Econometric Methods for Valuation Analysis

Transcription

1 Econometric Methods for Valuation Analysis Margarita Genius Dept of Economics M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

2 Correlation Analysis Simple Regression Outline We will examine the concept of correlation between two variables Calculate the simple correlation between two variables Understand how to use the simple regression model to explain the relationship between two variables Understand how to use the simple linear regression model to make forecasts of one variable Perform tests of hypothesis in the simple linear regression model Use Stata to perform linear regression using housing data M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

3 Correlation Analysis and Simple Regression Correlation Analysis Question Type 1: Are the number of patents a firm applies for correlated with the level of R&D expenditures? Is the price of a house correlated with the level of air pollution? Question Type 2: What will be the effect of an increase of R&D expenditures on the number of patents? What will be the effect of an increase in the level of pollution on the price of a house? The first type of questions can be answered using correlation analysis. While for the second type we need to postulate a model that explains the relationship between number of patents and R&D expenditures in the first case and house prices and measurements of air pollution in the second case. This model is the simple regression model. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

4 Correlation Analysis and Simple Regression Correlation Analysis: Application and data Linear Regression will be applied to a hedonic model (i.e a model that assumes that the price of a good is a function of its characteristics) The data come from Harrison and Rubinfeld (1978) who used a hedonic model to study how house values are affected by air pollution in Boston. The ultimate goal was to estimate willingness to pay for clean air. The data have been downloaded from Wooldridge, Harrison, D. and D.L. Rubinfeld (1978) Hedonic housing prices and the demand for clean air, Journal of Environmental Economics And Management, 5, The basic idea: if two houses are identical and only differ in pollution levels, then differences in their values should reflect willingness to pay for clean air. The data consists of 506 census tracts from the 1970 US census, so it is aggregate data and not individual houses data. Only the information of owner-occupied one-family houses were included. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

5 Correlation Analysis and Simple Regression Correlation Analysis: Scatter Plot Below you can see the scatter plot of price in $ versus nox (parts per 100 million). Where price is the median housing price, nox is nitrous oxide (NO2) median housing price, $ nit ox concen; parts per 100m Stata Command: scatter price nox Is there a strong relationship between price and nox? Are they positively or negatively related? M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

6 Correlation Analysis: The correlation coefficient The correlation coefficient measures the strength of the linear relationship between two variables. If we have a sample of size n of observations for two variables X and Y, the sample correlation coefficient r is computed as follows, Correlation Coefficient: r Note n i=1 r = (y i ȳ) (x i x) n i=1 (y i ȳ) 2 n i=1 (x i x) 2 the sample correlation coefficient is an estimate of the population correlation (ρ) 1 r 1 the value of r is independent of the units of measurement of both variables and therefore we can compare the values of r for different pairs of variables. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

7 y y y y Simple Regression Correlation Analysis y Figure: Scatter Plots for different values of r r= r= r= x x x r=0.6 r= x x M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

8 Correlation Analysis when r=1.0 we have perfect positive correlation, all points in the scatter plot lie on an upward sloping line. when r=0, there is no correlation between the two variables when r=-1.0 we have perfect negative correlation, all points in the scatter plot lie on a downward sloping line when r=0.6 we have weak positive correlation when r=-0.6 we have weak negative correlation. Note: If r=0 this means that there is no linear relationship between the two variables, IT DOES NOT MEAN THEY ARE INDEPENDENT M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

9 Correlation Analysis The existence of a significant correlation between two variables X and Y does not mean that there is a relationship of cause-effect between them. For instance beer sales are the highest when ice cream sales are highest because of the heat (summer months). We can not say that beer sales cause people buying ice creams or vice-versa. In this case we say that there exists a spurious correlation between beer sales and ice cream sales (i.e there is a third factor (summer heat) which drives both of them. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

10 The Simple Regression Model We saw from the scatter plot of price versus nox that there is a negative relation between the two variables although it is not very strong as we will see later that the correlation coefficient is equal to r= We would like now to find a model that explains the relationship between the two variables so that we can use it to make forecasts of prices or predict the effect of changes in the level of pollution. We are therefore assuming that there is cause-effect relationship between the two variables. So we are assuming that there is a variable y whose changes can be explained by changes in another variable x. If furthermore we assume that the relationship is linear, then we have THE SIMPLE REGRESSION MODEL y i = β 0 + β 1 x i + ɛ i M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

11 The Simple Regression Model Main Components of Simple Regression Model y i is the value of the dependent variable for observation i x i is the value of the independent or explanatory variable for observation i β 0 is the intercept of the regression line β 1 is the slope of the regression line. β 0 and β 1 are the parameters of the model. ɛ i is the error or disturbance term, i.e the difference between the value of y i and the value of E(y i ) Discussion in class: what does the error term capture? M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

12 Assumptions Assumption 1: The model is linear. Assumption 2: E(ɛ i ) = 0 for all i. Assumption 3: Var(ɛ i ) = σ 2, i.e the variance of the error term is the same for all observations. This means that the variance of Y is also σ 2 for all observations. Assumption 4: The observations of the error term are independent of one another and therefore the observations of Y are also independent. Assumption 5: The distribution of all ɛ i is Normal. This assumption will be needed when we test hypotheses and have small samples. When the sample size is large then by the Central Limit Theorem we will not need to make this assumption. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

13 Assumptions Using equation y i = β 0 + β 1 x i + ɛ i and Assumption 2 we have, The Population Regression Line E(y i ) = β 0 + β 1 x i It results that y i = E(y i ) + ɛ i So the observed value of y i can be decomposed in two parts, one part that can be predicted from x i, ie E(y i ) and a part that is random and is due to other factors that affect y i and is depicted by ɛ i. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

14 The Regression Coefficients The simple linear regression model has two regression coefficients: Regression Coefficients β 0 is the value of E(y) when x = 0, so it is the point where line intersects the vertical axis (intercept) β 1 measures the average change in the dependent variable when x increases one unit (slope of the regression line). Therefore it can be positive or negative depending on the relationship between x and y. The units of measurement of y and x will affect the interpretation of the values of β 0 and β 1 If the values of the regression coefficients were known to us we could use them in order to make forecasts for y but since they are unknown we have to estimate them using data for y and x. The estimation method we will use is called the Least Squares Method M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

15 The Method of Least Squares: OLS The objective is to find estimates of β 0 and β 1 (which will be denoted by b 0 and b 1 ) so that the estimated line represents in the best way the linear relationship between x and y. How? Let the estimated line be given by, ŷ i = b 0 + b 1 x i, ŷ i is the predicted value by the regression model The difference between the actual value and the predicted value is called the residual and denoted by e i. e i = y i ŷ i The least squares method determines a regression line (in other words a value of b 0 and b 1 ) that minimizes the sum of squared residuals ( n i=1 e2 i ). M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

16 The Method of Least Squares: OLS Sample Regression Line ŷ i = b 0 + b 1 x i The sample regression line can be used to forecast the value of y for different values of x M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

17 The Coefficient of Determination: R-squared The coefficient of determination measures what percentage of the total variation in the dependent variable is explained by its relationship with the independent variable or is explained by the regression model. It therefore measures how well the model fits the data and is computed after a model has been estimated. Total Variation Decomposition SST = SSR + SSE Total variation in Y = variation of Y explained by regression + variation that is not explained by the regression The coefficient of determination is given by Coefficient of Determination Note, 0 R 2 1 R 2 = SSR SST = 1 SSE SST The closer it is to 1 the better the fit In the simple regression model R 2 = r 2, the coefficient of determination is equal to the square of the correlation coefficient between x and y. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

18 Confidence Intervals and Testing Hypotheses about β 1 Confidence Intervals for β 1 A (1 α)% Confidence interval for β 1 is given by b 1 t n 2, α 2 S b1 < β 1 < b 1 + t n 2, α 2 S b1 where t n 2, α 2 is the value from the table of the t distribution with n 2 degrees of freedom and S b1 is the standard error of b 1. Testing Hypotheses about β 1 H 0 : β 1 = 0 H 1 : β 1 = 0 We use the t-statistic which follows a t distribution with n 2 degrees of freedom and is given by Test Statistic t = b 1 S b1 M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

19 Confidence Intervals and Testing Hypotheses about β 1 For an α% significance level test, We reject the null hypothesis if t statistic > t n 2, α 2 OR t statistic < t n 2, α 2 Alternatively we can use the p-value and reject the null hypothesis if p value < α If we accept the null hypothesis then we say that the coefficient is not statistically significant (or that x has no significant effect on y). If we reject the null hypothesis then we say that the coefficient is statistically significant (or that x has a significant effect on y). M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

20 Application: House Prices and Pollution-Data We are going to analyze regional (census tracts in Boston) median housing prices. The following variables appear in the data set in the given order: Variables: price crime nox rooms dist radial proptax stratio lowstat lprice lnox lproptax Variable labels: 1. price: median housing price, $ 2. crime: crimes committed per capita 3. nox: nitrous oxide, parts per 100 mill. 4. rooms: avg number of rooms per house 5. dist: weighted dist. to 5 employ centers 6. radial: accessibiliy index to radial hghwys 7. proptax: property tax per stratio: average student-teacher ratio 9. lowstat: % of people lower status 10. lprice: log(price) 11. lnox: log(nox) 12. lproptax: log(proptax) M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

21 Application: House Prices and Pollution-Summary Statistics Stata command summarize gives us the descriptive statistics of the data Table: Summary statistics Variable Mean Std. Dev. Min. Max. price crime nox rooms dist radial proptax stratio lowstat lprice lnox lproptax N 506 M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

22 Application: House Prices and Pollution-Correlations Stata has been used to produce the scatter plot we saw before and to compute the correlation coefficient and to estimate the simple regression model. The following table shows the bivariate correlations between price, nox and rooms. Comment the signs, the strength and the significance. Table: Cross-correlation table Variables price, $ nox rooms price, $ nox (0.000) rooms (0.000) (0.000) In stata: Select Statistics, Summaries, tables, and tests, Summary and descriptive statistics, Pairwise correlations then enter the names of the variables and tick Print significance level for each entry M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

23 Application: House Prices and Pollution-OLS estimates Results from least squares estimates for model price = β 0 + β 1 nox + ɛ Stata command regress price nox Table: Dep = price Variable nox Intercept Coefficient (Std. Err.) ( ) ( ) N 506 R F (1,504) Significance levels : : 10% : 5% : 1% M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

24 Application: House Prices and Pollution-Interpretation The estimated model is ˆ price = nox. Interpretation of estimated model. An increase in one unit of nox (1 part per 100 million) will decrease the average value of houses by $ Variable nox is significant at 1% level (this is denoted by ** on the table). The p-value reported in the program Stata is R-squared is only What would be the effect on the value of a house of implementing a policy that reduces pollution by 10%? Denoting with the subindex 0 the initial situation and with subindex 1 the situation after implementing the policy, we have, ˆ price 0 = nox 0 and nox 1 = 0.90nox 0 ˆ price 1 = nox 1 = nox 0 The change in price is given by P = nox 0 M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

25 Application: House Prices and Pollution We found that the effect of a 10% decrease in nox is an increase in price of nox$. This means that for an area with an average pollution level (nox=5.55) the average price would increase by $ Discussion: Should we maybe include other important factors that affect the house prices? We will discuss this in the next set of slides. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26

26 References Wooldridge, J. M. (2015). Introductory Econometrics: A Modern Approach. South-Western College Pub. Stock, J. H and Watson, M. W (2010). Introduction to Econometrics. Addison-Wesley. M. Genius (Univ. of Crete) Econometric Methods for Valuation Analysis Cagliari, / 26