tiểu luận kinh tế lượng tài chính factors that affect housing prices among location and region

Group 10
I.

Econometrics Report

INTRODUCTION..................................................................................................................3

1.

Overall about econometrics.............................................................................3

2.

Why choosing OLS?...................................................................................................4

II. QUESTION OF INTEREST...............................................................................................5
III.

ECONOMIC MODEL..........................................................................................................5

1.

Choosing the variables.......................................................................................5

2.

Embedding that target in a general unrestricted model (GUM)
8

IV. ECONOMETRICS MODEL....................................................................................................9
1.

Population regression function (PRF).....................................................9

2.

Sample regression function (SRF)...............................................................9

V.

DATA COLLECTION.........................................................................................................10

1.

Data overview..........................................................................................................10

2.

Data description...................................................................................................10

VI. ESTIMATION OF ECONOMETRIC MODEL...................................................................10
1.

Checking the correlation among variables:.......................................10

2.

Regression run........................................................................................................12

VII.

CHECK MULLTICOLLINEARITY AND HETEROSCEDASTICITY.........................15

1.

Multicollinearity.................................................................................................15

2.

Heteroskedasticity..............................................................................................16

VIII.

HYPOTHESES POSTULATED...................................................................................19

1.

The t test.................................................................................................................19

2.

Confidence Intervals.........................................................................................21

3.

P­ Value......................................................................................................................22

4.

Testing the overall significance: The F test................................23

IX. RESULT ANALYSIS AND POLICY IMPLICATION..................................................24
X.

CONCLUSION.....................................................................................................................24

XI. REFERENCES.....................................................................................................................25
Y
Figure
Figure
Figure
Figure

1........................................................7
2........................................................9
3.......................................................10
4.......................................................11
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Figure
Figure
Figure
Figure
Figure

5.......................................................13

6.......................................................15
7.......................................................16
8.......................................................18
9.......................................................21

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I. INTRODUCTION
1.Overall about econometrics
Econometrics   is   the   application   of   statistical   methods   to
economic data and is described as the branch of economics that
aims to give empirical content to economic relations. Precisely
speaking,   it   is   the   quantitative   analysis   of   actual   economic
problems,   based   on   the   concurrent   development   of   theory   and
observation, related by appropriate methods of inference. It is
understandable   that   economist   make   comparison   econometrics   is
like an effective tool to convert mountains of data into extract
simple relationships.
The reason why econometrics is effective is economics theory
use   statistical   theory   and   mathematical   statistics   to   evaluate
and   develop   econometrics   method.   In   reality,   econometrics   help
economists   to   assess   economic   theories,   developing   econometrics
model, analyzing and forecasting the economic history.
Aware of the importance of econometrics to economic phenomena,
our group decides to carry out a research of econometrics: “The

factors that have influence on median housing price” and aim to
analyze statistic and point out differences and their reason of
price level.
The data set has 506 observations with 12 variables in total.
We choose 6 variables: price, crime, nox, rooms, dist and proptax
to do the research in which price is dependent variable and the
other five are independent variables. The general method used in
this research is OLS (ordinary least squares). In addition, the
specialized method is estimate, running Stata software as well.
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During carrying out this research, our group is so lucky to
be guided thoroughly by Dr. Dinh Thi Thanh Binh. We are grateful
for everything you have taught us!
This is the first time our group carry out an econometrics
research, our performance is unavoidable to have many mistakes.
It would be a pleasure if we can receive the feedback from you to
better ourselves next time.

2.Why choosing OLS?
Ordinary   least   squares (OLS)   is   a   type   of linear   least
squares method for estimating the unknown parameters in a linear
regression model.   OLS   chooses   the   parameters   of   a linear
function of   a   set   of explanatory   variables by   the   principle
of least   squares:   minimizing   the   sum   of   the   squares   of   the

differences   between   the   observed dependent   variable in   the
given dataset and those predicted by the linear function. 
With the six selected variables, we use the OLS model because
all regressions variable are exogenous variables, the effects of
independent   variables   on   the   dependent   variable   are   linear
effects.   In   addition,   the   estimates   calculated   by   means   of   the
least squares OLS are linear estimates that are not deviate and
are better than others.
When using OLS, we have some basic assumptions:
1.
2.

The regression model is linear in the parameters
X values are fixed in repeated sampling, which means Xi

and ui are uncorrelated 
3.
Zero mean value of disturbance (E(ui)) =0)
4.
Homoscedasticity or equal variance of ui : var(ui) = 
5.
No correlation between disturbances
6.
The model is correctly specified.
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7.

II.

Econometrics Report
Number of observations must be greater than the number

of parameters to be estimated.
8.
X values in a given sample must not be the same.
9.
No perfect multicollinearity.
10. Normal distribution. 
QUESTION OF INTEREST
We   have   always   been   wondering   “Why   do   housing   prices   among

locations   and   regions   differ   so   much?”.   Housing   prices   are
affected   by   many   different   factors   such   as   structure,
neighborhood, accessibility, air pollution and so on. To seek the
answer to that question, our group is going to use the collected
data to build and run the regression model and then the results
are   going   to   be   analyzed   to   finally   answer   the   question   of
interest above.
 
III. ECONOMIC MODEL
According the provided data, the economic model used in this
report is an empirical one. Note that the fundamental model is
mathematical; with an empirical model, however, data is gathered
for the variables and using accepted statistical techniques, the
data are used to provide estimates of the model's values.
1. Choosing the variables
Having   described   data   via   the   command   “des”   in   file…   from

Stata software, we gain the result as following: 
. des
 obs:           506   
 vars:            12                            31 Oct 1996 
16:37
 size:        22,770  

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Variable name

storag

display

valu

e type

format

e

variable label

median housing 

price

float

%9.0g

 

price, $
crimes committed

crime

float

%9.0g

 

per capita
nit ox concen; 

nox

float

%9.0g

 

parts per 100m
avg number of 

rooms

float

%9.0g

 

rooms
wght dist to 5 

dist

float

%9.0g

 

employ centers
access. index to

radial

byte

%9.0g

 

rad. hghwys
property tax per

proptax

float

%9.0g

 

$1000

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Econometrics Report
average student­

stratio

float

%9.0g

 

teacher ratio
perc of people 

lowstat
lprice
lnox
lproptax
Figure 1

float
float
float
float

%9.0g
%9.0g
%9.0g
%9.0g

 
 
 
 

'lower status'
log(price)

log(nox)
log(proptax)

The above table reveal that this is the statistic of factors
which have influence in housing price via 506 observations. After
discussing   carefully,   our   group   jumped   into   a   conclusion   to
choose   a   dependent   variable   Y:   Price,   independent   variable
contains:






X1­crime
X2­nox
X3­rooms
X4­dist
X5­proptax

2. Embedding   that   target   in   a   general   unrestricted   model
(GUM)
In   its   simplest   acceptable   representation   (which   will   later   be
specified in the econometric model), the GUM of is determined to
be:

A brief description of each variable is given in Figure 1.

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Econometrics Report

Name
Dependent
Variable (Y)

Independent
Variables (X)

Meaning

Expected

Price

Median housing price

sign
+

Crime

Number of crimes

­

Nox

committed per capita
The amount of nitrogen

­

oxide concentrator parts
Rooms

in the air per 100m
The average number of

+

Dist

rooms
Weight distance to 5

­

Proptax

employ centers
Property tax per $1000

­

Figure 2
IV. ECONOMETRICS MODEL

1.  Population regression function (PRF)
PRF: 
2.  Sample regression function (SRF)
SRF:

where:
 0  is the intercept of the regression model

 i  is the slope coefficient of the independent variable  xi
  is the disturbance of the regression model

 
 
 

 is the estimator of   0
is the estimator of  i  

is the residual (the estimator of  i )
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V. DATA COLLECTION
1. Data overview
 This set of data is collected from a given source, therefore
it is a secondary one.

 The structure of Economic data: cross­sectional data
2. Data description
To get statistic indicators of the variables, in Stata, the
following command is used:
. sum 
Variab

Std.

le

Obs

Mean
22511.

Dev.
9208.85

Min

Max

price

506

51
3.6115

6
8.59024

5000
0.00

50001
88.97

crime

506

36
5.5497

7
1.15839

6

6

nox

506

83
6.2840

5
0.70259

3.85

8.71

rooms

506

51
3.7957

38
2.10613

3.56

8.78

dist
propta

506

51
40.823

7

16.8537

1.13

12.13

x

506

72

1

18.7

71.1

Figure 3
where:
Obs is the number of observations
Std. Dev is the standard deviation of the variable
Min is the minimum value of the variable
 
Max is the maximum value of the variable
VI.

ESTIMATION OF ECONOMETRIC MODEL
1. Checking the correlation among variables:

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Group 10

 

Econometrics Report

price

price
crime

crime

nox

rooms

dist

proptax

 

 

 

 

1  

 

 

 

1  

 

 

1  

 

1  
­0.3879

nox

­0.426

0.4212

rooms

0.6958

­0.2188

­0.3028

dist

0.2493

­0.3799

­0.7702

0.2054

­0.4671

0.5828

0.667

­0.2921

proptax
Figure 4

1  
­0.5344

1

First and foremost, the correlation of Price and nox, crime,
rooms,   dist,   proptax   is   checked   by   calculating   the   correlation
coefficient   among   these   variables.   The   correlation   coefficient
measures   the   strength   and   direction   of   a   linear   relationship
between two variables on a scatterplot. In Stata, the correlation
with matrix is generated the command:
corr price crime nox rooms dist proptax
We   can   see   from   the   matrix,   it   can   be   inferred   that   the
correlation between price and each of the independent variable is
decent enough to run the regression model. Specifically:
­

Correlation coefficient between price and crime is ­0.3879

=> price and crime have a moderate relationship.
­
Correlation coefficient between price and nox is ­0.426 =>
price and nox have a moderate relationship.
­
Correlation coefficient between price and rooms is 0.6958 =>
price and rooms  have a moderate relationship.
­
Correlation coefficient between price and dist is 0.2493 =>
price and dist have a weak relationship.
­
Correlation coefficient between price and proptax is ­0.4671
=> price and proptax have a moderate relationship.

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Independent   variables   including   Rooms   and   Dist   have
correlation   coefficient   larger   than   0,   which   means   they   are   in
directly   relationship   with   dependent   variable.   The   highest
coefficient is 0.6958 (between Rooms and Price) points out that
Rooms   have   the   strongest   impaction   on   Price.   When   rooms
increases, then price will increase much. On the other hands, the
correlation   coefficient   between   Price   and   Dist   is   0.2493.   It
implies that they have not strong connection. Even if the Dist
increases, Price increases but not much. 
In   addition,   all   variables   have   correlation   coefficient   not
larger   than   0.8   so   this   model   does   not   have   multicollinearity
problem. 

2. Regression run
Having   checked   the   required   condition   of   correlation   among
variables, the regression model is ready to run. In Stata, this
is done by using the command:
Reg price nox crime rooms dist proptax

Number 
of obs
F(  5,  

=

506

500)
Prob > F
R­

=
=

142.92
0

Source
Model

SS
2.52E+10

df
5

MS
5.04E+09

Residual

1.76E+10

500

35258403.7

squared
Adj R­

=

0.5883

Total

4.28E+10

505

84803032

squared

=

0.5842

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Root MSE
[95% 

Std. 
Err.
t
38.11571
410.7763
399.0772

P>t

=

5937.9

price
crime
nox
rooms

Coef.
­150.0703
­1737.66
7707.327

dist

­791.2588 197.9444

­4

0

1180.164
­

­402.3535

proptax

­89.95717 23.61555

­3.81

0
0.02

136.3551
­

­43.55923

_cons

­9060.303 3978.871

­2.28

3

16877.67

­1242.937

­3.94
­4.23
19.31

Conf.
Interval]
0 ­224.957
­75.18364
0 ­2544.72
­930.5992
0 6923.252
8491.402
­

Figure 5
From table above we have Sample Regression Function:
Price   =   ­9060.303   ­   1737.66*nox   +   7707.327*rooms
­89.95717*proptax
From the result, it can be inferred that
crime,   nox,   rooms,   dist,   proptax  all   have   statistically
significant effects on  price  at the 5% significant level (as all
p­values are smaller than 0.05). In particular, those effects can
be specified by the regression coefficients as follows:

 β0 = ­9060.303
  1 = ­1737.66 means that if nit ox concen per 100m increases by
one , average housing price will decrease by 1737.66 in condition
other factors do not change.
  2   =   ­150.0703   means   that   if   crimes   committed   per   capital
increases by one , average housing price will decrease by 150.0703
in condition other factors do not change.
  3 = 7707.327 means that if average number of rooms increases by
one, average housing price will increase by 7707.327 in condition
other factors do not change.

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  4 = ­791.2588 means that if weight distance to 5 employ centers
increases 1 unit, average housing price will decrease by 791.2588
in condition other factors do not change.
  5   =   ­89.95717   means   that   if   average   property   tax   per   $1000
increases by one, average housing price will decrease by 89.95717
in condition other factors do not change.


The   coefficient   of   determination   R­squared=0.5883:   all

independent   variables   (crime,   nox,   rooms,   dist,   proptax,)
jointly   explain   58.83%   of   the   variation   in   the   dependent

variable   (price);   other   factors   that   are   not   mentioned
explain the remaining 41.17% of the variation in the price.
Other indicators:
­ Adjusted coefficient of determination adj R­squared = 0.5842
­ Total Sum of Squares TSS = 4,28E+14
­ Explained Sum of Squares ESS = 2,52E+14
­ Residual Sum of Squares RSS =  1,76E+14
­ The degree of freedom of Model Dfm= 5
­ The degree of freedom of residual Dfr = 500


VII.  CHECK MULLTICOLLINEARITY AND HETEROSCEDASTICITY 
1. Multicollinearity

 Multicollinearity is the high degree of correlation amongst
the   explanatory   variables,   which   may   make   it   difficult   to
separate   out   the   effects   of   the   individual   regressors,
standard errors may be overestimated and t­value depressed.
 Detect multicollinearity
o
Method 1: Use cor command to examine multicollinearity
If   independent   variables   are   strongly   correlated   (r   >   0.8),
multicollinearity may occur.
 
price
crime

price
1.0000
­0.3879

crime
 
1.0000

nox
 
 

rooms
 
 

dist
 
 

proptax
 
 
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nox
rooms
dist
proptax
Figure 6

Econometrics Report
­0.426
0.6958
0.2493
­0.4671

0.4212
­0.2188
­0.3799
0.5828

1.0000
­0.3028
­0.7702
0.667

 
1.0000
0.2054
­0.2921

 
 
1.0000
­0.5344

 
 
 
1.0000

From   the   table   above,   we   can   easily   see   that   correlating
coefficient   among   independent   variables   are   pretty   low   and   all
smaller   than   0.8.   As   a   result,   we   can   conclude   that
multicollinearity does not occur in this model.
o
Method 2: Use variance inflation factor (VIF)
If VIF > 10, multicollinearity occurs.

Variable
nox
dist
proptax
crime
rooms
Mean VIF
Figure 7

VIF
3.24
2.49
2.27
1.54
1.13
2.13

1/VIF
0.308352
0.401709
0.440742

0.651256
0.888073
 

The table shows that all VIF value is smaller than 10, thus,
multicollinearity does not is occur in this model.
We   can   draw   a   conclusion   from   2   methods   above   that
multicollinearity   not   too   worrisome   a   problem   for   this   set   of
data.
2. Heteroskedasticity
Another   problem   that   our   model   can   suffer   from   when   being
examined is heteroskedasticity. Heteroskedasticity may result in
the   situation   that   some   least   squared   estimators   are   still
unbiased but are no longer effective, along with that, estimators
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of variances will become biased, thus lead to the reduction in
effectiveness of our model.
When   the   assumption   of   variance   of   each   error   term   Ui   is
unchanged when i moves from 1, 2 to n. It can also be rewritten
as:

Var (Ui) = Var (Uj)

i=1,2,3,…,n

j=1,2,3,…,n

When that assumption is violated, heteroskedasticity appears
 Causes
o
  Essence of economic phenomena: If economic phenomena
is examined on subjects having difference in scale or they
are examined under periods of time that are not similar in
fluctuation level.
o

  Model’s function is wrongly formatted, maybe because

appropriate   variables   are   missing   or   function   analysis   is
false.
o

    cannot   fully   and   correctly   reflect   the   essence   of

economic   phenomena.   For   example,   external   observations
appear.   Bringing   in   or   eliminate   these   observations   does
great impact on regression analysis.
o

 Error tends to decrease as data collecting, conserving

and processing techniques are improved
o

 Behaviors in the past are learnt.

Hypothesis:   
Using the command estat hettest in STATA:
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Breusch­Pagan / Cook­Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: fitted values of price
         chi2(1)      =    26.56
         Prob > chi2  =   0.0000
We can see that Prob > chi2 = 0.0000 < 0.05 => We reject H0,
accept H1
We   can   conclude   that   heteroskedasticity   does   occur   in   this
model
Correcting heteroskedasticity
We use command: 
reg price crime nox rooms dist proptax, robust
we have the result 

Number   of
obs
F(  

Robust
price

Coef.
 
 
crime
­150.0703
nox
­1737.66
rooms
7707.327
dist
­791.2588
proptax ­89.95717
_cons
­9060.303
Figure 8

Std. Err.
 
30.45247
389.6642
670.6304
175.744
26.84788
5398.964

t
 
­4.93
­4.46
11.49

­4.5
­3.35
­1.68

=

506
103.2

500)
Prob > F

=
=

2
0
0.588

R­squared

=

3
5937.

Root MSE

=

9

P>t
 
0
0
0
0
0.001
0.094

  5,

[95% Conf.
 
­209.9009
­2503.241
6389.726
­1136.546
­142.7057
­19667.75

Interval]
 
­90.23976
­972.0787
9024.928
­445.9712
­37.20862
1547.148

Note that comparing the results with the earlier regression,
none   of   the   coefficient   estimates   changed,   but   the   standard
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errors   and   hence   the   t   values   are   different,   which   gives
reasonably more accurate p values.
VIII. HYPOTHESES POSTULATED
1.  The t test
Hypothesis:  

c(500)0.025 = 1.965 < |ts | => Reject 
Conclusion:  Number   of   crimes   committed   per   capita 

has

statistically signifincant effect on median housing price. Higher
number of crimes commited per capita, lower median housing price
Hypothesis:  

c(500)0.025 = 1.965 < |ts | => Reject 
Conclusion:   nitrogen   oxide   concentrator   per   100m   has
statistically signifincant effect on median housing price. Higher
nitrogen oxide concentrator per 100m, lower median housing price.
Hypothesis:  

c(500)0.025 = 1.965 < |ts | => Reject 
Conclusion:   The   average   number   of   rooms   has   statistically
signifincant   effect   on   median   housing   price,   higher   average
number of rooms, higher median housing price.
Hypothesis:  
­4.5
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c(500)0.025 = 1.965 < |ts | => Reject 
Conclusion weight distance to 5 employ centers has statistically
signifincant   effect   on   median   housing   price,   higher  weight
distance to 5 employ centers, lower median housing price.

Hypothesis:  

c(500)0.025 = 1.965 < |ts | => Reject 
Conclusion  Property tax per $1000  has statistically signifincant
effect   on   median   housing   price,   higher  property   tax   per   $1000,
lower median housing price.
2. Confidence Intervals
Test the following hypothesis:

Variable
Const

X1
X2
X3
X4

Coefficient

Significant
Level
5%
5%
5%
5%
5%

Confidence Interval
(­19667.75  ;  1547.148)
(­209.9009  ; ­90.23976)
(­2503.241  ; ­972.0787)
(6389.726  ; 9024.928)
(­1136.546 ;­445.9712)
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5%

(­142.7057  ; ­37.20862)

Figure 9
We   can   see   that   for   all   coefficients,   0  doesn’t   belong   to   the
confidence interval, so we reject the hypotheses H0: , , , , 
Conclusion: Number of crimes committed per capita, nitrogen oxide
concentrator   per   100m,   the   average   number   of   rooms,  weight
distance to 5 employ centers and property tax per $1000 all have
statistically   signifincant   effect   on   median   housing   price  with
the confidence level of 95%. 
3. P­ Value
Hypothesis testing: 
P­value = 0.0004 < α = 0.05 => Reject H0
Number   of   crimes   committed   per   capita  has   statistically
signifincant   effect   on   median   housing   price.   Higher  number   of
crimes commited per capita, lower median housing price.
In   particular,   with   the   sample   we   have,   the   estimated   result
shows   that   one   more   crime   committed   decreases   median   housing
price by 150.07$, holding other factors fixed.
Hypothesis testing: 
P­value = 0.0004 < α = 0.05 => Reject H0
Nitrogen   oxide   concentrator   per   100m   has   statistically
signifincant   effect   on   median   housing   price.   Higher   nitrogen
oxide concentrator per 100m, lower median housing price
In   particular,   with   the   sample   we   have,   the   estimated   result
shows that one more unit in nitrogen oxide concentrator per 100m
decreases median housing price by 1737.66$, holding other factors
fixed.

Hypothesis testing: 
P­value = 0.0004 < α = 0.05 => Reject H0
The average number of rooms has statistically signifincant effect
on median housing price, higher average number of rooms, higher
median housing price.
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In   particular,   with   the   sample   we   have,   the   estimated   result
shows   that   one   more   room   added   in   the   house   increases   median
housing price by 7707.33 $, holding other factors fixed.
Hypothesis testing: 
P­value = 0.0004 < α = 0.05 => Reject H0
Weight   distance   to   5   employ   centers 

has   statistically

signifincant   effect   on   median   housing   price,   higher  weight
distance to 5 employ centers, lower median housing price.
In   particular,   with   the   sample   we   have,   the   estimated   result
shows that one more unit increased in weight distance to 5 employ
centers decreases median housing price by 791.25$, holding other
factors fixed.
Hypothesis testing: 
P­value = 0.0008 < α = 0.05 => Reject H0
Property tax per $1000  has statistically signifincant effect on

median housing price, higher property tax per $1000, lower median
housing price. 
In   particular,   with   the   sample   we   have,   the   estimated   result
shows   that   one   more   $   increased   in   property   tax   per   1000$
decreases median housing price by 89.96 $, holding other factors
fixed.
4.  Testing the overall significance: The F test
This   test   is   to   examine   if   the   parameters   of   the   independent
variable βi at the same time can be zero.
The hypothesis is as follows:

= 142.92   >    
As   a   result,   there   is   enough   evidence   to   reject   the   null
hypothesis and conclude that at least one independent variable in
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the subset does have explanatory or predictive power on price, so
we don’t reduce the model by dropping out this subset.
IX. RESULT ANALYSIS AND POLICY IMPLICATION
From data analysis in previous sections, we have gained an 
overall view of data set given in term of the satistical 
relationship between housing prices and each of the factors 
proposed. As mentioned at the beginning of this report, we aim to
learn how security of the neighborhood, the air pollution, the 
size of house, accessibility and the property tax are associated 

with housing price. In other words, we are concerned about what 
is the willingness of buyers to pay for these components. 
Following the analysis of data, regression model run and 
hypothesis testing, it can be concluded that security of the 
neighborhood, the air pollution, the size of house, accessibility
and the property tax statistically affect the housing prices. 
Therefore, tenants, investors or constructors should take all of 
these ingredients into account when making deals.
X. CONCLUSION
This   report   is   completed   on   the   dedicated   contribution   of
each   member   and   the   knowledge   from   our   study   in   Econometrics.
This research has provided us with a good opportunity to practice
what we have learned and to get a deeper understanding of data
analysis and relevant testing. From this useful application, we
hope   that   our   research   can   somehow   suggest   the   relationship
between the housing prices and some other factors.
Again, due to the limitation of understanding and resources, our
report may contain misinterpretations. We hope that teacher and
readers can give us constructive comments on the report so that
we would improve ourselves and do better in the future.
XI.

REFERENCES

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/>Feb_2011.pdf
1. />2. />doi=10.1.1.926.5532&rep=rep1&type=pdf
3. D.A. Belsey, E. Kuh, and R. Welsch, Regression Diagnostics:
Identifying Influential Data and Sources of Collinearity, New
York: Wiley (1990).

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