VIEWS: 13 PAGES: 13 POSTED ON: 8/24/2011
Introduction to Applied Econometrics National Graduate Institute for Policy Studies Associate Prof. Wade Pfau Computer Exercise #3 “Introducing EViews” Your Assignment: You should write solutions to the following questions. Though you can discuss the assignment with others, you must write the solutions by yourself. Do not share solutions! You should include the printed output for any EViews commands you used. I will inform you of the due date during our class meeting. Working with Macroeconomic Time Series Data in EViews Computer Exercise #3 and #4 will use the same dataset. We will build towards estimating a VAR model. A popular VAR model used by applied macroeconomists is the RMPY. This model includes R (interest rate), M (money supply), P (price level), and Y (real GDP). Monetary economists and macroeconomists have spent a lot of time developing economic theories to explain the relationships between these variables: The Aggregate Demand / Aggregate Supply Model, or the Phillip’s Curve model, provides a way to link all four variables. The IS-LM model modified to include inflation can also link all four variables. The Fisher equation (i=r+) implies a positive relationship between R (which is i in the Fisher equation) and P. The liquidity preference theory indicates that the money supply is inversely related to the interest rate. The VAR does not rely on theory when developing the econometric specification. The VAR only states that the four variables can be related to each other: each variable is explained by lags of itself and lags of the other variables. But we can use the results of the VAR estimation to comment about various economic theories: Milton Friedman’s theory that “Inflation is always and everywhere a monetary phenomena” (Does M Granger Cause P? Is M the only variable to Granger Cause P?) Is the real economy affected by inflation? (does P Granger Cause Y?) Do changes in money growth affect the business cycle? (does M Granger Cause Y? Note, Keynesian theory says yes, but Monetarist theory says no.) What is the relationship between M and R, between R and Y, between R and P? Page 1 The file RMPY.csv contains quarterly data on the variables for the US from 1947Q1 to 1992Q4. More specifically: R is the three month Treasury bill rate M is the money supply (M2) measured in billions of dollars P is the price level measured by the GDP deflator (a price index with 1987 = 1.00) Y is the real GDP measured in billions of 1987 dollars 1. Import the data into EViews and examine the data. How many observations are in the dataset? 2. Generate a variable, nomY, which is the nominal GDP measures in billions of dollars. 3. Generate the logarithms, differences, and percent changes for these four variables. 4. What is the average quarterly inflation rate during this time period? What is the inflation rate on an annualized basis? 5. Make time series graphs for real GDP, and for the real GDP growth rates. Comment about the appearance of these graphs. 6. Examine the autocorrelation functions (with 10 lags) for real GDP and real GDP growth rates. Comment about what you find. 7. Determine the appropriate AR(p) and whether a deterministic time trend should be included for each of the four variables in logged form. Page 2 (Optional introductory exercise) If you are taking “Monetary Economics – Money and Banking”, you may be interested in looking at this dataset in relation to the Mishkin textbook. The book includes a variety of graphs showing the relationships between these variables. Can you re-create the graphs? Can you develop a set of hypotheses to relate the cause and effect patterns between these variables? (a) Check for Stationarity. To work with a VAR, you must use stationary data. So the first step is to check if each of these variables has a unit root. But first, for each variable we need to decide on the appropriate AR(p) model and decide whether to include a deterministic trend. We also need to decide whether to use logged data. Find the appropriate specification for each variable (R, M, P, Y) and then test for a unit root. (b) Check for Stationarity in the differenced data. In (a) you should have found that unit roots exist for all of the variables. We will also assume that cointegration does not occur. So we will work with differenced data. Test to make sure that the data is indeed difference stationary. (c) Estimating a VAR. Calculate the VAR(1) model, including a time trend. -Do your results match those in Koop’s Table 11.4 (page 197). -Discuss Granger causality using the method described in Koop. -Then use the vargranger command to see Stata’s Granger Causality Tests. Does the method for testing Granger Causality discussed in Koop match the Stata procedure? (d) Estimating a VAR. Calculate the VAR(2), including a time trend. -Do your results match Koop’s Table 11.5? -Discuss Granger Causality using the Stata procedure. -Are there differences in Granger Causality between using a VAR(1) and VAR(2)? (e) Choosing the Optimal Lag Length. -Use the method described in Koop to determine the appropriate p for this VAR(p) model. Use a pmax of 8. -Use the varsoc command in Stata for determining the appropriate lag length with pmax of 8. What are the results of Stata? What lag length do you think we should use? (f) Forecasting. In order to match up our results with Koop, let’s work with the VAR(2) model. -Estimate the VAR(2) with time trend for the data through 1991:4. Then use Stata to forecast data for the year 1992. Compare the results to the actual data. -Do the same thing in Excel. (g) Impulse Response Functions. Examine the impulse response functions for the VAR(2) model. How can we interpret these results? What are the implications for monetary policy? Page 3 Comments on Computer Exercise #4 (a) First, you need to decide whether or not to use logged data. Usually, you should used logged data when working with macroeconomic time series variables. I will do the unit root tests first without taking the logs of the data, and then with taking the logs of the data. Hopefully, the results will be the same both ways. In Lecture 9, we discussed this procedure using a simple method. First we choose a reasonable pmax and include a deterministic trend, and then we check the significance of the final pth coefficient. If it is not significant, we re-estimate with p-1, and we repeat this process until we find a final coefficient that is significant. Once we decide on the value of p, we check to see if the deterministic trend is significant. If not, then we re-estimate the model without the deterministic trend. Then we are ready for the unit root test. This is the procedure we will follow in Stata. Remember, we are estimating an equation of the form: Yt Yt 1 1Yt 1 ... p1Yt p 1 t et which means that the number of differenced terms we include in the regression is one less than the AR(p) we are estimating. For example, if we created d_y=D.y and obs is our time trend, then an AR(8) model would be estimated as: reg d_y L.y L(1/7).d_y obs In the case of R, we find that the 7th differenced lag is significant, so we want an AR(8) model. The time trend is not significant (p-value is .444) so we do not include it. Then, we are ready to use the Dickey-Fuller test. We can see that we cannot reject the null hypothesis of a unit root at 5% significance, because the t-statistic for the unit root is -1.78, and the 5% critical level is -2.88, and we only reject the null hypothesis if the test statistic is more negative than the critical level. We have found evidence that R has a unit root. Following the procedure I just described, we obtain: Data in Levels # of AR(p) include differenced model t-stat unit variable time trend? terms for 5% tcv root? R no 7 8 -1.78 -2.88 yes M yes 7 8 -2.39 -3.44 yes P yes 3 4 -1.72 -3.44 yes Y yes 1 2 -2.16 -3.44 yes Again, the null hypothesis for the unit root test is that we have a unit root. We reject the null hypothesis when the t-statistic is more negative than the critical value. If we reject the null hypothesis, we have a stationary time series. If we cannot reject the null hypothesis, we have a unit root. Here are the unit root tests for the logged data: Page 4 Data in Logs # of AR(p) include differenced model t-stat unit variable time trend? terms for 5% tcv root? log_R no 6 7 -2.23 -2.88 yes log_M yes 5 6 -2.17 -3.44 yes log_P yes 2 3 -1.78 -3.44 yes log_Y yes 1 2 -2.28 -3.44 yes (Note) Since you found unit roots for all the variables, actually the next appropriate step is to check for cointegration among the variables. This topic is discussed in Koop, Chapter 10. We did not have time to discuss this in class. We will not check for cointegration, which means that we could potentially be ignoring some valuable information about the long-run relationships between the variables. We will work with differenced data, so our VAR will use can only describe short run relationships. The Stata code for completing everything so far would look something like this: clear insheet using "C:\rmpy.csv", clear gen obs=_n gen dates=obs-4*(1960-1947)-1 tsset dates, quarterly format dates %tq foreach name in r m p y { gen log_`name'=log(`name') gen d_`name'=d.`name' gen pc_`name'=100*d.log_`name' foreach num of numlist 7(-1)1 { * AR(p) model specification for data in levels: * reg d_`name' L.`name' L(1/`num').d_`name' obs * reg d_`name' L.`name' L(1/`num').d_`name' * AR(p) model specification for logged data: reg pc_`name' L.log_`name' L(1/`num').pc_`name' obs reg pc_`name' L.log_`name' L(1/`num').pc_`name' } } dfuller r, lags(7) reg dfuller m, lags(7) trend reg dfuller p, lags(3) trend reg dfuller y, lags(1) trend reg dfuller log_r, lags(6) reg dfuller log_m, lags(5) trend reg dfuller log_p, lags(2) trend reg dfuller log_y, lags(1) trend reg Page 5 (b) If we are thinking in terms of levels, we should use differenced data. If we are thinking in terms of logs, we should use percent changes. In order to match the method used by Koop in Chapter 11, let’s use percent changes (but notice that Koop follows his usual procedure of using the name of “differences” for percent changes). Data in percent changes include difference AR(p) t-stat unit variable time trend? terms model for 5% tcv root? pc_R no 2 3 -6.96 -2.88 no pc_M no 0 1 -4.71 -2.88 no pc_P no 1 2 -4.1 -2.88 no pc_Y no 0 1 -9.04 -2.88 no We can see that none of the variables in percent change terms have unit roots. They are all stationary. So we will be able to use OLS on each equation of the VAR (which is what Stata does). My Stata code for this part was: foreach name in r m p y { gen pc2_`name'=d2.log_`name' foreach num of numlist 2(-1)1 { reg pc2_`name' L.pc_`name' L(1/`num').pc2_`name' obs reg pc2_`name' L.pc_`name' L(1/`num').pc2_`name' } } reg pc2_m L.pc_m obs reg pc2_m L.pc_m reg pc2_y L.pc_m obs reg pc2_y L.pc_m dfuller pc_r, lags(2) reg dfuller pc_m, lags(0) reg dfuller pc_p, lags(1) reg dfuller pc_y, lags(0) reg (c) var pc_r pc_m pc_p pc_y, lags(1) exog(obs) Vector autoregression Sample: 1947q3 1992q4 No. of obs = 182 Log likelihood = -1252.744 AIC = 14.03015 FPE = 14.56671 HQIC = 14.20143 Det(Sigma_ml) = 11.18873 SBIC = 14.45266 Equation Parms RMSE R-sq chi2 P>chi2 ---------------------------------------------------------------- pc_r 6 13.5847 0.2277 53.65175 0.0000 pc_m 6 .542227 0.6574 349.2755 0.0000 pc_p 6 .56744 0.4533 150.9273 0.0000 pc_y 6 .920751 0.2026 46.23159 0.0000 Page 6 ---------------------------------------------------------------- ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- pc_r | pc_r | L1 | .221877 .0730766 3.04 0.002 .0786494 .3651046 pc_m | L1 | 3.390594 1.219471 2.78 0.005 1.000475 5.780714 pc_p | L1 | 1.778738 1.444631 1.23 0.218 -1.052686 4.610162 pc_y | L1 | 3.224263 1.098898 2.93 0.003 1.070462 5.378064 obs | -.0561754 .0214133 -2.62 0.009 -.0981447 -.014206 _cons | -3.518505 2.561094 -1.37 0.169 -8.538156 1.501147 -------------+---------------------------------------------------------------- pc_m | pc_r | L1 | -.012993 .0029168 -4.45 0.000 -.0187098 -.0072761 pc_m | L1 | .7494569 .0486745 15.40 0.000 .6540566 .8448572 pc_p | L1 | .0606078 .0576616 1.05 0.293 -.0524069 .1736225 pc_y | L1 | -.0315711 .0438619 -0.72 0.472 -.1175388 .0543967 obs | .0003412 .0008547 0.40 0.690 -.001334 .0020164 _cons | .3347869 .1022246 3.28 0.001 .1344303 .5351435 -------------+---------------------------------------------------------------- pc_p | pc_r | L1 | .0099351 .0030524 3.25 0.001 .0039524 .0159178 pc_m | L1 | .1206095 .0509378 2.37 0.018 .0207731 .2204458 pc_p | L1 | .5190142 .0603429 8.60 0.000 .4007444 .637284 pc_y | L1 | -.0387775 .0459015 -0.84 0.398 -.1287427 .0511878 obs | .0018118 .0008944 2.03 0.043 .0000587 .0035648 _cons | .1570825 .106978 1.47 0.142 -.0525906 .3667555 -------------+---------------------------------------------------------------- pc_y | pc_r | L1 | .0003811 .004953 0.08 0.939 -.0093266 .0100888 pc_m | L1 | .283097 .0826537 3.43 0.001 .1210987 .4450952 pc_p | L1 | -.1168863 .0979146 -1.19 0.233 -.3087955 .0750229 pc_y | L1 | .3085509 .0744815 4.14 0.000 .1625699 .4545319 obs | -.0031337 .0014514 -2.16 0.031 -.0059783 -.0002891 _cons | .5017515 .1735866 2.89 0.004 .1615279 .841975 ------------------------------------------------------------------------------ Compare carefully. We can see that, except for minor differences in rounding, these results do match Koop’s table 11.4. Regarding Granger Causality, we can see with the VAR(1) model that an explanatory variable Granger Causes the dependent variable with 5% significance if its coefficient is significant at the 5% Page 7 significance level. We have to do hypothesis tests to see Granger causality. Stata gives us the p-values for coefficients. Looking at p-values, we can see that, at 5 percent significance, in addition to each variable’s own lag being significant, M(+) and Y(+) Granger cause R R(-) Granger Causes M R(+) and M(+) Granger Causes P M(+) Granger Causes Y Using the “vargranger” command after running the above VAR model, we obtain the same results: vargranger Granger causality Wald tests +------------------------------------------------------------------+ | Equation Excluded | chi2 df Prob > chi2 | |--------------------------------------+---------------------------| | pc_r pc_m | 7.7305 1 0.005 | | pc_r pc_p | 1.516 1 0.218 | | pc_r pc_y | 8.6089 1 0.003 | | pc_r ALL | 24.214 3 0.000 | |--------------------------------------+---------------------------| | pc_m pc_r | 19.843 1 0.000 | | pc_m pc_p | 1.1048 1 0.293 | | pc_m pc_y | .51809 1 0.472 | | pc_m ALL | 26.92 3 0.000 | |--------------------------------------+---------------------------| | pc_p pc_r | 10.594 1 0.001 | | pc_p pc_m | 5.6064 1 0.018 | | pc_p pc_y | .71368 1 0.398 | | pc_p ALL | 15.251 3 0.002 | |--------------------------------------+---------------------------| | pc_y pc_r | .00592 1 0.939 | | pc_y pc_m | 11.731 1 0.001 | | pc_y pc_p | 1.4251 1 0.233 | | pc_y ALL | 12.222 3 0.007 | +------------------------------------------------------------------+ (Think about what these results imply for monetary theory) (d) Estimating the VAR(2) model with a time trend. We obtain: var pc_r pc_m pc_p pc_y, lags(1/2) exog(obs) Vector autoregression Sample: 1947q4 1992q4 No. of obs = 181 Log likelihood = -1204.875 AIC = 13.75553 FPE = 11.07257 HQIC = 14.0421 Det(Sigma_ml) = 7.113764 SBIC = 14.46238 Equation Parms RMSE R-sq chi2 P>chi2 ---------------------------------------------------------------- pc_r 10 12.0595 0.3421 94.1107 0.0000 pc_m 10 .533896 0.6767 378.8909 0.0000 pc_p 10 .547858 0.5047 184.4367 0.0000 pc_y 10 .907615 0.2471 59.39752 0.0000 Page 8 ---------------------------------------------------------------- ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- pc_r | pc_r | L1 | .3148494 .069509 4.53 0.000 .1786143 .4510845 L2 | -.3458243 .0709026 -4.88 0.000 -.4847909 -.2068577 pc_m | L1 | 2.823967 1.688882 1.67 0.095 -.4861806 6.134115 L2 | -2.20095 1.709912 -1.29 0.198 -5.552316 1.150416 pc_p | L1 | 3.049076 1.573373 1.94 0.053 -.0346791 6.132831 L2 | 1.163968 1.517259 0.77 0.443 -1.809805 4.137741 pc_y | L1 | 3.696047 1.005351 3.68 0.000 1.725596 5.666499 L2 | 1.085008 1.021413 1.06 0.288 -.9169245 3.086941 obs | -.0451783 .0198965 -2.27 0.023 -.0841747 -.0061818 _cons | -3.909845 2.388825 -1.64 0.102 -8.591857 .7721668 -------------+---------------------------------------------------------------- pc_m | pc_r | L1 | -.0167695 .0030773 -5.45 0.000 -.0228009 -.0107381 L2 | .0033744 .003139 1.07 0.282 -.0027779 .0095267 pc_m | L1 | .6552472 .0747697 8.76 0.000 .5087012 .8017932 L2 | .1574915 .0757008 2.08 0.037 .0091207 .3058623 pc_p | L1 | -.0196234 .069656 -0.28 0.778 -.1561466 .1168998 L2 | .0951263 .0671717 1.42 0.157 -.0365278 .2267804 pc_y | L1 | -.0506719 .0445086 -1.14 0.255 -.1379072 .0365634 L2 | .0356292 .0452197 0.79 0.431 -.0529999 .1242582 obs | -.0002324 .0008809 -0.26 0.792 -.0019588 .0014941 _cons | .2618382 .1057575 2.48 0.013 .0545574 .469119 -------------+---------------------------------------------------------------- pc_p | pc_r | L1 | .0093898 .0031578 2.97 0.003 .0032007 .0155789 L2 | -.0008624 .0032211 -0.27 0.789 -.0071756 .0054508 pc_m | L1 | .0856009 .076725 1.12 0.265 -.0647774 .2359792 L2 | .0249308 .0776804 0.32 0.748 -.12732 .1771817 pc_p | L1 | .3660449 .0714775 5.12 0.000 .2259515 .5061383 L2 | .2822199 .0689283 4.09 0.000 .147123 .4173169 pc_y | L1 | -.0097744 .0456726 -0.21 0.831 -.099291 .0797422 L2 | -.0462155 .0464023 -1.00 0.319 -.1371623 .0447313 obs | .0011716 .0009039 1.30 0.195 -.0006 .0029432 _cons | .1103431 .1085231 1.02 0.309 -.1023583 .3230445 -------------+---------------------------------------------------------------- pc_y | pc_r | L1 | .0023011 .0052313 0.44 0.660 -.0079522 .0125543 L2 | -.0095239 .0053362 -1.78 0.074 -.0199828 .0009349 pc_m | L1 | .3102452 .1271074 2.44 0.015 .0611192 .5593712 L2 | -.0937597 .1286902 -0.73 0.466 -.3459878 .1584684 Page 9 pc_p | L1 | .0738481 .1184141 0.62 0.533 -.1582393 .3059355 L2 | -.2328913 .1141908 -2.04 0.041 -.4567013 -.0090814 pc_y | L1 | .2698176 .075664 3.57 0.000 .1215189 .4181164 L2 | .153286 .0768729 1.99 0.046 .002618 .303954 obs | -.0025195 .0014974 -1.68 0.092 -.0054545 .0004154 _cons | .5185362 .1797861 2.88 0.004 .166162 .8709104 ------------------------------------------------------------------------------ . vargranger Granger causality Wald tests +------------------------------------------------------------------+ | Equation Excluded | chi2 df Prob > chi2 | |--------------------------------------+---------------------------| | pc_r pc_m | 2.802 2 0.246 | | pc_r pc_p | 8.1358 2 0.017 | | pc_r pc_y | 17.501 2 0.000 | | pc_r ALL | 31.675 6 0.000 | |--------------------------------------+---------------------------| | pc_m pc_r | 29.82 2 0.000 | | pc_m pc_p | 2.2976 2 0.317 | | pc_m pc_y | 1.5789 2 0.454 | | pc_m ALL | 39.122 6 0.000 | |--------------------------------------+---------------------------| | pc_p pc_r | 8.859 2 0.012 | | pc_p pc_m | 4.1246 2 0.127 | | pc_p pc_y | 1.2081 2 0.547 | | pc_p ALL | 15.169 6 0.019 | |--------------------------------------+---------------------------| | pc_y pc_r | 3.2269 2 0.199 | | pc_y pc_m | 8.4903 2 0.014 | | pc_y pc_p | 4.4325 2 0.109 | | pc_y ALL | 22.056 6 0.001 | +------------------------------------------------------------------+ Again, we are able to see a match with Koop Table 11.5, aside from some differences in rounding off numbers. Using Stata’s Granger Causality Tests, we find a few differences from before. It is still the case that only R Granger Causes M, and only M Granger Causes Y. But now, P and Y Granger Cause R, and only R Granger Causes P. By including a second lag, M lost its ability to Granger Cause R and P, once we hold other factors constant. Meanwhile, P gained the ability to Granger cause R. For your reference, here is a more complete list of Granger Causality results at the 5 percent significance level: VAR(1) VAR(2) VAR(3) VAR(5) Dependent Explanatory Hypothesized VAR(1) Granger Granger Granger Granger Variable Variable Sign sign Cause? Cause? Cause? Cause? DR DM - + Yes No No No DR DP + + No Yes No (close) No Page 10 DR DY + + Yes Yes Yes No DM DR none or - - Yes Yes Yes Yes DM DP unclear + No No No (close) No DM DY unclear - No No No No DP DR + + Yes Yes No No DP DM + + Yes No Yes No DP DY - - No No No No DY DR unclear + No No No No DY DM + or none + Yes Yes Yes No DY DP none or - - No No Yes Yes (e) Using a p of 8 and including a time trend, we do find a significant coefficient among the 8th lags. Thus, by Koop’s criteria, we should use a lag of 8. But Koop’s criteria is only a simple criteria that uses skills we have learned in the course. There are many more sophisticated techniques we could use. With a pmax of 8, we obtain the following: . varsoc pc_r pc_m pc_p pc_y, maxlag(8) exog(obs) Selection order criteria Sample: 1949q2 1992q4 Number of obs = 175 +---------------------------------------------------------------------------+ |lag | LL LR df p FPE AIC HQIC SBIC | |----+----------------------------------------------------------------------| | 0 | -1335.78 54.9273 15.3575 15.4162 15.5022 | | 1 | -1183.37 304.83 16 0.000 11.5545 13.7985 13.9745 14.2325* | | 2 | -1148.05 70.64 16 0.000 9.26895 13.5777 13.8711* 14.3011 | | 3 | -1131.59 32.91 16 0.008 9.22892* 13.5725 13.9833 14.5852 | | 4 | -1118.18 26.828 16 0.043 9.52047 13.602 14.1302 14.9041 | | 5 | -1099.47 37.424 16 0.002 9.25234 13.571* 14.2166 15.1625 | | 6 | -1089.35 20.226 16 0.210 9.93107 13.6383 14.4012 15.5191 | | 7 | -1075.85 27.013* 16 0.041 10.2675 13.6668 14.5471 15.837 | | 8 | -1064.67 22.36 16 0.132 10.9179 13.7219 14.7195 16.1814 | +---------------------------------------------------------------------------+ Endogenous: pc_r pc_m pc_p pc_y Exogenous: obs _cons Unfortunately, the results are mixed. Different criteria suggest optimal lag lengths of 1, 2, 3, 5, and 7. What should we do? It’s not clear. We will continue assuming that a lag length of 2 is optimal for the rest of the assignment. (f) Steps to forecast the VAR(2) model with time trend in Stata: 1. estimate the VAR(2) with time trend on the sample 1947:1 to 1991:4 var pc_r pc_m pc_p pc_y if dates<=q(1991q4), lags(1/2) exog(obs) 2. compute the forecasts for 1992 to be compared to the actual values of 1992. fcast compute var2_1991, step(4) Page 11 3. list the results list dates pc_* var2* if dates>=q(1991q1), sep(4) Actual data for 1992 Forecasts dates pc_r pc_m pc_p pc_y var2_199~r var2_19~m var2_19~p var2_199~y 1992q1 -15.36699 .7966042 .928919 .8651733 -11.158003 1.034592 .62643653 -.01968567 1992q2 -5.634439 .0760078 .6889954 .6984711 -4.6516364 1.2538859 .73161304 .21987159 1992q3 -17.69122 .2095222 .2891243 .8378029 -4.2362703 1.3529998 .86177913 .27538361 1992q4 -.4323006 .6717682 .8133203 1.392746 -5.6864829 1.4720951 .94033463 .27067321 We can compare with Koop for the variables he includes on page 202 of the 2nd edition.. pc_p and pc_y are the actual values for quarterly percent changes in 1992. var2_1991pc_p and var2_1991pc_y are the forecasts. We can see that the forecasts match Koop. To make the forecasts in Excel, we need to follow the forecasting approach described in Koop. Making forecasts in Excel is not straightforward. We have to make a new spreadsheet whenever we have a different number of endogenous and/or exogenous variables or a different VAR(p). I will show you a spreadsheet for the VAR(2) case that we have used. You can find the spreadsheet on the class webpage. You can use the calculation formulas in the spreadsheet as a model if you ever wish to forecast a VAR using Excel in the future. (i) The impulse response functions show you how a one-time positive shock to one of the endogenous variables effects not only that variable, but is also transmitted to all the other endogenous variables in the VAR through the lag structure of the model. If the error terms are uncorrelated with each other, then interpreting the impulse response is straightforward: a shock to the i-th error term will simply effect the i-th endogenous variable directly, and then feed through the whole system in subsequent periods. But the error terms across the VAR are usually correlated. Using linear algebra, a transformation (such as the Cholesky decomposition) can be made to the residuals so that they become uncorrelated. For our VAR(2) model with time trend, we obtain the impulse response functions below. I will include the graphs of impulse responses from EViews, because I think they look much nicer than the graphs from Stata. However, here is some Stata code to see impulse response functions in Stata: irf set results var pc_r pc_m pc_p pc_y, lags(1/2) exog(obs) irf create results irf graph oirf, impulse(pc_r pc_m pc_p pc_y) response (pc_r) title(How Increases in R M P Y Affect Interest Rates) irf graph oirf, impulse(pc_r pc_m pc_p pc_y) response (pc_m) title(How Increases in R M P Y Affect Money Growth) irf graph oirf, impulse(pc_r pc_m pc_p pc_y) response (pc_p) title(How Increases in R M P Y Affect Inflation) irf graph oirf, impulse(pc_r pc_m pc_p pc_y) response (pc_y) title(How Increases in R M P Y Affect Real GDP Growth) Page 12 Response to Cholesky One S.D. Innovations Response of DR to DR Response of DR to DM Response of DR to DP Response of DR to DY 16 16 16 16 12 12 12 12 8 8 8 8 4 4 4 4 0 0 0 0 -4 -4 -4 -4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Response of DM to DR Response of DM to DM Response of DM to DP Response of DM to DY .5 .5 .5 .5 .4 .4 .4 .4 .3 .3 .3 .3 .2 .2 .2 .2 .1 .1 .1 .1 .0 .0 .0 .0 -.1 -.1 -.1 -.1 -.2 -.2 -.2 -.2 -.3 -.3 -.3 -.3 -.4 -.4 -.4 -.4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Response of DP to DR Response of DP to DM Response of DP to DP Response of DP to DY .6 .6 .6 .6 .5 .5 .5 .5 .4 .4 .4 .4 .3 .3 .3 .3 .2 .2 .2 .2 .1 .1 .1 .1 .0 .0 .0 .0 -.1 -.1 -.1 -.1 -.2 -.2 -.2 -.2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Response of DY to DR Response of DY to DM Response of DY to DP Response of DY to DY 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Let’s just consider the “Response of DY to DM.” What is we see if that we increase money growth for one time period, at time period 0, then the growth rate of Y will start to increase, and Y will grow more quickly than other wise for about one year (four quarters). Then the effects of the expansionary monetary policy stimulus die out, or are even slightly negative. This is support for the Keynesian position that monetary policy can affect the real GDP. What other interesting results do you find? Page 13