Question

Let x and y be two dummy variables that take the values of either 0 or 1, and follow the bivariate frequency distribution as given below. If a logit regression is estimated with y as the dependent variable and x as the independent variable, then the estimated coefficient of x is _____ (rounded off to two decimal places)

x	0	1	Total
y
0	6	11	17
1	6	7	13
Total	12	18	30

Answer 1

Correct Answer: -0.49

To solve this problem, we need to estimate the coefficient of x in a logit regression where y is the dependent variable. Given the bivariate frequency distribution, the table summarizes joint counts of x and y values:

x	0	1	Total
y
0	6	11	17
1	6	7	13
Total	12	18	30

The logit model is represented as:

log(odds) = β0 + β1x

where odds = P(y=1|x=1)/P(y=0|x=1) when x=1 and odds = P(y=1|x=0)/P(y=0|x=0) when x=0. We calculate these probabilities and then find the coefficient.

• P(y=1|x=1) = 7/18
• P(y=0|x=1) = 11/18
• P(y=1|x=0) = 6/12
• P(y=0|x=0) = 6/12

Calculate the odds for each x:

• odds_x=1 = (7/18)/(11/18) = 7/11
• odds_x=0 = (6/12)/(6/12) = 1

Now, calculate the log-odds (logit) for both scenarios:

• logit_x=1 = log(7/11)
• logit_x=0 = log(1) = 0

The coefficient β₁ is the change in the log-odds, so:

β₁ = log(7/11) - 0 = log(7/11)

Compute this value:

β₁ = log(0.63636) ≈ -0.45198

Rounding to two decimal places, β₁ ≈ -0.45. Verify the range: -0.49 < -0.45 < -0.49

Therefore, the estimated coefficient of x is -0.45.