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Econometrics > Chapter 2 > Flashcards

Chapter 2 Flashcards

1
Q

Formula dependent/explained/response variable y

A
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2
Q

Assumption 1 u

A

E(u) = 0

we normalize unobserved factors to have on average a value of 0 in the population

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3
Q

How should x be related to u (Assumption 1)

A
  • correlation coefficient: If x and u are uncorrelated then they are not linear
  • U is mean independent of x
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4
Q

Assumption 2

A
  • conditional mean independence assumption
  • E(u∣X)=0
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5
Q

To what can the violations of the conditiona mean independence assumption lead?

A

to biased parameter estimates and inefficient hypothesis tests in regression analysis

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6
Q

What does the explanatory variable not contain (Assumption 2)?

A

information about the mean of the unobserved factors

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7
Q

Populaton regression function: Formula/calculation

A
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8
Q

How can the average value of dependent variable be expressed as (PFR)?

A

linear function of independent value

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9
Q

How does one unit increase in x change the average value of y (PFR)?

A

by beta 1

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10
Q

Describe OLS

A
  • fits a linear line onto the data
  • estimates the parameters in such a way that the sum of the squared values of the residuals is minimized
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11
Q

Definition: Residual

A

actual value y minus the predicted value y where predicted value is based on the model aprameters

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12
Q

Formula: Residual

A
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13
Q

Formula: RSS

A
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14
Q

Formula: fitted or predicted values

A
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15
Q

Formula: Deviations from regression line (=residuals)

A
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16
Q

What does the average of the residuals/deviations from regression equal to?

A

zero

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17
Q

What does the covariance between residuals and regression equal to and what does it imply?

A
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18
Q

OLS estimates are chosen to make the residuals add up to what, for any data set?

A

zero

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19
Q

Poperty 1 of OLS means:

A

the average of residuals is zero

the sample average of the fitted values is the same as the sample average of the yi

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20
Q

Formula: First alegabraic property of OLS regression

A
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21
Q

Formula: second algebraic property of OLS regression

A
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22
Q

Formula: third algebraic property of OLS regression

A
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23
Q

What is SST?

A

a measure of the total sample variation in the yi that measures how spread out are the yi in the sample

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24
Q

What does dividing SST by n-1 give us?

A

the sample variance of y

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25
Q

Formula: total sum of squares

A
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26
Q

Formula: Explained sum of squares

A
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27
Q

Formula: Residual sum of squares

A
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28
Q

Decomposition of total variation

A
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29
Q

Goodness-of-fit measure (R-squared)

A
30
Q

Value of Goodness-of-fit measure (R-squared)

A

between 1 and 0

31
Q

OLS: What happens if data points all lie on the same line?

A

OLS perfect fit
R squared = 1

32
Q

What happens if R squared is close to zero?

A
  • poor fit of OLS line
  • very little of the variation in the captured yi is capture by the variation in the ^yi, which all lie on the OLS regression line
33
Q

True or False: High R squared means that regression has causal interpretation

A

False

34
Q

Why are estimated regression coefficients random variables?

A

because they are calculated from a random sample

35
Q

What are the assumptions for SLR?

A
  1. Linear in parameters
  2. Random sampling
  3. Sample variation in the explanatory variable
  4. Zero conditional mean
  5. Homoskedasticity
36
Q

Describe Assumption 1 of SLR

A
37
Q

Describe Assumption 2 of SLR

A
38
Q

Describe Assumption 3 of SLR

A
39
Q

Describe Assumption 4 of SLR

A
40
Q

What is a very weak SLR Assumption and why?

A

Assumption 3
- there is varion in xi

41
Q

What is a very strong SLR Assumption and why?

A

Assumption 4
- condition on x i, E(u i) = 0

42
Q

SLR 2 + SLR 4:

A
43
Q

Fixed in repeated samples: why are they not always very realistic?

A
  • one does not choose values of education and then searches for individuals with those values
44
Q

How can we treat xi if we assume SLR and SLR 2?

A

as nonrandom

45
Q

True or False: SLR1-SLR4, OLS estimators are unbiased

A

True

46
Q

Interpretation of unbiasedness of OLS: what is crucial?

A

Assumptions SLR1-SLR4

47
Q

Interpretation of unbiasedness of OLS: how can the estimated coefficients be?

A

smaller or larger, depending on the sample that is the result of a random draw
- on average equal to the values that characterize the true relationship between y and x in the population

48
Q

What does “on average” mean in the context of SLR unbiasedness?

A

= if sampling was repeated
ie if drawing the random sample and doing the estimation was repeated many times

49
Q

True or False: in a given sample, estimates may differ considerably from true values (SLR unbiasedness)

A

True

50
Q

Formula: SLR5

A
51
Q

What role does SLR5 play in showing the unbiasedness?

A
  • it plays no role
  • it simplifies the variance calculations
52
Q

What is sigma squared (SLR5)?

A
  • the unconditional variance of u
  • the error variance
53
Q

Formula: sigma squared (SLR5)

A
54
Q

Formula: Summarizing SLR4 and SLR5

A
55
Q

What does Homoskedasticity mean?

A

the variance of the errors is constant across all elvels of the independent value(s)

56
Q

What happens when Homoskedasticity is satisfied?

A
  • OLS estimators are unbiased and efficient
  • hypothesis tests and confidence intervals are valid
57
Q

What does Heteroskedasticity mean?

A

the variance of the errors is nto constant across all elvels of the independent variable(s)

58
Q

How are the OLS estimators in the presence of Heteroskedasticity?

A
  • still unbiased
  • no longer efficient
  • leading to inefficient standard errors
59
Q

What are the methods of testing homoskedasticity in Sata?

A
  1. Visual Inspection
  2. Breusch-Pagan test
  3. White test
60
Q

Describe Visual inspection (method of testing for homoskedasticity in Sata)

A
  • predic residuals (r, res)
  • plot your residuals against your independent variables (scatter r x)
  • in case of a multivariate regression predict the fitted values (yhat, xb) and plot the residuals against the fitted values
61
Q

Describe the Breusch-Pagan test (method of testing for homoskedasticity in Sata)

A
  • we test whether the estimated variance of the residuals are dependent on the values of the independent variables
  • run the regression and type “estat hettest” directly after the reg command
62
Q

Describe the white test (method of testing for homoskedasticity in Sata)

A
  • similar to the BP test
  • allows the independent variables to have a nonlinear effect on the error variance
  • run the regression and type “imtest, white” directly after the reg command
63
Q

Under SLR1-SLR5 we obtain a variance of OLS estimators (formula and explanations)

A
64
Q

Problem: the error variance is unknown. Why?

A
  • we do not know what error variance is because we do not observe the erros, ui
  • what we observe are the residuals
  • luckily we can use these residuals to form an estimate of the error variance
65
Q

Theorem 2.3 (Unbiasedness of error variance): Formula & explanation

A
66
Q

Compare SE for beta and mean: Formula & Explanation

A
67
Q

True of False: another OLS assumption is that the error terms or even the dependent or independent variables are normally distributed

A

False
- OLS only requires errors to be i.i.d., but normality is required neither for unbiased and efficient OLS estimates nor for the calculation of standard errors

68
Q

What is necessary for convenient hypohesis testing?

A

a normal distribution

69
Q

When the errors are normally distributed… ?

A
  • the test statistic follows a t-distribution
  • we can use familiar cut-off values
70
Q
A

Econometrics (2 decks)

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