Definition of the Simple Regression Model The Simple Regression Model Deriving the Ordinary Least Squares Estimates Properties of Caio Vigo OLS on any Sample of Data The University of Kansas Units of Measurement Department of Economics and Functional Form Using the Natural Fall 2018 Logarithm in Simple Regression Expected Value of OLS These slides were based on Introductory Econometrics by Jeffrey M. Wooldridge (2015) 1 / 83
Topics Definition of the Simple Regression Model 1 Definition of the Simple Regression Model Deriving the Ordinary Least Squares 2 Deriving the Ordinary Least Squares Estimates Estimates Properties of OLS on any 3 Properties of OLS on any Sample of Data Sample of Data Units of Measurement 4 Units of Measurement and Functional Form and Functional Using the Natural Logarithm in Simple Regression Form Using the Natural Logarithm in Simple Regression 5 Expected Value of OLS Expected Value of OLS 2 / 83
Definition of the Simple Regression Model Definition of the Simple • What type of analysis will we do? Cross-sectional analysis Regression Model Deriving the • First step: Clearly define what is your population (in what you are interested Ordinary Least Squares to study). Estimates Properties of OLS on any • Second Step: There are two variables, x and y , and we would like to “study Sample of Data how y varies with changes in x .” Units of Measurement and Functional Form • Third Step: We assume we can collect a random sample from the population Using the Natural Logarithm in Simple of interest. Regression Expected Value of OLS Now we will learn to write our first econometric model, derive an estimator ( what’s an estimator again? ) and use this estimator in our sample. 3 / 83
Introduction Definition of the Simple Regression Model Deriving the We must confront three issues: Ordinary Least Squares 1 How do we allow factors other than x to affect y ? There is never an exact Estimates relationship between two variables. Properties of OLS on any Sample of Data 2 What is the functional relationship between y and x ? Units of Measurement and Functional Form 3 How can we be sure we a capturing a ceteris paribus relationship between y and Using the Natural Logarithm in Simple x ? Regression Expected Value of OLS 4 / 83
Introduction Definition of the Simple Regression Model Deriving the Consider the following equation relating y to x : Ordinary Least Squares Estimates y = β 0 + β 1 x + u, Properties of OLS on any Sample of which is assumed to hold in the population of interest. Data Units of Measurement • This equation defines the simple linear regression model (or two-variable and Functional Form regression model, or bivariate linear regression model ). Using the Natural Logarithm in Simple Regression Expected Value of OLS 5 / 83
Introduction Definition of the Simple Regression • y and x are not treated symmetrically. We want to explain y in terms of x . Model Deriving the Ordinary Least Squares Estimates x explains y Properties of OLS on any Sample of Data Units of x − → y Measurement and Functional Form Using the Natural Logarithm in Simple • Example: Regression Expected Value of OLS size of the city x , explains number of crimes ( y ) (not the other way around) . 6 / 83
Terminology for Simple Regression Definition of the Simple Regression Model Deriving the Ordinary Least y x Squares Estimates Dependent Variable Independent Variable Properties of OLS on any Explained Variable Explanatory Variable Sample of Data Resonse Variable Control Variable Units of Predicted Variable Predictor Variable Measurement and Functional Regressand Regressor Form Using the Natural Logarithm in Simple Regression Expected Value of OLS 7 / 83
The error term Definition of the Simple Regression Model Deriving the Ordinary Least y = β 0 + β 1 x + u Squares Estimates Properties of OLS on any This equation explicitly allows for other factors, contained in u , to affect y . Sample of Data Units of This equation also addresses the functional form issue (in a simple way). Namely, y Measurement and Functional is assumed to be linearly related to x . We call β 0 the intercept parameter and β 1 Form the slope parameter . These describe a population, and our ultimate goal is to Using the Natural Logarithm in Simple Regression estimate them. Expected Value of OLS 8 / 83
The simple linear regression model equation Definition of • The equation also addresses the ceteris paribus issue. In the Simple Regression Model y = β 0 + β 1 x + u, Deriving the Ordinary Least Squares all other factors that affect y are in u . We want to know how y changes when x Estimates changes, holding u fixed . Properties of OLS on any • Let ∆ denote “change. ”Then holding u fixed means ∆ u = 0 . So Sample of Data Units of Measurement and Functional ∆ y = β 1 ∆ x + ∆ u Form Using the Natural = β 1 ∆ x when ∆ u = 0 . Logarithm in Simple Regression Expected Value of OLS • This equation effectively defines β 1 as a slope, with the only difference being the restriction ∆ u = 0 . 9 / 83
The simple linear regression model equation Definition of the Simple Regression Model Deriving the Example: Yield and Fertilizer Ordinary Least Squares • A model to explain crop yield to fertilizer use is Estimates Properties of OLS on any yield = β 0 + β 1 fertilizer + u, Sample of Data Units of Measurement where u contains land quality, rainfall on a plot of land, and so on. The slope and Functional Form parameter, β 1 , is of primary interest: it tells us how yield changes when the amount Using the Natural Logarithm in Simple of fertilizer changes, holding all else fixed. Regression Expected Value of OLS 10 / 83
The simple linear regression model equation Definition of the Simple Regression Model Deriving the Ordinary Least Example: Wage and Education Squares Estimates wage = β 0 + β 1 educ + u Properties of OLS on any Sample of where u contains somewhat nebulous factors (“ability”) but also past workforce Data experience and tenure on the current job. Units of Measurement and Functional Form ∆ wage = β 1 ∆ educ when ∆ u = 0 Using the Natural Logarithm in Simple Regression Expected Value of OLS 11 / 83
The simple linear regression model equation Definition of We said we must confront three issues: the Simple Regression 1. How do we allow factors other than x to affect y ? Model Answer: u Deriving the Ordinary Least Squares 2. What is the functional relationship between y and x ? Estimates Answer: Linear ( x has a linear effect on y ) Properties of OLS on any Sample of Data 3. How can we be sure we a capturing a ceteris paribus relationship between y and Units of x ? Measurement and Functional Answer: Related with ∆ u = 0 Form Using the Natural Logarithm in Simple • We have argued that the simple regression model Regression Expected Value of OLS y = β 0 + β 1 x + u addresses each of them. 12 / 83
Relation between u and x Definition of the Simple Regression To estimate β 1 and β 0 from a random sample we also need to restrict how u and Model x are related to each other. Deriving the Ordinary Least Squares Estimates • Recall that x and u are properly viewed as having distributions in the population. Properties of OLS on any Sample of Data • What we must do is restrict the way in when u and x relate to each other in the Units of population . Measurement and Functional Form • First, we make a simplifying assumption that is without loss of generality: the Using the Natural Logarithm in Simple Regression average, or expected, value of u is zero in the population: Expected Value of OLS E ( u ) = 0 13 / 83
Relation between u and x Definition of the Simple • Normalizing u should cause no impact in the most important parameter: β 1 Regression Model Deriving the • The presence of β 0 in Ordinary Least Squares Estimates y = β 0 + β 1 x + u Properties of OLS on any Sample of allows us to assume E ( u ) = 0 . Data Units of Measurement • If the average of u is different from zero, we just adjust the intercept, leaving the and Functional Form slope the same. If α 0 = E ( u ) then we can write Using the Natural Logarithm in Simple Regression y = ( β 0 + α 0 ) + β 1 x + ( u − α 0 ) , Expected Value of OLS where the new error, u − α 0 , has a zero mean. 14 / 83
Relation between u and x Definition of the Simple Regression Model We need to restrict the dependence between u and x Deriving the Ordinary Least Squares Estimates • Option 1: Uncorrelated Properties of OLS on any Sample of Data We could assume u and x uncorrelated in the population: Units of Measurement and Functional Corr ( x, u ) = 0 Form Using the Natural Logarithm in Simple Regression Expected It implies only that u and x are not linearly related. Not good enough . Value of OLS 15 / 83
Relation between u and x Definition of the Simple Regression Model • Option 2: Mean independence Deriving the Ordinary Least The mean of the error (i.e., the mean of the unobservables) is the same across all Squares Estimates slices of the population determined by values of x . Properties of OLS on any Sample of We represent it by: Data Units of Measurement E ( u | x ) = E ( u ) , all values x, and Functional Form Using the Natural Logarithm in Simple Regression Expected And we say that u is mean independent of x Value of OLS 16 / 83
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