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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

y = b0 + b1x + u

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### 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

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### Some Terminology

• we typically refer to x as the

• Independent Variable, or

• Right-Hand Side Variable, or

• Explanatory Variable, or

• Regressor, or

• Control Variables

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### 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

• E(u|x) = 0

• E(y|x) = b0 + b1x

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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)

• 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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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

• 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

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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

• 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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### 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

• 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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### More terminology

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• 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

• 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

• 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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### 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

• 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

• 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 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

• 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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