The file COINT PPP.XLS contains monthly values of the Japanese, Canadian and Swiss consumer price levels and the bilateral exchange rates with the United States. The file also contains the U.S. consumer price level. The names on the individual series should be self-evident. For example, JAPANCPI is the Japanese price level and JAPANEX is the bilateral Japanese/U.S. exchange rate. The starting date for all variables is January 1974 while the availability of the variables is such that most near the end of 2013. The price indices have been normalized to equal 100 in January 1973 and only the U.S. price index is seasonally adjusted. (a) Form the log of each variable and pretest each for a unit root. Can the null hypothesis of a unit root be rejected for any of the series? How might you proceed if you found that the U.S. CPI was trend stationary? (b) Form the log of each variable. Estimate the long-run relationship between Japan the U.S. as log(japanex) = 9.97 (27.25) − −0.98 (0.104) log(japancpi) − 0.768 (−17.05) log(uscpi) i. Do the point estimates of the slope coefficients seem to be consistent with long-run PPP? ii. From the t-statistics, can you conclude that the Japanese CPI is not significant at the 5% level? (c) Let ut denote the residuals from the long-run relationship. Use these residuals to perform the Engle-Granger test for cointegration. If you use eleven lagged changes, you should find ∆ut = −0.025ut−1 + X 11 i=1 ai∆ut−1 + εt • The t-statistic on the coefficient for ut−1 is −3.44. From Table C, with three variables and 457 usable observations, the 5% and 10% critical values are about −3.760 and −3.464, respectively. Do you conclude that long-run PPP fails? (d) Repeat parts (i) and (ii) using Canada and Switzerland. If you use the residuals from the long-run equilibrium relationships you should find Canada (10 lags) ∆ut = −0.012ut−1 + Xai∆ut−i + εt ; t-stat = -1.89 Switzerland (10 lags) ∆ut = −0.027ut−1 + Xai∆ut−1 + εt ; t-stat = -3.02. 2 (e) Although (at conventional significance levels) we reject the null hypothesis of long-run PPP between Japan and the United States, estimate the error-correction model for ljapanext . If you use 11 lagged changes of each variable, you should find ∆ljapanext = −0.005 − 0.030ˆεt−1 + Σ∆β1ljapanext−i + Σ∆β2ljapancpit−i + Σ∆β3luscpit−i -.004 -.002 .000 .002 .004 .006 .008 2 4 6 8 10 12 14 16 18 20 22 24 LNUSCPI LNJAPANCPI LNJAPANEX Response of LNUSCPI to Innovations -.001 .000 .001 .002 .003 .004 .005 2 4 6 8 10 12 14 16 18 20 22 24 LNUSCPI LNJAPANCPI LNJAPANEX Response of LNJAPANCPI to Innovations -.02 -.01 .00 .01 .02 .03 .04 2 4 6 8 10 12 14 16 18 20 22 24 LNUSCPI LNJAPANCPI LNJAPANEX Response of LNJAPANEX to Innovations Response to Cholesky One S.D. (d.f. adjusted) Innovations Figure 1: (solid) U.S. Price Shock, (short dash) Japanese Price Shock, (long dash) Ex Rate Shock • where ˆεt−1 is the residual from the equilibrium relationship above and eleven lagged changes are used for each variable. The t-statistic on the error correction term is −3.54. Which of the variable(s) can be said to be weakly exogenous? (f) Obtain the impulse response functions using the ordering uscpit → ljapancpit → ljapanext . As in Figure above, you should find that the U.S. price shock has little effect on the exchange rate but that the shock to the Japanese price level causes the yen to depreciate. The response of the exchange rate to its own shock is immediate and permanent. (g) Are the results of the cointegration test sensitive to the normalization (i.e. which of the variables is used as the ’dependent’ variable) used in the equilibrium regression? 3 2. In the previous question, you were asked to use the Engle-Granger procedure test for PPP among the variables log(canex), log(cancpi), and log(uscpi). (a) Now use the Johansen methodology and constrain the constant to the cointegreating vector to obtain: Rank λi λmax λtrace 1 0.0535 25.647 35.987 2 0.0138 6.460 10.339 3 0.0083 3.879 3.879 • Use the table to show that there is a cointegrating vector. (b) Consider the estimated cointegrating vector: −0.949 log(canex) − 6.484 log(cancpi) + 1.600 log(uscpi) + 31.653 = 0 • Normalize with respect to the exchange rate. Does the long-run relationship seem to be consistent with PPP? 4 3. Consider the VAR(1): xt yt = 0.4 0.3 0.8 0.6 xt−1 yt−1 + ε1,t ε2,t where {εt} is a vector white noise process. (a) How can you verify that xt and yt are cointegrated? (b) Write this model in error correction form. (c) Compute the speed of adjustment coefficient α and the cointegrating vector β where the β on xt is normalized to 1. 5 4. Consider the VAR(1): xt yt = 0.625 −0.3125 −0.75 0.375 xt−1 yt−1 + ε1,t ε2,t where {εt} is a vector white noise process. (a) How can you verify that xt and yt are cointegrated? (b) Write this model in error correction form. (c) Compute the speed of adjustment coefficient α and the cointegrating vector β where the β on xt is normalized to 1. 6 5. Suppose you estimate π to be π = 0.60 −0.50 0.20 0.30 −0.25 0.10 1.20 −1.00 0.40 (a) Show that the determinant of π is zero. (b) Show that two of the characteristic roots are zero and that the third is 0.75. (c) β 0 = (3 − 2.51) be the single cointegrating vector normalized with respect to x3t . Find the (3 × 1) vector α such that π = αβ0 . How would α change if you normalized β with respect to x1t? (d) Describe how you could test the restriction β1 + β2 = 0. (e) Now suppose you estimate π to be π = 0.80 0.40 0.00 0.10 0.10 0.00 0.75 0.25 0.50 (f) Show that the three characteristic roots are 0.0, 0.5, and 0.9. (g) Select β such that β = 0.80 0.75 0.40 0.25 0.00 0.50 (h) Find the (3 × 2) matrix α such that π = αβ0 . 7 6. An econometrician estimates a vector autoregression for two variables x and y using one lag of each variable. The estimated VAR is yt xt = 0.8 0.1 −0.2 0.5 yt−1 xt−1 + εt et The variance covariance matrix of the shocks et and ut is known to be diagonal. (a) Write a computer program to, compute the impulse response functions out to ten periods for a shock to εt , as well as et . (Hint: Simulate the following difference equation) yt xt = A t 1 0 , and yt xt = A t 0 1 t = 1, 2, . . . , 10, where A = 0.8 0.1 −0.2 0.5 (b) Plot the four impulse response functions for t = 1, . . . , 10.