Econ 140 Lecture 71 Classical Regression Lecture 7

Econ 140 Lecture 72 Today’s Plan For the next few lectures we’ll be talking about the classical regression model –Looking at both estimators for a and b –Inferences on what a and b actually tell us Today: how to operationalize the model –Looking at BLUE for bi-variate model –Next lecture: Inference and hypothesis tests using the t, F, and  2 –Examples of linear regressions using Excel

Econ 140 Lecture 73 Estimating coefficients Our model: Y = a + bX + e Two things to keep in mind about this model: 1) It is linear in both variables and parameters Examples of non-linearity in variables: Y = a + bX 2 or Y = a + be x Example of non-linearity in parameters: Y = a + b 2 X OLS can cope with non-linearity in variables but not in parameters

Econ 140 Lecture 74 Estimating coefficients (3) 2) Notation: we’re not estimating a and b anymore We are estimating coefficients which are estimates of the parameters of a and b We will denote the coefficients as or and or We are dealing with a sample size of n –For each sample we will get a different and pair

Econ 140 Lecture 75 Estimating coefficients (4) In the same way that you can take a sample to get an estimate of µ y you can take a sample to get an estimate of the regression line, of  and 

Econ 140 Lecture 76 The independent variable We also have a given variable X, its values are known –This is called the independent variable Again, the expectation of Y given X is E(Y | X) = a + bX With constant variance V(Y) =  2

Econ 140 Lecture 77 A graph of the model (Y 1, X 1 ) Y Y X

Econ 140 Lecture 78 What does the error term do? The error termgives us the test statistics and tells us how well the model Y = a+bX+e fits the data The error term represents: 1) Given that Y is a random variable, e is also random, since e is a function of Y 2) Variables not included in the model 3) Random behavior of people 4)Measurement error 5)Enables a model to remain parsimonious - you don’t want all possible variables in the model if some have little or no influence

Econ 140 Lecture 79 Rewriting beta Our complete model is Y = a + bX + e We will never know the true value of the error e so we will estimate the following equation: For our known values of X we have estimates of , , and  So how do we know that our OLS estimators give us the BLUE estimate? –To determine this we want to know the expected value of  as an estimator of b, which is the population parameter

Econ 140 Lecture 710 Rewriting beta(2) To operationalize, we want to think of what we know We know from lecture 2 that there should be no correlation between the errors and the independent variables We also know Now we have that E(Y|X) = a + bX + E(  |X) The variance of Y given X is V(Y) =  2 so V(  |X)=  2

Econ 140 Lecture 711 Rewriting beta(3) Rewriting  –In lecture 2 we found the following estimator for  Using some definitions we can show: E(  ) = b

Econ 140 Lecture 712 Rewriting beta (4) We have definitions that we can use: Using the definitions for y i and x i we can rewrite  as So that We can also write

Econ 140 Lecture 713 Rewriting beta (5) We can rewrite  as where The properties of c i :

Econ 140 Lecture 714 Showing unbiasedness What do we know about the expected value of beta? We can rewrite this as Multiplying the brackets out we get: Since b is constant,

Econ 140 Lecture 715 Showing unbiasedness (2) Looking back at the properties for c i we know that Now we can write this as We can conclude that the expected value of  is b and that  is an unbiased estimator of b

Econ 140 Lecture 716 Gauss Markov Theorem We can now ask: is  an efficient estimator? The variance of  is Where How do we know that OLS is the most efficient estimator? –The Gauss-Markov Theorem

Econ 140 Lecture 717 Gauss Markov Theorem (2) Similar to our proof on the estimator for  y. Suppose we use a new weight We can take the expected value of E(  )

Econ 140 Lecture 718 Gauss Markov Theorem (3) We know that –For  to be unbiased, the following must be true:

Econ 140 Lecture 719 Gauss Markov Theorem (4) Efficiency (best)? We have where Therefore the variance of this new  is +  d i 2 +2  c i d i If each d i  0 such that c i  c’ i then So when we use weights c’ I we have an inefficient estimator

Econ 140 Lecture 720 Gauss Markov Theorem (5) We can conclude that is BLUE

Econ 140 Lecture 721 Wrap up What did we cover today? Introduced the classical linear regression model (CLRM) Assumptions under the CLRM 1)X i is nonrandom (it’s given) 2)E(e i ) = E(e i |X i ) = 0 3)V(e i )= V(e i |X i ) =  2 4)Covariance (e i e j ) = 0 Talked about estimating coefficients Defined the properties of the error term Proof by contradiction for the Gauss Markov Theorem