Reference no: EM132507536
ASB4408 Financial Econometrics - Bangor University
Assignment 1:
Part A
A1.An AR(2) process has a characteristic equation with two roots: z=-3 and z=1. What can you say about this series?
a. The series is stationary.
b. The series is non-stationary. It contains one unit root.
c. The series is non-stationary. It contains two unit roots.
d. The series is I(0).
A2. Which of the following statements is correct?
a. A stationary AR(1) model has a PACF which is significant for lag 1 and insignificant for other lags.
b. A non-stationary AR(1) model has an ACF which is significant for lag 1 and insignificant for other lags.
c. An MA(1) model has a PACF which is significant for lag 1 and insignificant for other lags.
d. A stationary AR(1) model has a PACF that decays geometrically.
A3. Among the following tests, which one is typically used to detect serial correlation in the residuals of a regression?
a. KPSS test.
b. Durbin-Watson test.
c. Phillips-Perron test.
d. None of the above.
A4. You have downloaded from Datastream historical share price data for a company. Which among of the following tests would be useful to understand whether the series is stationary or non-stationary?
a. Augmented Dickey-Fuller test.
b. Breusch-Godfrey test.
c. Durbin-Watson test.
d. None of the above.
A5. The time series y(t) must be differenced once in order to achieve a stationary transformation. We can say that:
a. The series is I(0).
b. The series is I(2), or integrated of order 2.
c. The series contains two unit roots.
d. None of the above.
A6. Which of the following statements is correct?
a. The product of a 2 x 5 matrix by a 5 x 3 matrix is a 2 x 3 matrix.
b. The product of a 5 x 2 matrix by a 3 x 5 matrix is a 2 x 3 matrix.
c. The product of a 2 x 5 matrix by a 5 x 3 matrix is not defined.
d. None of the above.
A7. What is the trace of the square matrix: A =
a. 6.
b. 5.
c. 12.
d. None of the above.
A8. You have to write the characteristic equation of the AR(2) process:
yt = 0.5yt-1 + 0.4yt-2 + ut
Which of the following equations is correct?
a. 1 + 0.6z - 0.3z2 = 0
b. 1 + 0.5z + 0.4z2 = 0
c. 1 - 0.5z - 0.4z2 = 0
d. None of the above.
A9. Which of the following statements about the Durbin-Watson test is correct?
a. There must be a constant term in the regression.
b. There must be no lags of the dependent variable in the regression (it should not be a dynamic model).
c. The Durbin-Watson statistic does not follow a Chi-square distribution.
d. All of the above statements are correct.
A10. You have run four bivariate Vector Autoregression (VAR) models, with 2, 3, 4, and 5 lags, respectively. The computed values for the Multivariate Akaike Information Criteria are as follows:
Lag MAIC
2 -17.844
3 -17.901
4 -18.856
5 -17.852
Which model would you prefer?
a. The model with 2 lags.
b. The model with 3 lags.
c. The model with 4 lags.
d. The model with 5 lags.
Question A-ii. Give a short definition of endogeneity
Part B:
B1. Explain the concept of cointegration in the context of spot and futures prices of a commodity. How could you test for cointegration using the Engle-Granger approach?
B2. Briefly explain how to test for non-stationarity in a time series using the Dickey-Fuller test.
B3. Explain the concept of spurious regression in the context of models with non-stationary time series.
B4. Briefly describe the Breusch-Godfrey (BG) test for serial correlation in the residuals of a regression.
B5. Explain how you can choose between a Random Effect and a Fixed Effect (also Within-Group) model using a Hausman test. If the Hausman test is rejected, what can you conclude?
B6. Briefly describe the main features of a Moving Average process and of an Autoregressive process.
B7. Compare the standard GARCH(1,1) model with the GJR-GARCH(1,1) model.
B8. Briefly discuss Vector Autoregression models.
Assignment 2:
Part A
A1. Which of the following is not a necessary assumption underlying Ordinary Least Squares (OLS) estimation of the coefficients of a linear regression?
A. The error term is random.
B. The explanatory variable is endogenous.
C. The error term is homoscedastic.
D. The error term is normally distributed.
E. Both B and D.
A2. In the estimation of a two-variable linear regression model using 20 observations, a researcher has obtaineβ ^2 = 3.34 and se(β ^2) = 1.86. Which of the following is TRUE?
A. is significantly greater than zero in a one-tail test at the 0.05 significance level.
B. is significantly different from zero in a two-tail test at the 0.05 significance
level.
C. The p-value for a test of H0:β2=0 against H1:β2>0 is smaller than 0.01.
D. The p-value for a test of H0:β2=0 against H1:β2≠0 is larger than 0.1.
E. Both B and D.
A3. You have fitted a cross-sectional regression using n=30 observations, and you have then estimated an auxiliary regression of the squared residuals on the squared fitted values. The R2 value in the auxiliary regression is 0.16. What do you infer from this result?
A. The residuals from the cross-sectional regression are homoscedastic.
B. The residuals from the cross-sectional regression are free of serial correlation.
C. The residuals from the cross-sectional regression are not normally distributed.
D. The residuals from the cross-sectional regression are heteroscedastic.
E. The residuals from the cross-sectional regression are serially correlated.
A4. The time series y_tmust be differenced twice in order to achieve a stationary transformation. We can say that:
A. The series is I(0).
B. The series is I(1), or integrated of order 1.
C. The series does not contain unit roots.
D. ?y_t is I(1), or integrated of order 1.
E. None of the above.
A5. You have downloaded data from the United Kingdom Household Longitudinal Survey for the 5 most recent waves. The data corresponds to the income of 5000 British households. Which among of the following tests would be useful to understand whether the dataset is stationary or non-stationary?
A. Augmented Dickey-Fuller test.
B. Phillips-Perron test.
C. KPSS test.
D. Im, Pesaran and Shin (IPS) test.
E. All of the above.
A6. When fitting a 2SLS regression, the properties that make the instruments strong are:
A. No serial correlation and efficiency.
B. Heteroscedasticity and exogeneity.
C. Exogeneity and relevance.
D. Normality and relevance.
E. None of the above.
A7. When the exogeneity assumption is invalid in the two-variable linear regression model , which of the following statements is TRUE?
A. ui is random but not normally distributed.
B. There is backward (reverse) causality from yi to xi.
C. There is backward (reverse) causality from yi to ui.
D. The OLS estimators of β1 and β2 are efficient.
E. The covariance between xi and ui is zero.
A8. The variables yt and xt are both non-stationary. You estimate a regression of yt on xt.using 120 observations, and then you save the residuals of the regression. You run an Augmented Dickey-fuller type test on the residuals and obtain a statistic τ=-3.85Which of the following is correct?
A. The residuals from the regression of yt on xt are normally distributed.
B. The residuals from the regression of yt on xt are free of serial correlation.
C. There is evidence of cointegration at the 5% level of significance.
D. There is no long-run statistically significant relationship between xt and yt at any relevant level of significance.
E. There is evidence of cointegration at the 1% level of significance.
A9. Among the following tests, which one is typically used to detect serial correlation in the residuals of a regression?
A. KPSS test.
B. Breusch-Godfrey test.
C. Phillips-Perron test.
D. Bera-Jarque test.
E. None of the above.
A10. You have the following data for Johansen's λmax rank test for cointegration between 3 international equity market indices:
r λmax 5% Critical Value
0 33.05 31.22
1 15.81 23.33
2 10.52 15.25
What do you conclude?
A. All equity market indices are stationary.
B. There are no cointegrating vectors.
C. There are two cointegrating vectors.
D. There is evidence of Granger-causality among the series.
E. None of the above.
Part B
B1. (a) Explain the following terms:
(i) Rank of a square matrix,
(ii) Vector Error CorrectionModel (VECM),
(iii) Cointegration,
(iv) Unit root,
(v) Akaike Information Criterion,
(vi) VectorAutoregression,
(b) With reference to the following VAR(1) model:
(¦(y_1t@y_2t ))=(¦(0.4&[email protected]&0.1))(¦(y_(1t-1)@y_(2t-1) ))+(¦(u_1t@u_2t ))
calculate the impulse responsefunction for the effect of a unit shock to y2t at t=1 on y1t and y2t at t=1,2,3,4 and 5.
(c) With reference to the following VAR(2) model:
explain briefly how you would test the null hypothesis that y2tdoes not Granger-cause y1t.
B2. (a) Explain the distinction between the fixed effects and random effects estimators of the ‘within groups' model?
(c) Describe a test you could use to determine the choice between the fixed effects and random effects estimators.
(e) Explain why it would be inappropriate to estimate a panel model that includes a lagged dependent variable using either the fixed effects estimator or the random effects estimator. Additionally, explain how the Arellano and Bond dynamic panel estimator overcomes these.
B3.
(a) Carefully discuss what features of financial data cannot be modelled using linear time series models and how these features can be used using a GARCH(1,1) process.
(b) Describe the procedure for determining whether ARCH affects are present in the residuals of an ARCH(3) model.
B4. Using 175 monthly observations for the period August 2002 - April 2017 inclusive on
yt = natural logarithm of the US consumer price index, a researcher has obtained the following results by estimating three ADF (Augmented Dickey Fuller) autoregressions:
I Δyt = μ0 + ρyt-1 + δ1Δyt-1 + ... + δ4Δyt-8 + γt + ut
= -0.0154 se( ) = 0.0157
II Δ2yt = μ0 + ρΔyt-1 + δ1Δ2yt-1 + ... + δ3Δ2yt-7 + ut
= -1.0944 se( ) = 0.1569
III Δ3yt = μ0 + ρΔ2yt-1 + δ1Δ3yt-1 +...+ δ2Δ3yt-6 + ut
= -2.4099 se( ) = 0.3234
where Δyt = yt - yt-1, Δ2yt = Δyt - Δyt-1 and Δ3yt = Δ2yt - Δ2yt-1.
(a) Using these results, determine the order of integration of the series yt.
(b) What are the likely consequences for the performance of the ADF test if an inappropriate number of lagged values of Δyt-j is included in the ADF autoregressions?
B5. (a) With reference to the linear regression model, describe briefly the implications for estimation and statistical inference if the distribution of the disturbance term is non-normal.
(b) Describe the Bera-Jarque skewness-kurtosis test for the validity of the normality assumption.
(c) For each of the following assumptions concerning the multiple regression model, describe one method you could use to test the validity of the assumption.
(i) Exogeneity,
(ii) Homoscedasticity.
B6. (a) You have obtained cross-sectional data on scores in a financial literacy test for a
group of individuals who have either participated or not participated previously in a financial literacy training program. The decision to participate is believed to depend on past educational attainment and other personal characteristics of the participants, which may also influence performance in the test, and which are also recorded in the data set.
Explain how a propensity score matching estimator could be used to evaluate the effectiveness of the financial literacy training program.
(b) Explain how difference-in-differences estimation using panel data can assist in evaluating the impact of an exogenous policy or regulatory change.
Assignment 3:
Part A
A1. Which of the following is the correct definition of the p-value of a hypothesis test?
A. The probability of rejecting the null hypothesis when the null is false.
B. The probability of accepting the null hypothesis when the null is true.
C. The minimum significance level at which the null hypothesis can be accepted.
D. The maximum significance level at which the null hypothesis can be accepted.
E. The minimum significance level at which the null hypothesis can be rejected.
A2. In the estimation of a two-variable linear regression model using 20 observations, a researcher has obtained and . Which of the following is TRUE?
A. is significantly greater than zero in a one-tail test at the 0.05 significance level.
B. is significantly different from zero in a two-tail test at the 0.05 significance level.
C. The p-value for a test of H0:β2=0 against H1:β2>0 is smaller than 0.01.
D. The p-value for a test of H0:β2=0 against H1:β2≠0 is larger than 0.1.
E. Both B and D.
A3. Which of the following statements is correct?
A. The product of a 3×5 matrix and a 2×5 matrix is a 3×2 matrix.
B. The product of a 3×5 matrix and a 5×2 matrix is a 5×5 matrix.
C. The product of a 3×5 matrix and a 5×2 matrix is not defined.
D. The product of a 3×5 matrix and a 2×5 matrix is not defined.
E. Both B and D.
A4. A multiple regression model with k=3 regression coefficients (including the intercept) was estimated using 30 observations. The following results were obtained: , , . Which of the following statements is TRUE?
A. The goodness-of-fit for this regression is 0.375.
B. The goodness-of-fit for this regression is 0.625.
C. An F-test for the joint significance of the explanatory variables produces a test statistic of 8.1, which is significant at the 0.05 level.
D. Both A and C.
E. Both B and C.
A5. Which of the following is not a necessary assumption underlying Ordinary Least Squares (OLS) estimation of the coefficients of a linear regression?
A. The error term is homoscedastic.
B. The true relationship between the explanatory variable and the dependent variable is linear.
C. The error term is random.
D. The explanatory variable is exogenous.
E. The error term is normally distributed.
A6. When the exogeneity assumption is invalid in the two-variable linear regression model , which of the following statements is TRUE?
A. ui is random but not normally distributed.
B. There is backward (reverse) causality from yi to xi.
C. There is backward (reverse) causality from yi to ui.
D. The OLS estimators of β1 and β2 are efficient.
E. The covariance between xi and ui is zero.
A7. The trace of the square matrix A = is:
A. 5
B. 10
C. 16
D. 21
E. 26
A8. The determinant of the square matrix A = is:
A. 5
B. 10
C. 16
D. 21
E. 26
A9. You have downloaded from Datastream historical share price data for a company. Which among the following tests would be useful to understand whether the series is stationary or non-stationary?
A. Breusch-Godfrey test
B. White test.
C. Skewness-kurtosis test.
D. Augmented Dickey-Fuller test.
E. Engle-Granger test.
A10. The variables yt and xt are both non-stationary, but the residuals from the regression of yt on xt are stationary. Which of the following is correct?
A. There is no long-run relationship between xt and yt.
B. The residuals from the regression of yt on xt are heteroscedastic.
C. The residuals from the regression of yt on xt exhibit serial correlation.
D. xt and yt are cointegrated.
E. None of the above.
Part B
B1. A researcher is developing a regression model to explain growth in per capita GDP over the period 1980-2000, using cross-sectional data for 35 countries. The variable definitions (for country i) are as follows:
dlypci = Log growth in per capita GDP (US dollars), 1980-2000
lypc80i = Log of per capita GDP in 1980 (US dollars)
lsecedi = Log proportion of secondary school age population enrolled at
secondary school in 1980
govgdpi = Ratio of government expenditure to GDP in 1980
openi = Ratio of (exports+imports) to GDP in 1980
crediti = Private credit by deposit money banks and other financial
institutions to GDP in 1980
The following model has been estimated:
dlypci = 2.2052 - 0.1891 lypc80i + 0.2843 lsecedi + 0.0103 govgdpi
(.8996) (.1004) (.1108) (.0095)
+ 0.0012 openi + 0.5353 crediti + ei
(.0027) (.2927)
= 4.4352 = 0.9686 = 3.4666
Standard errors of estimated coefficients are shown beneath in parentheses.
(a) Calculate the value of R2 (goodness-of-fit) for this regression.
(b) Test the null hypothesis that a country's growth in per capita income over the period 1980-2000 did not depend on the level of its per capita income at the start of this period in 1980.
(c) Use an F-test to evaluate the null hypothesis that the true values of all
slope coefficients in this regression are zero (H0:β2=β3=β4=β5=β6=0).
B2. (a) Explain what is meant by an endogeneity problem.
(b) With reference to instrumental variables regression, explain the following terms:
(i) Instrument relevance.
(ii) Instrument exogeneity.
(b) In a regression model where one of the explanatory variables is endogenous rather than exogenous, explain how an instrumental variables regression estimated using two stage least squares (2SLS) can be used to obtain consistent estimates of the regression coefficients.
B3. (a) Explain the following terms:
(i) Logit regression
(ii) Ordered probit regression
(b) A researcher has fitted a model to predict whether an application for a mortgage will be declined, in the form of a probit regression. The data set comprises cross-sectional data on 500 mortgage applications. The variable definitions are as follows:
DENY = 1 if the application was refused
0 if the was not refused
PIRATIO = Ratio of mortgage payments to income
SELF = 1 if lead applicant is self-employed, 0 otherwise
JOINT = 1 if application is joint (two persons), 0 if application is
single
The estimated model is as follows:
= -2.01 + 4.49 PIRATIO + 0.32 SELF - 0.68 JOINT
(0.21) (1.62) (0.20) (0.11)
DENY = 1 if > 0, DENY = 0 if ? 0 where ui ~ N(0,1)
Standard errors of estimated coefficients are shown in parentheses.
(i) What is the estimated probability of refusal for a self-employed single applicant with PIRATIO=0.25?
(ii) What is the estimated probability of refusal for a joint application with the same PIRATIO, if the lead applicant is not self-employed?
B4. Using 91 yearly observations for the period 1924-2014 inclusive on yt = natural logarithm of the UK retail price index, a researcher has obtained the following results by estimating three ADF (Augmented Dickey Fuller) autoregressions:
I Δyt = μ0 + ρyt-1 + δ1Δyt-1 + ... + δ4Δyt-4 + γt + ut
= -0.02385 se( ) = 0.00953 5% critical value for ADF test = -3.459
II Δ2yt = μ0 + ρΔyt-1 + δ1Δ2yt-1 + ... + δ3Δ2yt-3 + ut
= -0.16677 se( ) = 0.06533 5% critical value for ADF test = -2.897
III Δ3yt = μ0 + ρΔ2yt-1 + δ1Δ3yt-1 + δ2Δ3yt-2 + ut
= -1.4754 se( ) = 0.17885 5% critical value for ADF test = -2.897
where Δyt = yt - yt-1, Δ2yt = Δyt - Δyt-1 and Δ3yt = Δ2yt - Δ2yt-1.
(a) Using these results, determine the order of integration of the series yt.
(b) What are the likely consequences for the performance of the ADF test if an inappropriate number of lagged values of Δyt-j is included in the ADF autoregressions?
B5. (a) Explain the following terms:
(i) Spurious regression problem,
(ii) Cointegration.
(b) With reference to the following VAR(2) model
explain briefly how you would test the following hypotheses:
(i) y1tdoes not Granger-cause y2t.
(ii) y2tdoes not Granger-cause y1t.
B6. (a) Following Johansen's methodology, the following VECM (Vector Error Correction Model) has been proposed to model the relationship between the non-stationary time series variables y1t and y2t:
Explain carefully the implications of the rank of the coefficient matrix for the existence or non-existence of a cointegrating relationship between y1t and y2t.
(b) From the estimation of the above model, the researcher has obtained the following results for Johansen's trace and maximal eigenvalue statistics:
rank Trace Critical Maximal Critical
value eigenvalue value
0 23.3945 15.41 22.5794 14.07
1 0.8151 3.76 0.8151 3.76
You are asked to advise the researcher on the interpretation of these results.
B7. Consider the following AR(1)-GARCH(1,1) model for the series yt:
yt = μ0 + Φ1yt-1 + ut with ut ~ N(0, )
= a0 + a1 + b1
Using daily data from 01/01/2005 to 31/12/2014, a researcher has obtained the following estimated version of this model for yt = daily logarithmic return on the FTSE100 index:
yt = 0.00051 - 0.04144 yt-1 + ut
(.00015) (.02124)
= 0.0000 + 0.1107 + 0.8804
(.0000) (.0100) (.0102)
Standard errors of the estimated coefficients are shown in parentheses.
(a) Comment briefly on your interpretation of the estimated values of the coefficients
a1 and b1.
(b) Describe a test you could use to determine whether the fitted model adequately describes the daily DJIA logarithmic returns series.
(c) Describe how the specification of the AR(1)-GARCH(1,1) model could be adapted to allow for a tendency for the conditional variance of the disturbance term to increase more in response to negative shocks (‘bad news') than in response to positive shocks (‘good news').
B8. A researcher is considering estimating each of the following regression models using a panel data set:
‘Pooled' model:
yi,t = α + β1xi,t + ui,t
‘Within' model:
yi,t = αi + β1xi,t + ui,t
‘Between' model:
where yi,t = return on equity of bank i in year t
xi,t = log assets of bank i in year t
and
(a) Explain how you would interpret the information about the relationship between log assets and return on equity that is provided by the estimated versions of the ‘pooled' model, the ‘within' model, and the ‘between' model.
(b) Explain the distinction between the fixed effects and random effects estimators of the ‘within' model?
(c) Describe a test you could use to determine the choice between the fixed effects and random effects estimators.
B9. (a) Explain why it would be inappropriate to estimate a panel model that includes a lagged dependent variable using either the fixed effects estimator or the random effects estimator.
(b) Explain how the Arellano and Bond dynamic panel estimator overcomes the problems with the fixed effects and random effects estimation that you have identified in your answer to part (a).
Assignment 4:
Part A
A1. Which of the following is the correct definition of the p-value of a hypothesis test?
A. The probability of rejecting the null hypothesis when the null is false.
B. The probability of accepting the null hypothesis when the null is true.
C. The minimum significance level at which the null hypothesis can be
accepted.
D. The maximum significance level at which the null hypothesis can be
accepted.
E. The minimum significance level at which the null hypothesis can be
rejected.
A2. Which of the following statements is FALSE?
A. The product of a 3×5 matrix and a 5×3 matrix is a 3×3 matrix.
B. The product of a 3×5 matrix and a 5×2 matrix is a 5×5 matrix.
C. The product of a 3×5 matrix and a 3×5 matrix is not defined.
D. The sum of a 3×5 matrix and a 3×5 matrix is not defined.
E. Both B and D.
A3. Which of the following is NOT a necessary assumption underlying Ordinary Least Squares (OLS) estimation of the coefficients of a linear regression?
A. The error term is homoscedastic.
B. The error term is normally distributed.
C. The error term is random.
B. The true relationship between the explanatory variable and the dependent variable is linear.
D. The explanatory variable is exogenous.
A4. When the exogeneity assumption is invalid in the two-variable linear regression model , which of the following statements is TRUE?
A. There is backward (reverse) causality from yi to xi.
B. There is backward (reverse) causality from yi to ui.
C. The OLS estimators of β1 and β2 are efficient.
D. The covariance between xi and ui is zero.
E. ui is heteroscdastic and serially correlated.
A5. The determinant of the square matrix A = is:
A. 13
B. 19
C. 22
D. 32
E. 40
A6. The trace of the square matrix A = is:
A. 13
B. 19
C. 22
D. 32
E. 40
A7. You have fitted a cross-sectional regression using n=30 observations, and you have then estimated an auxiliary regression of the squared residuals on the squared fitted values. The R2 value in the auxiliary regression is 0.16. What do you infer from this result?
A. The residuals from the cross-sectional regression are homoscedastic.
B. The residuals from the cross-sectional regression are free of serial correlation.
C. The residuals from the cross-sectional regression are not normally distributed.
D. The residuals from the cross-sectional regression are heteroscedastic.
E. The residuals from the cross-sectional regression are serially correlated.
A8. You have downloaded from Datastream historical share price data for a company. Which among the following tests would be useful to understand whether the series is stationary or non-stationary?
A. Im, Pesaran and Shin (IPS) test.
B. White test.
C. Engle-Granger test.
D. Augmented Dickey-Fuller test.
E. Ljung-Box test.
A9. The asymmetric GARCH model is applicable in which of the following cases:
A. The distribution of the residuals is positively or negatively skewed.
B. A negative shock exerts a bigger influence on the conditional variance than a positive shock of the same absolute magnitude.
C. The conditional variance is non-stationary.
D. The explanatory variable is non-stationary, but the dependent variable is stationary.
E. The residuals exhibit volatility clustering and excess kurtosis.
A10. The variables yt and xt are both non-stationary, but the residuals from the regression of yt on xt are stationary. Which of the following is correct?
A. The residuals from the regression of yt on xt are normally distributed.
B. The residuals from the regression of yt on xt are free of serial correlation.
C. xt and yt are cointegrated.
D. There is no long-run relationship between xt and yt.
E. None of the above.
Part B
B1. (a) Explain what is meant by the following terms:
(i) Heteroscedasticity.
(ii) Serial correlation.
(b) With reference to the multiple regression model, explain how you could test for the presence of serial correlation.
(c) Explain what is meant by a spurious regression problem.
B2. (a) Explain what is meant by an endogeneity problem.
(b) With reference to instrumental variables regression, explain the following terms:
(i) Instrument relevance.
(ii) Instrument exogeneity.
(b) In a regression model where one of the explanatory variables is endogenous rather than exogenous, explain how an instrumental variables regression estimated using two stage least squares (2SLS) can be used to obtain consistent estimates of the regression coefficients.
B3. Using 53 yearly observations for the period 1953-2005 inclusive on yt = natural logarithm of the UK retail price index, a researcher has obtained the following results by estimating three ADF (Augmented Dickey Fuller) autoregressions:
I Δyt = μ0 + ρyt-1 + δ1Δyt-1 + ... + δ4Δyt-4 + γt + ut
= -0.0582 se( ) = 0.0181 5% critical value for ADF test = -3.497
II Δ2yt = μ0 + ρΔyt-1 + δ1Δ2yt-1 + ... + δ3Δ2yt-3 + ut
= -0.1353 se( ) = 0.0947 5% critical value for ADF test = -2.928
III Δ3yt = μ0 + ρΔ2yt-1 + δ1Δ3yt-1 + δ2Δ3yt-2 + ut
= -1.5634 se( ) = 0.2493 5% critical value for ADF test = -2.928
where Δyt = yt - yt-1, Δ2yt = Δyt - Δyt-1 and Δ3yt = Δ2yt - Δ2yt-1.
(a) Using these results, determine the order of integration of the series yt.
(b) What are the likely consequences for the performance of the ADF test if an inappropriate number of lagged values of Δyt-j is included in the ADF autoregressions?
B4. (a) Explain the following terms:
(i) White noise.
(ii) Integrated of order one.
(b) With reference to the following VAR(1) model:
calculate the impulse responsefunction for the effect of a unit shock to y2t at t=1 on y1t and y2t at t=1,2,3,4 and 5.
(c) With reference to the following VAR(2) model:
explain briefly how you would test the null hypothesis that y2tdoes not Granger-cause y1t.
B5. (a) (i) Describe the method developed by Engle and Granger to test for the
existence of a cointegrating relationship between two non-stationary time series variables.
(ii) Explain how a cointegrating relationship is incorporated into the specification of an Error Correction Model.
(b) Following Johansen's methodology, the following VECM (Vector Error Correction Model) has been proposed to model the relationship between the non-stationary time series variables y1t and y2t:
Explain carefully the implications of the rank of the coefficient matrix for the existence or non-existence of a cointegrating relationship between y1t and y2t.
B6. Consider the following AR(1)-GARCH(1,1) model for the series yt:
yt = μ0 + Φ1yt-1 + ut with ut ~ N(0, )
= a0 + a1 + b1
Using daily data from 01/01/2004 to 31/12/2013, a researcher has obtained the following estimated version of this model for yt = daily logarithmic return on the FTSE100 index:
yt = 0.0005 - 0.0567 yt-1 + ut
(.0002) (.0214)
= 0.0000 + 0.0953 + 0.8953
(.0000) (.0085) (.0091)
Standard errors of the estimated coefficients are shown in parentheses.
(a) Comment briefly on your interpretation of the estimated values of the coefficients
a1 and b1.
(b) Describe a test you could use to determine whether the fitted model adequately describes the daily FTSE100 logarithmic returns series.
(c) Describe how the specification of the AR(1)-GARCH(1,1) model could be adapted to allow for a tendency for the conditional variance of the disturbance term to increase more in response to negative shocks (‘bad news') than in response to positive shocks (‘good news').
B7. (a) You have obtained cross-sectional data on scores in a financial literacy test for a
group of individuals who have either participated or not participated previously in a financial literacy training program. The decision to participate is believed to depend on past educational attainment and other personal characteristics of the participants, which may also influence performance in the test, and which are also recorded in the data set.
Explain how a propensity score matching estimator could be used to evaluate the effectiveness of the financial literacy training program.
(b) Explain how difference-in-differences estimation using panel data can assist in
evaluating the impact of an exogenous policy or regulatory change.
B8. (a) Explain why it would be inappropriate to estimate a panel model that includes a lagged dependent variable using either the fixed effects estimator or the random effects estimator.
(b) Describe the construction of Arellano and Bond's Difference GMM (Generalized Method of Moments) dynamic panel estimator.
(c) Explain how the ImPesaran and Shin (IPS) test could be used to test a null hypothesis that nominal exchange rates are non-stationary, using a panel data set comprising monthly US dollar exchange rates for 20 other currencies over a five-year period.
Assignment 5:
Part A Answer ALL questions. Part A carries 40% of the total mark
A1. Which of the following is the correct definition of the power of a hypothesis test?
A. The minimum significance level at which the null hypothesis can be
accepted.
B. The maximum significance level at which the null hypothesis can be
rejected.
C. The minimum significance level at which the null hypothesis can be
rejected.
D. The probability of accepting the null hypothesis when the null is true.
E. The probability of rejecting the null hypothesis when the null is false.
A2. In the estimation of a two-variable linear regression model using 30 observations, a researcher has obtained and . Which of the following is TRUE?
A. is significantly greater than zero in a one-tail test at the 0.01 significance level.
B. is significantly different from zero in a two-tail test at the 0.05 significance
level.
C. is not significantly different from zero in a two-tail test at the 0.1 significance
level.
D. The p-value for a test of H0:β2=0 against H1:β2>0 is larger than 0.1.
E. The p-value for a test of H0:β2=0 against H1:β2≠0 is smaller than 0.01.
A3. Which of the following statements is correct?
A. The product of a 2×5 matrix and a 5×2 matrix is a 2×2 matrix.
B. The product of a 2×5 matrix and a 2×5 matrix is a 2×5 matrix.
C. The product of a 2×5 matrix and a 5×2 matrix is a 5×5 matrix.
D. The product of a 2×5 matrix and a 5×2 matrix is not defined.
E. Both B and D.
A4. A multiple regression model with k=7 regression coefficients (including the intercept) was estimated using 25 observations. The following results were obtained: , , . Which of the following statements is TRUE?
A. The goodness-of-fit for this regression is 0.25.
B. The goodness-of-fit for this regression is 0.75.
C. An F-test for the joint significance of the explanatory variables produces a test statistic of 9, which is significant at the 0.05 level.
D. Both A and C.
E. Both B and C.
A5. Which of the following is a necessary assumption underlying Ordinary Least Squares (OLS) estimation of the coefficients of a linear regression?
A. The error term is random.
B. The explanatory variable is endogenous.
C. The error term is heteroscedastic.
D. The error term is normally distributed.
E. Both B and D.
A6. When the homoscedasticity assumption is invalid in the two-variable linear regression model , which of the following statements is TRUE?
A. The OLS estimators of β1 and β2 are unbiased.
B. The OLS estimators of β1 and β2 are efficient.
C. There is non-zero covariance between consecutive values of ui.
D. There is non-zero covariance between ui and xi.
E. Both A and D.
A7. The determinant of the square matrix A = is:
A. 40
B. 35
C. 34
D. 13
E. 8
A8. The trace of the square matrix A = is:
A. 40
B. 35
C. 34
D. 13
E. 8
A9. Which among the following tests would be useful to understand whether the disturbance term of a regression model follows a normal distribution?
A. White test.
B. Augmented Dickey-Fuller test.
C. Engle-Granger test.
D. Skewness-kurtosis test.
E. Hausman test
A10. The series yt and Δyt are both non-stationary, but Δ2yt is stationary. Which of the following is correct?
A. yt is integrated of order one.
B. yt contains two unit roots.
C. yt necessarily contains a deterministic time-trend.
D. yt could contain a deterministic time-trend.
E. Both B and D.
Part B Answer THREE questions.
B1. A researcher is developing a regression model to explain the profitability of a sample of 100 small banks, using cross-sectional data for a particular year. The variable definitions are as follows:
ROA = Return on assets (%) = 100 x Net income / Total assets
LASSET = Natural logarithm of total assets
LOAASS = loans to assets ratio (%) = 100 x Total loans / Total assets
NONPER = Non-performing loans ratio (%)
= 100 x Non-performing loans / Total loans
The following model has been estimated:
ROA = -1.3773 + 0.1354 LASSET - 0.0053 LOAASS - 0.0695 NONPER + error
(1.0413) (.0562) (.0027) (.0571)
R2= 0.1054 = 21.0651 = 18.8438
Note: is the sum of squared deviations of the dependent variable, ROA, from its own mean. is the sum of squared residuals (error terms). Standard errors of estimated coefficients are shown in parentheses.
A smaller version of the model is also being considered, with LOAASS and NONPER excluded from the list of explanatory variables. From the estimation of this version, the researcher obtained R2= 0.0545 and = 19.9162.
Using hypothesis tests based on the estimation results that are reported above, you are asked to evaluate each of the following assertions:
(a) "Bank size, measured by assets, is irrelevant in determining bank profitability."
(b) "After allowing for the other controls that are included in the model, banks with a smaller loans-to-assets ratio were more profitable than banks with a larger loans-to-assets ratio."
(c) "Collectively, the LOAASS and NONPER variables are helpful in explaining the variation in profitability."
B2. (a) Explain what is meant by the following terms:
(i) Heteroscedasticity.
(ii) Serial correlation.
(b) With reference to a regression model to be fitted using time series data, explain how you could test for the validity of the no serial correlation assumption.
(c) In a regression model where one of the explanatory variables is endogenous rather than exogenous, explain how an instrumental variables regression estimated using two stage least squares (2SLS) can be used to obtain consistent estimates of the regression coefficients.
B3. (a) Explain the following terms:
(i) Probit regression
(ii) Ordered probit regression
(b) A researcher has fitted a model to predict whether an application for a mortgage will be declined, in the form of a probit regression. The data set comprises cross-sectional data on 746 mortgage applications. The variable definitions are as follows:
DENY = 1 if the application was refused
0 if the was not refused
PIRATIO = Ratio of mortgage payments to income
SELF = 1 if lead applicant is self-employed, 0 otherwise
JOINT = 1 if application is joint (two persons), 0 if application is
single
The estimated model is as follows:
= -1.95 + 6.42 PIRATIO + 0.27 SELF - 0.47 JOINT
(0.14) (1.05) (0.17) (0.09)
DENY = 1 if > 0, DENY = 0 where ui ~ N(0,1)
Standard errors of estimated coefficients are shown in parentheses.
(i) Comment briefly on the information this model provides about the factors that influence the probability of a mortgage application being refused.
(ii) What is the estimated probability of refusal for a self-employed single applicant with PIRATIO=0.25? What is the estimated probability of refusal for a joint application with the same PIRATIO if the lead applicant is not self-employed?
B4. Using 58 yearly observations for the period 1957-2014 inclusive on yt = natural logarithm of the UK real gross domestic product, a researcher has obtained the following results by estimating three ADF (Augmented Dickey Fuller) autoregressions:
I Δyt = μ0 + ρyt-1 + δ1Δyt-1 + δ2Δyt-2 + γt + ut
= -0.1345 se( ) = 0.0703 5% critical value for ADF test = -3.492
II Δ2yt = μ0 + ρΔyt-1 + δ1Δ2yt-1 + ut
= -0.6756 se( ) = 0.1249 5% critical value for ADF test = -2.924
III Δ3yt = μ0 + ρΔ2yt-1 + ut
= -1.1216 se( ) = 0.1318 5% critical value for ADF test = -2.924
where Δyt = yt - yt-1, Δ2yt = Δyt - Δyt-1 and Δ3yt = Δ2yt - Δ2yt-1.
(a) Using these results, determine the order of integration of the series yt.
(b) What are the likely consequences for the performance of the ADF test if an inappropriate number of lagged values of Δyt-j is included in the ADF autoregressions?
B5. (a) Explain the following terms:
(i) Endogeneity problem.
(ii) Spurious regression problem.
(b) Describe the method developed by Engle and Granger (1987) to test for the existence of a long-run equilibrium relationship between two non-stationary time series variables.
B6. (a) Explain the following terms:
(i) Rank of a square matrix.
(ii) Unit root.
(iii) Integrated of order two.
(b) Following Johansen's methodology, the following VECM (Vector Error Correction Model) has been proposed to model the relationship between the non-stationary time series variables y1t and y2t:
Explain carefully the implications of the rank of the coefficient matrix for the existence or non-existence of a cointegrating relationship between y1t and y2t.
B7. Consider the following AR(1)-GARCH(1,1) model for the series yt:
yt = μ0 + Φ1yt-1 + ut with ut ~ N(0, )
= a0 + a1 + b1
Using daily data from 01/01/2001 to 31/12/2010, a researcher has obtained the following estimated version of this model for yt = daily logarithmic return on the FTSE-100 index:
yt = 0.00038 - 0.04538 yt-1 + ut
(.00018) (.02208)
= 0.0000 + 0.0787 + 0.9150
(.0000) (.0075) (.0082)
Standard errors of the estimated coefficients are shown in parentheses.
(a) Comment briefly on your interpretation of the estimated values of the coefficients
a1 and b1.
(b) Describe a test you could use to determine whether the fitted model adequately describes the daily FTSE-100 logarithmic returns series.
(c) Describe how the specification of the AR(1)-GARCH(1,1) model could be adapted to allow for a tendency for the conditional variance of the disturbance term to increase more in response to negative shocks (‘bad news') than in response to positive shocks (‘good news').
B8. (a) Explain briefly what is meant by the following terms:
(i) Within groups model.
(ii) Between groups model.
(iii) Dynamic panel model.
(b) Explain the distinction between the fixed effects and random effects estimators of the ‘within groups' model?
(c) Describe a test you could use to determine the choice between the fixed effects and random effects estimators.
B9. (a) Explain why it would be inappropriate to estimate a panel model that includes a lagged dependent variable using either the fixed effects estimator or the random effects estimator.
(b) Explain how the Arellano and Bond dynamic panel estimator overcomes the problems with the fixed effects and random effects estimation that you have identified in your answer to part (a).
Assignment 6:
Part A
Which of the following sets of characteristics would usually best describe an autoregressive process of order 3 (i.e., an AR(3))?
(a) A slowly decaying acf, and a pacf with 3 significant spikes
(b) A slowly decaying pacf and an acf with 3 significant spikes
(c) A slowly decaying acf and pacf
(d) An acf and a pacf with 3 significant spikes.
2. A process, xt, which has a constant mean and variance, and zero autocovariance for all non-zero lags is best described as
(a) A white noise process
(b) A covariance stationary process
(c) An autocorrelated process
(d) A moving average process.
3. Which of the following conditions must hold for the autoregressive part of an ARMA model to be stationary?
(a) All roots of the characteristic equation must lie outside the unit circle
(b) All roots of the characteristic equation must lie inside the unit circle
(c) All roots must be smaller than unity
(d) At least one of the roots must be bigger than one in absolute value.
4. Which of the following could be viewed as a disadvantage of the vector autoregressive (VAR) approach to modelling?
(a) We do not need to specify which variables are endogenous and which are exogenous
(b) Standard form VARs can be estimated equation-by-equation using OLS
(c) VARs often contain a large number of terms
(d) VARs can be expressed using a very compact notation.
5. Which of the following are problems associated with the Engle-Granger approach to modelling using cointegrated data?
(i) The coefficients in the cointegrating relationship are hard to calculate
(ii) This method requires the researcher to assume that one variable is the dependent variable and the others are independent variables
(iii) The Engle-Granger technique can only detect one cointegrating relationship
(iv) The Engle-Granger technique does not allow the testing of hypotheses involving the actual cointegrating relationship.
(a) (i), (ii), (iii), and (iv)
(b) (ii), (iii), and (iv) only
(c) (ii), (iii) only
(d) (ii) and (iv) only.
6.A process, xt, which has a constant mean and variance, and zero autocovariance for all non-zero lags is best described as
(a) A white noise process
(b) A covariance stationary process
(c) An autocorrelated process
(d) A moving average process
7.Three characteristics of a weakly stationary process are
(I)
(II)
(III)
What do the mathematical expressions I, II and III imply?
(a) Constant variance, constant mean and constant autocovariance, respectively
(b) Constant autocovariance structure, constant mean and constant variance, respectively
(c) Constant mean, constant autocorrelation and constant autocovariance, respectively
(d) Constant mean, constant variance and constant autocovariance structure, respectively
8.Consider the following conditional variance equation for a GJR model.
ht = α0 + α1 +βht-1+γut-12It-1
whereIt-1 = 1 if ut-1< 0
= 0 otherwise
For there to be evidence of a leverage effect, which one of the following conditions must hold?
α0 positive and statistically significant
γ positive and statistically significant
γ statistically significantly greater than α0
α1+βstatistically significantly less than γ
9. Which of these is a type of panel estimator approach?
(I) Fixed effects
(II) Random effects
(III) Seemingly unrelated regression effects
(IV) Time-varying effects.
(a) I only
(b) I and II only
(c) I, II, and III only
(d) I, II, III, and IV.
10. Entity fixed effects models
(a)Allow the intercept in the regression model to differ cross-sectionally but not over time, while all of the slope estimates are fixed both cross-sectionally and over time
(b) Allow the slope in the regression model to differ cross-sectionally but not over time, while the intercept estimates are fixed both cross-sectionally and over time
(c) Allow the intercept in the regression model to differ over time, while all of the slope estimates are different both cross-sectionally and over time
(d) Any of the above could be true depending on the model specification.
Part B
Question 1
What kinds of variables are likely to be non-stationary? How can variables be made stationary?
Why is it general important test for non-stationarity in time series data before attempting to build an empirical model?
Define the following terms and describe the processes that they represent
Weak stationarity
Strict Stationarity
Deterministic Trend
Stochastic Trend
A researcher to test the order of integration of some time-series data. He decides to use the DF test. He estimates a regression of the form
?y_t=μ+ψy_(t-1)+u_t
And obtains the estimate ψ ^=-0.02 with standard error= 0.31.
What are the null and alternative hypotheses for this test?
Given the data, and a critical value of -2.88, perform the test.
What is the conclusion from this test and what should be the next step?
Question 2
Consider the following three models:
(i) yt = 0.3yt-1 + + 0.1yt-2+ ut
(ii) yt = ut + 0.6ut-1 + 0.2ut-2
(iii) yt = 0.6yt-1 + ut + 0.3ut-1
Each of these models can be described as ARMA(p,q). What are p and q in each case?
Discuss shortly the principles behind maximum likelihood estimation. .OLS and maximum likelihood are used to estimate the parameters of a standard linear regression model. Will they give the same estimates? Explain your answer.
3. Consider the following MA(2) process
X_t=u_t+θ_1 u_(t-1)+θ_2 u_(t-2)
whereu_t is a zero mean white noise process with variance σ^2.
Calculate the mean and variance of X_t.
Derive the autocorrelation function for this process
Question 3
Describe briefly the three hypothesis testing procedures that are available under maximum likelihood estimation. Which is likely to be calculated in practice and why?
What stylized fact of financial data cannot be explained using linear trend models?
Which of these features can be modelled using a GARCH (1,1) process?
Attachment:- Exam question.zip