Topics in Applied Econometrics

Question 1

How is an independently pooled cross section usually obtained?

Answer 1

Sampling randomly from a large population at different points in time, usually, but not necessarily, different years.

Question 2

When sampling from a population at different points in time, what can we likely say about the data obtained?

Answer 2

It may be independent but is unlikely to be I.I.D.

Question 3

What is Panel Data?

Answer 3

The same entities are tracked across time, be it families, cities etc.

A sample population is selected from the given entity and the selected individuals are reinterviewed at several subsequent points in time.

It has a cross sectional and time series dimension.

Question 4

What is another way to refer to panel data?

Answer 4

Longitudinal data.

Question 5

13.1 - Pooling Independent Cross Sections Across Time - Why is panel data useful?

Answer 5

As it measures the changes in given parameters in a specific area over time, there exist some unknown factors within a sample. As such, by re-questioning the same individuals, the relationship and the dependent variable and at least some of the independent variables will remain the same over time.

Question 6

13.1 - Pooling Independent Cross Sections Across Time - When the population has different distribution in different time periods, the intercept can differ across periods. How can we account for this?

Answer 6

Dummy variables can be included for all but one year, with the earliest year in the sample usually being the base year.

Question 7

13.1 - Pooling Independent Cross Sections Across Time - Heteroskedasticity?

Wealth example?

Answer 7

The circumstance in which the variability of a variable is unequal across the range of values of a second variable that predicts it. Variability can increase or decrease with time.

Annual income might be a heteroscedastic variable when predicted by age, because most teens aren’t flying around in G6 jets that they bought from their own income. More commonly, teen workers earn close to the minimum wage, so there isn’t a lot of variability during the teen years. However, as teens turn into 20-somethings, and 20-somethings into 30-somethings, some will tend to shoot-up the tax brackets, while others will increase more gradually (or perhaps not at all, unfortunately). Put simply, the gap between the “haves” and the “have-nots” is likely to widen with age.

Question 8

13.1 - Pooling Independent Cross Sections Across Time - When should Logarithms be used in graphs?

Answer 8

LOGS are just a way of writing exponential functions.

The first is to respond to skewness towards large values; i.e., cases in which one or a few points are much larger than the bulk of the data. The second is to show percent change or multiplicative factors.

https://www.forbes.com/sites/naomirobbins/2012/01/19/when-should-i-use-logarithmic-scales-in-my-charts-and-graphs/

Question 9

13.1 - Pooling Independent Cross Sections Across Time - Given the model:

log(wage) = β|0| + 𝛿|0| y85+ β|1|educ + 𝛿|1|y85.educ + β|2|female + 𝛿|2|y85.female + u

and given that the base year is 1975, what do the following equal:

β|0|
β|0| + 𝛿|0|
β|1| + 𝛿|1|
𝛿|1|

Answer 9

β|0| is the intercept for 1978.

β|0| + 𝛿|0| us the intercept in 1985

β|1| + 𝛿|1| is the return to education in 1985.

𝛿|1| measures how the return to another year of education has changed over the seven-year period.

Question 10

13.1 - Pooling Independent Cross Sections Across Time - Given the model:

log(wage) = β|0| + 𝛿|0| y85+ β|1|educ + 𝛿|1|y85.educ + β|2|female + 𝛿|2|y85.female + u

What is the differential between men and woman in the two periods?

How could we test the null hypothesis that nothing has happened to the gender differential over the seven year period?

Answer 10

β|2| is the log(wage) differential between men and woman.

The differential in 1985 is β|2|+ 𝛿|2|
H|0|: 𝛿|2| = 0
H|1|: 𝛿|2| > 0

Question 11

13.1 - Pooling Independent Cross Sections Across Time - When should one adjust nominal to real terms in a regression (use wage and log(wage) as an example)?

Answer 11

If we were using wage as the dependent variable, we would need to change the values to real. A year dummy should also be included to enable use to convert.

When using log(wage), with dummy variables for all periods (except the base period), the use of aggregate price deflators will only affect the intercepts; non of the slope estimates will change.

log(wage/P85) = log(wage) -log(P85).

where P(85) is the deflation factor for 1985 wages. You will note that while wage figures across people, P85 does not.

Question 12

13.1 - Pooling Independent Cross Sections Across Time - Simply, what is the Chow test?

Answer 12

An F test.

Question 13

13.1 - Pooling Independent Cross Sections Across Time - What is an F-Test?

Answer 13

F-Tests are used for testing multiple coefficients. e.g. H|0| = β|1|. β|2|, …, β|n| = 0.

We estimate the model using the independent variables of interest and find the estimate of the population error (u), which we call residuals.

Unrestricted Model: We then find the Sum of Squared Residuals (SSR) which will be the sum of the population error terms squared.

Restrecited Model: We find the SSR for y = α, where α is the mean, saying that the dependent variable doesn’t depend on the independent variables.

SSR of the unrestricted will always be greater than SSR for the restricted, as it will always go up as independent variables are added. We are asking whether it is significantly higher.

F stat = [SSR|restricted| - SSR|unrestricted|] / SSR|unrestricted| - We do need to adjust this based on n and the p value though, but it serves as a good base for understanding.

Question 14

13.1 - Pooling Independent Cross Sections Across Time - T-Test?

Answer 14

Uses a T-Distribution - this is basically a normal distribution but with fatter tails. If the sample SD were known, we would use the normal distribution, but when it is not known, we use the t-distribution which is calculated based on the sample population. with n-1 degrees of freedom. As n increases, the t distribution increases to the standard normal.

We test that the sample mean (μ) is equal to some hypothesised value (μ|0|)

If there is a large enough difference between the sample mean and the value you are testing against, relative to the standard deviation, we reject the null hypothesis.

The higher the t stat, the less likely the null is true

Question 15

13.1 - Pooling Independent Cross Sections Across Time - F-test vs T-test?

Answer 15

T-Tests are used for testing a restriction on a single variable of coefficient.

F-Tests are used for testing multiple coefficients. e.g. H|0| = β|1|. β|2|, …, β|n| = 0.

This would fail if any βi had was significant at a value greater than 0.

Question 16

13.1 - Pooling Independent Cross Sections Across Time - Chow Test use?

How can this be applied to Panel data?

Is it robust against Heteroskedasticity?

Answer 16

An F test that can be used to determine whether a multiple regression function differs across two groups.

This can be applied to testing in two different periods.

As with any F test based on sums of squared residuals or R-squareds, this
test is not robust to heteroskedasticity. To
obtain a heteroskedasticity-robust test, we must construct the interaction terms and do a
pooled regression.

Question 17

13.1 - Pooling Independent Cross Sections Across Time - How would we conduct a Chow Test?

Answer 17

Estimate the restricted model by doing a pooled regression allowing for different time intercepts; this gives SSRr. Then, run a regression
for each of the, say, T time periods and obtain the sum of squared residuals for each
time period. The unrestricted sum of squared residuals is obtained as SSRur = SSR1 +
SSR2 + … + SSR|T| .

If there are k explanatory variables (not including the intercept or the
time dummies) with T time periods, then we are testing (T 2 1)k restrictions, and there
are T 1 Tk parameters estimated in the unrestricted model.

Question 18

13.2 (Panel Data - Two Period) - In empirical economics, what has 𝛿|1| HAT become known as?

Answer 18

The difference-in-differences estimator.

It shows the difference in the dependent variable due to an independent variable in the two time periods.

It will be fond in the equation next to a dummy variable for the latter year and the aforementioned independent variable.

𝛿|1| = (y|2, group 1| - y|2, group 2| - (y|1, group 1| - y|1, group 2|)

See P.455 in Text book for more information.

Question 19

13.2 (Panel Data - Two Period) - What are β symbols used to represent and what are 𝛿 used to represent?

Answer 19

β are coefficients of explanatory/ independent variables.

𝛿 are coefficients used to allow multiple periods into an equation. The 𝛿 will be partnered with a dummy variable for a different year.

Question 20

13.2 (Panel Data - Two Period) - When creating a regression to test two different time periods, what tips should be followed?

Answer 20

𝛿|0| will be paired with the new period.
β|1| will be the new explanatory variable.
𝛿|1| will be paired with a dummy variable for the new year and be paired with the the new explanatory paired with β|1|.

Question 21

When would a test be deemed as statistically significant?

Answer 21

A test is deemed statistically significant if there’s a very low probability the result could have occurred by chance.

Question 22

13.3 - Two-Period Panel Data Analysis - In a scenario where we are looking at how a dependent variable, say crime rate, and an independent variable, say unemployment interact; what would be a solution if we find that the coefficient on unemployment is not statistically significant?

Answer 22

We can use panel data from two time periods. The likely reason for the variable not being statistically significant is due to omitted variables. Some factors can be controlled for by using the crime rate from the previous year to control for the fact that different cities have different levels of crime.

Question 23

13.3 - Two-Period Panel Data Analysis - See BA 1. What can each term be explained as?

What is this model called?

Answer 23

BA 1.

Unobserved / Fixed Effects Model

y: Dependent Variable
β: Intercept when t=1
𝛿 + β: Intercept when t=2
d2: a dummy variable equalling 0 if t=1 and 1 if t=2
x: the independent variable
a: the unobserved time-constant factors that effect y. AKA unobserved effect or fixed effect.
u: the unobserved time-varying factors that effect y.
i: the entity, e.g. person, city…
t: the time period
u: the idiosyncratic error or time varying error.

Question 24

13.3 - Two-Period Panel Data Analysis - What issue would exist for using OLS to estimate β|1| with two period panel data in the unobserved effects model?

y|it| = β|0| + 𝛿|0|d2|t| + β|1|x|it| + v|it|
t = 1,2

v|it| = a|i| + u|it|.

Answer 24

For pooled OLS to produce a consistent estimator of β|1| we would have to assume that the unobserved effect, a|I|, is uncorrelated with x|it|. This will be on top of the assumption that x|it| and u|it| are uncorrelated.

The former would result in heterogeneity bias, but it is really just bias caused from omitting a time-constant variable.

Question 25

13.3 - Two-Period Panel Data Analysis - Composite error?

Answer 25

v|it| = a|I| + u|it|.

This is the combination of the idiosyncratic error and the unobserved time-constant factors.

Question 26

13.3 - Two-Period Panel Data Analysis - Heterogeneity vs homogeneity?

Answer 26

Heterogeneity in statistics means that your populations, samples or results are different. It is the opposite of homogeneity, which means that the population/data/results are the same.

Question 27

13.2 (Panel Data - Two Period) - Heterogeneity Bias?

Answer 27

The bias in OLS due to omitted

heterogeneity (or omitted variables).

Question 28

13.3 - Two-Period Panel Data Analysis - How do we correct for serial correlation in the Unobserved Model?

y|it| = β|0| + 𝛿|0|d2|t| + β|1|x|it| + v|it|
t = 1,2

v|it| = a|i| + u|it|.

Answer 28

For a two period model, we expect the a|i| value to be correlated with the dependent variable (explanatory variable), x|it|. However, it is assumed to be constant.

As such, we can subtract the equation for the first interval from the equation in the second interval.

We find the FIRST-DIFFERENCE.

See BA 2 for an example of this.

Question 29

13.3 - Two-Period Panel Data Analysis - When will a first differenced equation hold?

Answer 29

When the idiosyncratic error, u|it| is uncorrelated with the explanatory variable in both periods.

I.e. Strict Exogeneity.

This rules out the case where x|it| is the lagged dependent y|i, t-1|.

Question 30

13.3 - Two-Period Panel Data Analysis - Strict Exogeneity?

Answer 30

E[U|t| | X|s| ] = 0
t=s ∀s

Cov(U|itj|, X|is|) = 0
for all i,t,j.

Intuition:
The expected error term given the independent explanatory variable is 0 for all periods. i.e. the error is independent of the explanatory variable in all periods.

Note: We in this model a|I| is correlated with one or more explanatory variables, so v|it| would also be correlated. This is why we use first difference equations, to remove the time invariant, (unobserved effect) a|I| component.

Question 31

13.3 - Two-Period Panel Data Analysis - What is the OLS estimator for β|1| using a first-differenced equation, what do we call the resulting estimator?

Answer 31

β|1| hat will be the first-differenced estimator.

Question 32

R Squared?

Answer 32

R-squared is a statistical measure of how close the data are to the fitted regression line.

It is the percentage of the response variable variation that is explained by a linear model. Or:

R-squared = Explained variation / Total variation

0% indicates that the model explains none of the variability of the response data around its mean.

100% indicates that the model explains all the variability of the response data around its mean.

Question 33

What is the effect of β
in the following cases?

Long(Y) = α +β X + …

And

Y = α +β LogX + …

Answer 33

Long(y) = α +β X + …

β will be the percentage increase in Y from a 1 unit increase in X

Y = α +β LogX + …

β will be the unit increase in Y from a 1 percent increase in X

Question 34

13.2 (Panel Data - Two Period) - What is the difference-in-differences estimator, 𝛿|1|, sometimes referred to as and why?

Answer 34

The average treatment effect, because it measure the effect of the “treatment” or policy on the average outcome of y.

Remember, it is always paid with a dummy variable , for the second period, and the dummy variable for the treatment.

See BA 3.

Question 35

13.3 - Two-Period Panel Data Analysis - a|i|?

Answer 35

The unobserved effect or fixed effect.

It is an effect that is fixed over time.

Sometimes also referred to as unobserved heterogeneity

Question 36

13.3 - Two-Period Panel Data Analysis - What is a reason to complete two-period panel data?

Answer 36

To view the unobserved factors affecting the dependent variable, a|I|.

This is called the unobserved effects model and requires:

1. Δx changes with time
2. Δu is uncorrelated with x.
3. Homoskedasticity assumption is maintained.

Question 37

13.3 - Two-Period Panel Data Analysis - What would the first difference of the following equation be:

Log (wage|it|) = β|0| + 𝛿|0|d2|t| + β|1|educ|it| + a|i| + u|it|

Answer 37

Log (wage|it|) = β|0| + 𝛿|0|d2|t| + β|1|educ|it| + a|i| + u|it|

first difference would be:

ΔLog (wage|i|) =Δ𝛿|0| + Δβ|1|educ|i| + Δu|i|

All subscript ‘t’s’ are removed as we now have a level change, not two periods. It is the change in a period.

Question 38

13.3 - Two-Period Panel Data Analysis - Would using the first difference to evaluate the following be useful?

Log (wage|it|) = β|0| + 𝛿|0|d2|t| + β|1|educ|it| + a|i| + u|it|

For and against?

Answer 38

It would seem to work well as inane ability would be an unobserved variable , a|i|, and this could be accounted for using first difference.

However, the majority of workers’ education does not change with time. Therefore, only a small number of our sample would have Δeduc different from 0, making it hard to get a precise estimate of Δβ|1| without a very large sample.

Question 39

13.4 Policy Analysis with Two-Period Panel Data -

Scrap|it| = β|0| + 𝛿|0| y88|t|+ β|1|grant|it| + a|i| + u|it|

scrap|it|: number of items per 100 that are scrapped due to defects.
grant|it|: dummy variable equal to 1 if a grant was received.

What could a|i| be in this instance?

Answer 39

Average employee ability
Managerial skill
Capital

Any other non time dependent (fixed) effects.

Question 40

13.4 Policy Analysis with Two-Period Panel Data -

Scrap|it| = β|0| + 𝛿|0| y88|t|+ β|1|grant|it| + a|i| + u|it|

scrap|it|: number of items per 100 that are scrapped due to defects.
grant|it|: dummy variable equal to 1 if a grant was received.

Why could grant|it| and a|i| are correlated, intuitively?

(3)

What would the outcome be?

Answer 40

1. Administrators may give priority to firms with lower skilled employees.
2. Administrators may give the grant to firms with more productive individuals.
3. If it is a first come first served grant system, more productive firms may apply quicker.

Simple pooling or using a single cross section may therefore lead to biased and inconsistent estimators.

Question 41

Why use a Log(y)?

Answer 41

i) Sometimes, a linear relationship between Y and the independent variables is not relevant. If we have a multiplicative relationship logs would be used to fit the data.

Using log(wage) = β|0| + β|1|educ + β|2|female + u,

this is the equivalent of y = e^ β|0| * e^β|1|educ * e^β|2|female *e^u

ii) Another reason occurs when we have a limited dependent variable. With the above example, we will not have negative wages, and thus we should change the variable for theoretical soundness. The log of a dependent variable has a maximum value of infinity and a minimum value of minus infinity, which enables us to reach all values, despite y not being negative at any point.

iii) Heteroskedasticity: A higher variance in y as x increases.
The log has a decreasing marginal returns to xi (x increases at a lower and lower rate). This therefore counters heteroskedasticity. - We Log independent and dependent variables in this case.

iV) Making the data (error) more normal. The error can cause a skewness in the normal curve. Taking the logs supresses skewness and makes it more normal. This is better to deal with.

Question 42

Heteroskedasticity?

Answer 42

A higher variance in y as x increases - cone like outfit.

Question 43

Levels regression?

Answer 43

This is the normal regression we evaluate. It looks at changes in y and associated changes in x. We call it levels as the ‘level’ of y and x changes.

Question 44

13.2 (Panel Data - Two Period) - Control Group and Treatment Group?

What are the Implications on the data and what is therefore the intuition behind 𝛿|1|?

Answer 44

When we have a natural experiment, such as a policy change, we need to make comparisons with respect to a control group who do not experience the policy change.

This enables us to separate the changes due to the policy change from those that are due to other factors. E.g. if we are looking at whether a law reduces traffic fatalities, we would compare states that adopted the law and states that didn’t. There may be a reduction in fatalities even in states that didnt adopt the law. Without taking a control group we would have overestimated the effect of the new law.

The Treatment Group is effect by the policy change.

Implications?
We end up with 4 sets of data.

𝛿1 will be the change difference in the two groups at time period 2 minus that of the two group in the first time period. This thereby shows the effect of the policy or change.

Question 45

13.4 Policy Analysis with Two-Period Panel Data - What will be the equivalent of the panel version of the difference in difference esimator used in two period pooled cross sectional data (𝛿|1|)?

Show the equation and explain the intuition when we have a change in in some policy between two periods?

Answer 45

β|1|.

With Panel Data, we use first differenced equations to get rid of any time invariant omitted variables, and as such, the change in β|1| (so the effect of a change from period 1 to period 2) will be evaluated as difference changes:

β|1| = (Δy|2,t| - Δy|2,c|) - (Δy|1,t| - Δy|1,c|)

As we stated that the change only occurred between the first and second difference equation, the second part of the equation on RHS is 0, so:

β|1| = (Δy|2,t| - Δy|2,c|)

(𝛿|1|) would be the level change, and wouldn’t include the delta sign.

Question 46

Pooled vs Panel data?

Answer 46

Pooled Data - Sampling randomly from a large population at different points in time, usually, but not necessarily, different years.

Panel Data - The same entities are tracked across time, be it families, cities etc.

Panel data is more useful when controlling for time-constant unobserved features.

Question 47

Asymptotically?

Answer 47

Evaluation of data as n approaches infinity. Think of the law of large numbers.

Question 48

Homoskedastic?

Answer 48

Variance of errors remains constant over time.