Homework 6 100 points There are two problems in this homework Problem 1 should be submited in the standard format we ve used many times The first pages should contain your answers to all the questions along with showing any key algebraic equations or explanations you need to use along the way After that include a printout of the output from the regressions you executed in support of your answers Highlight any numbers in this output that you used in the first section You are encouraged to save paper here you may print this section with a small font double sided and or with 2 up format Last include a copy of the DO file that contains the commands you asked STATA to execute Be sure you organize these in a way that will be clear to the reader Problem 2 will be presented in a slightly different more polished style Please refer to that specific problem for details Problem 1 40 points total Part 1 The dataset Phillips dta contains information on the civilian unemployment rate and the cpi inflation rate The data is available from 1948 through 2010 and was downloaded via FRED a import the dataset and let STATA know it is time series with the command tsset year b Estimate an AR 1 model fot the unemployment rate That is regress unemp on a single lag of unemp Do this for observations of the year 1951 You should use the lag operator in STATA to do this L unemp c Calculate the BIC with the command estat ic d Repeat b and c for an AR 2 and AR 3 e 5 points Use the BIC to determine what is the optimal lag length q for unemployment in this AR q The AR 2 seems optimal as it has the lowest BIC 182 799 f 5 points What is the predicted unemployment rate for 2011 according to the AR 1 model Use the coefficients and the data to calculate the expected value for 2011 What is it for the AR 2 and AR 3 models Predicted 2011 unemployment from the models using the data on unemployment from 2010 2008 in the forecast AR 1 1 229 796653 9 6 8 877 Similarly AR 2 8 597 AR 3 8 282 AR 1 9 8 877 0123 AR 2 402 AR 3 718 Part 2 g 5 points The actual unemployment rate in 2011 was 9 0 What is the difference between the predicted values and actual for the AR 1 AR 2 and AR 3 models a 5 points Add the lag of the cpi inf rate to the AR 2 regression Continue to execute with year 1951 Is the lag inflation statistically significant The coefficient is positive 1547 and it is signficant with a p value of 001 b 5 points Predict the unemployment rate 2011 in using the ADL model from 2a Compare it to the actual level ADL 2 1 Using the coefficients from the ADL 2 1 we can find the predicted unemployment 1 53 935 9 6 9 3 298 1 6 155 7 99 The difference between the predicted and actual is 1 01 c The inflation adjusted Phillips curve proposes that the unemployment rate falls when we encounter unexpected inflation Let s propose that the expected value of inflation for time t is the actual value of inflation for time t 1 This means the difference between actual inflation and expected inflation for time t will be inft inft 1 Create a variable called d inf that equals the change in inflation Us the syntax D cpi inf to help you do this While you re at it also create d unemp that equals the change in unemployment d 5 points Regress unemp on d inf Report the coefficient and p value Is it of the expected sign The coefficient is 2247 and the p value is 054 Marginal significance The sign is indeed negative as expected by the theory alluded to above Part 3 a 5 points At a level of 05 do unemp and cpi inf have unit roots Execute Dickey Fuller tests for each as we did in class Note that the critical value for a 5 test for a dataset of our size is approximately 2 92 For unemployment the Dickey Fuller test yields a t value of 2 55 this is smaller in absolute value than the critical value Thus we cannot reject a unit root for unemployment For inflation the Dickey Fuller tests yields a t value of 3 44 which exceeds the critical value in absolute value and thus we reject the hypothesis of a unit root b 5 points It appears you cannot reject a unit root for unemp Let s try an Augmented Dickey Fuller test for unemployment That is include a lag of the change in unemployment in your Dickey Fuller regression we created this above it s called d unemp Can you reject a unit root now Including the value of the lagged change in unemployment yields a t value of 2 90 in the Dickey Fuller specification This is really close to our critical value Techincally we still can t reject the unit root hypothesis but you would have a reasonable case to treat unemployment as a stationary variable A student can execute this test in two ways First the command dfuller unemp lags 1 will do it Alternatively one could manually run the regression I asked for d unempt t tunempt 1 2d unempt 1 Then calculate the test statistic on 1 Part 1 tsset year time variable year 1948 to 2010 delta 1 unit regress unemp L unemp if year 1951 robust Linear regression Number of obs 60 F 1 58 81 68 Prob F 0 0000 R squared 0 5734 Root MSE 1 0609 Robust unemp Coef Std Err t P t 95 Conf Interval unemp L1 7966533 0881462 9 04 0 000 6202095 9730971 cons 1 229359 5236268 2 35 0 022 1812062 2 277512 estat ic Model Obs ll null ll model df AIC BIC 60 113 2242 87 66675 2 179 3335 183 5222 Note N Obs used in calculating BIC see R BIC note regress unemp L 1 2 unemp if year 1951 robust Linear regression Number of obs 60 F 2 57 54 95 Prob F 0 0000 R squared 0 6063 Root MSE 1 0281 Robust unemp Coef Std Err t P t 95 Conf Interval unemp L1 9993643 1284622 7 78 0 000 7421231 1 256605 L2 2925832 1560901 1 87 0 066 6051482 0199819 cons 1 724678 5849248 2 95 0 005 5533866 2 89597 estat ic Model Obs ll null ll model df AIC BIC 60 113 2242 85 25845 3 176 5169 182 7999 Note N Obs used in calculating BIC see R BIC note regress unemp L 1 3 unemp if year 1951 robust Linear regression Number of obs 60 F 3 56 37 04 Prob F 0 0000 R squared 0 6128 Root MSE 1 0286 Robust unemp Coef Std Err t P t 95 Conf Interval unemp L1 1 025975 1305874 …