Week 3 and Week 4

Question 1

What is a short model?

Answer 1

We replace some variables with a new variable, ex. Beta:2X2 + U becomes V and replace beta for alpha

Question 2

if the short model satisfies the exogeneity assumption E[V | X1] = 0 then

Answer 2

αˆ_1 ≈ α_1 = β_1

Question 3

What 2 condition can confirm that cov(X1, V ) = 0

Answer 3

1 of these has to hold:
• the variable X2 does not affect the outcome, β2 = 0,
• the regressors X1 and X2 are uncorrelated, cov(X1, X2) = 0.

Question 4

What is omitted variable bias /OV bias and how does the formula look?

Answer 4

The term (cov(X1,X2)/var(X1))β2 is called the omitted variable bias of αˆ1.

This bias indicates by how much the estimator αˆ1 deviates systematically from its estimand β1.

Question 5

When is OV bias not zero?

Answer 5

The bias is different from zero if
• the variable X2 does affects the outcome, β2≠ 0 and
• the regressors X1 and X2 are correlated, cov(X1, X2)≠ 0.

Question 6

Can we reduce OV bias by adding more regressors?

Answer 6

No.

Question 7

When dealing with measurement error what is omitted variable bias called?

Answer 7

(cov(X1∗, W )/var(X1∗))β2 is called the attenuation bias.

Question 8

What does it mean if αˆ1 is biased toward zero.

Answer 8

Therefore, in large samples, αˆ1 is a scaled-down version of the true effect β1, it has the same sign but is smaller in absolute value. In other words, αˆ1 estimates a value that is closer to zero than the true effect. We say that αˆ1 is biased toward zero.

Question 9

if the variance of the measurement error is small relative to the variance of X1 the attenuation factor will be …

Answer 9

close to 1 and the attenuation bias will be small.

• Conversely, if the variance of the measurement error is large relative to the variance of X1 the attenuation factor will be close to 0 and the attenuation bias will be large.

-In particular, we may estimate the effect of X1 to be close to zero even if its true effect is substantially different from zero.

Question 10

What does exogeneity mean?

Answer 10

That we can’t predict U from the regressors.

Question 11

Transforming a long model to a short model might create an issue

Answer 11

Endogeneity, we can then predict U from ex. B2

Question 12

Is the correlation (covariance) never zero in the short model.

Answer 12

It is never zero.

Question 13

If we have a positive B1, Can a VERY negative OV-bias flip the sign of B^1?

Answer 13

Yes. A very negative can do that.

Question 14

classical measurement error assumptions

Answer 14

1. E[w]=0 measures correctly on average
2. W is independent of X1 and U, no systematic MISmeasurement.
3. var(w)>0 measurement error exist

Question 15

Attenuation bias formula, can we switch B2 to -B1

Answer 15

Yes.

Question 16

What does RCT stand for?

Answer 16

Randomized controlled trial

Question 17

3 examples for when exogeneity isn’t fulfilled

Answer 17

1. omitted variables
2. measurement error exists
3. equilibrium conditions

Question 18

OLS function for B^_1

Answer 18

E^[Y|X_1=1]-E^[Y|X_1=0] / E^[X|X_1=1] - E^[Y|X_1=0]

Question 19

IV regression for B^_1

Answer 19

E^[Y|group1]-E^[Y|group 2] / E^[X|group1]-E^[X|group 2]

Question 20

What does endogenous sorting do

Answer 20

reveals ceteris paribus effect horizontally.

Question 21

Instrumental variable in IV

Answer 21

Z, binary or dummy

Question 22

Instrumental exogeneity

Answer 22

E[U|Z]=0

Question 23

Biv for instrumental variable

Answer 23

E^[Y|Z=1]-E^[Y|Z=0] / E^[X|Z=1]-E^[X|Z=0]

Question 24

What does Instrumental exogeneity and instrument relevance mean and imply?

Answer 24

Instrument exogeneity: E[U|Z]=0
Instrument relevance: E^[X|Z=1]≠E^[X|Z=0]
both assures that we are only moving horizontally in graph.

Question 25

OLS characteristics

Answer 25

• for all X1
• B^1= Cov^ (Y,X) / var (X)

for x binary
B^_1 = E^[Y|X=1]-E^[Y|X=0] / E^[X|X=1]-E^[X|X=0]

• slope coefficient: B^_1 is the estimated change in y and X when Z increases by one unit

Question 26

IV characteristics

Answer 26

• for all X Z
• B^1= Cov^ (Y,Z) / cov ^ (X, Z)

for binary instrument
B^_1 = E^[Y|Z=1]-E^[Y|Z=0] / E^[X|Z=1]-E^[X|Z=0]

• slope coefficient: delta / fi is the estimated change in y and X when X´Z increases by one unit

Question 27

What differs between the first stage and second stage regression in 2SLS

Answer 27

The regression at the second stage deviates from the OLS regressions that we have considered so far in that one of the regressors is an estimated quantity.

Question 28

instrument relevance assumption

Answer 28

cov(X1,X2)≠0

Question 29

Is this true: A linear model that may be suitable for causal inference may not be a good choice for prediction and vice versa.

Answer 29

True.

Question 30

instrument exogeneity assumption

Answer 30

x1 can’t predict U

Question 31

In the context of prediction what do we refer a long linear model as?

Answer 31

Complex model

Question 32

passing from a linear model with only a few variables to a complex model with many control variables tends to …

Answer 32

inflate the variance of estimated (causal) marginal effects.

Question 33

By constructing an IV estimator of the short linear model we INCREASE OR DECREASE the variance of the estimated slope coefficients compared to direct OLS estimation of the short linear model.

Answer 33

Increase

Question 34

A high variance means that we will have large confidence intervals and therefore the power will

Answer 34

be lacking on the tests on hypothesis about the true causal effect.

Question 35

Does using a less complex model guarantee a well-behaved error term?

Answer 35

No, using a less complex model does not guarantee a well-behaved error term,
- it will decrease variance but allow the possibility of systematic bias of unknown (and in general unbounded) size.

Question 36

What is the bias-variance trade-off of prediction?

Answer 36

we want to avoid specifying a very complex model that is difficult to estimate, i.e., for which we estimate parameter values (think slope coefficients) with very large variances.

Question 37

Difficult or easy to find a model that is valid for causal inference?

Answer 37

Difficult

Question 38

In the context of prediction what is this called:
- a random sample (Yi, X1,i, . . . , Xk,i) Ni=1 from our population then we can
estimate these coefficients by the OLS estimators βˆ0, . . . , βˆk.

Answer 38

training sample –> estimating model parameters from this is called model training.

Question 39

Are we interested in the predictions of the training sample

Answer 39

No

Question 40

training sample version of the mean-squared error is called a

Answer 40

training error

Question 41

What does a low/high training error mean in R2

Answer 41

• low training error corresponds to a high R2 value (close to one),
• large training error corresponds to a low R2 value (close to zero).

Since the training error is not a good measure of predictive power, neither is the R2.

Question 42

How close our prediction is to the realized outcome Y oos(ω1) depends on three factors:

Answer 42

1. How close the estimates βˆ0(ω0),…,βˆk(ω0) are to the population coefficients b∗∗,…,b∗. This factor exists because of randomness of the training sample (uncertainty about ω0).
2. The realized values of the predictors x1,…,xk. The linear prediction rule will typically work better for some realizations than for others. What values we see depends on the randomness of the out-of-sample draw (uncertainty about ω1).
3. How the part of Y oss that is not predictable from the predictors realizes. This is determined by how the out-of-sample draw realizes (uncertainty about ω1).

Question 43

what does EPE take account to?

Answer 43

both uncertainty about the realization of the training sample and uncertainty about the realization of the out-of- sample draw.

ex ante measure of the cumulative errors

Question 44

EPE has three errors, which are these?

Answer 44

1. irreducible error (U)
2. bias (approximation error)
3. variance (estimation error)

Question 45

To choose a prediction model with a low EPE we have to optimally trade off two effects. Which ones?

Answer 45

Bias and variance, bias-variance trade-off

Question 46

why is training error not a good estimate of EPE?

Answer 46

training error estimates only bias but ignores the variance.

-making a model more complex by adding additional predictors will never increase (and in practice almost always strictly decrease) the training error. However, as we add more and more predictors the variance component of the EPE is expected to dominate eventually and, unlike the training error, the EPE will increase.

Question 47

what is sample splitting?

Answer 47

To make sure that there is both a training and a test sample the common approach is to randomly split the available data (of size m + n) into a training and test sample (of size n and m, respectively)

Question 48

What is overfitting

Answer 48

OLS usually overfits, fit too close to the true functional form.

Usually a problem if the researchers keeps adding new predictors in order to decrease the training error (equivalently increase the R2) even further.

Question 49

Ridge regression

Answer 49

Ridge regression improves on our previous approach by shrinking different slope coefficients by different factors.

Question 50

What does OLS only care about?

Answer 50

It would only care about reducing the bias component of the EPE and would tend to overfit.

Question 51

Lasso regression

Answer 51

Lasso tends to produce models that are of low complexity

Question 52

If predictors are correlated then Ridge regression will not …

Answer 52

apply the same amount of shrinkage to all coefficients. This distinguishes Ridge regression from the na ̈ıve shrinkage method discussed above.

Question 53

discrete time series

Answer 53

Often, a time series can only be observed at pre-defined discrete points in time.

Question 54

Forecast

Answer 54

Predictions about the future

Question 55

nowcasting

Answer 55

Predicting the current period yt (or recent past periods such as yt−1) from the data that is available to the econometrician in period t

Question 56

Can we do statistical inferenceon a sample of size one?

Answer 56

Observing the time series in only a single state means that we have a sample of size one. SO NO

Question 57

Stationarity

Answer 57

requires that these two segments have an identical UNCONDITIONAL distribution.

-Under stationarity, Y1 and Ys2 have the same distributions and in particular
E[Y1] =E[Ys2 ]
var(Y1) = var(Ys2 ).

Question 58

Weak dependence

Answer 58

Weak dependence restricts the information about the time series that becomes available dynamically as time passes and more and more periods of the time series are observed.

Question 59

Serial- / autocorrelation

Answer 59

Serial correlation means that observations of the time series at different points in time are correlated. One important example of serial correlation is auto-correlation. This refers to correlation between two subsequent time periods.

Question 60

Weak time dependence

Answer 60

Time Yt isn’t affected by Yt-1

Question 61

sample independently and identically

Answer 61

• We already understand that weak dependence ensures that every period reveals new information. (independent)
• in a time series we observe many k segments and under stationarity (drum roll) they are all the same (=they have same distribution). (identical)

Question 62

forecasting model left and right side

Answer 62

(outcome on left-hand side)
(predictors on right-hand side).

Question 63

cross-sectional data

Answer 63

Cross-sectional data can be represented in a spread sheet format where each row represents an observed unit and each column describes a unit characteristic.

Question 64

wide vs long model

Answer 64

contains more columns and is therefore wider than the table in Figure 1 Therefore, the representation of the data in Figure 2 is called the “wide” and Figure 1 is called the long model.

Question 65

fixed effect

Answer 65

Suppose that At does not change over time. In that case, we can write At = A. Such an A describes the total effect of unobserved unit characteristics that do not change over time.

Question 66

fixed effect transformation

Answer 66

fr =(fr1982 + fr1988)/2, (average year)
tax =(tax1982 + tax1988)/2, (average tax)
U ̄ =(U1982 + U1988)/2. (Average U)

fr = β1tax + A + U
subtract averages. from the original model ?

gives us frt = β1taxt + Ut t = 1982, 1988. where the fixed effect is removed and B1 is preserved

Question 67

clusters

Answer 67

Computing standard errors under the assumption that certain blocks of observations exhibit correlation is called “computing standard errors with clustering”. The blocks of correlated observations are called clusters. For panel data, it is often sensible to assume that all the observations of one unit form a cluster.

Question 68

Errors depend on:

Answer 68

Bias: good estimation with more Xs
Variance: good estimation with less Xs

Question 69

Training error

Answer 69

-Does not measure EPE
-Estimated idiosyncratic + bias

Question 70

Test error

Answer 70

-measures EPE

Question 71

cross section sampling

Answer 71

Random sampling
identical & independent
imposed by sampling design

Question 72

time series sampling

Answer 72

stationary and weakly time dependence
property of economic environment
difficult to verify empirically
often fail

Question 73

first difference transformation

Answer 73

delta all variables