---
title: "Functional Forms"
---
{{< include _setup.qmd >}}
> **Reading.** SW §8.1–8.2, HGL §2.8, 4.3–4.6
So far the regression line has been *straight*: $\E(y \given x) = \beta_1 +
\beta_2 x$. But the world rarely is. Food spending rises with income, but at a
decreasing rate; test scores climb steeply with district income and then
flatten; cost curves are U-shaped. A single straight line forces *one* slope
everywhere, and economic theory almost never wants that.
The key realization that frees us is this: "**linear** regression" means linear
in the **parameters** $\beta$ — *not* in the variables. We are free to feed OLS
*transformed* variables. We can put a squared or logged regressor on the right,
$$
y = \beta_1 + \beta_2\,\underbrace{f(x)}_{\text{e.g. } x^2,\ \ln x} + e,
$$
or transform the dependent variable on the left,
$$
\underbrace{g(y)}_{\text{e.g. } \ln y} = \beta_1 + \beta_2 x + e .
$$
It is still OLS, still BLUE under the [usual assumptions](07-ols-properties.qmd)
— only the **interpretation** of $\beta_2$ changes. This chapter lays out the
menu of functional forms — quadratics, logs, elasticities — explains when to
reach for each, and shows how to read a transformed coefficient.
## Linear in parameters, not variables {#sec-linear-in-params}
Economic theory frequently predicts a **changing** slope. Food spending rises
with income, but *at a decreasing rate* — the marginal propensity to spend
falls as you get richer. Test scores rise with district income, steeply at
first and then flattening out. Cost curves are U-shaped and total-product
curves are S-shaped. In every case a straight line is the wrong tool: it forces
one slope everywhere. Transforming $x$ or $y$ lets the slope — and the
**elasticity** — change from point to point, as in @fig-curvature.
```{r}
#| label: fig-curvature
#| fig-cap: "A relationship that increases at a decreasing rate: the slope is steep at low income and gentle at high income."
#| fig-width: 5
#| fig-height: 3.4
xs <- seq(0.3, 9.5, length.out = 200)
dat <- data.frame(x = xs, y = 2.2 * log(1 + xs))
ggplot(dat, aes(x, y)) +
geom_line(color = ucla$blue, linewidth = 1) +
annotate("segment", x = 1, y = 1, xend = 3, yend = 4.0,
linetype = "dashed", color = ucla$red) +
annotate("segment", x = 7, y = 4.4, xend = 9, yend = 5.0,
linetype = "dashed", color = ucla$red) +
annotate("text", x = 2.6, y = 1.1, label = "steep slope",
color = ucla$red, size = 3.2) +
annotate("text", x = 7.7, y = 5.6, label = "gentle slope",
color = ucla$red, size = 3.2) +
scale_x_continuous(breaks = NULL) +
scale_y_continuous(breaks = NULL, limits = c(0, 8)) +
labs(x = "income x", y = expression(E(y * "|" * x)))
```
Almost everything in this chapter is built from just two transformations.
::: {.keyidea title="Powers"}
$x^2,\ x^3,\ 1/x,\dots$ give quadratics (U or $\cap$ shapes) and cubics
(S-shapes). They capture turning points and acceleration — a slope that grows
or reverses.
:::
::: {.keyidea title="Natural logarithm"}
$\ln(x)$ and $\ln(y)$ convert **changes into percentage changes**. Logs tame
right-skewed money variables (income, wages, prices) and deliver
**elasticities** — unit-free, comparable across studies.
:::
There is one habit that prevents nearly every mistake with these forms.
::: {.keyidea title="The one habit that prevents every mistake"}
Whenever you transform a variable, the **slope and elasticity formulas
change**. Before interpreting any coefficient, ask: *is each side in levels, or
in logs, or squared?* The answer dictates the wording.
:::
## Polynomials {#sec-polynomials}
The simple quadratic regression puts a *squared* regressor on the right:
$$
y = \beta_1 + \beta_2 x^2 + e
\qquad\Longrightarrow\qquad
\text{slope} = \frac{dy}{dx} = 2\beta_2 x .
$$
The slope is **no longer constant** — it grows in magnitude with $x$. If
$\beta_2 > 0$ the curve sweeps upward ever more steeply, exactly the curvature a
single straight line cannot express.
::: {.example title="House prices (Baton Rouge)"}
Fitting house price on squared floor area gives
$$
\widehat{\text{PRICE}} = 55{,}776 + 0.0154\,\text{SQFT}^2 .
$$
The estimated price of one more square foot is $2(0.0154)\,\text{SQFT}$, which
*depends on how big the house already is*:
$$
\text{at } 2000\text{ sqft}: \$61.69, \qquad
\text{at } 4000\text{ sqft}: \$123.37 .
$$
Bigger homes command a **higher price per added square foot**.
:::
We can reproduce these numbers from the `br` (Baton Rouge) data:
```{r}
#| label: house-quadratic
#| code-fold: false
fit_house <- lm(price ~ I(sqft^2), data = br)
coef(fit_house)
# marginal effect dy/dx = 2 * beta2 * sqft, evaluated at 2000 and 4000 sqft
b2 <- coef(fit_house)[["I(sqft^2)"]]
2 * b2 * c(2000, 4000)
```
@fig-house-quad plots the fitted curve over the data: it bends upward, so the
gap in price between a 4000- and a 3900-square-foot home is larger than the gap
between a 1100- and a 1000-square-foot home.
```{r}
#| label: fig-house-quad
#| fig-cap: "House price against floor area with the fitted quadratic. The curve steepens, so the marginal price per square foot rises with size."
#| fig-width: 5
#| fig-height: 3.4
grid <- data.frame(sqft = seq(min(br$sqft), max(br$sqft), length.out = 200))
grid$price <- predict(fit_house, newdata = grid)
ggplot(br, aes(sqft, price)) +
geom_point(color = ucla$gray, alpha = 0.25, size = 0.7) +
geom_line(data = grid, aes(sqft, price), color = ucla$blue, linewidth = 1) +
scale_y_continuous(labels = dollar) +
labs(x = "SQFT", y = "PRICE")
```
A cubic term goes one step further, capturing S-shapes (total cost, total
product) or growth that accelerates:
$$
y = \beta_1 + \beta_2 x^3 + e,
\qquad \text{slope} = 3\beta_2 x^2 .
$$
::: {.example title="Wheat yield over time"}
A straight line in $\text{TIME}$ left U-shaped residuals — it missed the
acceleration in yield from technological progress. Using
$\text{TIMECUBE} = (\text{TIME}/100)^3$ instead:
$$
\widehat{\text{YIELD}} = 0.874 + 9.682\,\text{TIMECUBE},
\qquad R^2: 0.649 \to 0.751 .
$$
The cubic both fits better and respects the residual pattern the line ignored.
:::
```{r}
#| label: wheat-cubic
#| code-fold: false
wa_wheat$timecube <- (wa_wheat$time / 100)^3
fit_line <- lm(greenough ~ time, data = wa_wheat)
fit_cube <- lm(greenough ~ timecube, data = wa_wheat)
coef(fit_cube)
c(linear = summary(fit_line)$r.squared,
cubic = summary(fit_cube)$r.squared)
```
::: {.callout-note appearance="simple"}
Here a polynomial uses *one* transformed regressor, so it still fits inside
simple regression. The richer form $y = \beta_1 + \beta_2 x + \beta_3 x^2$ has
*two* regressors ($x$ and $x^2$) — that needs [multiple
regression](13-multiple-regression.qmd), where we can also *test* whether the
curve is needed.
:::
How do you read a polynomial? A polynomial coefficient has **no standalone
interpretation** — "$\beta_2$ holding $x^2$ fixed" is meaningless, since $x$ and
$x^2$ move together. Instead, do one of two things. **Plot** the fitted curve
over the data and describe its shape, as in @fig-house-quad. Or **evaluate the
slope** $dy/dx$ at a few interesting values of $x$ — low, median, high — and
report those marginal effects, exactly as we did with house prices at 2000
versus 4000 square feet.
::: {.keyidea title="Marginal effects are local now"}
The whole point of a nonlinear form is that "the effect of $x$ on $y$" is no
longer a single number. Always quote it *at a stated value of $x$*.
:::
## Logarithms: the three cases {#sec-logs}
One fact lies behind every log interpretation: for a *small* change,
$$
\ln(x + \Delta x) - \ln(x) \approx \frac{\Delta x}{x} = \%\Delta x \;/\; 100 .
$$
A *difference in logs* is approximately a **percentage change**. That single
identity is why economists reach for logs constantly. Many relationships are
naturally **proportional** ("a 1% price rise cuts quantity by $\eta$%"). Money
variables — income, wages, prices, sales — are **right-skewed**, and logging
them pulls the long tail in toward normality, which helps the normality
assumption (SR6) hold. And logs deliver **unit-free** elasticities you can
compare across studies. @fig-skew-log shows the effect on a skewed income
variable: the raw distribution piles up at low values with a long right tail,
while the logged version is far more symmetric.
```{r}
#| label: fig-skew-log
#| fig-cap: "Family income in levels (left) is right-skewed; in logs (right) it is far more symmetric — which is why logging a money variable helps the normality assumption."
#| fig-width: 5.6
#| fig-height: 3.2
inc <- cps5_small$faminc[cps5_small$faminc > 0]
df_raw <- data.frame(v = inc, panel = "income")
df_log <- data.frame(v = log(inc), panel = "log(income)")
both <- rbind(df_raw, df_log)
ggplot(both, aes(v)) +
geom_histogram(bins = 30, fill = ucla$blue, color = ucla$darkblue,
linewidth = 0.2) +
facet_wrap(~ panel, scales = "free") +
labs(x = NULL, y = "count")
```
The three log models differ only in *where* the log sits. We take them in turn.
### Case 1 — log-linear: $\ln(y) = \beta_1 + \beta_2 x$
Only the *dependent* variable is logged. A **one-unit** change in $x$ is
associated with a **$100\beta_2\%$** change in $y$ — a *constant growth rate*
interpretation. (This requires $y > 0$.) The sign of $\beta_2$ sets the
direction; in the *levels* of $y$ the curve rises, or falls, at a changing rate.
::: {.example title="Returns to education (cps5_small)"}
Regressing the log wage on years of schooling,
$$
\widehat{\ln(\text{WAGE})} = 1.597 + 0.0988\,\text{EDUC} .
$$
Each extra year of education raises the wage by about
$100(0.0988) \approx \mathbf{9.9\%}$ (a 95% confidence interval of roughly
$8.9\%$ to $10.9\%$). A *percentage* return — not a fixed dollar amount — which
matches how the labor market actually works.
:::
```{r}
#| label: returns-education
#| code-fold: false
fit_wage <- lm(log(wage) ~ educ, data = cps5_small)
coef(fit_wage) # 100 * slope is the % return per year
confint(fit_wage, "educ") # CI on the slope -> ~8.9% to 10.9%
```
### Case 2 — linear-log: $y = \beta_1 + \beta_2 \ln(x)$
Only the *regressor* is logged. A **1% change in $x$** is associated with a
**$\beta_2/100$-unit** change in $y$. (This requires $x > 0$.) When
$\beta_2 > 0$ the curve rises at a *decreasing* rate — ideal for the food
example, where each extra dollar of income buys less additional food.
::: {.example title="Food expenditure, linear-log"}
$$
\widehat{\text{FOOD\_EXP}} = -97.19 + 132.17\,\ln(\text{INCOME}),
\qquad R^2 = 0.357 .
$$
A 1% income rise adds about $132.17 / 100 = \$1.32$ to weekly food spending. The
marginal effect of \$100 more income *shrinks* as income grows — \$13.22 per
\$100 at \$1{,}000/wk versus \$6.61 per \$100 at \$2{,}000/wk — exactly the
declining marginal propensity to consume that theory predicted.
:::
```{r}
#| label: food-linearlog
#| code-fold: false
fit_food <- lm(food_exp ~ log(income), data = food)
coef(fit_food)
# marginal effect of $100 more income = beta2/100 * (100/income),
# i.e. beta2 / income, evaluated at income = 10 and 20 (hundreds of $)
b2_food <- coef(fit_food)[["log(income)"]]
b2_food / c(10, 20)
```
@fig-food-loglinearlog overlays the linear-log fit on the food data, with the
straight-line fit for contrast: the linear-log curve bends, flattening at high
income just as the marginal-effect numbers say it should.
```{r}
#| label: fig-food-loglinearlog
#| fig-cap: "Weekly food expenditure against income. The linear-log fit (blue) bends and flattens at high income; the straight line (gold, dashed) cannot."
#| fig-width: 5
#| fig-height: 3.4
fit_food_lin <- lm(food_exp ~ income, data = food)
g <- data.frame(income = seq(min(food$income), max(food$income),
length.out = 200))
g$loglog <- predict(fit_food, newdata = g)
g$lin <- predict(fit_food_lin, newdata = g)
ggplot(food, aes(income, food_exp)) +
geom_point(color = ucla$gray, alpha = 0.5, size = 1) +
geom_line(data = g, aes(income, lin), color = ucla$gold,
linewidth = 1, linetype = "dashed") +
geom_line(data = g, aes(income, loglog), color = ucla$blue,
linewidth = 1) +
labs(x = "INCOME", y = "FOOD_EXP")
```
### Case 3 — log-log: $\ln(y) = \beta_1 + \beta_2 \ln(x)$
Both sides are logged. Now $\beta_2$ *is* the **elasticity** of $y$ with respect
to $x$ — and it is **constant** along the whole curve. (This requires
$x, y > 0$.)
$$
\beta_2 = \frac{\%\Delta y}{\%\Delta x}
= \text{elasticity of } y \text{ w.r.t. } x .
$$
So a **1% change in $x$** produces a **$\beta_2\%$ change in $y$**. This is why
log-log is the workhorse for **demand curves** (price elasticity) and
**production functions** (constant-returns checks) — the elasticity *is* the
parameter, read straight off the output with no further calculation.
::: {.example title="Test scores (Stock & Watson)"}
$$
\widehat{\ln(\text{TestScore})} = 6.336 + 0.0554\,\ln(\text{Income}) .
$$
A 1% rise in district income is associated with about $0.0554\%$ higher test
scores — a small, constant elasticity across the whole income range.
:::
### The master table
Collecting all six forms in one place makes the pattern visible: the slope
formula and the wording follow mechanically from where the transformations sit.
| Name | Model | Slope $dy/dx$ | Interpretation of $\beta_2$ |
|------|-------|---------------|------------------------------|
| Linear | $y = \beta_1 + \beta_2 x$ | $\beta_2$ | 1-unit $\Delta x \to \beta_2$-unit $\Delta y$ |
| Quadratic | $y = \beta_1 + \beta_2 x^2$ | $2\beta_2 x$ | slope changes with $x$ |
| Cubic | $y = \beta_1 + \beta_2 x^3$ | $3\beta_2 x^2$ | slope changes with $x$ |
| Log-linear | $\ln y = \beta_1 + \beta_2 x$ | $\beta_2 y$ | 1-unit $\Delta x \to 100\beta_2\%\ \Delta y$ |
| Linear-log | $y = \beta_1 + \beta_2 \ln x$ | $\beta_2 / x$ | 1% $\Delta x \to \beta_2/100$-unit $\Delta y$ |
| Log-log | $\ln y = \beta_1 + \beta_2 \ln x$ | $\beta_2\,y/x$ | 1% $\Delta x \to \beta_2\%\ \Delta y$ (elasticity) |
: The six functional forms, their slopes, and how to read $\beta_2$. {.striped}
::: {.keyidea title="Decode any specification by the location of the logs"}
**$\ln$ on the left** means the effect is in *percent of $y$*. **$\ln$ on the
right** means the cause is a *percent of $x$*. **Logs on both** sides means
$\beta_2$ is an *elasticity*. Keep this map and you can read any of them.
:::
## Choosing and interpreting {#sec-choosing}
How do you pick a form in the first place? Three guideposts, in order.
::: {.property title="Three guideposts for choosing a functional form"}
1. **Theory first.** Pick a shape consistent with the economics — declining
MPC, constant elasticity, U-shaped cost. Decide *before* looking at the data
whether the slope should vary, and how.
2. **Flexibility.** The form must be able to bend the way the data bend; a
residual plot reveals a missed curve (more in [model
specification](18-model-specification.qmd)).
3. **Assumptions.** Prefer a form under which the regression assumptions
SR1–SR6 look reasonable — for instance, logging a skewed $y$ often tames
heteroskedasticity and non-normal errors.
:::
::: {.callout-note appearance="simple"}
We never know the "true" form — every choice is an approximation. The goal is a
form that is theoretically sensible, fits, and respects the assumptions.
:::
When comparing fit across forms, there is a trap to avoid with $R^2$.
::: {.warningbox title="R² is comparable only across models with the same dependent variable"}
You may compare $R^2$ across models **only when they share the same dependent
variable**. A linear-$y$ model, a linear-log model, and a quadratic in $x$ all
have $y$ on the left, so their $R^2$'s are on the same scale — for food, linear
$R^2 = 0.385$ versus linear-log $0.357$ are comparable, and nearly tied. But a
$y$-model versus a $\ln(y)$-model is an **invalid** comparison: the two
dependent variables explain different "total variation," so their $R^2$'s are
not on the same scale. Choose between them with **theory**, not $R^2$.
:::
When you must summarize fit for a logged-$y$ model, use the *generalized*
$R^2$, computed on the *original* $y$ scale:
$$
R^2_g = \bigl[\,\mathrm{corr}(y, \hat y)\,\bigr]^2 ,
$$
where $\hat y$ is the model's prediction transformed back to levels. Because it
lives on the original scale of $y$, it can be compared across a level-$y$ model
and a log-$y$ model on equal footing.
### A bonus: logs and growth rates
Log-linear models fall straight out of **compound interest**. If $y$ grows at a
constant rate $g$ per period, $y_t = y_0(1 + g)^t$, then taking logs of both
sides gives
$$
\ln(y_t) = \underbrace{\ln(y_0)}_{\beta_1}
+ \underbrace{\ln(1 + g)}_{\beta_2}\,t,
\qquad \beta_2 = \ln(1 + g) \approx g .
$$
So the slope on $t$ in a log-linear time trend **is the growth rate**.
::: {.example title="Wheat-yield growth"}
$$
\widehat{\ln(\text{YIELD})} = -0.343 + 0.0178\,t
\quad\Longrightarrow\quad \hat g \approx 1.78\% \text{ per year}
$$
from technological progress.
:::
```{r}
#| label: wheat-growth
#| code-fold: false
fit_growth <- lm(log(greenough) ~ time, data = wa_wheat)
coef(fit_growth) # slope on time ≈ annual growth rate
100 * coef(fit_growth)[["time"]] # ≈ 1.78% per year
```
::: {.callout-note appearance="simple"}
One caveat for later: to predict $y$ itself from a log-linear model,
$\exp(b_1 + b_2 x)$ slightly *under*-predicts. A correction factor
$e^{\hat\sigma^2/2}$ fixes it (HGL §4.5.1). Mind the level-versus-log scale when
forecasting.
:::
## Recap {#sec-recap}
The big idea is that "linear" means linear in $\beta$, not in $x$. We can
transform with **powers** and **logs** and the model is still OLS, still BLUE —
but the slope and elasticity now vary point to point, so marginal effects must
be quoted at a stated value of $x$.
For **polynomials**, the quadratic $y = \beta_1 + \beta_2 x^2$ has slope
$2\beta_2 x$ (house prices rise faster per square foot for bigger homes); read
them by plotting the curve or evaluating the slope at chosen values of $x$.
The **three log cases** are summarized by where the log sits:
| Form | One-line reading | Example |
|------|------------------|---------|
| Log-linear $\ln y = \beta_1 + \beta_2 x$ | 1-unit $\Delta x \to 100\beta_2\%\ \Delta y$ | wage $\approx 9.9\%$ per year of school |
| Linear-log $y = \beta_1 + \beta_2 \ln x$ | 1% $\Delta x \to \beta_2/100$-unit $\Delta y$ | food spending |
| Log-log $\ln y = \beta_1 + \beta_2 \ln x$ | $\beta_2$ is the elasticity | demand, production |
: The three log forms at a glance. {.striped}
For **choosing** a form, combine theory, flexibility, and the assumptions — and
remember that $R^2$ is comparable only when the dependent variable is the same.
**Next time:** one regressor is rarely enough. The [multiple regression
model](13-multiple-regression.qmd) adds $X_2, X_3, \dots$ to control for
confounders and finally give *ceteris paribus* real teeth — starting with Big
Andy's Burgers (SALES, PRICE, ADVERT).
