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

We choose 6 variables: price, crime, nox, rooms, dist and proptax to do the research in which price is dependent variable and theother five are independent variables.. Name Meaning Expec

I 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

Figure 3 10

Figure 4 11

Figure 5 13

Figure 6 15

Figure 7 16

Figure 8 18

Figure 9 21

I INTRODUCTION

1.Overall about econometrics

Econometrics is the application of statistical methods toeconomic data and is described as the branch of economics thataims to give empirical content to economic relations Preciselyspeaking, it is the quantitative analysis of actual economicproblems, based on the concurrent development of theory andobservation, related by appropriate methods of inference It isunderstandable that economist make comparison econometrics islike an effective tool to convert mountains of data into extractsimple relationships

The reason why econometrics is effective is economics theoryuse statistical theory and mathematical statistics to evaluateand develop econometrics method In reality, econometrics helpeconomists to assess economic theories, developing econometricsmodel, 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: “Thefactors that have influence on median housing price” and aim toanalyze statistic and point out differences and their reason ofprice 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 theother five are independent variables The general method used in

this research is OLS (ordinary least squares) In addition, thespecialized 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 gratefulfor everything you have taught us!

This is the first time our group carry out an econometricsresearch, our performance is unavoidable to have many mistakes

It would be a pleasure if we can receive the feedback from you tobetter ourselves next time

2.Why choosing OLS?

Ordinary least squares (OLS) is a type of linear leastsquares method for estimating the unknown parameters in a linearregression model OLS chooses the parameters of a linearfunction of a set of explanatory variables by the principle

of least squares: minimizing the sum of the squares of thedifferences between the observed dependent variable in thegiven dataset and those predicted by the linear function

With the six selected variables, we use the OLS model becauseall regressions variable are exogenous variables, the effects ofindependent variables on the dependent variable are lineareffects In addition, the estimates calculated by means of theleast squares OLS are linear estimates that are not deviate andare better than others

When using OLS, we have some basic assumptions:

1. The regression model is linear in the parameters

2. X values are fixed in repeated sampling, which means Xiand ui are uncorrelated

3. Zero mean value of disturbance (E(ui)) =0)

4. Homoscedasticity or equal variance of ui : var(ui) = σ2

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

8. X values in a given sample must not be the same

9. No perfect multicollinearity

10. Normal distribution

We have always been wondering “Why do housing prices amonglocations 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 tothat question, our group is going to use the collected data tobuild 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 thisreport is an empirical one Note that the fundamental model ismathematical; with an empirical model, however, data is gatheredfor the variables and using accepted statistical techniques, thedata are used to provide estimates of the model's values

1 Choosing the variables

Having described data via the command “des” in file… fromStata software, we gain the result as following:

des

obs: 506 vars: 12

31 Oct 1996 16:37 size: 22,770

l variable label

median housingprice, $

crimes committed per capita

wght dist to 5employ centers

average student-teacher ratio

perc of people'lower status'

lnox float %9.0g   log(nox)

Figure 1The above table reveal that this is the statistic of factorswhich have influence in housing price via 506 observations Afterdiscussing carefully, our group jumped into a conclusion tochoose a dependent variable Y: Price, independent variablecontains:

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

A brief description of each variable is given in Figure 1

Name Meaning Expected

signDependent

Variable (Y)

Price Median housing price +

IndependentVariables (X)

Crime Number of crimes

committed per capita

-Nox The amount of nitrogen

oxide concentrator parts

in the air per 100m

1 Population regression function (PRF)

PRF:

Price=β0+β1× crime +β2× nox+β3×rooms+ β4× dist +β5× proptax+u i

2 Sample regression function (SRF)

SRF:

Price= ^ β0+ ^β1× crime+^ β2× nox+ ^ β3×rooms+ ^ β4× dist + ^ β5× proptax

where:

is the intercept of the regression model

is the slope coefficient of the independent variable

is the disturbance of the regression model

8.590247

0.00688.976

nox 506

5.549783

1.15839

rooms 506

6.284051

0.70259

38 3.56 8.78

dist 506

3.795751

2.10613

7 1.13 12.13propta

40.82372

16.8537

Figure 3where:

Obs is the number of observationsStd Dev is the standard deviation of the variableMin is the minimum value of the variable

Max is the maximum value of the variable

1 Checking the correlation among variables:

dist 0.2493 -0.3799 -0.7702 0.2054 1  proptax -0.4671 0.5828 0.667 -0.2921 -0.5344 1Figure 4

First and foremost, the correlation of Price and nox, crime,rooms, dist, proptax is checked by calculating the correlationcoefficient among these variables The correlation coefficientmeasures the strength and direction of a linear relationshipbetween two variables on a scatterplot In Stata, the correlationwith matrix is generated the command:

corr price crime nox rooms dist proptax

We can see from the matrix, it can be inferred that thecorrelation between price and each of the independent variable isdecent 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 havecorrelation coefficient larger than 0, which means they are indirectly relationship with dependent variable The highest

coefficient is 0.6958 (between Rooms and Price) points out thatRooms have the strongest impaction on Price When roomsincreases, then price will increase much On the other hands, thecorrelation coefficient between Price and Dist is 0.2493 Itimplies that they have not strong connection Even if the Distincreases, Price increases but not much

In addition, all variables have correlation coefficient not largerthan 0.8 so this model does not have multicollinearity problem

2 Regression run

Having checked the required condition of correlation amongvariables, the regression model is ready to run In Stata, this

is done by using the command:

Reg price nox crime rooms dist proptax

136.3551 -43.55923

0.02 3

16877.67 -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 bythe regression coefficients as follows:

β0 = -9060.303

1 = -1737.66 means that if nit ox concen per 100m increases byone , average housing price will decrease by 1737.66 in conditionother factors do not change.

2 = -150.0703 means that if crimes committed per capitalincreases 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 byone, average housing price will increase by 7707.327 in conditionother factors do not change

4 = -791.2588 means that if weight distance to 5 employ centersincreases 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 $1000increases by one, average housing price will decrease by 89.95717

in condition other factors do not change

 The coefficient of determination R-squared=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 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

1 Multicollinearity

 Multicollinearity is the high degree of correlation amongstthe explanatory variables, which may make it difficult toseparate out the effects of the individual regressors, standarderrors 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

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 1.0000Figure 6

From the table above, we can easily see that correlatingcoefficient among independent variables are pretty low and allsmaller than 0.8 As a result, we can conclude thatmulticollinearity does not occur in this model

o Method 2 : Use variance inflation factor (VIF)

If VIF > 10, multicollinearity occurs

Variable VIF 1/VIF

2 Heteroskedasticity

Another problem that our model can suffer from when beingexamined is heteroskedasticity Heteroskedasticity may result inthe situation that some least squared estimators are stillunbiased but are no longer effective, along with that, estimators

of variances will become biased, thus lead to the reduction ineffectiveness of our model

When the assumption of variance of each error term Ui isunchanged when i moves from 1, 2 to n It can also be rewrittenas:

Var (Ui) = Var (Uj) i=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 theyare examined under periods of time that are not similar influctuation level

o Model’s function is wrongly formatted, maybe becauseappropriate variables are missing or function analysis isfalse

o cannot fully and correctly reflect the essence ofeconomic phenomena For example, external observationsappear Bringing in or eliminate these observations doesgreat impact on regression analysis

o Error tends to decrease as data collecting, conservingand processing techniques are improved

o Behaviors in the past are learnt.

Hypothesis: { H0:the variance is homogenous

H1:the variance is not homogenous

Using the command estat hettest in STATA:

Breusch-Pagan / Cook-Weisberg 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 thismodel

Correcting heteroskedasticity

We use command:

reg price crime nox rooms dist proptax, robust

we have the result

R-squared =

0.5883

Root MSE =

5937

9

Robustprice Coef Std Err t P>t [95% Conf Interval]

crime -150.0703 30.45247 -4.93 0 -209.9009 -90.23976nox -1737.66 389.6642 -4.46 0 -2503.241 -972.0787rooms 7707.327 670.6304 11.49 0 6389.726 9024.928

dist -791.2588 175.744 -4.5 0 -1136.546 -445.9712proptax -89.95717 26.84788 -3.35 0.001 -142.7057 -37.20862_cons -9060.303 5398.964 -1.68 0.094 -19667.75 1547.148Figure 8

Note that comparing the results with the earlier regression,none of the coefficient estimates changed, but the standard errorsand hence the t values are different, which gives reasonably moreaccurate p values

VIII HYPOTHESES POSTULATED

Conclusion: Number of crimes committed per capita hasstatistically signifincant effect on median housing price Highernumber of crimes commited per capita, lower median housing price