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ECOM30001/ECOM90001: Basic Econometrics
Creation date：2024-05-10 17:08:26
Belonging category：经济Economics
Basic Econometrics
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ECOM30001/ECOM90001: Basic Econometrics

This tutorial reviews some concepts using the econometrics software package R. Specifi-
cally, the tutorial reviews:
- methods for estimating and interpreting the Linear Probability Model (LPM) in R
- methods for estimating and interpreting the Probit Model in R
- calculating marginal (partial) effects for the Probit Model in R
This tutorial requires one (1) data file:
- tut12 churn.csv
This file can be obtained from the Canvas subject page.
In addition, the R script file tut12.R provides the program code necessary to complete
the tutorial. This R script file uses the following package(s) which need to be installed
prior to running the R script file:
stargazer : for easily producing output tables in R
ggplot2 : for easily producing graphs in R
car : for easily conducting hypothesis tests in R
lmtest : for easily conducting the Ramsey RESET test in R
sandwich : for easily calculating robust (Huber-White) heteroskedasticty consistent
standard errors in R
margins : for easily calculating marginal effects in R
These can be installed directly in RStudio from the packages tab or by using the com-
mand install.packages() and inserting the name of the package in the brackets.
1
Question 1
Suppose you have been engaged as a consultant for a large telecommunications provider.
Your current role is to investigate the characteristics of customers who decide to switch
to another competitor.
Consider the following econometric model for customer churn in a clearly defined narrow
geographic segment of the market:
churn∗i = β0 + β1 partneri + β2 dependentsi + β3 tenurei + β4 tenure
2
i
+ β5 monthlychargesi + β6 contract2i + β7 contract3i + β8 paperlessi + εi (1)
where ε|Xi ∼ N (0, σ2ε) and:
churn∗ = latent variable determining whether a customer switched
providers in the last month
partner = 1 if customer has a partner, 0 otherwise
dependents = 1 if customer has dependents, 0 otherwise
tenure = number of months customer has stayed, 0 otherwise
monthlycharges = monthly account charges, in dollars
contract1 = 1 if customer does not have a contract (‘month-to-month’), 0 otherwise
contract2 = 1 if customer has a 12-month contract, 0 otherwise
contract3 = 1 if customer has a 24-month contract, 0 otherwise
paperless = 1 if customer receives bills electronically, 0 otherwise
Note that contract1 is the omitted category.
a) Provide a brief interpretation of the latent variable churn∗
Solution: Suppose, each month, customers weigh up the advantages and disadvan-
tages of remaining with the provider. They would trade-off the cost of the service,
against the quality and reliability of the service. In principle, customers (implicitly)
calculate a score that determines their decision to remain with the provider. Let
this ‘customer dissatisfaction’ score be denoted by churn∗i . Customers that perceive
the benefits of remaining with the provider exceed the disadvantages will choose to
remain with the provider for another month (churn∗i sufficiently low). Similarly, cus-
tomers that perceive the benefits of remaining with the provider do not exceed the
disadvantages will choose to switch providers (churn∗i sufficiently high). There will
be some implied threshold level where customers are indifferent to remaining with
the provider, or switching providers. For example, there might be some switching
rule as follows:
churni =
{
1 if churn∗i ≥ H
0 if churn∗i < H
where H represents the threshold ‘cut-off’ level for switching providers, based upon
2
the sufficiently high level of the ‘customer dissatisfaction’ variable churn∗i .
b) The data set tut12 churn contains 7, 043 observations on the population of cus-
tomers for this large telecommunications provider in a specific geographic segment.
This data contain the following indicator variable:
churni =
{
= 1 if churn∗i ≥ 0
= 0 if churn∗i < 0
This suggests the following econometric model:
churni = β0 + β1 partneri + β2 dependentsi + β3 tenurei + β4 tenure
2
i
+ β5 monthlychargesi + β6 contract2i + β7 contract3i + β8 paperlessi + εi
(2)
3
Estimate model (2) by Ordinary Least Squares (OLS).
Solution: The estimation results are provided in Figure 1:
i) What is the interpretation of your estimate for β5? What is your interpreta-
tion of your estimate of β8?
Solution: The econometric model (2) is a linear model. Recall also that for
a binary dependent variable:
E[churni|Xi] = Pr[churni = 1|Xi]
so the interpretation of β5: the effect on Pr[churni = 1|Xi] associated with
a one dollar increase in the monthly charges, holding all other variables con-
stant. Note that the magnitude of the marginal effect is in terms of probability
points—a one dollar increase in monthly charges, holding all other variables
constant, changes the probability of churn by 100 ∗ β5 probability points. The
results in column (1) of Figure 1 provide the estimate b5 = 0.0032 so a one dol-
lar increase in monthly charges, holding all other variables constant, changes
the probability of churn by 0.32 probability points.
The parameter β8 is the coefficient attached to the paperless indicator vari-
able. Consequently, it has the ‘usual’ interpretation for a linear model:
β8 = Pr[churni = 1|paperlessi = 1,Xi]− Pr[churni = 1|paperlessi = 0,Xi]
so the interpretation of β8 is the difference in Pr[churni = 1|Xi] for customers
who receive electronic bills, relative to customer who do not receive electronic
bills, holding all other variables constant. Note that the magnitude of the
marginal effect is in terms of probability points—holding all other variables
constant, the probability of churn for customers who receive electronic bills
is 100 ∗ β8 different to that for customers who do not receive electronic bills.
The results in column (1) of Figure 1 provide the estimate b8 = 0.0717 so,
holding all other variables constant, customers who receive electronic bills,
have a probability of churn that is higher by 7.17 probability points, relative
to customers who receive paper bills.
ii) At the 5% level, test for the presence of heteroskedasticity using White’s test.
Since there are numerous indicator variables in model (2), use the ‘no cross
products’ form of White’s test. Is there any evidence of heteroskedasticity?
Solution: The OLS estimation results are presented in column 1 of Figure 1
The null hypothesis is that the errors are homoskedastic against the alternative
hypothesis that the errors are heteroskedastic. The test statistic is calculated
from the auxiliary regression:
eˆ2i = α0 + α1 partneri + α2 dependentsi + α3 tenurei + α4 tenure
2
i
+ α5 monthlychargesi + α6 contract2i + α7 contract3i + α8 paperlessi
+ α9 tenure
4
i + α10 monthlycharges
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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