AAEC4510/6510: Hedonic Pricing Homework Assessment

Creating Distance Variables and Reporting Averages

A new variable miles_cbd is created by converting the original cbd (feet) into miles by dividing by 5,280.
Similarly, miles_incinerator is created by converting dist (feet) into miles.

After generating these two variables, the mean distance to the CBD (in miles) and the mean distance to the incinerator (in miles) are reported.
(Insert your computed averages here.)

Graphing Housing Price vs Distance to CBD

A graph is generated using housing price (in dollars) on the y-axis and miles_cbd on the x-axis.

Interpretation:
The plotted relationship indicates whether housing prices rise or fall with increased distance from the CBD.
You may find either:

• Prices decrease with distance (typical in compact cities), or

• Prices increase with distance (typical where suburban homes are larger and more expensive).

(Insert your interpretation based on your actual graph.)

Graphing Housing Price vs Distance to Incinerator

A second graph is produced using housing price (dollars) as the dependent variable and miles_incinerator as the independent variable.

Interpretation:
Normally, housing prices increase as distance from an incinerator increases due to environmental disamenities.
(Insert your interpretation from your graph.)

Hedonic Regression with Log Housing Price

A log-level hedonic regression is run with the natural logarithm of housing price as the dependent variable and the following explanatory variables:

• age of house

• number of bathrooms

• log of square footage (house)

• log of lot size

• distance to CBD (miles)

• distance to incinerator (miles)

You must report estimated coefficients, standard errors, t-values, and p-values.
(Insert your regression outputs as sentences instead of tables.)

Interpretation of Regression Results

a) Percentage of Variation Explained

The R-squared value indicates how much of the variation in log housing prices is explained by the included attributes.
(Insert your R⊃2; interpretation.)

b) Variables with Statistically Significant Positive Effects

List all variables whose coefficients are positive and have p-values less than 0.05.
Typical examples include bathrooms and house size.

c) Variables with Statistically Significant Negative Effects

List variables with negative coefficients and p-values < 0> Common findings may include:

• age

• closeness to incinerator (if negative)

d) Does the Coefficient on Distance to CBD Match Graph (2)?

Explain whether the regression coefficient aligns with the simple graph.

Often, the graph shows higher suburban home prices, but regression controls for:

• bigger houses

• larger lots

• newer construction

Once these are controlled, the relationship can flip.
Thus, the regression provides the marginal effect, not the average relationship.

e) Interpretation of Each Coefficient

Interpret all coefficients correctly according to log–log or log–level formats:

• Log–log coefficients (e.g., log house size): % change in price for % change in size.

• Log–level coefficients (e.g., distance variables): % change in price for 1-unit change in distance.

• Level–log coefficients: absolute dollar change for % change in explanatory variable.

Also note whether signs are intuitive (e.g., bathrooms ↑ → price ↑, age ↑ → price ↓).

Linear Regression

a) Report the Regression Results

You regress raw housing price on:

• age

• bathrooms

• rooms

• house area

• lot area

• distance to CBD

• distance to incinerator

(Insert your outputs in paragraph form.)

b) Interpretation of Distance to Incinerator Coefficient

Explain whether increasing distance from the incinerator increases or decreases price and whether this variable is statistically significant at the 5% level.

c) Predicting Sales Price Using Given House Attributes

Plug the following values into your estimated regression equation:

• age = 20

• bathrooms = 3

• rooms = 5

• house area = 3,000 sq ft

• lot area = 30,492 sq ft (0.7 acre)

• distance to CBD = 4 miles

• distance to incinerator = 1 mile

Insert your calculated predicted price here.

d) Predicted Price When Distance to Incinerator = 2 Miles

Repeat the calculation with distance = 2 miles.

Explain whether the change in price equals the estimated coefficient.
It usually will not because:

• the regression has multiple interacting independent variables

• the effect depends on the full model structure

Hedonic Analysis of Forestland 

Report average values for:

• nracre

• ppt

• tmean

• tmax

• tmin

• goodsoil

Regression of Net Return on Mean Temperature

The regression is:
net return = α + β × mean temperature

Interpret β:
It represents the change in net return (in dollars per acre) from a one-degree Celsius increase in mean annual temperature.

Explain whether hotter regions have higher or lower forest returns.

Quadratic Temperature Regression

The model becomes:
net return = α + β × tmean + γ × tmean⊃2;

Interpretation rules:

• β > 0 and γ < 0>

• Returns increase with temperature until the turning point.

State whether your results follow this shape.

Graph of Net Return vs Temperature (Quadratic)

Describe your graph:

• upward curve then flattening

• or monotonic increase

• or decreasing shape

Explain the curvature based on your coefficients.

Adding Precipitation to the Model

Model:
net return = α + β × tmean + γ × tmean⊃2; + δ × ppt

Interpret δ:

• how much net return changes with one-inch increase in precipitation

• whether δ is statistically significant at the 5% or 10% level

Comment on whether β and γ change substantially after adding precipitation.

Adding Soil Quality

Model:
net return = α + β × tmean + γ × tmean⊃2; + δ × ppt + θ × goodsoil

Interpret θ:

• A positive θ means counties with higher good-soil share have higher forest returns.
  Discuss whether soil quality has a meaningful economic impact.

Log-Linear Specification

Dependent variable becomes log(nracre):
log(nracre) = α + β × tmean + γ × tmean⊃2; + δ × ppt + θ × goodsoil

Interpret θ:

• It is a semi-log coefficient

• θ × 100 ≈ % change in net return for a 1-unit change in goodsoil

Research Questions the Model Could Address

Example questions:

1. How will climate change (rising temperatures) affect forest profitability across U.S. regions?

2. Does soil quality or precipitation play a larger role in determining economic value of pine forestlands?

Additional Relevant Variables to Improve the Model

One example variable:
Elevation – affects temperature, rainfall patterns, species growth, and profitability.
Higher elevation forests may produce lower or higher net returns depending on weather and soil conditions.

Brief Summary of Assessment Requirements

This homework has two main parts (total 100 points):

Part 1 Hedonic analysis of residential properties (50 points)
Using the provided hprice3 dataset, students must:

• Create two new distance variables (miles_cbd, miles_incinerator) by converting feet → miles and report their sample means.

• Produce two scatter/relationship graphs: housing price (dollars) vs miles_cbd, and housing price vs miles_incinerator (label axes; describe linear or non-linear patterns).

• Run a log-level hedonic regression: ln(price) on age, bathrooms, ln(house area), ln(lot area), miles_cbd, and miles_incinerator; report coefficient estimates, SEs, t-values, and p-values.

• Interpret the regression: R⊃2;, which variables have statistically significant positive/negative effects at 5%, whether the miles_cbd coefficient confirms the simple graph, and interpretation of each coefficient (pay attention to log–log, log–level interpretation).

• Run a level (non-logged) regression of price (dollars) on age, bathrooms, rooms, house area, lot area, miles_cbd, miles_incinerator; report results, interpret miles_incinerator coefficient and its significance, predict price for a specified house profile, re-predict when miles_incinerator = 2 and comment on changes relative to the coefficient.

Part 2 Hedonic analysis of forestland (50 points)
Using the loblolly dataset (995 plots), students must:

• Report means for nracre, ppt, tmean, tmax, tmin, and goodsoil.

• Regress nracre on tmean; interpret coefficient (unit meaning).

• Extend model to quadratic in temperature (tmean and tmean⊃2;); characterize shape (e.g., inverse U) and explain turning point intuition.

• Plot net return vs temperature using the quadratic fit and describe curvature.

• Add ppt to the quadratic model and interpret the precipitation coefficient (statistical significance at 5%/10% and effect on β, γ).

• Add goodsoil share and interpret its effect on net returns.

• Re-specify with ln(nracre) as dependent variable and interpret soil coefficient in semi-log terms.

• Propose two research questions the final model could address and suggest at least one additional variable (e.g., elevation) with brief justification.

Deliverables & format: final answers in Word/PDF, include intermediate code/files (R script, Excel sheets or STATA do file). No tables required for this submission is acceptable but numerical outputs must be provided where applicable.

How the Academic Mentor Guided the Student 

The mentor followed a scaffolded, hands-on approach so the student learned both the econometric reasoning and the technical execution.

Data preparation & creation of distance variables

• What mentor taught: Converting units correctly (feet ÷ 5,280 = miles), naming conventions, and verifying new variables with summary() or simple descriptive checks.

• Student task: Create miles_cbd and miles_incinerator, run simple means and histograms to check distributions.

• Why it matters: Correct unit scaling is essential for interpretable coefficients and for matching the units used in predictions.

Exploratory graphs 

• What mentor taught: How to make clear scatter plots with axis labels, trend lines (linear and loess/smoothed), and how to read visual patterns (direction, curvature, heteroskedasticity).

• Student task: Produce two plots, note whether relationships appear linear/nonlinear, and prepare a concise visual interpretation.

• Why it matters: Visuals reveal raw associations and potential nonlinearity or influential observations before formal modeling.

Log-level hedonic regression 

• What mentor taught: Rationale for log transformation (stabilize variance, interpret coefficients as percent changes), model specification, and how to obtain and report coefficient, standard error, t-stat, and p-value.

• Student task: Estimate ln(price) = α + β1 age + β2 bathrooms + β3 ln(house area) + β4 ln(lot area) + β5 miles_cbd + β6 miles_incinerator. Export coefficient estimates and diagnostics.

• Why it matters: The log model gives elasticities for logged regressors and percent effects for level regressors; it often produces better behaved residuals.

Interpreting regression outputs

• What mentor taught: (a) interpreting R⊃2; for ln(price), (b) identifying statistically significant positive/negative predictors at the 5% level, (c) reconciling graph vs. regression graphs show unconditional relationships while regression gives conditional (marginal) effects after controlling for other attributes, and (d) the precise interpretation of each coefficient depending on log/level status.

• Student task: Write clear sentences stating which variables are significant and the direction of effects; provide intuition for any discrepancies between plot and coefficient (e.g., omitted variable confounding such as house size).

• Why it matters: Students learn causal thinking and why multivariate controls change apparent relationships from raw plots.

Level regression with rooms included & predictions

• What mentor taught: How to run the level model (price in dollars) including rooms, correctly interpret the miles_incinerator coefficient (dollars per mile), test significance at 5% level, and how to use estimated coefficients to compute predicted prices manually or via predict() function.

• Student task: Estimate the model, evaluate miles_incinerator p-value, compute predicted price for the specified house profile, recompute for 2 miles, and explain why the change in predicted price may not exactly equal the coefficient (due to intercept and other covariates and model structure).

• Why it matters: Practical application of regression for prediction and understanding marginal vs. total effect.

Forestland: descriptive statistics & simple regression

• What mentor taught: How to compute means for nracre, ppt, tmean, tmax, tmin, goodsoil and why those summaries matter; then run nracre = α + β tmean and interpret β as $/acre per 1°C.

• Student task: Report means and interpret the sign and magnitude of β in practical units.

• Why it matters: Grounding econometric coefficients in real units makes results actionable for policy and management.

 Quadratic temperature term and plotting

• What mentor taught: Adding tmean⊃2; to capture nonlinear marginal effects, how to interpret β and γ jointly, compute turning point (−β/(2γ) when γ<0>

• Student task: Estimate quadratic model, compute turning point, and produce a plot of predicted nracre vs tmean.

• Why it matters: Many environmental responses are nonlinear; the quadratic captures diminishing returns and optimal temperature ranges.

Adding precipitation and soil quality

• What mentor taught: How to sequentially add ppt and goodsoil to see marginal contributions, check statistical significance (5% and 10% thresholds), and evaluate how coefficients for temperature terms change when covariates are included.

• Student task: Estimate expanded models, interpret δ (ppt) and θ (goodsoil), and comment on economic significance (not just statistical).

• Why it matters: Model expansion reduces omitted variable bias and reveals the relative importance of climatic and soil factors.

Log-linear specification and interpretation

• What mentor taught: Benefits of log(nracre) dependent variable (percent interpretation), and that θ×100 approximates the percent change in nracre for a one-unit change in goodsoil.

• Student task: Estimate semi-log model and write a short interpretation for θ.

• Why it matters: Choosing functional form affects interpretability and can stabilize variance.

Research questions and additional variables

• What mentor taught: How to derive researchable questions from the final model (e.g., climate change impacts, optimal site selection) and propose useful additional covariates such as elevation, age of stand, tree density, or market access.

• Student task: Provide two focused research questions and motivate one additional variable with a brief rationale.

• Why it matters: This connects empirical exercises to policy and future research.

Final Outcome and Learning Objectives Covered

Final outcome: The student completed a structured hedonic analysis for both residential housing and forestland, produced descriptive summaries and graphs, estimated log and level regressions, performed prediction exercises, implemented nonlinear specifications for environmental data, and interpreted statistical and economic significance in plain language. The student also produced reproducible code/workfiles (R/Excel/STATA) as required.

Core learning objectives achieved:

1. Data handling and variable construction (unit conversion, variable creation).

2. Exploratory data analysis and visualisation (scatter plots, fitted trends).

3. Model selection and functional-form reasoning (log vs level, inclusion of quadratic terms).

4. Estimation and reporting of regression results with statistical diagnostics.

5. Interpreting coefficient estimates correctly in terms of percent or absolute changes.

6. Predictive calculation and understanding of marginal vs total changes.

7. Managing multivariate confounding and understanding why conditional estimates differ from unconditional graphs.

8. Conducting sequential model building to assess robustness and omitted variable bias.

9. Translating econometric output into meaningful research questions and policy implications.

10. Reproducible workflow management (scripts, outputs, and documentation for submission).

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