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Econometric Analysis An Applied Approach To Business And Economics 1st Edition Sharif Hossain - Solutions

Find the variance-covariance matrix of ˆ????if the random error terms are autocorrelated with order one.
Show that the OLS estimator ˆ????will be a consistent estimate of ???? even if the disturbance terms are autocorrelated.
What are the consequences of autocorrelation?
Find the variance-covariance matrix of ????T????1 if the random error terms are autocorrelated with order 1.
Find the mean, variance, covariance and autocorrelation of AR(1) autoregressive random error terms.
Write different sources of autocorrelation with an example of each.
What is meant by autocorrelation? Explain with an example.
The moment matrices and all other necessary information for the regression equation of per capita real GDP (Y, in thousand USD) on corruption perception index (X) for high-income and low-income countries are given below: (in each case the matrix X includes a constant and one independent
Let the regression equation of total income (Y, in million $) on money supply (X1, in million $), and total investment (X2, in million $) of the banking sector of a developing country be given by t 1 1t 2 2t t Y = ????+????X +???? X +???? , t t t ????Y = x???????? + ???? , where t t E(???? |x ) =
Let the regression equation between per capita income (X) and per capita consumption expenditure (Y) of a developed country be given by i i i Y = ????+????X +???? , where i i E(???? |x ) = 0, and the form of heteroscedasticity is 2 2 i i i i Var(???? |x ) = ???? = ???? x . The moment matrices and
Discuss the technique to estimate a heteroscedastic model with variances constant within subgroups of observations.
For the regression equation given in problem 4-25, if the form of heteroscedasticity is 2 ????i i i Var(???? |x ) = ???? z , where i z is a non-stochastic variable and which will be identical or a function of X’s, discuss the ML method to estimate the heteroscedastic model.
For the regression equation given in problems 4-25, the form of heteroscedasticity is 2 i i Var(???? |x ) = ???? ????, and i j E(???? , ???? ) = 0, ???? i ???? j. Given a sample of observations on Y and X’s, then discuss the maximum likelihood (ML)method to estimate the heteroscedastic model.
Let the regression equation between Y and X’s be given by, i 0 1 1i k ki i Y = ???? +???? X +........+???? X +???? , i i i ???? Y = x????????+???? , where i i E(???? |x ) = 0 and 2 2 ???? ???? 2 ???? ????i i i 1 1i 2 2i p pi Var(???? |x ) = ???? = ???? exp ???? z +???? z +......+???? z = ???? exp
Consider a linear model to the explain monthly beer consumption i 0 1 1i 2 2i 3 3i i Y = ???? +????X +???? X +???? X +???? , i i i ???? Y ???? x???????? ???? ???? , where the variable Y indicates consumption of milk, X1 is the income level of a consumer, X2 is the price per litre of beer, and X3 is
Discuss the method to estimate a heteroscedastic model with correlated disturbances.
Let the regression equation between Y and X’s be given by i 0 1 1i 2 2i k ki i Y = ???? +????X +???? X +.......+???? X +???? , i i i ???? Y ???? x???????? ???? ???? , where i i E(???? |x ) = 0, and 2 2 i i i i Var(???? |x ) = ???? = ???? w , where i w is a non-stochastic variable which will be
Let the regression equation between Y and X is given by i i i Y = ????+????X +???? , where i i E(???? |x ) = 0, and the form of heteroscedasticity is 2 i i i Var(???? |x ) = ???? , i = 1, 2,....,n. Given a sample of observations on Y and X, discuss the technique to estimate the equation. If the
If the regression model is i 0 i y = ???? +???? , for the structural form of heteroscedasticity given in problems 4-19, answer the following questions.(i) What is the most efficient estimator of ????0 ? What is its variance?(ii) What is the OLS estimator of ????0 and what is the variance of the OLS
Let the regression model between Y and X be given by i i i y = ????x +???? , where i i E(???? |x ) = 0, the form of heteroscedasticity is 2 2 i i i Var(???? |x ) = ???? x , i = 1, 2,....,n, and i j Cov(???? ,???? ) = 0, ????i ???? j. Here, the variables y and x are measured in the deviation form
Discuss the WLS method and GLS method to estimate a heteroscedastic model.
Assume that the regression equation indicate the relationship between expenditure and income of ith industry ( i= 1, 2, 3, 4, 5, 6 7). The results for the maximum likelihood estimate of 2 ???? for the combined sample is s2 =19. Also, the results for the maximum likelihood estimate of 2????i ( i =
Discuss the likelihood ratio test for testing heteroscedasticity. Assume that the variable Y indicate the corruption perception index and X indicates the economic growth (per capita real GDP) of all the countries of this globe. Let all the countries be divided into two groups namely (i) low-income
Let the regression model between the corruption perception index (CPI) and per capita GDP (PGDP, constant 2015 USD) of high-income and low-income countries be given by ij 0 i ij ij CPI = ???? + ???? PGDP +???? , (i=1, 2, j = 1, 2,….., i n ), where i i E(???? |PGDP ) = 0, the form of
Discuss the following tests for detecting the presence of heteroscedasticity:(i) The Goldfeld and Quandt test, (ii) The Glejser test, (iii) The Spearman’s rank correlation test, (iv) The likelihood ratio (LR) test, (v) The Breusch-Pagan test, (vi) The Lagrange multiplier test, (vii) The White
Let the regression equation between Y and X be given by i i i Y = ????+????X +???? and the form of heteroscedasticity be 2 2 vi i i i Var(???? ) = ???? = ???? X???? e , i = 1, 2,.........,n. Test the null hypothesis 0 H : ???? = 2 against an appropriate alternative hypothesis.
Discuss the Park test for detecting the presence of heteroscedasticity.
Discuss the Bartlett test for detecting the presence of heteroscedasticity in the data set in the case of simple and multiple linear regression equations.
Discuss the graphical method for detecting the presence of heteroscedasticity in the case of simple and multiple linear regression equations.
Let the heteroscedastic model between Y and X’s be i 1 1i 2 2i k ki i y = ???? x +???? x +...... +???? x +???? , where the variable Y and all the independent variables are measured in the deviation form from their mean value. The equation can also be written as i i i y = x ???????? ???? ???? or y
Discuss the consequences, if we apply the OLS method to the heteroscedastic model.
Write the important sources of heteroscedasticity and discuss the nature of heteroscedasticity.
Distinguish between multiplicative and additive forms of heteroscedasticity with an example of each.
Let the heteroscedastic model between Y and X be i i i Y = ????+????X +???? and the form of heteroscedasticity be 2 2 2 i i i Var(???? ) = ???? = ???? X , i = 1, 2,.......,n, and i j Cov(???? ,???? ) = 0, ????i ???? j. Obtain the variance co-variance matrix of ???? where???? is a (n×1) matrix of
Let ????n????1 be a vector of random error terms of a multiple linear regression equation. Find the variance-covariance matrix of ????n????1 if the random error terms are not correlated and if they are correlated.
Explain the meaning of heteroscedasticity. Distinguish between homoscedasticity and heteroscedasticity with an example of each.
Define a Cobb-Douglas production function and explain all the terms of the Cobb-Douglas production function.Find the output elasticities with respect to labour and capital of the Cobb-Douglas production function and also find their marginal products.The estimation of the Cobb-Douglas production of
Test the following null hypothesis using the Wald test, F-test and LM test . B-B+B,+B4-B,=1 Ho BB B=B4
Let the non-linear multiple regression equation be given by ????1 ????2 ????i i 0 1i 2i Y = ???? X X e , where Yi is the ith value of the output variable Y, X1i is the ith value of the input variable X1 (labour), X2i is the ith value of the input variable X2(capital) , and ???? is the random error
Let the functional relationship of private investment with GDP and interest rate be given by????1 ????2 ????t t t t pinv = AGDP ir e , where, the variable pinv indicates private investment, the variable ir indicates interest rate ???? is the random error term. The private investment (in billion
Data on the quantity demanded of a commodity (in kg), the price of that commodity (in USD) and the level of income (in USD) are given below:Let the demand function be given by, ????1 ????2 ????i i i i q = Ap in e , where q = per capita consumption of a commodity, p = price per unit of the
Assume that the variable Y indicates the level of outputs in log (Yi = lnPi, the variable P indicates the level of output), the variable X1 indicate the number of labours in log ( i.e., X1i = ln(Li), the input variable L indicates the number of labours ) and the variable X2 indicates capital in log
Suppose you estimate the following regression equation to evaluate the effects of various firm-specific factors on the returns of a sample of 150 firms.i 0 1 i 2 i 3 i 4 i i RET =???? +???? PE +???? MBR +???? S +???? BETA +????Where i RET is the annual return in the percentage of the ith firm, i PE
Let the variable Y indicate the quantity demanded of a commodity Q (in kg), the variable 1 X indicate the per unit price of that commodity (in $), and the variable 2 X indicate per capita disposable income (in $). For sample observation the equation is given byTotal Sum of Squares (TSS) =
Let the variable Y indicate per capita consumption of chickens (in kg), the variable 1 X indicates the real retail price of chicken per kg (in $) and the variable 2 X indicates per capita real disposable income (in $). Let for the sample observations the equation iTotal Sum of Squares (TSS) =
Let the variable Y indicate private investment (in million USD), the variable 1 X indicate GDP (in million USD)and the variable 2 X indicate interest rate (in USD). Here, given that,Requirements:(i) Obtain the regression equation of private investment on GDP and interest rate and comment on your
Which would you expect to be bigger: the unrestricted residual sum of squares (UESS) or the restricted residual sum of squares (RESS) and why?
Let us consider a multiple linear regression equation of type t 0 1 1t 2 2t 3 3t 4 4t t Y =???? +???? X +???? X +???? X +???? X ???? ???? . Which of the following hypotheses can be tested using the t-test? Which of them can be tested using an F test, an LM test, and the Wald test? In each case,
Discuss the testing procedure for non-linearity of a multiple regression equation.
Discuss the technique to estimate the Cobb-Douglas production function of the type ????1 ????2 ????i i 0 i i q =A L K e and explain the meaning of all the terms of this function.
Define a double log model with an example. Define the Cobb-Douglas production function with an example.Explain the meaning of all the terms of a Cobb-Douglas production function.
Define multivariate non-linear relationships with an example of each.
Discuss the test procedure for testing the equality of parameters of two multiple linear regression equations.
Discuss the F-test, Wald test, and Lagrange multiplier (LM) test procedures for testing a general null hypothesis in case of multiple linear regression equations.
Discuss the technique to test the joint null hypothesis 0 1 2 h H : ???? = ???? ???? .... ???? ???? against an appropriate alternative hypothesis where h < k, k is the number of independent variables.
Discuss the test of significance of parameter estimates of a multiple linear regression equation.
Define analysis of variance. Construct the ANOVA tables for the unadjusted and adjusted multiple linear regression equations.
Show that 2 2 yyˆ r ???? R , where yyˆ r is the correlation coefficient between y and ˆy.
Discuss different steps that are involved in fitting a multiple linear regression equation with and without distributional assumptions of the random error term.
Explain the meaning of the coefficient of multiple determination ( R2 ). Discuss the techniques to obtain it.Discuss the important characteristics of R2 .
Discuss the important properties of residuals.
Discuss the important properties of Yˆ .
Find the sampling distribution of s2 = ????ˆ 2 and also find its mean and variance.
Show that 2 ESS????is distributed as Chi-square with degrees of freedom (n-k-1).
Find the expected value of the components’ sum of squares.
Show that RSS and ESS are independent.
Find the quadratic forms of the regression sum of squares (RSS), residual sum squares (ESS) and total sum of squares (TSS).
Show that the total sum of squares can be partitioned into components sum of squares.
Show that 1 1 2 2 k k RSS = ????ˆ SP(X ,Y)+????ˆ SP(X ,Y)+......+????ˆ SP(X ,Y) , where RSS is the regression sum of squares and j SP(X ,Y) is the sum of products between Y and j X (j=1, 2,…,k).
Show that Yˆ ~N(X????, ????2P), where P is a symmetric and idempotent matrix.
Show that 2 n Y~N(X????, ???? I ), where n I is a unit matrix of order n.
Show thatis a consistent estimator of 2 ???? where e is the residual vector, n is the sample size and k is the number of independent variables in a multiple linear regression equation. s= S e'e n-k-1
Show that the OLS estimator s2 is an unbiased estimator of 2 ???? but the ML estimator of 2 ???? is biased.
Discuss all the important properties of the MLE of 2 ???? .
Discuss the ML method to estimate a multiple linear regression equation.
Discuss the important properties of the OLS estimators of a multiple linear regression equation.
Write a multiple linear regression equation in the matrix form, and discuss the OLS method to estimate a multiple linear regression equation.
Define a multiple linear regression equation with an example. Explain the meaning of all the terms of a multiple linear regression equation.
Define the coefficient of determination in case of a three-variable linear regression equation. Discuss the techniques to obtain it.
Let three-variable linear regression equation be given by: i 0 1 1i 2 2i i Y = ???? +????X +???? X +???? ;(i) Explain all the terms of the given equation.(ii) Discuss the technique to estimate this equation.(iii) Obtain the variance of the estimators 0ˆ????, 1ˆ????, and 2ˆ????.(iv) Discuss the
What are the basic assumptions of a multiple linear regression equation?
Distinguish between three-variable linear and non-linear regression equations.
Define a multiple regression and multiple linear regression equation with an example of each.
Let the functional relationship between private investment and GDP be given by ???? ????t t 0 t pinv = A GDP e , where t pinv indicates private investment at the time t, t GDP indicates the gross domestic product at time t and ????t is the random error term corresponding to the tth set of
Obtain the semi-log model for the GDP of Bangladesh and then compare it with that of India using the real data.
Given below are data showing the price per unit in $ and consumption in kilograms of a commodity:Obtain the reciprocal model and comment on your results. Also, obtain the price elasticity of demand. Price Quantity Demanded 30 112 48 55 25 120 55 40 40 75 20 125 35 98 50 45 36 90 45 65 42 4449 70 40
Estimate the linear log model for the production of rice in Bangladesh and then compare it with India using the real data.
The obtained points of 12 students on midterm (X) and final exam. (Y) are as follows:Requirements:[i] Obtain the regression equation of y on x.[ii] Test the null hypothesis 0 H : ???? = 0 against a suitable alternative hypothesis.[iii] Calculate the value of the coefficient of determination.[iv]
The following data indicate total sales revenue ( in billions USD) and profits (in millions USD) for 28 companies in a developed country.Requirements:[i] Obtain the regression equation of profits on sales revenue and comment on your estimated results.[ii] Find the 95% confidence interval for the
Given below are the data of the average productivity (X) and average salary (Y) of 10 different districts.[i] Obtain the regression equation i i i Y = ????+????X +???? and comment on your results.[ii] Obtain the ˆ var(????) and hence the standard error of ˆ????.[iii] Do you think the relationship
Let the variable Y indicate the amount of profits in million $ and the variable X indicate the amount of sales revenue in million $. Here, given thatTotal Sum of Squares (TSS) = 3299, Residual Sum of Squares (ESS) = 332 Requirements:[i] Obtain the regression equation i 0 i i Y = ???? +????X +????
Let the variable Y indicate the level of savings (in $) and the variable X indicate the level of income (in $) of 26 households of city A. Here, given thatTotal Sum of Squares (TSS) = 99870.087, and Residual Sum of Squares (ESS) = 23248.298 Requirements:[i] Obtain the regression equation i i i Y =
Let the variable Y indicate the per capita expenditure ( in USD) on public schools and the variable X indicate the per capita income (in USD) by districts of a country for a particular year. The following information is givenTotal Sum of Squares (TSS) = 78670.857 and Residual Sum of Squares (ESS) =
Write the regression equation for each of the following relationships along with the importance of the parameters of each of the models. Give an example of each model that will be applicable to deal with real problems. (i) Level-level relationship, (ii) Log-level relationship, (iii) Level-log
Explain the significance of the parameter estimates of different types of non-linear regression equations.
What is meant by a non-linear relationship? Define different types of non-linear regression equations and discuss their estimation techniques.
Define the capital asset pricing model (CAPM) with an example. Derive the regression equations of the CAPM.Estimate the CAPM using the real data.
Show that the square of the correlation coefficient is given by =and = ; where i=1
How do we estimate the elasticity of the dependent variable y with respect to the independent variable x from the regression coefficient? Explain.
Discuss the important characteristics of the coefficient of determination.

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