Post on 10-Jul-2020
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