Logit Model Exercises

Question 1

a. Describe ROC Curve

ROC curves usually look like a bow reaching out to the top left corner of the chart. The more the curve bow out from the diagonal, the better the credit scoring model.

b. Estimate of the Probability of External Financing

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mdl2_zi <- \(def, ass, iag, p_fin){-0.72 + 0.02*def + 0.0003*ass - 0.002*iag + 0.79*p_fin}
mdl2_prob <- \(z){exp(z)/(1+exp(z))}
prob_firm1 <- mdl2_prob(mdl2_zi(1.10, 1.00, 0.00, 0.00))

The probability of external financing of Firm 1 using Model 2 is 33.23.

c. Marginal Effect Explanation

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mdl2_me_pf <- \(z){(exp(z)/(1+exp(z))^2)*0.79}
me_f1 <- mdl2_me_pf(mdl2_zi(1.10, 1.00, 0.00, 0.00))
me_f2 <- mdl2_me_pf(mdl2_zi(0.13, 1.00, 0.00, 0.50))

The marginal effect generally varies across observations because it depends not only on the estimated coefficients but also the evaluation point through \(z_i\), i.e. \(f(z_i)\). For instance, the marginal effects shrink at extreme probabilities as \(f(z_i)\) becomes smaller.

  • For firm 1, marginal effect is 0.1753 percentage point change in probability for an incremental change of 1 percentage point in \(PF_1\).
  • For firm 2, marginal effect is 0.1925 percentage point change in probability for an incremental change of 1 percentage point in \(PF_2\).

Question 3

a. Model Type

The probability model proposed by Jim

$$ p_i = \frac{\exp(\alpha_0 + \alpha_1 Lev_i)}{1 + \exp(\alpha_0 + \alpha_1 Lev_i)} $$

is a logit probability model.

b. Compute the Probability of a Take-over Bid

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jim_mdl <- \(lev){exp(-3 - 0.54*lev)/(1+exp(-3 - 0.54*lev))}

prob_tov <- jim_mdl(2)
# prob_tov

The probability of a specific firm with a leverage level of 2 is 1.66.

c. Coefficient Justification

Yes, the estimated coefficients of the proposed model align with Jim’s prior expectations. As \(Lev_i\) increases, the exponential term \(\exp(\alpha_0 + \alpha_1 Lev_i)\) reduces accordingly, implying a lower probability of takeover.

Examples:

  • A firm with leverage level of 3 has the probability of being taken over of 0.9757%.
  • A company having leverage level of 5 indicates a take-over probability of 0.3335%.

d. Marginal Impact

The marginal effects are measured through \(ME = f(z_i)\beta_j = \frac{\exp(x_i'\beta)}{[1 + \exp(x_i'\beta)]^2}\beta_j\). With \(Lev_i = 2\), the marginal effect is \(\frac{\exp(-3 - 0.54*2)}{[1 + \exp(-3 - 0.54*2)]^2}*(- 0.54) =\) -0.008829.

As a result, a percentage point increase in leverage decreases the probability of a takeover bid by approximately -0.0088 percentage points at a leverage level of 2.

e. Model Aquedacy Evaluation

With the AUROC of 0.58, Jim’s model is minorly better than a random guess (AUROC = 0.5). Imagine that the lower right triangle of the TPR vs. FPR is a random guess, Jim model’s ROC just expands a little from the diagonal line, hence the AUROC is merely bigger than the triangle itself.

Panel Data DiD Exercises

Question 3

a. Pooled Regression Model

A typical pooled regression model with the given regressors is

$$ L_{it} = \alpha_0 + \beta_1 Q_{it} + \beta_2 T_{it} + \beta_3 C_{it} + \varepsilon_{it} $$

However, this “classical” model may disregard the within firm-specific effect \(\alpha_i\) or common time trend across firms \(\gamma_t\). Consequently, we would like to control for firm-specific characteristics using:

$$ L_{it} = \alpha_i + \beta_1 Q_{it} + \beta_2 T_{it} + \beta_3 C_{it} + \varepsilon_{it} $$

b. Fixed & Random Effect

Definitions:

  • (Firm) Fixed effects: (firm) individual-specific effects that are time-invariant, e.g. credit availability, business model complexity, market position, etc.
  • Random effects: random firm-specific effects that “fall from heaven”, which are also constant over time, e.g. risk management ability, technology adoption and adaptation capabilities, and so forth.

Decision Rules:

  • Choose fixed effect model if:
    • Able to access to the entire population of interest
    • Unobserved entity effects is correlated with the regressors
    • Concern about endogeneity and consistency
  • Apply random effect model if:
    • Can only acquire a random sample of the population
    • Strictly assume \(u_i\) and regressors are uncorrelated
    • Want to evaluate time-invariant variables
    • Expect a higher level of efficiency (if the assumptions hold)

d. Clustered Standard Errors Explanation

Clustered standard errors (SE) are the adjusted SEs that correct the underestimated standard SEs due to serial correlation \(\text{Cov}(\varepsilon_{it}, \varepsilon_{is}) \neq 0\) for \(t \neq s\), or any other within-cluster correlation (e.g. cross-sectional correlation, spatial correlation, etc.). The clustered variance is calculated as:

$$ \hat{V}{cluster} \approx \frac{\sum{i=1}^{N} \left(\sum_{t=1}^{T} \hat{\varepsilon}_{it}\right)^2}{NT \cdot NT} $$

Why we need clustered SEs: If within-cluster correlation exists, traditional SEs are underestimated. Thus, it is more likely to reject the null hypotheses and we get too many false positives, creating incorrect inference.