Question

Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and Mcclain (1995) , the following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist): $$\begin{aligned} \widehat{\log (\text {price})} &=9.40+0.312 \log (\text {dist}) \\ n &=135, R^{2}=0.162. \end{aligned}$$ i. Interpret the coefficient on log (dist). Is the sign of this estimate what you expect it to be? ii. Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist? (Think about the city's decision on where to put the incinerator. iii. What other factors about a house affect its price? Might these be correlated with distance from the incinerator?

Short answer

i. The coefficient 0.312 means a 1% increase in distance increases price by 0.312%. Yes, the sign is expected. ii. Simple regression may not be unbiased due to omitted variable bias. iii. Factors like house size, age, and neighborhood affect price and might correlate with distance.

Step-by-step solution

Understanding the Coefficient on log(dist)

In the regression equation \( \widehat{\log (\text{price})} = 9.40 + 0.312 \log (\text{dist}) \), the coefficient 0.312 on \( \log (\text{dist}) \) represents the elasticity of the housing price with respect to the distance from the garbage incinerator. This means a 1% increase in distance from the incinerator is associated with a 0.312% increase in the housing price, all else being equal. The positive sign indicates that houses further away from the incinerator are more expensive, which is expected as people often prefer to live further from waste facilities due to potential nuisances.

Analyzing Bias in Simple Regression

Simple regression may not provide an unbiased estimator of the elasticity of housing price with respect to distance from the incinerator due to omitted variable bias. The location of an incinerator isn't randomly chosen; it might be placed in less desirable areas, which could naturally have lower property values irrespective of distance. Thus, there could be other factors influencing both property prices and location decisions, leading to endogeneity.

Considering Other Influential Factors

Other factors affecting house prices include size, age, amenities, neighborhood quality, schools, and local taxes. These factors could be correlated with the distance to the incinerator. For example, larger houses might be built further from industrial sites, such as incinerators, potentially influencing price and confounding the relationship if not controlled for in the regression.

Key concepts

Regression Analysis

Regression analysis is a statistical method used to understand the relationship between variables. In our given exercise, the regression equation \[\widehat{\log (\text{price})} = 9.40 + 0.312 \log (\text{dist})\]is employed to analyze how the distance from a garbage incinerator affects housing prices. Here, \(\log(\text{price})\) is the dependent variable, and \(\log(\text{dist})\) is the independent variable. The goal is to see how much a change in one variable can affect the other.

• The coefficient 0.312 is a key part of understanding this relationship as it measures the elasticity, or responsiveness, of the housing price to changes in distance.
• An \(R^2\) value of 0.162 indicates that roughly 16.2% of the variation in housing prices is explained by the model. This tells us that other factors might influence housing prices alongside the distance from the incinerator.

Elasticity

Elasticity measures the responsiveness of one variable to changes in another. In economics, it is a critical concept because it gives insight into how a change in one factor can affect an outcome. In the context of housing prices, elasticity is reflected in the coefficient 0.312 from the regression equation.

• This means that if the distance from the incinerator increases by 1%, housing prices are expected to increase by 0.312%, assuming all other factors remain unchanged (ceteris paribus).
• A positive elasticity value in this case aligns with the assumption that people prefer to reside further from undesirable utilities, which drives up housing prices at greater distances.

Omitted Variable Bias

Omitted variable bias occurs when a regression model leaves out relevant variables, such that the estimated effects of included variables are biased. In our exercise, the regression's aim is to assess the impact of distance from an incinerator on housing prices.

• If other factors like neighborhood quality or house size that affect housing prices are left out, and these factors are correlated with distance from the incinerator, then the estimates become skewed.
• This bias creates unreliable results, indicating an observed effect that is actually due to the influencing hidden variables, not solely the distance from the incinerator.

Endogeneity

Endogeneity is a major challenge in regression analysis. This happens when an independent variable is correlated with the error term in the model, usually due to omitted variable bias or simultaneous causality. Regarding the location of the incinerator from our example:

• The placement of an incinerator is likely not random; it may be purposely located in areas perceived as less desirable, which could inherently affect property values irrespective of distance.
• This strategic location decision suggests a two-way causality between distance and residential appeal, creating an endogeneity issue. Simply put, external factors influencing the positioning of an incinerator also impact the housing prices.

Housing Market Analysis

Analyzing the housing market involves looking at all variables influencing property values. While proximity to undesirable features like an incinerator is significant, many other factors also come into play.

• Key factors are house size, age, amenities, quality of schools, crime rates, and access to transportation. These factors can have a significant correlation with the distance from industrial sites like a garbage incinerator.
• For instance, larger homes or neighborhoods with better amenities might be located further away from industrial areas, potentially confounding the relationship observed if such factors are not considered in the model.

Understanding these broader elements is crucial for a comprehensive housing market analysis, ensuring all critical variables are accounted for, thus providing more precise and reliable estimates.