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Year 2 Economics Term 1 > Econometrics > Flashcards

Econometrics Flashcards

1
Q

Explain what it means for an estimator to be consistent

A

An estimator of a parameter is consistent if its distribution gets more and more concentrated around the true value of the parameter as the sample size increases. This answer would get almost full points. For maximum points also give the mathematical definition of a consistent estimator

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2
Q

When is an unbiased estimator consistent?

A

An unbiased estimator is consistent when its variance goes to 0 as the sample size increases, because in that case the distribution of the estimator gets more and more concentrated around the true value of the parameter as the sample size increases.

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3
Q

In the linear regression model, what are the assumptions required for the OLS estimators of the regression coefficients to be consistent?

A

see lecture notes, Topic 5. MLR1-4 are sufficient for consistency of OLS. A complete answer would also say that MLR4 can be replaced by MLR4’ which is weaker than MLR4. Always remember to explain in an answer what these assumptions mean, remember that your answers must be understandable by someone who is not necessarily familiar with our notation.

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4
Q

Carefully explain the meaning of the two entries “F(2,30)=33.12” and
“Prob>F=0.0000” in the STATA output above.

A

F(2,30)=33.12 is the sample realization of the F test statistic for the overall significance of the regression. That is the test statistic is a a random variable with an F(2,30) distribution, and its realization in the given sample is 33.12. The null hypothesis is β1=β2=0 against the alternative that β1 and β2 are not both zero. Prob > F gives the p-value, that is the probability that a random variable with an F(2,30) distribution is larger than 33.12. In this case the p-value is 0.0000 (i.e. 0 up to 4 decimal places), meaning there is a lot of evidence against the null hypothesis.

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5
Q

What is the exact formula used for the log Lin model?

A

(exp(𝛽̂)−1)100

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6
Q

Log-lin?

A

e.g. ln(Y) = Beta0 + Beta1(X). This will be a constant semi-elasticity model. Estimate: Increasing X by one unit increases y by (100x Beta1) Percent, Ceterus paribus.

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7
Q

Log-log model?

A

ln(Y) = Beta0 + Beta1(X). Regression coefficients are interpreted as constant elasticities (model). Estimate: Increasing X by 1% increases Y by (Beta1)%, Ceterus paribus

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8
Q

Lin-Lin model?

A

Linear regression model

e.g. Y = Beta0 + Beta1(X). Increasing X by 1 unit, increases Y by Beta1 units. Ceterus paribus

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9
Q

When should the R squared comparison not be used?

A

When the two models have different dependent variables.

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10
Q

Causal effect of x on y?

A

How does variable y change if variable x is changed

but all other relevant factors are held constant“

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11
Q

The zero conditional mean independence assumption implies what? and what does this mean?

A

E(y|x) = E( Beta0 + Beta1X + u|x)
=Beta0 + Beta1X + E(u|x)
= Beta0 + Beta1X.

E(u|x)=0
This means that the conditional mean of the dependent variable can be expressed as a linear function of the explanatory variable.
Stronger than saying u and x are uncorrelated but less string than saying they are independent.

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12
Q

Difference between dependent and independent variable?

A

Dependent (y axis) is being measured, Independent (x axis) is being changes.

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13
Q

SST?

A

Total sum of squares, represents total variation in the dependent variable (y)

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14
Q

SSE?

A

Explained sum of squares, represents the variation explained by regression.

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15
Q

SSR?

A

Residual sum of squares, represents the variation not explained by regression.

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16
Q

Relationship between SSR, SST, SSE? What are they known as?

A

SST = SSR + SSE, Total variation = explained part + unexplained part. Known as the measures of variation.

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17
Q

R squared measure and its meaning?

A

R squared = SSE/SST = 1 - SSR/SST. Measures the fraction of the total variation explained by the regression, the higher R squared, the closer Yi are to the regression line, so more linearity. Says nothing about casualty though.

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18
Q

Can can R squared be shown to equal?

A

the square of corr(x,y).

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19
Q

Assumptions in the SLR model? Explain

A 
SLR.1 (Linear in parameters)
SLR.2 (Random sampling)
SLR.3 (Sample variation in explanatory variable)
SLR.4 (Zero conditional mean)
SLR.5 (Homoskedasticity)
Look at sheet for equation.
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20
Q

Theorem 2.1?

A

Unbiasedness of OLS. SLR1 - 4

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21
Q

Theorem 2.2? Explain

A

Variances of OLS estimators. SLR1-5. The sampling variability of the estimated regression coefficients is higher the larger the variability of the unobserved factors and is lower the higher the variation off the explanatory variable

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22
Q

Explain MLR.1 (Linear in parameters).

A

In the population, the relationship between y and the explanatory variables is linear

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23
Q

Explain MLR.2 (Random sampling)

A

We have a random sample of size n from the population.
The data is a random sample drawn from the population
MLR.2 implies the data are a representative sample from the population

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24
Q

Explain MLR.3 (No perfect collinearity)

A

In the sample (and therefore in the population), none
of the explanatory variables is constant and there are
no exact linear relationships among the explanatory variables

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25
Q

Explain MLR.4 (Zero conditional mean)

A

The value of the explanatory variables must contain no information about the mean of the unobserved factors

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26
Q

Explain MLR.5 (Homoskedasticity)

A

The value of the explanatory variables must contain no information about the variance of the unobserved factors

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27
Q

Root MSE?

A

The standard error of the regression

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28
Q

What is a negative of focusing on the R squared value when adding regressors?

A

Can never fall when a regressor is added, so might lead to the including f silly regressors.

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29
Q

The differences between SLR and MLR assumptions?

A

all the same aside from MLR3 (No perfect collinearity)

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30
Q

Why does MLR3 differ from SLR3?

A

In the sample (and therefore in the population), none
of the explanatory variables is constant and there are
no exact linear relationships among the explanatory variables.

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31
Q

Remarks on MLR.3

A

The assumption only rules out perfectcollinearity/correlation bet-ween explanatory variables; imperfect correlation is allowed
If an explanatory variable is a perfect linear combination of other explanatory variables it is superfluous and may be eliminated
In practice violations of MLR.3 are rare unless a mistake has been made in specifying the model. Stataand other regression packages will indicate a problem.

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32
Q

Theorem 2.3?

A

(Unbiasedness of the error variance)

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33
Q

Gauss-Markov assumption?

A

MLR1-MLR5

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34
Q

Why is MLR4 more likely to hold than SLR4?

A

More independent variables mean it is less likely t=something will end up in the error variable.

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35
Q

How does the error variance affect the variance of an independent variable?

A

A high error variance increases the sampling variance because there is more „noise“ in the equation
A large error variance necessarily makes estimates imprecise
The error variance does not decrease with sample size
Feta squared

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36
Q

How does the sample variation affect the variance of an independent variable?

A

More sample variation leads to more precise estimates
Total sample variation automatically increases with the sample size
Increasing the sample size is thus a way to get more precise estimates
SSTj

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37
Q

How do Linear relationships among the independent variables affect the variance of an independent variable?

A

The higher Rj (the better explanatory variable xj can be linearly explained by other independent variables), the higher the sample variance.
1-R squared

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38
Q

MLR.6 ?

A

Normality of error terms. It is assumed that the unobserved factors are normally distributed around the population regression function. The form and the variance of the distribution does not depend on
any of the explanatory variables.

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39
Q

Discussion of normality.

A

Error term is the results of many unobserved factors, with these factors being normal distributed by the CLT. Normality is often a questionable assumption.

40
Q

When will normality not hold? How do we resolve this?

A
  1. Indicator variables
  2. No Non negative variables
  3. Integers.
    To resolve, we could use log wage instead of wage, to allow non negatives.
41
Q

How do we resolve the idea of normality

A

Replace the assumption with a large sample size instead.

42
Q

CLM?

A

Classical Linear assumptions. MLR1-MLR6

43
Q

Theorem 4.1?

A

MLR1-6
Normal sampling distributions.
The estimators are normally distributed around the true parameters with the variance that was derived earlier.

44
Q

Theorem 4.2

A

MLR1-6
t-distribution for standardized estimators.
If the standardization is done using the estimated standard deviation(= standard error), the normal distribution is replaced by a t-distribution

45
Q

SD and standard error?

A

The standard error is the estimated standard deviation. it is unbiased is divided by n-k-1 rather than n.

46
Q

What does the t-stat measure?

A

how many estimated standard deviations the estimated coefficient is away from zero.

47
Q

When do you reject a null hypothesis?

A

When the t-stat is greater than the critical value at the given significance level.

48
Q

How would you discuss a variable that is statistically significant?

A

The fact that a coefficient is statistically significant does not necessarily mean it is economically or practically significant!
If a variable is statistically and economically important but has the „wrong“ sign (with respect to what you would expect from economic reasoning), the regression model might be misspecified

49
Q

P-value?

A

The smallest significance level at which the hypothesis is still rejected. The smaller the p-value, the more evidence there is against the hypothesis.

50
Q

What is the idea behind the F-test and what is it?

A

Tests a restricted model vs an unrestricted model to establish whether the model would change significantly to when some of the variables are dropped. F tests only take on positive values to reflect that SSR will only increase when moving from a unrestricted to a restricted model.

51
Q

Key points about F-test?

A

Fq,n-k-1

SSR will always increases in the restricted model, but is it statistically significant?

52
Q

How do you manipulate the F-test to swap out SSR from the equation?

A

R-sqaured = 1 - SSR/SST. Therefore, (1 - R-squared)SST = SSR. If the restricted and unrestricted models have the same dependent variable, SSTur=SSTr, so we can input the above formula.

53
Q

If 2 or more independent variables are not statistically significant individually, but cannot be rejected after performing an F-test, how would you explain this?

A

The variables are likely to be jointly significant. This is probably due to multicollinearity between them.

54
Q

What makes F-tests valid?

A

For all tests and confidence intervals, validity of assumptions MLR.1 –MLR.6 has been assumed. Tests may be invalid otherwise.

55
Q

Consistency?

A

MLR.1-4
An estimator is consistent for a parameter (Geta) if it converges in probability to feta.
Interpretation: Consistency means that the probability that the estimate is arbitrarily close to the true population value can be made arbitrarily high by increasing the sample size.
Consistency is a minimum requirement for sensible estimators

56
Q

Theorem 5.1?`

A

Consistency of the OLS.

57
Q

What is the special case for the simple regression model with respect to consistency?

A

One can see that the slope estimator is consistent if the explanatory variable is exogenous, i.e. un-correlated with the error term

58
Q

What can MLR4 be replaced with?

A

The weaker assumption, Cov(xj,u) = 0.

E.g. all explanatory variables must be uncorrelated with the error term.

59
Q

Theorem 5.2 (Asymptotic Normality of the OLS), and its implications?

A

In large samples the standardises estimates are normally distributed.
Implication: So T-test and F-test (which were both enabled by this result) will still be able to be used without assumption number 6, providing that n is large. BUT, the tests are only approximate

60
Q

what is the problem that arises when we include to many regressors?

A

Main problem is a higher sample variance. Estimators will still be unbiased despite it not being a true model.

61
Q

When might the MLR1-mlr4 assumption of unbiasedness not hold?

A

if there are omitted variables.

62
Q

What is omitted variable bias and when would this not occur?

A

bias when there is correlation between the variables. If we omit a variable that is correlated with the other one, we will then overestimate the significance of the first variable, be it the slope, error or intercept.

63
Q

When would omitting a variable not be a problem?

A

if the variable omitted, say beta1 = 0. So, the omitted variable was irrelevant.
If the omitted variable and the kept variable are uncorrelated.

64
Q

What is the outcome of the omitted variable bias?

A

It is better to start with more variable and omitted them by using the F or t-tests.

65
Q

Advantages of using logs and when not to use them?

A

Convenient elasticity interpretation
Models with log(y) as the dependent variable often more closely satisfy the classical linear model assumptions. For example, the model has a better chance of being linear, homoskedasticityis more likely to hold, and normalityis often much more plausible (These are observations based on experience)
Variables measured in units such as years should not be logged. We often want to know what the effect of increasing those variables by one year (not by a certain percentage)
Logs must not be used if variables take on zero or negative values
In most cases, taking the log greatly reduces the variability of a variable, making OLS estimates less prone to outlier influence.

66
Q

Why might the OLS estimator be prone to outlier influence?

A

because we square the numbers, so outliers will have more impact.

67
Q

Meaning of linear regression?

A

Model must be linear in parameters, not necessarily in its variables. so Betas must be linear

68
Q

When might a model not be linear in variables?

A

If a model that has some increasing and decreasing effect.

69
Q

When is an interaction term used?

A

if you want the partial effect of one variable to depend on the level of another variable. For instance, if the price of a house increases with number of bedrooms, but the increase also depended on the size of the bedrooms.

70
Q

Adjusted R squared?

A

imposes a penalty on adding another regressor (as normally this would always guarantee and increase in R squared).

71
Q

When would the adjusted r squared increase?

A

if, and only if, the t-statistic of a newly added regressor is greater than one in absolute value

72
Q

What is the dummy trap?

A

Including the base category as well as the dummy variable, leading to perfect collinearity.

73
Q

What is a dummy variable?

A

A dummy variable is a qualitative variable, one that instead of being intrinsically numerical, captures a given characteristic,

74
Q

What changes with dummy variables?

A

The intercept will change but the slope will stay the same

75
Q

Slope dummies?

A

By introducing an interaction term, we can alter the slope of the regression line as well. It allows for completely independent wage equations between the two different qualitative measures.

76
Q

What do assumption MLR1-4 imply?

A

unbiasedness and consistency

77
Q

What is required for the Gauss-markox theorem?

A

MLR1-5, so heteroskedasticity, namely that

78
Q

BLUE

A

Best in linear unbiased estimators. Under the Gauss Markov assumptions. Meaning it is the estimator with the lowest variance

79
Q

BUE?

A

Best unbiased estimator. MLR1-6, with 6 being normality in error terms. The estimator with the lowest variance

80
Q

SLR 3?

A

Sample variation in the explanatory variable. The values of the explanatory variables arenot all
thesame (other wise it would be impossible to study how different values of the explanatory variable
lead to different values of the dependent variable)

81
Q

What are the implications of heteroskedasticity?

A

Estimator is no longer BLUE, and t-tests and F-tests are no longer valid.

82
Q

Heteroskedasticity robust/ consistent standard erros?

A

Also called White/ Eicker standard errors. They involve the squared residuals from the regression and from a regression of xj on all other explanatory variables.

83
Q

What is available with Heteroskedasticity robust/ consistent standard erros?

A

Using these formulas, the usual t-test is valid asymptotically. T tests are available to use, but the usual F-statistic does not.

84
Q

If we can compute standard errors and test statistics that work with or without homoskedasticity, how come we bother with the usual standard errors at all?

A

The heteroskedasticity-robust test statistics and CIs only have asymptotic justification, even if the full set MLR1-MLR6 of CLM (Classical Linear Model) assumptions hold.
With smaller sample sizes, the heteroskedasticity-robust statistics need not be well behaved. In some cases, they can have more bias than the usual statistics.
•Some researchers, especially with large sample sizes, only report the heteroskedasticity-robust statistics.
•It is not a bad idea to compute, and even report, both sets of standard errors, often with the robust standard errors below the usual standard errors.

85
Q

Differences between using heteroskedasticity robust vs non robust counterparts? SE, F-stat

A

Robust SE may be larger or smaller than non robust, differences likely to be small.
F-stat often not to different, but If there is strong heteroskedasticity, differences may be larger. To be on the safe side, it is advisable to always compute robust standard errors.

86
Q

Why is it so important to tell whether the OLS is heteroskedastic?

A

To know whether the OLS is BLUE

87
Q

Breusch-Pagan test for heteroskedasticity

A

Look at slide and recite. Null hypotheses will be homoskedasticity

88
Q

Extra thing White test does over the BP test? and what else it detects?

A

The White test for heteroskedasticity includes the explanatory variables, as with the B-P test, but also squares and interactions of all explanatory variables.

89
Q

Key information about the White test?

A

The White test detects more general deviations from heteroskedasticity than the Breusch-Pagan test. Requires a large sample size though.

90
Q

Disadvantages of the first form of white test?

A

Including all squares and interactions leads to a large number of estimated parameters (e.g. k=6 leads to 27 parameters to be estimated) –loss of precision.

91
Q

Alternative White test version?

A

Similar in spirit to RESET in which we use predicted values of yin the auxiliary regression

92
Q

How do we make a better estimator than the OLS if MLR5 fails?

A

A weighted least squares estimation
Start by assuming we know the form of the heteroskedasticity.
Even if we do not correctly specify the form of heteroskedasticity, sometimes we can do better than OLS by using an incorrect variance function.
Suppose heteroskedasticity is known up to a multiplicative constant.
The outcome will be a transformed model that is homoskedastic, and providing the Gauss Markov assumptions hold as well, OLS applied to the model BLUE.

93
Q

Why is weighted least squares WLS more efficient than OLS in the original model?

A

Observations with a large variance are less informative than observa-tions with small variance and therefore should get less weight
WLS is a special case of generalized least squares (GLS).
WLS estimates have smaller standard errors

94
Q

FGLS?

A 
For an unknown heteroskedasticity function
Feasible GLS, Assumed general form of heteroskedasticity; exp-function is used to ensure positivity 
FeasibleGLS is  consistent. and asymptotically more efficient than OLS.
Multiplicative error (assumption: independent of the explanatory variables).
95
Q

What if the assumed heteroskedasticity function is wrong?

A

If the heteroskedasticity function is misspecified, WLS is still consistent under MLR.1 –MLR.4, but robust standard errors should be computed
WLS is consistent under MLR.4 but not necessarily under MLR.4‘
If OLS and WLS produce very different estimates, this typically indi-cates that some other assumptions (e.g. MLR.4) are wrong
If there is strong heteroskedasticity, it is still often better to use a wrong form of heteroskedasticity in order to increase efficiency

96
Q

Theorem 3.4, Gauss,

A

(Gauss-Markov Theorem)
Under assumptions MLR.1 -MLR.5, the OLS estimators are the best linear unbiased estimators (BLUEs) of the regression coefficients, i.e.

Year 2 Economics Term 1 (4 decks)

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