Basic Excel MCMC
Transcript of Basic Excel MCMC
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Teaching basic econometric concepts using Monte Carlosimulations in Excel
Genevieve Briand a,*, R. Carter Hill b
a Instructor, School of Economic Sciences, Washington State University, Pullman, Washington 99164-6210, United StatesbOurso Family Professor of Econometrics and Thomas Singletary Professor of Economics, Economics Department,
Louisiana State University, Baton Rouge, Louisiana 70803, United States
1. Introduction
Monte Carlo experiments rely on repeated random sampling to simulate and compute results of
interest to researchers, instructors or students. Undergraduate econometrics textbooksmake use of
Monte
Carlo
simulations
tohelp teach
basiceconometric
concepts.
In Gujarati
andPorter
(2009), the
authors state that the reader will be asked to conduct Monte Carlo experiments using different
statistical packages (p.12). Hill et al. (2011) illustrate the sampling properties of the least squares
and interval estimators in the beginning chapters of their textbook (pp. 88–93 and pp. 127–129). In
later chapters, they make use of Monte Carlo simulations to explore the properties of the least
International Review of Economics Education 12 (2013) 60–79
A R T I C L E I N F O
Article history:
Available online 8 April 2013
Keywords:
Teaching
Econometrics
Excel
Monte Carlo simulations
A B S T R A C T
Monte Carlo experiments can be a valuable pedagogical tool for
undergraduate econometrics courses. Today this tool can be used in
the classroom without the need to acquire any specialized
econometrics software. This paper argues that Microsoft Excel,
which is already available at many office and home computerstations, offers the opportunity to run meaningful Monte Carlo
simulations and to successfully teach students basic econometric
concepts. The reader is guided, step-by-step, through two different
exercises. The first one is a repeated sampling exercise showing that
least squares estimators are unbiased. The second one expands on
the first to explain the true meaning of confidence interval
estimates of least squares estimators.
2013 Elsevier Ltd. All rights reserved.
* Corresponding author.
E-mail address: [email protected] (G. Briand).
Contents lists available at SciVerse ScienceDirect
International Review of Economics
Educationjournal homepage: www.elsevier.com/locate/iree
1477-3880/$ – see front matter 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.iree.2013.04.001
http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.iree.2013.04.001http://dx.doi.org/10.1016/j.iree.2013.04.001mailto:[email protected]://dx.doi.org/10.1016/j.iree.2013.04.001
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squares estimator
in
the case of
random independent variables
(pp. 273–4, 280) and limited
dependent variables (pp. 442–4). Murray (1999) points out that ‘‘Monte Carlo techniques offer a
rich device for discovery learning’’ (p. 308–9). He further states that ‘‘as I explored the potential of
Monte Carlo techniques for learning econometrics, I realized how much Monte Carlo methods
highlight the central role of sampling properties in econometrics’’ (p. 309). In his preface for
teachers, Murray
(2006) explains that
his
textbook ‘‘starts out with
the Monte
Carlo
approach to
estimators’’
and ‘‘returns
to Monte Carlo
analyses to
facilitate
learning
about
heteroskedasticity,
errors in variables, and consistency’’ (p. xxvii and p. xxx). Kennedy (2008, 1998a, 1998b) is a strong
advocateof Monte Carlo experiments as a pedagogical tool for undergraduate econometricscourses.
He suggests the use of ‘‘explain how to do a Monte Carlo study’’ problems to teach student the
sampling-distribution concept which, he argues, is the ‘‘statistical lens’’ allowing students to make
sense
of
the statistics
world
(1998a, 1998b). He
proposesan
arrayof
such
problems in
Appendix
D
of
his textbook (2008).
Kennedy recommends that instructors do not ask students to actually do a Monte Carlo study
(1998a).
He
cautions
of
the
high
opportunity
cost
of
having
them
learn
how
to
program
(1998b).
Although Murray (1999) ‘‘eagerly champions’’ using computers to teach econometrics and allow ‘‘a
hands-on, discovery mode of learning’’, he also warns that a computer classroom can be very costly in
faculty time and institutional dollars (p. 308).
We, on the other hand, following Judge (1999) and Craft (2003), argue that Microsoft Excel offers
the
means
to
run
meaningful
Monte
Carlo
simulations
and
to
successfully
teach
students
basic
econometric
concepts,
at
a
relatively
low
opportunity
cost.
Cahill
and
Kosicki
(2000
p.
771)
offer
this
perfect summary of arguments for using Microsoft Excel:
From a practical perspective, spreadsheet software such as Excel is a natural choice to use in
exploring economic models because it is widely available on most campuses. This availability
eliminates the task of seeking funding for the purchase and support of specialized software
packages. In addition, spreadsheet software is relatively easy to use, and its flexibility makes it
useful
in
many
different
courses at all
levels of
the
traditional economics curriculum.
Most economics students almost certainly will use it after graduation in both career and personal
settings.
Most
important, it
minimizes black-box features that
characterize
much
computer-
assisted learning software.
Our paper differs from Judge (1999) and Craft’s (2003) in that we restrict ourselves to presenting
how Monte Carlo simulations can be run in Excel. Instructors will decide for themselves how to
incorporate
them
in
their
econometrics
courses.
Our
exposition
includes
many
screen
shots
and
provides
step-by-step
instructions
for
using
Excel
to
run
the
Monte
Carlo
simulation
exercises.
The first Monte Carlo simulation exercise we go through is a repeated sampling exercise showing
that least squares estimators are unbiased. The second one expands on the first to explain the true
meaning of confidence interval estimates of least squares estimators.
2. Repeated sampling and unbiasedness
Following Hill et al. (2011) and Briand and Hill (2012), the examples that follow are developed
around
the
idea
of
studying
the
relationship
between
household
weekly
income
and
their
corresponding weekly expenditure on food. We consider the experiment of randomly selecting
households from a population, and subsequently estimating the simple linear regression model:
y=b1+b2 x+e, where y represents weekly food expenditure and x represents weekly income. In aMonte Carlo experiment the repeated sampling properties of estimators and tests are observed
directly, by creating many samples of data, applying an estimator or test to each sample, recording the
outcomes, and then summarizing the outcomes. Samples are created using a specific data generating
process
(DGP ) by
which
we
create
a
sample
of
N
values.
The
ingredients
of
the
DGP
include
(i)
choosingthe
sample
size
N ,
(ii)
choosing
x
values
(which
may
be
fixed
in
repeated
samples,
or
not)
(iii)
selecting
parameter
values,
and
(iv)
randomly
selecting
values
of
the
regression
error
terms
e
from
a
probability
distribution with a given mean and variance. Given these elements we can ‘‘create’’ an outcome y via a
process that resembles a controlled experimental outcome.
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We will work with random samples of N =40 households: 20 of them with weekly income
x = $1000 and 20 of them with weekly income x = $2000. These values will remain fixed in repeated
samples.
To illustrate the unbiasedness of the least squares estimators, the actual values of the regression
parameters
b1 and b2, and the variance of the regression error s2 do
not
matter.
We
choose
b1=100and
b2=0.10. Then E ( y| x=1000)=$200, and E ( y| x=2000)=$300. In keeping with the assumptions of the normal linear regression model, the random errors e will be chosen to have independent normal
distributions with mean 0 and constant, homoskedastic, variance s2=2500. Because x is fixed inrepeated
samples,
the
distribution
of
y
is
normal
with
mean
E ( y| x=1000or2000)
and
variance
s2=2500. These distributions and the regression function E ( y| x)=b1+b2 x are shown in Fig. 1.The objective of the first Monte Carlo exercise is to show that if we draw many samples of size
N =40 using the specified data generation process, the average value of the least squares estimates b1and b2will be close to their true parameter values b1 and b2. If, for example, we use 1000 Monte Carlosamples,
then
the
sample
average
ð1=1000Þ P1000s¼1 b2s ¼ b2, where b2s is the least squares estimate of b2 in the s’th Monte Carlo sample, will be close to b2=0.10. The expected value of the least squaresestimator is based on an infinite number of repetitions. If we could compute an infinite number of least
squares estimates b1 and b2, their average value would equal their parameter values b1 and b2. AMonte Carlo simulation is only based on a finite number of repetitions, and thus the sample average b2will not exactly equal E (b2)=b2, however the sample average b2 will converge towards b2 as thenumber of Monte Carlo samples is increased.
To
begin
the
Monte
Carlo
simulation
exercise,
we
first
enter
the
following
labels,
values
andformulas
in
cells
A1:B3,1 D1:E3,
A5:B6
and
A26, as
shown
in
Table
1.
We
then Copy the cell reference from A6 into A7:A25 and the cell reference from A26 into A27:A45.
Table
1
Monte Carlo experiment parameters.
A B C D E
1 N= 40 s= 502 x 1= 1000 b1= 100
3 x 2= 2000 b2= 0.104
5 x y
6 =$B$2
. . . . . .
26 =$B$3
Fig.
1. Probability distribution functions of food expenditure given income level and linear relationship between expected food
expenditure and income.
1 A1:B3 refers to the range of cells between A1 and B3, inclusively. For more on Excel basic skills, please see Appendix A.
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Wehave
specified
the
sample
size
N , the
x
values
and
the
parameter
values
of
the
data
generating
process, b1, b2 and s. Next, we randomly select error terms e and generate a random sample of households’ food expenditure values, y’s.
2.1. Generating a random sample
We
use
Excel
functions2 NORMINV and RAND to generate random values using what is called the
‘‘inversion method.’’ This technique, briefly described in Appendix B,3 can be used to generate random
deviates from a probability distribution for a continuous random variable for which the cumulative
distribution function, or cdf , has an inverse. Let u be a uniformly distributed random variable on the
interval [0,1] and let F ( x) denote the cdf for a N (m,s2) random variable X , such that P ( X x)=F ( x). Then arandom
value
x
from
the
distribution
N (m,s2) is created by solving the equation u=F ( x) for x as x=F 1(u), where F 1 denotes the inverse function. In Excel the function RAND creates a uniform
random value, u, and NORMINV is the inverse function for the normal distribution, F 1. Thus by
nesting
these
two
functions
we
can
create
a
random
value
from
a
N (m,s2) distribution. The statisticalbasis for this result is discussed in Appendix B.
The general syntax of the NORMINV function is:
=NORMINV(probability, m, s) Fct. (1)
The NORMINV function computes the value x of a normally distributed variable X with mean m,
standard
deviation s, and with probability =P ( X x), where 0 probability 1.
We obtain random error values from the N (m,s2) distribution by specifying the probability argument of the NORMINV function to be the RAND function. The general syntax of the RAND function
is as follows:
=RAND() Fct. (2)
The RAND function4 returns a uniformly distributed random number u, with 0u
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drawn
from
the
probability
distribution
function
of
food
expenditure
for
income
level
x=$1000
(in
the
front in Fig. 1). The second group of points from the scatter plot (on the right in Fig. 2) was drawn from
the probability distribution function of food expenditure for income level x=$2000 (in the back inFig.
1).
The
fitted
regression
line,
which
runs
between
the
two
groups
of
points
in
Fig.
2, is
an
estimate
of the true regression line depicted in Fig. 1
Screen Shot 1:
Fig. 2. Scatter plot of random sample.
Table 2
Specifying the regression function.
B
6 =$E$2+$E$3*$B$2+NORMINV(RAND(),0,$E$1)
. . .
26
=$E$2+$E$3*$B$3+NORMINV(RAND(),0,$E$1)
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2.2. Generating least squares estimates
Next,
use
the LINEST function to obtain the least squares estimates for the intercept and slope
parameters, based on the random sample just drawn. For this purpose, the general syntax of the
LINEST function is as follows:
=LINEST(y’s, x’s) Fct. (4)
The
first
argument
of
the LINEST function specifies the y values, and the second argument specifies
the x values, the least squares estimates are based on. In our case, we specify7:
=LINEST(B6:B45,A6:A45) Fct. (5)
The LINEST function creates a table where the least squares estimates are stored in Excel, first the
slope coefficient estimate, and then the intercept coefficient estimate. The estimates are reported as
shown in Table 3.
We nest the LINEST function in the INDEX function to get the estimated coefficients, one at a time.
The INDEX function returns values from within a table. In the case of a table with only one row, the
INDEX function general syntax is as follows:
=INDEX(table of results, column_num) Fct. (6)
The first argument of the INDEX function specifies the source table. In our case, this is the table of
results generated by the LINEST function above. So, replace ‘‘table of results’’ by ‘‘LINE-
ST(B6:B5,A6:A5)’’. The second argument indicates from which column of the table to retrieve the
result of interest. If we want to retrieve the estimate of the intercept coefficient, b1, from the table
above,
we
would
indicate
that
it
can
be
found
in
column
2
by
replacing
‘‘column_num’’
by
‘‘2’’.
We
report
the
estimated
coefficients
at
the
bottom
of
our
worksheet.
In
cell A47:B48 enter the
labels and equations shown in Table 4.
The estimates of the intercept and slope coefficients we obtain from our sample of data are:
Screen Shot 2:
Table
4
How to report the estimates.
A B
47 b1= =INDEX(LINEST(B6:B45,A6:A45),2)
48 b2= =INDEX(LINEST(B6:B45,A6:A45),1)
Table 3
How LINEST reports parameter estimates.
column 1 column 2
row 1 b2 b1
7 Note that because we will nest the LINEST function in the INDEX function, we will effectively be working with regular
formulas, as opposed to the more complex array formulas. In addition, we choose to work with the LINEST function instead of
the SLOPE and INTERCEPT functions so we can also generate standard errors estimates (see Section 3.1).
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Recall
that
each
random
sample
is
different
and
will
yield
different
estimates,
which
may
or
may
not be close to the true parameter values. The property of unbiasedness is about the average values of
b1 and b2 if many samples of the same size are drawn from the same population. In the next section, we
thus
repeat
our
sampling
and
least
squares
estimation
exercise.
Screen Shot 3 presents the generation of a random sample and its least squares estimates—
accompanied by Table 5 of the formulas used and addresses where the formulas were copied to.8
Screen Shot 3:
2.3. Repeated sampling
We
would
like
to
draw
9
additional
random
samples.
For
that, Copy the formula from B6 into
C6:K25
and
the
formula
from
B26
into
C26:K45.Next, before copying the formula to obtain coefficient estimates for the new samples, transform
the Relative cell reference A6:A45 into an Absolute cell reference $A6:$A45—this retains the same
Table 5
Key cell formulas to report estimates.
Cell Formula Copied to
A6 =$B$2 A7:A25
A26
=$B$3
A27:A45B6 =$E$2+$E$3*$B$2+NORMINV(RAND(),0,$E$1) B7:B25
B26 =$E$2+$E$3*$B$3+NORMINV(RAND(),0,$E$1) B27:B45
B47 =INDEX(LINEST(B6:B45,A6:A45),2) –
B48 =INDEX(LINEST(B6:B45,A6:A45),1) –
8 This presentation is drawn from Ragsdale (2008) Managerial Decision Modeling textbook.
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x-values for the next 9 rounds of least squares estimations. Copy the formulas from B47:B48 into
C47:K48. In cells K50:K51, we compute the AVERAGEs of the estimates from the 10 samples by
entering the formulas shown in Table 6.
The estimates and average values that we obtain for the 10 samples are:
Screen
Shot
4:
Screen Shot 5 presents the repeated sampling of 10 random samples and their least squares
estimates—accompanied by Table 7 of the formulas used and addresses where the formulas were
copied to.
Screen Shot 5:
Taking the averages of estimates from many samples, the averages will approach the true
parameter values b1 and b2. To show that this is the case, we repeated the exercise again—and you
Table 6
Average the estimates.
A B K
47 b1= =INDEX(LINEST(B6:B45,$A6:$A45),2) . . .
48
b2=
=INDEX(LINEST(B6:B45,$A6:$A45),1) . . .
50 =AVERAGE(B47:K47)
51 =AVERAGE(B48:K48)
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could easily ask your students to do the same.9We encourage a reader who has so far been replicating
the Excel steps along with us to continue reading, and abstain for now from repeating the exercise. The
cell references that are given next in this paper assume that this is the case. In Table 8 are the average
values
of
b1 and b2 obtained when the number of samples is increased from 10 to 100, and finally to
1000.
Note that for 1000 random samples, or even 100, the average of the least squares estimates are veryclose
to
the
true
values.
3. Repeated sampling and interval estimation
In the second Monte Carlo exercise we will illustrate the meaning of 95% ‘‘level of confidence’’ in
interval
estimation.
A
95%
interval
estimator
is
bk t (0.975,N K )se(bk), where bk is the least squaresestimator of bk, t (0.975,N K ) is the 97.5 percentile from a t -distribution with N K degrees of freedom andse(bk) is the standard error of bk. In a large number of repeated samples from the same population, 95% of
interval estimates will contain the true underlying population parameter.
This
section
provides
step-by-step
instructions
for
constructing
a
Monte
Carlo
simulation
template
in Excel. If the following exercise were proposed to students, it would require of them to spend moretime familiarizing themselves with new Excel functions as well as figuring how to use them to
properly design a Monte Carlo simulation template. The authors believe that the time spent on those
two activities will enhance their grasp of basic econometric concepts as well as present an opportunity
for
them
to
refine
their
spreadsheet
software
skills.
This time, we would like to draw 90 additional random samples. Copy the formula from K6 into
L6:CW25 and the formula from K26 into L26:CW45.
3.1. The LINEST function revisited
The LINEST function will obtain the least squares estimates and their standard errors with one
additional
option.
The
general
syntax
of
the
LINEST
function
is:
Table 7
Key cell formulas: 10 random samples.
Cell Formula Copied to
A6 =$B$2 A7:A25
A26
=$B$3
A27:A45B6 =$E$2+$E$3*$B$2+NORMINV(RAND(),0,$E$1) B7:B25, C6:K25
B26 =$E$2+$E$3*$B$3+NORMINV(RAND(),0,$E$1) B27:B45, C26:K45
B47 =INDEX(LINEST(B6:B45,$A6:$A45),2) C47:K47
B48 =INDEX(LINEST(B6:B45,$A6:$A45),1) C48:K48
L47 =AVERAGE(B47:K47) –
L48
=AVERAGE(B48:K48)
–
Table 8
Monte Carlo average values.
Number of samples 10 100 1000 Parameter Values
Average value of b1 89.87141 97.52929 99.09254 100 Average value of b2 0.105656 0.101557 0.100546 0.1
9 As an alternative to the method used here, see Barreto and Howland (2005) add-in: http://www3.wabash.edu/
econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/MCSim.htm.
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=LINEST(y’s, x’s,,TRUE) Fct. (7)
The first argument of LINEST function specifies the y values; the second argument specifies the x
values; we ignore the third argument by putting a spacebetween the second and third commas; and the
fourth
argument, TRUE, indicates that we would like LINEST to return additional regression statistics.
The LINEST function creates a table where it stores the least squares and standard errors estimates.
The order in which they are reported is shown in Table 9.
We nest the LINEST function in the INDEX function to get the estimated coefficients, one at a time.
The INDEX function returns values from within a table. The INDEX function general syntax is as
follows:
=INDEX(table of results, row_num, column_num) Fct. (8)
The first argument of the INDEX function specifies the source table. The second argument and third
argument indicate the intersection of a row and a column at which the result of interest can be found.
The nested commands are:
b1: =INDEX(LINEST(y-values,x-values, TRUE),1,2) Fct. (9)
se(b1): =INDEX(LINEST(y-values,x-values, TRUE),2,2) Fct. (10)
b2:
=INDEX(LINEST(y-values,x-values,
TRUE),1,1)
Fct.
(11)se(b2): =INDEX(LINEST(y-values,x-values, TRUE),2,1) Fct. (12)
3.2. The simulation template
The
template
shown
in
Table
1010 reports
estimated
coefficients,
standard
errors,
t -percentile
values
and
limits
of
the
interval
estimates
(Lower
Limit:
LL
and
Upper
Limit:
UL). Next,
we
count
how
many of the 100 interval estimates contain the true parameters’ values. Finally, we compute summary
statistics for our estimated coefficients and standard errors. Specify cells A47:B68 as shown in Table 10
(some cells are outlined in different shades of gray only to distinguish groups of similar or related cells
which
we
comment
on
shortly).In
cells A47:B48, the sample size (N ) and a value are specified, for a 100(1a)% confidence
interval. The t -distribution degrees of freedom (d.f.) and t -percentile value (t c ) are computed and
reported in cells A49:B50.
Cells A51:B52 and A60:B61 are used to report and compute coefficient estimates (bk, k=1,2) and
standard errors (se(bk), k=1,2). In the formula typed in cells B51:B52 and B60:B61, the cell references
to the x values are in Absolute format, $A6:$A45, as opposed to Relative format, as we will be using
the same x values for all 100 repetitions.
Cells A53:B54 and A62:B63 are used to compute and report interval estimates. The t -critical value
t c will be the same over all repetitions, so its cell reference in the formulas of the intervals limits is
specified
in Absolute format, $B$50.
In
cells
A55:B55
and
A64:B64, we
establish
and
report
whether
the
true
parameters’
values
arecontained in the interval estimates. In cells A56:B56 and A65:B65, we keep track of the number of
interval estimates that do contain the true parameters values.
Table 9
How LINEST reports estimates and standard errors.
column 1 column 2
row 1 b2 b1
row
2
se(b2)
se(b1)
10 Available from the authors upon request: [email protected].
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In cells A57:B58 and A66:B67 we compute and report the average and standard deviation of the
100
estimates
of
the
intercept
and
slope
coefficients.
The
average
of
the
parameter
estimates
should
be
close
to
the
true
parameter
values
because
the
least
squares
estimator
is
unbiased.
The
standard
deviations of the estimates are the actual finite sample variability of estimates in the Monte Carlo
experiment. In cells A59:B59 and A68:B68 we compute and report the average of the 100 standard
errors of the intercept and slope coefficients, which should be close to the estimates’ standard
deviations. This provides an opportunity to remind students that standard errors reflect the sampling
variability
of
the
estimates.
3.3. The TINV function
The
TINV
function
returns
100(1a/2) percentile values for a t -distribution (t -critical values) withspecified degrees of freedom. The syntax of the TINV function is:
=TINV(a, degrees of freedom) Fct. (13)
where a is the two-tail probability.
3.4. The IF function
Use the IF and OR logical functions to indicate, for each interval estimate, whether or not it contains
the true parameter value. The general syntax for the IF function is
IF(logical_test,value_if_true,value_if_false) Fct. (14)
where:
Logical_test is any value or expression that can be evaluated to be TRUE or FALSE. In this
exercise we want to determine whether or not the true parameter value, bk, is within theestimated interval [LL,UL], where LL = bk t c se(bk) and UL= bk+ t c se(bk). The logical expression we
Table 10
The simulation template.
A B
47 N = 40
48
a
=
0.0549 d.f. = =B47-2
50 tc = =TINV(B48,B49)
51 b1 = =INDEX(LINEST(B6:B45,$A6:$A45, TRUE),1,2)
52 se(b1) = =INDEX(LINEST(B6:B45,$A6:$A45, TRUE),2,2)
53 LL = =B51-$B$50*B52
54 UL = =B51+$B$50*B52
55 b1 in CI =IF(OR(100B54),‘‘No’’, ‘‘Yes’’)56 Yes’ =COUNTIF(B55:CW55, ‘‘Yes’’)
57 average b1’s = =AVERAGE(B51:CW51)
58 std. dev. (b1’s) = =STDEV(B51:CW51)
59 average se(b1)’s = =AVERAGE(B52:CW52)
60 b2 = =INDEX(LINEST(B6:B45,$A6:$A45, TRUE),1,1)
61 se(b2) = =INDEX(LINEST(B6:B45,$A6:$A45, TRUE),2,1)
62 LL = =B60-$B$50*B6163 UL = =B60+$B$50*B61
64 b2 in CI =IF(OR(0.1B63),‘‘No’’, ‘‘Yes’’)
65 Yes’ =COUNTIF(B64:CW64, ‘‘Yes’’)
66 average b2’s = =AVERAGE(B60:CW60)
67 std. dev. (b2’s) = =STDEV(B60:CW60)
68 average se(b2)’s = =AVERAGE(B61:CW61)
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use is: if
bkUL. If bk is outside [LL,UL], then this expression is TRUE. Otherwise, theexpression is FALSE.
Value_if_true is the value that is returned if logical_test is TRUE. For example, if this argument is
the text string ‘‘No’’ and the logical_test argument is TRUE, then the IF function displays the text ‘‘No’’.
Value_if_false is the value that is returned if logical_test is FALSE. For example, if this argument is the
text
string
‘‘Yes,’’
and
the logical_test argument is FALSE, then the IF function displays the text ‘‘Yes’’.
3.5. The OR function
Use the OR function to write the logical_test. The general syntax of the OR function is
OR(argument_1,argument_2) Fct. (15)
If the first logical expression, argument_1, or the second logical expression, argument_2, is TRUE,
then
the OR function returns TRUE. It returns FALSE only if both arguments are FALSE.
The
general
syntax
for
the
OR
function,
nested
in
the
IF
function,
is:
IF(OR(argument_1,argument_2),value_if_true,value_if_false) Fct. (16)
Applied to our exercise, the nested function looks like this (which is what we have in cells B55 and
B64):
IF(OR(bk UL),‘‘No’’,‘‘Yes’’) Fct. (17)
If
bk is outside [LL,UL], then the logical_test bkUL is TRUE, and ‘‘No’’ is returned toindicate that bk is not in the estimated confidence interval. Otherwise, the logical expression is FALSE,and ‘‘Yes’’ is returned to indicate that bk is in the estimated confidence interval.
3.6. The COUNTIF function
Finally, we use the COUNTIF function to count the number of times bk is within the intervalestimate
[LL,UL]. The COUNTIF function is a statistical function that counts the number of cells within
a
range
that
meet
a
given
criteria.
Its
general
syntax
is:
COUNTIF(cell_range,criteria) Fct. (18)
Cell_Range
is
one
or
more
cells
to
count.
Criteria
is
the
number,
expression,
cell
reference,
or
textthat defines which cells will be counted. Since we are interested in counting how many interval
estimates, among all the ones we construct, actually contain the true parameter value, we count the
‘‘Yes’’ that are generated following the application of our logical_test (this is what we do in cells B56
and B65):
COUNTIF(cell_range,‘‘Yes’’) Fct. (19)
Review
with
students
the
meaning
of
the
formulas
and
values
in B47:B68. Copy the content of
B51:B55 to C51:CW55 and copy the content of B60:B64 to C60:CW64.
3.7.
Results
Screen Shot 6 presents the repeated sampling of 100 random samples and 95% confidence interval
estimates—the accompanying table of the formulas used and addresses where the formulas were
copied to can be found in Appendix C.
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Screen
Shot
6:
Wefind that 94 out of our 100 confidence intervals contain b1, and 95 out of 100 interval estimatescontained b2 (see Screen Shot 6). Note that each replication will result in different random samples,different interval estimates and thus a different number of intervals that will contain the true
parameters values.
With our simulation of 1000 samples, we find that 954 out of 1000 confidence intervals contained
the
true
parameter
value,
both
for
the
intercept
and
slope
coefficients.
With
our
simulation
of
10,000samples
we
find
that
95%
of
both
the
intercept
and
slope
coefficients
interval
estimates
contained
the
true
parameters
values.
Finally by computing the summary statistics of the Monte Carlo estimates we can illustrate that the
standard errors se(bs,k), s=1, . . ., S and k=1,2, measure the sampling variation in the estimates bs,k. The
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average
of
the
standard
errors
for
each
coefficient
is
close
to
the
standard
deviation
of
S
estimates
of
each coefficient (B59 versus B58, and B68 versus B67 on Screen Shot 6).
3.8. Extensions
The
exercise
developed
thus
far
can
be
modified
or
extended
to
investigate
additional
concepts.
Some
suggestions
are:
Effect of changes in variation of the x values on se(bk); Effect
of
random
errors
from
alternative
distributions
on
the
sampling
distribution
of
the
bkestimators, as shown in Appendix B;
Effect of rescaling x and/or y values on bk estimates; and Effect of changes in sample size on se(bk).
4. Conclusion
This paper presented a step-by-step guide to Microsoft Excel fundamentals and two Monte Carlo
simulations exercises. The first Monte Carlo simulation is a repeated sampling exercise showing that
least squares estimators are unbiased. The second one expands on the first to explain the true meaning
of confidence interval estimates of least squares estimators. In these two exercises, the sampling
distribution
concept
is
presented
in
the
context
of
the
regression
analysis,
as
Kennedy
(2001)
suggests
it should be introduced.
The use of Excel requires students to understand what they are doing. Wehope that the experience
of
using
Excel
in
an
undergraduate
econometrics
course
will
be
akin
to
that
of
programming
procedures in a graduate econometrics text such as in Mittelhammer et al. (2000 p.713). The way we
suggest to use Excel in an undergraduate econometrics course is different than what Barreto and
Howland (2006) propose. They ‘‘use Excel workbooks powered by Visual Basic macros’’ that ‘‘enable
Monte Carlo simulations to be run by students with a click of a button’’ (p. i). While Barreto andHowland
use
of
Excel
would
seem
paramount
to
what
Day
(1987)
refers
to
as
a
‘‘canned
program’’
in
the
context
of
macro-economic
simulation
exercises,
ours
would
rather
presents
the
advantages
of
a
‘‘student-built model’’ (p. 351), giving students active learning opportunities.
Becker and Greene (2001) point out that ‘‘the starting point for any course in statistics and
econometrics is the calculation and use of descriptive statistics and mastery of basic spreadsheets
skills (emphasis added)’’ (p. 173). Some students are still not familiar with Excel, and a few are even
reluctant to use it; but at the end of a course using Excel to teach undergraduate econometrics,
students and instructors alike, come out of the experience with a stronger understanding of core
econometrics concepts and with better Excel skills.
Acknowledgement
We are grateful to anonymous referees for helpful comments and suggestions.
Appendix A. Microsoft Excel fundamentals (Using Excel 2007)
A.1. Starting Excel
Find the Excel shortcut on your desktop and double click on it to start Excel (left clicks).
Screen Shot A1:
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Alternatively,
left-click
the
Windows
Start
button
at
the
bottom
left
corner
of
your
computer
screen.
Screen Shot A2:
Slide
your
mouse
over All programs, Microsoft Office, and finally Microsoft Office Excel 2007.
Left-click on the last one to start Excel. To create a shortcut instead, right-click on it; slide the mouse
over Send to, and then select (i.e. drag the mouse over and left-click on) Desktop (create shortcut).
From
the
Windows Start button, an easier way yet to start Excel is to select Run, type Excel in the
Open window of the Run dialog box and select the OK button or simply press your Enter key.
Screen Shot A3:
Excel
opens
to
a
new
file,
titled
Book1.
Find
the
name
of
the
open
file
on
the
very
top
of
the
Excel
window,
on
the Title bar . An Excel file like Book1 contains several sheets. By default, Excel opens to
Sheet1 of Book1. Determine which sheet is open by looking at the Sheet tabs found in the lower left
corner of the Excel window.
Your screen might look slightly different than the one shown below. If your computer screen is
bigger, Excel will automatically display more of its available options. For example, in the Styles group
of
command,
instead
of
the Cell styles button, there might be a colorful display of cell styles.
Screen Shot A4:
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The
Active cell
is
surrounded by a
border and is
in
Column
A
and
Row
1; its
Cell
reference is
A1. Below the title bar is a Tab list. The Home tab is the one Excel opens to. Under each tab we
find groups of commands. Under the home tab, the first one is the Clipboard group of commands,
named after the tasks it relates to. The wide bar including the tab list and the groups of commands
is referred to as the Ribbon. The content of the Active cell shows up in the Formula bar (as shown
above,
there is
nothing
in
it). Perhaps
themost important
of
all of
this is
to
locate
the Help button
on
the
upper
right
corner of
the Excel
window.
Finally, we
can use the Scroll bars and the arrows
around them to navigate up-down and right-left in the worksheet. And we potentially have a long
way to go: each worksheet in Microsoft Excel 2007 contains 1,048,576 rows and 16,384
columns!!!!
A.2. Entering data and carrying out calculations
To enter labels and data into an Excel worksheet move the cursor to a cell (i.e. drag our mouse over
and left-click on) and begin typing. Type X in cell A1, then press the Enter key to get to cell A2 or
navigate
by
moving
the
cursor
with
the
mouse,
or
use
the
Arrow
keys
(to
move
right,
left,
up
or
down).Fill
in
the
rest
of
the
worksheet
as
shown
below
in
Table
A1.
Excel’s primary usefulness is to carry out repeated calculations. We can add, subtract, multiply and
divide; and apply mathematical and statistical functions to the data in a worksheet. To illustrate, we
are
going
to
compute
the
squares
of
the
numbers
we
just
entered.
There
are
two
main
ways
to
perform
calculations
in
Excel.
One
is
to
write
formulas
using
arithmetic
operators,
which
we
demonstrate
below; the other is to write formulas using mathematical functions—these and other functions will be
used in the main section of this paper.
Place the cursor in cell B2. We want to compute the square of the value from cell A2. Let us
emphasize that the trick to using Excel efficiently is NOT to re-type values already stored in the
worksheet,
but
instead
to
use
references
of
cells
where
the
values
are
stored.
So,
to
compute
the
square
of 10, which is the value stored in cell A2, instead of typing the formula =10*10, type the formula
=A2*A2 or =A2^2 (the asterisk and the caret can be typed by simultaneously pressing the Shift key and
the * or ̂ key) as shown in Table A2.
Press Enter . Note that: (1) a formula always starts with an equal sign; this is how Excel
recognizes it is a formula, and (2) formulas are not case sensitive, so we could also have typed
=a2^2 instead.
The way Excel understands the instructions we gave in cell B2 is ‘‘square the value found at the
address A2’’. It is important to fully understand how Excel interprets ‘‘address A2’’. To Excel ‘‘address
A2’’ means ‘‘from where you are at, go left by one cell’’—because this is where A2 is located vis-à-vis
B2. In other words, an address gives directions: left-right, up-down, and distances: number of cells
away—all in reference to the cell where the formula is entered.
Table
A2
Carrying out calculations.
A B
1 x y
2 10 =A2^2
3 20
Table
A1
Entering data.
A B
1 x y
2 10
3 20
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To
copy
the
formula
in
cell B2 to cell B3 place the cursor back into cell B2, and move it to the south-
east corner of the cell, until the fat cross turns into a skinny one, as shown below:
Screen Shot A5:
Then left-click, hold it, drag it down to the next cell below, and release!
Excel has copied the formula we typed in cell B2 into the cell below. The formula entered in cell B2
instructed Excel to collect the value stored one-cell away from its left, and then square it—those exact
same instructions are now found in cell B3. Place your cursor back into B3, and look at the Formula
bar .
You
can
see
that,
in
this
cell,
these
same
instructions
translate
into
‘‘=A3^2’’,
and
the
value
computed
is
thus 400.
Screen Shot A6:
Now, if we had wanted the values in B2 and B3 to be equal to the square of the value stored in cell
A2, then we would transform the cell reference in cell B2 from a Relative Cell Reference into an
Absolute Cell Reference, before copying it to cell B3. This ensures that an address does NOT change
when
we
copy
a
formula
from
one
cell
to
another.
A Relative Cell Reference is made into an Absolute Cell Reference by preceding both the row
and column references by a dollar sign. Place the cursor back in cell B2 (i.e. move the mouse over
and left-click), and in the Formula bar , place the cursor before the A, insert a dollar sign (by
pressing the Shift key and the $ key at the same time); then move the cursor before the first 2 and
insert another dollar sign; and finally, place the cursor at the end of the formula and press Enter .
See
Table A3.
Copy the formula from cell B2 to cell B3. Place the cursor back into B3, and look at the Formula bar .
You can see that, this time, the formula is still =$A$2^2—which means the instruction (or address)
given
to
Excel
has
not
changed.
The
value
computed
is
thus 100.
Appendix B. The inversion method for generating random deviates
Using the inversion method is an opportunity to explain to students the usefulness of change-of-
variable
techniques
and
cumulative
distribution
functions.
Table A3
Absolute cell reference.
A B
1 x y
2 10
=$A$2^23 20
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B.1.
Distributions
of
functions
of
random
variables
Let X be a continuous random variable with probability density function, pdf , f ( x). Let Y = g ( X ) be a
function that is strictly increasing or strictly decreasing. This condition ensures that the function is
one-to-one,
so
that
there
is
exactly
one
Y
value
for
each
X
value,
and
exactly
one
X
value
for
each
Y value.
The
importance
of
this
condition
on
g ( X )
is
that
we
can
solve
Y = g ( X )
for
X .
That
is,
we
can
find
an
inverse function X =w(Y ). Then the pdf for Y is given by
hð yÞ ¼ f ½wð yÞ dwð yÞdy
(A1)
where j j denotes the absolute value. This is called the change-of-variable technique.11 As an example,let X be a continuous random variable with pdf f ( x)=2 x for 0
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with
1
degree
of
freedom,
implying
that
it
has
mean
1
and
variance
2.
For
the
regression
random
errors to have mean 0 and variance s2 use e ¼ s ð y2 1Þ= ffiffiffi 2
p Extreme value random errors. The extreme value density is
f ð yÞ ¼ expð yÞ expðexpð yÞÞ (A2)
which
has
cdf
F ( y)=exp(exp( y)). Despite its imposing form, we can obtain random values fromthis distribution using the inversion method, since y=F 1(u)= ln( ln(u)) where u is a uniformlydistributed
random
value.
The
mean
and
variance
of
this
distribution
are
0.57722
and p2/6.13 For the
regression random errors to have mean 0 and variance s2 use e ¼ s ð y 0:57722Þ= ffiffiffiffiffiffiffiffiffiffiffi p2=6
p
Random
values
from
many
distributions
can
be
similarly
formed.
Books
of
statistical
distributions
provide ‘‘formulas’’ for generating random deviates given uniform random values.
With Monte Carlo studies based on these alternative non-normal distributions, it is easy to
illustrate
using
a
histogram
how
effective
the
Central
Limit
Theorem
can
be,
resulting
in
nearly
normally distributed least squares estimators in even moderately sized samples. Furthermore the
performance
of
interval
estimators,
or
hypothesis
tests,
can
be
studied
under
these
alternativeassumptions as we have illustrated in Section 3.
Appendix C. Repeated sampling and interval estimation
Screen Shot 6 accompanying table of the formulas used and addresses where the formulas were
copied to.
Key cell formulas
Cell formula Copied to
A6 =$B$2 A7:A25
A26 =$B$3 A27:A45B6 =$E$2+$E$3*$B$2+NORMINV(RAND(),0,$E$1) B7:B25, C6:CW25
B26 =$E$2+$E$3*$B$3+NORMINV(RAND(),0,$E$1) B27:B45, C26:CW45
B49 =B47-2 –
B50 =TINV(B48,B49) –
B51 =INDEX(LINEST(B6:B45,$A6:$A45,TRUE),1,2) C51:CW51
B52 =INDEX(LINEST(B6:B45,$A6:$A45,TRUE),2,2) C52:CW52
B53 =B51-$B$50*B52 C53:CW53
B54 =B51+$B$50*B52 C54:CW54
B55 =IF(OR(100B54),‘‘No’’,‘‘Yes’’) C55:CW55
B56 =COUNTIF(B55:CW55, ‘‘Yes’’) –
B57 =AVERAGE(B51:CW51) –
B58 =STD(B51:CW51) –
B59 =AVERAGE(B52:CW52) –
B60 =INDEX(LINEST(B6:B45,$A6:$A45,TRUE),1,1) C60:CW60
B61 =INDEX(LINEST(B6:B45,$A6:$A45,TRUE),2,1) C61:CW61
B62 =B60-$B$50*B61 C62:CW62
B63 =B60+$B$50*B61 C63:CW63
B64 =IF(OR(0.1B63),‘‘No’’, ‘‘Yes’’) C64:CW64
B65 =COUNTIF(B64:CW64, ‘‘Yes’’) –
B66 =AVERAGE(B60:CW60) –
B67 =STD(B60:CW60) –
B68 =AVERAGE(B61:CW61) –
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