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

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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

Figure 1........................................................7
Figure 2........................................................9
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Figure
Figure
Figure
Figure
Figure
Figure
Figure

3.......................................................10
4.......................................................11
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
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this research is OLS (ordinary least squares). In addition, the
specialized method is estimate, running Stata software as well.

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.
function of
of least

a

set

squares:

differences

OLS

chooses

the

of explanatory
minimizing

between

the

the

parameters

variables by
sum

of

the

observed dependent

of

a linear

the

principle

squares

of

variable in

the
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:
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1.

The regression model is linear in the parameters

2.

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) = σ 2

5.

No correlation between disturbances

6.

The model is correctly specified.

7.

Number of observations must be greater than the number

of parameters to be estimated.

II.

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

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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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valu
e

Variable name

storag

display

labe

e type

format

l

variable label
median housing

price

float

%9.0g

 

price, $
crimes
committed per

crime

float

%9.0g

 

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

radial

byte

%9.0g

 

to rad. hghwys
property tax

proptax

float

%9.0g

 

per $1000
average
student-

stratio

float

%9.0g

 

teacher ratio
perc of people

lowstat

float

%9.0g

 

'lower status'

lprice

float

%9.0g

 

log(price)
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lnox

float

%9.0g

 

log(nox)

lproptax

float

%9.0g

 

log(proptax)

Figure 1
The above table reveal that this is the statistic of factors
which have influence in housing price via 506 observations. After
discussing
choose

carefully,

a

dependent

our

group

variable

jumped

Y:

Price,

into

a

conclusion

independent

to

variable

contains:


X1-crime



X2-nox



X3-rooms



X4-dist



X5-proptax
Price=f (x )

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:

Price=f (crime , nox , rooms , dist , proptax)

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

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Name

Meaning

Expected
sign

Dependent

Price

Median housing price

+

Crime

Number of crimes

-

Variable (Y)

committed per capita
Nox

The amount of nitrogen

Independent

oxide concentrator parts

Variables (X)

in the air per 100m
Rooms

The average number of

-

+

rooms
Dist

Weight distance to 5

-

employ centers
Proptax

Property tax per $1000

-

Figure 2
IV.

ECONOMETRICS MODEL

1. Population regression function (PRF)
PRF:
Price=β 0 + β 1 × crime + β 2 × nox+ β 3 ×rooms+ β 4 × dist + β 5 × proptax+u i

2. Sample regression function (SRF)
SRF:
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^

Price= β^0 + ^
β 1 × crime + ^
β 2 × nox+ β^3 ×rooms+ ^
β 4 × dist + β^5 × proptax

where:
is the intercept of the regression model
is the slope coefficient of the independent variable
is the disturbance of the regression model
^
β 0 is the estimator of
^
β iis the estimator of
μi is the residual (the estimator of
^

)

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: cross-sectional data
2. Data description

To get statistic indicators of the variables, in Stata, the
following command is used:

. sum
Variab

le

price

crime

Std.
Obs

506

506

Mean

Dev.

Min

Max

22511.

9208.85

51

6

5000

50001

3.6115

8.59024

0.00

88.97

36

7

6

6
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nox

5.5497

1.15839

83

5

6.2840

0.70259

51

38

3.7957

2.10613

51

7

40.823

16.8537

72

1

506

rooms

506

dist

506

propta
x

506

3.85

8.71

3.56

8.78

1.13

12.13

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:

 
price
crime

price

crime

nox

rooms

dist

proptax

 

 

 

 

1  

 

 

 

1  

 

 

1  

 

1  
-0.3879

nox

-0.426

0.4212

rooms

0.6958

-0.2188

-0.3028

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dist
proptax

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0.2493

-0.3799

-0.7702

0.2054

-0.4671

0.5828

0.667

-0.2921

1  
-0.5344

1

Figure 4
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.

Independent

variables

including

Rooms

and

Dist

have

correlation coefficient larger than 0, which means they are in
directly

relationship

with

dependent

variable.

The

highest
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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

=

506

500)

=

142.92

Prob > F

=

0

=

0.5883

F(
Source

SS

df

MS

Model

2.52E+10

5

5.04E+09

5,

RResidual

1.76E+10

500

35258403.7

squared

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Adj R-

Total

4.28E+10

505

84803032

Std.

squared

=

0.5842

Root MSE

=

5937.9

[95%

price

Coef.

Err.

t

P>t

Conf.

Interval]

crime

-150.0703 38.11571

-3.94

0

-224.957

-75.18364

nox

-1737.66 410.7763

-4.23

0

-2544.72

-930.5992

rooms

7707.327 399.0772

19.31

0

6923.252

8491.402

dist

-791.2588 197.9444

-4

0

1180.164

-402.3535

proptax

-89.95717 23.61555

_cons

-9060.303 3978.871

-3.81

-2.28

0

136.3551

0.02

-

3

16877.67

-43.55923

-1242.937

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 p-values are
smaller than 0.05). In particular, those effects can be specified by
the regression coefficients as follows:

β0 = -9060.303

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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.
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

independent
jointly
variable

of

variables

explain

58.83%

(price);

determination
(crime,
of

other

nox,

the

R-squared=0.5883:
rooms,

variation

factors

that

dist,
in

are

all

proptax,)

the

dependent

not

mentioned

explain the remaining 41.17% of the variation in the price.


Other indicators:

- Adjusted coefficient of determination adj R-squared = 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
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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 t-value 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

nox

rooms

dist

proptax

price

1.0000

 

 

 

 

 

crime

-0.3879

1.0000

 

 

 

 

nox

-0.426

0.4212

1.0000

 

 

 

rooms

0.6958

-0.2188

-0.3028

1.0000

 

 

dist

0.2493

-0.3799

-0.7702

0.2054

1.0000

 

proptax

-0.4671

0.5828

0.667

-0.2921

-0.5344

1.0000

Figure 6
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.

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Variable

VIF

1/VIF

nox

3.24

0.308352

dist

2.49

0.401709

proptax

2.27

0.440742

crime

1.54

0.651256

rooms

1.13

0.888073

Mean VIF

2.13

 

Figure 7

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
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
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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

Hypothesis:

Behaviors in the past are learnt.

{

H 0 :the variance is homogenous
H 1 :the variance is not homogenous

Using the command estat hettest in STATA:
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
Ho: Constant variance
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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(

506

5,

103.2

500)

=

2

Prob > F

=

0
0.588

R-squared

=

3
5937.

Root MSE

=

9

Robust
price

Coef.

Std. Err. t

P>t

[95% Conf. Interval]

 

 

 

 

 

 

 

crime

-150.0703 30.45247

-4.93

0

-209.9009

-90.23976

nox

-1737.66

389.6642

-4.46

0

-2503.241

-972.0787

rooms

7707.327

670.6304

11.49

0

6389.726

9024.928
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dist

Econometrics Report
-791.2588 175.744

-4.5

0

-1136.546

-445.9712

proptax -89.95717 26.84788

-3.35

0.001

-142.7057

-37.20862

_cons

-1.68

0.094

-19667.75

1547.148

-9060.303 5398.964

Figure 8
Note that comparing the results with the earlier regression,
none of the coefficient estimates changed, but the standard errors
and hence the t values are different, which gives reasonably more
accurate p values.

VIII. HYPOTHESES POSTULATED
1. The t test
Hypothesis:
t s=

{

H 0 : β 1=0
H 1 : β1≠ 0

^
β1 −0
=−4.93
se( β^1 )

c(500)0.025 = 1.965 < |ts | => Reject H 0
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:

t s=

{

H 0 : β 2=0
H 1 : β2≠ 0

^
β2 −0
=−4.46
se( β^2 )

c(500)0.025 = 1.965 < |ts | => Reject H 0

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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:

t s=

{

H 0 : β 3=0
H 1 : β3≠ 0

^
β3 −0
=11.49
se( β^ )
3

c(500)0.025 = 1.965 < |ts | => Reject H 0

Conclusion:

The

average

number

of

rooms

has

statistically

signifincant effect on median housing price, higher average number
of rooms, higher median housing price.

Hypothesis:

t s=

{

H 0 : β 4 =0
H 1 : β4 ≠ 0

^
β 4−0
=¿ -4.5
se( β^ )
4

c(500)0.025 = 1.965 < |ts | => Reject H 0

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.
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Hypothesis:

t s=

Econometrics Report

{

H 0 : β 5=0
H 1 : β5≠ 0

^
β5 −0
=−3.35
se( β^ )
5

c(500)0.025 = 1.965 < |ts | => Reject H 0

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:

{
{
{
{

H 0 : β 1=0
H 1 : β1≠ 0
H 0 : β 2=0
H 1 : β2≠ 0

H 0 : β 3=0
H 1 : β3≠ 0
H 0 : β 4 =0
H 1 : β4 ≠ 0

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{
Variable

H 0 : β 5=0
H 1 : β5≠ 0

Significant

Coefficient

Level

Confidence Interval

Const

β0

5%

(-19667.75

;

X1

β1

5%

(-209.9009

; -90.23976)

X2

β2

5%

(-2503.241

; -972.0787)

X3

β3

5%

(6389.726

X4

β4

5%

(-1136.546 ;-445.9712)

X5

β5

5%

(-142.7057

1547.148)

; 9024.928)

; -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: H 0 : β 1=0, β 2=0 ,
β 3=0 , β 4 =0, β 5=0

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:

{

H 0 : β 1=0
H1 : β1≠ 0

P-value = 0.0004 < α = 0.05 => Reject H0
Number

of

signifincant

crimes
effect

committed
on

median

per
housing

capita
price.

has

statistically

Higher

number

of

crimes commited per capita, lower median housing price.
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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:

{

H 0 : β 2=0
H1 : β2≠ 0

P-value = 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:

{

H 0 : β 3=0
H 1 : β3 ≠ 0

P-value = 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.
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:

{

H 0 : β 4 =0
H1 : β4 ≠ 0

P-value = 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
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centers decreases median housing price by 791.25$, holding other
factors fixed.

Hypothesis testing:

{

H 0 : β 5=0
H 1 : β5 ≠ 0

P-value = 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:

¿

F qs =

As

R 2 (n−k )

( 1−R2 ) ( k −1 )
a

= 142.92

result,

there

>
is

500
F 5;
0,05 =2.23

enough

evidence

to

reject

the

null

hypothesis and conclude that at least one independent variable in
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
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