econometrics i: ols
TRANSCRIPT
Econometrics I: OLS
Dean Fantazzini
Dipartimento di Economia Politica e Metodi Quantitativi
University of Pavia
Overview of the Lecture
1st EViews Session I: Convergence in the Solow Model
Econometrics I: OLS
Dean Fantazzini2
Overview of the Lecture
1st EViews Session I: Convergence in the Solow Model
2nd EViews Session II: Estimating a money demand equation
Econometrics I: OLS
Dean Fantazzini2-a
Overview of the Lecture
1st EViews Session I: Convergence in the Solow Model
2nd EViews Session II: Estimating a money demand equation
3rd EViews Session III: Monte Carlo Simulation
Econometrics I: OLS
Dean Fantazzini2-b
EViews Session I: Convergence in the Solow Model
a) Opening the workfile. EViews is build around the concept of an
object. Objects are held in workfiles. Thus open the workfile GROWTH.WMF.
Remember that the variables are defined as follows:
Variable Definition
gdp60 GDP per capita in real terms in 1960
gdp85 GDP per capita in real terms in 1985
pop60 Population in millions in 1960
pop85 Population in millions in 1985
inv Average from 1960 to 1985 of the ratio of
real domestic investment to real GDP
geetot Average from 1970 to 1985 of the ratio of
nominal government expenditure on education to nominal GDP
oecd Dummy for OECD member
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EViews Session I: Convergence in the Solow Model
... this is the Barro Data Set:
• Cross-sectional Data Set containing macro information of 118
Countries
• Collected in 1985
• OECD: Organisation for Economic Cooperation and Development
• 24 members in 1985: Australia, Austria, Belgium, Canada, Denmark,
Germany, Finland, France, Great Britain, Greece, Iceland, Ireland,
Italy, Japan, Luxembourg, Netherlands, New Zealand, Norway,
Portugal, Spain, Sweden, Switzerland, Turkey, USA
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EViews Session I: Convergence in the Solow Model
b) Sample
• The sample is the set of observations to be included in the analysis
• Often we analyse only subset of this set of observations
• Example: OECD countries
• Parameter values are missing for observations 3 8 15 23 26 27 53
115-117
• smpl 1 2 4 7 9 14 16 22 24 25 28 52 54 114 118 118
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EViews Session I: Convergence in the Solow Model
c) Generating new series. We need some additional variables for our
analysis. Generate the series l gdp60, l gdp85, l pop60 and l pop85 which
contain the logarithm of the original series . You can follow two ways : u
• Use the GENR button and type l gdp60=log(gdp60), and do the same
for the other series.
• Use the commanding line:
series l gdp60 = log(gdp60)
Furthermore we need the growth rates of the GDP and the population
(GENR - gr gdp=(l gdp85-l gdp60)/25, GENR -
gr pop=(l pop85-l pop60)/25).
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EViews Session I: Convergence in the Solow Model
Theory (Solow Model): All countries converge against a steady- state for
the per capita earnings → Poor countries grow faster as rich countries (see
also Barro, Sala-i-Matin, Economic growth)
d) Analyzing absolute convergence by means of a scatter plot.
Plot l gdp60 against gr gdp in a scatter plot. What does the plot tells you
about absolute convergence?
-.04
-.02
.00
.02
.04
.06
.08
-2 -1 0 1 2 3
L_GDP60
GR
_GD
P
Figure 1: Example: Scatter plot
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EViews Session I: Convergence in the Solow Model
e) Analyzing absolute convergence by means of a regression.
Regress gr gdp on l gdp60 and a constant (click on gr gdp - press
STRG and click on l gdp60 - click on one of the series - OPEN
EQUATION - OK). Using the online help become acquainted with the
regression output (HELP - SEARCH - regression output). What does the
regression output tells you about absolute convergence?
LS GR GDP c L GDP60
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EViews Session I: Convergence in the Solow Model
f) Conditional convergence *. Regress gr gdp on l gdp60, gr pop, inv,
geetot and a constant. Interpret the regression output.
g) Setting the sample*. One of the important concepts in EViews is the
sample of observations. The sample is the set of observations to be
included in the analysis. Set the sample to OECD countries.
• SAMPLE - type in if-box: oecd=1.
• smpl if OECD=1
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EViews Session I: Convergence in the Solow Model
h) Redoing the analysis for OECD countries.
Repeat d), e) and f) with the new sample.
What changes? Interpret your results.
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EViews Session II: Estimating a money demand equation
Some Premises...
Detection of Heteroscedasticity :
• Residual Plot;
• White Test;
• Breusch - Pagan test;
Detection of Autocorrelation:
• Residual Plot;
• Durbin-Watson test (... remember that X have to be deterministic);
d=2 → no autoc., d<2 → positive autoc., d>2 → negative
autocor.
• Breusch - Godfrey test
1. OLS to get the residuals et
2. et = γ1et−1 + . . . + γpet−p + x′
tβ + δt
3. H0 = n · R2∼ χ2
p
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EViews Session II: Estimating a money demand equation
What to do in case of 1) Heteroscedasticity , 2) Autocorrelation?
• GLS (if Ω known), or FGLS
• OLS with 1) White VC matrix, or 2) Newey - West VC matrix
Empirical Example: Money demand.
• Transaction demand → GDP
• Speculation Demand → Bond rate, Money market rate
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EViews Session II: Estimating a money demand equation
a) Open the workfile MONEY.WF1. You find the following variables:
Variable Description
bondr bondrate
gdp GDP (real, seasonally adjusted)
m3 M3 (seasonally adjusted)
monrat money market interest rate
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EViews Session II: Estimating a money demand equation
b) Generate the following variables...
l_gdp = log(gdp), l_m3 = log(m3).
c) ... and estimate the following equation (you take into account the
transaction demand):
l_m3 = β0 + β1 · l_gdp + ε.
Save it as eq1.
d) Has β1 the sign you expected? Interpret.
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EViews Session II: Estimating a money demand equation
e) Now you consider the speculation demand as well and estimate the
following equation:
l_m3 = β0 + β1 · l_gdp + β2 · monrat + β3 · bondr + ε.
Save it as eq2.
f) Have the coefficients the signs you expected? How do you interpret β1,
β2 and β3?
g) Now look at the residual plot. (View -> Actual, Fitted, Residual
-> Residual Graph). Do you find evidence of the presence of
heteroscedasticity and/or autocorrelation?
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EViews Session II: Estimating a money demand equation
h) Consider the Durbin-Watson statistic:
1. Is there evidence of autocorrelation? If yes, positive or negative
correlation?
2. Which assumptions does an application of the Durbin-Watson test
require?
3. Are these assumptions valid in this case?
i) Run a Breusch-Godfreyserial correlation test including 4 lags.
1. What is the null hypothesis?
2. Procedure in EViews: View -> Residual Tests
-> Serial Correlation LM Test - 4.
3. Do you reject the null hypothesis on a 5% significance level?
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EViews Session II: Estimating a money demand equation
j) Estimate the second equation, now using the Newey-West-VC-Matrix
(click on Options -> Heteroscedasticity -> Newey-West in the
specification window). Save the result as eq3. How do the t-values change
compared to equation 2?
k) Based on equation 3, test the following hypothesis on a 5% significance
level: i) β2 = 0, ii) β1 = 2 and iii) β3 < 0. Address the following points:
1. What are the null and the alternative hypothesis?
2. Calculate an appropriate test statistic.
3. Find the 5% critical value.
4. Do you reject the null hypothesis?
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EViews Session II: Estimating a money demand equation
l) Test if the additional parameters in equation 2 (and equation 3) are
jointly significant.
Calculate the appropriate F statistic using two different ways:
1. Use the F test that is implemented in EViews.
2. Calculate the F test by hand using both regression outputs.
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EViews Session III: Monte Carlo Simulation
A typical Monte Carlo simulation exercise consists of the following steps:
1. Specify the “true” model (data generating process) underlying the
data.
2. Simulate a draw from the data and estimate the model using the
simulated data.
3. Repeat step 2 many times, each time storing the results of interest.
4. The end result is a series of estimation results, one for each repetition
of step 2. We can then characterize the empirical distribution of these
results by tabulating the sample moments or by plotting the
histogram or kernel density estimate.
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EViews Session III: Monte Carlo Simulation
• Step 2 typically involves simulating a random draw from a specified
distribution. → EViews provides built-in pseudo-random number
generating functions for a wide range of commonly used distributions;
• The step that requires a little thinking is how to store the results from
each repetition (step 3). There are two methods that you can use in
EViews:
1. Storing results in a series;
2. Storing results in a matrix.
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EViews Session III: Monte Carlo Simulation
1) Storing results in a series (or group of series): The difficulty with
this approach is that series elements are most easily indexed by a sample in
EViews. To store the result from each replication as a different observation
in a series, you must shift the sample every time you store a new result.
Moreover, the length of the series will be constrained by the size of your
workfile sample: for example, if you wish to perform 1000 replications on a
workfile with 100 observations, you will not be able to store all 1000 results
in a series since the latter only has 100 observations.
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EViews Session III: Monte Carlo Simulation
2) Storing results in a matrix (or vector):
The matrix is indexed by its row-column position and its size is
independent of the workfile sample. For example, you can declare a matrix
with 1000 rows and 10 columns in a workfile with only 1 observation.
→ The disadvantage of the matrix method is that the matrix object does
not have as much built-in functions as a series object. For example, there
is no kernel density estimate view out of a matrix (which is available for a
series object).
For didactic purposes, I will illustrate both approaches. (My own
recommendation is a mixed approach. Store all the results in a matrix. If
you need to do further processing, convert the matrix into a group of
series.)
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EViews Session III: Monte Carlo Simulation
As a concrete example, consider the following exercise (3.26) from
Damodar Gujarati Basic Econometrics, 3rd edition:
Refer to the 10 X values of Table 3.2 (which are: 80, 100, 120, 140, 160,
180, 200, 220, 240, 260). Let Beta(1) = 2.5 and Beta(2) = 0.5. Assume
that the errors are distributed N(0, 9), that is, the errors are normally
distributed with mean 0 and variance 9. Generate 100 samples using these
values, obtaining 100 estimates of Beta(1) and Beta(2). Graph these
estimates. What conclusions can you draw from the Monte Carlo study?
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EViews Session III: Monte Carlo Simulation
1) Store monte carlo results in a series
’ set workfile range to number of monte carlo replications
wfcreate mcarlo u 1 100
’ create data series for x
’ NOTE: x is fixed in repeated samples
’ only first 10 observations are used (remaining 90 obs missing)
series x
x.fill 80, 100, 120, 140, 160, 180, 200, 220, 240, 260
’ set true parameter values
!beta1 = 2.5
!beta2 = 0.5
’ set seed for random number generator
rndseed 123456
’ assign number of replications to a control variable
!reps = 100
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EViews Session III: Monte Carlo Simulation
’ begin loop
for !i = 1 to !reps
’ set sample to estimation sample
smpl 1 10
’ simulate y data (only for 10 obs)
series y = !beta1 + !beta2*x + 3*nrnd
’ regress y on a constant and x
equation eq1.ls y c x
’ set sample to one observation
smpl !i !i
’ and store each coefficient estimate in a series
series b1 = eq1.@coefs(1)
series b2 = eq1.@coefs(2)
next
’ end of loop
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EViews Session III: Monte Carlo Simulation
’ set sample to full sample
smpl 1 100
’ show kernel density estimate for each coef
freeze(gra1) b1.kdensity
’ draw vertical dashline at true parameter value
gra1.draw(dashline, bottom, rgb(156,156,156)) !beta1
show gra1
freeze(gra2) b2.kdensity
’ draw vertical dashline at true parameter value
gra2.draw(dashline, bottom, rgb(156,156,156)) !beta2
show gra2
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EViews Session III: Monte Carlo Simulation
2) Store monte carlo results in a Matrix
’ set workfile range to number of obs
wfcreate mcarlo u 1 10
’ create data series for x
’ NOTE: x is fixed in repeated samples
series x
x.fill 80, 100,120, 140, 160, 180, 200, 220, 240, 260
’ set seed for random number generator
rndseed 123456
’ assign number of replications to a control variable
!reps = 100
’ declare storage matrix
matrix(!reps,2) beta
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EViews Session III: Monte Carlo Simulation
’ begin loop
for !i = 1 to !reps
’ simulate y data
series y = 2.5 + 0.5*x + 3*nrnd
’ regress y on a constant and x
equation eq1.ls y c x
’ store each coefficient estimate in matrix
beta(!i,1) = eq1.@coefs(1) ’ column 1 is intercept
beta(!i,2) = eq1.@coefs(2) ’ column 2 is slope
next
’ end of loop
’ show descriptive stats of coef distribution
beta.stats
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