linear regression with one regressor

Question 1

what is the population regression line?

Answer 1

it is the expected value of Y given X ie E(Y/X)

Question 2

what is the slope of the population regression line?

Answer 2

the slope is the difference in the expected values of Y, for two values of X that differ by one unit.

Question 3

what can the estimated regression be used for?

Answer 3

casual inference (learning about the casual effect on Y of a change in X)
prediction ( predicting the value of Y given X for an observation not in the data set)

Question 4

what is the problem of statistical inference for linear regression?

Answer 4

the problem of statistical inference for linear regression is, at a general level, the same as for estimation of the mean or of the differences between two means

Question 5

what does the statistical or econometric inference about the slope entails?

Answer 5

1) estimation - how should we draw a line through the data to estimate the population slope
2) hypothesis testing - how to test whether the slope is zero
3) confidence interval - how to construct a confidence interval for the slope?

Question 6

what is the equation for the population regression line?

Answer 6

test score = β0 + β1STR where β1 = slope of population regression line

Question 7

Why are β0 and β1 “population” parameters?

Answer 7

• We would like to know the population value of β1.
• We don’t know β1, so must estimate it using data

Question 8

what is the equation of the population linear regression model?

Answer 8

Yi = β0 + β1Xi + ui, i = 1,…, n
* We have n observations, (Xi, Yi), i = 1,.., n.
* X is the independent variable or regressor
* Y is the dependent variable
* β0 = intercept
* β1 = slope* ui = the regression error

Question 9

what does the regression error consist off?

Answer 9

The regression error consists of omitted factors. In general, these omitted factors are other factors that influence Y, other than the variable X. The regression error also includes error in the measurement of Y.

Question 10

how can you estimate the β0 and β1 from data?

Answer 10

we can use the ordinary least squares or OLS estimator of the unknown parameters β0and β1. The OLS estimator solves: min(b0,b1) ∑ [Yi - (b0 + b1Xi)]^2

Question 11

what is the equation for 𝛽1 ?

Answer 11

r(xy)*[Sy/Sx] where r(xy) is the sample correlation and Sy and Sx is the sample standard deviations

Question 12

what is the equation for 𝛽0?

Answer 12

𝛽0 = mean of Y - 𝛽1*mean of X

Question 13

what does the regression R^2?

Answer 13

the regression R^2 measures the fraction of the variance of Y that is explained by X; it is unitless and ranges between zero (no fit) and one (perfect fit)

Question 14

what does the standard error of regression(SER) measure?

Answer 14

the standard error of the regression measures magnitude of a typical regression residual in the units of Y

Question 15

what is the equation for R^2?

Answer 15

R^2 = ESS/TSS = [∑(Ŷ-mean of Ŷ)^2]/[∑(Y-mean of Y)^2]

Question 16

what is the standard error of regression(SER)?

Answer 16

The SER measures the spread of the distribution of u. The SER is(almost) the sample standard deviation of the OLS residuals

Question 17

what is the equation for the SER?

Answer 17

SER= [1/(n-2) *∑û^2]^(1/2)

Question 18

what is the definition of casual effect on Y of a unit change in X?

Answer 18

it is the expected difference in Y as measured in a randomized controlled experiment

Question 19

for a binary treatment, what is required for a difference in means to measure a causal effect?

Answer 19

it requires random assignment or as-if random assignment. random assignment ensures that the treatment (X) is uncorrelated with all other determinents of Y, so that there are no confounding variables

Question 20

what are the least squares assumptions for causal inference?

Answer 20

1) the conditional distribution of u given X has mean zero, that is E(u\X=x)=0
2) (X Y), i=1,….,n, are i.i.d. this is true if (X, Y) are collected by simple random sampling and delivers the sampling distribution of β0 and β1
3) large outliers in X and/or Y are rare. technically X and Y have finite fourth moments. outliers can result in meaningless values of β1