1 Multicollinearity may be present in your model if A. some or all of your tratios are individually small (cannot reject individual slopes being zero), but the Ftest value is large (rejects all slopes simultaneously zero) B. the tratios and the Ftest value are both small C. the maximized value of the loglikelihood is large D. The Rsquared value for the fitted model is large 2 One way of resolving heteroskedasticity is A. by applying least squares with weights that are given by the White’s test for detecting heteroskedasticity. B. by applying the method of unweighted least squares. C. by using the weighted transformation that Breusch and Godfrey proposed. D. by applying weights on the residuals so that none of them spreads unequally anymore. 3 If your dataset has serial correlation, but you completely ignore the problem and use a plain OLS command, you will: A. get biased parameter estimates. B. you get OLS estimators that are no longer BLUE. C. you get Fstatistics that are no longer correct.
ECONOMETRICS Allowable materials: Non programmable calculator Unmarked, non-electronic Foreign Language dictionary PART I: MULTIPLE-CHOICE QUESTIONS (30*2=60 marks) Answer space is provided below Multicollinearity may be present in your model if A some or all of your t-ratios are individually small (cannot reject individual slopes being zero), but the F-test value is large (rejects all slopes simultaneously zero) B the t-ratios and the F-test value are both small C the maximized value of the log-likelihood is large D The R-squared value for the fitted model is large One way of resolving heteroskedasticity is A by applying least squares with weights that are given by the White’s test for detecting heteroskedasticity B by applying the method of unweighted least squares C by using the weighted transformation that Breusch and Godfrey proposed D by applying weights on the residuals so that none of them spreads unequally anymore If your dataset has serial correlation, but you completely ignore the problem and use a plain OLS command, you will: A get biased parameter estimates B you get OLS estimators that are no longer BLUE C you get F-statistics that are no longer correct Heteroscedasticity in your data is a problem because: A ordinary OLS assumes that the data are homoscedastic and calculates the point estimates of regression parameters accordingly B ordinary OLS assumes that the data are homoscedastic and calculates the standard error estimates of the parameters accordingly C it biases the parameter point estimates Look at the regression results, you find the Durbin-Watson stat is 2.000186 You understand that A There is evidence of positive serial correlation because the DW statistic is very close to B There is evidence of negative serial correlation because the DW statistic is very close to C There is no evidence of serial correlation because the DW statistic is very close to Multicollinearity exists if there is a high correlation between your one (or more) of your 10 11 explanatory variables A True B False Dummy variables are categorical variables, with no natural numerical values We usually assign D=0 for the base or numeraire group, which is typically the majority group, and D=1 for the alternative, or special group But we could just as easily it the other way around, or we could assign D=-5 for the numeraire and D=12 for the alternative group A True B False Auxiliary R-squared regressions consist of A Y regressed on each of the X-variables in turn B each of the X-variables regressed on each other X variable, one at a time C each of the X-variables regressed on all other X variables at once D each of the X-variables regressed on all possible subsets of X-variables One possible way to check for evidence of heteroskedasticity informally is: A by plotting the residuals of a multiple regression model against the dependent variable and all the regressors B by plotting the squared residuals of a multiple regression model against the dependent variable and all the regressors C by plotting the residuals of a multiple regression model against the dependent variable only D by plotting the squared residuals of a multiple regression model against all the regressors Multicollinearity exists if there is a high correlation between your dependent variable and any one (or more) of your explanatory variables A True B False How you test whether there is significant quarterly seasonality in a time- series of quarterly data? A Estimate a model with three quarterly dummy variables and conduct a joint F- test of whether the coefficients on the three dummy variables are simultaneously equal to zero B Put in dummy variables for each quarter one at a time in a set of four regressions C Estimate a model with four quarterly dummy variables and no intercept 12 Perfect multicollinearity is highly dangerous because the estimated betas you will get from OLS will be biased and inefficient A True B False 13 If you wish to use a set of dummy variables to capture six different categories of an explanatory variable, you should use six different dummy variables, each equal to one when the observation is a member of a specific category and equal to zero otherwise A True B False 14 Multicollinearity A compromises the goodness-of-fit of your regression model B can make it difficult to distinguish the individual effects on the dependent variable of one regressor from that of another regressor C causes the values of estimated coefficients to be insensitive to the presence or absence of other variables in the model D causes t-ratios to be high and F-tests to be low 15 Look at the regression results, you find the Durbin-Watson stat is 0.370186 You understand that A There is evidence of positive serial correlation because the DW statistic is very small B There is evidence of negative serial correlation because the DW statistic is very small 16 Which of the following equation is AR(2) A ut = ρ 1.ut-1 + vt B ut = ρ 1.ut-1 + ρ 1.ut-2 + vt C ut = ρ 1.ut + ρ 2.ut-2 + vt 17 The best way to detect multicollinearity in a multiple regression model is to A look at plots of each regressor against all other regressors B to find any high pairwise correlations C look for low t-ratios D look for high R-squared values E conduct auxiliary regressions and look for high R-squareds 18 By heteroskedasticity we mean A that the residuals of a regression model are not independent B that the residuals of a regression model are related with one or more of the regressors C that the squared residuals of a regression model are not equally spread D that the variance of the residuals of a regression model is not constant for all observations 19 In your data set, your explanatory income variable, INC, is coded as 1= income between and 10,000 per year, 2=income between 10,000 and 25,000 per year, 3= income between 25,000 and 50,000 per year, 4= income between 50,000 and 75,000 per year, and 5= income greater than 75,000 per year The best way to handle these data is to: A recognize that INC is not perfect, but use it as an explanatory variable anyway B use the midpoints of each income interval, divided by 10,000 (to keep the parameter point estimates from being too tiny) C create a set of dummy variables, where each takes on the value of one for one category and zero for all others; use this full SET of dummy variables as your income variable D use both INC and INC-squared as your explanatory variables, not just INC alone 20 For the regression equation Y = 10 - 3X + 2.5X2, which of the following statements is true? A When X1 decreases by units, Y decreases by unit B When X2 decreases by 2.5 units, Y decreases by unit C When X1 decreases by units, Y increases by unit 21 In comparing linear models with log-log models, we take logarithms of all nonnegative continuous variables, but we not use the logs of dummy variables This is because: A log of is zero and log of is undefined B Adam Smith decreed that economists should never log a dummy variable C only dummies attempt to log dummy variables 22 In the estimated regression equation Y = 2.5 - X, which of the following is NOT a correct interpretation of the slope coefficient? A The dependent variable increases by units if X decreases by unit B The dependent variable declines by -2 units if X increases by unit C If the value of X is zero, the value of Y will be -2 23 Heteroscedasticity happens when which of the following OLS assumption is not fulfilled? ( ) A E u i X i = ( ) 2 B E u i X i = σ ( ) C Cov u i , u j = with i ≠ j 24 Which of the following is NOT an assumption of linear regression analysis? A The Y values are statistically independent B The normal distributions are within standard deviations from the line regression C Y values are normally distributed D The means of the distribution of Y values all lie on the straight line of regression 25 By autocorrelation we mean A that the residuals of a regression model are not independent B that the residuals of a regression model are related with one or more of the regressors C that the variance of the residuals of a regression model is not constant for all observations 26 In a two-variable model Yi = β1 + β X 2i + u i , β and β are known as: A partial regression coefficients B slope coefficient and intercept respectively C intercept and slope coefficient respectively D intercept and partial regression coefficient respectively 27 Which sentence given below is Incorrect? A The disturbance term is a random variable that has well-defined probabilistic properties B The disturbance term can take negative value only C The disturbance term is an unobservable random variable D The disturbance term can take positive or negative values 28 In the case of existing less than perfect multicollinearity problem, OLS estimators’ variances will appear to be……………………than normally A smaller B larger 29 Given a two-variable model Yi = β1 + β X 2i + u i , the estimator β in the presence of autocorrelation is: A a BLUE B a linear unbiased but not the best estimator C a best linear but biased estimator D a best unbiased but non-linear estimator 30 Which indicator shows how well a regression line fits through the scatter of data points? A F-test B R2 C t-test Answer Space 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 PART II: PROBLEM (40 marks: questions*4pts=36, except for question 2: points and for question 6: points) Consider the following demand function for money in the United States for the period 1980:1 – 2003:4: β β ut (1) t t t where M = real money demand (in billions USD), using the M2 definition of money (in billions USD), Y = real GDP (in billions USD), and R= interest rate (in %) M =βY R e Preliminary Question 1.1 Your data gives you the nominal quantities; can you describe how to convert nominal quantities into real quantities? 1.2 Equation (1) is a linear regression model or non-linear model? Why? If this is nonlinear, the transformation so that we can estimate it using OLS method 1.1 To convert nominal quantities into real quantities, one needs to divide the nominal money M and nominal GDP by CPI (Consumer Price Index) or Deflator There is no need to divide the interest rate variable by CPI or Deflator 1.2 Equation (1) is not a linear regression model as all the parameter betas are in power form We need to transform the non-linear regression function at the beginning into a linear regression form by taking the ln(natural log) the equation: (2) Equation (2) is our regression function Here is the regression result: Dependent Variable: LOG(M) Method: Least Squares Date: 12/10/08 Time: 18:57 Sample(adjusted): 1980:1 2003:3 Included observations: 95 after adjusting endpoints Variable Coefficient Std Error t-Statistic C 1.407849 0.217476 (1)6.473596 LOG(Y) 0.549451 (2)0.046806 11.73902 LOG(R) -0.073395 0.014350 (3)-5.114764 R-squared 0.873860 Mean dependent var Adjusted R-squared (5)0.871118 S.D dependent var S.E of regression (6)0.050179 Akaike info criterion Sum squared resid 0.231653 Schwarz criterion Log likelihood 150.9795 F-statistic Durbin-Watson stat 0.044972 Prob(F-statistic) Prob 0.0000 0.0000 0.0000 3.607830 0.139775 -3.115358 -3.034709 (4)318.6748 0.000000 Write the estimated regression function and briefly interpret the coefficients LOG(M)^= 1.407849+0.549451 LOG(Y) -0.073395 LOG(R) Beta2=0.549451, meanings that if the real GDP (Y) increases by 1% on average in this sample, the real demand for money (in term of M2) increase of 0.55%, given other variables remains constant (Remember in the log-log functional form, the value of the coefficient beta is exactly the value of elasticity) Beta3=-0.073395, meanings that if the interest rate increases by 1% then the real demand for money will decrease of 0.073%, given other variables remains constant Beta1=1.407849, meanings that if the real GDP (Y) equal to one and interest rate equal to one (or log equals to zero), then the demand for money is exp(1.407849)=3increases by 1% on average in this sample, the real demand for money (in term of M2) increase of 0.55% (remember in the log-log functional form, the value of the coefficient beta is exactly the value of elasticity) Calculate the missing values in the regression results and explain its significations Value (1) t-stat1=beta1/std(beta1)= 1.407849/0.217476=6.473596, measures the individual significant of the beta1 Value (2) standard error2= beta2/ t-stat2=0.549451/11.73902=0.046806 measures the standard deviation of the beta2 Value (3) t-stat3=beta3/std(beta3)= -0.073395 /0.014350=-5.1147 Value (4) F stat =R2 *(n-k)/(1- R2)(k-1)= 0.873860*(95-3)/(1-0.873860)*(3-1)=318.6748 Value (5) R2 adjusted =1-(1/ R2)*(n-1)/(n-k)=1-(1-0.873860)*(95-1)/(95-3)= 0.871118 measures the effectiveness influence of independent variables, Value (6) S.E of regression= (Sum squared resid/(n-k))^0.5=(0.231653/(95-3))^0.5=0.050179 It measures the standard deviation of the residual Kim Anh a: this part, you can look at the book for more details Which of the coefficients are individually statistically significant and not significant at 5% level? Do the t-test, given the t-critical t c is equal to Construct the interval of confidence for the coefficients with the given t-critical H0: betai =0, the coefficient betai is individually not significant H1: betai#0, the coefficient betai is individually significant /t-statistic/=(betai^- betai)/std betai compare with the t-critical=2 If the /t-statistic/ is higher than t-critical, we reject H0, then… In this model, all the three are statistically significant as … For the interval construction, the formula is provided, so no problem… Can you establish a test of the overall significance of the regression? It is just the conventional F-test : statement of H0, H1, calculate F-statistic as in the previous question, F critical Students not need the F critical to conclude the overall significant First reason: the F-statistic is very high 318.6748 Second reason: all the t-statistics are statistically significant Third reason: look at the R2 and R2 adjusted, they are high Four reason: they may know that the F critical is only 3.1 Do you suspect that there is Multicollinearity in this regression? I not suspect any Multicollinearity problem in this regression as all the tstatistics are significant and the R2 are not very very high Based on D-W d-test, what can you conclude about the model’s autocorrelation at 5%? Given the dL and dU respectively are 1.489 and1.573, explain how to find these critical values for this particular model (assume that we have D.W d table) Before the test, observe the d=0.044972 lower than 2, so we suspect the positive autocorrelation H0 : ρ = H1 : ρ > no autocorrelation yes, positive autocorrelation D.W-d table to find the critical value k’=2 and n=95: dL =1.489 and dU =1.573 d=0.044972 < dL =1.489: reject Ho, there exists a positive autocorrelation in this model 8 Here is part of the test for higher-order autocorrelation, what can we conclude, given the critical value is 9.48775 Explain how to find the critical value Breusch-Godfrey Serial Correlation LM Test: F-statistic 469.5326 Probability Obs*R-squared 90.74799 Probability Test Equation: Dependent Variable: RESID Variable Coefficient Std Error t-Statistic C -0.082657 0.051620 -1.601266 LOG(Y) 0.017122 0.011068 1.546997 LOG(R) 0.006113 0.003412 1.791842 RESID(-1) 1.034548 0.107355 9.636736 RESID(-2) 0.020620 0.154456 0.133501 RESID(-3) -0.039154 0.155146 -0.252368 RESID(-4) -0.042845 0.107836 -0.397316 R-squared 0.955242 Mean dependent var Adjusted R-squared 0.952190 S.D dependent var 0.000000 0.000000 Prob 0.1129 0.1255 0.0766 0.0000 0.8941 0.8013 0.6921 5.64E-16 0.049643 Critical value, Chi-square, alpha =5%, p=4, number of lag, order of lag order to find 9.48775 Ho: No autocorrelation, there is a no higher-order autocorrelation H1: Autocorrelation, there is a higher-order autocorrelation than p The Chi-square statistic= Obs*R-squared=95*0.955242=90.74799 higher then critical value 9.48775 at p=4, so we reject H0, there is a higher order autocorrelation than The White Heteroscedasticity test is also provided Explain how to compute the chi-squared value based on provided information and tell us the conclusion of the test? White Heteroskedasticity Test: F-statistic 12.84273 Obs*R-squared 34.52080 Test Equation: Dependent Variable: RESID^2 Included observations: 95 Variable Coefficient C -0.774543 LOG(Y) 0.367448 (LOG(Y))^2 -0.043342 LOG(R) -0.001272 Probability Probability Std Error 0.133687 0.063099 0.007417 0.001284 0.000000 0.000001 t-Statistic -5.793708 5.823390 -5.843390 -0.991102 Prob 0.0000 0.0000 0.0000 0.3243 (LOG(R))^2 R-squared Adjusted R-squared 0.000515 0.363377 0.335082 0.000480 1.072134 Mean dependent var S.D dependent var 0.2865 0.002438 0.001899 Critical value, Chi-square, alpha =5%, number of variables k’=4, so critical value is the same as the previous one, so Chi-square critical is 9.48775 Ho: homoscedasticity Var ( ui ) = σ2 H1: heteroscedasticity Var ( ui ) = σi2 The Chi-square statistic= Obs*R-squared=95*0.363377=34.52080 higher then critical value 9.48775 at k’=4, so we reject H0, there is a heteroscedasticity problem in this model, or we can say that the variance of errors terms are not constant 10 What is your final conclusion of the model? (Synthetic) - This model explains the real demand of money as a positive function of real GDP (Y) and the interest rate as follows the Keynesian model The empirical data of the GDP and interest rate of US explains for 87.36% of the variations of the demand for money M2 - The model has all coefficients statistically significant, and no problem of multicollinearity (for informal test) implying that there is no correlation between the real GDP and the interest rate - This model however has the problem of autocorrelation as well as the problem of heteroskedasticity, imply that the coefficient estimators may not be the BLUE (still unbiased but not having the minimum variance) - We need to correct the problem of heteroskedasticity, as it may influence the tstatistic of the coefficients of GDP and interest rate… Bref: open question for econometrics conclusion or on economical problem Formulae and Free Space 10 ... tstatistic of the coefficients of GDP and interest rate… Bref: open question for econometrics conclusion or on economical problem Formulae and Free Space 10 ... dummy variables This is because: A log of is zero and log of is undefined B Adam Smith decreed that economists should never log a dummy variable C only dummies attempt to log dummy variables 22 In... critical to conclude the overall significant First reason: the F-statistic is very high 318.6748 Second reason: all the t-statistics are statistically significant Third reason: look at the R2 and