APPLIED MACROECONOMETRICS Licenciatura Universidade Nova de Lisboa Faculdade de Economia. FINAL EXAM JUNE 3, 2004 Starts at 14:00 Ends at 16:30

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1 APPLIED MACROECONOMETRICS Licenciatura Universidade Nova de Lisboa Faculdade de Economia FINAL EXAM JUNE 3, 2004 Starts at 14:00 Ends at 16:30 I In Figure I.1 you can find a quarterly inflation rate series (INFLATION). Unit root tests confirm that INFLATION is I(1). Based on several selection criteria, two ARIMA(p,1,q) models were estimated. Results appear in Tables I.1 and I. a) Write down the mathematical form of the first estimated ARIMA model. In your formulas use the values of the estimated mean and variance parameters. b) Write down the mathematical form of the second estimated ARIMA model. In your formulas use the values of the estimated mean and variance parameters. 1

2 c) In Figures I and I.3 appear the correlograms of the residuals and of the squared residuals from the first estimated model. What can you conclude? d) In Table I.3 appears the output from the estimation of an EGARCH model. i) Write down the mathematical form of the estimated model. In your formulas use the values of the estimated mean and variance parameters. ii) What can you conclude from it? 2

3 e) In Table I appears the output from the estimation of a GARCH in mean model. i) Write down the mathematical form of the estimated model. In your formulas use the values of the estimated mean and variance parameters. ii) Several studies have confirmed Friedman s hypothesis that higher inflation rates are associated with higher inflation uncertainty. What can you say about that hypothesis in the present case? 3

4 IN FLATION Figure I.1. Quarterly inflation rate series Table I.1. First estimated ARIMA model Dependent Variable: D(INFLATION) Method: Least Squares Sample: 1951:4 1996:1 Included observations: 178 Convergence achieved after 4 iterations Backcast: 1951:3 Variable Coefficient Std. Error t-statistic Prob. C MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots 0 Table I. Second estimated ARIMA model Dependent Variable: D(INFLATION) Method: Least Squares Sample: 1951:4 1996:1 Included observations: 178 Convergence achieved after 3 iterations Variable Coefficient Std. Error t-statistic Prob. C AR(1) AR(2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots i i 4

5 Figure I. Correlogram of the residuals from the first estimated ARIMA model Figure I.3. Correlogram of the squared residuals from the first estimated ARIMA model 5

6 Table I.3. Estimated EGARCH model Dependent Variable: D(INFLATION) Method: ML - ARCH (Marquardt) Sample: 1951:4 1996:1 Included observations: 178 Convergence achieved after 21 iterations MA backcast: 1951:3, Variance backcast: ON Coefficient Std. Error z-statistic Prob. C MA(1) Variance Equation C RES /SQR[GARCH](1) RES/SQR[GARCH](1) EGARCH(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots 5 Table I. Estimated GARCH in mean model Dependent Variable: D(INFLATION) Method: ML - ARCH (Marquardt) Sample: 1951:4 1996:1 Included observations: 178 Convergence achieved after 23 iterations MA backcast: 1951:3, Variance backcast: ON Coefficient Std. Error z-statistic Prob. GARCH C MA(1) Variance Equation C ARCH(1) GARCH(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots.52 6

7 II In Figures II.1 and II you can find the graphs of private consumption expenditure at constant 1995 market prices (CONS) and of the consumer price index (PRICE). Unit root tests confirm that these series are I(1). Consumers are subject to money illusion when their decisions on a real variable are affected by the level of prices. To test for this possibility, the regression in Table II.1 was estimated. 1. What can you conclude from the results in Table II.1? Explain. 2. In Figure II.3 appears the actual, fitted and residual graph. What can you conclude from it? Explain. 7

8 CO NS Figure II.1 Consumption graph PRICE Figure II Price graph 8

9 Table II.1. Regression of LOG(CONS) on LOG(PRICE) Dependent Variable: LOG(CONS) Method: Least Squares Sample: Included observations: 24 Variable Coefficient Std. Error t-statistic Prob. C LOG(PRICE) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) R es idual Ac tual Fitted Figure II.3. Actual, fitted and residual graph from regression of LOG(CONS) on LOG(PRICE) 9

10 III A. The following questions are based on estimated VAR models with 2 lags for the three-month T-bill rate (TBILL), and the 3-year and 10-year rates on government bonds (R3 and R10 respectively). All models are estimated in the first differences. 1. In Figures III.1 and III you can find the impulse response functions obtained using two different Cholesky orderings. What ordering is implicit in each of these? 2. In Figure III.3 appear the impulse response functions obtained using a structural VAR. Explain in words the restrictions behind this model. 3. Two estimated structural VAR models are presented in Tables III.1 and III. a) Explain in words the structure of each model. 10

11 b) Is it possible to test the validity of the structural restrictions of these models? If yes, do it. 4. Suppose you want to identify the 3 structural shocks in a structural VAR model for the first differences of TBILL, R3 and R10 as follows. The first one is a shock to D(R10). The second is a shock to the medium-to-long-term spread defined as the difference D(R10) D(R3). The third is a shock to the short-to-medium-term spread defined as the difference D(R3) D(TBILL). Explain how you could accomplish this in EViews. Specifically, which restrictions would you need to impose? 11

12 B. In Figure III appear the accumulated impulse response functions of the VAR model estimated using the first differences of the variables as in question III.A. In Figure III.5 appear the impulse response functions of a VAR model with 3 lags estimated using the levels of the variables (i.e. without taking the first differences). 1.The impulse response functions in these figures do not converge to zero. Why? 2. Which of these two estimated models (the VAR in levels or the VAR in first differences) imposes less restrictions? Explain. 3. Assuming that all interest rates (TBILL, R3 and R10) are I(1), under what conditions is it valid to estimate the impulse response functions using the accumulated impulse response functions from the VAR model in first differences? 12

13 C. A VECM model was estimated and the results appear in Table III Write down the estimated cointegration relations. 2. Write down the mathematical form of the estimated model for the three-month T- bill rate equation. In your formulas use the estimated parameters values. 13

14 Response to Cholesky One S.D. Innovations ± 2 S.E. 1 Response of D(TBILL) to D(TBILL ) 1 Response of D(TBILL) to D(R3 ) 1 Response of D(TBILL) to D(R10) Response of D(R3) to D(TBILL ) Response of D(R3) to D(R3) Response of D(R3) to D(R10) Response of D(R10) to D(TBILL ) Response of D(R10) to D(R3) Response of D(R10) to D(R10) Figure III.1. Impulse response functions using a Cholesky ordering 14

15 Response to Cholesky One S.D. Innovations ± 2 S.E. 1 Response of D(TBILL) to D(TBILL ) 1 Response of D(TBILL) to D(R3 ) 1 Response of D(TBILL) to D(R10) Response of D(R3) to D(TBILL ) Response of D(R3) to D(R3) Response of D(R3) to D(R10) Response of D(R10) to D(TBILL ).5 Response of D(R10) to D(R3).5 Response of D(R10) to D(R10) Figure III. Impulse response functions using a different Cholesky ordering 15

16 Response to Structural One S.D. Innovations ± 2 S.E. 2 Response of D(TBILL) to Shock1 2 Response of D(TBILL) to Shock2 2 Response of D(TBILL) to Shock Response of D(R3) to Shock1 Response of D(R3) to Shock2 Response of D(R3) to Shock Response of D(R10) to Shock1 Response of D(R10) to Shock2 Response of D(R10) to Shock Figure III.3. Impulse response functions from a structural VAR 16

17 Table III.1. Estimates of a structural VAR Structural VAR Estimates Sample: 1961:1 1991:4 Included observations: 124 Estimation method: method of scoring (analytic derivatives) Convergence achieved after 12 iterations Structural VAR is over-identified (1 degrees of freedom) Model: Ae = Bu where E[uu']=I Restriction Type: short-run text = = = C(5)*@u3 represents D(TBILL) represents D(R3) represents D(R10) residuals Coefficient Std. Error z-statistic Prob. C(1) C(3) C(2) C(4) C(5) Log likelihood LR test for over-identification: Chi-square(1) Probability 0000 Estimated A matrix: Estimated B matrix: Table III. Estimates of another structural VAR Structural VAR Estimates Sample: 1961:1 1991:4 Included observations: 124 Estimation method: method of scoring (analytic derivatives) Convergence achieved after 10 iterations Structural VAR is just-identified Model: Ae = Bu where E[uu']=I Restriction Type: short-run text = = = C(4)*@e1+C(5)*@e3+C(6)*@u3 represents D(TBILL) represents D(R3) represents D(R10) residuals Coefficient Std. Error z-statistic Prob. C(1) C(4) C(5) C(2) C(3) C(6) Log likelihood Estimated A matrix: Estimated B matrix:

18 Accum ulated Response to Cholesky One S.D. Innovations ± 2 S.E. 1 Accumulated Response of D(TBILL) to D(TBILL ) 1 Accumulated Response of D(TBILL) to D(R3) 1 Accumulated Response of D(TBILL) to D(R10 ) Accumulated Response of D(R3) to D(TBILL) 1 Accumulated Response of D(R3) to D(R3) 1 Accumulated Response of D(R3) to D(R10) Accumulated Response of D(R10) to D(TBILL ) Accumulated Response of D(R10) to D(R3) Accumulated Response of D(R10) to D(R10) Figure III. Accumulated impulse response functions from a VAR in first differences 18

19 Response to Cholesky One S.D. Innovations ± 2 S.E. 1 Response of TBILL to TBILL 1 Response of TBILL to R3 1 Response of TBILL to R Response of R3 to TBILL Resp on se of R3 to R3 Response of R3 to R Response of R10 to TBILL Response of R10 to R3 Response of R10 to R Figure III.5. Impulse response functions from a VAR in levels 19

20 Table III.3. Estimated VECM model Vector Error Correction Estimates Sample(adjusted): 1960:3 1991:4 Included observations: 126 after adjusting endpoints Standard errors in ( ) & t-statistics in [ ] Cointegrating Eq: CointEq1 CointEq2 R10(-1) R3(-1) TBILL(-1) (09694) (06338) [-11563] [-17304] C (07382) (04057) [-02286] [-08157] Error Correction: D(R10) D(R3) D(TBILL) CointEq ( ) (01929) (08002) [ 32397] [ ] [ 38110] CointEq (03642) ( ) (00837) [-39163] [ ] [ ] D(R10(-1)) ( ) (02328) ( ) [ 04342] [ ] [ ] D(R3(-1)) ( ) (03704) ( ) [ 19100] [ 19230] [ 13733] D(TBILL(-1)) ( ) ( ) (01583) [-18590] [ ] [ ] R-squared Adj. R-squared Sum sq. resids S.E. equation F-statistic Log likelihood Akaike AIC Schwarz SC Mean dependent S.D. dependent Determinant Residual Covariance Log Likelihood Log Likelihood (d.f. adjusted) Akaike Information Criteria Schwarz Criteria

21 IV In Figures IV.1 and IV you can find the graph of two series: Y and X. In Tables IV.1 to IV appear the results of several unit root tests for these two variables. Classify each series in terms of: a) stationarity; b) stochastic and/or deterministic trends; 21

22 Y Figure IV.1. Graph of Y X Figure IV. Graph of X 22

23 Table IV.1. Unit root test on Y Null Hypothesis: Y has a unit root Exogenous: Constant Lag Length: 1 (Automatic based on SIC, MAXLAG=5) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y) Method: Least Squares Sample(adjusted): Included observations: 22 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. Y(-1) D(Y(-1)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Table IV. Another unit root test on Y Null Hypothesis: Y has a unit root Exogenous: Constant, Linear Trend Lag Length: 2 (Automatic based on SIC, MAXLAG=5) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y) Method: Least Squares Sample(adjusted): Included observations: 21 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. Y(-1) D(Y(-1)) D(Y(-2)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

24 Table IV.3. Unit root test on X Null Hypothesis: X has a unit root Exogenous: Constant Lag Length: 5 (Automatic based on SIC, MAXLAG=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(X) Method: Least Squares Sample(adjusted): 1960:3 2001:1 Included observations: 163 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. X(-1) D(X(-1)) D(X(-2)) D(X(-3)) D(X(-4)) D(X(-5)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Table IV. Another unit root test on X Null Hypothesis: X has a unit root Exogenous: Constant, Linear Trend Lag Length: 5 (Automatic based on SIC, MAXLAG=12) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(X) Method: Least Squares Sample(adjusted): 1960:3 2001:1 Included observations: 163 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. X(-1) D(X(-1)) D(X(-2)) D(X(-3)) D(X(-4)) D(X(-5)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)