ECON 166

Lecture 9Introduction to Regression Analysis in Urban Economics

continued

J. M. Pogodzinski

Some Regression Diagnostics

R2 – the proportion (on a scale from zero to one) of the variation in y

(the dependent variable) explained by the regression (i.e., all the x’s)

Adjusted R2 – R2 including a penalty for using “too many” variables,

i.e., for creating a non-parsimonious model

t-statistic – a statistic associated with each coefficient estimate of a

regression model. The t-statistic has the same sign as the

coefficient estimate. A critical value for the absolute value of a t-

statistic is 1.96 (or approximately 2). We say of coefficient estimates

whose t-statistics are greater than or equal to the critical value that

these coefficient estimates are statistically significantly different from

zero.

Empirical Testing of Monocentric City Model

Intensity of Land Use Population density = people/land area

Declining density as function of distance from city center.

Assume this follows the equation

From McDonald and McMillen, Urban Economics and Real Estate

Empirical Testing of Monocentric City Model

Logarithmic transformation:

Second equation is “linear in the logarithms”; so it can be estimated by OLS; estimates (and t-statistics and R2) appear in the following table.

From McDonald and McMillen, Urban Economics and Real Estate

Population Density Gradients in 2000

Metro Area

Population

(millions) Gradient T-Value R2

New York 17.6 0.124 44.518 0.345

Los Angeles 13.8 0.051 19.124 0.134

Chicago 8.6 0.087 31.009 0.365

Washington DC 6.8 0.073 17.498 0.228

SF-SJ-Oakland 6.1 0.069 15.008 0.203

Philadelphia 6.7 0.080 18.460 0.200

Boston 4.5 0.100 23.424 0.406

Detroit 4.6 0.072 21.446 0.287

Dallas 5.0 0.046 9.154 0.089

Houston 4.5 0.082 13.030 0.183

Atlanta 4.2 0.069 14.789 0.278

Miami 3.6 0.028 5.892 0.059

Seattle-Tacoma 3.4 0.059 11.170 0.165

Phoenix 3.1 0.064 7.694 0.083

Minneapolis-St. Paul 2.9 0.111 18.978 0.342

Cleveland 3.1 0.063 13.611 0.196

San Diego 2.8 0.057 9.472 0.145

St. Louis 2.5 0.088 12.288 0.241

Denver 2.6 0.085 10.341 0.158

Tampa-St. Pet. 3.1 0.021 3.295 0.019

Pittsburgh 2.7 0.088 16.680 0.298

Portland, OR 2.2 0.091 9.007 0.165

Cincinnati 2.5 0.080 11.424 0.222

Sacramento 2.5 0.079 8.304 0.153

Kansas City 1.9 0.074 7.345 0.102

From McDonald and McMillen, Urban Economics and Real Estate

Empirical Testing of Monocentric City Model

Evaluating the Density Gradient

•t-statistics: all above 2

•R2 “relatively low”; indication of (a kind of) specification error

•Alternative functional forms (e.g., polynomials:

lnD(x)=α+g1x+g2x2 +g3x

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J. M. Pogodzinski

Hedonic Regression

Basic Concepts

the term “hedonic” – pleasure (+) and pain

(-)

consumers demand characteristics

consumption of an item is consumption of a

bundle of characteristics

there is an explicit market in the item, but no

explicit (only implicit) markets in characteristics

Examples: pocket calculators, houses

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Hedonic Regression Example

hedonic_example.xls

• Source of data:

http://www.sfgate.com/homes/

• Variables included in the analysis

– House price (asking price)

– Number of bedrooms

– Number* of bathrooms

– Square feet of floor area

– Year built

* “half” or “partial” bath entered as one-half

Dependent (LHS)

variable

Independent

(RHS) variables

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Expectations of Sign and Significance

of Variables in Hedonic Equation

House prices (dependent or LHS variable) is expected to be:

• Positively related to the number of bedrooms

– but WATCH OUT! Remember we are holding “other factors” (like floor area) constant.

So if consumers prefer floor plans that break up space into smaller subspaces, rather

than preferring larger open areas, the number of bedrooms will be positively related to

house price. Basically, this expectation is reasonable if people have large families and

each person in the family wants his or her “private space”

• Positively related to the number of bathrooms

– but WATCH OUT! In this and similar situations there may not be much variation in the

variable, which makes the coefficient estimates less efficient

• Positively related to square feet of floor area

– The mother of all housing hedonic variables – bigger houses cost more

• Positively related to Year built

– Usual expectation is that the greater is the year built (the younger is the structure – other

factors held constant), the greater the house price. But WATCH OUT! Year Built may

be a proxy variable for “vintage” or character of the neighborhood

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How Good are the Data?

• Because it comes from real estate listings, the prices are asking prices not transaction prices

• Not all listings included square feet. This may introduce a subtle bias in the sample, if the listings not including square feet differ systematically from those including square feet.– Different stories of bias can be told

• No indication of how long any particular house has been on the market and whether the asking price has changed

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Results of Hedonic Regression

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Interpreting the Coefficient

Estimates• Number of bedrooms is negative, statistically

significant, and large– Interpretation: each additional bedroom reduces the

value of the house by about $140,000

• Number of bathrooms is negative, not statistically significant, and large– Interpretation: each additional bathroom reduces the

value of the house by about $124,000, but we have little confidence that this effect is statistically valid

• Square feet of floor area is positive, statistically significant, and large– Interpretation: Each additional square foot of floor area

increases the value of the house by almost $580

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Transforming Variables

Suppose the functional form of a relationship between x and y is assumed to be:

y=Axα

which is a non-linear equation.

However, by taking the natural logarithm of both sides of the equation gives

ln(y)=ln(A) + αln(x)

This expression that is linear in the logarithms of the transformed variables

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Transforming Variables

Why transform variables?

If we use the functional form y=Axα the

parameter α represents an elasticity – the

elasticity of y with respect to x.

Elasticity is a measure of how sensitive one

variable is to changes in another variable –

expressed in terms of percentages

α is the percent change in y for a 1% change in

x

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Elasticity

What’s so good about elasticity?

It is scale-free: it does not depend on the units in which x and y are measured.

Some common elasticities

own-price elasticity of demand

cross price elasticity of demand

income elasticity of demand

Some uncommon elasticities

elasticity of house price with respect to floorarea

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Hedonic Regression Transformed

Sometimes debatable about which

variables should be transformed.

Teaser question: should you make a

logrithmic transformation of a

dummy variable?

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(Re)-Visit the Bathroom

Introduce

quadratic term

into bathroom

relationship

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Hedonic Regression with Quadratic Term

fro Bathrooms

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Bathroom Quadratic

Could also introduce a

cubic (3rd degree) term

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Applied Regression Analysis

Example

Determinants of public transit use for

work (number of riders)

• Number of workers

• Median household income

• Distance to CBD

Percent of workers using public transit

Percent of workers using public transit