Introductory Econometrics For Finance 3rd Edition Chris Brooks - Solutions

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5. Re-open the ‘fail xls’ spreadsheet for modelling the probability of MSc failure and do the following:(a) Take the country code series and construct separate dummy variables for each country. Re-run the probit and logit regression above with all of the other variables plus the country dummy
4. (a) Explain the difference between a censored variable and a truncated variable as the terms are used in econometrics.(b) Give examples from finance (other than those already described in this book) of situations where you might meet each of the types of variable described in part (a) of this
3. (a) Describe the intuition behind the maximum likelihood estimation technique used for limited dependent variable models.(b) Why do we need to exercise caution when interpreting the coefficients of a probit or logit model?(c) How can we measure whether a logit model that we have estimated fits
2. Compare and contrast the probit and logit specifications for binary choice variables.
1. Explain why the linear probability model is inadequate as a specification for limited dependent variable estimation.
4. (a) What are the advantages and disadvantages of conducting unit root tests within a panel framework rather than series by series?(b) Explain the differences between panel unit root tests based on a common alternative hypothesis and those based on heterogeneous processes.
3. Find a further example of where panel regression models have been used in the academic finance literature and do the following:● Explain why the panel approach was used.● Was a fixed effects or random effects model chosen and why?● What were the main results of the study and is any
2. (a) Explain how fixed effects models are equivalent to an ordinary least squares regression with dummy variables.(b) How does the random effects model capture cross-sectional heterogeneity in the intercept term?(c) What are the relative advantages and disadvantages of the fixed versus random
1. (a) What are the advantages of constructing a panel of data, if one is available, rather than using pooled data?(b) What is meant by the term ‘seemingly unrelated regression’? Give examples from finance of where such an approach may be used.(c) Distinguish between balanced and unbalanced
4. (a) Re-open the exchange rate returns series and test them for day-of-the-week effects.(b) Re-open the house price changes series and determine whether there is any evidence of seasonality.
3. A researcher suggests that the volatility dynamics of a set of daily equity returns are different:● on Mondays relative to other days of the week● if the previous day’s return volatility was bigger than 0.1% relative to when the previous day’s return volatility was less than
2. (a) What is a switching model? Describe briefly and distinguish between threshold autoregressive models and Markov switching models. How would you decide which of the two model classes is more appropriate for a particular application?(b) Describe the following terms as they are used in the
1. A researcher is attempting to form an econometric model to explain daily movements of stock returns. A colleague suggests that she might want to see whether her data are influenced by daily seasonality.(a) How might she go about doing this?(b) The researcher estimates a model with the dependent
6. Using EViews, estimate a multivariate GARCH model for the spot and futures returns series in ‘sandphedge.wf1’. Note that these series are somewhat short for multivariate GARCH model estimation. Save the fitted conditional variances and covariances, and then use these to construct the
5. (a) What is a news impact curve? Using a spreadsheet or otherwise, construct the news impact curve for the following estimated EGARCH and GARCH models, setting the lagged conditional variance to the value of the unconditional variance (estimated from the sample data rather than the mode
4. Suppose that a researcher is interested in modelling the correlation between the returns of the NYSE and LSE markets.(a) Write down a simple diagonal VECH model for this problem. Discuss the values for the coefficient estimates that you would expect.(b) Suppose that weekly correlation forecasts
3. (a) Distinguish between the terms ‘conditional variance’ and ‘unconditional variance’. Which of the two is more likely to be relevant for producing:i. one-step-ahead volatility forecasts ii. twenty-step-ahead volatility forecasts.(b) If ut follows a GARCH(1,1) process, what would be the
2. (a) Discuss briefly the principles behind maximum likelihood.(b) Describe briefly the three hypothesis testing procedures that are available under maximum likelihood estimation. Which is likely to be the easiest to calculate in practice, and why?(c) OLS and maximum likelihood are used to
1. (a) What stylised features of financial data cannot be explained using linear time series models?(b) Which of these features could be modelled using a GARCH(1,1) process?(c) Why, in recent empirical research, have researchers preferred GARCH(1,1) models to pure ARCH(p)?(d) Describe two
9. (a) What issues arise when testing for a unit root if there is a structural break in the series under investigation?(b) What are the limitations of the Perron (1989) approach for dealing with structural breaks in testing for a unit root?
8. In EViews, open the ‘currencies.wf1’ file that will be discussed in detail in the following chapter. Determine whether the exchange rate series (in their raw levels forms) are non-stationary. If that is the case, test for cointegration between them using both the Engle–Granger and Johansen
7. Compare and contrast the Engle–Granger and Johansen methodologies for testing for cointegration and modelling cointegrated systems. Which, in your view, represents the superior approach and why?
6. (a) Suppose that a researcher has a set of three variables, yt(t = 1, . . . , T), i.e.yt denotes a p-variate, or p × 1 vector, that she wishes to test for the existence of cointegrating relationships using the Johansen procedure.What is the implication of finding that the rank of the
5. (a) Briefly outline Johansen’s methodology for testing for cointegration between a set of variables in the context of a VAR.(b) A researcher uses the Johansen procedure and obtains the following test statistics (and critical values):r λmax 95% critical value 0 38.962 33.178 1 29.148 27.169 2
4. (a) Consider a series of values for the spot and futures prices of a given commodity. In the context of these series, explain the concept of cointegration. Discuss how a researcher might test for cointegration between the variables using the Engle–Granger approach. Explain also the steps
3. Using the same regression as for question 2, but on a different set of data, the researcher now obtains the estimate ˆψ = −0.52 with standard error = 0.16.(a) Perform the test.(b) What is the conclusion, and what should be the next step?(c) Another researcher suggests that there may be a
2. A researcher wants to test the order of integration of some time series data.He decides to use the DF test. He estimates a regression of the formyt = μ + ψyt−1 + ut and obtains the estimate ˆψ = −0.02 with standard error = 0.31.(a) What are the null and alternative hypotheses for this
1. (a) What kinds of variables are likely to be non-stationary? How can such variables be made stationary?(b) Why is it in general important to test for non-stationarity in time series data before attempting to build an empirical model?(c) Define the following terms and describe the processes that
5. Define carefully the following terms● Simultaneous equations system● Exogenous variables● Endogenous variables● Structural form model● Reduced form model.
4. Consider the following vector autoregressive model yt = β0+k i=1βiyt−i + ut (7.99)where yt is a p × 1 vector of variables determined by k lags of all p variables in the system, ut is a p×1 vector of error terms, β0 is a p×1 vector of constant term coefficients and βi are p × p
3. Explain, using an example if you consider it appropriate, what you understand by the equivalent terms ‘recursive equations’ and ‘triangular system’. Can a triangular system be validly estimated using OLS? Explain your answer.
2. Consider the following system of two equations y1t = α0 + α1y2t + α2X1t + α3X2t + u1t (7.97)y2t = β0 + β1y1t + β2X1t + u2t (7.98)(a) Explain, with reference to these equations, the undesirable consequences that would arise if (7.97) and (7.98) were estimated separately using OLS.(b) What
1. Consider the following simultaneous equations system y1t = α0 + α1y2t + α2y3t + α3X1t + α4X2t + u1t (7.94)y2t = β0 + β1y3t + β2X1t + β3X3t + u2t (7.95)y3t = γ0 + γ1y1t + γ2X2t + γ3X3t + u3t (7.96)(a) Derive the reduced form equations corresponding to (7.94)–(7.96).(b) What do you
13. Select two of the stock series from the ‘CAPM.XLS’ Excel file, construct a set of continuously compounded returns, and then perform a time series analysis of these returns. The analysis should include(a) An examination of the autocorrelation and partial autocorrelation functions.(b) An
12. (a) Briefly explain any difference you perceive between the characteristics of macroeconomic and financial data. Which of these features suggest the use of different econometric tools for each class of data?(b) Consider the following autocorrelation and partial autocorrelation coefficients
11. (a) Explain what stylised shapes would be expected for the autocorrelation and partial autocorrelation functions for the following stochastic processes:● white noise● an AR(2)● an MA(1)● an ARMA (2,1).(b) Consider the following ARMA process.yt = 0.21 + 1.32yt−1 + 0.58ut−1 + ut
10. You have estimated the following ARMA(1,1) model for some time series data yt = 0.036 + 0.69yt−1 + 0.42ut−1 + ut Suppose that you have data for time to t−1, i.e. you know that yt−1 = 3.4, and ˆut−1 = −1.3(a) Obtain forecasts for the series y for times t, t+1, and t+2 using the
9. (a) You obtain the following sample autocorrelations and partial autocorrelations for a sample of 100 observations from actual data:Lag 1 2 3 4 5 6 7 8 acf 0.420 0.104 0.032 −0.206 −0.138 0.042 −0.018 0.074 pacf 0.632 0.381 0.268 0.199 0.205 0.101 0.096 0.082 Can you identify the most
8. ‘Given that the objective of any econometric modelling exercise is to find the model that most closely ‘fits’ the data, then adding more lags to an ARMA model will almost invariably lead to a better fit. Therefore a large model is best because it will fit the data more closely.’Comment
7. How could you determine whether the order you suggested for question 6 was in fact appropriate?
6. A researcher is trying to determine the appropriate order of an ARMA model to describe some actual data, with 200 observations available. She has the following figures for the log of the estimated residual variance (i.e. log(ˆσ 2)) for various candidate models. She has assumed that an order
5. You obtain the following estimates for an AR(2) model of some returns data yt = 0.803yt−1 + 0.682yt−2 + ut where ut is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity.
4. (a) Describe the steps that Box and Jenkins (1976) suggested should be involved in constructing an ARMA model.(b) What particular aspect of this methodology has been the subject of criticism and why?(c) Describe an alternative procedure that could be used for this aspect.
3. Consider the following three models that a researcher suggests might be a reasonable model of stock market prices yt = yt−1 + ut (6.190)yt = 0.5yt−1 + ut (6.191)yt = 0.8ut−1 + ut (6.192)(a) What classes of models are these examples of?(b) What would the autocorrelation function for each of
2. Why might ARMA models be considered particularly useful for financial time series? Explain, without using any equations or mathematical notation, the difference between AR, MA and ARMA processes.
1. What are the differences between autoregressive and moving average models?
14. (a) Explain the term ‘measurement error’.(b) How does measurement error arise?(c) Is measurement error more serious if it is present in the dependent variable or the independent variable(s) of a regression? Explain your answer.(d) What is the likely impact of measurement error on tests of
13. Re-open the ‘macro.wf1’ and apply the stepwise procedure including all of the explanatory variables as listed above, i.e. ersandp dprod dcredit dinflation dmoney dspread rterm with a strict 5% threshold criterion for inclusion in the model. Then examine the resulting model both financially
12. Explain why it is not possible to include an outlier dummy variable in a regression model when you are conducting a Chow test for parameter stability. Will the same problem arise if you were to conduct a predictive failure test? Why or why not?
11. Why is it desirable to remove insignificant variables from a regression?
10. For the same model as above, and given the following results, do a forward and backward predictive failure test:1981M1–1995M12 rt = 0.0215 + 1.491 rmt RSS = 0.189 T = 180 (5.87)1981M1–1994M12 rt = 0.0212 + 1.478 rmt RSS = 0.148 T = 168 (5.88)1982M1–1995M12 rt = 0.0217 + 1.523 rmt RSS =
9. (a) Explain the term ‘parameter structural stability’?(b) A financial econometrician thinks that the stock market crash of October 1987 fundamentally changed the risk–return relationship given by the CAPM equation. He decides to test this hypothesis using a Chow test.The model is estimated
8. (a) Why is it necessary to assume that the disturbances of a regression model are normally distributed?(b) In a practical econometric modelling situation, how might the problem that the residuals are not normally distributed be addressed?
7. What might Ramsey’s RESET test be used for? What could be done if it were found that the RESET test has been failed?
6. Calculate the long-run static equilibrium solution to the following dynamic econometric modelyt = β1 + β2x2t + β3x3t + β4yt−1 + β5x2t−1+β6x3t−1 + β7x3t−4 + ut (5.82)
5. (a) What do you understand by the term ‘autocorrelation’?(b) An econometrician suspects that the residuals of her model might be autocorrelated. Explain the steps involved in testing this theory using the Durbin–Watson (DW) test.(c) The econometrician follows your guidance (!!!) in part
4. (a) State in algebraic notation and explain the assumption about the CLRM’s disturbances that is referred to by the term ‘homoscedasticity’.(b) What would the consequence be for a regression model if the errors were not homoscedastic?(c) How might you proceed if you found that (b) were
3. A researcher estimates the following model for stock market returns, but thinks that there may be a problem with it. By calculating the t-ratios and considering their significance and by examining the value of R2 or otherwise, suggest what the problem might be.ˆyt = 0.638 + 0.402x2t −
2. What pattern(s) would one like to see in a residual plot and why?
1. Are assumptions made concerning the unobservable error terms (ut) or about their sample counterparts, the estimated residuals ( ˆut)? Explain your answer.
13. A researcher wishes to examine the link between the returns on two assets A and B in situations where the price of B is falling rapidly. To do this, he orders the data according to changes in the price of B and drops the top 80%of ordered observations. He then runs a regression of the returns
12. What are quantile regressions and why are they useful?
11. What are the units of R2?
10. Re-open the Macro file and apply the same APT-type model to some of the other time series of stock returns contained in the CAPM-file.(a) Run the stepwise procedure in each case. Is the same sub-set of variables selected for each stock? Can you rationalise the differences between the series
9. Re-open the CAPM Eviews file and estimate CAPM betas for each of the other stocks in the file.(a) Which of the stocks, on the basis of the parameter estimates you obtain, would you class as defensive stocks and which as aggressive stocks?Explain your answer.(b) Is the CAPM able to provide any
8. A researcher estimates the following two econometric models yt = β1 + β2x2t + β3x3t + ut (4.58)yt = β1 + β2x2t + β3x3t + β4x4t + vt (4.59)where ut and vt are iid disturbances and x3t is an irrelevant variable which does not enter into the data generating process for yt. Will the value of
7. A researcher estimates the following econometric models including a lagged dependent variable yt = β1 + β2x2t + β3x3t + β4yt−1 + ut (4.56)yt = γ1 + γ2x2t + γ3x3t + γ4yt−1 + vt (4.57)where ut and vt are iid disturbances.Will these models have the same value of (a) The residual sum of
6. You estimate a regression of the form given by (4.54) below in order to evaluate the effect of various firm-specific factors on the returns of a sample of firms. You run a cross-sectional regression with 200 firms ri = β0 + β1Si + β2MBi + β3PEi + β4BETAi + ui (4.54)where: ri is the
5. You decide to investigate the relationship given in the null hypothesis of question 2, part (c). What would constitute the restricted regression? The regressions are carried out on a sample of 96 quarterly observations, and the residual sums of squares for the restricted and unrestricted
4. Which would you expect to be bigger – the unrestricted residual sum of squares or the restricted residual sum of squares, and why?
3. Which of the above null hypotheses constitutes ‘THE’ regression F-statistic in the context of (4.53)? Why is this null hypothesis always of interest whatever the regression relationship under study? What exactly would constitute the alternative hypothesis in this case?
2. Which of the following hypotheses about the coefficients can be tested using a t-test? Which of them can be tested using an F-test? In each case, state the number of restrictions.(a) H0 : β3 = 2(b) H0 : β3 + β4 = 1(c) H0 : β3 + β4 = 1 and β5 = 1(d) H0 : β2 = 0 and β3 = 0 and β4 = 0 and
1. By using examples from the relevant statistical tables, explain the relationship between the t- and the F-distributions.For questions 2–5, assume that the econometric model is of the form yt = β1 + β2x2t + β3x3t + β4x4t + β5x5t + ut (4.53)
10. Using EViews, select one of the other stock series from the ‘capm.wk1’ file and estimate a CAPM beta for that stock. Test the null hypothesis that the true beta is one and also test the null hypothesis that the true alpha (intercept)is zero. What are your conclusions?
9. Are hypotheses tested concerning the actual values of the coefficients (i.e. β)or their estimated values (i.e. βˆ) and why?
8. Form and interpret a 95% and a 99% confidence interval for beta using the figures given in question 7.
7. The analyst also tells you that shares in Chris Mining plc have no systematic risk, in other words that the returns on its shares are completely unrelated to movements in the market. The value of beta and its standard error are calculated to be 0.214 and 0.186, respectively. The model is
6. The capital asset pricing model (CAPM) can be written as E(Ri) = R f + βi[E(Rm) − R f ] (3.44)using the standard notation.The first step in using the CAPM is to estimate the stock’s beta using the market model. The market model can be written as Rit = αi + βiRmt + uit (3.45)where Rit is
5. Which of the following models can be estimated (following a suitable rearrangement if necessary) using ordinary least squares (OLS), where X, y, Z are variables and α, β, γ are parameters to be estimated? (Hint: the models need to be linear in the parameters.)yt = α + βxt + ut (3.39)yt =
4. What five assumptions are usually made about the unobservable error terms in the classical linear regression model (CLRM)? Briefly explain the meaning of each. Why are these assumptions made?
3. What is an estimator? Is the OLS estimator superior to all other estimators?Why or why not?
2. Explain, with the use of equations, the difference between the sample regression function and the population regression function.
1. (a) Why does OLS estimation involve taking vertical deviations of the points to the line rather than horizontal distances?(b) Why are the vertical distances squared before being added together?(c) Why are the squares of the vertical distances taken rather than the absolute values?
28. The covariance between two variables is 0.99. Are they strongly related?Explain your answer.
27. Which is a more useful measure of central tendency for stock returns – the arithmetic mean or the geometric mean? Explain your answer.
26. Explain the differences between the mean, mode and median. Which is the most useful measure of an average and why?
25. What is the central limit theorem and why is it important in statistics?
24. (a) Explain the difference between a pdf and a cdf(b) What shapes are the pdf and cdf for a normally distributed random variable?
23. Expand the parentheses as far as possible for the following expressions(a) E(ax + by) for x,y variables and a,b scalars(b) E(axy) for x,y independent variables and a a scalar(c) E(axy) for x,y correlated variables and a a scalar
22. (a) Add2 −1−7 4to−3 0 7 −4(b) Subtract−3 0 7 −4from2 −1−7 4(c) Calculate the inverse of3 −1−4 2(d) Does the inverse of the following matrix exist? Explain your answer3 2 3 2
21. Suppose that we have the following four matrices A =1 6−2 4, B =−3 −8 6 4,C =1 2 3 4 5 6,D =6 −2 0 −1 3 0(a) Which pairs of matrices can be validly multiplied together? For these pairs, perform the multiplications.(b) Calculate 2A, 3B, 12 D(c) Calculate Tr(A), Tr(B), Tr(A + B)
20. Construct an example not used elsewhere in this book to demonstrate that for two conformable matrices A and B, (AB)−1 = B−1A−1.
19. Find the minima of the following functions. In each case, state the value of the function at the minimum(a) y = 6x2 − 10x − 8(b) y = (6x2 − 8)2
18. Solve the following using logs and a calculator(a) 4x = 6(b) 42x = 3(c) 32x−1 = 8
17. Use the result that log(8) is approximately 2.1 to estimate the following(without using a calculator):(a) log(16)(b) log(64)(c) log(4)
16. Solve the following(a) log x4 − log x3 = log 5x − log 2x(b) log(x − 1) + log(x + 1) = 2 log(x + 2)(c) log10 x = 4
15. Simplify the following as far as possible(a) log 27 − log 9 + log 81(b) log 8 − log 4 + log 32
14. Write the following as simply as possible as sums of logs of prime numbers(a) log 60(b) log 300
13. Express the following logarithms using powers(a) log5 3125 = 5(b) log49 7 = 12(c) log0.5 8 = −3
12. Evaluate the following (without using a calculator)(a) log10 10000(b) log2 16(c) log10 0.01(d) log5 125(e) loge e2

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Introductory Econometrics For Finance 3rd Edition Chris Brooks - Solutions

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