Introduction to Econometrics
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Instructions to candidates
Indicative time to complete: 4 hours
You must submit your answers in a single Word or PDF document via the Assignment Submission link provided on 212ECN Introduction to Economics 2021JANMAY Virtual Learning Environment page (i.e. AULA) in the Assessment folder.
Answer any 2 questions out of 4
All questions carry equal marks
Non programmable calculators may be used
Statistical tables are provided
The variables to be used in Question 1 are:
Coke – Dummy variable which takes the value of 1 if Coke is chosen and value of 0 if Pepsi is chosen
Pr_pepsi – Price of Pepsi
Pr_coke – Price of Coke
Disp_pepsi – Dummy variable which takes the value of 1 if Pepsi is displayed at the time of purchase and 0 otherwise
Disp_coke – Dummy variable which takes the value of 1 if Coke is displayed at the time of purchase and 0 otherwise
Pratio – Price of Coke relative to price of Pepsi
Table I
Model 1: Logit, using observations 1-1140
Dependent variable: coke
Standard errors based on Hessian5
Coefficient Std. Error z Slope*
const 0.0486209 1.24674 0.03900
pr_pepsi 0.830053 0.931183 0.8914 0.204315
pr_coke 0.361987 0.975300 0.3712 0.0891019
disp_pepsi −0.500906 0.179637 −2.788 −0.121531
disp_coke 0.642171 0.180811 3.552 0.158037
pratio −1.74091 1.18131 −1.474 −0.428518
Mean dependent var 0.447368 S.D. dependent var 0.497440
McFadden R-squared 0.102718 Adjusted R-squared 0.095064
Log-likelihood −703.3436 Akaike criterion 1418.687
Schwarz criterion 1448.920 Hannan-Quinn 1430.104
*Evaluated at the mean
Number of cases 'correctly predicted' = 767 (67.3%)
f(beta'x) at mean of independent vars = 0.497
Likelihood ratio test: Chi-square(5) = 161.033 [0.0000]
Predicted
0 1
Actual 0 505 125
1 248 262
Data represent a random selection of 1140 individuals deciding between
Coke and Pepsi.
Table II
Table II presents the restricted version of the Model 1 in Table I.
Model 2: Logit, using observations 1-1140
Dependent variable: coke
Standard errors based on Hessian
Coefficient Std. Error z Slope*
const −0.211309 0.0595658 −3.547
Mean dependent var 0.447368 S.D. dependent var 0.497440
McFadden R-squared 0.000000 Adjusted R-squared NA
Log-likelihood −783.8603 Akaike criterion 1569.721
Schwarz criterion 1574.759 Hannan-Quinn 1571.623
*Evaluated at the mean
Number of cases 'correctly predicted' = 630 (55.3%)
f(beta'x) at mean of independent vars = 0.497
Predicted
0 1
Actual 0 630 0
1 510 0
Some information:
We should rather focus on sign of the coefficient
An increase in x increases/decreases the likelihood that y=1 (makes the outcome MORE/LESS LIKELY). In other words, an increase in x makes the outcome of 1, MORE or LESS LIKELY.
The slope is the marginal effect
Marginal effects reflect the change in the probability of y=1 given a unit change in an independent variable, x. The marginal effect depends on x so we need to estimate effects at a specific value of x.
****************************************************************
Question 1
i. Estimate by how much the probability of choosing Coke changes based on Model 1 if we assume that both Coke and Pepsi are displayed at the time of purchase, Pratio is 1, Pepsi price is $2.20 and Coke price changes from $2.20 to $2.15. Please refer to Table I.
[20 marks]
LOGIT Model:
Calculate z1 for the case both Coke and Pepsi are displayed, Pratio is 1, Pepsi price is $2.20 and Coke price is $2.20:
P_1=Prob(D_1=1)=1/(1+e^(-z_1 ) )=e^(z_1 )/(1+e^(z_1 ) ) (1)
π§1 = b1 + b2pr_pepsi + b3pr_coke + b4disp_pepsi + b5disp_coke + b6pratio = 0.0486209 + 0.830053*2.20 + 0.361987*2.20 − 0.500906*1 + 0.642171*1
− 1.74091*1 = 1.0714639
π1 = ππππ(Coke1 = 1) = 1/(1+e^(-(1.0714639)) ) = 0.7449=74.49%
There is a 74.49% of probability that an individual will choose Coke over Pepsi if both Coke and Pepsi are displayed at the time of purchase, Pratio is 1, Pepsi price is $2.20 and Coke price is $2.20.
Calculate z2 for the case both Coke and Pepsi are displayed, Pratio is NOW 0.978, Pepsi price is $2.20 and Coke price is $2.15:
Pratio: Price of Coke relative to price of Pepsi: 2.15/2.20=0.978
P_2=Prob(D_2=1)=1/(1+e^(-z_2 ) )=e^(z_2 )/(1+e^(z_2 ) ) (2)
π§2 = b1 + b2pr_pepsi + b3pr_coke + b4disp_pepsi + b5disp_coke + b6pratio = 0.0486209 + 0.830053*2.20 + 0.361987*2.15 − 0.500906*1 + 0.642171*1
− 1.74091*0.978 = 1.09166457
π2 = ππππ(Coke2 = 1) = 1/(1+e^(-(1.09166457)) ) = 0.7487=74.87%
There is a 74.87% of probability that an individual will choose Coke over Pepsi if both Coke and Pepsi are displayed at the time of purchase, Pratio is 0.978, Pepsi price is $2.20 and Coke price is $2.15.
Absolute Change:
βπ = π2 − π1 = 0.7487 – 0.7449 = 0.0038
Answer is 0.38π. π. (not %)
Probability of choosing Coke is increased by 0.38 percentage points if the price of Coke declines from $2.20 to $2.15
Relative Change:
βπ =(π2 − π1)/ π1 =(0.7487− 0.7449)/ 0.7449 = 0.0051
Answer is 0.51%.
Probability of choosing Coke increases by 0.51% if the price of Coke changes from $2.20 to $2.15
ii. Is the coefficient on pr_coke variable in Model 1 statistically significant? This requires you to: state the null and alternative hypotheses used, determine a critical value at the 5% significance level; calculate z statistic and present your decision. Please refer to Table I.
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