testing for cointegration.

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Integrated Processes

• A unit root /non-stationary/ Difference
  Stationary/ Integrated of order one I(1)
• A process will be I(1) if its auto regressive
  coefficients sum to one
• Random walk
• I(1) Impulse response functions
• Temporary shock has permanent effects
• Important economic implications

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What is Stationarity?

• A stochastic process is said to be stationary if
  its mean and variance are constant over time and
  the value of the covariance between the two time
  periods depends only on the distance or gap
  between the two time periods and not the actual
  time at which the covariance is computed

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Multivariate Time Series Models

• We first dealt with multivariate time series
  models in the section on distributed lags.
• We noted that the Koyck model contained an
  autogregressive component as well as as a
  multivariate component.
• Key issue is whether or not the relationship
  between variables is cointegrated or spurious.

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Beware Spurious Regressions

• Remember the TSP example
• Two completely unrelated variables trending
  upward over time may look as if they are related
• This is also something to keep in mind when you
  are thinking about distributed lag models
• A test for cointegration can be thought of a
  pre-test to avoid spurious regression
  situations Granger (1986)

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Co-integrated Processes

• Spurious Regression Two I(1) variables not
  related but OLS says that they are
• Cure include lags of both in regression
• Two I(1) variables could be truly related
• co-integrated
• Test
• Run the co-integrating regression
• Calculate the residuals
• Test residuals are I(1)
• If I(0) then the two variables are co-integrated

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Test (1) Engle-Granger Tests

• Test both series to see if I(1)
• Run OLS on the two variables with no lags
• If cointegrating relationship exists (Z) then OLS
  will find it
• If it doesnt exist then spurious regression
• To see which it is, test Z to see if I(1)
• Z will be the residual from the OLS regression
• use DF or ADF test
• critical values are different from standard DF!
• if Z is I(1) then variable are not cointegrated

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Test (2) Cointegrating Regression Durbin-Watson
test (CRDW)

• Remember the DW test for Autocorrelation
• The null was that the relations between the error
  terms is zero i.e. d2
• Here the null is that there is a unit root so
  that d 2(1-1) 0
• If the DW test exceeds the critical values we can
  accept the hypothesis of cointegration
• Asymptotically a unit root in ?t ? DW statistic
  is zero
• Critical values for this test depend on thedata
  generating process

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Conclusions

• Do you understand what stationary stochastic
  processes are?
• Do you know how this relates to unit roots?
• Do you know how unit roots are related to
  integration?
• Do you know how to test for unit roots?
• Do you understand what is meant by cointegration
  and why it is important?
• Do you know how to test for cointegration?

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6. Panel Econometrics

• Describe panel data as opposed to time series or
  cross-sectional data.
• Explain how you would estimate a panel model.
• Explain the difference between fixed effects and
  random effects models.
• How would you chose between the two (N.B. The
  Hausman Test).

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Fixed Effects or Random Effects

• IF N is large and T is small, and if the
  assumptions underlying RE hold, the RE are more
  efficient estimators.
• Use Fixed Effects if the errors and the
  observations are correlated (e.g. countries).
• The Hausman test is distributed Chi-Squared
  Asymptotic around the null hypothesis that Random
  Effects is appropriate.

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Hausman Test

• Hausman (1978).
• The null hypothesis is that the FE and RE do not
  differ substantially.
• Test is distributed asymptotically chi-squared.
• FE is consistent under both the null and the
  alternative.
• RE is consistent under the null and inconsistent
  under the alternative.
• We can test the appropriateness of RE using
  critical values.

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