The simple regression model
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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.

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The Simple Regression Model

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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

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We also assume

We also assume

  • E(u|x) = 0

  • E(y|x) = b0 + b1x

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The simple regression model

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

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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

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The simple regression model

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

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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


The simple regression model

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

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One approach to estimate coefficients

One approach to estimate coefficients

  • 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:

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It could be shown that estimated coefficient is

It could be shown that estimated coefficient is

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Summary of ols slope estimate

Summary of OLS slope estimate

  • 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

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Algebraic properties of ols in english

Algebraic Properties of OLS: in 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

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Algebraic properties of ols in mathematics

Algebraic Properties of OLS: In mathematics:

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More terminology

More terminology

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Notations alert

Notations Alert

  • 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.

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Goodness of fit

Goodness-of-Fit

  • 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

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Ols regressions

OLS regressions

  • 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?

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Under some conditions ols esimated coefficients are unbiased

Under some conditions, OLS esimated coefficients are unbiased.

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Unbiasedness summary

Unbiasedness Summary

  • 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

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Variance of the ols estimators

Variance of the OLS Estimators

  • 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

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Variance of ols estimators

Variance of OLS estimators

  • 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

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Variance of ols estimator

Variance of OLS estimator

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Variance of ols summary

Variance of OLS Summary

  • 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

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Estimating the error variance

Estimating the Error Variance

  • 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

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Estimating the error variance1

Estimating the Error Variance

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Estimating standard error of coefficients estimate

Estimating Standard Error of coefficients Estimate

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