Dr Hena Oak Not for circulation outside class or private distribution.

31st March 2022 Only meant for students of BA(Hons) Economics Miranda House.

Questions

1. A researcher computes Jarque-Bera statistic for a large sample as 7.378. Does it provide
evidence in favor of normality of the error term at 5% LoS?

2. Given ∑ 𝑌 = 2220, ∑ 𝑋 = 3400, ∑ 𝑋𝑌 = 411000, ∑ 𝑋 2 = 644000, ∑ 𝑌 2 = 264200,

n=10. Obtain the intercept and slope coefficient in a two variable model

3. Let Y be output, 𝑋2 be unskilled labour and 𝑋3 be skilled labour in the following relationship
𝑌𝑖 = 𝐵1 + 𝐵2 𝑋2𝑖 + 𝐵3 𝑋3𝑖 + 𝐵4 𝑋4𝑖 + 𝑢𝑖 where 𝑋4𝑖 = 𝑋2𝑖 + 𝑋3𝑖 . Can the parameters of this
model be uniquely estimated as per OLS? Explain.

4. Consider 𝑌𝑖 = 𝐵1 + 𝐵2 𝑋2𝑖 + 𝐵3 𝑋3𝑖 +𝑢𝑖 . The following auxiliary regression is run to check for
multicollinearity 𝑋̂2𝑖 = 2.456 + 0.7952𝑋3𝑖
𝑠𝑒 (0.56) (1.598) 𝑅22 = 0.90
(a) Compute VIF. Do you find evidence of multicollinearity?
(b) Would multicollinearity necessarily result in high standard errors of the OLS estimators
of the above model?

̂ 𝑖 = 21.045 + 0.0545𝐺𝐷𝑃𝑖 + 1.864𝐺𝑜𝑣𝑖𝑛𝑑𝑒𝑥𝑖

5. Consider 𝐹𝐷𝐼
𝑡 (1.232) (0.744) (1.005) 𝑅 2 = 0.9667

where FDI and GDP are in billions of dollars and Govindex is governance index, with a higher
value indicating better governance. Is there evidence of multicollinearity? Why?

6. A researcher finds evidence of heteroscedasticity in the regression model 𝑌𝑖 = 𝐴 + 𝐵𝑋𝑖 + 𝑢𝑖 .

How will you modify the original regression to solve for heteroscedasticity, given the
following assumptions about the error variance?
1/3
(a) 𝐸(𝑢𝑖2 ) = 𝜎 2 𝑋𝑖2 (b) 𝐸(𝑢𝑖2 ) = 𝜎 2 𝑋𝑖9 𝐸(𝑢𝑖2 ) = 𝜎 2 𝑋𝑖

7. Using data for 120 individuals, the following model of wage determination was estimated:
𝑊𝑎𝑔𝑒𝑖 = 𝐵1 + 𝐵2 𝐼𝑄2𝑖 + 𝐵3 𝑃𝐺𝑅𝐴𝐷3𝑖 + 𝑢𝑖 where Wage is hourly wages in rupees, IQ is the
intelligent quotient measured on a scale of 70-130, PGRAD is dummy variable equal to 1 if
the individual is a postgraduate and 0 if the individual is a undergraduate. The estimated
model is 𝑊𝑎𝑔𝑒𝑖 = 224.8438 + 5.0766𝐼𝑄2𝑖 + 498.0493𝑃𝐺𝑅𝐴𝐷3𝑖
𝑠𝑒 (66.6424) (0.6624) (20.0768) 𝑅 2 = 0,4540
(a) Write the equations for postgraduates and undergraduates.
(b) Test the statistical significance of the dummy variable at 5% LoS. What conclusion can
you draw from it?

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Dr Hena Oak Not for circulation outside class or private distribution.
31st March 2022 Only meant for students of BA(Hons) Economics Miranda House.

(c) If PGRAD was defined to take values (0, 2) instead of (0, 1), will the estimated value of
𝐵3 and its standard error change? What about its statistical significance?

8. Suppose 𝑌𝑖 = 𝐵1 + 𝐵2 𝑋𝑖 + 𝑢𝑖 has heteroscedasticity. How would the model be re-estimated if

𝜎2
it is assumed that the error variance is proportional to the reciprocal of 𝑋𝑖 , 𝐸(𝑢𝑖2 ) = 𝑋𝑖
? Show

that the transformed model is homoscedastic. Can you compare the 𝑅 2 of the original and
transformed model?

9. In a regression of average wages (W) on the no. of employees (N) for a random sample of 30
firms, the following regressions were obtained:
̂ = 7.5 + 0.009 𝑁
Regression 1: 𝑊
𝑡 (16.10) ( ) 𝑅 2 = 0.9
̂
𝑊 1
Regression 2: 𝑁
= 0.008 + 7.8 𝑁

𝑡 (14.43) (76.58) 𝑅 2 = 0.99

(a) Interpret the two regressions.
(b) What might be the reason for transforming Regression 1 to Regression 2? What
assumption has been made about the error variance in going from Regression 1 to
Regression 2?
(c) Can you relate the slope and intercept coefficients of the two models?
(d) Can you compare the 𝑅 2 of the two models? Why?

10. Suppose Earnings = f(Skill of worker, Work experience).

(a) Define dummy variables to capture whether workers are skilled or not. Take workers
being unskilled as reference category.
(b) Develop a model that is linear in parameters that shows Earnings = f(Skill of worker,
Work experience). Interpret the model.
(c) Assume that there is an interaction between skill of the worker and their work experience.
How would the model in (b) change? Interpret the new model.

11. The following results are obtained from a cross-section of 30 households. V is the
consumption expenditure in 1000s of Rs and X is income in 1000s of Rs. To check for
heteroscedasticity, the observations are arranged in increasing order of magnitude of X.
Regression is run separately for the 1st 11 observations (Group-1) and the last 11 observations
(Group-2). Following results are reported:

Group-1: 𝑌̂𝑖 = 1.0533 + 0.876𝑋𝑖

𝑠𝑒 (0.616) (0.038) 𝑅 2 = 0.9851 𝑅𝑆𝑆1 = 0.475 ∗ 105
Group-2: 𝑌̂𝑖 = 3.279 + 0.835𝑋𝑖

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Dr Hena Oak Not for circulation outside class or private distribution.
31st March 2022 Only meant for students of BA(Hons) Economics Miranda House.

𝑠𝑒 (3.443) (0.096) 𝑅 2 = 0.9585 𝑅𝑆𝑆2 = 3.154 ∗ 105

(a) Do the Goldfeld-Quandt test and state the hypothesis clearly.
(b) List the assumptions made related to the disturbance term in the above test.

12. Consider the following models where Y=f(time) for the period 1949-1964:

𝑀𝑜𝑑𝑒𝑙 𝐴: 𝑌̂𝑖 = 0.453 − 0.0041𝑡 𝑅 2 = 0.5284 𝑑 = 0.8252

𝑡 − 𝑣𝑎𝑙𝑢𝑒 (−3.961)
𝑀𝑜𝑑𝑒𝑙 𝐵: 𝑌̂𝑖 = 0.479 − 0.0127𝑡 + 0.0005𝑡 2 𝑅 2 = 0.6629 𝑑 = 1.82
𝑡 − 𝑣𝑎𝑙𝑢𝑒 (−3.272) (2.778)
(i) Is there serial correlation in Model A and Model B?
(ii) What accounts for serial correlation?

13. Suppose 𝑃𝑡 = 𝑓(𝑊𝑡 , 𝑋𝑡 , 𝑀𝑡 , 𝑀𝑡−1 , 𝑃𝑡−1 ) such that d=2.54 and n=18.

P=Price of final output, W=Wages, X=GDP, M=Import price. “Since for 18 observations and
5 explanatory variables, the 5% lower and upper d values are 0.71 and 2.06, the estimated d
value of 2.54 indicates that there is no positive autocorrelation” Comment.
If the d statistic is extremely small, then we may run the regression in first difference form.
Comment.

14. A researcher estimated the following demand function for money for 101 quarters using data
for the period Q1:1986-87 to Q2: 2011-12. The regression results in logs are

̂ 𝑖 = 2.603 − 0.402 𝑙𝑛𝑅𝑡 + 0.59 𝑙𝑛𝑌𝑡 + 0.524 𝑙𝑛𝑀𝑡−1 𝑅 2 = 0.9165 𝑑 = 0.65

𝑙𝑛𝑀
𝑠𝑒 (1.24) (0.36) (0.34) (0.02)
where 𝑀𝑡 is real cash balance, 𝑅𝑡 is the interest rate, 𝑌𝑡 is real national income.
(i) use Durbin’s h statistic to check for autocorrelation.
(ii) Can we use Durbin-Watson d statistic to check for autocorrelation? Why?

15. The following regression is given for the Phillips curve for the period 1958-1969:

1
𝑌̂𝑖 = −0.2594 + 20.588 ( ) 𝑅 2 = 0.6594 𝑑 = 0.6394
𝑋𝑖
𝑡 (−0.257) (4.3996)
(i) Interpret the regression. Is there evidence of autocorrelation?
(ii) If there is autocorrelation, estimate the coefficient of autocorrelation.
1 1
(iii) If the transformation for X variable is given by (𝑋 ) = (𝑋 ) (1 − 𝑒) and the error term
𝑡 𝑡−1

follows AR(1) scheme, then how would we use GLS to correct for autocorrelation?

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Dr Hena Oak Not for circulation outside class or private distribution.
31st March 2022 Only meant for students of BA(Hons) Economics Miranda House.

References:

• 10 year question papers of Delhi University.

• Basic Econometrics, Damodar Gujarati, 4th edition, The McGraw−Hill Companies, 2004