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The Simple Regression Model

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The Simple Regression Model

y = b0 + b1x + u

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- 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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- 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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- The average value of u, the error term, in the population is 0. That is,
- E(u) = 0

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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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- 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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