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.

G. Briand, R.C. Hill / International Review of Economics Education 12 (2013) 60–79 61


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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.

G. Briand, R.C. Hill / International Review of Economics Education 12 (2013) 60–7962


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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.


mailto:[email protected]:[email protected]:[email protected]:[email protected]


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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].


mailto:[email protected]:[email protected]:[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)



12/20

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.



13/20

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



14/20

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:



15/20

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:



16/20

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



17/20

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



18/20

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


19/20

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


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) –




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’’) –


B67 =STD(B60:CW60) –


References

Barreto, H., Howland, F.M., 2005. Excel Add-Ins for Introductory Econometrics using Monte Carlo Simulation with MicrosoftExcel. Cambridge University Press, New York Available at: http://www3.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/index.htm.

13 For example, Forbes et al. (2011).


http://www3.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/index.htmhttp://www3.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/index.htmhttp://www3.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/index.htmhttp://www3.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/index.htm


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Available

at:http://en.wikipedia.org/wiki/Random_number_generation.

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