The simple regression model
Download
1 / 28

The Simple Regression Model - PowerPoint PPT Presentation


  • 98 Views
  • Uploaded on

The Simple Regression Model. y = b 0 + b 1 x + u. Some Terminology. In the simple linear regression model, where y = b 0 + b 1 x + u , we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or. Some Terminology.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' The Simple Regression Model' - zahina


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
The simple regression model

The Simple Regression Model

y = b0 + b1x + u

FIN357 Li


Some terminology
Some Terminology

  • In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as the

    • Dependent Variable, or

    • Left-Hand Side Variable, or

    • Explained Variable, or

FIN357 Li


Some terminology1
Some Terminology

  • we typically refer to x as the

    • Independent Variable, or

    • Right-Hand Side Variable, or

    • Explanatory Variable, or

    • Regressor, or

    • Control Variables

FIN357 Li


A simple assumption
A Simple Assumption

  • The average value of u, the error term, in the population is 0. That is,

  • E(u) = 0

FIN357 Li


We also assume
We also assume

  • E(u|x) = 0

  • E(y|x) = b0 + b1x

FIN357 Li


E(y|x) as a linear function of x, where for any x

the distribution of y is centered about E(y|x)

y

f(y)

.

E(y|x) = b0 + b1x

.

x1

x2

FIN357 Li


Ordinary least squares ols
Ordinary Least Squares (OLS)

  • Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population

  • For each observation in this sample, it will be the case that

  • yi = b0 + b1xi + ui

FIN357 Li


Population regression line, sample data points

and the associated error terms

y

E(y|x) = b0 + b1x

.

y4

{

u4

.

u3

y3

}

.

y2

u2

{

u1

.

}

y1

x2

x1

x4

x3

x

FIN357 Li


Basic idea of regression is to estimate the population parameters from a sample
Basic idea of regression is to estimate the population parameters from a sample

  • Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible.

  • The residual, û, is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point

FIN357 Li


Sample parameters from a sample regression line, sample data points

and the associated estimated error terms (residuals)

y

.

y4

{

û4

.

û3

y3

}

.

y2

û2

{

û1

}

.

y1

x2

x1

x4

x3

x

FIN357 Li


One approach to estimate coefficients
One approach to estimate coefficients parameters from a sample

  • Given the intuitive idea of fitting a line, we can set up a formal minimization problem

  • That is, we want to choose our parameters such that we minimize the following:

FIN357 Li


It could be shown that estimated coefficient is
It could be shown that estimated coefficient is parameters from a sample

FIN357 Li


Summary of ols slope estimate
Summary of OLS slope estimate parameters from a sample

  • The slope estimate is the sample covariance between x and y divided by the sample variance of x

  • If x and y are positively correlated, the slope will be positive

  • If x and y are negatively correlated, the slope will be negative

FIN357 Li


Algebraic properties of ols in english
Algebraic Properties of OLS: parameters from a samplein English

  • The sum of the OLS residuals is zero

  • Thus, the sample average of the OLS residuals is zero as well

  • The sample covariance between the regressors and the OLS residuals is zero

  • The OLS regression line always goes through the mean of the sample

FIN357 Li


Algebraic properties of ols in mathematics
Algebraic Properties of OLS: parameters from a sampleIn mathematics:

FIN357 Li


More terminology
More terminology parameters from a sample

FIN357 Li


Notations alert
Notations Alert parameters from a sample

  • The notation SSR (Sum of Squared Residuals) in this handout and my other lecture slides= ESS (Error Sum of Squares) in our textbook.

  • The notation SSE (Sum of Squared Explained) in this handout and my other lecture slides = RSS (Regressed Sum of Squares) in our textbook.

FIN357 Li


Goodness of fit
Goodness-of-Fit parameters from a sample

  • How do we think about how well our sample regression line fits our sample data?

  • Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression

  • R2 = SSE/SST = 1 – SSR/SST

FIN357 Li


Ols regressions
OLS regressions parameters from a sample

  • Now that we’ve derived the formula for calculating the OLS estimates of our parameters, you’ll be happy to know you don’t have to compute them by hand

  • Regressions in GRETL are very simple.

  • Have you installed the software yet?

FIN357 Li



Unbiasedness summary
Unbiasedness Summary unbiased.

  • The OLS estimates of b1 and b0 are unbiased

  • Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter

FIN357 Li


Variance of the ols estimators
Variance of the OLS Estimators unbiased.

  • Now we know that the sampling distribution of our estimated coefficient is centered around the true parameter

  • Want to think about how spread out this distribution is

  • Assume Var(u|x) =Var(u) = s2

FIN357 Li


Variance of ols estimators
Variance of OLS estimators unbiased.

  • s2 is called the error variance

  • s, the square root of the error variance is called the standard deviation of the error

  • E(y|x)=b0 + b1x and Var(y|x) = s2

FIN357 Li


Variance of ols estimator
Variance of OLS estimator unbiased.

FIN357 Li


Variance of ols summary
Variance of OLS Summary unbiased.

  • The larger the error variance, s2, the larger the variance of the slope estimate

  • The larger the variability in the xi, the smaller the variance of the slope estimate

  • Problem that the error variance is unknown

FIN357 Li


Estimating the error variance
Estimating the Error Variance unbiased.

  • We don’t know what the error variance, s2, is, because we don’t observe the errors, ui

  • What we observe are the residuals, ûi

  • We can use the residuals to form an estimate of the error variance

FIN357 Li


Estimating the error variance1
Estimating the Error Variance unbiased.

FIN357 Li



ad