ECON 450 Development Economics

Transcription

1 ECON 450 Development Economics Statistics Background University of Illinois at Urbana-Champaign Summer 2017

2 Outline 1 Introduction

3 Introduction Regression analysis is one of the most important statistical tools used in economics. Whenever we cannot benefit from experimental settings or quasi-experiments, we need to rely on observational studies. In these situations, regression plays a significant role.

4 Outline 1 Introduction

5 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

6 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

7 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

8 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

9 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

10 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

11 Suppose you are interested in the relationship of some variable Y and a range of other variables X 1, X 2,..., X k. For example, suppose you are a Real Estate agent and is interested in predicting the price of houses Y in a given location. What are the variables that affect Y? Size # of bedrooms Neighborhood Physical conditions

12 Population Sample Regression Model The population regression model is Y = β 0 + β 1 X 1 + β 2 X β k X k + ε Notation: Y is the dependent variable X i s are the independent variables or covariates β 0 is the intercept β i s are the slope terms ε is the error term

13 Population Sample Regression Model The β s are population unknown parameters. We estimate these parameters by applying the regression analysis tools on a sample from the population.

14 Population Sample Regression Model Our task is to find the (estimated) sample regression model Y = b 0 + b 1 X 1 + b 2 X b k X k + ɛ Notice that the b i s are estimates of population parameters based on sample data. Therefore, they are sample statistics.

15 Outline 1 Introduction

16 Estimating the Model Coefficients Suppose you want to estimate the following population model Y = β 0 + β 1 X + ε How to proceed to estimate the above model? Select a sample from the population Collect data on both Y and X Then what?

17 Estimating the Model Coefficients Suppose you want to estimate the following population model Y = β 0 + β 1 X + ε How to proceed to estimate the above model? Select a sample from the population Collect data on both Y and X Then what?

18 Estimating the Model Coefficients Suppose you want to estimate the following population model Y = β 0 + β 1 X + ε How to proceed to estimate the above model? Select a sample from the population Collect data on both Y and X Then what?

19 Estimating the Model Coefficients Suppose you want to estimate the following population model Y = β 0 + β 1 X + ε How to proceed to estimate the above model? Select a sample from the population Collect data on both Y and X Then what?

20 Estimating the Model Coefficients Suppose you want to estimate the following population model Y = β 0 + β 1 X + ε How to proceed to estimate the above model? Select a sample from the population Collect data on both Y and X Then what?

21 Estimating the Model Coefficients Below is an example of a plot of a given sample from some population.

22 Estimating the Model Coefficients There are a number of ways which could be used to formulate the line that best characterizes those data points. We will rely on the most commonly used and most powerful method, which is to select the line that minimizes the sum of squared vertical differences between the points and the line. This method is called "Ordinary Least Squares" or OLS for short.

23 Estimating the Model Coefficients Compare two lines, the first upward sloping, the second horizontal: Line 1 (upward sloping): Sum of the squared differences = (2 1) 2 + (4 2) 2 + (1.5 3) 2 + (3.2 4) 2 = 6.89 Line 2 (horizontal): Sum of the squared differences = (2 2.5) 2 + (4 2.5) 2 + ( ) 2 + ( ) 2 = 3.99

24 Estimating the Model Coefficients We can write Recall and e i = Y i Ŷi Y i = b 0 + b 1 X i + e i Ŷ i = b 0 + b 1 X i We want to minimize the sum of the squares of the residuals. Thus, we need to solve n Min ei 2 = i=1 n (Y i Ŷi) 2 = i=1 n (Y i b 0 + b 1 X i ) 2 i=1

25 Estimating the Model Coefficients Applying calculus technique, we can simultaneously find both b 1 and b 0. cov(x, y) xi y i n xȳ b 1 = sx 2 = x 2 i n x 2 b 0 = Ȳ b 1 X

26 Estimating the Model Coefficients In multiple regression, we use the same idea to estimate the coefficients. The computational aspect becomes more challenging as it involves matrix algebra and more advanced calculus techniques. It suffices to grasp the main ideas behind the OLS method of estimating some population model.

27 Outline 1 Introduction

28 Assessing the goodness of fit Once our model is estimated, the next step is to assess how successful we have been in accomplishing our objective. We make use of some statistics and hypothesis testing to address this issue. R 2 and adjusted R 2 F test t test for the slope coefficients

29 The Coefficient of Determination R 2 When we want to measure the strength of the linear relationship, we use the coefficient of determination. R 2 = [cov(x, y)]2 s 2 xs 2 y or R 2 = SSR SST = 1 SSE SST

30 The Coefficient of Determination R 2 Recall Variation in Y (SST) = SSR + SSE Therefore, R 2 measures the proportion of the variation in Y that is explained by the variation in X. R 2 takes on any value between zero and one. R 2 = 1: Perfect match between the line and the data points. R 2 = 0: There is no linear relationship between x and y.

31 The Adjusted Coefficient of Determination Adj-R 2 The "Adjusted" Coefficient of Determination is defined as: AdjR 2 = 1 SSE/(n k 1) SST /(n 1) Adj-R 2 penalizes a small amount for each additional independent variable you add. The new variable must significantly contribute to explaining SST, before Adj-R 2 will go up.

32 The F test for Overall Validity of the Model The question addressed in this test is: "Is there at least one independent variable linearly related to the dependent variable?" We perform the following test: H 0 : β 1 = β 2 =... = β k = 0 H 1 : At least one β i is not equal to zero We reject H 0 in favor of the alternative if the test statistic F = SSR/k SSE/(n k 1) is greater than the critical value F α,k,n k 1.

33 Hypothesis Test to Evaluate Slope term When no linear relationship exists between two variables, the regression line should be horizontal.

34 Hypothesis Test to Evaluate Slope term The set of hypotheses for each individual slope β i are given by H 0 : β i = 0 H 1 : β i 0 The test statistic is where s bi = t = b i β i s bi s ε (n 1)s 2 x If the error variable is normally distributed, the statistic is Student t distributed with d.f. = n-k-1.

35 Implicit Assumptions About the Error Term It is worth mentioning the implicit assumptions we need to make in order to make inference about the population parameters. 1 The probability distribution of ε is normal, with mean 0 2 The standard deviation of ε is σ ε for all values of x 3 The set of errors associated with different values of y are all independent. 4 No unnecessary outliers 5 No serious multicolinearity

36 Regression as a Powerful Tool Finally, notice that regression is a powerful and flexible tool since it allows a variety of mathematical models such as 1 linear relationships 2 many independent variables 3 curvilinear relationships 4 qualitative variables

37 Outline 1 Introduction

38 Suppose you are interested in finding the relationship between years and education and income. Returns to education is one of the most important relationships studied in economics. It is extremely useful for economic policies. The true relationship is assumed to be given by Income = β 0 + β 1 Educ + ε

39 A sample of 20 individuals is shown below:

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