ECON2515 · Intermediate Applied Econometrics Ii
Goodness of Fit and Functional Forms
Goodness of Fit and Functional Forms (Week 3) does two separate jobs. First it measures how well the regression line fits using the variation identity SST = SSE + SSR and the coefficient of determination R² = SSE/SST — the share of variation in y that the model explains. Second it chooses what shape the relationship takes: level-level, log-linear, linear-log, log-log or quadratic forms all keep the model linear in the βs but change how each slope is read — as a unit effect, a percentage effect, an elasticity, or a turning point. The exam rewards interpretation over recall, so the marks are in reading coefficients correctly and computing marginal effects, not in deriving formulas.
What this chapter covers
- 011. Variation decomposition — SST = SSE + SSR from yᵢ − ȳ = (ŷᵢ − ȳ) + (yᵢ − ŷᵢ)
- 022. Total, explained and residual sums of squares — anchor on the words, not the SSE/SSR letters
- 033. R² = SSE/SST = 1 − SSR/SST — the fraction of variation in y explained (0 ≤ R² ≤ 1)
- 044. In simple regression R² = r² (the squared correlation); a low R² is common and not fatal
- 055. Functional forms — level-level, log-linear, linear-log, log-log; still linear in β
- 066. Interpretation rules — unit vs percentage vs elasticity vs semi-elasticity
- 077. Marginal effect ∂y/∂x and elasticity (∂y/∂x)·(x/y); constant vs varying with x
- 088. Quadratic terms — effect = β₁ + 2β₂x and turning point x* = −β₁/(2β₂)
Interpret a log-log demand model and its R²
- +2(a) Both variables are logged, so this is a log-log model and the slope is the elasticity directly: a 1% rise in price is associated with about a 1.25% fall in sales, ceteris paribus. Because the magnitude exceeds 1, demand is price-elastic.
- +2(b) R² = 0.58 means 58% of the variation in ln(SALES) is explained by ln(PRICE); the remaining 42% is left in the residual. Note this R² is for ln(SALES), so it cannot be compared with an R² from a model whose dependent variable is SALES in levels.
- +2(c) With a constant elasticity of −1.25, %ΔSALES ≈ −1.25 × 4% = −5%. A 4% price rise lowers sales by about 5%.
Key terms
- Variation decomposition (SST = SSE + SSR)
- Each point's deviation from the mean splits into an explained part and a residual part: yᵢ − ȳ = (ŷᵢ − ȳ) + (yᵢ − ŷᵢ). Squared and summed this gives Total = Explained + Residual sum of squares, the foundation of R².
- R² (coefficient of determination)
- The share of the variation in y that the model explains: R² = SSE/SST = 1 − SSR/SST, always between 0 and 1. In simple regression it equals the squared correlation r². Report it as a percentage of variation explained.
- Functional form
- How x and y enter the (still linear-in-parameters) model — level-level, log-linear, linear-log, log-log or quadratic. The chosen form fixes how the slope and elasticity are interpreted.
- Marginal effect
- The change in y per one-unit change in x, ∂y/∂x. It is the constant β₁ only in a pure linear model; in log and quadratic forms it depends on x, so you must evaluate it at a stated value of x.
- Elasticity
- The percentage change in y for a 1% change in x, (∂y/∂x)·(x/y). In a log-log model it is read directly as the slope β₁, which is why log-log forms are popular for demand and cost relationships.
- Semi-elasticity
- The log-linear special case: with ln y = β₀ + β₁x, a one-unit change in x is associated with a 100·β₁ percent change in y — a percentage on y but a unit on x (e.g. β₁ = 0.078 on years of schooling ≈ 7.8% higher wage per year).
- Quadratic term & turning point
- Adding β₂x² lets the marginal effect change with x: the effect is β₁ + 2β₂x, and it switches sign at the turning point x* = −β₁/(2β₂). A negative β₂ gives a hump (rise then fall); a positive β₂ gives a U-shape.
- Adjusted R²
- A version of R² that penalises extra regressors, R̄² = 1 − (1 − R²)(n − 1)/(n − k − 1). Because raw R² can only rise as variables are added, adjusted R² (or an F-test) is what you use to compare models of different size.
Goodness of Fit and Functional Forms FAQ
What is the difference between SSE and SSR — the letters keep swapping?
The letters are not standardised across textbooks and software, which is exactly the trap. In Adelaide's WWLJ convention SSE is the Explained sum of squares Σ(ŷᵢ − ȳ)² and SSR is the Residual sum of squares Σ(yᵢ − ŷᵢ)², and SST = SSE + SSR is the total Σ(yᵢ − ȳ)². Rather than memorise the letters, anchor on the words: one piece is the variation the line explains, the other is what it leaves over. Then R² = explained/total is unambiguous.
Does a low R² mean my model is useless?
No. A low R² only says the model explains a small share of the variation in y, which is common with cross-sectional or noisy data — a classic example regresses CEO salary on firm return and gets R² ≈ 0.013, yet the relationship can still be real and statistically significant. Goodness of fit (R²) and significance (the t-test on the slope) are separate questions, so quote both and never dismiss a model on R² alone.
How do I decide between level, log and quadratic forms?
Choose by theory, by the shape of the marginal effect, and by fit. Ask whether the effect of x should be a constant number of units (level-level), a percentage / elasticity (a log form), or rise then fall (a quadratic). If you want an elasticity directly, use log-log; if you want a percentage change in y per unit of x, use log-linear. Fit helps within one dependent variable, but you cannot compare R² across a y-model and a ln y-model.
Why can't I just compare R² to pick the better model?
Because raw R² can only rise (or stay the same) when you add regressors, even useless ones, so it will always favour the larger model — that is not evidence of improvement. Use adjusted R², which penalises extra variables and can fall, or run an F-test on the added terms. And never compare R² across models with different dependent variables (y versus ln y), since they explain the variation in different quantities.
For a quadratic like y = β₀ + β₁x + β₂x², what is 'the effect of x'?
It is β₁ + 2β₂x, not just β₁ — and you must plug in the value of x the question gives you, because the effect is no longer constant. If asked when the relationship turns (peaks or bottoms out), set the effect to zero and solve: the turning point is x* = −β₁/(2β₂). A negative β₂ means a hump (the effect starts positive and becomes negative past x*).
How is this topic tested in the exam?
It appears in both parts of the closed-book final. Part A multiple-choice items ask you to interpret a log, quadratic or interaction coefficient (getting the percentage-versus-unit direction right). Part B worked-answer questions make you recover R² from sums of squares, compute a marginal effect at a stated x, find an elasticity, or solve for a turning point — always finishing with a plain-English sentence in the variables' real units.
Exam move
Treat Week 3 as two skills that are examined separately: measuring fit and choosing form. For fit, be able to move fluently between SSE, SSR, SST and R² = SSE/SST = 1 − SSR/SST in either direction, and always attach a one-sentence interpretation ('the model explains X% of the variation in y') — that sentence carries marks. For form, build a small table you can reproduce from memory — level-level (unit per unit), log-linear (100·β % per unit), linear-log (β/100 units per 1%), log-log (β = elasticity) and quadratic (effect β₁ + 2β₂x, turning point −β₁/2β₂) — and drill saying the percentage-versus-unit direction out loud before writing a number, because flipping it is the single most common error. Practise the recurring Part-B moves end to end: recover R² from a partial ANOVA table, interpret each coefficient with the correct rule, differentiate a quadratic and evaluate the marginal effect at the stated x, and compute an elasticity as (∂y/∂x)·(x/y). Remember the two guardrails throughout: raw R² only rises with more regressors (use adjusted R² or an F-test to compare sizes), and R² is never comparable across models whose dependent variable differs.
