Functional Forms of Regression Models

How the exam asks it: four core exam templates

In midterm exams, functional form questions almost invariably test one of four canonical structures. Master these four worked solutions, then follow the cheat-sheet and 4-step method below.

Exam template 1: Log-lin growth model with a, b, c parts

Question: An econometrician regresses the natural logarithm of personal consumption expenditure on services ($\text{EXS}_t$, in billion dollars) against quarterly time $t$, yielding: $\widehat{\ln Y}_t = 7.789 + 0.00743 t$ a) What is the instantaneous growth rate of services expenditure per quarter?
b) What is the compound growth rate of services expenditure per quarter?
c) What was the estimated expenditure on services at the beginning of the period ($t = 0$)?

Exam template 2: Lin-log model (Engel curve)

Question: An Engel curve estimated for 55 households in India is: $\widehat{\text{Food}} = -1283 + 257 \ln(\text{Total})$ where $\text{Food}$ is food expenditure and $\text{Total}$ is total household expenditure (both measured in rupees). Interpret the coefficient $257$. What is the effect of a 1% rise in total expenditure?

Exam template 3: Log-log demand elasticity

Question: An estimated demand function for an agricultural commodity is: $\ln Q = 3.2 - 0.8 \ln P$ where $Q$ is quantity demanded and $P$ is unit price. Interpret the slope coefficient $-0.8$ and state whether demand is price-elastic or price-inelastic.

Exam template 4: Model selection for GDP growth

Question: “You want to estimate the continuous percentage growth rate of a nation’s GDP over the past 40 years. Which regression model should you specify, and why?”

Cheat-sheet & the 4-step method

Types of change and elasticity

Before analyzing non-linear functional forms, we must distinguish three mathematical definitions of change in a variable $Y$:

1. Absolute change: the simple difference between the current value and the previous value: $\Delta Y = Y_t - Y_{t-1} \quad \text{or} \quad dY$
2. Relative change: the absolute change divided by the base value: $\frac{\Delta Y}{Y_{t-1}} \approx \frac{dY}{Y} = d(\ln Y)$ A change in the natural logarithm of a number is approximately equal to the relative change of that number.
3. Percentage change: the relative change multiplied by 100: $\% \Delta Y = \left( \frac{\Delta Y}{Y_{t-1}} \right) \times 100 = 100 \times d(\ln Y)$

Defining elasticity

Elasticity measures the responsiveness of one variable to changes in another in percentage terms:

Notice the vital connection: $\text{Elasticity} = \text{Slope} \times \left( \frac{X}{Y} \right)$

In a standard linear model ($Y = \beta_1 + \beta_2 X$), the slope $\beta_2$ is constant, but the elasticity $\beta_2(X/Y)$ varies with every point $(X, Y)$ on the line. Different functional forms allow us to model constant elasticities, constant growth rates, or asymptotic limits.

Log-log model (double-log): constant elasticity

Suppose economic theory suggests an exponential relationship between quantity demanded $Y$ and price $X$: $Y_i = \beta_1 X_i^{\beta_2} e^{u_i}$

This function is non-linear in parameters. However, taking the natural logarithm of both sides transforms it into a linear regression model: $\ln Y_i = \alpha + \beta_2 \ln X_i + u_i \quad (\text{where } \alpha = \ln \beta_1)$

This is known as the log-log, double-log, or log-linear model.

Slope and elasticity in log-log

Differentiating $\ln Y$ with respect to $\ln X$: $\frac{d(\ln Y)}{d(\ln X)} = \frac{dY / Y}{dX / X} = \frac{dY}{dX} \cdot \frac{X}{Y} = \beta_2$

The slope of the line in logarithmic space is the elasticity of $Y$ with respect to $X$. In this model, the elasticity is completely constant everywhere across all ranges of $X$ and $Y$.

The slope in original units is: $\text{Slope} = \frac{dY}{dX} = \beta_2 \left( \frac{Y}{X} \right)$

Slide worked example 1: Cobb–Douglas production function

A classic log-log application is the Cobb–Douglas production function: $Y = A L^{\beta_2} K^{\beta_3} \implies \ln Y = \beta_1 + \beta_2 \ln L + \beta_3 \ln K$ From the lecture slides (Slide 17): $\widehat{\ln Y} = -3.3384 + 1.4988 \ln L + 0.4899 \ln K$

Slide worked example 2: Durable goods consumption

Lecture Slide 18 presents personal consumption expenditure on durable goods ($\text{EXDUR}$) versus total personal consumption expenditure ($\text{PCEX}$), measured in 1992 billions of dollars: $\widehat{\ln \text{EXDUR}}_t = -9.6971 + 1.9056 \ln \text{PCEX}_t$ $\text{se} = (0.4341) \quad (0.0514) \quad t = (-22.3370)^* \quad (37.0962)^* \quad r^2 = 0.9849$

Log-lin model: growth rates (semi-log)

In a log-lin model, the dependent variable $Y$ is logarithmic, but the regressor $X$ is linear: $\ln Y_i = \beta_1 + \beta_2 X_i + u_i$

When the regressor is time ($X = t$), this is the workhorse model for estimating growth rates of economic time series (GDP, population, inflation, sales).

Slope and growth rate in log-lin

Differentiating with respect to $X$: $\frac{d(\ln Y)}{dX} = \frac{1}{Y} \frac{dY}{dX} = \beta_2 \implies \frac{dY}{dX} = \beta_2 Y$

$\text{Relative change} = \frac{dY}{Y} = \beta_2 dX$ $\text{Percentage change} = 100 \times \left( \frac{dY}{Y} \right) = (100 \times \beta_2) dX$

Slide worked example: Expenditure on services

Lecture Slide 22 estimates the quarterly growth rate of expenditure on services ($\text{EXS}$) from 1993:1 to 1998:3: $\widehat{\ln \text{EXS}}_t = 7.7890 + 0.00743 t$ $\text{se} = (0.0023) \quad (0.00017) \quad t = (3387.619)^* \quad (44.2826)^* \quad r^2 = 0.9894$

Linear trend model comparison

In Slide 24, instead of regressing the log of $Y$ on time, services expenditure is fitted with a linear trend model: $\widehat{\text{EXS}}_t = 2405.848 + 19.6920 t \quad (r^2 = 0.9843)$

Lin-log model: absolute change from relative change

In the lin-log model, the regressand $Y$ is in level form, but the regressor $X$ is logarithmic: $Y_i = \beta_1 + \beta_2 \ln X_i + u_i$

Slope, elasticity and the divide-by-100 rule

Differentiating with respect to $X$: $\text{Slope} = \frac{dY}{dX} = \beta_2 \left( \frac{1}{X} \right)$ $\text{Elasticity} = \frac{dY}{dX} \cdot \frac{X}{Y} = \beta_2 \left( \frac{1}{Y} \right)$

To understand the change in $Y$: $dY = \beta_2 \left( \frac{dX}{X} \right) = \frac{\beta_2}{100} \left( 100 \frac{dX}{X} \right) = \left( \frac{\beta_2}{100} \right) \times (\% \Delta X)$

Slide worked example 1: Food expenditure in India

Lecture Slide 26 revisits food expenditure for 55 families in India: $\widehat{\text{FoodExp}}_i = -1283.912 + 257.2700 \ln \text{TotalExp}_i$ $t = (-4.3848)^* \quad (5.6625)^* \quad r^2 = 0.3769$

Slide worked example 2: Money supply and GNP

Lecture Slide 27 shows a monetary model relating M2 money supply to GNP (measured in billion TL): $\widehat{\text{GNP}}_t = -16329 + 2584.8 \ln \text{M2}_t$

Reciprocal models: asymptotes and the Phillips curve

A reciprocal model takes the form: $Y_i = \beta_1 + \beta_2 \left( \frac{1}{X_i} \right) + u_i$

Although non-linear in the variable $X$, the model is strictly linear in the parameters $\beta_1$ and $\beta_2$. Therefore, it is a valid linear regression model that can be estimated by standard OLS simply by defining a new variable $Z_i = 1/X_i$.

Asymptotic property

As $X$ increases indefinitely ($X \to \infty$), the term $1/X \to 0$. Therefore: $\lim_{X \to \infty} E(Y_i) = \beta_1$ The constant term $\beta_1$ is the horizontal asymptote or limiting value that the dependent variable approaches as $X$ grows without bound.

Slope and elasticity

$\text{Slope} = \frac{dY}{dX} = -\beta_2 \left( \frac{1}{X^2} \right)$ $\text{Elasticity} = \frac{dY}{dX} \cdot \frac{X}{Y} = -\beta_2 \left( \frac{1}{XY} \right)$

The three reciprocal shapes

Lecture Slide 29 illustrates three distinct shapes depending on the signs of $\beta_1$ and $\beta_2$:

1. $\beta_1 \gt 0, \beta_2 \gt 0$ (Falling to positive asymptote): curve starts high and falls at a decreasing rate toward the positive horizontal asymptote $\beta_1$. (Example: child mortality).
2. $\beta_1 \gt 0, \beta_2 \lt 0$ (Rising to positive asymptote): curve starts at $X = -\beta_2 / \beta_1$ (where $Y = 0$) and rises at a decreasing rate toward $\beta_1$ from below.
3. $\beta_1 \lt 0, \beta_2 \gt 0$ (Phillips curve): curve falls from positive $Y$, crosses the horizontal axis at the natural rate $X = -\beta_2 / \beta_1$, and approaches the negative horizontal asymptote $\beta_1$.

Slide worked example 1: Child mortality vs per capita GNP

In Slide 30–31, child mortality ($\text{CM}$, deaths per 1000) is regressed on per capita GNP ($\text{PGNP}$): $\widehat{\text{CM}}_i = 81.79436 + 27273.17 \left( \frac{1}{\text{PGNP}_i} \right)$ $\text{se} = (10.8321) \quad (3759.999) \quad t = (7.5511)^* \quad (7.2535)^* \quad r^2 = 0.4590$

Slide worked example 2: The Phillips curve

Lecture Slide 32 models the wage/inflation Phillips curve relating the change in inflation $(\pi_t - \pi_{t-1})$ to the unemployment rate $\text{UN}_t$: $\widehat{(\pi_t - \pi_{t-1})} = -3.2514 + 18.5508 \left( \frac{1}{\text{UN}_t} \right)$ $t = (-2.9715)^* \quad (3.0625)^* \quad r^2 = 0.2067$

Log-reciprocal model (log hyperbola)

A log-reciprocal model takes the natural log of $Y$ and the reciprocal of $X$: $\ln Y_i = \beta_1 - \beta_2 \left( \frac{1}{X_i} \right) + u_i \quad (\beta_1 \gt 0, \beta_2 \gt 0)$

Properties and S-shaped curve

1. Upper horizontal asymptote: As $X \to \infty$, $1/X \to 0$, so $\ln Y \to \beta_1$. Exponentiating gives: $\lim_{X \to \infty} Y = e^{\beta_1}$ The dependent variable approaches an upper ceiling $e^{\beta_1}$.
2. Slope: $\frac{d(\ln Y)}{dX} = \frac{1}{Y} \frac{dY}{dX} = \beta_2 \left( \frac{1}{X^2} \right) \implies \frac{dY}{dX} = \beta_2 \left( \frac{Y}{X^2} \right) \gt 0$ The slope is strictly positive throughout.
3. Inflection point (S-curve): Setting the second derivative $d^2Y/dX^2 = 0$ yields: $X_{\text{inflection}} = \frac{\beta_2}{2}$
   • For $X \lt \beta_2 / 2$: $Y$ increases at an increasing rate (accelerating growth).
   • At $X = \beta_2 / 2$: inflection point occurs.
   • For $X \gt \beta_2 / 2$: $Y$ increases at a decreasing rate (diminishing returns, tapering toward ceiling $e^{\beta_1}$).

Summary: comparison of functional forms

Choice of functional form: the 5 rules

Lecture Slide 35 outlines the five fundamental principles governing the choice between models:

1. Underlying economic theory: Theory must guide the formulation. If theory dictates constant elasticity, use log-log. If theory dictates an asymptotic ceiling, use reciprocal or log-reciprocal.
2. Know the signs of the coefficients: Before estimation, establish expected economic signs (e.g., negative price elasticity in demand). A model with the wrong sign violates economic theory regardless of high fit.
3. Plausibility of slope and elasticity: Examine what happens as $X$ grows large. If a linear model predicts negative consumption or impossible levels at realistic values, switch to a non-linear form.
4. Data transformations: When logging variables, all values must be strictly positive ($X > 0, Y > 0$).
5. Never choose solely based on $R^2$:
   • You CANNOT compare $R^2$ across models with different dependent variables (e.g., comparing $R^2$ of a model with $Y$ to one with $\ln Y$ is strictly invalid because total sum of squares $\sum y_i^2$ differs).
   • A higher $R^2$ never compensates for theoretically nonsensical coefficients or violated assumptions.

Exam practice

Summary and review

• Linearity: OLS requires models to be linear in parameters, not necessarily linear in variables.
• Log-log: $\ln Y = \beta_1 + \beta_2 \ln X$. Slope is $\beta_2 (Y/X)$; elasticity is constant $\beta_2$.
• Log-lin: $\ln Y = \beta_1 + \beta_2 t$. Instantaneous growth is $100 \beta_2\%$; compound growth is $[e^{\beta_2} - 1] \times 100\%$; initial value is $Y_0 = e^{\beta_1}$.
• Lin-log: $Y = \beta_1 + \beta_2 \ln X$. Absolute change in $Y$ for 1% rise in $X$ is $\beta_2 / 100$.
• Reciprocal: $Y = \beta_1 + \beta_2 (1/X)$. Asymptote is $\beta_1$; slope is $-\beta_2/X^2$.
• Log-reciprocal: $\ln Y = \beta_1 - \beta_2 (1/X)$. S-shaped curve; inflection point at $X = \beta_2/2$; ceiling at $e^{\beta_1}$.