Appunti per esame di Economic assessment of urban transformations

OLS

from those of the real one, minimizes the sum of the squares of the

+ , + .

=

differences between the observed dependent variable and the linear function of the independent

variables . �

Hats on variables indicate those that are estimated = − .



(Considerando una funzione lineare infinita, l’OLS fornisce stime efficienti dei parametri minimizzando la somma dei quadrati

dei residui, ovvero la somma dei quadrati delle differenze tra i valori osservati e i valori predetti dal modello. Trova quindi i

migliori coefficienti per descrivere al meglio la relazione tra le variabili indipendenti e dipendenti)

2

(̂ ̂ ̂ ̂

′

=1

Residual Sum of Squares ∑

) ≡ − = ( − )′( − )

� �

A matrix is an …

11 12 1

… …

ordered array of 21 2

≡� �

… … … …

numbers with the …

following form: 1 2

2

�̂ ̂ ̂ ̂ ̂

′ ′ ′ ′ ′

≡ � − = ⋯ = − 2 +

� � �

=1

(̂ ) 2′̂

′

= −2 +

̂

̂ ̂ ̂ ̂

′ ′ ′ ′ ′ ′

OLS estimator analytical formula: −2 + 2 = 0 → 2 = 2 → = → =

′ −

(

= ) ′

Classical hypotheses/assumptions of the linear model:

1) Regression function is correctly specified Data-Generating Process DGP

(|) = 

has linear nature 2

2) Full column rank of if is an matrix, then Each variable (vector of is

: () = )



not a linear combination of other model variables, and no variable is redundant

3) [|] = [ |]

1

[ |]

2

[|] = = 0

� �

 …

|]

[

′ 2

4) Homoscedasticity: |] =

[|] = [ 2

0 0 0

1

⎡ ⎤

This hypothesis requires uncorrelated (with one another) 2

0 0 0

⎢ ⎥

2

residuals. The variance-covariance matrix generated by the [|] = ⎢ ⎥

2

0 0 0

regression must therefore take on this form: ⎢ ⎥

2

⎣ 0 0 0 ⎦

5) is a non-stochastic matrix vectors of variables are measurable observations of an



economic phenomenon 2

residuals are normally distributed:

6) Model |~(0, )

is a measure of the OLS estimator’s goodness-of-fit:

2

=1

- ∑ (

= − �) ′

= − = −



�̂ ̂ ̂ ̂

′ ′ ′ ′

- = − 2 +

� ′

Where 0 ≤ ≤ 1:

2

- measures the proportion of variance in due to the variance in the regressors

2

- Since is a non-decreasing function of the number of regressors (explanatory variables),

(−1)

���

�

2

usually we calculate the so called adjusted : 2

= 1 − (1 − )

(−)

2

- Adjusted increases only if the contribution of the new variable to the regression’s “fit” more

than offsets the loss in degrees of freedom, −

A fit of an equation via OLS:

Gauss-Markov Theorem: ′ −1

Considering the OLS estimator and the generic estimator the OLS estimator is

( ) ′ = 

BLUE (Best Linear Unbiased Estimator). Linear by construction and unbiasedness because of:

′ −1 ′ ′ −1 ′ ′ −1 ′ ′ ′

( (

(|) = ( ) = ( ) ) ) + ( ) = + ( ) =

+ ) = ( 3

(|) = 0



OLS beats all other linear estimators: considering the generic linear estimator

“Best” and

=

 ′ −1 which is possible iff

then

defining the matrix as ( = 0 = 0.

) ′, () = → �

+ �

2. The hedonic price method

The hedonic price method HPM evaluates the impact of a characteristic of a good without a direct

market (pollution, car crash risk…) onto the market price of a good regularly traded on markets. Used

to evaluate impacts on residential land prices (land rent) on changes in environmental characteristics

because a good regularly traded price is linked to its intrinsic features (characteristic), or the

services/benefits it offers (externalities). Individual characteristics of a good can be “priced”



observing how prices of that good changes when others (structural characteristic, neighborhood

quality, accessibility to attractive places…) vary.

Method born in Detroit by the economist Andrew Court because characteristics of car change yearly

so to price them, one must control for each of those features holding them “constant” in an empirical

analysis: individual characteristic, time trend, and residual

= + + ⋯ + +



Residential good as multi-dimensional good The price of an apartment/building is very complex



because all the characteristics (not “substitutable” among them, so they don’t compete) may in turn

be decomposed into other determinants.

Accessibility of a residential good is priced through externalities.

Ex-ante issues: the HPM analysis can revolves around:

- Indirect costs (ex. acoustic pollution due to a large expressway close by…)

- Indirect benefits (ex. higher accessibility, proximity to positive sources such as public park…)

Considering the following linear function (must be linear?): = + + ⋯ + +

- The 10 room of an apartment has a different value of the 3 one

th rd

- The hedonic price of a characteristic: = (impatto sul Prezzo di equilibrio alla variazione della

caratteristica)

- Market prices of the composite good show simultaneously supply and demand (WTS/WTB)

- The coefficients are not stable over time because of trends and unpredictable events

- The characteristics of buyers and sellers are indirectly considered (reflected) in the price

Formally:

Implicit price function: )

= ( , … ,

1

Hedonic price of the characteristic: =

= =

Consumer utility maximization: ) )]

ℒ = (, , … , , … ,

+ [ − − (

1 1

- utility

- budget

o income

o prices of each vector

o prices for the characteristics

Estimated hedonic prices = Revealed utility Roses (1974) first suggested that hedonic prices



represent the simultaneous envelope of business proposal from both the demand and the supply side.

Logical step: 4

1) Data collection on land rent values within the area and limited time span: location and sold

properties, apartments/neighborhood/accessibility-related/environmental characteristics

(available on institutional and public sources).

2) Statistical (econometric) analysis of the relation among house prices and the characteristics

associated, with coefficient with the same unit of measure of price Regression analysis

.

estimates the impact of a characteristic on the price ex. , , , … )

= (,



= + ∗ + ∗ + ∗ + ⋯ +

1 2 3

the fictitious constant of the highest price with all the best characteristics. Regression is

is

based on the logs of variables to interpret estimated coefficients as elasticities (%).

) )

ln(

ln( = ⋯ + + ⋯ +

3

Estimated represents the role of each factor (as percentage) in the final price houses.

represents the value attributed (on average) by the market to the Air Quality, considering the

3

other variables at the same time Ceteris Paribus.



HPM works only with cardinal meaningful characteristics (measurable), otherwise

contingent evaluations must be integrated.