Regression Models Depend on the type of Dependent Variables
Which regression model should you use that depends on the character of the dependent variable.
In economics, the choice of regression model is driven by the specific nature of the economic decision or outcome being measured. Here are the same models:
1. Linear Regression
Economic DV: Continuous, unbounded, often in logs (e.g., log wages, log prices).
Example (Labor):
Predicting an individual’s log hourly wage (USD) based on years of education, experience, and gender.
Why linear? Wages are continuous and roughly normal after log transformation. Linear regression gives the familiar Mincer equation.
2. Tobit Regression
Economic DV: Continuous but corner solution (many observations at a boundary, usually zero).
Example (Consumer demand):
Household expenditure on alcohol per month. Many households spend zero (abstainers), but among spenders, the amount is continuous.
Why not linear? Linear would predict negative spending. Tobit estimates both the probability of any purchase and the intensity given purchase, treating zero as a corner solution.
3. Poisson Regression
Economic DV: Count of events (non-negative integers), often rare.
Example (Health economics):
Number of job-related injuries per worker in a factory per year.
Why Poisson? Count of injuries is integer, mostly 0, sometimes 1–2. Poisson handles the skewness naturally.
4. Negative Binomial Regression
Economic DV: Overdispersed counts (variance much larger than mean).
Example (Innovation economics):
Number of patents filed by a firm in a year. Most firms have zero, a few have 1–5, some (e.g., tech companies) have 50+ — far more spread than Poisson allows.
Why not Poisson? Poisson would severely underestimate standard errors, making R&D spending look much more significant than it truly is.
5. ZIP / ZINB / Hurdle Regression
Economic DV: Counts with excess zeros from two distinct economic processes.
Example (Labor economics):
Number of job training applications submitted by an unemployed worker.
Always-zero group: Workers who are not searching (discouraged, retired)
At-risk group: Active searchers who may apply 0, 1, or more times
ZIP or ZINB models separate the “participation” decision (search vs. not) from the “intensity” decision (how many applications).
6. Binary Logistic Regression
Economic DV: Binary economic choice.
Example (Consumer choice):
Will a household buy organic milk (Yes/No) given price, income, and education?
Why logistic? Linear probability model can predict probabilities >1 or <0. Logistic maps to [0,1] and is standard for discrete choice.
7. Multinomial Logistic Regression
Economic DV: Unordered categorical choice among ≥3 options.
Example (Transport economics):
Commuter’s primary mode to work: Car / Bus / Subway / Bicycle. No natural order.
Why not ordered? You cannot rank “Car > Bus > Bicycle” meaningfully. Multinomial logit is the workhorse for such choices.
8. Ordered Logistic Regression
Economic DV: Ordered categorical outcome, often from survey ratings or bond ratings.
Example (Financial economics):
Corporate bond credit rating (AAA, AA, A, BBB, BB, B, CCC, D). Order is clear, but distances between ratings are unknown.
Why not linear? Treating AAA=8, AA=7… imposes equal spacing, which is false. Ordered logistic respects the rank order only.
9. Cox / Survival Regression
Economic DV: Duration until an event, with censoring.
Example (Labor economics – unemployment duration):
Weeks until an unemployed worker finds a job (event). Some workers are still unemployed when the study ends (censored).
Why Cox? Unemployment spells are often right-censored. Cox models the hazard of re-employment without assuming a parametric distribution (e.g., exponential, Weibull). Often used with “unemployment benefits exhaustion” as a key covariate.
10. Multivariate Probit Regression
Economic DV: Multiple correlated binary choices made simultaneously.
Example (Household finance):
A household owns a home (Yes/No) and holds stocks (Yes/No). The error terms may be correlated because unobserved wealth or risk tolerance drives both.
Why not separate logits? Separate logits ignore the correlation, leading to inefficient estimates. Multivariate probit estimates the correlation coefficient between the two decisions (e.g., ρ > 0 suggests wealth drives both).
In accounting research (archival financial accounting, managerial accounting, auditing, tax), the dependent variables often reflect firm disclosures, auditor decisions, financial reporting outcomes, or earnings management. These have very distinct characteristics that drive model choice.
Below are the same regressions, but with accounting-specific examples:
1. Linear Regression
Character of DV: Continuous, unbounded (often scaled or logged).
Accounting Example (Earnings management):
Firm 's discretionary accruals (e.g., from the Jones or modified Jones model) in year , regressed on leverage, growth, and corporate governance metrics.
Why linear? Discretionary accruals are continuous, can be positive or negative, and are approximately normally distributed after winsorizing outliers.
2. Tobit Regression
Character of DV: Continuous but censored (pile-up at zero or a boundary).
Accounting Example (Tax avoidance):
Cash effective tax rate (ETR) – defined as cash taxes paid divided by pretax income. Many firms pay little or no tax (ETR near 0), but ETR cannot go below 0. Also, firms with losses are often excluded or censored.
Alternative: R&D expenditures – many firms report zero R&D, but among those that do, it is continuous.
Why not linear? Linear regression would predict negative ETRs. Tobit respects the lower bound and models the decision to have positive tax liability separately from the amount.
3. Poisson Regression
Character of DV: Counts (non-negative integers), variance ≈ mean.
Accounting Example (Auditing):
Number of audit adjustments (proposed or booked) during a financial statement audit. Most audits have few adjustments (0–3), some have more.
Why Poisson? Count data, not continuous. Linear regression could predict fractional adjustments and violates normality.
4. Negative Binomial Regression
Character of DV: Overdispersed counts (variance > mean).
Accounting Example (Disclosure intensity):
Number of footnote disclosures in the annual report (e.g., related to leases, pensions, fair value). Some firms have very long, complex footnotes (variance >> mean).
Why not Poisson? Poisson would understate standard errors. Negative binomial handles the extra spread common in disclosure studies.
5. ZIP / ZINB / Hurdle Regression
Character of DV: Counts with excess zeros from two distinct processes.
Accounting Example (Tax strategy):
Number of tax haven subsidiaries owned by a multinational firm.
Always-zero group: Domestic-only firms with no tax haven presence (structural zeros)
At-risk group: Multinationals that may have 0, 1, 5, or 10+ tax haven subs
ZINB is common here because overdispersion is also present. The model estimates separately: (1) probability of being a domestic-only firm, and (2) count of tax havens given the firm is multinational.
6. Binary Logistic Regression
Character of DV: Dichotomous outcome.
Accounting Example (Auditing – going concern opinion):
Does the auditor issue a going-concern modified opinion (Yes/No) for a financially distressed firm? Predictors include leverage, current ratio, prior year loss.
Why logistic? Linear probability model can predict probabilities outside [0,1] and assumes constant marginal effects, which is unrealistic for rare events like going-concern opinions.
7. Multinomial Logistic Regression
Character of DV: Unordered categorical (≥3 categories).
Accounting Example (Audit opinion types):
Audit opinion issued: Unmodified / Qualified / Adverse / Disclaimer. These are unordered categories (Adverse is not "between" Qualified and Disclaimer).
Why not ordered? There is no agreed ranking. A disclaimer (auditor could not do the work) is different from an adverse (financial statements are materially misstated) – they are distinct types, not a single ordered scale.
8. Ordered Logistic Regression
Character of DV: Ordered categorical.
Accounting Example (Credit ratings):
Firm's credit rating (AAA, AA, A, BBB, BB, B, CCC, D). Used in accounting studies of debt contracting and earnings quality.
Why ordered logit? The categories are inherently ordered by creditworthiness, but the distances between categories are unknown and likely unequal. Ordered logit (proportional odds) respects the ordering without imposing equal spacing.
9. Cox / Survival Regression
Character of DV: Time until an event, with censoring.
Accounting Example (Restatements / auditor turnover):
Months until a firm restates its financial statements (event). Firms that never restate during the study period are right-censored.
Another example: Months until a firm switches auditors (Big N to non-Big N, or vice versa).
Why Cox? Accounting restatements are irregularly spaced over time. Cox regression handles censored observations (firms that leave the sample or never restate) without assuming a parametric distribution of survival time.
10. Multivariate Probit Regression
Character of DV: Multiple correlated binary outcomes.
Accounting Example (Corporate governance / disclosure):
Does a firm (1) voluntarily disclose internal control weaknesses (Yes/No) and (2) voluntarily disclose segment information (Yes/No)? The errors may be correlated because unobserved transparency culture drives both.
Why not separate probits? Ignoring the correlation leads to inefficient estimates. Multivariate probit allows testing whether the two disclosure decisions are jointly determined (ρ ≠ 0).
In marketing research (quantitative marketing, consumer behavior, retail analytics, digital marketing), the dependent variable often reflects consumer choices, purchase quantities, website interactions, or customer lifetime value. These have distinct characteristics—counts, durations, shares, or repeated binary choices—that drive model selection.
Below are the same regressions with marketing-specific examples:
1. Linear Regression
Character of DV: Continuous, unbounded (often logged or transformed).
Marketing Example (Sales response):
Log of unit sales for a product (e.g., detergent) in a given week, regressed on price, display, and advertising expenditure.
Why linear? After log transformation, sales are approximately normal. Linear regression gives interpretable elasticities.
2. Tobit Regression
Character of DV: Continuous but censored (many zeros or a corner solution).
Marketing Example (Household purchase amount):
Household expenditure on yogurt per week. Many households buy zero (non-buyers). Among buyers, spending is continuous.
Why not linear? Linear would predict negative spending. Tobit models the corner solution (zero) and positive amounts jointly.
Alternative use: Coupon redemption value – many households receive coupons but redeem $0.
3. Poisson Regression
Character of DV: Counts (non-negative integers), variance ≈ mean.
Marketing Example (Store visits):
Number of times a customer visits a specific store (e.g., Target) in a month. Most customers visit 0–3 times.
Why Poisson? Count data, often small numbers. Linear regression would predict fractional visits and violate distributional assumptions.
4. Negative Binomial Regression
Character of DV: Overdispersed counts (variance > mean).
Marketing Example (Online purchases):
Number of items purchased on an e-commerce website (Amazon) by a customer over a year. Most buy 0–2 items, but heavy users buy 50+ (far more spread than Poisson allows).
Why not Poisson? Poisson would severely underestimate standard errors, making marketing interventions (e.g., email campaigns) appear falsely significant.
5. ZIP / ZINB / Hurdle Regression
Character of DV: Counts with excess zeros from two distinct processes.
Marketing Example (Email click-throughs):
Number of email links clicked by a customer in a campaign.
Always-zero group: Customers who never opened the email (structural zeros)
At-risk group: Customers who opened but may click 0, 1, 2+ links
Use ZINB if overdispersion also exists. The model separately estimates (1) probability of opening the email, and (2) clicks given opened.
6. Binary Logistic Regression
Character of DV: Dichotomous outcome.
Marketing Example (Purchase incidence):
Did the customer purchase the product (Yes/No) on a given shopping trip? Predictors: price promotion, feature ad, competitor price.
Why logistic? Logit is the foundation of discrete choice models in marketing. Linear probability model gives nonsense probabilities (<0 or >1).
7. Multinomial Logistic Regression
Character of DV: Unordered categorical choice among ≥3 options.
Marketing Example (Brand choice):
A customer chooses one brand of ketchup: Heinz / Hunt's / Del Monte / Store Brand. No natural order among brands.
Why multinomial logit? This is the classic conditional logit (McFadden, 1974) used in marketing for brand choice models, often with alternative-specific attributes (price, feature display) and consumer characteristics.
8. Ordered Logistic Regression
Character of DV: Ordered categorical.
Marketing Example (Customer satisfaction / star ratings):
Customer product rating on a 5-star scale (1 to 5 stars). Order is clear, but distance between "4 stars" and "5 stars" is not the same as between "1 star" and "2 stars".
Why ordered logit? Linear regression would incorrectly assume equal spacing (e.g., that the jump from 3→4 is the same utility gain as from 4→5).
9. Cox / Survival Regression
Character of DV: Time until an event, with censoring.
Marketing Example (Customer churn / retention):
Months from subscription start until a customer cancels their streaming service (e.g., Netflix, Spotify). Customers still active at the end of the study are censored.
Why Cox? Survival analysis handles right-censoring and time-varying covariates (e.g., price changes, competitor entry). Used heavily in customer lifetime value (CLV) and retention modeling.
10. Multivariate Probit Regression
Character of DV: Multiple correlated binary outcomes.
Marketing Example (Cross-category purchase):
On a shopping trip, does a household purchase (1) beer and (2) diapers (both Yes/No)? The errors are correlated due to unobserved household traits (e.g., parental stress, party planning).
Why not separate probits? Ignoring correlation gives inefficient estimates. Multivariate probit estimates the correlation () – famously, the "beer and diapers" association in market basket analysis.
Comments
Post a Comment