Econometrics By Example 1st Edition Damodar Gujarati - Solutions

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11. Express the following in logarithmic form(a) 53 = 125(b) 112 = 121(c) 64 = 1296
10. Factorise the following expressions(a) x2 − 7x − 8(b) 5x − 2x2(c) 2x2 − x − 3(d) 6 + 5x − 4x2(e) 54 − 15x − 25x2
9. Differentiate the following functions partially with respect to x and(separately) partially with respect to y(a) z = 10x3 + 6y2 − 7y(b) z = 10xy2 − 6(c) z = 6x(d) z = 4
8. Differentiate the following functions twice with respect to x(a) y = 6x(b) y = 3x2 + 2(c) y = 4x3 + 10(d) y = 1x(e) y = x(f ) y = 7(g) y = 6x−3 + 6 x3(h) y = 3 lnx(i) y = ln(3x2)(j) y = 3x4 − 6x2 − x − 4 x3
7. Write the equations for each of the following lines(a) Gradient = 3, intercept = −1(b) Gradient = −2, intercept = 4(c) Gradient = 12, crosses y-axis at 3(d) Gradient = 12, crosses x-axis at 3(e) Intercept 2 and passing through (3,1)(f ) Gradient 4 and passing through (−2,−2)(g) Passes
6. Write out all of the terms in the following and evaluate them(a)3j=1 j(b)5j=2( j2 + j + 3)(c)ni=1 x with n = 4 and x = 3(d)3j=1 x with x = 2(e)6i=3 i
5. Solve the following equations(a) 3x − 6 = 6x − 12(b) 2x − 304x + 8 = x + 9 − 3x + 4(c) x+3 2= 2x−6 3
4. Write each of the following as a prime number raised to a power(a) 9(b) 625(c) 125−1
3. Solve the following(a) 1251/3(b) 641/3(c) 161/4(d) 93/2(e) 92/3(f ) 811/2 + 641/2 + 641/3
2. Simplify the following as much as possible(a) 4x5 × 6x3(b) 3x2 × 4y2 × 8x4 ×−2y4(c) (4p2q3)3(d) 6x5 ÷ 3x2(e) 7y2 ÷ 2y5(f )3(xy)3 × 6(xz)4 2(xy)2x3 (g) (xy)3 ÷ x3y3 (h) (xy)3 − x3y3
1. (a) If f (x) = 3x2 − 4x + 2, find f (0), f (2), f (−1)(b) If f (x) = 4x2 + 2x − 3, find f (0), f (3), f (a), f (3 + a)(c) Considering your answers to the previous question part, in general does f (a) + f (b) = f (a + b)? Explain.
4. The following table gives annual, end of year prices of a bond and of the consumer prices index Year Bond value CPI value 2006 36.9 108.0 2007 39.8 110.3 2008 42.4 113.6 2009 38.1 116.1 2010 36.4 118.4 2011 39.2 120.9 2012 44.6 123.2 2013 45.1 125.4(a) Calculate the simple returns(b) Calculate
3. What are the key features of asset return time series?
2. Present and explain a problem that can be approached using a time series regression, another one using cross-sectional regression, and another using panel data.
1. Explain the difference between the following terms:(a) Continuous and discrete data(b) Ordinal and nominal data(c) Time series and panel data(d) Noisy and clean data(e) Simple and continuously compounded returns(f) Nominal and real series(g) Bayesian and classical statistics
19.9 Consider the following model:Y =B+BX + (1) where Y = monthly changes in the AAA bond rate, X = monthly change in the three month Treasury bill rate (TB3), and u stochastic error term. Obtain monthly data on these variables from any reliable source (e.g. the Federal Reserve Bank of St. Louis)
19.8 Consider the model given in Equations (19.35) and (19.36). Obtain data on the crime rate, law enforcement spending and Gini coefficient for any country of your choice, or for a group of countries, or for a group of states within a country, and esti- mate the two equations by OLS. How would you
19.7 In his article, "Instrumental-Variable Estimation of Count Data Models: Appli- cations to Models of Cigarette Smoking Behavior", Review of Economics and Statistics (1997, pp. 586-93), John Mullahy wanted to find out if a mother's smoking during pregnancy adversely affected her baby's birth
19.6 Continue with the wage function discussed in the text. The raw data contains in- formation on several variables besides those included in Eq. (19.39). For example, there is information on marital status (single, married, and divorced), ASVAB scores on arithmetic reasoning and word knowledge,
19.5 Return to the wage regression discussed in the text. Empirical evidence shows that the wage-work experience (wexp) profile is concave - wages increase with work experience, but at a diminishing rate. To see if this is the case, one can add the wexp variable to the wage function (19.39). If
19.4 Verify Eq. (19.29).
19.3 Verify Eq. (19.12).
19.2 Verify Eq. (19.11).
19.1 Prove that Ex;X/Ex? =1, where x = X; -X.
18.4 The book by Klein and Moeschberger gives several data sets from the fields of bi- ology and health. 12 These data can be accessed from the website of the book. Pick one or more data sets from this book and estimate the hazard function using one or more probability distributions discussed in
18.3 The Kleinbaum text cited in this chapter gives several data sets on survival analy- sis in Appendix B. Obtain one or more of these data sets and estimate appropriate SA model(s) so that you are comfortable in dealing with duration models.
18.2 Which of the regressors given in Section 18.1 are time-variant and which are time-invariant? Suppose you treat all the regressors as time-invariant. Estimate the ex- ponential, Weibull and PH survival models and comment on your results.
18.1 Using Durat as the dependent variable, estimate an OLS regression in relation to the regressors given in Table 18.1 and interpret your results. How do these results compare with those obtained from the exponential, Weibull and PH models?
17.3 From the website of the Frees book cited earlier, obtain panel data of your liking and estimate the model using the various panel estimation techniques discussed in this chapter.
17.2 Expand the LSDV model by including the time dummies and comment on the results.
17.1 Table 17.8 gives the LSDV estimates of the charity example. If you examine the raw data given in Table 17.1, can you spot some pattern regarding individuals that have significant intercepts? For example, are married taxpayers likely to contribute more than single taxpayers?
16.7 Refer to the 3-month and 6-month Treasury Bills example discussed in the text. Suppose you also want to include the Federal Funds Rate (FFR) in the model. Obtain the data on FFR for comparable time period and estimate a VAR model for the three variables. You can obtain the data from the
16.6 Collect data on housing starts and develop a suitable ARIMA model for forecast- ing housing starts. Explain the procedure step by step.
#!# 16.5 Develop a suitable ARIMA model to forecast the labor force participation rate for females and males separately. What considerations would you take into account in developing such a model? Show the necessary calculations and explain the various di- agnostic tests you use in your analysis.
16.4 Suppose you want to forecast employment at the national level. Collect quarterly employment data and develop a suitable forecasting model using ARIMA methodol- ogy. To take into account seasonal variation, employment data are often presented in seasonally adjusted form. In developing your
16.3 Replicate your model used in the preceding exercise using more recent data and comment on the results
16.2 Refer to the IBM stock price ARIMA model discussed in the text. Using the data provided, try to come up with an alternative model and compare your results with those given in the text. Which model do you prefer, and why?
16.1 Estimate regression (16.1) using the logs of the variables and compare the results with those obtained in Table 16.2. How would you decide which is a better model?
15.1 Collect data on a stock index of your choice over a period of time and find out the nature of volatility in the index. You may use ARCH, GARCH, or any other member of the ARCH family to analyze the volatility.
14.3 Suppose you have data on real GDP for Mexico and the USA. A priori, would you expect the two time series to be cointegrated? Why? What does trade theory have to say about the relationship between the two? Obtain quarterly data on the two time series and analyze them from the perspective of
14.2 Repeat the steps in Exercise 14.1 to analyze the Treasury Bill rates, but make sure that you use the quadratic trend model. Compare your results with those discussed in the chapter.
14.1 Consider the relationship between PCE and PDI discussed in the text. (a) Regress PCE on an intercept and trend and obtain the residuals from this re- gression. Call it S. (b) Regress PDI on an intercept and trend and obtain residuals from this regres- sion. Call it S2. (c) Now regress S; on
13.7 If a time series is stationary, does it mean that it is a white noise series? In the chapter on autocorrelation, we considered the Markov first-order autoregressive scheme, such as: where u is the error term in the regression model, p is the coefficient of autocorrelation, ande, is a white
13.6 Repeat 13.5 for the Consumer Price Index (CPI) for the USA.
13.5 Would you expect quarterly US real GDP series to be stationary? Why or why not? Obtain data on the quarterly US GDP from the website of the Federal Reserve Bank of St Louis to support your claim.
13.4 Suppose in Eq. (13.7) B3 = 0. What is the interpretation of the resulting model?
13.3 For the IBM stock price series estimate model (13.7) and comment of the results.
13.2 Verify Eqs. (13.17) and (13.18).
13.1 Verify Eqs. (13.13) and (13.14).
12.3 Use the data in Table 12.1. What is the mean number of patents received by a firm operating in the computer industry in the USA with an LR value of 4.21? (Hint: Use the data in Table 12.4). For your information, a firm with these characteristics in our sample had obtained 40 patents in 1990.
12.2 Table 12.8 (see the companion website) gives data on the extramarital affairs of 601 people and is obtained from Professor Ray Fair's website: http://fairmodel.econ.yale.edu/rayfair/pdf/1978ADAT.ZIP. The data consists of: y = number of affairs in the past year zl = sex z2 = age z3 = number of
12.1 Table 12.1 also gives data on patents and other variables for the year 1991. Replicate the analysis discussed in this chapter using the data for 1991
11.3 The data given in Table 11.1 includes many more variables than are used in the illustrative example in this chapter. See if adding one or more variables to the model in Tables 11.4 and 11.6 substantially alters the results given in these tables.
11.2 Expand the models discussed in this chapter by considering interaction effects, for example, education and family income.
11.1 Include the Faminc-squared variable in both the censored and truncated regression models discussed in the chapter and compare and com
1.1 Several data sets are available on the websi tes of the books listed in the footnotes in this chapter. Access the data of your interest and estimate the various models discussed in this chapter so that you can be comfortable with the techniques discussed in the preceding pages.
8.4 Divide the smoker data into 20 groups. For each group compute Pb the probability of smoking. For each group compute the average values of the regressors and estimate the grouped logit model using these average values. Compare your results with the ML estimates of smoker log it discussed in
8.3 For the smoker data discussed in the chapter, estimate the count R2.
8.2 Table 8.8 (available on the companion website) gives data on 78 homebuyers on their choice between adjustable and fixed rate mortgages and related data bearing on the choice. 17 The variables are defined as follows: Adjust = 1 if an adjustable mortgage is chosen, 0 otherwise. Fixed rate fixed
8.1 To study the effectiveness of price discount on a six-pack of soft drink, a sample of 5,500 consumers was randomly assigned to 11 discount categories as shown in Table 8.7.15 (a) Treating the redemption rate as the dependent variable and price discount as the regressor, see whether the logit
7.9 Consider the following simultaneous equation model: Yu ==Al +A2Y2t +A3X U +uu Y2t ==BI +B2YU +B3X 2t +u2t (1) (2) In this model the Ys are the endogenous variables and the Xs are the exogenous variables and the us are stochastic error terms. (a) Obtain the reduced form regressions. (b) Which
7.8 The error term in the log of wages regression in Table 7.7 was found to be non-normally distributed. However, the distribution oflog of wages was normally distributed. Are these findings in conflict? If so, what may the reason for the difference in these findings?
7.7 In the cancer data we have 43 observations for each type of cancer, giving a total of 172 observations for all the cancer types. Suppose you now estimate the following regression model: Ci ==Bl +B2Cigj +B3Lungi +B4KidneYi +B5Leukemiai +Uj where C = number of deaths from cancer, Cig number of
7.6 Continue with Exercise 7.5. Are there any outliers in the cancer data? If there are, identify them.
7.5 Use the data given in Table 7.8 to find out the impact of cigarette smoking on bladder, kidney, and leukemia cancers. Specify the functional form you use and present your results. How would you find out if the impact of smoking depends on the type of cancer? \Vhat may the reason for the
7.4 What other variables do you think should be included in the wage determination model? How would that change the models discussed in the text?
7.3 In the chapter on heteroscedasticity we discussed robust standard errors or White's heteroscedasticity-corrected standard errors. For the wage determination models, present the robust standard errors and compare them with the usual OLS standard errors.
7.2 In the various wage determination models discussed in this chapter, how would you find out if the error variance is heteroscedastic? If your finding is in the affirmative, how would you resolve the problem?
7.1 For the wage determination model discussed in the text, how would you find out if there are any outliers in the wage data? If you do find them, how would you decide if the outliers are influential points? And how would you handle them? Show the necessary details.
6.4 Re-run the regression in Table 6.7 using In INC( -1) as a regressor in place ofLC( -1), and compare the results with those in Table 6.7. What difference, if any, do you see?\X'hat may be logic behind this substitution? Explain.
6.3 Repeat Exercise 6.2 for the model given in Eq. (6.15) and comment on the results.
6.2 Reestimate regression (6.1) by adding time, t, as an additional regressor, t taking values of 1, 2, ... , 54. t is known as the trend variable. (a) Compare the results of this regression with those given in Table 6.2. Is there a difference between the two sets of results? (b) If the coefficient
6.1 Instead of estimating model (6.1), suppose you estimate the following linear model: (6.16) (a) Compare the results of this linear model with those shown in Table 6.2. (b) What is the interpretation of the various coefficients in this model? What is the relationship between the A coefficients in
5.5 Refer to Table 5.5. Assume that the error variance is related to the square of income instead of to the square of ABORTIONF. Transform the original abortion rate function replacing ABORTIONF by income and compare your results with those given in Table 5.5. A p1'iori, would you expect a
5.4 Refer to any textbook on econometrics and learn about the Park, Glejser, Spearman's rank correlation, and Goldfeld-Quandt tests ofheteroscedasticity. Apply these tests to the abortion rate, wage rate, and hours of work regressions discussed in the chapter. Find out if there is any conflict
5.3 Do you agree with the following statement: "Heteroscedasticity has never been a reason to throw out an otherwise good model"?21
5.2 Refer to the hours worked regression model given in Table 4.2. Use log of hours worked as the regressand and find out if the resulting model suffers from heteroscedasticity. Show the diagnostic tests you use. How would you resolve the problem of heteroscedasticity, if it is present in the
5.1 Consider the wage model given in Table 1.2. Replicate the results of this table, using log of wage rates as the regressand. Apply the various diagnostic tests discussed in the chapter to find out if the log wage function suffers from heteroscedasticity. If so, what remedial measures would you
exercise show? And how would you describe the residuals e; and e2?
4.8 Refer to Exercise 4.6. First regress Yon X3 and obtain the residuals from this re- gression, saye. Then regress X2 on X3 and obtain the residuals from this regression, say e2. Now regress e; on e2. This regression will give the partial regression coeffi- cient given in Eq. (4.2). What does this
4.7 The following data describes the manpower needs for operating a US Navy bachelor officers' quarters, consisting of 25 establishments. The variables are described below and the data is given in Table 4.9,12 which can be found on the companion website: Y: Monthly manhours needed to operate an
4.6 Consider the sets of data given in the following two tables: Table 1 Table 2 Y X2 X3 Y X2 X3 1 2 4 1 2 4 2 0 2 2 0 2 3 4 12 3 4 0 4 6 0 4 6 12 5 8 16 5 8 16 The only difference between the two tables is that the third and fourth values of X3 are interchanged. (a) Regress Yon X2 and X3 in both
4.5 Run the 15 auxiliary regressions mentioned in the chapter and determine which explanatory variables are highly correlated with the rest of the explanatory variables.
4.4 In the three-variable model, Yand regressors X2 and X3, we can compute three partial correlation coefficients. For example, the partial correlation between Yand X2, holding X3 constant, denoted as r12.31 is as follows: where the subscripts 1, 2, and 3 denote the variables Y, X 2, and X3,
4.3 Continuing with Exercise 4.1, find out the partial correlation coefficients for the variables included in Table 4.2, using Stata or any other software you have. Based on the partial correlations, which variables seem to be highly correlated?
4.2 Do you agree with the following statement and why? Simple correlations between variables are a sufficient but not a necessary condition for the existence of multicollinearity.
4.1 For the hours example discussed in the chapter, try to obtain the correlation matrix for the variables included in Table 4.3. Eviews, Stata, and several other programs can compute the correlations with comparative ease. Find out which variables are highly correlated.
3.14 Estimate the effects of ban and sugar_sweeCcap on diabetes using the data in Table 3.16, which can be found on the companion website, 12 where diabetes diabetes prevalence in country ban = 1 if some type of ban on genetically modified goods is present, 0 otherwise sugar_sweeCcap domestic
3.13 Continue with the preceding exercise. Estimate the sales series after removing the seasonal and trend components from it and compare your analysis with that discussed in the text.
3.12 Refer to the fashion sales example discussed in the text. Reestimate Eq. (3.10) by adding the trend variable, taking values of 1, 2, and so on. Compare your results with those given in Table 3.10. What do these results suggest?
3.11 Continuing with Exercise 3.10, how would you find out if there are structural breaks in the relationship between PCE and DPI? Show the necessary calculations.
3.10 Collect quarterly data on personal consumption expenditure (PCE) and disposable personal income (DPI), both adjusted for inflation, and regress personal consumption expenditure on personal disposable income. If you think there is a seasonal pattern in the data, how would you deseasonalize
3.9 Regress Sales, RPDI, and CONF individually on an intercept and the three dummies and obtain residuals from these regressions, say Sl, S2, S3' Now regress Sl on S2 and S3 (no intercept term in this regression)l1 and show that slope coefficients of S2 and S3 are precisely the same as those of
3.8 Refer to the fashion data given in Table 3.10. Using log of sales as the dependent variable, obtain results corresponding to Tables 3.11, 3.12, 3.13, 3.14, and 3.15 and compare the two sets of results.
3.7 Suppose somebody suggests that in the semi-log wage function instead of using 1 and 0 values for the dummy variables, you use the values 10 and 1. What would be the outcome?
3.4 Besides the variables included in the wage regression in Tables 3.1 and 3.5, what other variables would you include? 3.5 Suppose you want to consider the geographic region in which the wage earner resides. Suppose we divide US states into four groups: east, south, west, and north. How would you
3.3 Suppose you regress the log of the wage rate on the logs of education and experience and the dummy variables for gender, race, and union status. How would you interpret the slope coefficients in this regression?
3.2 Replicate Table 3.4, using log of wage rate as the dependent variable and compare the results thus obtained with those given in Table 3.4.
3.1 How would you compare the results of the linear wage function given in Table 3.1 with the semi-log wage regression given in Table 3.5? How would you compare the various coefficients given in the two tables?
2.6 Table 2.15 gives cross-country data for 83 countries on per worker GDP for 1997 and Corruption Index for 1998.26 (a) Plot the index of corruption against per worker GDP. (b) Based on this plot what might be an appropriate model relating corruption index to per worker GDP? (c) Present the

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Econometrics By Example 1st Edition Damodar Gujarati - Solutions

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