This overview opens the second issue of Foundations, the journal’s department of concept entries. Its context is the department’s own: it says what the eight entries and the supplement that follow are about, for whom they are written — practicing appraisers first — and in what order to read them.

The inaugural issue of Foundations built the hedonic vocabulary: heterogeneous goods, characteristics space, the hedonic price function, implicit prices, hedonic regression, and latent variables. It ended with a regression that missed seven of its eight Pacifica comparables by no more than $18,000 and the eighth — Comp H — by $46,455, and with the explanation that the miss was a latent attribute: the dataset had been built with a recent kitchen remodel on Comp H, worth about $60,000, that the regression’s variable list left out.

That loose end is this issue’s starting point, though no longer its whole extent. Eight entries appear here, in two movements. The first four address the statistical concepts issue 1 used without naming. Each is a foundational idea that working appraisers rely on constantly and almost never state, and each has a direct and demonstrable consequence for whether the value contribution a regression assigns to a feature, and so the adjustment derived from it, can be defended. The remaining four are legal-conceptual rather than statistical; they were not planned alongside the four, and this issue’s survey of recent court decisions forced them. They are introduced after the four statistical entries, on which they draw.

The four entries that follow use ordinary regression, the kind that gives one rate for each feature, so many dollars per square foot across the whole market. The Journal’s own work uses MARS, which does the same job with one difference: it lets the rate change where the sales show the price of the feature changing, so it may give $300 a foot up to 1,500 square feet and $275 a foot above. Either way the appraiser ends up with the same kind of result: a rate for each feature and a value contribution built from it, and the adjustment is the subject’s contribution less the comparable’s, the amount added to the comparable’s sale price for that feature. Whatever can go wrong with a rate goes wrong the same way whether there is one rate or several: a rate can carry the value of a feature that was left out, it can be distorted when the feature and the price push on each other, and the sales may simply not contain enough contrast to pin it down. Those are the three failures the entries describe, and they are failures of the rates, not of the method that produced them. So everything said here about ordinary regression applies to MARS, feature by feature, and within each stretch of a feature’s values that has its own rate.

For the reader who wants the mechanism.Underneath, MARS is ordinary least squares fit on a set of hinge terms, each zero on one side of a knot (the point where the rate may change) and linear on the other. The program first adds knots one at a time, keeping each that best improves the fit, and then removes those that do not earn their place (pruning), ordinarily by generalized cross-validation (GCV): a score that rewards a closer fit to the sales but charges for every term kept, so a knot stays only if it improves the fit by more than it costs. Once the terms are chosen, the fit is a least-squares fit, so the algebraic results of the four entries (what a rate estimates, how an omitted feature shifts it, whether the sales can pin it down) hold for it term by term. Standard errors and significance tests computed as if the terms had been fixed in advance do not, because the same sales chose the terms. The supplement The Barrier to Human Understanding of MARS, bound next, works this through on one model. What MARS adds is that a rate may differ between ranges and may depend on another feature; what it does not change is the list of features the model was given or the market the sales came from, and those are what the entries are about.

The first movement: why these four, in this order

The sequence is a chain. The first link names what a regression estimates; each of the other three is a different kind of failure.

Conditional Expectation (007) names what a regression is trying to estimate. Issue 1 read coefficients as implicit prices without saying what market quantity they estimated. The answer is the conditional expectation function — the expected price at each point in characteristics space (the probability-weighted average of all the prices houses with exactly those features could sell for, not the average of the sales in hand) — and naming it makes precise the questions the next three entries ask. A linear regression recovers it exactly only when it is linear, and otherwise recovers its best straight-line approximation. The entry also establishes the decomposition \(P = E[P \mid \mathbf{Z}] + \varepsilon\) with \(E[\varepsilon \mid \mathbf{Z}] = 0\), which always holds (\(P\) is the sale price, \(\mathbf{Z}\) the list of measured features, \(E[P \mid \mathbf{Z}]\) the expected price given those features, and \(\varepsilon\) what is left of the price once that expected price is taken away); the later entries then ask whether a regression’s rates can be read as the market’s price for each feature. That depends on three things: the equation it fits, the features it is given, and the sales it has.

Omitted Variable Bias (008) is the formal name for what happened to issue 1’s coefficients. In the Pacifica case the information needed could have been obtained (here, the remodel, which a permit search or inspection would reveal) and the model does not use it; a missing feature no one could measure biases the coefficients the same way. The entry gives Theil’s formula¹ and verifies it to the cent against the Pacifica data.

Endogeneity (009) is the general condition of which omitted variable bias is one special case. The entry adds simultaneity (price and a feature each acting on the other), selection (which houses sell, or are remodeled, depends on things the model cannot see) and measurement error, and makes the point that measuring the omitted variable cures the omission without necessarily curing the endogeneity: owners choose to remodel, and the choice depends on what the model cannot see.

Parameter Identification (010) asks whether a rate (a parameter of the model) could be recovered at all, by anyone, from data of this kind. It is logically prior to the other three. Whereas the problem of Omitted Variable Bias (008) was fixed by including a variable, Parameter Identification (010) exhibits a set of sales in which including the same variable fixes nothing, because the information is absent from the data relied upon rather than unused by the model.

Read together, the four supply a diagnostic ladder for any value contribution a regression assigns to a feature, and therefore for any adjustment, an adjustment being the subject’s contribution for a feature less the comparable’s, added to the comparable’s sale price. Is the estimand (the quantity to be estimated) named? Is a feature that affects price missing from the model and correlated with a feature that is in it? Is any regressor (an explanatory variable in the regression) correlated with the structural error — the price minus what the causal relationship says it should be — for any reason? And — prior to all of it — could these data have answered the question at all? The entries are printed in the order the ideas are easiest to learn; in practice the questions are asked in reverse, the last one first.

The second movement

The four remaining entries answer questions the ladder does not ask. This issue’s survey of recent court decisions crossed enough jurisdictions, purposes, and languages that four conceptual questions outgrew the survey’s footnotes: what the reference project in a compulsory acquisition (the taking of property by the state or a body given the power by statute — what United States practice calls eminent domain or condemnation) is, how valuation practice should be classified at its top level, what happens to valuation vocabulary at a language border, and how the kind of value an assignment requires differs from the approach that reaches it. One entry answers each. They are not rungs on the ladder; the survey, not issue 1’s regression, is their occasion.

The Scheme, and Effects on Value (011) is the hinge between the movements. In compulsory acquisition, value is commonly measured against a world in which the project for which the property is taken, called the scheme, was never undertaken — the rule United States practice calls the project-influence, or scope-of-the-project, rule. The entry states this in the terms 007 (Conditional Expectation) and 010 (Parameter Identification) establish: a value effect is a counterfactual quantity — the difference between the value in the world as it is and the value in a world that did not happen (here, the world without the scheme); the governing law fixes the form of the counterfactual and a tribunal (in the United States, the court, or the condemnation commission or jury) fixes the scheme’s extent in time and scope; and the identification question — whether scheme-world data can pin down no-scheme value — stays with the appraiser.

The Purposes of Valuation (012) sets out the journal’s classification of valuation practice: eight purposes at the top, from purchase and sale through insurance; beneath each, the property interest, the governing law and the basis of value (in USPAP terms, the type and definition of value), settled together; beneath those, methodology. The quarter’s instruments (in the wide sense of any formal legal text in force: a statute, decree or regulation, or a treaty), each verified against its official text, are placed under their purposes in comparative tables, and the purposes for which the survey produced nothing are reported empty rather than filled; a statutory war-damage compensation regime is placed under damages and compensation, because its objective is to measure the owner’s loss.

Terminological Divergence (013) classifies the recurring ways the valuation vocabularies of different legal systems fail to correspond: five forms of mismatch, evidenced by statutory vocabulary in twelve languages; terms already checked against the statute are kept apart from terms still awaiting that check. Its finding of record is that no statute surveyed uses a general abstract noun as a term of art for the scheme’s effect on value; some speak only directionally, through a directional pair (separate words for increase and decrease), others use a neutral noun for a change in value without making it a term of art, and some carry the idea in a verb.

The Basis of Value and the Approach (014), the fourth, separates the basis of value — the definition of value an assignment must produce — from the premise of value (the assumed circumstances of the sale: open market, orderly liquidation, forced sale, continued use) and from the approach (sales comparison, cost, income), and shows that no approach determines the basis of the number it returns.

The running example

Issue 1’s eight Pacifica comparables return, extended to sixteen and with the remodel indicator (1 if remodeled, 0 if not), which Issue 1 disclosed but did not record as a variable, now measured.

The extension rests on a fact about that dataset worth recording: once the remodel that Issue 1 disclosed is included, the dataset is exactly noiseless — every price equals a fixed formula of the features, with nothing left over. Regressing its eight prices on gross living area (GLA, square feet), lot size (square feet), view (1 = view, 0 = none), the condition score (0.00 to 10.00: the percentage of houses in the market area in worse condition, divided by ten, as estimated by the appraiser; average 5.00, good 7.00, excellent 9.00; Conditional Expectation gives the full definition), and a remodel indicator (1 for Comp H, 0 for the other seven) returns residuals of zero at every observation, and the coefficients

\[P = -30{,}000 + 600 \cdot z_{\text{GLA}} + 25 \cdot z_{\text{lot}} + 100{,}000 \cdot z_{\text{view}} + 20{,}000 \cdot z_{\text{cond}} + 60{,}000 \cdot z_{\text{remodel}} ,\] where \(P\) is the sale price (not Comp P).

Comp H’s $46,455 residual in issue 1 was the shadow of the $60,000 remodel term redistributed across the other coefficients by the omission. The example puts condition into the regression, as traditional appraisal does and as Issue 1 did. Traditional appraisal methodology is not the Journal’s method, RCA-based appraisal, which keeps condition and quality out of the regression; Conditional Expectation (007) says how it differs. The eight new comparables (I through P) are generated from the same function, so comps A–H keep their published attributes and prices unchanged and the two issues are exactly consistent.

Noiselessness is a deliberate pedagogical choice rather than a claim about real markets. It means that every gap between a coefficient fitted to the Pacifica data and its generating value is bias or a failure of identification, never noise. 008 (Omitted Variable Bias) can therefore verify the omitted-variable-bias formula exactly rather than approximately, and 010 (Parameter Identification) can exhibit, unambiguously, a set of sales in which two features always move together, so that no method can separate their rates (a rank deficiency). Real sales data never behave this way, and the entries say so where it matters.

The dataset ships as data/pacifica_comps_16.csv in the directory of each first-movement entry (Conditional Expectation through Parameter Identification) and as in the issue’s knowledge base (a set of Prolog files, a language in which facts and rules can be checked by machine), which carries the generating function as true_price/2 so that every price can be verified against it.

How to read this issue

The first movement is ordered by dependency and is best read in sequence; each entry uses vocabulary the previous one establishes. A reader who has not read issue 1 will still be able to follow 007 (Conditional Expectation), which restates what it needs, but the worked example in Omitted Variable Bias (008) refers back to Issue 1’s regression and will be richer with it in hand, and that of Parameter Identification (010) builds on its sixteen sales. The second movement is printed in one order, but some of its entries rely on entries printed after them: The Scheme, and Effects on Value (011) assumes Conditional Expectation and Parameter Identification; The Purposes of Valuation (012) leans on the basis of value defined in The Basis of Value and the Approach (014), and Terminological Divergence (013) on the scheme and the value effect of The Scheme and on The Basis of Value as well, so a reader new to the vocabulary may prefer to read The Scheme and The Basis of Value before The Purposes of Valuation and Terminological Divergence.

Practitioners looking for the immediately applicable material should read the Where appraisers encounter it and Why it matters for defensibility sections of 008 (Omitted Variable Bias) and 010 (Parameter Identification) first. Those two entries carry the results most directly usable in a report: that fit statistics (such as \(R^2\), the share of the variation in prices the model reproduces) are not evidence against bias, that a biased model can predict well and still yield unusable adjustments, and that the bracketing requirement answers two questions, not one: whether the sales contain contrast in a feature, varying apart from the others (identification) and whether the subject lies inside the sales (support); bracketing secures only the second.

Editorial posture

Two points of posture carry over from Issue 1, and a third is added.

The entries are not a textbook chapter. Each concept was selected because appraisers genuinely encounter it, usually under another name or under none, and because naming it changes what can be said in a report. Concepts with no practice contact are deferred regardless of their standing in the literature.

The department takes no position on whether hedonic methods should displace traditional grid analysis. The claim throughout is narrower and, we think, harder to dispute: the grid is already a statistical procedure, and the concepts here describe what it is doing whether or not the appraiser using it thinks in these terms.

Finally, the entries are meant to be citable. Each is a separate submission with its own DOI, its own Prolog galley encoding its claims in machine-checkable form, and a stable identifier in the journal-wide dictionary. Corrections are made as new versions, not as silent edits.

The supplement and the eight entries in this issue:

1. The Barrier to Human Understanding of MARS — a supplement, bound between this overview and 007 (Conditional Expectation), that shows on one worked model how the terms a MARS program prints become the rates, value contributions and adjustments the four statistical entries reason about.

2. Conditional Expectation — what a regression actually estimates.

3. Omitted Variable Bias — Theil’s formula, verified to the cent on the Pacifica data.

4. Endogeneity — omitted variables, simultaneity, selection and measurement error as one condition.

5. Parameter Identification — what the data could reveal, prior to any estimator.

6. The Scheme, and Effects on Value — the acquiring project as the reference for attributing value effects; the hinge to the ladder.

7. The Purposes of Valuation — eight purposes at the top of the journal’s classification of practice.

8. Terminological Divergence — five recurring forms of cross-language mismatch in valuation vocabulary.

9. The Basis of Value and the Approach — which value an assignment must produce, and why the technique never settles it.

Looking ahead. Issue 3 (2027 Q1) takes up the spatial cluster: submarkets, the spatial weights matrix, the modifiable areal unit problem, and kriging. The dependency on the present issue is direct — spatial dependence is, among other things, a structured form of the omitted-variable problem developed here.