Econometrics Midterm Set 2

Front 1

What are the 3 rules for accepting an Instrumental variables?

Back 1

1. The instrument must be correlated with xi
2. The instrument must be uncorrelated with ui
3. The instrument must not have any explanatory power on Yi, own its own

Front 2

What is an endogenous variable? What about an exogenous variable?

Define the endogenous and exogenous variables in the following model:

crime_ratei= β1+ β2policei+ β3Density+ ui

policei= α1+ α2crime_ratei+ α2tax_revenue+ ui

Back 2

Determined within the system?

Exogenous:
Determined externally, taken as given in the structural model

In the model:
Endogenous: Crime rate, police
Exogenous: density, tax revenue

Front 3

Suppose we have the following model:

logPricei = β1 + β2Foodi + β3Servicei + β4Decori + β5logCapacityi + ui

Set up the following:

You expect the service quality in a restaurant to have a positive effect on the price. Using the results of your estimation, test the appropriate hypothesis at 1% level.

Given:
tstat= 7.07
df= 363

Find tcrit and make a decision.

Back 3

One-sided test

HO: β_3≤0
HA: β_3>0

t-statistic: 7.07; degrees of freedom: 363; tcrit: 2.33

|7.07| > 2.33

We reject the null hypothesis. Service quality in a restaurant has a positive effect on the price at the 1% significance level.

Front 4

Suppose we have the following model:

logPricei = β1 + β2Foodi + β3Servicei + β4Decori + β5logCapacityi + ui

Set up the following:

Test whether the quality ratings for food, service and decor are jointly significant.

Given:
SSR_U=19.93; SSR_R=58.86; m= 5; k= 2; N= 368
Find Fcrit and make a decision.

Back 4

F-test

HO: β2= β3= β4=0
HA: β2 or β3 or β4≠0

SSR_U=19.93; SSR_R=58.86; m= 5; k= 2; N= 368
[(58.86-19.93)/(5-2)]/[19.93/(368-5)]= 236.35

Fcrit: (3, 368-5)= 2.63
|236.35|> 2.63

We reject the null hypothesis. The quality ratings for food, service and decor are jointly significant at the 5% significance level.

Front 5

Suppose we have the following model:

logPricei = β1 + β2Foodi + β3Servicei + β4Decori + β5logCapacityi + ui

Set up the following:

A restaurant owner claims the total effect of service quality and decor on price is about 10%. Test this claim.

What is the restricted model? What does it simplify to?

Given:
SSR_U=19.9314; SSR_R=19.9368; m= 5; k= 4; N= 368
Find Fcrit and make a decision.

Back 5

F-test

HO: β2+ β3= .10
HA: β2+ β3≠.10

Restricted Model:
logPricei = β1 + β2Foodi + β3Servicei + (0.10 − β3)Decori + β5logCapacityi + ui

SIMPLIFIES TO:
logPricei − 0.10Decori = β1 + β2F oodi + β3(Servicei − Decori) + β5logCapacityi + ui

[(19.9368-19.9314)/(5-4)]/[19.931/(368-5)]= 0.0983

Fcrit: (1, 368-5)= 3.87

|0.0983|< 3.87

We fail reject the null hypothesis. A restaurant owner claims the total effect of service quality and decor on price is about 10% and he is right.

Front 6

What should base your answer on when you have a question like this:

Jack and Jane both live in Chicago. Jack thinks that restaurants in central Chicago are ridiculously expensive just because they are located in downtown. Jane argues that the reason for higher prices in downtown restaurants is not just their location; she says that most restaurants in downtown have better service, higher food quality and nicer ambiance compared to restaurants in other parts of the town. Which one of the two Chicagoans do you agree? Why?

Back 6

We need to include the dummy variables and other quantitative variables. This implies that the coefficients we get on dummies will represent the effect of the qualitative variables on the dependent variable holding other quantitative variables constant.

Look at the variable has a statistically significant coefficient even after accounting for other variables that may affect the dependent variable (the price in this case)

Front 7

Suppose we have the following model:

Pricei = β1 + β2Foodi + β3Servicei + β4Decori + β5*North + ui

Set up the following:
A restaurant owner on the north side of Chicago argues that having a higher rating for decor is not as much valued by customers in North as it is in other parts of the town.

Do your findings agree with what the restaurant owner says?

Given:
P-Value= 0.000
5% significant level

Back 7

HO: β_5≤0
HA: β_5>0

P-Value= 0.000

Test @ 5% significant level:
0.000<0.05
We reject the null that having a higher rating for decor is as much valued by customers in North as it is in other parts of the town. The restaurant owner is right.

Front 8

Suppose we have the following model:

Pricei =β1 +β2Decori +ui

Do you think β2 would represent the effect on price of only the decor rating? Or other things too? (i.e. Would you be worried about omitted variables bias? Why? Or why not?)

Back 8

Any factor that is correlated with both decor rating and price that is not accounted for in the model will result in an omitted variable bias. So, in this model, omitted variable bias would be a concern because restaurants with nice decor tend to also have better service and higher quality food. If we do not include food and service in our regression, the coefficient estimate for decor will also reflect the effect of these factors.

Front 9

Suppose we have the following model:

Pricei = β1 + β2Decori + β3Foodi + β4Servicei + ui

When you estimate this model, you will find that the coefficient on food rating is not statistically significant. This suggests that the food quality has no effect on price! Explain why this might be a result of multicollinearity between variables.

What are the consequences of multicollinearity?

Back 9

It wouldn’t be surprising if there is a high correlation between food quality and service quality (or decor) in a restaurant. If there is a high correlation between food rating and any one of the other two variables, then it would result in high standard errors and low t test statistics. It might be that we fail to reject that the food coefficient is not different from zero, although in reality it is different from zero, because of multicollinearity.

Front 10

Suppose we have the following model:

Pricei = β1 + β2Decori + β3Foodi + β4Servicei + ui

Do at least one diagnostic test (e.g. looking at pairwise correlation between variables, comparing individual t test and joint significance test results etc.) for multicollinearity. Do you see a concern for multicollinearity? If yes, what would you do about it? (do nothing? remove variables from regression? etc.)

Given:
P-Value= 0.000
5% significant level

Back 10

0.000>0.05 Fail to reject null that food and services are jointly significant.

Since food and service are jointly significant, that is a sign of multicollinearity.

It is wise to do nothing about multicollinearity in this case. Because neither food nor service is the variable of interest here; we are not concerned about the efficiency of their coefficients. The coefficient that we are primarily interested in estimating is the coefficient of decor, and if we remove food and service from the model, the coefficient of decor is likely to suffer from omitted variable bias.

Front 11

Solve this:

The table gives data on government recurrent expenditure, G, investment, I, gross domestic product, Y, and population, P, for 30 countries in 1997.

G, I ̧and Y are measured in U.S.$ billion and P in million. A researcher investigating whether government expenditure tends to crowd out investment fits the regression (standard errors in parentheses):

Iˆ = 18.10 –1.07G + 0.36Y R2 = 0.99
(7.79) (0.14) (0.02)

She sorts the observations by increasing size of Y and runs the regression again for the 11 countries with smallest Y and the 11 countries with largest Y. RSS for these regressions is 321 and 28101, respectively. Perform a Goldfeld–Quandt test for heteroscedasticity.

Back 11

Goldfeld-Quant Test

HO: SSR_HIGH= SSR_LOW (No Heteroscedasticity)
HA: SSR_HIGH≠ SSR_LOW (Heteroscedasticity)

Test Statistic= 28,101/321 = 87.542; n’= 11; k=3; Fcrit: F(11-3), (11-3)= 3.44

|87.542| > 3.44

We reject the null hypothesis. That means there is heteroscedasticity.

Front 12

GIVEN THIS:
The table gives data on government recurrent expenditure, G, investment, I, gross domestic product, Y, and population, P, for 30 countries in 1997.

G, I ̧and Y are measured in U.S.$ billion and P in million. A researcher investigating whether government expenditure tends to crowd out investment fits the regression (standard errors in parentheses):

Iˆ = 18.10 –1.07G + 0.36Y R2 = 0.99
(7.79) (0.14) (0.02)

She sorts the observations by increasing size of Y and runs the regression again for the 11 countries with smallest Y and the 11 countries with largest Y. RSS for these regressions is 321 and 28101, respectively.

SOLVE THIS:

The researcher saves the residuals from the full-sample regression in Exercise 7.1 and regresses their squares on G, Y, their squares, and their product. R2 is 0.9878. Perform a White test for heteroscedasticity.

Back 12

White Test
HO: No Heteroscedasticity
HA: Heteroscedasticity

N=30; R2= .9878

Test Statistic= 30*(.9878)= 29.63; P= 6; X_CRIT^2: X_((6-1))^2= 11.0705

|29.63| > 11.0705

We reject the null hypothesis. That means there is heteroscedasticity.

Front 13

Suppose we have the following model:

Passengersi = β1 + β2Populationi + ui

Set up the following:
Perform Goldfeld-Quandt test to see if there is heteroscedasticity of this specific form.

Given:
Test Statistic= 2,451.085/145.744 = 16.817; n’= 36; k=2;
(You have to find the Fcrit)

Back 13

HO: SSR_HIGH= SSR_LOW (No Heteroscedasticity)
HA: SSR_HIGH≠ SSR_LOW (Heteroscedasticity)

Test Statistic= 2,451.085/145.744 = 16.817; n’= 36; k=2; Fcrit: F(36-2), (36-2)= 1.77

|16.817| > 1.77

We reject the null hypothesis. That means there is heteroscedasticity.

Front 14

Suppose we have the following model:

Passengersi = β1 + β2Populationi + β3Incomei + ui

Set up the following:
Test this model for heteroscedasticity using White’s test

Given:
Test Statistic=17.74; P= P= 6;
(You have to find the Chi_crit)

Back 14

HO: No Heteroscedasticity
HA: Heteroscedasticity

Test Statistic=17.74; P= P= 6; X_CRIT^2: X_((6-1))^2= 11.0705

|17.74| > 11.0705

We reject the null hypothesis. That means there is heteroscedasticity.